Introduction

Magnetic materials have been used since the beginning of the computing era to store information in magnetic bits (up or down orientation of the magnetization in ferromagnetic medias). The discovery of the giant magneto-resistance effect in 1988 1 , contributed to the advent of a new domain of condensed matter Physics based on the manipulation and detection of spin polarized currents: Spintronic. The main phenomena such as the magnetoresistance effect and the spin transfer torque have major applications in hard-drives and fast memories MRAMs. These discoveries and applications have driven the study of magnetic materials to unprecedented levels in the past twenty years holding keys to unlock many of the challenges facing information technologies 2 . In ordered magnetic materials, the excitation of the magnetic order gives rise to collective behaviors, uniform or propagative, that have all the typical properties of a wave: a frequency (in the GHz range), a wavevector and a relation between the two: the dispersion relation. These elementary excitations are consequently called spin-waves (wave representation) or magnons (particle representation). Their properties make them good candidates for applications as information carriers at radio-frequencies. Indeed, contrary to electronics, spin-waves can carry the information without movement of charges and thus without any Joule effect, one of the main bottlenecks of current CMOS processors. Additionally, the frequency and wavelength can be tuned by applying a static magnetic field. The devices can thus operate with sub-µm wavelengths in the practical GHz frequency bands and take full advantage of wave-based computing schemes. Although this was known since the 1970's, it is only in the early 2000's, with the advent of spintronic and the perspective of an end of Moore's law, that the domain of magnonic has flourished.

One of the main issues of spin-wave based systems is the existence of magnetic losses that takes the form of a damping term. This damping, which depends on the ferromagnetic material, leads to the exponential decay of the spin-wave precession amplitude on a typical scale of a few µm in thin films, even for the ferromagnet with the lowest magnetic losses Y3Fe5O12 (Yttrium Iron Garnet or YIG). Furthermore, spin-waves are mainly excited and detected using inductive antennas, whose efficiency depends on the ferromagnetic volume. Thus, it is currently impossible to scale down magnonic circuits if frequent detections and re-emissions of spinwaves are required. A possible solution is to combine spintronic and magnonic to more efficiently control, detect and excite spin-waves using DC currents. In the early 2000's, it was demonstrated that spinwaves can be excited using a spin polarized DC current through the spin transfer torque 3,4 . However, this powerful torque could only be applied in metallic ferromagnets. In 2010, using pure spin current generated by the spin Hall effect in a simple adjacent Pt layer 5 , Kajiwara et al 6 . demonstrated that a similar torque, called spin orbit torque (SOT), could also be applied to insulating ferromagnets like YIG. Another advantage of the spin orbit torque is that it can be applied much more efficiently on large areas, which is critical even for small magnonic circuits. These discoveries spurred an intense research effort on the control of the magnetization dynamics using spin currents 7,8 . In particular, by adjusting the current polarity and the magnetic field orientation, the spin orbit torque can be anti-parallel to the damping torque 9 , thus providing an electrical control of the magnetic relaxation. At a critical current value, a full compensation of the damping torque was demonstrated in discs 10 and 1D nanostructures 11 . Above the critical current, the magnetization auto-oscillates due to a negative effective damping. The magnons propagation length can thus be significantly increased by applying a DC current to a YIG/Pt waveguide 12 . However, at the critical current, where the effective damping of the uniform mode is zero, the decay length of propagating magnons is finite (24 µm), sign that their effective damping is strictly positive. Furthermore, above the critical current, the propagation length values collapse and the propagating magnons are scattered by the auto-oscillations. The effect is attributed to the non-linear coupling with incoherent magnons modes that is strongly enhanced when the damping is compensated. The loss of energy of the propagating magnon mode to the incoherent magnon modes through these relaxation channels can be seen as a nonlinear damping term. Two years later, Soumah et al. grew ultra-thin films of Bi-doped YIG with ultra-low damping 13 in which the perpendicular magnetic anisotropy (PMA) can be controlled to compensate the shape anisotropy (vanishing effective magnetization) and thus obtain a circular magnetization precession. They demonstrated very high amplitude auto-oscillations as well as the coherent emission of spin-waves in full BiYIG film using SOT 14 . It was later confirmed that an elliptical precession of the magnetization is a predominant source of nonlinear damping 15 whose contribution can be greatly reduced by using a ferromagnet with PMA. After these findings, the ensuing question was: "Is the suppression of the leading non-linear damping term in BiYIG enough to enable the lossless propagation of spin-waves using SOT ?". It is the question that I will answer in Part 1 of the present manuscript. In the first chapter, I'll describe the properties of BiYIG films that I measured during my thesis. In particular, I found a non-standard temperature dependence of the uniaxial anisotropy term in BiYIG, leading to a strong thermal stability of the ferromagnetic resonance frequency for films with vanishing effective magnetization. We show that the thermal stability is instrumental to the results obtained in Ref 14 , as well as in the experiments presented in the ensuing chapters of Part 1. In the second chapter, the auto-oscillation regime in a BiYIG/Pt waveguide is described. Using the enhanced magneto-optical coupling of BiYIG, the auto-oscillation transient regime is precisely measured for the first time using time-resolved micro Brillouin Light Scattering microscopy (µ-BLS). We find that the auto-oscillations are well described by the Slavin & Tiberkevich non-linear auto-oscillator model 16 in the low amplitude regime. In the third chapter, rf-excited spin-waves are propagated when the SOT is applied continuously and in 250ns pulses. Clear evidence of a propagating spin-wave amplification is observed in the pulsed SOT experiment.

Another important topic is to functionalize the magnonic circuits to realize either computing or signal processing operations. The advent of high quality ultra-thin films 2 made the nanostructuration of ferromagnetic materials possible using standard lithography techniques. In the meantime, the emergence of GPU-based micromagnetic simulation software like MuMax3 19 significantly enhanced our capacity to simulate large magnonic systems. This has led to many propositions of magnonic components based on patterned ferromagnetic films in the last 5 years (see review 20 ). To cite just a few, components like majority gates [21][22][23][24][25] , magnonic diodes 26 , cross-elements for 2D magnonic circuits 27 (that can also serve as a tunable power splitter, a frequency separator or a multiplexer), spectrum analysers 28 or simple nanoscale conduits 29,30 have been proposed and were, for some of them, experimentally demonstrated (Ref 23,[25][26][27]30 ). In this thesis, we have studied magnonic filters as this component is at the center of any signal processing stage, governing its sensitivity (signal/noise ratio) and its dynamic range. It is known that propagating waves in a periodic medium (called an artificial crystal) generates constructive and destructive interferences. Consequently, for certain wavelengths, the waves are strongly attenuated and filtered when they go through the artificial crystal. This principle was applied to light in the early 2000's in photonic crystals [START_REF] Joannopoulos | Photonic Crystals: Molding the Flow of Light[END_REF] , allowing the frequency filtering (hundreds of THz) of light-carried electromagnetic signals on small distances. Similarly, magnonic crystals are micron-sized components that can filter radiofrequencies (1-100GHz). Moreover, the frequency of the rejection band can be tuned by varying the applied magnetic field. In 2009, Kim et al. [START_REF] Kim | [END_REF] demonstrated by simulation that a nanometric Py waveguide (30nm-wide) could open GHz-size frequency gaps. It led to many experimental demonstrations (see review 33 ) using different materials, mainly thick YIG films and metallic ferromagnets. However, none was carried on ultra-thin YIG films (tens of nm thick) which is so far the most promising material thanks to its long magnon lifetime and its compatibility with SOT control. The long magnon lifetime is particularly important for magnonic crystals, as the spin-wave effectively perceives a higher number of periods, making the filtering effect more efficient. Beyond their filtering properties, magnonic crystals were also shown to be very versatile systems that could be used to create a wide range of components for magnonic circuits (see 33 and references therein). Other interesting effects like chiral edge states that are unidirectional and topologically protected against defects, are expected in magnonic crystals [34][35][36] . In photonic crystals, the prediction 37,38 and observation 39,40 of topological edge state has spurred a large research effort (see reviews 41,42 ), with significant fundamental and applied perspectives. The ability to design, simulate and measure efficient magnonic crystal in thin films with long magnon lifetime is consequently a key topic in modern magnonics. In Part 2 of the present manuscript, two different magnonic crystal based on 20nm-thick YIG films are studied. The first system is a width-modulated waveguide in which we demonstrated strong filtering thanks to µ-BLS and evidenced the important role of defects and of the antenna position thanks to full-scale micromagnetic simulations. The second system is a thickness modulated waveguide measured using an all-electrical method called propagating spin-wave spectroscopy (PSWS). This design showed exceptional results, with almost no extra losses at frequencies outside the rejection band compared to the unpatterned waveguide. Similar results were obtained synchronously in different groups 43,44 bringing understanding and context to our experimental demonstration.

Finally, turning a laboratory proof of concept into an industrial prototype is not a straightforward task. By identifying a high frequency magnonic delay line as a potential need for Thales units, we looked into the process of making a simple but efficient magnonic component, using modern lithography and modelling techniques. The results of this ongoing project are presented in Part 3.

An introduction to the theoretical context of this thesis is first provided in the Part 0 of the manuscript. The experimental methods are also presented, from the magnetic films' growth and characterization, to the fabrication of magnonic devices and the measurement of the spin-waves properties.

Part 0: Analytical and experimental methods

Since the discovery of ferromagnetic resonance, many different formalisms have been used to solve the Landau-Lifshitz-Gilbert equation that describes the magnetization dynamics phenomena in ferromagnets. In the first part, I'll mainly use two canonical magnonic references, the Gurevich "Magnetization oscillations and Waves" [START_REF] Gurevich | Magnetization Oscillations and Waves[END_REF] and both Kalinikos-Slavin [START_REF] Kalinikos | [END_REF]47 articles that accurately derives the spin-wave dispersion relation for the general cases relevant to the present manuscript. The more recent concepts of spin currents and spin orbit torque (SOT) will then be introduced. The spin orbit torque represents a very promising way to control the magnetization dynamics using DC currents. The SOT is used extensively in Part 1 of the present thesis. In particular, we will be interested in the non-linear magnetization dynamics generated by the SOT and its interaction with propagating spin-waves. Finally, the experimental methods used in this thesis are described. In particular, we detail how our samples are grown, characterized and nanostructured. The two main spin-waves observation techniques used in this work are finally presented.

I. Magnetization dynamics in ferromagnets: Theory and formalisms

The study of magnetization dynamics is a complex multiscale problem. Spins in the lattice mainly interact through two very different forces: the dipolar and the exchange interactions. The dipolar interaction is a long-range interaction governed by classical Maxwell equations while the exchange interaction is a quantum interaction emerging from electronic orbitals overlap (few Å range). The interaction energy due to the exchange mechanism exceeds by far all the other magnetic energies contribution in a ferromagnet, the magnetic moments are therefore aligned on the exchange interaction length scale. Hence, neighboring spins tends to have almost the same orientation. Meaning that for the sizes of our magnetic structures (0.1-100 µm), the magnetization in our material can be described as a continuous function 𝑴(𝑟 ⃗).

This assumption is at the foundation of the micromagnetic theory and will be used in the rest of this manuscript.

Let's define our coordinate systems for our thin film ferromagnet.

-(𝑥 ', 𝑦 ', 𝑧) is the static coordinate system of the thin film. 𝑧̂ is always normal to the film and 𝑥 ' is the waveguide axis when the film is patterned. -(𝑒̂-, 𝑒̂., 𝑒̂/) is the magnetization coordinate system. 𝑒̂/ is along the magnetization direction and 𝑒. lies in the (𝑥 ', 𝑦 ') plane. One can go from one coordinate system to the other using the rotation matrix 𝑅 1 (𝜑) and 𝑅 3 (𝜃) defined by: 𝑅 1 (𝜑) = 6 cos(𝜑) -sin(𝜑) 0 sin(𝜑) cos(𝜑) 0 0 0 1 ? ; 𝑅 3 (𝜃) = 6 cos(𝜃) 0 sin(𝜃) 0 1 0 -sin (𝜃) 0 cos(𝜃) ?

Figure 1: (𝑥 ', 𝑦 ') is the thin film plane of normal 𝑧. Canonical change of coordinate system from (𝑥 ', 𝑦 ', 𝑧) to the magnetization coordinate system (𝑒-, 𝑒., 𝑒/) is achieved by rotating the frame by an angle 𝜃 around the y-axis (a) and then by an angle 𝜑 around the z-axis (b).

-Other coordinate systems can be defined in a similar way. For instance, the external magnetic field is not necessarily aligned with the magnetization and can be assigned a coordinate system (ℎ L -, ℎ L . , ℎ L / ) with the angles (𝜃 M , 𝜑 M ). Similarly, uniaxial and cubic anisotropy axis have their own frame (𝑢 ' -, 𝑢 ' . , 𝑢 ' / ), (𝑎 ' -, 𝑎 ' . , 𝑎 ' / ) with angles (𝜃 P , 𝜑 P ) and (𝜃 Q , 𝜑 Q ) respectively.

Most interactions have a simple expression in their own frame but putting them all together for arbitrary angles can lead to quite heavy equations. We will either resort to the rotation matrix we introduced or give the analytical formulas only for particular cases. The system free energy governs the static configuration (the direction of equilibrium magnetization 𝑀 SS⃗ T minimizes the free energy) and the dynamic (in the Landau-Lifshitz equation presented in the next part). Let's describe all the terms intervening in the free energy of a ferromagnetic film and introduce the concept of effective field. The total free energy (that 𝑀 SS⃗ T (𝑟 ⃗) minimizes) is:

𝐸 VWV = X 𝜖 VWV Z [
𝑑𝑉 with the volume free energy 𝜖 VWV , at each point 𝒓 SS⃗, being composed of the following terms:

𝜖 VWV (𝑟 SS⃗) = 𝜖 _DD`Qa + 𝜖 cD`+ 𝜖 DC + 𝜖 Q + 𝜖 WVMDde (3) We can express it as an effective field, felt by the magnetic moments at each point 𝒓 SS⃗:

𝐻 S S⃗ Dgg (𝑟 SS⃗) = - 1 𝜇 T 𝜕𝜖 VWV 𝜕𝑀 SS⃗ (4)
𝐻 S S⃗ Dgg (𝑟 SS⃗) = 𝐻 S S⃗ + 𝐻 S S⃗ cD`+ 𝐻 S S⃗ DC + 𝐻 S S⃗ Qa + 𝐻 S S⃗ WVMDde

Let's explicitly write the volume energies and effective field terms for all the interactions in our system.

The Zeeman energy -the magnetic moments tend to align to the externally applied field. 𝜖 _DD`Qa = -𝜇 T 𝑀 SS⃗ ⋅ 𝐻 S S⃗

The dipolar energy -due to the mutual magnetostatic interactions of the spins.

𝜖 cD`= -𝜇 T 𝑀 SS⃗ ⋅ ∇ S S⃗ A∇ S S⃗ ⋅ X 𝑀 SS⃗ (𝑟 l ) |𝑟 ⃗ -𝑟 ⃗′| o 𝑑𝑟′B (7) 𝐻 S S⃗ cD`( 𝑟 SS⃗) = ∇ S S⃗ A∇ S S⃗ ⋅ X 𝑀 SS⃗ (𝑟 l ) |𝑟 ⃗ -𝑟 ⃗′| o 𝑑𝑟′B

The Exchange energy -favoring the alignment of neighboring spins.

𝜖 DC = 𝐴 DC A ∇ S S⃗ ⋅ 𝑀 SS⃗ 𝑀 e B . ( 9
)
𝐻 S S⃗ DC = 𝛬 DC . ∇ S S⃗ . 𝑀 SS⃗ (10) where 𝛬 DC = r The Anisotropy energy -favoring the alignment of the magnetization along a given axis 𝑢 ' / (uniaxial anisotropy 𝝐 𝒖 ) or along any axis of an orthogonal coordinate system (𝑎 ' -, 𝑎 ' . , 𝑎 ' / ) (cubic anisotropy 𝝐 𝒂 ).

𝜖 P = 𝐾 P sin . (𝜃 P )

𝐻 S S⃗ P = 2𝐾 P 𝜇 T 𝑀𝑠 𝑀 P/ 𝑀 e 𝑢 ' / (12) With 𝜃 P the angle between the magnetization and the uniaxial anisotropy axis 𝑒̂" and 𝑀 P/ = 𝑀 SS⃗ ⋅ 𝑢 ' / . Hence, the effective field created by anisotropy, in the case where 𝑀 SS⃗ T ∥ 𝑢 ' / is: 𝐻 S S⃗ P = 𝐻 P 𝑢 ' / (13) With 𝐻 P defined as:

𝐻 P = 2𝐾 P 𝜇 T 𝑀 e (14)
NB: In the Kalinikos-Slavin model 47 , 𝐻 P = † ‡ v w x y is used as a convention, so one should be careful when translating the result to the more widely adopted convention (14).

The cubic anisotropy has the following expression at first order:

𝜖 Q = 𝐾 ˆ(𝛼 - . 𝛼 . . + 𝛼 . . 𝛼 / . + 𝛼 / . 𝛼 - . )

𝛼 -= 𝑀 - 𝑀𝑠 = cos(𝜙 Q ) sin(𝜃 Q ) ; 𝛼 . = 𝑀 . 𝑀𝑠 = sin(𝜙 Q ) sin(𝜃 Q ) ;

𝛼 / = 𝑀 / 𝑀𝑠 = cos(𝜃 Q ) (16)
And its effective field expression can be found in Ref [START_REF] Prabhakar | Spin Waves Theory and Applications[END_REF] .

B) Landau-Lifshitz equation and resonance conditions

The magnetization dynamics is described by the well-known Landau-Lifshitz equation. The effective field applies a conservative torque to the magnetization along the 𝑀 SS⃗ × 𝐻 S S⃗ Dgg direction.

𝑑𝑀 SS⃗ 𝑑𝑡 = -𝛾𝑀 SS⃗ × 𝐻 S S⃗ Dgg (17) This equation is non-linear since the effective field contain terms that depend on 𝑀 SS⃗ . However, in the small oscillations limit, where a harmonic response is expected, we can adopt a linearization approach. Thus, we consider magnetization states of the type:

𝑀 SS⃗ (𝑡) = 𝑀 SS⃗ T + 𝛿𝑚 SS⃗(𝑡) with |𝛿𝑚 SS⃗| ≪ |𝑀 SS⃗ T | (18) Which implies that at first order:

𝛿𝑚 SS⃗ ⋅ 𝑀 SS⃗ T = 0 (19)
i.e. the direction of the perturbation must be orthogonal to 𝑀 SS⃗ T .

Similarly, we can express the effective field as a static part and a dynamic part:

𝐻 S S⃗ Dgg T = 𝐻 S S⃗ T + 𝐻 S S⃗ cDT + 𝐻 S S⃗ Qa T + 𝐻 S S⃗ DC T (20) The variation of the exchange field can be neglected as long as the magnetization direction is almost uniform on the scale of the exchange length. This may no longer be true close to the edges in confined structures or in non-trivial magnetic textures such as vortices or skyrmions.

The static demagnetization and anisotropy fields can be put in the tensorial form, such that:

𝐻 S S⃗ Dgg T = 𝐻 S S⃗ T -𝑁 I 𝑒 𝑀 SSS⃗ 0 = 𝐻 S S⃗ T -•𝑁 I + 𝑁 I 𝑎𝑛 -𝑀 SSS⃗ 0 (21) with the demagnetizing tensor 𝑁 I being of trace equal to one and depending on the shape of the sample. In the case where our sample is an ellipsoid whose principal axis are parallel to (x, y, z), 𝑁 I can be expressed as:

𝑁 I = ˜𝑁C 0 0 0 𝑁 3 0 0 0 𝑁 1 ™ C ',3 ',1̂ (22)
NB: For a thin film of normal 𝑧, 𝑁 C = 𝑁 3 = 0 and 𝑁 1 = 1. This remains true for waveguide of width 𝑤 and thickness 𝑡, as long as 𝑤 ≫ 𝑡.

In the case of uniaxial anisotropy, along the vector û / :

𝑁 I Qa = 𝑁 I P = • 0 0 0 0 0 0 0 0 -𝐻 P 𝑀 e ž P Ÿ E ,P Ÿ F ,P Ÿ G (23) The anisotropy tensor has a simple expression in the (𝑢 ' -, 𝑢 ' . , 𝑢 ' / ) basis. To express it in another basis we use the 𝑇 ¡ P matrix we defined using our rotation matrices (2) with angles (𝜃 P , 𝜑 P ).

•𝐻 S S⃗ P -C ',3 ',1̂= 𝑇 ¡ P •𝐻 S S⃗ P -P Ÿ E ,P Ÿ F ,P Ÿ G = 𝑇 ¡ P 𝑁 I P 𝑇 ¡ P |-𝑇 ¡ P •𝑀 SS⃗ -P Ÿ E ,P Ÿ F ,P Ÿ G = 𝑁 I C ',3 ',1P •𝑀 SS⃗ -C ',3 ',1̂

with 𝑁 I C ',3 ',1P = 𝑇 ¡ P 𝑁 I P 𝑇 ¡ P |--expressed in (𝑥 ', 𝑦 ', 𝑧) basis (24) For cubic anisotropy, the tensor expression in the (𝑎 ' -, 𝑎 ' . , 𝑎 ' / ) basis is a bit more complex and can be found in Ref [START_REF] Prabhakar | Spin Waves Theory and Applications[END_REF] (page 86-87).

The dynamic part of the effective field can be quite complex. If we assume uniform precession, then we simply have:

𝛿𝐻 S S⃗ Dgg = -𝑁 I D 𝛿𝑚 SS⃗ (25)

1) Equilibrium magnetization

One can notice that the equilibrium magnetization 𝑀 SS⃗ T satisfies:

𝑀 SS⃗ T × 𝐻 S S⃗ Dgg T = 0 (26)
i.e. the magnetization 𝑀 SS⃗ T is aligned with the direction of the effective field.

Still assuming that the exchange field can be neglected, the equation (26) implies that 𝐻 S S⃗ Dgg T can be written in the basis 𝐸 L = (𝑒-, 𝑒., 𝑒/ ∥ 𝑀 SS⃗ T ) as:

𝐻 S S⃗ Dgg T = 6 0 0 𝐻 / T -𝑁 // D 𝑀 e ? ¢ L (27)
This gives an easy way to compute the total free energy:

Ε VWV = -𝜇 T ∫ 𝐻 S S⃗ Dgg ⋅ 𝑑𝑀 SS⃗ = -𝜇 T 𝑀 e (𝐻 / T -𝑁 // D 𝑀 e 2 ) (28)
The equilibrium magnetization is then such that the free energy is minimized. Hence (𝜃, 𝜑) satisfies:

𝜕𝜖 VWV 𝜕𝜃 = 0 and 𝜕𝜖 VWV 𝜕𝜑 = 0 ( 29
)
The equilibrium is stable if:

𝜕 . 𝜖 VWV 𝜕 . 𝜃 > 0 and 𝜕 . 𝜖 VWV 𝜕 . 𝜑 > 0 (30)

2) Uniform precession

We linearize equation ( 17) by neglecting 2 nd order term in 𝛿𝑚 SS⃗: 𝑑 𝑑𝑡 𝛿𝑚 SS⃗ = -𝛾𝑀 SS⃗ T × 𝛿𝐻 S S⃗ Dgg -𝛾𝛿𝑚 SS⃗ × 𝐻 S S⃗ Dgg T (31) As shown in the previous part, in the case of uniform precession the effective field can be expressed for arbitrary directions and intensity of anisotropy as:

𝐻 S S⃗ Dgg T = 𝐻 S S⃗ T -𝑁 I D 𝑀 SS⃗ T (32)
𝛿𝐻 S S⃗ Dgg = -𝑁 I D 𝛿𝑚 SS⃗ (33) Equation (31) gives us the dynamic equations in the (𝑒-, 𝑒.) plane:

𝜕 𝜕𝑡 ¨𝛿𝑚 - 𝛿𝑚 .
Where: 𝜔 « = 𝛾(𝐻 / T -𝑁 // D 𝑀 e ) ; 𝜔 x = 𝛾𝑀 e (36) The solutions to the system (35) are such that: 𝛿𝑚 ∝ exp(𝑖𝜔𝑡) and the condition of existence of such solutions writes:

𝜔 . = (𝜔 « + 𝜔 x 𝑁 .. D )(𝜔 « + 𝜔 x 𝑁 -- D ) -𝜔 x . 𝑁 -. D 𝑁 .- D (37) This is the well-known Kittel law, expressing the ferromagnetic resonance frequency 𝜔 as a function of the magnetic parameters of the material and the external magnetic field.

Let's derive the explicit formulas for certain important cases:

By symmetry, only the angle 𝜃 is playing a role, and we obtain:

𝑁 I D = ³ sin . 𝜃 0 -sin 𝜃 cos 𝜃 0 0 0 -sin 𝜃 cos 𝜃 0 cos . 𝜃 ´DÊ,DF,DĜ

Hence, the resonance is given by: 𝜔 . = 𝜔 « (𝜔 « + 𝜔 x sin . 𝜃)

And the equilibrium condition is given by the equation (along 𝑒̂-):

𝜕𝜖 VWV 𝜕𝜃 = 0 ⇒ 𝐻 T sin(𝜃 -𝜃 « ) -𝑀 e sin 𝜃 cos 𝜃 = 0

With 𝜖 VWV = -𝜇 T 𝑀 e ¶𝐻 / T -𝑁 // D x y . • = -𝜇 T 𝑀 e (𝐻 T cos(𝜃 -𝜃 « ) -

x y

. cos . 𝜃)

The equilibrium positions can be found easily using a numeric solver. And the stability conditions can then be checked (Eq. ( 30)).

For simple cases, the stable equilibrium position can be determined analytically:

-If 𝜃 « = .: (in-plane magnetic field) ⇒ 𝜃 = 𝜃 « = .

-

If
Hence the resonance condition is:

𝜔 . = 𝜔 « ¶𝜔 « + 𝜔 x t½½ sin . 𝜃• (43)
Where the effective magnetization is defined as: 𝑀 Dgg = 𝑀 e -𝐻 P (44) The equilibrium condition writes:

𝜕𝜖 VWV 𝜕𝜃 = 0 ⇒ 𝐻 T sin(𝜃 -𝜃 « ) -𝑀 Dgg sin 𝜃 cos 𝜃 = 0

With 𝜖 VWV = -𝜇 T 𝑀 e ¶𝐻 / T -𝑁 // D x y . • = -𝜇 T 𝑀 e (𝐻 T cos(𝜃 -𝜃 « ) -

x t½½ . cos . 𝜃)

Hence the equilibrium and resonance conditions are similar to the case of the isotropic ferromagnet by replacing 𝑀 e by 𝑀 Dgg . The only difference comes from the fact that 𝑀 Dgg can be negative, which only modifies the previously established case:

-If 𝜃 « = . (in-plane magnetic field):

o If 𝐻 T > -𝑀 Dgg : 𝜃 = . is stable o If 𝐻 T < -𝑀 Dgg : 𝜃 = arcsin(-« w

x t½½ )

Meaning that for negative 𝑀 Dgg , the magnetization is pointing out-of-plane for small magnetic fields. The anisotropy is strong enough to overcome the shape anisotropy of the thin film.

NB:

It is not possible to disentangle 𝑀 e from 𝐻 P in these equations. When the precession is uniform, only the effective magnetization can be determined by measuring the resonance frequency.

3) Spin-waves

Deriving the dispersion relation for an arbitrary wavevector 𝑘 is a much harder task since the dynamic effective field is much more complex than in the 𝑘 = 0 case (Eq. ( 25)). This problem was fully solved in Ref [START_REF] Kalinikos | [END_REF] for an isotropic ferromagnet and in Ref 47 for an anisotropic ferromagnet.

The formula is extremely useful (if not mandatory) to design or analyze any magnonic device. Knowing how and when to use it is very important, but knowing its complete derivation is less so. However, the techniques used in Refs [START_REF] Kalinikos | [END_REF]47 are very powerful tools to solve these kinds of problems. Hence, they were used to generalize the results to more complex systems, such as periodic structures [START_REF] Grigoryeva | Magnonics[END_REF] or nanoscopic waveguides that go beyond the thin film approximation 29 .

The eigenvectors for the dynamic magnetization in the linear regime can be written as:

𝑚 SS⃗(𝑟 ⃗, 𝑡) = 𝑚 SS⃗ T (r ⃗)exp (𝐾 S S⃗ ⋅ 𝑟 ⃗ -𝑖𝜔𝑡) (46) Like electromagnetic waves in a waveguide, or electrons in a quantum well, when a spin-wave is confined in a ferromagnetic waveguide, its k-vector gets quantized in the directions of confinement.

𝐾 . = 𝐾 ¿ + 𝜅 . = 𝑘 C . + 𝑘 a . + 𝜅 . (47) Where 𝐾 ¿ is the k-vector in the plane, and the quantization conditions read: 𝜅 `= 𝑚𝜋 𝑡 ; 𝑚 = 0,1, … and 𝑘 a = 𝑛𝜋 𝑤 ; 𝑛 = 1,2, …

The dispersion relation is then given by: 𝜔 a . = (𝜔 « + 𝛬 . 𝜔 x 𝐾 . )(𝜔 « + 𝛬 . 𝜔 x 𝐾 . + 𝜔 x 𝐹 ``) (48) With:

𝐹 ``= 𝑃 ``+ sin . 𝜃 [1 -𝑃 ``(1 + cos . 𝜑 AE ) + 𝜔 x 𝑃 ``(1 -𝑃 ``) sin . 𝜑 AE 𝜔 « + 𝛬 . 𝜔 x 𝐾 . ]

And:

𝑃 ``= 𝐾 ¿ . 𝐾 . (1 - 𝐾 ¿ . 𝐾 . 2 1 + 𝛿 T`1 -(-1) `𝑒| † È V 𝐾 ¿ 𝑡 ) (50)
𝜑 AE = 𝜑 -𝜑 † (51) With 𝜑 † being the angle of 𝐾 S S⃗ in the (𝑥 ', 𝑦 ', 𝑧) basis. 𝜑 † = arcos

É u É u F ÊÉ Ë F NB:
In the thin film approximation 𝑡 ≪ 𝑤, quantization starts at 0 for the z-component (𝜅 `ÌT = 0 represents a uniform magnetization along the thickness), while the dynamic dipolar field imposes n to start at one (𝑘 aÌ-= Í).

NB2: For wavelengths small compared to the thickness: 𝐾 ¿ 𝑡 ≪ 1, ie

-|D ÎÏ È Ð † È V ≈ 1 - † È V
. . Thus:

𝑃 TT ≈ 1 - 𝐾 ¿ 𝑡 2 (52)
For thin films, higher order thickness modes are much higher in frequency. Hence, in this thesis we mainly deal with the 𝑚 = 0 mode that is uniform in the film thickness. Thus in the considered magnonic waveguide, the main modes are the quantized along the waveguide width and are represented in

4) Relaxation processes

The magnetic degrees of freedom are coupled to other degrees of freedom (phonons, electrons, etc…) creating relaxation channels which effectively damp the magnetization oscillations. These relaxation processes can be expressed phenomenologically as a damping torque in the Landau-Lifshitz equation, which becomes the Landau-Lifshitz-Gilbert equation:

𝜕𝑀 SS⃗ 𝜕𝑡 = -𝛾𝑀 SS⃗ × 𝐻 S S⃗ Dgg + 𝛼 𝑀 e A𝑀 SS⃗ × 𝜕𝑀 SS⃗ 𝜕𝑡 B ( 53
)
𝛼 is the dimensionless Gilbert damping constant. The Gilbert constant is frequency invariant and depends on the material composition, structure and interfaces. Its value typically ranges from about 10 -2 in metallic thin films (Fe, Co, Py) to about 10 -4 for Yttrium Iron Garnet films (Y / Fe ä O -. ). As a consequence, in all these materials where 𝛼 . ≪ 1, the damping torque is much smaller than the conservative torque, allowing us to rewrite the LLG equation as:

𝜕𝑀 SS⃗ 𝜕𝑡 = -𝛾𝑀 SS⃗ × 𝐻 S S⃗ Dgg - 𝛼 𝑀 e 𝛾(𝑀 SS⃗ × (𝑀 SS⃗ × 𝐻 S S⃗ Dgg )) (54)
The damping torque is then orthogonal to both the magnetization and the conservative torque. It will eventually bring the magnetization back to its equilibrium position.

Adding the damping term in the linearized LLG equation yields:

¨𝑖𝜔-𝜔 x 𝑁 .- D -𝜔 « -𝑖𝛼𝜔 -𝜔 x 𝑁 .. D 𝜔 « + 𝑖𝛼𝜔 + 𝜔 x 𝑁 -- D 𝑖𝜔 + 𝜔 x 𝑁 -. D © ¨𝛿𝑚 - 𝛿𝑚 . © = 0 (55)
This is equivalent to substituting 𝜔 « with 𝜔 ae « = 𝜔 « + 𝑖𝛼𝜔 in the linearized LL equation. This additional small complex contribution to the precession frequency defines a decay of the oscillations. Using a Taylor's expansion, we can find the typical decay at first order in 𝛼:

One can define the relaxation time 𝜏, defined as the time for the uniform magnetization oscillation amplitude to decrease by a factor e.

1 𝜏 = 𝛼𝜔 𝜕𝜔 𝜕𝜔 « = 𝛼𝜔 d (57)
For a spherical ferromagnet or an infinite medium 𝜔 = 𝜔 « and the magnon lifetime is simply:

𝜏 T = 1 𝛼𝜔 (58)
For a propagating spin-wave, using that

𝜔 ïð ïð ñ = - . ïð F ïð ñ
, we obtain:

Γ Ø = 1 𝜏 = 𝛼 ¶𝜔 « + 𝛬 . 𝜔 x 𝐾 . + 𝜔 x 2 𝐹 ``• (59)
This relaxation time can also be seen as the lifetime of the pseudo particle associated with the elementary excitation of the magnetic order (magnon).

This limited lifetime results in the broadening of the ferromagnetic resonance which can be measured using an FMR setup. This is the main way to deduce the Gilbert damping of ferromagnetic films. When exciting the magnetization with a small harmonic magnetic field ℎ S⃗ dg , linearizing the LLG equation (as in ( 31)-( 36)) yields:

¨𝑖𝜔-𝜔 x 𝑁 .- D -𝜔 « -𝑖𝛼𝜔 -𝜔 x 𝑁 .. D 𝜔 « + 𝑖𝛼𝜔 + 𝜔 x 𝑁 -- D 𝑖𝜔 + 𝜔 x 𝑁 -. D © ¨𝛿𝑚 - 𝛿𝑚 . © = ¨0 -𝜔 x 𝜔 x 0 © A ℎ - dg ℎ . dg B (60)
Hence, one can express the Kittel susceptibility tensor (reaction of the magnetic system to a harmonic excitation) as:

𝑚 SS⃗ = 𝜒̅ † ℎ S⃗ dg = õ 𝜒 -𝑖κ -𝑖𝜅 𝜒 . ÷ ℎ S⃗ dg (61)
With:

𝜒 " = 𝜔 " 𝜔 x 𝜔 T . -𝜔 . ; κ = 𝜔𝜔 x 𝜔 T . -𝜔 . (62)
And:

𝜔 T = 𝜔 -𝜔 . -(𝜔 x 𝑁 -. D ) . ; 𝜔 " = 𝜔 ae « + 𝜔 x 𝑁 "" D ; 𝜔 ae « = 𝜔 « + 𝑖𝛼𝜔 (63)
NB: It is important not to confuse the Kittel susceptibility tensor, that relates the dynamic magnetization to the applied rf field (hence the tensor that we directly measure during an FMR experiment) and the Polder susceptibility tensor relating the dynamic magnetization and the total dynamic field (ℎ S⃗ dg + ℎ S⃗ `QøaDV"1QV"Wa ). The Polder susceptibility tensor is useful to solve the Maxwell equations.

Both real part and imaginary part of the tensor component are resonant, and their Lorentzian linewidth (width at half maximum) can be inferred:

Δ𝐻 = 2𝛼𝜔 T 𝛾 (64)
Δ𝜔 = 2𝛼𝜔 d (65)
The Kittel susceptibility tensor component 𝜒 -is plotted in Figure 4.

In a ferromagnetic film, the spread of the magnetic properties and the scattering of FMR mode due to inhomogeneities broadens the linewidth. These loses introduce an additional term Δ𝐻 T in the linewidth called inhomogeneous broadening:

Δ𝐻 = Δ𝐻 T + 2𝛼𝜔 T 𝛾 (66)
In YIG, Δ𝐻 T is relatively frequency invariant, allowing to extract the damping constant 𝛼 from simple linear fit of the linewidth as a function of the frequency. However, in Bi-doped YIG, such as the one used in this thesis, the spread of magnetic anisotropy introduces a field-dependent inhomogeneous broadening, making it harder to directly extract the damping constant from FMR experiments 13 . Inhomogeneous broadening must be taken into account to calculate the spin-wave propagation length [START_REF] Gladii | [END_REF] (via an effective damping constant for instance) and in spin-pumping experiments 52 to determine the threshold current of auto-oscillations (via the full relaxation rate).

𝛼 Dgg = 𝛼 + 𝛾 Δ𝐻 T 2𝜔 T (67)
Γ d = Γ ú + 𝛾 ∆𝐻 T 2 𝜔 d 𝜔 T (68)
Inhomogeneous broadening was found to be greatly reduced in µm-size devices due to the higher homogeneity at the µm-scale. Though values at the film level tend to reflect the local homogeneity of the sample and thus remain a strong indication of the quality of the dynamic properties of the film.

5) Going further

One can also be confronted to experiments that land outside the assumptions of these formalisms and need a different approach. For instance, a formalism was recently developed for nanometer-scale confined waveguides 29 in which the thin film conditions are not valid anymore: the exchange interaction can suppress the dipolar pinning phenomenon which strongly modifies the dispersion relation. One can also expand previously developed formalisms to describe the non-linear magnetization dynamics 16,53,54 . These recent major theoretical progresses are spurred by the growth of the field of magnonics and reflect its current trends towards confined nanostructures and non-linear phenomena. Finally, the recent advent of GPU based micromagnetic simulation software (like MuMax3 19 and now OOMMF 55 ) is a crucial addition to the scientific tools at our disposal that must not be overlooked. MuMax3 is used in this thesis to analyze the spin-waves modes properties in a fullscale periodic structure in Part 2 Chapter 1.

II. Spin current and spin orbit torque

In this section, the interactions between spin-currents and the magnetization dynamics at the interface between a ferromagnet (FM) and a normal metal (NM) are discussed. The concept of pure spin-current is first introduced. In particular, we describe the spin-Hall effect, that allows the generation of pure spin-current using heavy metals (HM) with strong spin-orbit coupling (such Pt or Ta). The transfer of angular momentum at the NM/FM interface is then presented in light of the spin torque (𝑁𝑀 → 𝐹𝑀) and spin pumping (𝐹𝑀 → 𝑁𝑀) phenomena. We show how the spin torque can exert a large anti-damping torque on the magnetization. When using pure spin currents, the torque is called spin orbit torque (SOT), in opposition to the spin transfer torque (STT) where the spin-polarized current goes through the metallic FM. In particular, the spin angular momentum of a pure spin current can be transferred to insulating ferromagnets such as YIG. SOT is an efficient and versatile way to control the magnetization state and dynamic using a simple DC current and has consequently become the subject of intense research 56 , with applications in magnetic memories, nano-oscillators, racetrack memories and novel spin logic devices. The advent of SOT has been particularly important to the field of magnonic enabling the application of ST to magnetic insulators like YIG and over extended areas, to generate, detect and control the propagation of spin-waves 7,8 .

A) Spin current

1) Definition

Each electron carries a charge and a spin angular momentum. The spin current is used to describe the motion of spin angular momentum, just like the electrical current is used to describe the motion of charges. As spins also have an orientation, the spin current density is described by a tensor:

𝚥 e I = " 𝑗 eC C 𝑗 e3 C 𝑗 e1 C 𝑗 eC 3 𝑗 e3 3 𝑗 e1 3 𝑗 eC 1 𝑗 e3 1 𝑗 e1 1 $ (69)
Where 𝑗 e3 C is the current density of spins polarized in the y-direction, flowing in the x-direction.

In general, only one direction of the spin is relevant, due to the preferential orientation of the FM magnetization, for instance along the y-axis. We can then define the current density of spin angular momentum having eigenvalues 𝑆 3 = ±ℏ/2 as:

𝒋 e = " 𝑗 e3 C 𝑗 e3 3 𝑗 e3 1 $ = - ℏ 2𝑒 (𝒋 ↑ -𝒋 ↓ ) (70)
Where 𝒋 ↑ (resp. 𝒋 ↓ ) is the charge current of spins with eigenvalue +ℏ/2 (resp. -ℏ/2). The total charge current can thus be written as:

𝒋 ˆ= 𝒋 ↑ + 𝒋 ↓ (71)
The charge current of up and down spins can be written:

𝒋 ↑(↓) = 𝜎 ↑(↓) 𝛁µ ↑(↓) 𝑒 ( 72
)
Where σ is the conductivity (in S.m -1 ) of the up and down spin population and µ is the electrochemical potential (in Joule):

µ ↑(↓) = ℰ ↑(↓) / -𝑒𝜙 (73)
ℰ ↑(↓)

/

is the Fermi energy of the up or down spin population and 𝜙 the electric potential.

Thus, a spin-current can be induced in a metal if:

-𝜎 ↑ ≠ 𝜎 ↓ , which creates spin-polarized currents -𝜇 ↑ ≠ 𝜇 ↓ , which creates pure spin-currents

2) Spin polarized current and pure spin-current

For a spin-polarized current, it is assumed that the electrochemical potential is identical (𝜇 ↑ = 𝜇 ↓ ) but the conductivity of electrons with spin up and spin down is different. It is the case in transition metals like Fe, Ni and Co where their band structure is such that the 3𝑑 ↑ and 3𝑑 ↓ are split, creating a difference in density of states and mobility at the Fermi level. This property is widely used in spintronic where the spin information is carried by electrons. The charge current is non-zero and carries a non-zero spin angular momentum current due to the difference of conductivity:

𝒋 𝒄 = (𝜎 ↑ + 𝜎 ↓ )𝛁µ 𝑒 (74)
And

𝒋 𝒔 = - ℏ 2𝑒 . (𝜎 ↑ -𝜎 ↓ )𝛁µ (75)
And the polarization of the spin current, often expressed as a percentage, is defined by:

𝑃 = - 2𝑒 ℏ 𝑗 e 𝑗 ˆ= (𝜎 ↑ -𝜎 ↓ ) (𝜎 ↑ + 𝜎 ↓ ) (76)
For a pure spin current, the conductivity of both populations is equal 𝜎 ↑ = 𝜎 ↓ (like in a normal metal), but 𝜇 ↑ ≠ 𝜇 ↓ . In this case, the charge current writes:

𝒋 𝒄 = 𝜎 𝛁µ ↑ + 𝛁µ ↓ 𝑒 = 0 (77)
The conservation of the number of charge carriers imposes 𝛁µ ↑ = -𝛁µ ↓ . Thus, a net spin angular momentum can be transferred, with a zero-charge current. The difference of chemical potential 𝜇 e = 𝜇 ↑ -𝜇 ↓ , called the spin accumulation, generates a pure spin-current equal to:

𝒋 e = - ℏ𝜎 2𝑒 . 𝛁µ 𝐬 (78)
3) Spin Hall effect

The spin Hall effect is an efficient way to generate a pure spin-current. For instance, the spinorbit coupling (soc), naturally present in heavy metals like Pt or Ta, can convert a proportion of a charge current into a transverse pure spin current. Indeed, the electrons are deviated by the soc, in opposite direction depending on their spin orientation. Thus, a spin accumulation is created at the interface of the HM layer and a spin current, orthogonal to the interface is generated.

The efficiency of the conversion is phenomenologically described by the spin-Hall angle 𝜃 3«¢ , such that:

𝒋 e = 4 𝑗 e(5 6 ×C ') C 𝑗 e(5 6 ×3 ') 3 𝑗 e(5 6 ×1) 1 7 = ℏ 2𝑒 𝜃 3«¢ (𝑒¿ × 𝒋 ˆ) (79)
Where 𝑒̂¿, is the spin polarization direction. 𝜃 3«¢ is expressed as a percentage. The values of 𝜃 3«¢ varies for the same material. For instance, for the Pt, values of 0.37 to 11% have been measured in the literature 57 . The reciprocal effect, called inverse spin Hall effect (ISHE), describes the conversion of a pure spin-current to a charge current thanks to the spin orbit coupling. It is a very practical way to measure spin currents.

𝒋 𝒄 = - 2𝑒 ℏ 𝜃 3«¢ (𝑒̂¿ × 𝒋 3 ) (80)
In particular, for a current flowing in the plane of a Pt layer (x-direction for instance):

𝒋 e = " 0 𝑗 e1 3 -𝑗 e3 1 $ = ℏ 2𝑒 𝜃 3«¢ 𝑗 Ĉ 6 0 1 -1 ? (81)
NB: In the present work, we use 7nm Pt layer to generate spin currents. Although, there exist other ways than resorting to heavy metals to generate a spin hall effect (see review by Sinova 58 ), Pt remains a particularly simple and efficient way to induce a spin current at the (Bi)YIG interface.

B) Spin pumping and spin torque

The spin torque phenomenon was predicted by Slonczewski 4 and Berger 3 in 1996 as the response of a ferromagnet to a spin-polarized current. The resulting torque on the magnetization (called spin transfer torque) is ascribed to the direct transfer of spin angular momentum between the flowing electrons and the local magnetization. The torque was shown to be large enough to induce the switching 59 or drive the auto-oscillation 60 of the magnetization in metallic ferromagnets and is consequently at the heart of many spintronic applications 61 . Reciprocally, the magnetization precession induces a transfer of spins through the FM|NM interface. This spin pumping effect was theoretically described by Tserkovnyak 62 in 2002.

The same concepts can be applied using pure spin currents (for instance using the SHE in a Pt layer). We speak of spin orbit torque (SOT) in this case. Contrary to STT, SOT can be used with insulating ferromagnets, as shown by Kajiwara 6 in 2010 with YIG|Pt bilayers. This result is particularly interesting for magnonic, as YIG (and BiYIG), the most popular ferromagnet due to its very low damping, is an insulator.

In this section, we briefly introduce the spin pumping and spin torque phenomena. Then we will introduce the Slavin & Tiberkevich model 16 to describe the auto-oscillation regime. In this regime, the applied DC current is such that the spin torque is opposite to the damping torque and larger in magnitude. We derive simple results that will be useful to understand our experimental observations on BiYIG|Pt waveguides in Part 1, where we propagate spin-waves in the auto-oscillation regime and demonstrate the possibility to amplify spin-waves using SOT.

1) Spin pumping

In a FM|NM system, the precession of the magnetization induces a transfer of spin angular momentum through the interface. In the case of the devices used in this thesis (BiYIG(20nm)|Pt(7nm)), the expression of the induced spin-current at the interface is 63 :

𝑱 𝒔 𝑭𝑴→𝑵𝑴 = " 𝑗 eC 1 𝑗 e3 1 𝑗 e1 1 $ = - ℏ 4𝜋𝑀 e . 𝑔 ↑↓ A𝑀 SS⃗ × 𝑑𝑀 SS⃗ 𝑑𝑡 B ( 82
)
Where 𝑔 ↑↓ is the spin-mixing conductance, a real number expressed in 𝑚 |. , that characterizes the transmission of spins at the interface.

NB: 𝑱 𝒔 𝑭𝑴→𝑵𝑴 is the spin-current, flowing orthogonally to the interface (z-direction). Its components represent the spin-polarization of the current contrary to 𝒋 e where the components represent the current flow direction.

The Pt layer acts as a spin sink and absorbs part of the angular momentum of the magnetization precession. This spin pumping (SP) effect can be described as a damping torque acting on the magnetization. Indeed, the torque induced by the spin-current writes:

𝜯 𝑺𝑷 = 𝛾 𝑡 /x (-𝑱 𝒔 𝑭𝑴→𝑵𝑴 ) (83)
Thus, the LLG equation is now:

𝜕𝑀 SS⃗ 𝜕𝑡 = -
Where the extra damping caused by the Pt layer is:

𝛼 3AE = 𝛾ℏ 𝑡 /x 4𝜋𝑀 e 𝑔 Dgg ↑↓ (86)
In practice, part of the spin-current can be reflected back which is taken into account by using an effective spin-mixing conductance that is equal to 63,64 ;

𝑔 Dgg ↑↓ = 𝑔 ↑↓ 1 + 2𝑔 ↑↓ /𝑔 @x (87)
Where 𝑔 @x is the spin conductance of the NM layer:

𝑔 @x = A B yC D F
M (with 𝜆 ec the spin diffusion length). In practice, the Pt is an excellent spin sink 63,65 ie 𝑔 @x ≫ 𝑔 ↑↓ , so that the 𝑔 Dgg ↑↓ ≈ 𝑔 ↑↓ .

For the YIG(20nm)|Pt(7nm) systems, such as the one previously studied at the CNRS-Thales lab 12,52 , the extra damping is typically in the range of 𝛼 3AE ≈ 1.5 × 10 |/ , much larger than the Gilbert damping of YIG (𝛼 EFú ≈ 2 -4 × 10 |ý ). The typical spin-mixing conductance of 𝑔 Dgg ↑↓ = 3.6 ± 0.5 × 10 |-G m |. is rigorously extracted in Ref 52 . The 𝑔 Dgg ↑↓ is not significantly modified when using Bi-doped YIG, as shown in Ref 13 , with typical value 𝑔 Dgg ↑↓ = 3.9 ± 0.5 × 10 |-G m |. . In general, the 𝑔 Dgg ↑↓ value depends on the quality of the interface. Thus, the growth method of the garnet as well as the deposition technique used for the Pt can impact the transparency of the interface as noted in Ref 66 .

In the work presented in this thesis, the BiYIG is grown using pulsed laser deposition and the Pt is deposited by sputtering, as in the previously mentioned studies 12,13,52 .

2) Spin torque

When applying a spin-current at the interface FM|NM, a spin torque (ST) is produced and is equal to:

𝑻 𝑺𝑻 = 𝛾 𝑡 /x 𝑀 e . 𝐽 e @x→/x ¶𝑀 SS⃗ × •𝑀 SS⃗ × 𝑒̂¿-• (88)
Where, 𝑒¿ is the spin polarization direction of the spin current (𝑒¿ = 𝑱 𝒔 𝑵𝑴→𝑭𝑴 J𝑱 𝒔 𝑵𝑴→𝑭𝑴 J

).

In the limit of weakly dissipative ferromagnets (𝛼 < 0.01), the Gilbert term is very small compared to the conservative torque, and the damping term can be expressed as:

𝜕𝑀 SS⃗ 𝜕𝑡 = -𝛾𝑀 SS⃗ × 𝐻 S S⃗ Dgg - 𝛼 𝑀 e 𝛾(𝑀 SS⃗ × (𝑀 SS⃗ × 𝐻 S S⃗ Dgg )) (89)
The spin torque being perpendicular to 𝑀 SS⃗ , it has a component along the conservative torque (field-like) and a component along the damping torque (damping like). At the current density used, the field-like torque is much smaller than the conservative torque and can thus be neglected. On the other hand, the damping torque is much smaller than the conservative torque and can be of the same order of magnitude as the spin torque.

The efficiency of the damping-like spin torque is maximal when 𝑒¿ is parallel to 𝐻 S S⃗ Dgg . For instance, for a charge current flowing in the x-direction, the polarization of the perpendicular spin-current is 𝑒̂¿ = 𝑦 '. The maximal efficiency is achieved when the equilibrium magnetization is oriented along 𝑦 '. On the contrary, no torque is applied on the magnetization if its equilibrium position is along 𝑥 ' or 𝑧.

Another crucial property is that the sign of the spin torque is controlled by the sign of 𝐽 e @x→/x , itself depending on the sign of the charge current in the Pt. Thus, depending on the sign of the current, the SOT can either increase or decrease the total damping torque. Above a certain critical current, the effective damping can thus be negative which sets off the auto-oscillation of the magnetization layer. The auto-oscillation is described in the next section using the nonlinear auto-oscillator model developed by Slavin and Tiberkevich 16 .

Let's now relate the spin-current going through the interface 𝑱 e @x→/x to the spin current 𝒋 e 1 flowing in the Pt:

𝑱 e @x→/x = T𝒋 e 1 ( 90
)
Where T is the interface transparency (unitless). In the case where the NM is an efficient spinsink (thickness larger than the spin-diffusion length λ ec ), the transparency writes 64,67 :

T = 2𝑔 Dgg ↑↓ 𝑔 @x (91)
In the case of a charge current in the 𝑥 ' direction, 𝑱 e @x→/x is such that:

𝑱 e @x→/x = -T ℏ 2𝑒 𝜃 3«¢ 𝑗 Ĉ 𝑦 ' (92)
Typical value of T = 0.2 ± 0.05 is found for systems similar to the one presented in this thesis (Ref 52 ). Finally, the spin orbit torque applied in our system can be written as:

𝑻 𝑺𝑶𝑻 = - T𝜃 3«¢ 𝛾ℏ 2𝑒 𝑡 /x 𝑀 e . 𝑗 Ĉ ¶𝑀 SS⃗ × •𝑀 SS⃗ × 𝑦 '-• (93)
We can compare it to the spin transfer torque as written in the canonical paper from Slavin & Tiberkevich 16 :

𝑻 𝑺𝑻𝑻 = 𝜖 𝛾ℏ 2𝑒 𝑡 /x 𝑀 e . 𝑗 1 ¶𝑀 SS⃗ × •𝑀 SS⃗ × 𝑦 '-• (94)
With 𝜖 the current polarization efficiency (|𝜖| ≤ 1). It is similar to the expression for SOT obtained using the SHE in Pt, with the effective charge to spin conversion efficiency T𝜃 3«¢ replacing the polarization efficiency 𝜖. This comparison allows us to easily translate the results obtained in Ref 16 . We see here an advantage of using SOT in magnonics to inject spin current over extended areas. Indeed the charge current for the spin transfer torque must be applied over the whole area S, ie 𝐼 Q¿¿O"Dc

3PP

= 𝑗 1 × 𝑆, while the current applied for the SOT is such that 𝐼 Q¿¿O"Dc = 𝑗 1 × 𝑆 AEV , with 𝑆 AEV the cross-section of the 7nm-thick Pt nanowire. The goal is to be able to comfortably reach the critical current density, for which the SOT compensates the damping torque. The maximum current that can be applied is indeed limited by the Joule heating which scales quadratically with the current.

3) Non-linear auto-oscillation regime

When the applied current is large enough, the spin torque compensates the damping torque and the magnetization in the ferromagnetic layer starts to auto-oscillate. The LLG equation being non-linear, the system can be described as a non-linear auto-oscillator. The powerful formalism developed by Slavin & Tiberkevich in Ref 16 allowed to understand and predict the experimental behavior of spin torque oscillators, based on magnetic nanopillars or nanocontacts driven by spin transfer torque. The model not only describes the non-linear dynamic of the FM layer, but can also be extended to solve other problems, such as the external and mutual phase lock-in of spin torque oscillators, or to characterize the influence of the thermal noise on the output power and linewidth. Another derivation, using the so-called Hamiltonian formalism, was developed around the same time by Krivosik and Patton to describe the non-linear spin-wave dynamics 53 in the presence of spin torque. It was later extended by Slavin, Tiberkevich and Verba 54 to antisymmetric interactions such as the Dzyaloshinskii-Moriya interaction. Both derivations (Ref 16 and Ref 53 ) are based on the Holstein-Primakoff transformation 68 to elegantly simplify the nonlinear LLG equation. However, the formalism used in Ref 16 is more accessible and is consequently the one we will be using in this section.

As we have shown in the previous section, the SOT studied in this thesis (Eq. ( 141)) is similar to the STT of Ref 16 (Eq. ( 94)) and so are the general assumptions.

The goal is to solve the LLG with the spin-torque term, without simply linearizing it, as we are interested in the non-linear behavior. The full LLG equation writes.

𝑑𝑀 SS⃗

𝑑𝑡 = 𝑇 S⃗ ˆ+ 𝑇 S⃗ | + 𝑇 S⃗ Ê (95)
Where 𝑇 S⃗ ˆ is the conservative torque, that is much larger than all the other torques: 𝑇 S⃗ ˆ= -𝛾𝑀 SS⃗ × 𝐻 S S⃗ Dgg (96) The Gilbert damping and the spin pumping torque have a similar expression and can be treated similarly:

𝑇 S⃗ Ê = 𝑇 S⃗ ú + 𝑇 S⃗ 3AE = 𝛼 + 𝛼 3AE 𝑀 e ¶𝑀 SS⃗ × •𝑀 SS⃗ × 𝑦 '-• (97)
With the damping induced by the spin-pumping:

𝛼 3AE = Qℏ V R[ ý¸x y 𝑔 Dgg ↑↓ .
The SOT is used as an anti-damping torque and is labelled:

𝑇 S⃗ | = 𝜅𝑗 Ĉ 𝑀 e ¶𝑀 SS⃗ × •𝑀 SS⃗ × 𝑦 '-• with 𝜅 = - Û± SñT Qℏ .D V R[ x y
.

The respective orientations of the torques are schematically represented in Figure 5.

The first step is to define the complex dimensionless spin-wave amplitude:

𝑐 = 𝑀 1 -𝑖𝑀 C r2𝑀 e •𝑀 e + 𝑀 3 - (98)
The magnetization can be entirely expressed as a function of 𝑐:

𝑀 SS⃗ = 𝑀 T (1 -2|𝑐| . )𝑦 ' + 𝑀 T V1 -|𝑐| . •(𝑐 + 𝑐 * )𝑧̂+ 𝑖(𝑐 -𝑐 * )𝑥 '- (99)
When |𝑐| ≪ 1, the precession amplitude is small and we are in the linear regime. When the amplitude increases, both the dynamic components 𝑀 1 and 𝑀 C increase, while the static component 𝑀 3 decreases. At |𝑐| . = -.

, the 𝑦 '-component of the magnetization is zero: the cone angle is .. For |𝑐| . > -. , 𝑀 3 is negative and the cone angle is larger than .. Later, |𝑐| . will be called 𝑝. The cone angle expression is given by:

cos(𝜃 ˆ) = 1 -2|𝑐| . (100)
Rewriting the LLG equation using the variable 𝑐 allows us to express each component as a Taylor expansion in |𝑐| .É , and study the impact of the spin-wave amplitude on the dynamics at first order (|𝑐| . ) or even second order (|𝑐| ý ) if necessary. For instance, the resonance frequency of the auto-oscillation depends on its amplitude. Using 𝑐, we can derive the non-linear frequency shift by rewriting the conservative part of the LLG equation:

¨𝑑𝑐 𝑑𝑡 © ˆWaeDdXQV"XD = -𝑖𝜔(|𝑐| . ) 𝑐 (101)
with the precession frequency depending on the spin wave amplitude:

𝜔(|𝑐| . ) = 𝜔 T + 𝑁|𝑐| . (102)
with 𝜔 T = 𝜔(|𝑐| . ), the FMR frequency computed by linearizing the LLG equation (Kittel law Eq. ( 37)).

𝑁 is the first order non-linear frequency shift coefficient. The expression is given in Ref 16 , for in-plane magnetization and for a uniform precession:

𝑁 = - ð ñ ð w ð ñ Êð [ /ý ð ñ Êð [ /. 𝜔 x (103)
In our case, the BiYIG has a uniaxial anisotropy field 𝐻 P , which can be fully taken into account for in-plane magnetization and uniform precession by replacing 𝜔 x = 𝛾𝑀 e by 𝜔 x t½½ = 𝛾(𝑀 e -𝐻 P ). In this case:

𝑁 = - 𝜔 « 𝜔 T 𝜔 « + 𝜔 x t½½ /4 𝜔 « + 𝜔 x t½½ /2 𝜔 x t½½ (104)
with 𝜔 T = 𝜔 « ¶𝜔 « + 𝜔 x t½½ • and 𝜔 « = 𝛾𝐻 T .

NB:

The normalization choice of 𝑐 is arbitrary. The non-linear frequency shift coefficient depends on the chosen expression for 𝑐 as stressed in Ref 53 .

NB2:

The rf-power emitted by spin torque oscillator is proportional to |𝑐| . which is very practical to directly measure the relative value of |𝑐| . . In this thesis, the spin-wave power is measured using micro Brillouin Light Scattering microscopy (µ-BLS, see the experimental methods). The BLS signal (S) is proportional to the out-of-plane dynamic field 𝑀 1 squared 69 , ie 𝑆 ∝ |𝑐| . (1 -|𝑐| . ).

The auto-oscillation amplitude dynamics depends on the damping and anti-damping torques. By rewriting the damping part of the LLG equation, we get:

¨𝑑𝑐 𝑑𝑡 © cQ`¿"aø = -Γ Ê (|𝑐| . ) 𝑐 (105)
with the Γ Ê , the non-linear damping rate:

Γ Ê (|𝑐| . ) = Γ d (1 + 𝑄|𝑐| . ) (106)
with the relaxation rate

Γ d = Γ ú + Γ 3AE + Γ Z« w = 𝛼 Dgg ¶𝜔 « + ð [ t½½ .
•. And the effective damping coefficient:

𝛼 Dgg = 𝛼 ú + 𝛼 3AE + 𝛾 Δ𝐻 T 2𝜔 T (107)
The contribution of the inhomogeneous broadening was shown to be significant at low frequencies in Ref 52 . In the experiment, the total damping (𝛼 ú + 𝛼 3AE ) of the BiYIG/Pt bilayer is determined using broadband FMR.

The 𝑄 factor expression is very complex and depends on the non-linearity of the Gilbert damping itself 70 :

𝛼 = 𝛼 ú (1 + 𝑞 -|𝑐| . + ⋯ ) (108)
where 𝑞 -is usually considered a phenomenological parameter. Therefore, Q is also generally considered a phenomenological parameter, that can be quite difficult to obtain experimentally.

Finally, the SOT contribution to the LLG can be written:

Γ | (|𝑐| . ) = 𝜅𝑗 Ĉ (1 -|𝑐| . ) (109)
with 𝜅 = -

Û± SñT Qℏ .D V R[ x y
and 𝑗 Ĉ the charge current density flowing in the Pt layer in the +𝑥 ' direction.

The LLG equation can thus be expressed as:

𝑑𝑐 𝑑𝑡 + 𝑖𝜔(|𝑐| . ) 𝑐 + Γ Ê (|𝑐| . ) 𝑐 -Γ | (|𝑐| . ) 𝑐 = 0 (110)
It is possible to write this equation as a system of two real equations for the power 𝑝 = |𝑐| . and the phase 𝜙 = arg(𝑐 ):

𝑑𝑝 𝑑𝑡 = -2 [Γ Ê (𝑝) -Γ | (𝑝)]𝑝 𝑑𝜙 𝑑𝑡 = -𝜔(𝑝) (111)
From the power equation, the stationary power of our system under the application of the DC current can be derived. The first trivial state is 𝑝 = 0 (no precession). At 𝑝 = 0, the dynamic is such that:

𝑑𝑝 𝑑𝑡 = -2 [Γ Ê (0) -Γ | (0)]𝑝 (112)
Thus, the state is stable as long as Γ Ê (0) > Γ | (0), ie the damping rate is higher than the antidamping rate of the SOT. Above a certain threshold current 𝑗 VM , where Γ Ê (0) = Γ | (0), the state is no longer stable, and the spin-wave power increases exponentially:

Γ Ê (0) = Γ | (0) ⟺ 𝑗 VM = Γ d 𝜅 (113)
It will then reach the stationary state, with power 𝑝 T where Γ Ê (𝑝 T ) = Γ | (𝑝 T ), which is stable in most cases.

It is practical to introduce the supercriticality coefficient: 𝜁 = . By replacing 𝛼 Dgg Γ ú by 𝜅𝑗 VM , the dynamical equation for 𝑝 ≈ 0 is such that:

𝑑𝑝 𝑑𝑡 = 2Γ d [𝜁 -1]𝑝 (114)
The solution is given by:

𝑝 = 𝐴 ⋅ exp(Γ(𝜁)𝑡) (115)
with

Γ(𝜁) = 2Γ d [𝜁 -1].
The growth rate of the exponential is positive if and only if 𝜁 -1 > 0 ⟺ 𝑗 > 𝑗 VM .

III. Experimental setups

This section is dedicated to the experimental methods used during the thesis. The usual cycle to fabricate and measure a magnonic sample is presented. First, the ferromagnetic material is grown in the lab and the films thickness and magnetic parameters are characterized. Magnonic devices are then nanofabricated using various lithographic techniques. Samples usually include the definition of nanostructures such as waveguides or discs, the deposition of rf-antennas and DC current lines. Finally, the two main techniques used in this thesis to measure the propagation of spin-waves are presented. The first is the Propagating Spin-Waves Spectroscopy 71,72 (PSWS), a fully electrical method, where the spin-waves properties are deduced from the measured impedance spectra. The second is the micro Brillouin Light Scattering microscopy (µ-BLS), a direct optical technique based on the inelastic scattering of light by the spin-waves.

A) Growth techniques

1) Pulsed Laser Deposition (PLD) of YIG and BiYIG

The Pulsed Laser Deposition (PLD) is the main technique used in this thesis to fabricate high quality garnet thin films. In a chamber, under controlled O2 atmosphere and temperature, high energy laser pulses are focused on a polycrystalline target of the desired garnet. The species are vaporized and ejected as a plasma onto a heated substrate facing the target. Depending on many parameters, the species will arrange themselves in the desired complex structure. Optimizing these parameters (O2 pressure, target distance, laser power, growth temperature, pre-treatment of substrate and target, cooling conditions…) is a very difficult and heavy work. Hence all the films used in this thesis were grown by Lucile Soumah and Diane Gouéré whose successive PhDs were focused on the growth of these high-quality garnet thin-films. However, I performed the dynamical characterizations of many films myself in order to find those that were best suited for my experiments. Understanding the physics at the "film-level" was also very important to anticipate and control the fabrication and measurement of devices on said films. I can only encourage magnonic experimentalists to stay in close contact with the films they want to use and the people growing them.

Let's review here the important step to produce high-quality Bi-doped films 13 since it is one of the main garnet I used during my thesis. Figure 6 illustrates schematically the growth procedure.

First, impurities are cleaned from the surface of the substrate (GGG or sGGG(111)) prior to the growth. The substrate must be polished on both sides to accommodate BLS experiments where a probing laser is focused through the substrate. The substrate is cleaned with an acetone bath, rinsed with propanol and dried with N2. It is then heated at 700°C under 0.4mbar of O2 pressure for 15-20 minutes to get rid of the last contaminants. It is glued on a holder using silver paste.

The silver paste must be as homogeneous as possible to ensure a uniform good thermal contact.

However, silver paste residues on the back of the sample can be problematic for BLS measurements. Thin residues can be removed using a sharp blade with little damage to the surface transparency, but large ones can hinder optical measurements.

The growth chamber contains a carrousel holding 4 different targets. Changing a target requires to expose the chamber to the air. Thus, after every change, the target is pre-ablated to get rid of atmospheric contaminants that might have deposited.

The substrate is then put in the chamber with 0.25mbar of O2 pressure, ensuring stoichiometric growth. The growth temperature is set typically between 400°C and 500°C for Bi-YIG (600°C for YIG). Temperature gives kinetic energy to the atoms to reorganize once they are on the surface. It was found to be the key parameter controlling the uniaxial perpendicular anisotropy (PMA) in our Bi-YIG films for a given thickness. The PMA emerges mainly as a result of Bi substitution in preferential dodecahedral sites during the growth. Our hypothesis is that increasing the temperature give the possibility to the Bi atoms to be redistributed more uniformly in the dodecahedral sites, hence reducing the PMA. While a certain variability of the PMA is still observed for films with similar growth conditions, one can expect 20nm-thick films to be out-of-plane for growth temperature below 410°C and in-plane for temperature above 470°C. The frequency tripled YAG laser (𝜆 = 355𝑛𝑚) is used to ablate the BiYIG target. The focus spot of about 4.2mm 2 is fixed and the target is scanned over a 1x1cm 2 region, keeping its erosion uniform. The typical laser energy is 45mJ, controlled by an attenuator and its pulse rate is 2.5Hz. The substrate position is chosen to be at a 45mm distance from the target. The energy of the laser and the distance between the substrate and the target control the energy of the atoms hitting the surface and the deposition rate (typically 2 to 3Å/s). At the end of the growth process the sample is cooled down to room temperature (about 30 minutes) under 300 mbar of O2 to provide enough oxygen to the structure during the cooling procedure. There is a significant sample variability even for constant growth parameters. This can come from different sources. For example, the laser is shone through a window in the growth chamber. The transparency of the window is affected by the successive growths and can only be cleaned by opening the chamber. Hence the effective energy of the laser on the target varies from growth to growth. Similarly, the temperature of the substrate is measured through a window that can get soiled, the amount and repartition of silver paste can impact the heat dissipation processes during the growth, the target can also be slightly off-centered. One effective way to get stable results is to perform series of samples multiple days in a row. This reduces the impact of growths of other materials by other groups in the lab and keeps most variables from significantly drifting in time.

Additionally, one of the main drawbacks of PLD is the poor spatial homogeneity of the samples.

During my thesis, we pushed the fabrication size to get to 10x10 mm 2 samples. While the center is homogeneous, the edges have very inhomogeneous magnetic properties. This was observed in FMR, Kerr and BLS experiments. However, since the coating of resist for lithography is inhomogeneous on the edges, it allows us to have a perfectly even coating in the central homogeneous region (area ranging from 6x6mm 2 to 8x8mm 2 ).

2) Other techniques

Phase liquid epitaxy (LPE) is another growth technique. It is less flexible than PLD but outperforms PLD grown films in terms of damping in the case of YIG and BiYIG. The rotating substrate is dipped in a liquid bath made of different chemical solutions: reactants that bring the Y, Fe and O2 needed and other elements that regulate the reaction. The main advantage is the reproducibility of the technique and the stellar magnetic damping that are obtained. The main drawbacks are the thickness inhomogeneities. The presence of heavy ions like Pb that can make the YIG slightly conductive at high temperature [START_REF] Thiery | [END_REF] . Very good BiYIG films have been grown in the team of J. Ben Youssef at the LabSticc in Brest.

The minimal thickness obtained is 120nm as of now. The growth speed is still at the moment really high (~10-20nm/s), making it impossible to create ultra-thin (20-30nm) films.

YIG films are now also being grown using off-axis sputtering. In a low pressure Ar-atmosphere, species are stripped from a YIG target using ionized Ar bombarding and deposit amorphously on a substrate. After the deposition, the crystallization is done by multiple annealing processes under controlled oxygen pressure.

B) Characterization techniques

Once the films are grown, their structural and magnetic properties are characterized. It is crucial to know our films well, both to select the ones that are best suited for certain magnonic devices and also to provide a solid basis to understand more complex magnonic experiments. I mainly performed dynamic characterizations, so I will only briefly describe both structural and static magnetic characterizations.

1) X-ray Diffraction and Reflectivity

Crystalline structures are ordered on the atomic scale therefore they are diffracting X-Ray light in which 𝜆~Å. Hence our main structural characterization is X-ray Diffraction (XRD) and Reflectivity (XRR). The latter gives access to the film thickness with a precision of less than 1nm while the XRD gives more information about the crystal structure like the interatomic planes spacing, and the epitaxial strain. More details can be found in the thesis of Lucile Soumah [START_REF] Soumah | Pulsed Laser Deposition of Substituted Thin Garnet Films for Magnonic Applications[END_REF] .

2) Ferromagnetic resonance (FMR) setup

During my thesis, I was mainly interested in the magnetic properties of the films and my main characterization tool was broadband ferromagnetic resonance as it quickly provides both the effective magnetization, damping and inhomogeneous broadening of the film. The sample is placed on a radiofrequency strip line. The derivative of the power absorbed by the spin precession is precisely measured as a function of the external magnetic field for a constant frequency. This configuration allows the external magnetic field to be modulated at low frequency (𝑓 T ≈ 73 𝐻𝑧) to perform a homodyne detection and consequently achieve a significant reduction of the noise using a low-pass filter.

The voltage obtained through this lock-in detection scheme is proportional to the derivative of the imaginary part of the Kittel susceptibility tensor 𝜒 † ll . The ferromagnetic resonance field 𝐻 dDe and linewidth 𝛥𝐻 can thus be obtained by an anti-Lorentzian fit. By repeating the measurement for different frequencies and plotting 𝜔(𝐻 dDe ) one can extract the effective magnetization from the Kittel Law (Eq. ( 37) for general case, Eq. ( 43) for a BiYIG film). Plotting 𝜔(𝛥𝐻) gives access to the Gilbert damping constant and inhomogeneous broadening (Eq. ( 66)) in the case where the field dependence of the inhomogeneous broadening is small enough. The results of temperature dependent FMR measurements study, conducted on 20nm-thick BiYIG films, are presented in Part. 1 Chapter. 1 of the present manuscript.

C) Nanofabrication techniques

Once the ferromagnetic film is characterized, standard nanofabrication techniques are used to define magnetic nanostructures, rf-antennas and DC conducting lines.

In this thesis, both laser and e-beam lithography were used. Although the resolution of e-beam (~10 nm) is much smaller than of laser lithography (~500 nm), it was not as accessible (and stable) as the laser lithography, so both techniques were used. The operating principle is identical for both methods. An organic resist is spin-coated on the magnetic sample. The resist is photo-sensitive (laser lithography) or electro-sensitive (e-beam). The resist is prepared for insolation using baking and chemical baths depending on the recipe used. It is then locally insolated using a laser beam or an ion beam. Using a developer (base), the resist is removed in the locations where it was insolated (resp. not insolated) for so-called positive resist (resp. negative resist). Metals can then be deposited on the sample and will only stick in the resist holes after plunging the sample in an acetone bath (lift-off process). The metallic nanostructures can then be used as such (rf-antennas, DC lines) or as a hard mask to protect the magnetic layer from etching. In the latter case, the metal can be removed after the etching process using a specific chemical etching. Insulating materials can also be deposited using chemical vapor deposition (CVD) or atomic layer deposition (ALD).

We now describe in detail the fabrication of the sample studied in Part 1 of this manuscript, as it was the most complex sample fabricated and is representative of most of the fabrications done during this thesis. This section is mainly aimed at the next generation of Ph.D students as it records the specific steps and good practice.

Step0: Make sure that the surface of the sample is clean -The sample can be cleaned with an acetone bath (few minutes to 24h if necessary), rinsed with propanol and properly dried with N2.

-If some residues are left, it is possible to clean it with a cold O2-plasma (30s to 10minutes if necessary).

Step1: Alignment marks and numbering (Figure 7-a)

-Opposite dual L-shape marks are used. This type of alignment marks are well visible, give the sample orientation (asymmetric shapes) and give a good alignment even when the pattern is overdosed or underdosed. -Square marks (8µm-wide) are also used for the automatic recognition software of the ebeam. -For each device, it is important to have multiple square marks in order to have spare ones for the whole fabrication process. -Alignment marks can be defined using laser lithography, but if an alignment better than 100 nm is necessary, then e-beam should be used. -One can typically use Ti(20nm)/Pt(80nm) composition or Ti(20nm)/Pt(40nm)/Ti(40nm) if a long etching process is planned. The Pt is useful as a heavy element to have a high contrast for the e-beam while the hard Ti protects the mark from etching. The bottom Ti layer is used as a sticking layer (absorption of oxygen atoms forming Titanium oxide). -Numbering devices with large patterns is very useful for experiments, typically 25 to 50 µm long numbers are used.

NB:

The electronic charges of the ion-beam need to be evacuated. Thus, the use of a special conductive e-resist is necessary on e-beam steps on insulating samples (like BiYIG).

Step2: Pt patterns:

-Pt is used to apply SOT on the ferromagnetic layer. Pt constrictions can be defined to inject high current density locally (as in Ref 14 ). E-beam lithography can be used to define subµm constrictions. In this sample however, the Pt was deposited on the full film. -The typical Pt thickness deposited is 7nm, which is very easy to lift-off. The Pt is deposited using sputtering. A soft 30" O2-plasma is done in situ to clean the surface of any resist residue just before the deposition. -Pt being a heavy atom, it can modify the electronic dose received by the sample. When a full layer of Pt is present, a lower dose must be used.

Step3: Magnetic nanostructures (Figure 7-b)

-
The goal is to define BiYIG/Pt waveguides (or other shapes like nanodiscs). The magnetic nanostructures are protected either using negative resist or by a hard mask (metal deposited in areas defined by positive resist). The unprotected parts are etched away using an Ar ion beam (IBE). -To avoid "bunny-ears", etching residues along the resist flanks that can be hundreds of nm high and cause bad connections when depositing metals on top, we use a hard-mask method. The mask thickness is small which limits the redepositing of matter. -The typical etching times for standard IBE parameters are given in Figure 8. For instance, if one wants to etch a BiYIG(20nm)/Pt(7nm) film, the hard mask should protect the defined structure for at least: 𝑡 DVˆM = 2′30 + 28′′ ≈ 3′. It is always important to take a bit of margin.

For instance, let's take a 1′ margin. In this case, one can use 15nm of Ti (~2'15) and 40nm of Al (~1 '45). It is important that all the Ti is removed by the etching, because it cannot be removed by selective chemical etch. Thus, the etching time must be at least 2'30 long to make sure the Ti is gone. Then, one can monitor the GGG peak in the SIMS to stop the Step 1

Step 3

Step 4

Step 5 etching. In any case, the maximum etching time should be 4'. It is better to have some residues of damaged BiYIG on the side than risking etching the waveguide. -After the etching, the hard mask is removed using chemical etching. Developer MF319 is used to remove Al. It etches the aluminum oxide very quickly, but the oxide needs time to form. We usually leave it for a day. -When the YIG or BIYIG is bare (no Pt on top), it is important to protect its surface from elements that tend to pump oxygen, like Ti or Al. In this case, we start by depositing 10nm of gold by sputtering (evaporation would not stick to the surface) as a neutral chemical barrier between Ti (or Al) and the YIG. To remove the gold, the sample in plunged 25'' in a potassium ionide bath (KI . ) and abundantly rinced with EDI (deionized water) for 2'30". For this reason, it is not recommended to use gold in the alignment marks but rather Pt.

Step4: Insulating the Pt line (Figure 7-c)

-DC current flowing in the Pt layer must not leak to the rf-antennas that will be deposited on top of the Pt. Therefore, an insulating layer is deposited on top of the Pt. -To avoid inhomogeneous stress on the BiYIG waveguides, the insulating layer is applied all over the BiYIG active parts while making sure that a part of Pt is still accessible to contact it to DC pads. In this sample, we chose wide 10 µm × 30µm rectangles.

NB:

There must be no resist left at the bottom of the squares before depositing the insulating layer. The presence of resist residues can generate huge stress on the BiYIG/Pt that can interrupt the electrical connection through the Pt layer.

-The insulating layer is either composed of 30nm-thick SiO2 deposited at 100°C with Chemical Vapor Deposition (CVD), or 10-30nm Al2O3 deposited by Atomic Layer Deposition (ALD) at 100°C. ALD is a softer method, but Al2O3 is sensitive to basic developer (like MF319 or MF CD-26), meaning that these products must not be used in the future steps. Recently, an HfO2 ALD recipe was developed by B. Dlubak at the lab. It is a high dielectric insulator and is not sensitive to developers. -Both ALD and CVD are conformal, thus it is safer to use small thicknesses and wide patterns to ensure an easy lift-off. In this sample 30 µm × 20µm rectangles of 30 nm-thick SiO2 were lifted.

Step5: Define rf-antennas (Figure 7-d)

-Using e-beam lithography, 200nm-wide antennas are defined and Ti(20nm)/Au(80nm) is evaporated.

Step6: Contact DC and rf-antennas -Using laser lithography, we contact the DC and rf-lines to large pads. The use of a µm-thick resist with stooped resist sidewalls (recipe with chlorobenzene) makes the lift-off very simple. Typically, Ti(20nm)/Au(200nm) is deposited.

D) Measuring spin-waves

1) Propagating Spin-Wave Spectroscopy

Propagating Spin-Wave Spectroscopy (PSWS) simply consists in the excitation and detection of spin waves in a ferromagnetic waveguide thanks to metallic inductive antennas deposited on top (see Figure 9). By flowing an rf-current 𝐼(𝜔) in the exciting antenna, a local rf magnetic field is created in the vicinity. For frequencies within the dispersion relation at the applied static magnetic field, spin-waves will be excited with an efficiency depending on the overlap between the rf-field distribution and their wavelength. Spin-waves then propagate until they reach the detection antenna through which they generate a varying flux, i.e. an rf-voltage 𝑉(𝜔).

The properties of the spin-waves can then be deduced by analyzing the inductance relating 𝐼(𝜔) to 𝑉(𝜔). As each antenna can serve for both excitation and detection, an inductance matrix is actually measured:

By measuring the inductance matrix out of resonance, at a reference field where no spin-waves exist at the frequencies of interest, typically 1 to 20 GHz, and subtracting it to 𝔏, one can isolate the spin-waves contribution to the inductance:

𝛥𝔏 = ¨𝛥𝐿 --𝛥𝐿 -. 𝛥𝐿 .-𝛥𝐿 .. © = A 𝐿 ---𝐿 -- dDg 𝐿 -. -𝐿 -. dDg 𝐿 .--𝐿 .- dDg 𝐿 .. -𝐿 .. dDg B (117)
From the analysis of the 𝛥𝔏 matrix, the spin-wave propagation properties can be extracted. The group velocity is deduced from the 𝛥𝐿 .-phase, while the attenuation length is derived from the ratio between the transmission and the reflection spectra amplitudes as described in Ref 72 . Detailed analysis of spin-wave induced inductance spectra measured at room temperature can be found in Part 2 chapter 2 and in Part 3 of this manuscript.

2) Brillouin Light Scattering

Micro-focus Brillouin Light Scattering (µ-BLS) is an optical method that allows to directly image the local dynamic magnetization of a magnetic sample. The monochromatic laser light is focused on a magnetic sample in a diffraction limited spot and the light is inelastically scattered by the magnons. The inelastically scattered light signal is analysed using a Tandem Fabry Pérot Interferometer (TFPI) and a single photon detector. The photon count is eventually proportional to the number of magnons, and contains the information concerning their frequency and phase. The technique also takes advantage of the ultra-fast character of light so that time resolved measurements can be performed with a resolution down to 0.2 ns. Consequently, µ-BLS is the ultimate experimental tool for magnonics. It is however not suited for an integrated detection of spin-waves as a µ-BLS setup currently necessitates a few m . in a temperature-controlled room.

The interaction between photons and magnons is first described. Then the role of the Fabry Pérot Interferometer (TFPI) is detailed. The TFPI is critical to the sensitivity and the frequency resolution of the detection. Finally, the full setup used in this thesis is presented.

Inelastic scattering of light by magnons:

The principle of BLS relies on the inelastic scattering of light by spin-waves under the conservation of energy and momentum. The frequency and wavevector of the scattered photon (𝜔 e , 𝑘 e ) are thus related to the frequency and wavevector of the magnon (𝜔 `, 𝑘 `) and of the incident photon (𝜔 " , 𝑘 " ) by :

𝜔 e = 𝜔 " ± 𝜔 ` (118)
𝑘 e = 𝑘 " ± 𝑘 `

The plus-minus indicates either the creation of a magnon (-) or the annihilation of a magnon (+) in the process. The spectrum of the reflected light is thus composed of a large peak of elastically scattered light at frequency 𝜔 " and two side peaks, called Stokes (𝜔 " -𝜔 `) and anti-Stokes (𝜔 " + 𝜔 `) peaks. Classically, the inelastic scattering of light on spin-waves can be described as the interaction of the light with the spin-wave modulated dielectric tensor. The frequency shift is interpreted as the doppler shift due to the spin-wave phase velocity. The intensity of the scattered light is proportional to the spin-wave intensity 75 . In the case of µ-BLS, where the incident and scattered laser beam is orthogonal to the surface, only the out-of-plane dynamic component contributes to the intensity of the Stokes and anti-Stokes peaks.

It is important to note that the magnons only interact with the off-diagonal components of the dielectric tensor (Faraday-effect or magnetic birefringence effects). Thus, the light polarization is shifted by 90° after interacting with magnons. This effect allows us to dissociate the magnonscattered photons from the photons scattered by other sources like phonons by sending polarized light on the sample and using a second polarizer to select the magnon-scattered photons.

Every magnetic material has a different spectrum of absorption and a different Faraday rotation angle. For instance, it is better to use a blue (470nm) light laser for YIG because the magnetooptical effect is extremely small for green light (532nm). The BLS signal is proportional to the Faraday rotation angle (expressed in °/µm). The use of Bi1YIG strongly enhances the Faraday rotation of YIG by nearly two orders of magnitude for visible light in PLD grown thin-films 13 , while keeping the other parameters (𝑀 e and absorption) similar to YIG, leading to a large increase of the BLS sensitivity. The high sensitivity of BiYIG was instrumental to perform unprecedented time-resolved measurements in Part 1 of this thesis.

Tandem Fabry Pérot Interferometer:

When the light goes through a Fabry Pérot Interferometer (FPI), it interferes, and the transmission is maximal for wavelength such that:

𝜆 = 2𝑑 𝑛 (120)
Where d is the distance between the two mirrors.

After going through the FPI a single time, the transmitted intensity is given by:

𝐼 V = 1 1 + 4𝐹 . 𝜋 . sin . ¶ 2𝑑 𝜆 𝜋• 𝐼 (121)
with 𝐹, the finesse that depends on the mirror reflectivity (𝑅 ≈ 0.95 -0.97):

𝐹 = 𝜋√𝑅 1 -𝑅 ( 122
)
where 𝐹 is typically of the order of 80-130. Let's consider the light coming from the elastic scattering, with wavelength 𝜆 = 532 nm and frequency 𝑓 T = 𝑐/𝜆. When scanning the mirror distance of a FPI between a|-. λ and aÊ-

. 𝜆, we see the maxima of intensity at the transmission order 𝑛 -1, 𝑛 and 𝑛 + 1 (Figure 10).

A frequency at 𝑓 T ± 𝑓 `, will appear by scanning the mirror in the same transmission order as 𝑓 T , if:

𝛿𝑑 < 𝜆 2 (123)
As 𝛿𝑑 = g h g w 𝑑, we obtain that the frequency difference 𝑓 ` must satisfy:

𝑓 `< ∆𝑓

/3f = 𝑐 2𝑑 = 150 𝑑 [𝑚𝑚] GHz (124)
where ∆𝑓 /3f is the Free Spectral Range (FSR). The larger the distance between mirrors, the smaller the FSR. For instance, in a usual BLS experiment, the magnon frequencies are of the order of 0 to 10GHz. Thus, setting the mirror distance at 10mm (∆𝑓 /3f = 10 GHz) allows to differentiate between two magnon frequencies by scanning the mirror distance around the transmission order 𝑛 (𝑛 = l .c

B m).

The transmission function introduces a certain broadening. The width at half maximum is given by solving 𝑇(𝑓 ± ∆𝑓 /n«x /2) = -.

:

∆𝑓 /n«x = ∆𝑓 /3f 𝐹 (125)
Thus, it is possible to resolve only ⌊𝐹⌋ frequencies between 0 and ∆𝑓 /3f . A small ∆𝑓 /3f leads to a better frequency resolution.

In practice, in the µ-BLS setup used in this thesis, the light goes six times through an interferometer (see Figure 11). The contrast, defined as the ratio between the maximum intensity (1) and minimum intensity ¶1 + ý/ F ¸F •, is enhanced if the light goes multiple times through the interferometer. For an N-pass interferometer, the transmission is such that:

𝑇 @ = 𝐼 V 𝐼 = 𝑇 @ (126)
And the contrast scales as:

𝒞 @ = 𝒞 @ ∝ 𝐹 .@ (127)
The FSR remains the same, but to find the ∆𝑓 /n«x , one must solve

T(𝑓 ± ∆𝑓 /n«x /2) = ¶ - . • E r : 4𝐹 . 𝜋 . sin . ¨∆𝑓 /n«x 2∆𝑓 /3f 𝜋© = ( √2 r -1) (128) ⇒ ∆𝑓 /n«x = ∆𝑓 /3f 𝐹 @
with the finesse, enhanced by the multiple passage:

𝐹 @ = 𝐹 V √2 r -1 (129)
For 𝑁 = 6, the finesse is enhanced by a factor 2.85.

The BLS measurements presented in this thesis were performed with mirrors distance of 𝑑 = 10 mm, with ∆𝑓 /3f = 15 GHz and ∆𝑓 /n«x = 50 MHz (finesse ~300) and with 𝑑 = 20 mm to reach a ∆𝑓 /n«x = 25 MHz resolution.

To suppress the contribution from the other transmission orders, two interferometers are used in tandem, with an angle 𝛼 as represented in Figure 11 (taken from Ref [START_REF] Wang | Linear and Nonlinear Spin Waves in Nanoscale Magnonic Structures for Data Processing[END_REF][START_REF] Hillebrands | Light Scatt. Solids VII Cryst.-Field Magn. Excit[END_REF] ). In this configuration [START_REF] Lindsay | [END_REF] the distance between the mirrors obeys the following conditions: As the elastic peak is very bright compared to the magnon-scattered signal, there will always be a "ghost" peak visible at the FSR frequency. However, the scheme successfully erases the high order magnon peaks, so that the frequency of the peak within the FSR is unambiguously defined.

𝑑 -= 𝑑 - T + ∆𝑑 𝑑 . = 𝑑 . T + ∆𝑑 cos(𝛼) 𝑑 - T ≈ 𝑑 . T cos(α) .c E w B = 𝑛 -; .c F w B = 𝑛 . (130)
µ-BLS setup:

The full µ-BLS setup is shown in Figure 13. The light emitted from a laser (green at 532 nm or blue 473 nm) is expanded by a beam expander and then spatially filtered with a pin hole aperture to keep a single spatial mode. The laser can then be focused onto the magnetic structure in a diffraction limited spot. The utilized objective lens 1 with a numerical aperture of 0.85 and a magnification x100 enables a focal diameter down to 250 nm. A polarizer is introduced to filter the 90° shifted light scattered by the magnons. The scattered light is brought to the tandem 1 Here we use a 2-pass finesse of 35 for clarity. But the suppression is much stronger for higher finesse such as the one used in the real setup. The elastic peak is also much brighter than both inelastic peaks in practice.

LCPLFLN-LCD by Olympus

Δ𝑓 - /3f n 1 +1 Transmision order n 2 n 2 -1 n 2 +1 𝜔 Light spectrum 𝜔 + 𝜔 ` 𝜔 -𝜔 ` 𝑓 ` 𝑓 ` Δ𝑓 - /3f
interferometer2 using 𝜆/2 wave-plate. The stage of the TFPI is continuously scanned, so that the magnon frequency is known. The signal intensity is measured using a semiconductor single photon detector 3 .

The sample is placed on a piezoelectric stage, illuminated by an orange light source. The illumination light is brought back to a CCD camera and used to position the desired structure under the laser spot.

A LabVIEW program controls the setup. In particular, because of thermal and mechanical drifts, the TFPI mirrors' position and orientation must be adjusted constantly. To do that, the program uses the reference elastic peak. Thus, when the TFPI scans the frequencies, it must always go through zero to measure the elastic peak. It is possible to define a region of interest (ROI) where the stage will scan more slowly, thus increasing the accumulation time in the area. The stage must scan the elastic peak approximately every second. The TFPI is less stable when the mirror distance is increased (30mm maximum distance). The LabVIEW program also takes care of the measurement's orchestration, sweeping the magnetic field, DC current, positions, rf-current frequency and power, time and frequency resolution.

Part 1: SOT in BiYIG/Pt bilayers Chapter 1. BiYIG unique properties

For a long time, magnonics was envisioned as the future of RF components. For these applications, Yttrium Iron Garnet (YIG -Y3Fe5O12) was used as it features the lowest damping of any known ferromagnet by at least an order of magnitude. However, at room temperature YIG's magnetization is sensitive to temperature drifts due to its relatively low Curie temperature 79 (560K). Hence, temperature regulation is necessary for all real-world applications, which contributes to severely hinder the development of magnonic rf-components and other applications.

With the progress in materials characterization and growth techniques, the design of new materials has been critical to unlock new applications and unveil new phenomena in all areas of physics in the past twenty years. For instance, magnonics has benefited from the development of high quality CoFeB films for spintronic application. It is now a widely used metallic ferromagnet with a low Gilbert damping 17,80 (~ 4 10 -3 ), highly tunable properties and high Curie temperature. However, due to it large magnetization, the high Curie temperature (1300-1400K) of CoFeB is not sufficient to guarantee a better thermal stability than YIG above 8 GHz. Indeed, as the magnetization is much smaller in YIG than in CoFeB, its temperature dependence is less impactful at high frequencies where the resonance frequency is mainly defined by the external magnetic field in YIG.

The temperature dependence of CoFeB magnetic properties has attracted a lot of interest (see 81- 84 and references therein) for spintronic applications 85 . In particular, for thin films, a strong perpendicular magnetic anisotropy (PMA) arises from interfaces. More importantly, it is possible to modulate the PMA and its temperature dependence by changing the materials at the interface and the film thickness. These complex dependences unlock the possibility to engineer the anisotropy to solve complex problems. For instance, it can be used to obtain an easy cone anisotropy, leading to faster and more deterministic switching of the FM layer 82 . Is it possible to engineer an ultra-low damping ferromagnet like YIG to obtain better thermal properties as it was is with CoFeB?

One way to modulate the YIG magnetic properties is to dope it. The effect of doping has been studied since the 1980s 86 . For example, doping with Bi 79 is used to bring a large magneto-optic effect and PMA. Recently, Lucile Soumah, former Ph. D student at the CNRS-Thales lab, showed that it induces large and controllable perpendicular magnetic anisotropy also in extrathin films (20nm) while preserving the very small damping of YIG 13 and inducing a large magneto-optic effect (about 50 times larger than YIG). In this thesis, the temperature dependence of the magnetic properties of BiYIG ultra-thin films (PLD and LPE grown) is studied for the first time. It reveals the effective magnetization (𝑀 Dgg ≝ 𝑀 e -𝐻 P ) can be tailored to either increase or decrease with temperature. We also evidence that for films with vanishing effective magnetization (𝑀 Dgg ≈ 0), high thermal stability is achieved on a very wide temperature range: from 260K to 400K, where the frequency thermal drift is 50 times smaller than that of YIG. These properties can be leveraged for new applications where the temperature dependence of YIG is detrimental, while keeping the extremely low damping and insulating character of YIG 13 . For instance, we show that this was crucial in order to demonstrate the excitation of coherent spin-waves using SOT described in Ref 14 .

In this chapter, additional peculiar properties of BiYIG observed during this thesis will be discussed. This include spin-waves propagation properties (attenuation and phase conservation) and the effect of patterning on the magnetic properties.

I. Temperature dependent FMR measurements

In this part, we measure a series of PLD-grown ultra-thin Bi1Y2IG film (20 nm < t < 30 nm) using temperature dependent FMR setup between 260K and 400K. Depending on the growth conditions, the magnetization easy axis is varied, from in-plane to out-of-plane. We observe a non-standard temperature dependence of the magnetic anisotropy. An 80nm LPE grown sample has been also measured and shows similar features, suggesting that the observed properties are general in BiYIG film with PMA dominated by growth anisotropy.

A) Experimental results:

Broadband ferromagnetic resonance measurements are performed in the in-plane configuration with the field-modulation technique using a PPMS setup from Quantum Design at the UMPhy laboratory.

In Figure 14 (a), the FMR signal (∝ cw Ï c« ) at 300K and at 10 GHz is plotted as a function of applied field for a Bi-YIG film. The resonance field 𝐻 dDe and the FMR linewidth Δ𝐻 are extracted by fitting the resonance with the derivative of a Lorentzian:

𝑑𝜒 𝑑𝐻 = 2(𝐴 + 𝐵)Δ𝐻 (𝐻 -𝐻 dDe ) Δ𝐻 ((𝐻 -𝐻 dDe ) . + (Δ𝐻) . ) . -

𝐵(𝐻 -𝐻 dDe ) Δ𝐻 ((𝐻 -𝐻 dDe ) . + (Δ𝐻) . ) + 𝐶 WggeDV

The rf-frequency is then varied, and the resonance fields are extracted from the measured resonance curves (see inset Figure 14-a). In Figure 14-b, the extracted resonance fields are plotted as a function of the rf-frequency for three different films with different anisotropies at room temperature. For an in-plane film, extrapolating the linear part yields a positive intercept (blue curve), while for out-of-plane easy axis film we get a negative intercept (red curve). The green line represents the case of a film close to compensation of anisotropy where the resonance field is proportional to the rffrequency and the intercept goes through zero. These points follow the Kittel formula from which the effective magnetization (𝑀 Dgg ) and gyromagnetic (𝛾) ratio can be extracted:

𝑓 dDe = 𝛾 ⋅ r𝐻 dDe ⋅ (𝐻 dDe + 𝑀 Dgg )
The result of the Kittel formula fit (dotted lines in Figure 14-b) is shown in the table (see inset).

The in-plane (resp. out-of-plane) sample features a positive (resp. negative) effective magnetization. The sample for which 𝑀 Dgg ≈ 0 is called compensated (or close to compensation) as the shape anisotropy is compensated by perpendicular magnetic anisotropy. While the gyromagnetic ratio found for sample with low anisotropy (in-plane) is the one of YIG, a progressive increase of 𝛾 is observed for films with increasing perpendicular anisotropy. This was previously observed in transition metallic ferromagnets in presence of anisotropy 80,87,88 , confirming theoretical predictions 89,90 . In these materials, the anisotropy is related to the asymmetry in the orbital moment 𝜇 z , which is directly linked to the Landé gfactor:

ø|. . = v { v S
(with 𝜇 3 = 𝜇 | the spin moment). As discussed in the following paragraphs, the physical origin of the anisotropy is not as clear in Bi-doped YIG films 79 . The measured increase in the Landé g-factor suggests that the induced anisotropy is also related to an asymmetry in the orbital moment 𝜇 z in BiYIG films. Obtaining a more quantitative understanding of the gyromagnetic ratio would require more precise FMR measurements and analysis as stressed in Ref 91 .

𝜇T𝑀Dgg (mT) 𝛾 (GHz/T)
This fitting procedure is repeated for temperatures from 260K to 400K every 20K. The extracted effective magnetizations are plotted as a function of temperature in Figure 15 for seven BiYIG films grown by PLD on sGGG substrate, whose thicknesses vary from 20nm to 30nm. These films were growns using different growth conditions during the thesis of Lucile Soumah. They were not selected for any particular properties other than being Bi1YIG films of thickness between 20nm and 30nm and having a wide range of anisotropy values. To our knowledge these films are equally strained.

On first approximation, samples with 𝑀 Dgg ≈ 0 show no dependence on temperature. Thus, at first order, 𝐻 P (𝑇) = 𝑀 e (𝑇). In-plane easy axis samples of BiYIG have a similar 𝑀 e (𝑇). dependence as YIG.

Figure 16 shows the dramatic reduction of the resonance field variation due to the temperature obtained for the films near compensation and compared to a "YIG-like" in-plane film. The resonance curve at 8GHz at 260K and 400K are compared for an in-plane film and a film close to compensation. While the in-plane sample resonates at a field nearly 30mT higher, the resonance is shifted by 1mT between 260K and 400K for the film close to compensation. The shift gets even smaller at high frequency, on average, the field shift is 50 times smaller for the measured frequencies.

In particular, the linewidth being of the order of 1mT, the resonances overlap from 260K to 400K. Hence operating a magnonic devices on this kind of films would not necessarily require any temperature control within this wide temperature range. Note that the "multi-peaks" shape of the in-plane sample is simply due to the spatial inhomogeneity of the PMA [START_REF] Soumah | Pulsed Laser Deposition of Substituted Thin Garnet Films for Magnonic Applications[END_REF] . The measured samples are 10mm×10mm and their resonance properties are not always homogeneous over the whole sample, especially at the edges and corners (see II.A). In Figure 17, the temperature dependence of the effective magnetization of a 20 nm thick PLD grown Bi1YIG films (green curves) is compared to an 80nm LPE grown Bi1.45YIG (black curve). The similar behaviors suggest a general temperature behavior for BiYIG films whose anisotropy is dominated by bulk contribution (independence on thickness and the exact Bi content).

B) Discussion

It has been established that the dominating PMA contribution in PLD-grown Bi1YIG/sGGG films comes from growth anisotropy 13,[START_REF] Soumah | Pulsed Laser Deposition of Substituted Thin Garnet Films for Magnonic Applications[END_REF] , often ascribed to the hypothesis that Bi 3+ atoms occupy preferential dodecahedral sites, resulting in a symmetry breaking that induces a magnetic anisotropy contribution. This hypothesis is confirmed by the recent work conducted in the laboratory by Diane Gouéré (to be published). Full control of the PMA is achieved by modulating the growth temperature while maintaining a constant film composition. Hence two equally strained films can have completely different anisotropies, confirming that the main term comes from growth anisotropy. Similarly, Bi1.45YIG/sGGG LPE films were chosen in order to perfectly match the sGGG lattice constant, thus minimizing the magneto-elastic anisotropy contribution.

The measured behaviors seem to indicate that growth anisotropy varies proportionally to the magnetization at first order ie:

𝐾 P (𝑇) 𝐾 P (0) ∝ A 𝑀 e (𝑇) 𝑀 e (0) B . ( 131
)
This unusual dependence was observed in FePt L10 ordered films 92,93 with an exponent 𝛾 = 2.1 that deviates from the 𝛾 = 3 established from the Callen-Callen single ion theory 94 . The conclusion of those studies was that the induced moment on the Pt atoms adds a dominating two-ion anisotropy term in the Hamiltonian, responsible for a 𝛾 = 2 contribution 95 (yielding the 𝛾 = 2.1 exponent that includes both contributions: the single-ion and two-ion anisotropies).

Mryasov et al. point out "We believe that this mechanism is common for various 3d-5d/4d ordered alloys having 5d/4d nominally non-magnetic elements with large s-o coupling and Stoner enhanced susceptibility"

The precise mechanisms of growth induced anisotropy are still unclear despite the significant research effort that was conducted on garnets in 1980's and 1990's spurred by bubble memory and optics applications. XCMD experiment 96 revealed in 2009 that despite being diamagnetic the Bi 3+ atoms carry an induced orbital moment and its direct influence on Fe 3+ electronic state dominates the indirect mechanisms from the O 2-neighbors. These mechanisms were initially thought 97 to be the main mechanism responsible for the growth induced anisotropy since the Bi 3+ was supposed to carry no orbital moment. This two-ion mechanisms (Fe 3+ -Bi 3+ ) could be the reason for the temperature dependence, observed in Ref 79 and the present work, not matching the single ion theory as pointed out in ref 97 . We believe that the mechanism at play in our films is similar to the one exposed by Mryasov 95 despite Bismuth not being in the 5d/4d transitional metal category. Indeed, YIG was substituted with many different cations 86 and large growth induced anisotropies were reported only when doping with transitional metals and with Bismuth. This could mean that the induced moment on Bi 3+ and the hybridization of Fe 3+ 3d orbitals with the Bi 3+ 6p orbital makes Bismuth play a similar role as transition metals in the process described by Mryasov.

For films close to compensation, the temperature dependence of smaller anisotropy term is likely to play a role:

-Like YIG, BiYIG films has a small cubic anisotropy term (𝐻 Q ~ 6 mT). As a single ion term, it is expected to vary with a 𝛾 = 3 exponent (as shown in Ref 94 for YIG) -The Bi1YIG films are not perfectly matched with the sGGG substrate (0.27% mismatch 13 ). This creates an additional small magneto-elastic anisotropy estimated at a few tens of mT in Ref 13 (see supplementary materials). -Additional terms can arise from depositing metals on top of our films. For instance, adding a thin Pt layer modifies the effective magnetization by approximatively -10 mT (supplementary of Ref 14 ). As no induced moment nor DMI was found to emerge at the interface, we attribute this term to anisotropy. Most probably, the Pt induces a relaxation of the strain, thus lowering the magneto-elastic anisotropy term. The presence of a surface anisotropy term cannot be presently ruled out however.

These terms might ultimately prevent achieving a perfect temperature stability over a wide temperature range. It seems however possible to engineer this kind of stability on a smaller temperature range.

Controlling the temperature dependence of BiYIG resonance by the engineering of anisotropy opens up new opportunities in the field. An example of that is the possibility to emit coherent propagating magnons using SOT 14 . Indeed, injecting DC current through a Pt layer results in significant heating when reaching current densities needed for the auto-oscillation threshold current (the temperature increase is estimated to be between 50K and 100K in Ref 7,12 ). For YIG and other typical ferromagnets, this creates a potential well under the Pt area, confining the auto-oscillations and preventing the excitation of propagating spin-waves 7 . By using a compensated, slightly out-of-plane BiYIG, the creation of this potential well is prevented. The auto-oscillations can radiate coherent spinwaves outside of the excitation area as their frequency now lies within the dispersion relation of the full film. The fabrication simplicity (a single step of lithography) contrast with the one of Non-Local Spin Injection devices proposed in Ref 7 or other implementations of SOT emitters. It is also free of the problems associated with inductive antennas. Namely, it needs no complex rf circuits, its excitation efficiency is independent of the size of the emitter, and it can efficiently generate magnons with very small wavelengths. By using a sample with higher anisotropy, one could control the excited frequency with the DC current thanks to the positive temperature frequency shift.

II. Other properties of ultra-thin BiYIG films

In this part, other material-related properties observed in ultra-thin BiYIG films during the thesis are presented. In particular, the presence of PMA does not simply add a constant term in the dispersion relation but has implications on the spin-wave propagation in plain and patterned films.

A) Spin-wave propagation in non-structured BiYIG films

One drawback of high PMA ferromagnets is the inhomogeneity of the PMA amplitude distribution, though the homogeneity of our BiYIG films is much smaller than the one of metallic alloys thin films like CoNi 4 . One direct repercussion of PMA inhomogeneity is the broadening of the linewidth due to the distribution of resonance conditions. Consequently, in anisotropic ferromagnets, the linewidth is often dominated by the PMA enhanced inhomogeneous broadening 13,88,98 . It can also lead to multiple peaks in the FMR linewidth [START_REF] Soumah | Pulsed Laser Deposition of Substituted Thin Garnet Films for Magnonic Applications[END_REF] . Hence, one way to select the best films for SW propagation is still to choose the ones exhibiting the lowest FMR linewidths as it direct reflects the PMA uniformity in low damping materials. Good PLD samples currently feature 1 to 1.5 mT inhomogeneous linewidth while some exceptional films can go down in the 0.5-0.8 mT range. To extract the Gilbert damping, a solution has been to perform an angle dependent study to get rid of the PMA induced inhomogeneous broadening 13 .

Local PMA inhomogeneities can be evidenced by measuring the local spectrum of thermal magnons using µ-BLS. An example of such measurement for a 15µm x 15µm area of a 20 nm thick BiYIG film is shown in A phase-resolved experiment carried out by our collaborators V. Demidov and B. Divinskiy from Münster University showed the absence of phase coherence for propagating rf-excited spin-waves in a non-structured BiYIG film with an inhomogeneous linewidth ΔH T ~ 1.5 ± 0.5 mT. The coherence length is much smaller than the attenuation length (few microns) of rfexcited spin-waves contrary to the case of YIG, meaning that the information encoded in the phase cannot be transmitted over long distances for now. Additional studies will be conducted on LPE grown BiYIG films whose inhomogeneous linewidth can reach values as low as 0.2 mT 5 .

B) Spin-wave propagation in BiYIG magnonic waveguides

5 from personal communications with J. Ben Youssef (July 2019)

As it is not possible to perform phase resolved µ-BLS studies on the devices based on BiYIG PLD films, we will hereafter limit the presented results to amplitude µ-BLS studies. We observe that in magnonic waveguides spin-waves propagate according to expectations. In

C) Effect of patterning on magnetic parameters

The most comprehensive results on the effect of patterning in Bi-YIG nanostructures was carried out by our collaborator G. De Loubens and his Ph.D student Igor Ngouagnia, using Magnetic Resonance Force Microscopy (MRFM). They showed that patterning leads to the quasi-suppression of the inhomogeneous broadening in small structures (similarly to what they observed in YIG nanodisks in Ref 99 ). However, the Gilbert damping parameter increases for nano-disks below 300nm. By studying the resonance frequency, they were able to model the PMA profile inside the disks and show that patterning affects the PMA over several tenths of nanometer from the edges. This effect is strengthened when Ti (from Ti/Au antennas) is directly in contact at the interface. The use of chemically neutral isolation layer (like SiO2 or Al2O3) is thus recommended to preserve the dynamic properties of BiYIG in small structures.

In Bi-YIG/Pt waveguides, I observe unpredictable FMR values. For instance, we measure the FMR in large squares connected to a thin waveguide (0.5µm or 1µm wide) and compare it to the FMR inside the waveguide when the field is applied perpendicularly to the waveguide axis.

The FMR frequency inside the waveguide should always be lower than in the large square due to the demagnetizing field. However, we observe multiple instances where it is higher, showing that patterning slightly changes the magnetic properties. Additionally, by looking at the local thermal spectrum of magnons using µ-BLS, we notice that the edge mode resonance frequency is fluctuating on a micro-meter scale, with GHz-size

Chapter 2. Time and frequency domain study of auto-oscillations

The time-resolved auto-oscillations properties in a Bi-YIG/Pt waveguide are studied using µ-BLS microscopy. With a Bi-YIG film whose 𝑀 Dgg is close to 0, we obtain large amplitude autooscillations. Using the high temporal resolution of µ-BLS, 0.8ns in this study, and the large magneto-optical coupling of Bi-YIG, we obtain for the first time a clear picture of the transient regime of auto-oscillations. The transient regime is decomposed in different phases and found to be in agreement with the non-linear auto-oscillator model developed by Slavin and Tiberkevich 16 . These new insights on the magnetization dynamics with SOT will be crucial to devise a spin-wave amplification scheme using DC current, described in the next chapter.

I. Auto-oscillations in a BiYIG waveguide in continuous regime

The system studied in this chapter is shown in Figure 20. We use a 20 nm-thick Bi1YIG film, grown on sGGG by PLD, with a 7nm-thick Pt layer sputtered on top. An 0.5µm-wide waveguide is defined using e-beam and IBE. An SiO2 insulating square is then deposited over the whole waveguide and a 300nm-wide stripe Au rf-antenna is ultimately evaporated. The magnetic field is applied transverse to the waveguide axis, in the so-called Damon Eshbach (DE) configuration 100 (Figure 20 (b)) with a magnitude 𝐻 T = 0.2 T. The materials parameters are the same as the one described in Figure 19. By injecting the DC current in the adequate direction, the SOT is opposite to the damping torque and an increase of the thermal magnons spectrum amplitude is observed (see Figure 22-a)). By increasing the current above 0.85 mA, the waveguide magnetization starts to auto-oscillate with an amplitude 400 times larger than the thermal magnon level without current. At 1.1mA, the auto-oscillations amplitude is more than 1000 times larger than the thermal level and starts to saturate. We show a similar measurement performed in YIG 12 in Figure 21 in a 1µm-wide waveguide. The auto-oscillation are approximatively 5 times smaller in YIG, with an amplitude 100 times larger at critical current than the thermal magnon level without current and a factor 200 at saturation. This is in agreement with what was measured in CoNi disks in Ref 101 , where choosing 𝑀 Dgg close to zero allowed the observation of high amplitude auto-oscillation compared to metallic ferromagnets without anisotropy (like Py). By introducing an anisotropy term that compensates the shape anisotropy in the dynamic magnetization term, a near circular precession is obtained in thin films instead of an elliptical one (as in YIG). The circular precession suppresses the pumping field from the parallel component of the magnetization at twice the frequency (𝑚 3 (2𝜔)), thereby reducing the coupling between different magnon modes. The enhanced coupling due to the parallel pumping can be seen as a non-linear damping term of the main mode, redistributing its energy and limiting its maximum amplitude. The inverse integral BLS intensity has a linear dependence with current for 𝐼 < 0.8 mA (Figure 22-a). The fit intercepts the x-axis at a current I = 1 mA. However, it is pretty clear from the intensity variation (Figure 22-a) that the magnetization is already auto-oscillating at 0.9mA. Hence, contrary to the YIG experiment in Ref 12 , linearly fitting the inverse integral intensity is not a reliable method to precisely deduce the threshold current of the system.

A) High amplitude auto-oscillations

B) Evidence for vanishing current induced frequency shift

The peak frequency of the thermal spectrum is plotted as function of current in (Figure 22(b))

and is found to be almost constant despite the Joule heating below critical current, with a slight blue shift of the order of 50MHz. The frequency shifts more strongly after the onset of autooscillations. It is thus dominated, above critical current, by the non-linear frequency shift resulting from the opening of the precession angle.

On the contrary, in the YIG experiment's case, the effect of the temperature is clearly visible below critical current (see Figure 21-b), with a power law decrease of the resonance frequency. The frequency shift is of the order of 500 MHz below the critical current, an order of magnitude larger than for BiYIG despite applying similar current densities in the Pt. The thermal stability of the BiYIG waveguide is consistent with the temperature dependent FMR measurements of BiYIG films with small negative 𝑀 Dgg discussed in Chapter 1.

While the effect of the Oersted field produced by the current in the Pt could be neglected in the YIG experiment, it is no longer the case with BiYIG. The Oersted field is proportional to the applied current and opposite to the direction of the static magnetic field for positive currents.

Using the Biot-Savart law for the geometry of our waveguide, a field of 1.2 mT per mA applied is deduced, equivalent to a down frequency shift of about 30 MHz per mA applied.

The auto-oscillations profile is uniform along the waveguide. The BLS spatial resolution is not small enough to reliably probe the auto-oscillation profile across the waveguide. However, in a 1µm wide waveguide, measured on the same sample, the auto-oscillation profile has the shape of a first width mode. The auto-oscillation frequency is also stable along the entire waveguide, with the frequency resolution of the BLS setup being 50 MHz per channel.

C) Temperature dependence of the Pt resistance

Before showing the time-resolved experiment where the current is applied in short pulses using DC voltage pulses, we stress that the resistance of the Pt layer depends on the temperature. Thus, the applied current is not constant during the pulse. The resistance of Pt is measured at 𝑅 AEV T = 2280 Ω at 0.1 mA and 𝑅 AEV = 2388 Ω at 1 mA in continuous regime. This means that the applied current varies by about 4% between the beginning and the end of the voltage pulse. For the experiment presented in this chapter, 2µs long pulses have been used. The temperature in the Pt was computed in a very similar structure (500nm wide YIG/Pt waveguides with 7nm Pt) in Ref 102 and was shown to saturate after approximately 150 ns (see Supplementary Fig. 10). Hence, in this chapter the current in the Pt is mostly constant during the pulse and the current values are directly given in mA. However, for shorter pulses (see next chapter), we will specify the voltages instead to clearly signifies that the current is not constant.

II. Time-resolved study of the auto-oscillations transient regime

Now we apply the DC current in 2µs pulses to perform a time and frequency resolved measurement of the auto-oscillation transient regime. The repetition rate is set at 5µs to reduce the Joule heating. The laser is focused in one point at the center of the waveguide. The timeresolution is 0.8ns and the SW frequencies are probed between 4.4GHz and 5.4GHz (20 channels of 50MHz). The integrated BLS signal (signal summed over all the frequency channels) is shown in Figure 23 for subcritical, critical and supercritical currents. The amplitude is normalized to the thermal magnon spectrum integral intensity before the beginning of the pulse. For all currents, the curves can be decomposed in distinct phases. Right at beginning of the pulse, the spectrum amplitude increases sharply for all currents. The intensity is multiplied by a factor of 3 on a 50ns timescale (1 st phase). Then the intensity slowly increases in an exponential fashion for supercritical current (2 nd phase). It remains constant for subcritical currents. It then reaches a plateau, where the auto-oscillation power is stable and maximal (3 rd phase). Finally, at the end of the DC pulse, the signal sharply decreases as the auto-oscillation decays (4 th phase).

During the first 50ns (phase 1), fitting the BLS intensity logarithm with a linear function (𝑆 ∝ exp(Γ -𝑡)) for 9 different currents yields typically Γ -= 21 ± 3 rad. µs |-(see Figure 24-a). During phase 1, the spectrum intensity increases uniformly over the magnon spectrum. Thus we attribute this observation to the sudden increase of thermal magnon fluctuations 103 due to the compensation of the damping. In this case, all the magnons contribute incoherently to the BLS signal and 𝜏 -= 1/Γ -is the typical growth time of the spin-wave modes amplitude (𝜏 -= 48 ± 8 ns).

During the second phase, the thermal fluctuations slowly saturate (see the blue curve in Figure 23) and only the growth of the auto-oscillation amplitude contributes to the increase of the BLS intensity. The growth rate Γ . (I €• ) of the auto-oscillation (𝑆 ∝ exp(Γ . 𝑡)) is again extracted by fitting the logarithm of the BLS intensity in the dedicated time window and is plotted as a function of the DC current in Figure 24-b. Γ . (I €• ) increases linearly with current. The linear fit intercepts the x-axis at the critical current 𝐼 ˆ= 0.845 mA for which the growth rate should be zero. The supercriticality coefficient 𝜁 = 𝐼/𝐼 ˆ (dimensionless) can now be computed. From Ref 16 , assuming the auto-oscillation is coherent on the spatial scale of the laser spot, its intensity follows:

𝑑𝑝 𝑑𝑡 = -2[𝛤 Ê (𝑝) -𝛤 | (𝑝)]𝑝 ( 132
)
where 𝑝 is the auto-oscillation power (proportional to the BLS intensity), defined using the same convention as Ref 16 as, with y the direction of the static magnetic field:

Γ d = Γ ú + Γ 3AE + Γ ∆« w = 𝛼𝜔 d + 𝛾 ∆𝐻 T 2 𝜔 d 𝜔 T ( 134
)
with 𝜔 d = 𝜔 « + 𝜔 x t½½ 2 ⁄ ; 𝜔 « = 𝛾𝐻 T + 𝛬 . 𝜔 x 𝐾 . ; 𝜔 x t½½ = 𝛾𝜇 T 𝑀 Dgg ; 𝜔 x = 𝛾𝜇 T 𝑀 e . As this experiment was carried out only at 𝐻 T = 0.2 T, we cannot distinguish between the Gilbert damping and the inhomogeneous broadening contribution. Therefore, we will describe a relaxation rate Γ d using an effective Gilbert damping Γ d = 𝛼 Dgg 𝜔 d . However, we expect the inhomogeneous broadening contribution to be small as observed in submicron-sized BiYIG disks (see chapter 1 on the effect of patterning). Thus, we anticipate that this effective damping is close to the actual damping value in our waveguide. In the small auto-oscillation regime (𝑝 ≪ 1), the solution to equation ( 132) is an exponential growth of the auto-oscillations:

𝑝 ∝ exp[2(𝜁 -1)Γ d 𝑡] (135)
This is exactly what we observe during the second phase, where the growth rate Γ . is proportional to (𝜁 -1). From equation ( 135 At the beginning of the third phase, the BLS intensity reaches a plateau. The third phase happens within the 2µs pulse only if the growth rate is large enough. For (𝜁 -1) ≈ 3%, the intensity is barely multiplied by 10 every 1.5 µs. During phase 2, the net damping rate 𝛤 Ê (𝑝) -𝛤 | (𝑝) is strictly negative (growth). At the plateau, the intensity is constant as the autooscillation has reached a stable orbital where the auto-oscillation power 𝑝 is such that the nonlinear damping and anti-damping rate are equal (Γ Ê (𝑝) = Γ | (𝑝)).

At the beginning of the fourth phase, the DC current pulse ends, and the auto-oscillation starts to decay with the full damping rate 𝛤 Ê (𝑝) and the incoherent thermal fluctuations are no longer enhanced. Both phenomena occur on comparable timescales contrary to phase 1 & 2. Thus, it is not possible to disentangle them as we did during the growth regime. The decay rate is relatively constant for all currents above critical currents and found to be Γ ‡ˆë‰Š ≈ 67 ± 2 rad. µs |-. The BLS intensity decreases by a factor of 10 in about 30ns, much faster than in the growth phase (factor of 10 in few hundreds of ns).

III. Time-evolution of the magnon spectrum

The µ-BLS setup gives us access to time and frequency resolved measurements. We now focus on the evolution of the shape and position of the magnon spectrum recorded in a single point of the waveguide by µ-BLS, during the application of the 2µs DC current pulse. The time-evolution of the magnon spectrum during a DC current pulse of amplitude 𝐼 = 1 mA is presented in Figure 25. Each spectrum at a given time is normalized by its maximum amplitude. Before the DC pulse (t<0), the amplitude of the thermal spectrum is small and dominated by instrumental noise when using a 4 ns integration constant (moving average). At the start of the DC pulse, the fluctuations of the thermal magnons spectrum are enhanced. The wide thermal spectrum can be clearly distinguished even with the 4ns integration constant (phase 1), with the bottom of the spectrum being near 4.9 GHz

After few hundred nanoseconds, the narrow auto-oscillation peak dominates the spectrum and we observe the non-linear frequency shift of the peak frequency (phase 2). Once the autooscillation reaches its stable orbital (see Figure 23), the peak frequency stabilizes (phase 3).

When the DC pulse ends, the spectrum remains dominated by a large amplitude and fine spectral linewidth state (phase 4). The central frequency of this state decreases very sharply, on a few nanoseconds time-scale, and ends up below 4.9 GHz (see a zoom in Figure 25-b). About 80ns after the end of the DC pulse, the amplitude has fully decayed, and the spectrum is once again dominated by the thermal magnons. In this smooth transition, we can see that the autooscillation occupies the bottom of the thermal spectrum near 4.9 GHz. Indeed, for our material parameters and shape, the bottom of the spectrum is occupied by the FMR of the first width mode. This confirms our earlier spatial observation of the auto-oscillation, consistent with a uniform first width-mode.

The auto-oscillation frequency shift mainly comes from three different effects: Joule heating (variation of 𝑀 e and 𝐻 P ), the non-linear frequency shift due to the high amplitude precession and the Oersted field generated by the DC current flowing in the Pt. Once again, those phenomena may occur on different time scales which allows us to observe them separately. In Figure 26 we plot the auto-oscillation frequency as a function of its intensity for four different currents during the auto-oscillation growth (phase 2 in Figure 26-a) and decay (phase 4 Figure 26-b). Both the frequency and intensity are obtained by a Lorentzian fit of the spectrum at times where the auto-oscillation signal dominates.

During the first phase, the thermal magnons dominate the spectra and fitting it with a Lorentzian would be inaccurate. The temperature is expected to saturate in about 150ns 102 and the Oersted field generated by the current is then stable. Therefore, after a few hundreds nanoseconds, when the auto-oscillation emerges, and the center frequency starts being recorded, both the temperature and the Oersted field are already established and stable. A linear dependence of the frequency with the auto-oscillation power is observed (dotted line in Figure 26-a). This linear dependence is found to be independent on the current. The effect of the temperature and Oersted field is visible as a frequency offset between different currents, confirming the quick vanishing of temperature variations in the structure. The formula linking the pulsation (𝜔) to the auto-oscillation power (𝑝) for an elliptical precession is quite complex (see Ref 16 ), but can still be derived exactly for our BiYIG waveguide:

𝜔(𝑝) = 𝜔(𝑝 = 0) + 𝑁𝑝 with 𝑁 = - 𝜔 « 𝜔 T 𝜔 « + 𝜔 x t½½ /4 𝜔 « + 𝜔 x t½½ /2 𝜔 x t½½ (136)
Hence, the linear fit of 𝜔 versus intensity in ). We stress that our cone angle is measured at a current 30% higher than the threshold. Near the threshold current (𝐼 = 0.85 mA), the frequency shift is much smaller with a large uncertainty (∆𝑓 VM ~ 1 -5MHz → 𝜃 VM ~ 3 -8°). This method is also imprecise at small angles due to the vertical slope of the 𝑝 ↦ arcos(1 -2𝑝) function near 𝑝 = 0. Hence, making meaningful comparison between these two experiments is difficult. The results could be confirmed using a more direct technique like MRFM (the change in the 𝑀 3 component mechanically excites a cantilever 10 ) or by time resolved magneto-optical effects.

When the current is switched off (phase 4), the temperature, the Oersted field and the autooscillation amplitude quickly drop. As a result, both terms in Eq. ( 136) decrease simultaneously.

Interestingly, for all measured currents (only 4 shown in Figure 26-b for clarity), the 𝜔 versus S curves overlap (within the experimental resolution). A consequence, we believe, of the temperature and auto-oscillation power decaying on similar time scales. It is at the moment difficult to tell whether it is more than a coincidence. About 3 nanoseconds after switching off the DC current, the 𝜔(𝑝) goes slightly up by 5-10 MHz. We attribute this effect to the vanishing of the Oersted field. The Biot-Savart law gives us an Oersted field of 1.2 mT at 1 mA, oriented in the opposite direction of the static magnetic field in this geometry. This results in a 30MHz up frequency shift when switching off the DC current pulse at 1 mA. However, both the amplitude and the temperature drop the most sharply right after the end of the pulse, leading to a downward frequency shift of comparable scale. The thermalization of incoherent magnons typically happens on a 5-10 nanoseconds scale 102 . A similar or even quicker reaction is expected in our case, since our BLS signal is dominated by a coherent auto-oscillation.

IV. Discussions and perspectives

A) Discussion

Using this straightforward experiment, we studied both the thermal fluctuations and the autooscillations transient regime under the application of SOT. By comparing subcritical to supercritical currents, we have been able to show that those two phenomena take place on different time scales. The growth rate of the thermal magnons' fluctuations seems to be equal to the magnetic relaxation rate of the system Γ d and independent of the applied current. An experiment with a longer accumulation time should be conducted to reduce the large uncertainty as the experiment was initially calibrated to study high intensity auto-oscillations. The only other time-resolved study of thermal magnons fluctuations was performed in Py/Pt disks in Ref 103 . The increase of thermal fluctuations is too steep to be precisely measured with the 20 ns resolution used in that study. This is explained by the fact that the natural relaxation rate of Py is much higher than the one of YIG or Bi-YIG.

In a second phase, the auto-oscillation growth rate is found to perfectly match the theoretical expectations of the low auto-oscillation amplitude regime: Γ(𝜁) = 2(𝜁 -1)Γ d described in the seminal theoretical work of Slavin and Tiberkevich. By fitting the obtained growth rate for many currents, one can obtain a precise estimation of the critical current and of the magnetic relaxation rate. This BLS amplitude measurement is local (laser focused on a 300x300 nm 2 region). By rastering the laser along the waveguide, a variation of the growth rate is observed. For instance, at 1mA we typically obtain a growth rate: Γ . = 6.8 ± 0.6 rad. µs |-; a 10% variability that is well above our measurement uncertainties. However, this space-resolved (long) experiment has not been conducted for enough currents and long enough pulses to tell if the observed variance only comes from a change in the local damping or if the critical current also depends on the position.

One very puzzling fact is that the magnon lifetime obtained from this method (47ns) and the one obtained from the analysis of the propagation length of rf-excited spin-waves without DC current (13ns), at the same field, do not match. The first method suggests an effective damping of the order of 𝛼 Dgg ≈ 6.8 × 10 |ý while the second indicates 𝛼 Dgg ≈ 2.3 × 10 |/ which is similar to the values found in similar YIG/Pt 99 system (deduced from a magnetic field dependence study of the threshold current and confirmed by broadband FMR measurements). One hypothesis is that the damping obtained via the local time-resolve study is indeed local, with a small inhomogeneous contribution, and that its variation along the waveguide explains the clear difference observed for the effective damping felt by propagating spin-waves. Another hypothesis is that auto-oscillations in the waveguide may not be coherent on a 300x300nm 2 scale. In this case, the BLS intensity would be proportional to 𝑆 ∝ 𝑝 Ž with 𝛽 = 1 for fully coherent auto-oscillation and 𝛽 = 0.5 for incoherent auto-oscillation 104 . However, even for fully incoherent auto-oscillations, the effective damping (𝛼 Dgg "aˆ≈ 1.4 × 10 |/ ) would still be significantly lower than expected. Repeating these measurements at different applied magnetic field (as in Ref 99 ) allows to separate the Gilbert damping contribution (proportional to frequency) to the inhomogeneous broadening effects (independent of frequency) and to settle on the degree of coherency of the auto-oscillations in our waveguide.

The observed steady exponential growth of the auto-oscillations demonstrates that the onset of non-linear damping effects happens only very close to the saturation point of the autooscillation. While the auto-oscillations grow, the amplitude is small enough to have a net negative damping torque while avoiding the saturation effects. This observation is at the core of the next chapter in which we design a pulse scheme to propagate and amplify rf-excited spinwaves at supercritical currents before the onset of auto-oscillation, thereby circumventing the non-linear scattering issues encountered in Ref 12 in YIG.

B) Perspectives on the influence of spin-current on the magnon distribution

The detailed study of the magnon spectrum during the pulse showed that we can quantitatively dissociate the role of temperature and non-linear frequency shift by looking at the evolution of the auto-oscillation frequency. From this, we have estimated the opening angle to be 36 ± 2°, much larger than what was obtained in micron-sized YIG/Pt disks driven by SOT. Additionally, the k-vector of the auto-oscillation mode lies at the bottom of the SW spectrum as clearly observed when the auto-oscillation decays. In our specific case (DE configuration), the bottom of the spectrum is occupied by the FMR of the first width mode (𝑘 = 0).

Very recently 102 , M. Schneider et al. have brought strong evidence of the formation of a magnon BEC at room temperature for the first time in 70nm-thick YIG films. They used a very similar 0.5µm-wide YIG/Pt waveguide to form a BEC by the process of rapid cooling. We stress that the voltages applied in our study are at least twice lower than the threshold voltage needed to obtain the critical cooling rate to form the BEC. Increasing the voltage to reach this threshold would allow us to study the influence of spincurrent on the emergence of the rapid cooling BEC state. Thanks to our ultra-thin BiYIG film, we can reach currents where both auto-oscillations and rapid cooling BEC can be formed contrary to the 70nm-thick YIG used in Ref 102 . By sweeping the field orientation in-plane, the SOT can be gradually turned on or off while the rapid cooling remains identical. The interaction between the decaying auto-oscillation and the emergent BEC could inform us on the similarities and differences of these two states. In particular, to call a magnon state a BEC one has to provide a thermodynamic description of the magnon population. It can be done by probing the magnon population measured by BLS. However, to reliably derive the thermodynamic parameters the spectrum must be wide enough in frequency, using either ferromagnets with large group velocities (like 10nm Py in ref 105 ) or by studying higher order thickness modes for low saturation magnetization ferromagnets (70nm YIG in ref 102 ). We observe the second thickness mode in our 20nm BiYIG films around 40GHz, but the signal is quiet low. Going to thicker films would significantly enhance the BLS intensity of those higher order modes to check whether the auto-oscillations can be described thermodynamically as in Ref 105 , and whether the chemical potential can reach the lowest energy state, that would be considered as a signature that a BEC is formed using SOT. At the time when this manuscript is being put together, preliminary data do indeed point toward a SOT induced BEC in our unpatterned films.

Chapter 3. Propagation of rf-excited spin-waves with SOT

The exponential growth and stabilization of auto-oscillations in our BiYIG waveguide demonstrate that both negative and zero effective damping are achieved using SOT for the uniform first width mode. What about other magnon modes? In this part, we will study the influence of SOT on propagating magnon modes of a 0.5µm wide BiYIG/Pt waveguide. Indeed, being able to propagate magnons without losses or to directly re-amplify a magnonic signal is a crucial step toward any magnon-based information technology. In 2017, M. Collet, former PhD from CNRS/Thales, showed that the decay length can be very efficiently modulated in YIG/Pt waveguides, from 3µm without current to 24µm when the threshold current for auto-oscillations is applied 12 . Above the threshold current, the spin-wave coherence rapidly collapses. Thus, non-linear phenomena, that remain not fully identified today, open relaxation channels for the propagating magnon modes, preventing a lossless propagation (and amplification) of rf-excited spin-waves in YIG when the SOT is applied continuously. In this chapter, the propagating magnons' decay length is characterized in the presence of SOT in the same BiYIG/Pt waveguide described in the previous chapter. First, the DC current is applied continuously in the Pt. Large decay lengths are obtained for magnons with frequencies close to FMR when applying currents near the threshold. Surprisingly, the damping compensation is found to be gradually less efficient when the frequency is increased by only a few percent. Additionally, strong non-linear phenomena are emerging when the DC current is continuously applied above threshold, preventing us from discriminating the propagating magnons from the auto-oscillation. To tackle this issue, we describe in a second part experiments for which the dc current is applied using a 250 ns voltage pulse. In this case, we show that the rf-excited spin-waves are now unaffected by the auto-oscillations. The frequency dependent efficiency of the damping compensation is however also observed. For large enough DC currents pulses, we demonstrate for the first time the observation of a lossless and an amplified spin-wave propagation in a magnonic waveguide using SOT.

I. Propagation of rf-excited spin-waves with continuous SOT

The propagating magnons are locally excited using a 300nm-wide strip antenna designed on top of a 0.5µm-wide BiYIG waveguide (the same as in the previous chapter). For all the experiments presented in this chapter, an external magnetic field of 0.2 T is applied perpendicularly to the waveguide axis (DE configuration), so that the SOT induced antidamping torque is the largest. By analyzing the decay length through µ-BLS as a function of the current amplitude, the efficiency of the damping compensation is measured for propagating magnon modes.

A) Spin-wave propagation at subcritical currents

For this series of experiments, spin-waves are excited at frequencies 𝑓 = 4.9, 4.925, 4.95, 4.975 and 5.0 GHz and DC current is applied continuously with values 𝐼 = 0, 0.6, 0.75, 0.8, 0.85 mA. By applying the rf power in the form of µs pulses (repetition rate of 260 µs), the contribution from rf-excited spin-waves can be dissociated from the contribution of SOT-enhanced thermal magnons. The BLS intensity measured when the rf-power is off, containing the contribution from thermal spin-waves, is subtracted from the BLS intensity measured with the rf-power on, containing both contributions. The rf-excited spin-wave intensity is extracted using this method on the condition that there is no interaction between the thermally excited magnons and the rf spin-waves. The decay length is extracted for every combination of frequency and current by fitting the spin-wave intensity as a function of the distance to the antenna. In order to compare the results at different currents, the offset of the applied rf frequency with respect to the FMR is computed at each current. Indeed, the magnon spectrum shifts when DC current is applied, due to Joule heating and the Oersted field produced by the DC current. In the case YIG waveguides, the application of dc currents near the threshold results in a 500 MHz shift at 4.8 GHz 12 , and thus the excitation frequencies must be changed to follow the downward shift of the magnon spectrum. In our case with Bi-YIG having a small effective magnetization, the position of the spectrum is stable enough that the excitation frequencies can be kept constant and only re-adjusted in the analysis. From the thermal magnons spectra measured when applying the DC current, the position of the FMR is extracted for the five currents used in the experiment and presented in a table in Figure 27. The effects of the SOT on the rf-excited spinwaves can hence be compared for frequencies that are equally distant from FMR.

The spin-wave intensity for frequencies 50 ± 10 MHz above FMR is plotted for the five DC currents in Figure 27-(a). Without applied DC current, the rf-excited spin-waves have a typical decay length of 1.7 µm, which corresponds to a damping α = 2.2 ± 0.3 × 10 |/ . As the current is increased, the attenuation length gets larger. At 𝐼 = 0.8 mA, about 5% below the critical current, the attenuation length is 19.5µm, 11 times larger than without current. Near the critical current 𝐼 = 0.85 mA, the measured attenuation length is 87 ± 25 µm. The large error bar reflects the fact that it is difficult to precisely evaluate large attenuation lengths over a distance of 8 µm. We can simply state that the damping modulation reaches at least a factor 30, which is 3 times better than what was observed in YIG for spin-waves with 5 µm wavelengths 12 . For spin-waves at 50 ± 10 MHz above FMR, their wavelength is expected to be 1.3 ± 0.2 µm, using the dispersion relation given by the Kalinikos-Slavin formula 47 and the parameters given in Figure 27. At a frequency 100 ± 10 MHz above FMR (Figure 27-b), the decay length at zero current is slightly larger at 1.8 µm. However, for all positive currents, the decay lengths are much smaller than at 50 MHz. At 𝐼 = 0.8 mA, the decay length is only twice larger than at zero current and only 5 times near the critical current. The decay lengths for all measured frequencies are shown in Figure 27-(c). The trend is very clear: at a given current the decay length gets continuously smaller for frequencies far from FMR. The solid curves represent the predicted decay lengths assuming a linear compensation of damping. The damping at zero current was taken as 𝛼 = 2.2 ± 0.3 × 10 |/ and critical current 𝐼 ˆ= 0.85 ± 0.02 mA (the impact of the uncertainties is shown by the colored shaded areas). While for frequencies close to FMR, the damping compensation is close to the linear theory, it is clearly inefficient at frequencies farther from FMR.

At 𝐼 = 0 mA, no such trend is observed when the frequency is increased. The spin-waves propagate further because their group velocity increases. Thus, when the DC current is applied, the decrease of the attenuation length with frequency is due to a decrease of the magnon lifetime and not a change of the group velocity. Moreover, if the group velocity was divided by a factor of six due to the SOT, the thermal spectrum would radically change, and it is not the case. Furthermore, the temperature can only minorly impact the group velocity as it reduces both 𝑀 e and 𝐻 P . Finally, we emphasize that a total frequency variation of only 2% (100MHz at 5GHz) increases the damping by 500%. Thus, the observed inefficiency is not due to the linear increase of the relaxation rate with the frequency as in Ref 15 . The reduction of the propagating magnon lifetime with increasing frequencies means that their effective damping is not zero at the threshold current. We suspect that the SOT enhances the non-linear coupling to other modes, which increases the damping of the propagating magnons. We can speculate that this additional non-linear damping depends on the frequency or the wavevector.

In the YIG sample studied in 2017, it was not possible to perform such frequency dependence experiments. Due to a 3µm wide antenna, only small k-vector could be excited. Hence, only a frequency close to FMR was investigated. For now, it is not possible to tell whether this effect is controlled by the frequency splitting compared to the main mode frequency or by the wavevector of the spin-waves. Repeating this experiment by varying the external field might help us discriminate its origin. The presence of other modes at the same frequencies might be important. However, in this 0.5µm waveguides, the 3 rd width mode is at least 125 MHz above the 1 st width mode, and the 2 nd width mode nearly overlaps the 1 st (but cannot be excited efficiently by the antenna). Thus, it seems difficult to explain the continuous decrease of the magnon lifetime from the frequency position of those modes. In any case, building a model to explain the lifetime of the different magnon modes under SOT seems necessary and we have shown that significant results can already be obtained at subcritical currents.

B) Strong interaction with auto-oscillations at supercritical currents

When supercritical currents are applied, a strong interaction between the auto-oscillations and the rf-excited spin-waves is observed. Indeed, the measured BLS intensity is sometimes larger when the rf current is off (with only the contribution from the auto-oscillation) rather than when it is on. If there was no interaction the signal from the rf-excited spin-waves would simply add up to the auto-oscillation signal. But in certain spots of the waveguides the signal decreases (or increases) very strongly when the rf-power is on, without any clear pattern. It is certainly different from what was observed in the YIG experiment, but we cannot say for sure whether the rf-excited spin-waves ends up propagating on long distances or not. Further study is required to elucidate the exact nature of this interaction. With future samples, we hope to simplify this study by creating a localized auto-oscillation using a constriction in the Pt layer and send the rf-excited spin-wave on it, so that the interaction is localized, and far from the influence of the rf-antenna.

Conclusion and perspectives:

Going back to our initial goal of propagating rf-excited spin-waves without losses, we showed that below the critical current the propagation lengths are finite as the damping is not fully compensated for most frequencies. Hence, supercritical currents are needed to achieve zero or negative effective damping for propagating magnons. However, going above critical current triggers strong non-linear interactions between auto-oscillations and rf-excited spin-waves, preventing the direct observation of the spin-waves and considerably complexifying the analysis.

Our observations raise important questions on the effect of SOT on propagating magnon modes.

The SOT induced phenomena such as the coupling between different modes and the frequency dependent lifetime, are promising for new magnonic devices that uses non-conventional computing schemes relying on non-linear effects. But in order to achieve our goal of lossless propagation, staying in the linear regime while applying supercritical currents seems to be the right direction to explore. In the previous chapter, we saw that during its slow growth, the autooscillation can be described by the low amplitude approximation until it reaches its saturation due to additional non-linear damping terms. Using a pulse scheme at supercritical dc currents, we propagate rf-excited spin-waves before the saturation of the auto-oscillations, and thus before the onset of the strong non-linear damping terms.

II. Propagation of rf-excited spin-waves with pulsed SOT

In this section, we report the first experimental observation of the amplification of coherent propagating magnons by SOT. We start by detailing the modus operandi of the experiment.

Then the results are presented and discussed.

A) Experiment and methods

Our goal is to propagate rf-excited spin-waves at supercritical currents before the saturation of auto-oscillations. In the previous chapter (see Figure 23), we have shown that upon applying 1.1 mA to the Pt, the auto-oscillation saturates after about 400 ns. During the transient regime, the auto-oscillation grows exponentially in agreement with linear theory. Thus, by applying a short DC voltage pulse (250 ns chosen in this experiment) we ensure that our system is in the low amplitude regime, far from the strong non-linear effects that are triggered near the saturation. During the DC pulse, a 200ns rf-current pulse is applied to the antenna and the spinwaves propagate in the waveguide. By rastering the laser along the waveguide, the timeresolved BLS signal is recorded in the center of the waveguide as a function of time and distance to the antenna. The spin-wave propagation is measured for four voltages 𝑈 = 1.7, 2.2, 2. To single out the rf-excited spin-wave intensity and get rid of thermal magnon and autooscillation contribution to the BLS signal, we repeat the measurement in a second run (Figure 30-b), where the rf power is set to 𝑃 dg = -90 dBm. The intensity measured in this second run gives us directly the auto-oscillation and thermal magnon contribution to the BLS signal. It is spatially uniform, contrary to the rf-excited spin-waves that propagate away from the antenna. By subtracting this background to the intensity measured at 𝑃 dg = -30 dBm, we extract the pure intensity of rf-excited spin-waves in the presence of SOT. The result of this subtraction is shown in Figure 29. Before 𝑡 = 0 ns, no DC and no rf voltage are applied. The noise level is low. At 𝑡 = 0 ns, the DC voltage pulse starts. No signal is visible since we subtract the contribution from the auto-oscillation. The noise level is simply increasing as the auto-oscillation signal grows. At 𝑡 = 50 ns the rf voltage is applied. Spin-waves are emitted under the antenna, and the large amplitude signal is clearly visible, propagating from the left to the right of the waveguide. By fitting the BLS intensity rising edge, the spin-wave group velocity is extracted. At 𝑡 = 250 ns, both pulses end and the spin-wave signal decays on a short time scale of the order of 30ns.

The procedure is repeated at frequencies 𝑓 = 4.925, 4.95, 4.975 and 5.0 ± 0.05 GHz with a low rf-power 𝑃 = -30 dBm, chosen to be well into the linear regime of excitation by the strip antenna. As the rf pulse is 200 ns long, the uncertainty on the applied frequency is 5 MHz.

B) Evidence of propagating spin-wave amplification

The results are put together in Figure 31 where the colormaps for frequencies 𝑓 = 4.925, 4.95 and 5.0 GHz and DC voltage 𝑈 = 1.7, 2.2, 2.4 V are presented. We first check the behavior at subcritical DC current for different frequencies (left column). We note that the decay length is dramatically reduced for high frequencies, similarly to what was observed in the continuous regime. Furthermore, the group velocity increases with the frequency, as expected from the dispersion relation. Hence the low decay length at high frequency is not due to a collapse of the group velocity but rather a decrease of the magnon lifetime, inconsistent with a fully compensated damping at 𝐼 ˆ= 0.85 mA. Additionally, we notice that the excitation efficiency of the rf excitation is not constant although we apply the same rf power for all frequencies (see color scales in Figure 31). This is due to the fact that our antenna excites small wavevectors more efficiently.

Then we focus on the description of the effect of SOT in the case where supercritical currents are applied i.e. when the voltage is increased to 2.2 V (center column) and 2.4 V (right column).

We notice that the spin-waves propagate much further although general shapes of the colormaps remain similar. At 5.0 GHz, the spin-wave signal is undistinguishable from the noise after a few microns for 𝑈 = 1.7 V while it clearly propagates the full 12µm at 2.4 V. At low frequencies the propagation shows no sign of decay at supercritical currents.

In order to quantitatively compare the attenuation lengths at different currents, the actual position of the FMR frequency must be determined precisely for the different applied voltages.

A striking characteristic of the color maps in Figure 31 is that the FMR seems independent of the applied DC voltage. Indeed, the extracted group velocities are independent of the voltage and does not vary during the initial propagation phase (~100 ns). In the previous experiment, the FMR was shifting with the applied DC current. The thermal frequency shift due to the temperature increase was strong enough to dominate over the downward shift from the Oersted field. In the experiment presented, the repetition rate is 250 ns every 2 µs instead of 2 µs every 5 µs previously, yielding an average thermal power that is 3.2 times smaller. In consequence, we conclude that the thermal shift must be of the same order as the Oersted field in this range of applied voltage, as the observed small variation of the group velocity is a signature that the FMR frequency variations are limited. Moreover, we also stress that the non-linear frequency shift from the auto-oscillation amplitude is negligible here since the auto-oscillation amplitude remains small. The expected group velocity, computed from the dispersion relation, is plotted as the blue curve in Figure 32. The comparison with the group velocities found experimentally indicates that the FMR must be stable around 4.885 ± 0.01 GHz during the rf pulse for the investigated voltages. Thus, contrary to the continuous regime case, the frequencies do not need to be re-aligned to the voltage dependent FMR and we can directly compare the effect of the different currents at a given applied rf frequency. We then describe how to extract the propagation length from the BLS intensity maps. In the steady state regime, at any position in the waveguide, the dynamic magnetization would reach a stable amplitude. Provided that there is no reflection of the spin-waves, taking a snapshot of this "steady state regime" magnetization reflects the actual decay of the spin-waves propagating in the waveguide. However, the spin-wave amplitude at a given point in the waveguide is not constant with time in all the above graphs. This is not surprising given the time needed for the magnetization amplitude to stabilize under the antenna, the propagation time (~100ns) and the eventual reflections at the end of the waveguide or the scattering events on defects in the waveguide. Thus, although the vertical shapes materializing in the color maps are the sign of an emerging temporal stability, the permanent regime is not reached during our 200 ns rf-pulse. Therefore, in order to properly extract the attenuation length, we go back to its basic definition. When a magnon is emitted at a given point and with a given velocity, how far will it go? The crucial point is that we have estimated velocity (𝑣 ø ) of the propagating magnons with relatively good precision. Consequently, we can follow the amplitude at position 𝑥 = 𝑥 -of a magnon emitted at 𝑥 = 𝑥 T and time 𝑡 = 𝑡 T , by looking at the amplitude at time 𝑡 -= 𝑡 T + (𝑥 --𝑥 T )/𝑣 ø . Hence, instead of looking at a snapshot (equivalent to a horizontal cross-section of a colormap at 𝑡 = 𝑡 eaQ¿eMWV ), the true amplitude of the propagating magnon is extracted by taking a crosssection of equation 𝑡(𝑥) = 𝑡 T + 𝑥/𝑣 ø . In the experiment, the time and space are discrete, so instead of extracting along a line, we are extracting along a "stripe" of height 0.8 ns. To reduce the noise, it is reasonable to integrate over a stripe that is several ns high as illustrated in Note that, doing so, the extracted intensity depends on the estimated group velocity. However, integrating over a large area reduces the impact of the uncertainty on the group velocity value, as a deviation from the real value only slightly modifies the probed area. For instance, the area represented in Table Figure 34: The decay lengths obtained from the fit are given in the following table. A negative value indicates that the slope is positive. Its absolute value therefore represents an amplification length. First, we verify that the values of attenuation lengths obtained for subcritical currents (blue curves in the four panels of Figure 34) are consistent with the ones obtained in the continuous regime. We find 9.6 µm, 10.4 µm, 6.2 µm and 2.4µm for frequencies 40 MHz, 65 MHz, 90 MHz, 115 ± 10 MHz above FMR (respectively). To compare, in the continuous regime, we found 10 µm, 9.9µm, 5µm and 3.5µm for frequencies 50 MHz, 75 MHz, 100 MHz and 125 ± 10 MHz above FMR. The excellent agreement validates the chosen method to extract the intensity of the propagating magnons. Importantly, it also supports the FMR frequency value determined using the group velocity in Figure 32. At 4.925 GHz (Figure 34-a), the spin-waves intensity has a positive slope for all voltages corresponding to supercritical currents. The growth is nicely exponential. This suggests that spin-waves are amplified along their propagation by the SOT. We define the amplification length (𝐿 Q`¿ ) as the counter part of the decay length: the distance needed for the amplitude to be increased by a factor e. It corresponds to the absolute value of a negative decay length, computed as twice the inverse of the slope of the linear fit of the BLS intensity logarithm (dotted lines in Figure 34).

2.2 V 2.3 V 2.4 V 1.7 V 2.2 V 2.4 V 1.7 V 2.2 V 2.3 V 2.4 V 1.7 V 2.2 V 2.3 V 2.4 V 1.7 V a) b) c) d)
𝐋 𝐚𝐭𝐭 (𝛍𝐦) U = 1.7 V U = 2.2 V U = 2.3 V U = 2.
As the voltage is increased, the amplification length decreases, showing not only that the effective damping is negative for modes other than the auto-oscillation mode but that their amplification length can be controlled by the magnitude of the applied current. At 4.95 GHz, the spin-waves intensity has a negative slope at 2.2 V, a nearly zero slope at 2.3 V and a positive slope at 2.4 V. This suggests that for this specific frequency, the damping compensation point is reached near 2.3 V. At 4.975 GHz, a 63 µm attenuation length is measured at 2.4 V. The damping compensation voltage is hence higher for this frequency. At 5.0 GHz, the spin-waves are still strongly attenuated at 2.4 V (𝐿 QVV = 17 µm). This is consistent with what was observed in the continuous subcritical regime in which we find that it steadily gets harder to achieve low effective damping as the applied frequency is increased.

To strengthen our claim that rf-excited spin-waves are indeed amplified by the SOT during their propagation, we extract the spin-wave intensity along the areas defined in Figure 33-a and plot them in Figure 35. Using this extraction method with the first area (magnons emitted at the beginning of the pulse at 𝑡 = 60 ns) ensures that no reflections can interfere with the forward propagating magnons and potentially influence the result. As the initial amplitude is still varying a lot, the choice of the group velocity is very impactful. Thanks to the sharp rise of the BLS intensity the group velocity is precisely determined at 𝑣 ø = 105 ± 5 m/s for the initial propagation. Later during the pulse, the uncertainty on the group velocity is larger but the amplitude has stabilized (see Figure 35-b) and even a 15 m/s error makes very little difference on the extracted intensity curves. The exponential fit yields an amplification length of 34 µm for the initial propagation and 42 µm later in the pulse. Those values are consistent with the 44 µm value found previously. We stress that in the worst case (initial group velocity 𝑣 ø = 110 m/s) the amplification length is still clearly positive (70 µm < 𝐿 Q`¿ < 250 µm).

Finally, we emphasize that the excitation efficiency of the antenna is gradually enhanced by the SOT during the first tens of nanoseconds of the pulse (see the difference between the blue curve and the red curve intensities in Figure 35). Indeed, 10 ns into the pulse (𝑡 = 60 ns) at 1.7 V, the maximum intensity is already stable at around 200 arbitrary units. Meanwhile, at 2.4 V, the amplitude still grows after 10ns. It is multiplied by 4 in 40 ns and stabilizes around 10 3 a.u. Note that this variation is not due to a change of wavevectors since the group velocity remains constant. The SOT-driven reduction of the damping leads to an increase of the susceptibility However predicting the susceptibility value when the effective damping is negative is a tricky question and getting a more quantitative look at the susceptibility can also be difficult in such structures. Indeed, the BLS signal under the antenna is often quite messy due to the interactions of the different rf-fields. A confined structure such as a disk, under a uniform rf-field as in Ref 52,99 , would offer simpler conditions to study the susceptibility of magnetic systems with zero or negative effective damping.

C) Conclusion perspectives

The experimental results presented in this section sheds new light on the influence of SOT on propagating magnons in the supercritical regime. The use of a short DC pulse allowed us to keep the system in the unsaturated regime, leading to clear results: (a) Propagating magnons can be amplified using SOT. (b) The current required to reach the spin-wave amplification is increased as the excitation frequency gets further from the FMR in our BiYIG waveguide.

These observations raise numerous new interrogations, most of which will require further investigations to be answered, with a new sample. Indeed, the devices presented in this manuscript unfortunately broke down during the summer 2020. Hence, we were not able to test everything we wanted. The data measured from march to summer still provides us with a clear path towards understanding the spin-wave amplification using SOT.

Emitted at t=60ns

Emitted at t=110ns For instance, we cannot currently predict the amplification at higher currents. With the present data, plotting the decay constants as function of the current does not reveal any clear linear trend as expected in the subcritical regime. If the behavior is different at supercritical currents, it will require more than 3 points (that are in fact squares due to the error bars) at one frequency (4.925 GHz) to identify a clear trend. Another open question is the frequency dependence of the damping for propagating magnons with SOT. The design of an experiment in which the FMR position is very stable during the dc pulse is a crucial step toward obtaining reliable results. Extending the range of explored currents is also important to uncover the trends. The only constraint is to keep a good track of the FMR frequency. We point out that a good way to monitor the frequency shift due to heat in the system is to use a frequency close to FMR, as the group velocity vs frequency curve (see Figure 32) gets steeper in this region, so that a small variation of FMR leads to a large and detectable change of the group velocity. Varying the external magnetic field is also an important step as it could clarify the impact of the frequency on the damping compensation.

a) b) 2.4 V 1.7 V 1.7 V 2.4 V
We stress that the use of BiYIG was critical in this study. In general, the advent of high quality BiYIG films is a real leap forward for BLS experiments in garnets. As the BiYIG generates a BLS signal about 50 times larger than YIG, a week of measurement with BiYIG is equivalent to a year of accumulation time with YIG. For instance, the data used in this section took a few weeks of accumulation time. Although the clever planning of an experiment can somewhat reduce it, conducting an exploratory work with pulses in YIG is a daunting task. The role of anisotropy was also instrumental in this study with SOT. First, because it provides temperature stability on a wide temperature range. Had it been YIG, the shift of FMR during the pulse would make the analysis much more complex (change of k-vector, change of group velocity during the propagation). Tilting the magnetization easy-axis between in-plane and out-of-plane to suppress the thermal frequency shift is possible, but it further decreases the BLS sensitivity (as only the out-of-plane component of the dynamic magnetization is probed) and the SOT amplitude. Secondly, the anisotropy greatly reduced the elliptical precession, allowing us to stay in the unsaturated regime for a longer time. All those BiYIG properties made possible this very challenging experimental demonstration. More work is also necessary to evaluate the potential of SOT for spin-wave amplification in applications. Indeed, BiYIG is still a material intended for research. Currently, its low coherence length limits its applications as no information can be encoded in the spin-waves phase. Furthermore, it is currently not possible to predict how much of an amplification can be obtained at high current and for frequencies close to FMR, nor under which condition these performances could be transposed to other ferromagnets. In our view, the main interest of this study remains fundamental: getting a better understanding of the effect of SOT on magnons and how to use it in the field of magnonics. Although we took many precautions to stay in the unsaturated regime in this section, the study of SOT in BiYIG also represents a unique opportunity to unveil the rich non-linear Physics such as magnons BEC.

Part 2: Frequency filtering using magnonic crystals

When a wave propagates in a periodic medium, part of it is reflected back at each period and can interfere constructively or destructively with the incoming wave. Around the Bragg's conditions 𝑘 a = 𝑛𝜋/𝑎, the interference is destructive, and a band gap is opened in the dispersion relation. Inside the band gap, the waves are strongly attenuated, while outside the gap their propagation is quasi unaltered. The frequencies corresponding to the band gap are thus filtered. This phenomenon has been exploited to control the dispersion relation and propagation of light in periodic dielectric media called photonic crystals [START_REF] Joannopoulos | Photonic Crystals: Molding the Flow of Light[END_REF] . For spin-waves, they are called magnonic crystals (see review 33 ). Different implementations have been proposed during the past years, either based on µm-thick YIG films 106,107 or on thin metallic ferromagnetic films 33 . However, µm-thick YIG films are incompatible with modern nanofabrication techniques, while metallic films have large magnetic losses that affect the spin-waves propagation at all frequencies. In this thesis, I demonstrate the efficient filtering of radiofrequencies using nanostructured magnonic waveguides based on ultra-thin (20nm) YIG with low damping. First, micromagnetic simulations are performed to understand the experimental µ-BLS data previously obtained during the Ph.D of M. Collet on a width-modulated YIG waveguide. This two techniques combined shed new light on the behavior of spin-waves in a magnonic crystal and demonstrate the power of GPU-based micromagnetic simulations tools 19 for the development of optimized magnonic systems. In a second part, a thickness modulated waveguide is studied using an allelectric characterization method. We show a strong suppression of the losses generated by the nano-structuration and thus demonstrate an efficient implementation of radiofrequency filtering using a 1D magnonic crystal.

Chapter 4: Width-modulated ultra-thin YIG waveguide -BLS and micromagnetic simulations

I. Fabrication and measurement setup

The 20nm thick YIG film used in this experiment is grown by pulsed laser deposition on GGG (111) substrate. Broadband ferromagnetic-resonance measurements (FMR) performed on extended films typically yielded a Gilbert damping of 𝛼= 3.4 10 -4 with an extrinsic linewidth µ0∆H0 = 0.2 mT and an effective magnetization 𝜇 T 𝑀 Dgg = 0.215 mT. The magnonic crystal (MC) is a 1µm-wide YIG waveguide whose width is periodically modulated to 0.8µm in a crenel like shape (see Figure 36 (a) inset). The wide parts are 0.5µm long while the narrow parts are 1µm long, which translates to periodicity a=1.5µm.

The MC shape is defined using a single step of e-beam lithography on positive resist. Gold is first deposited using dc sputtering then an aluminum layer is evaporated. Au is used as an oxygen barrier to avoid the Aluminum from altering the YIG surface stoichiometry. A lift-off is then performed, defining a hard mask that protects the YIG from the Ar etching process. The hard mask is later removed using selective chemical etching of Al (MF-319 developer for a night) and KI/I2 etch for Au (one minute). In a second lithography step, an inductive antenna consisting of a 600nm narrow strip (150 nm thick) is deposited across the MC. The antenna excites the spin-waves locally. The phaseresolved detection of the SWs is performed using the micro-focus BLS technique. The probing laser light with the wavelength of 473 nm and the power of 25 µW is focused through the sample substrate onto the surface of the YIG film into a diffraction-limited spot (Figure 36-a). A static magnetic field H0 = 1000 Oe is applied in the plane of the thin film and perpendicularly to the axis of the waveguide defining the propagation configuration of Damon-Eshbach SWs. Due to the demagnetization effects, the width modulation leads to the periodic spatial modulation of the internal static magnetic field in the waveguide (inset in Figure 36-a). Additionally, the finite width of the waveguide results in the quantization of the SW transverse k-vector that differs between the wide and narrow parts. Both effects cause noticeable modification of the SW dispersion between the wide and the narrow segments. To illustrate this fact, we show in Figure 36 (b) the dispersion curves for the 0.8 and 1 µm wide YIG stripes constituting the magnonic crystal calculated using the approach developed in Ref 69 that takes into account the demagnetizing field at the center of the waveguide calculated at first order. The validity of calculations is confirmed by the excellent agreement between the calculated curve and that obtained experimentally for a straight 1 µm wide YIG reference waveguide (displayed as symbols in Figure 36 (b).

II. Observation of the filtering effect

Figure 37 (a) shows representative examples of SW intensity maps recorded by rastering the probing laser spot over the waveguide for two different SW frequencies corresponding to the two wavelengths l = 3 and 2 µm. The maps clearly show that the SW with l = 3 µm that is equal to 2𝑎 forms a well-pronounced standing wave exhibiting a very fast spatial decay, as expected for waves with the wave vector 𝑘´ = 𝑘 | = 𝜋/𝑎 (Brillouin wavevector) propagating in a periodic potential with the period a, due to Bragg reflections. In contrast, the SW with l = 2 µm exhibits a much larger decay length. By plotting the propagation-coordinate dependence of the BLS intensity (see Figure 37-b) and fitting the data by the exponential function (note the log-linear scale), we determine the SW decay constant k´´ (imaginary part of the wave vector). For this intensity plot, the point x = 0 is shifted by 2 µm with respect to the edge of the antenna to exclude any influence of its near field on the results of measurements. The following analysis will only take into account the data measured between 2 µm and 12 µm (x = 0-10 µm) from the antenna. The real part of the wave vector 𝑘´ is extracted from the Fourier analysis of the phase maps (see Figure 37-c) recorded simultaneously with the intensity maps by using the phase-resolved BLS technique 108,109 . The phase maps reflect the spatial dependence of sin(f), where f is the phase accumulated by the SW during its propagation from the antenna to the detection point. Therefore, the spatial period of the phase map is equal to the wavelength of the SW at the given excitation frequency. By performing the above-described measurements at different excitation frequencies 𝑓, we directly determine the frequency dependence of the real and the imaginary parts of the wave vector 𝑘´(𝑓) and 𝑘´´(𝑓). As expected, the experimentally found dispersion curve 𝑘´(𝑓) (symbols in Figure 38-a) is located between the two curves calculated for the straight waveguides with the width of 1 (red curve) and 0.8µm (blue curve). In the vicinity of the Brillouin wavevector kB = p/a = 2.1µm -1 the slope of the measured dispersion curve exhibits an abrupt increase, indicating the formation of the magnonic band gap.

Additionally, the imaginary part of the wave vector k´´ is expected to abruptly increase in the band gap frequency range Δ𝑓 due to the destructive interferences between Bragg reflexions. Figure 38 (b) shows the measured dependence k´´(f) (symbols) together with the reference dependence (solid curve) calculated for the straight waveguide by using Eq. (6.80) from Ref. [START_REF] Gurevich | Magnetizaton Oscillation and Waves[END_REF] with a damping parameter 𝛼 = 3.4 × 10 |ý for a 0.8 µm wide waveguide. Indeed, the data of Figure 38 (b) confirm a strong increase of k´´. While for low and high k-vectors the decay constant is no different from a straight waveguide, k´´ spikes in the band gap as expected. This translates as almost a factor 4 decrease in the propagation length inside the gap. This increase span even outside the gap in a 60 MHz range. Note that these data of Figure 38 do not reveal any peculiarity at the frequency of 5 GHz, where the second band gap is expected to appear at 𝑘 = 2𝑘 | .

III. Other features of the gap

To get a sharper look at the filtering behaviour, BLS measurements are performed for frequencies close to the magnonic band gap with a frequency resolution of 2MHz. The obtained dispersion curve shown in Figure 39 (a) allows precise identification of the band gap edges i.e. f ›oe•ˆž = 4.874 GHz and f Ÿ ˆž = 4.896 GHz. Indeed, outside the gap, the dispersion relation progresses at constant group velocity. Right before transitioning to a vertical degeneracy, the group velocity is slightly decreased at f=flower. Similarly, at the upper edge of the gap, before transitioning back to an almost constant group velocity, the dispersion curve is flattened at f=fupper. In Figure 39 (b), we display the increased attenuation within the gap. Contrarily to the expected physical picture, the attenuation curve seems to be off-centered compared to the previously identified band gap by about 4MHz. The maximum of k´´, indicated by the cyan dotted line, is not located at the center of the band gap, but is slightly shifted toward its upper half, at a frequency 𝑓 ˆ= 4.89 GHz. In Figure 39 (c), we present a color map of the normalized intensity of the SWs as a function of frequency obtained by the high-resolution scan. Beside demonstrating the clear attenuation of SWs frequencies near 𝑓 ˆ, it exhibits a pronounced spatial beating of the amplitude of the SWs at all frequencies in the vicinity of the gap. The dotted black lines are meant as a guide to the This spatial beating is due to the standing waves present in the periodic structure. For frequencies inside the band gap, the k-vector is constant, only the amplitude can bring the degeneracy in frequency. The waves concentrate in the lowest energy region at the lower edge of the bandgap and shifts to the highest energy region at the upper edge. In our case, the lowest energy region is located in the middle of the narrow part (see Figure 38 (a)) due to the higher demagnetizing field while the highest energy is in the middle of the wide region.

Additionally, the maximum amplitude of spin-waves in much higher for frequencies below the gap than above the gap. For instance, the spin wave amplitude is plotted at frequency 𝑓 ˆ-4MHz and 𝑓 ˆ+ 4MHz (Figure 39 (d)). One can see that the maximum amplitude is already about six times smaller, which cannot be explained by the decrease of the efficiency of the antenna with increasing k-vector since both frequencies share similar k-vector (𝑘 = 𝑘 | ) nor by the decay of the spin-waves, since the 𝑘 ll is higher at 𝑓 = and 𝑓 ˆ-4MHz. Hence, the position of the standing waves must be playing a role in the efficiency of the coupling with the antenna. IV. Full-scale micromagnetic simulation of the MC Micromagnetic simulations are performed on the full-scale magnonic crystal using Mumax3 software 19 . Our aim is to compare the simulation with the experimental measurements in order to understand what are the driving parameters that allow for the optimization of a micron-scale magnonic filter. It will also allow us to test our understanding of the physics at stake. Using a 10x10x20 nm mesh, we simulate an 81.92 µm long, 1 µm wide and 20 nm thick MC with the same geometry and magnetic properties than our YIG magnonic crystal (no adjustable parameters). We use the MC nominal design, i.e. we did not consider any microfabrication induced imperfections like corner rounding and edge roughness.

SWs are excited at the desired frequency as harmonic response to a harmonic magnetic field, applied in a 20nm x 1µm x 20 nm region. The Gilbert damping is gradually increased in a 5 µm region on both ends of the 81.92 µm long structure to avoid unwanted SW reflections. After 0.5 µs, the steady state regime is established at the considered frequencies, hence the phase and intensity of the SWs in the MC can be extracted by plotting the out-of-plane component of the dynamic magnetization 𝑚 1 (𝑥, 𝑦, 𝑡). To mimic the size of the BLS probing spot, a 300x300 nm² sliding average is used for the intensity maps. Finally, the point 𝑥 = 0 was shifted by 2µ𝑚 from the simulated antenna as we did in the experimental analysis. More details about the method used and the testing made here can be found in the Supplementary Information in Ref 111 . the experiment, the maximum of 𝑘" is perfectly centred in the gap. The maximum value of 𝑘" is 0.16 𝜇𝑚 |-which is 30% less than in the experiment. The values of 𝑘" are also smaller outside the gap (0.06 𝜇𝑚 |-versus 0.12 𝜇𝑚 |-in the experiment), which tend to show that additional scattering processes are playing a role in the real magnonic crystal with edge roughness and defects. This additional scattering should also broaden the gap, which is what we observe with a simulated gap width of 12MHz compared to the 20MHz measured experimentally. Despite those small differences, the Bragg band gap of the MC is quantitatively reproduced by the simulation with no adjustable parameter. As for gaps stemming from mode crossings as described in Ref 112 , their position were found to be heavily dependent on materials parameters, making it very hard to precisely engineer such gaps and fabricate them reliably.

Figure 41 (a) represents a color map of the SWs amplitude (not normalized) as a function of frequency. At excitation frequencies smaller than 𝑓 ˆ, the spatial beating is well defined and its position is independent of frequency. The position of the amplitude maxima and minima shifts when the frequency is driven across the band gap. Above the gap, the SW amplitudes are smaller by about a factor of 2 and the pattern of the spatial beating tends to get more complex. Those observations are in good qualitative agreement with the experimental results shown in Figure 39 (c).

To get more insight on this asymmetry, color maps of the normalized amplitude of the SWs over the full width of the MC are plotted in Figure 41 (b,c,d) at three different frequencies (NB: log color scale). Below the gap (𝑓 = 𝑓 -), the propagation is single mode, the symmetry is reminiscent of a 𝑛 = 1 width mode living in the narrow part of the MC. The maxima of the standing SW amplitudes are located near the centre of the narrow region. On the contrary, above the gap (𝑓 = 𝑓 . ), the maxima of standing SW intensity are located at the centre of the wide part. Inside the bandgap (𝑓 = 𝑓 ˆ), the maxima of amplitude are located at an intermediate position but clearly not at the transition between wide and narrow part. Additionally, the SW beam gets wider as frequency is increased. It seems to feature complex self-focusing patterns 113 above the gap (𝑓 = 𝑓 . ), which are signs of appearance of 3 rd width mode in all parts of the MC. Finally, the antenna position was varied within the magnonic cell. We denote 𝜂 the ratio of spin-wave amplitude at frequency 𝑓 -over the one at frequency 𝑓 . . Placing the antenna in the middle of the narrow part results in the highest amplitude difference with 𝜂 ≈ 2.5, while when placed in the middle of the wide part, almost no difference is observed. Between these positions, the ratio varies continuously. This shows that the coupling with the antenna is strongly influenced by the position of the standing waves maxima relative to the antenna. Moreover, the ratio never gets lower than 1, indicating that there is already a difference of coupling due to the widening of the SW beam at frequencies above the gap. In the fabricated MC, surface roughness and defects are also expected to scatter more efficiently the higher order modes and the edge modes than the fundamental mode that dominates below the gap. This might result in the higher asymmetry observed in the experimental data (see Figure 39-d) where 𝜂 DC¿Dd"`DaV ≈ 4.

V. Conclusions

We have demonstrated a microscopic magnonic crystal based on ultra-thin YIG films with 100nm-size patterns. Using both phase-resolved µ-BLS and full-scale micromagnetic simulations we bring for the first time a detailed look at the modes propagating inside magnonic crystals. We have more specifically revealed the critical impact of standing waves on the coupling with inductive antennas. We also unveiled some effects due to the multimode propagation. Our results tend to show that modes crossing heavily depends on materials parameter such as the exchange constant which is difficult to precisely evaluate for a microstructure in an ultra-thin film. Hence, we believe Bragg reflection induced band gaps should be the primary focus for real-world application. With the advent of GPU-accelerated micromagnetic software, we have shown that reliable characterization can be achieved for this type of band gaps. This opens the prospect of quick performance optimizations of these structures, moving closer to applications. The study of a width-modulated magnonic crystal showed the potential of ultra-thin YIG for radiofrequency filtering. However, the width-modulation was shown to be lossy outside the band gap for two reasons. First, the spin-waves experience reflections, for any k-vector, when going from a wide region to a narrow region because of the strong difference in dispersion relations. And secondly, the design is sensitive to fabrication induced defects, which increases losses beyond what is computed in full-scale simulations.

In this chapter, we show that modulating the thickness gets rid of both problems. Indeed, the influence over the dispersion relation is greatly reduced, and the main effect is now a small modulation of the internal magnetic field due to the demagnetizing field. Additionally, we choose to modulate the thickness with extra thin grooves (150 nm wide) to reduce the impact of defects. The goal is to barely affect the spin-wave propagation outside the gap, while the resonant Bragg conditions creates a clear and effective filtering effect inside. Thickness modulated magnonic crystals were successfully implemented in µm-thick YIG films in Ref 106,107 and showed small losses. The impact of the number of grooves and the grooves depth was investigated. However, we will see in this chapter that the physics is quite different in thin films, resulting in an ultra-low loss magnonic crystal.

I. Fabrication

Using a similar device to the one characterized in 114 , a periodic pattern of thin-grooves is gradually etched in YIG waveguides (see Figure 42) and the impact of the grooves depth on the spin-wave induced inductance spectra is studied.

A schematic of the obtained thickness modulated magnonic crystal is presented in Figure 42. On a 23 nm thick YIG film, fifty 2.5 µm wide waveguides, separated by 1.5 µm gaps, are designed with laser lithography. Two asymmetric U-shaped gold antennas are evaporated on top for excitation and detection. It is composed of a 1.5 µm wide signal line and a 10 µm wide ground line, separated by a 2µm gap (also done by laser lithography). Using e-beam lithography, ten 150 nm grooves are defined in the electronic resist with a periodicity a = 3.14 µm. The sample is then successively etched using IBE and the inductance spectra are measured by Propagating Spin-Waves Spectroscopy (PSWS -see part 0) for grooves depths 𝑑 = 1, 5, 8, 11, 22 nm.

Electrically measuring spin-wave propagation in ultra-thin YIG films is a challenging task as YIG has a relatively weak saturation magnetization (𝜇 T 𝑀 e ≈ 0.175 𝑇) compared to metallic ferromagnets 72,115 . Probing fifty parallel identical waveguides allows us to increase the magnetic volume under the antennas and obtain a large signal amplitude using standard PSWS technique.

II. Inductance spectra for unetched waveguides

Though the unetched waveguides were measured in 114 using PSWS and the study completed with phase-resolved µ-BLS, some additional features were found during our initial PSWS measurements. They will be instrumental to the analysis of the etched waveguides data.

The measurement is carried out using micro-probes that connect the rf-antennas to a 2-port Vector Network Analyser (VNA). The VNA successively sweeps both ports in frequency, injecting rf current in port i (i=1,2) and measuring the induced reflection (port i) or transmission (port j). The result is given as an S-parameters matrix which is converted in an inductance matrix using:

𝔏(𝜔) = 𝑍 T 𝑗𝜔 (𝐼 . -𝑆(𝜔)) |-(𝐼 . + 𝑆(𝜔)) (137)
Where Z T is the impedance at the VNA ports (50 Ohm) and I . is the 2×2 identity matrix.

In this experiment, the frequency is swept between 0.6 and 2.6 GHz. The external magnetic field is swept from 5 to 30 mT in the DE configuration (Figure 43). A reference measurement is taken at 𝐻 dDg ≈ 40 mT, where no spin-wave are propagating (FMR = 2.8 GHz). By subtracting the reference, we can isolate the spin-wave contribution to the inductance spectra. Typical SW-induced inductance spectra (∆L11 and ∆L21) are shown in Figure 43 at different fields.

The imaginary part of the reflection spectra (∆L11 -dashed lines in Figure 43) peaks at the ferromagnetic resonance. From the position of the peaks, the Kittel law can be fitted and the effective magnetization 𝜇 T 𝑀 Dgg = 0.213 T found in Ref 114 is recovered.

The transmission spectrum (∆L21 -continuous lines in Figure 43) is shaped as a double lobe, modulated by an oscillating phase. Let's decompose the main terms that determine the transmission spectrum shape. Finally, the spectrum normalized amplitude is given in Figure 44(b) (black curve) as a function of the frequency using the dispersion relation obtained with the Kalinikos-Slavin theory 47 for an external field 𝐻 T = 6 mT and the parameters given in Figure 45.

To obtain the imaginary part of the inductance spectrum, the amplitude must simply be multiplied by the sine of the phase, accumulated by the spin-waves while travelling the effective distance 𝐷 Dgg between the two antennas:

𝜙 = 𝑘𝐷 Dgg [2𝜋] (138)
This yields an approximate expression for the complex valued ∆L21(𝜔) expressed as a function of k.

Δ𝐿 .-(𝜔) ∝ °𝐽± (𝑘) 𝐼 °. exp ¨-𝐷 Dgg 𝐿 QVV (𝑘) © 𝑒 uÉ" t½½ (139)
where

² ³ ±( É) F ² .
is the excitation efficiency of the antenna that depends on its geometry and defined in equation ( 140).

The approximations made to derive the above expression are detailed here:

-Only the first width mode is considered. Indeed it is the main mode contributing to the signal as the efficiency goes as 1/𝑛 . for odd modes and is very low for even modes 72,108 . -Additionally, the damping is very low in our YIG film, hence a one-to-one correspondence between frequency and k-vectors can be assumed. This correspondence is given by the dispersion relation. -The non-reciprocity of the excitation efficiency for DE spin-waves is not considered.

-The effective distance is assumed to be k-vector independent. This simple model is already useful to understand our data (in Figure 43).

For instance, a striking feature in Figure 43 is that the oscillating phase in the transmission signal no longer starts at 𝑘 = 0 𝜇𝑚 |-when the field is increased. At 𝐻 T = 27 𝑚𝑇 (red curve), the beginning of the oscillating lobe comes nearly 150 MHz later than the FMR peak in reflection (indicating 𝑘 = 0 𝜇𝑚 |-) while it starts at the FMR frequency for 𝐻 T = 6 𝑚𝑇. Moreover, the main lobe should be composed of about 20 oscillations (⌊𝑘 DCV 𝐷 Dgg /2𝜋⌋). This is the case for 𝐻 T = 6 mT but at 𝐻 T = 27 mT this number is down to 14, meaning that only kvectors from 𝑘 ≈ 1.4 𝜇𝑚 |-to 𝑘 ≈ 4.2 𝜇𝑚 |-propagate normally. In particular, this means that the Bragg k-vector of our periodic system (𝑘 | ≈ 1 𝜇𝑚 |-), is not properly detected above 𝐻 T = 15 mT. The issue probably originates from a difference in the YIG parameters below the antennas and in between antennas. As the field is increased, the difference in dispersion relation between both regions becomes larger. This can create standing waves at the excitation and detection volumes, lowering the transmission amplitude and destroying the phase coherence. The broad and sometime split FMR peaks detected in reflection is an indication of that. Multiple hypothesis can be formulated to explain this alteration of the YIG properties under the antennas:

-The Ti adherence layer used to make the gold antenna stick to the YIG is absorbing oxygen from YIG , thus locally changing the stoichiometry and stress. -The deposition of metals on top of a ferromagnets is known to affect the dispersion relation 116 . The presence of the 10µm wide ground line might cause problems that would not appear with a stripe antenna. The influence of heat due to the current flowing in the excitation antenna was checked and is found to be negligible at the excitation power used: 𝑃 = -20 dBm.

Another consequence of our simple model is that we can probe the dispersion relation from the measured phase using Eq. ( 138) and ( 139). The dispersion relation is extracted from the unwrapped phase of ΔL .-and is compared to Kalinikos-Slavin 47 theory computed for the first width mode in Figure 45.

As phase is a relative value, the blue curve was adjusted so that the first extinction of the antenna (𝑘 = 4.2 𝜇𝑚 |-) matches the frequency 1.8 GHz where the transmission amplitude goes to zero for the first time. As shown in Figure 46, the phase information is difficult to extract close to FMR and close to the extinction as the spin-wave signal amplitude goes to 0 (vertical parts). The theory matches pretty well with our measurement, using no fitting parameters despite the strong dependence of the dispersion relation on all the parameters. Some values were changed from 114 , as the thickness of the film is actually 𝑡 = 23.5 nm (from X-ray reflectometry) instead of 20 nm (which was estimated from the growth time). The effective magnetization was decomposed using the standard magnetization of YIG (µ T 𝑀 e = 176 mT) and a negative uniaxial anisotropy field (µ T 𝐻 P = -37 mT). The effective distance traveled by spin-wave was taken as the edge to edge distance plus the width of the signal line (as typically done in 72 ) 𝐷 Dgg = 31.5 µm.

By using an effective width in the Kalinikos-Slavin formulas, an almost perfect fit is found for w ˆ´´= 1.6 µm. The deviation from the nominal value could be due to the low external magnetic field applied and the demagnetization effects. The presence of multiple waveguides in parallel is also expected to have a small influence for small k-vectors. Being able to probe the dispersion relation is very valuable as it is a direct measurement of the spin-waves properties such as the k-vector or the group delay. It is more reliable than the amplitude that also depends on the coupling of the antenna with the main spin-wave mode.

III. Observation of the magnonic band gap

The waveguides are etched a first time, introducing a periodicity 𝑎 = 3.14 µ𝑚 in the system. The grooves depth is estimated at 2 nm, but any resist initially left at the bottom of the pattern might reduce this depth. The new inductance spectrum is compared to the unetched one in No second order gap seems to be detected. Given the small size and strength of the first order, the second order gap might simply be too small for detection as previously observed in 117 and later calculated in Ref [START_REF] Grigoryeva | Magnonics[END_REF] .

The phase and the amplitude are analyzed to quantify the properties of the observed band The asymmetry of the attenuation gap (wider on the right side) could be a sign that the imaginary k-vector is maximal in the center of the gap as was found in the simulations in the previous part. In the gap, the attenuation increases with the groove depth up to a 4dB attenuation for 8 nm grooves. Outside the gap, the attenuation is never higher than 0.4 dB on a 200 MHz range. The attenuation remains close to 0 dB even when the grooves depth is more than a third of the total film thickness. However, the effect is very pronounced for 𝐻 T = 12.5 mT for 5nm and 8nm grooves and the gap size is estimated at 13±2 MHz. In our limited magnetic field range, the gap opening seems to get qualitatively stronger as the external field is increased. However, as the group velocity decreases with the field, the gap width is expected to eventually decrease as found in Ref 43,118 . We note that a small effect is undeniably present already for the unetched waveguides. We attribute this effect to the e-beam writing that may have slightly altered the YIG properties and introduced the periodicity before any etching. The dispersion relation outside the gap, between 𝑘 = 1.4 and 1.8 µm |-, is shown in Figure 48 (c) for 𝐻 T = 12.5 mT. The dispersion is straight and is completely insensitive to the subsequent etchings. This graph confirms that our thickness modulation preserves the spin-waves dispersion everywhere except at the Bragg conditions where destructive interferences take place.

-0 nm -2 nm -2 nm -5 nm -2 nm -8 nm
The inductance spectra for a groove depth 𝑑 = 11 nm are very similar to the ones measured at 𝑑 = 8 nm, inside and outside the gap. We decided not to add them to lighten the above graphs Finally, the waveguides are almost fully etched. After the measurement, the resist is removed and the height is checked using an Atomic Force Microscope (see Figure 49 (b,c)). The 22nm±2nm depth means that less than 2 nm of YIG remains at the bottom of each grooves. Despite the waveguides' discontinuities, the measured spin-waves spectra amplitudes are barely affected. When comparing a 2 nm grooves depth at 𝐻 T = 11.3 𝑚𝑇 (Figure 49), we simply notice a decrease of the amplitude at high k-vectors and near the Bragg k-vector. Moreover, the gap seems to be closing compared to 5, 8 and 11 nm grooves depth.

Micromagnetic simulations were carried out and confirmed that spin-waves transmit were well through 150nm discontinuities for 𝜆 ≫ 150 nm. It was also experimentally observed in 260nm thick YIG films in Ref 43 .

-2 nm -22 nm

IV. Discussion

Thickness modulated magnonic crystals were previously studied for 5 to 15 µm thick YIG film in Ref 106,107 and for 260nm thick YIG films in Ref 43,118 . In thick films, the relative groove depth is found to be the leading parameter for the MC optimization. The depth is only modulated on 0-15% of the film total thickness, as the spinwave signal almost vanishes at 15% for wide grooves (𝑤 = 10% of 𝑎). Using grooves width 𝑤/𝑎 = 3.3% significantly reduces the reflection from a single groove. Consequently, the pass band losses are reduced, and the band gap losses are decreased even more strongly. In our case, (𝑤 = 5% of 𝑎), the losses outside the gap remain imperceptible even in the fully etched case. We infer that the reflection from a single groove is much smaller in our thin film in the DE configuration than in a thick film in the BW configuration. This backscattering immunity was discovered simultaneously in simulation in Ref 44 . By looking at the spin-wave power transmitted when going through a single groove defect, Pirro et al. showed that the surface waves are very minorly affected by deep grooves in thin films. They showed 44,119 that this property is intrinsic to surface waves in thin film as volume waves are strongly scattered by deep grooves. The width of the grooves is an important parameter. By increasing the single groove reflection, one can increase the gap width (see Ref 43,118 ) and depth (see Ref 106 ). A compromise must be found between losses outside the gap and the filtering efficiency depending on the specifications.

Increasing the number of grooves (N) is found to slightly decrease the gap width and significantly increase its depth (in Ref 106 ). However, as the filtering effect originate from destructive interferences in the lattice, the lattice size (𝑁 × 𝑎) must remain comparable with the decay length which is the case in our MC (𝑁 × 𝑎 = 31 µm -𝐿 QVV ≈ 25 µm).

Finally, we point out that the magnonic crystal filters the wavelengths. By scaling down the MC, the ratio ∆𝑘/𝑘 | remains roughly constant. This means that the gap width ∆𝑓 ≈ ∆𝑘 × 𝑣 ø (𝑘 | ) should increase when 𝑘 | is increased. Additionally, the group velocity and the attenuation length increase with k-vector, allowing to significantly increase the number of grooves for small lattice constants. All those elements lead us to think that the filtering performance of our system can be easily improved by scaling it down. The ultra-low losses observed outside the gap is expected to still be present in scaled down MC as the backscattering immunity is still valid for exchange dominated surface waves 44 as long as they remain in the volume spin waves gap.

Part 3: Delay line demonstrator -a step toward integrated magnonics

When the first magnonic based signal processing systems were developed in the 1980's, they quickly got outperformed by the acoustic-waves based systems and CMOS based-digital signal processing stages. However, both become very inefficient at high frequency (above 10GHz), where numerous systems currently operate (radars, satellite communication) and will operate in the future (high bandwidth 5G networks, IOT devices).

In particular, delaying a signal at those frequencies is either extremely lossy (50dB) or occupies large volumes 120 . In this context, developing a magnonic delay line is relevant as a first step towards integrating magnonic systems into real-world applications. Similarly to the PSWS technique (presented in Part 0 and Part 2), the rf-signal is converted to spin-waves via an inductive transducer. The typical group velocity of spin-waves being of the order of 1km/s, one nanosecond delay is achieved every micron of propagation. The spin-waves signal is then picked up by a second transducer and converted back to an electrical signal.

One objective of this thesis is first to progress toward a prototype of a fully integrated magnonic delay line, using standard lithography techniques. And second, to create a modelling tool able to predict the obtained performance and to optimize future iterations.

If the values obtained are competitive in terms of losses and dimensions, future developments could be conducted in collaboration with the Thales business units to design systems answering their specific needs and leveraging the advantages of magnonic based systems, namely: the tunability of the operating frequency, the micron-sized wavelengths at GHz frequencies, or the immunity of ferromagnetic materials to radiation.

The operating frequency of the prototype was chosen to be 10 to 20 GHz with delays between 10ns and 50ns.

Chapter 6. Delay line prototype: Results and modeling

Let's first derive the general magnonic properties needed for a magnonic delay line. The material used is YIG. Indeed, at 15GHz, the lifetime of spin-waves in high-quality YIG (𝛼 = 10 |ý ) is about 𝜏 ≈ 100 ns, which guarantees low propagation losses for delays of tens of nanoseconds.

NB: Metallic ferromagnets, such as CoFeB (𝛼 = 4 × 10 |/ in Ref 84 ), are consequently ruled out for delays exceeding the nanosecond. The easy growth of CoFeB films on Si substrate is an advantage if very small delays are desired.

A typical sample is shown in Figure 50. Two rf-antennas are deposited on top of an array of narrow YIG waveguide (4µm). The GSG antennas signal line is 4 µm wide, while the ground lines are 2µm wide. Other devices with a single wide waveguide (30µm) were also patterned and measured.

The film is chosen to be about 𝑡 = 300 nm thick. Indeed, the SW induced impedance is proportional to the film thickness while ohmic losses are independent on the film thickness. Hence a higher share of the impedance is dedicated to spin-wave transduction in thick films. However, the goal is to remain compatible with standard lithography and patterning techniques. The 300nm thickness marks the limit where thickness modes are split enough so that the delay line operates in the single thickness mode regime (see Figure 51). As such, it seems like a reasonable starting point to investigate the performance that can be obtained, but nothing prevents us to increase the thicknesses if needed or on the contrary push toward thinner films.

RFantenna

𝒘 𝒔𝒊𝒈𝒏𝒂𝒍 = 𝟒 𝝁𝒎 𝒘 𝒈𝒓𝒐𝒖𝒏𝒅 = 𝟐 𝝁𝒎 𝒘 𝐠𝐚𝐩 = 𝟐 𝝁𝒎
The magnetic field orientation is chosen to be in plane. Indeed, the limit for single mode operation would be 150 nm for forward volume waves (out-of-plane magnetic field) and would necessitate a higher field to operate to compensate the shape anisotropy. The properties will be measured both for a magnetic field orthogonal to the waveguide axis, called the Damon-Eshbach (DE) configuration 100 , and parallel to it, called the backward volume spin-waves (BW SWs) configuration. Both configurations offer interesting properties. Hence, we will analyze the delay line performance for both.

The YIG film is patterned in waveguides to avoid unwanted scattering and reflections at the antenna's edges and pads. The typical group velocities are of the order of 1000 m/s for narrow waveguides (such that 𝑤 ~ 𝜋/𝑘 QaV ) and 500 m/s in wide waveguides (such that 𝑤 ≫ 𝜋/𝑘 QaV ). This corresponds to a 20 µm (resp. 10 µm) and 50 µm (resp. 25 µm) distance between the antennas for 20 ns and 50 ns delays. Finally, the largest k-vector that can be comfortably obtained using laser lithography is typically 𝑘 QaV = 𝜋/5µm ≈ 0.6 rad/µm. That makes a standard ground-signal-ground (GSG) antenna 12µm-wide. Smaller k-vectors result in wider antennas for which the delay must be high enough to fit the antennas.

The different delay lines are measured using Propagating Spin-Wave Spectroscopy (PSWSsee part 0). The patterned rf-antennas are connected to a VNA via rf micro-probes and the Sparameters are recorded. The main specifications of the delay line are deduced from the Sparameters spectra: the delay, the insertion losses and the return losses. By analyzing their frequency dependence, we test the existing models and look for optimal designs and performance.

The analysis is decomposed in three parts. First, by analyzing the reflection spectra (voltage induced by spin-waves on the exciting antenna), the spectra shapes and amplitudes are discussed and compared to analytical models. These reflection spectra characterize the coupling between the spin-waves and the rf current through a spin-wave induced radiation resistance term. In a second part, the transmission spectra are used to derive the delay and losses generated by the propagation of the spin-waves. Finally, the full circuit is considered. This subsection goal is to understand where the injected power goes and how to maximize the transfer of energy through the magnonic circuit. We discuss the ways of solving the impedance matching problem, ie ways to optimize both the insertion and return losses.

I. Spin-wave excitation and detection: analysis of the reflection spectra

The rf-current circulating in the antenna generates a dynamic magnetic field that can excite spin-waves in the ferromagnet. At the resonance conditions, part of the rf-power goes into the magnonic degrees of freedom, effectively creating a frequency dependent radiation resistance. Such spin-wave induced impedance is measured by reflection measurement: the rf-power is injected in a port and the voltage is probed on the same port. It corresponds to the diagonal terms of the S-parameters matrix for a two-ports measurement. Predicting the spin-wave induced impedance spectrum is key to model the Delay Line (DL) performance as it quantifies the DL bandwidth, defined as the width at half-maximum of the main excitation lobe, and the share of the power that can be injected into the magnonic system and converted back to a voltage. The analytical computation of the spectrum linewidth and shape is shown in the first subsection as it is simply derived from the current distribution in the antennas. Predicting the absolute value of the SW induced Radiation Resistance (RR) is a more complex task. A brief review of the existing model is presented and two of them are chosen to analyze our results. Finally, the SW induced RR is extracted from the VNA measurements. Due to the presence of a strong RLC resonance in our transducers, a lumped RLC model is developed to extract the non-resonant spin-wave induced impedance in order to compare it to the analytical models' predictions.

A) Excitation spectrum

The shape of the antenna defines the efficiency with which the spin-waves are excited as schematically represented in Figure 52. Indeed, the spin-wave impedance is proportional to the squared norm of the Fourier transform of the current distribution along the propagation direction. The square comes from the fact that both the excitation and the detection efficiency depend on the norm of the Fourier transform.

For a single line of width 𝑤 `e, assuming a uniform distribution of the current (crenel function

³(C) F = - Í hy
for 𝑥 ∈ [0, 𝑤 `e]), the Fourier transform is given by:

°𝐽± (𝑘 C ) 𝐼 °= sinc ¨𝑘C 𝑤 `e 2 © (140)
In a microstrip, the current distribution is given by the relation: with ℬ T , the zeroth order Bessel function.

In the case of a more complex antenna, using the linearity of the Fourier transform, we can addup each line contribution provided that the correct phase is used and that the current distribution is identical in each line (NB: the norm of the Fourier transform is not a linear transformation). Let's call 𝐽 T the single line Fourier transform of the current distribution.

In the case of the antenna used in this study, the total Fourier transform is given by:

°𝐽± (𝑘 C ) 𝐼 °= °𝐽T ¨𝑘C 𝑤 e 2 © -𝐽 T A 𝑘 C 𝑤 øQ¿ 2 B 𝑒 5É u (Í ª‹È Ê Í y . Ê Í ª †AE ‡«C . ) + 𝐽 T ¨𝑘C 𝑤 e 2 © 𝑒 5É u (.Í ª‹È ÊÍ y ÊÍ ª †AE ‡«C ) ° (143)
with the width of the signal line, ground line and the gap given by 𝑤 e = 2 µm; 𝑤 ødWPac = 4 µm and 𝑤 øQ¿ = 2 µm. The minus sign comes from the current flowing in the opposite direction in the ground line compared to the signal lines.

Both hypothesis on the current distribution inside the microstrips have been used (𝐽 T = sinc in Ref 72,121 or

𝐽 T = ℬ T in Ref 122
). The results of both hypothesis for our antennas are shown in Figure 52. It is important to notice that the impact on the main lobe amplitude and position is small. It is not surprising that at first order the efficiency is defined by the periodicity of the metallic lines. The first order lobe is the main contribution to the spin-wave induced impedance. Thus, the DL operates with wavevectors that are comprised between 0 and 1 rad. µm |-, with a maximum contribution at 𝑘 T = 0.6 rad. µm |-and a width at half-height ∆𝑘 = 0.45 rad. µm |-. This spectral range corresponds to the bandwidth of our DL. As the group velocity is of the order of 1000 m/s, the frequency range is about ∆𝑓 ≈ X ª ∆É .¸, ie a few tens of MHz to 100 MHz.

B) Computing the spin-wave induced impedance

Predicting the spin-wave induced impedance is a complex task because of the complexity of the electromagnetic problem. Many different models were proposed, with different degree of generality and complexity and were mostly applied on thick YIG films [START_REF] Prabhakar | Spin Waves Theory and Applications[END_REF][122][123][124][125] . Good agreement were also obtained in metallic ferromagnet in Ref 72 . As the delay line is operated at wavevectors much higher than the exchange length, the exchange interaction can be neglected when solving the electromagnetic problem. Hence most models solve the Maxwell equations and the appropriate boundary conditions using formalisms with different degrees of complexity and generality.

The main model we consider is the one described in Ref 72 by Vlaminck and Bailleul. It uses a formalism introduced in Ref 123 by Emtage that elegantly solves the Maxwell equations and exposes the physical processes at play. Another advantage of the model is that it includes the derivation for both reflection and transmission impedance, as well as the effect of the damping.

As we work with an ultra-low damping YIG film the impact of damping on the amplitude is very small, but it allows the model to directly give both the real and imaginary part of the impedance, which we need to model the delay line. Finally the model was extensively compared to experiments in the out-of-plane in Ref 72 and the in-plane configurations in Ref 71 . Both experiments were conducted on few tens of nanometers thick permalloy. The model does not consider the back-action of the spin-waves on the current distribution in the antenna. Such "self-consistent" models 122,125,126 were developed to analyze experiments on µm-thick YIG films. However, the impact remains modest as shown in Ref 122 . Hence, to achieve our goal of evaluating the delay line range of performance, the retroaction of spinwaves on the current is neglected. Similarly, the impact of the metallic structures (also discussed in Ref 122 ) is neglected.

The model from Kalinikos (Ref 122 ), under its simplest assumptions (no back-action of SW, no damping, no impact of metallic structures), is very simple to implement. We will use it to control that we have implemented the Vlaminck & Bailleul's model correctly.

Both articles (Ref 72 and Ref 122 ) are very clear, exhaustive and full of insights. Thus, the full derivation of the models is not given in this manuscript as there would be very little added value. The notations used in Ref 122 are very close to the one used in this manuscript.

C) Comparing with the experiment

In order to compare the models' prediction to the experiment, the spin-waves contribution to the impedance must be extracted from the S-parameters. The Delay Line impedance matrix 𝑍 Ç is computed using [START_REF] Gurevich | Magnetization Oscillations and Waves[END_REF]127 :

𝑍 Ç(𝜔) = (𝐼 . -𝑆 È(𝜔)) |-⋅ (𝐼 . + 𝑆 È(𝜔))𝑍 T (144)
with 𝑍 T = 50 Ω, the characteristic impedance, and 𝐼 . the 2 × 2 identity matrix.

To isolate the spin-wave contribution, the reference impedance matrix is measured at zero field where no spin-waves propagate. It is then usually subtracted to the impedance matrix measured at the desired field:

Δ𝑍 Ç(𝐻, 𝜔) = 𝑍 Ç(𝐻, 𝜔) -𝑍 Ç dDg (𝜔) (145)
where 𝑍 Ç dDg (𝜔) = 𝑍 Ç(𝐻 = 0, 𝜔).

However, this method is only valid if the spin-waves impedance is in series with the total transducer impedance. It is usually the case at low frequencies where the resistive impedance of the antenna dominates. But in our specific case, the inductive and capacitive effects cannot be ignored, and the method is no longer valid.

In Figure 53, the transducer auto-impedance spectrum is plotted for the 30µm long antenna in the absence of spin-waves (zero magnetic field). A clear resonance is observed at 12GHz and the resistance peaks at 250 Ω, much higher than the DC resistance (see at 1GHz) of 8.5 Ω. This type of resonance was previously observed in Ref 71 , although the resonance was occurring at higher frequency so that it could be neglected in the analysis of the results. In Ref 71 , the resonant behavior is modeled using a distributed RLC circuit. In our case, the length of the antenna is small compared to the electromagnetic wavelength, so that a lumped RLC model can be used. The transducer impedance is given by:

𝑍 -- dDg = 𝑉 - 𝐼 - = 1 1 𝑍 z + 1 𝑍 } (146)
with 𝑍 z = 𝑅 + 𝑗𝐿𝜔 and 𝑍 } = 𝑅 -5 }ð .

𝑍 z is the inductive component of the impedance (linear behavior of the imaginary part at low frequencies) and 𝑍 } the capacitive component. At low frequency, 𝑍 } is very large and all the current goes through the inductive part. On the contrary, at high frequency, 𝑍 z is very large and the current goes to the capacitive branch.

In the case of the 30 µm long antenna, we extract the following values: 𝑅 = 8.2 Ω; 𝐶 = 0.2 pF; 𝐿 = 0.87 nH. We stress that this resonance gets flatter when the resistance is increased. Indeed, for a similar 120µm-long antenna, the inductance and conductance remain in a similar range, but the higher resistance (𝑅 = 21 Ω) significantly reduces the resonance peak amplitude (see Figure 53-b). The agreement at high frequencies between the model and the experiment is slightly worse. The presence of a second resonance at a higher frequency could modify the curvature. We observe such second resonance in the case of a U-shape antenna (not studied extensively because of its poor cross-talk properties) where the first resonance is at 8 GHz and a second one near 22GHz. The impact of the RLC resonance on the impedance 𝑍 "" is shown in Figure 54. Using Eq. ( 148), the change of impedance (Δ𝑍(𝐻)) induced by the presence of spin-waves is computed for experimental spectra at different fields. The amplitude is clearly enhanced by the RLC resonance, strongly peaking at 12 GHz. The Δ𝑍(𝐻) phase undergoes a gradual 360° shift at the resonance (Figure 54-b). At the resonance, the shift is 180° and the presence of spin-waves reduces the antenna resistance. A similar effect is observed for the transmission spectrum (blue curves).

The effect of spin-waves can be assimilated to a radiation resistance (RR), that is distributed along the antenna length, similarly to the ohmic resistance R. Thus, the transducer is modeled using the lumped parallel RLC model depicted in Figure 55, where the spin-wave induced autoimpedance is called 𝑍 -- 3n . In a non-resonant circuit, the change of impedance Δ𝑍 --would be equal to 𝑍 -- 3n . However, in our circuit, the impedance measured by the VNA is given by: 𝑍

--= 𝑉 - 𝑖 - = 1 1 𝑍 z + 𝑍 -- 3n + 1 𝑍 } + 𝑍 -- 3n (147) Δ𝑍 --= 1 1 𝑍 z + 𝑍 -- 3n + 1 𝑍 } + 𝑍 -- 3n - 1 1 𝑍 z + 1 𝑍 } ≠ 𝑍 -- 3n (148)
Thus, Eq. ( 148) is not appropriate to extract 𝑍 -- 3n at all frequencies.

Let's find a way to derive 𝑍 -- 3n and 𝑍 .. 3n (𝑍 "" 3n ; 𝑖 = 1,2) from a measurement at a magnetic field 𝐻 (𝑍 "" (𝐻, 𝜔)):

1 𝑍 "" (𝐻, 𝜔) = 1 𝑍 z + 𝑍 "" 3n + 1 𝑍 } + 𝑍 "" 3n (149) ⇔ (𝑍 "" 3n + 𝑍 z )• 𝑍 "" 3n + 𝑍 } --𝑍 "" •𝑍 z + 𝑍 } + 2𝑍 "" 3n -= 0 (150) ⇔ 𝑍 "" 3n . + (𝑍 z + 𝑍 } -2𝑍 "" )𝑍 "" 3n + 𝑍 z 𝑍 } -𝑍 "" (𝑍 z + 𝑍 } ) = 0 (151)
We find a quadratic equation in 𝑍 "" 3n whose solutions are:

𝑍 "" 3n = -(𝑍 z + 𝑍 } -2𝑍 "" ) ± r4𝑍 "" . + (𝑍 } -𝑍 z ) . 2 (152)
with

± = sign•Re(𝑍 z + 𝑍 } -2𝑍 "" )- (153)
The square root convention impose that the real part of the square root is positive. The sign definition ensures that if we take 𝑍 "" = 𝑍 "" dDg (Eq. ( 146)), we correctly obtain 𝑍 "" 3n = 0.

Hence, we have defined an unambiguous way to correctly extract the spin-wave autoimpedance from an experimentally measured spectrum.

In practice, when applying the formula on a spectrum measured at zero magnetic field (𝑍 "" 3n (𝐻 = 0)), we do not exactly get zero but rather a background spectrum that is present in all 𝑍 "" 3n spectra. It is due to other contributions to the impedance such as the cross-talk (see the bump at 13.5 GHz in Figure 53). Thus, the pure spin-wave induced impedance is given by:

∆𝑍 "" 3n (𝐻) = 𝑍 "" 3n (𝐻) -𝑍 "" 3n (𝐻 = 0) (154)
An example of our data processing scheme is given in Figure 56-a) for the reflection spectrum.

The method can also be applied to the transmission spectrum (Figure 56-b), and will be described in section II. After subtracting the background, a typical spin-wave spectrum is shown in Figure 56-c).

The inconvenience of such method is that it depends on the fitted value (𝑍 z , 𝑍 } ) and on the accuracy of the model (absence of secondary resonance for instance). From an analysis standpoint, such RLC resonant effects should be avoided, either by tuning down the C and L values or by using more resistive antennas. The value of C and L seem to depend on the shape of the antennas. The substrate thickness might also be important. The resistance of the antenna can be controlled by its thickness, length and width. From a performance stand point, the impact of the resonant transducer on the insertion and return losses is analyzed in section III. B).

The experimental spin-wave induced auto-impedance spectra are now compared to the analytical magnonic models.

1) Surface Spin-waves in a 30µm-wide waveguide

The impedance spectra are measured at different external magnetic field for a 30µm wide waveguide in the DE configuration 100 (magnetic field transverse to the propagation direction). We are interested in the SW RR as it indicates the interaction strength between the rf-current and the spin-wave degree of freedom. As a result, the maximum of the real part of the SW induced auto-impedance is plotted in Figure 57 as a function the resonance frequency. The experimental RR is extracted using both the "naïve" method (red dots), described by Eq. (148), and the method using our RLC model (Eq. ( 152)) (green squares). One can notice that using the naïve method gives rise to an artificial increase of the radiation resistance that peaks at the transducer resonance. On the contrary, using the RLC modeling gives a monotonous increase of the radiation resistance as expected from the models (dotted and dashed black line).

Real value Real value

We use two models to make sure that their implementation is correct. Indeed, the Kalinikos model 122 is simple to implement using its basic assumptions (ie neglecting the spin-wave backaction on the current distribution, the impact of metal layers and assuming zero damping). Bailleul's model is a bit more complex but includes the effect of damping in a natural way. Both models are identical at low frequency, where the magnon lifetime is still large compared to the excitation time (∝ 𝑤 QaVDaaQ /𝑣 ø ). But at large frequencies, the effect of a small damping (𝛼 = 5 × 10 |ý ) is clearly visible. For both models of Figure 57, no renormalization constants were used. However, in most experiments in Ref 72 , the introduction of a renormalization constant 𝑐 ≈ 0.65 -0.9 is necessary to accurately reproduce the experimental spectra. As explained in Ref 72 , this mainly comes from the fact that using a 2D model implies that dynamic magnetization profile is uniform along the waveguide width, which is not the case. The overlap integral gives a theoretical constant 𝑐 = 8/𝜋 . ≈ 0.81. The second uncertainty comes from the z-distribution of the current. Here, we assume that the current flows in the middle plane of the conductor.

This assumption impacts the predicted amplitude depending on the spin-wave k-vector due to the z-dependence of the magnetostatic potential ψ(𝑥, 𝑧, 𝑡) ~ 𝑒 |É1 exp 𝑖(𝜔𝑡 -𝑘𝑥). extracted from the experiment (symbols) using the naive method (Eq. ( 148) -in red) and the distributed RLC model (Eq. ( 152) -in green) and computed from models (lines).

The impact of the damping on the RR is shown in Figure 58. The RR at a given resonance frequency is observed to increase because of the inductive character of the impedance (proportional to the frequency) and the decrease of group velocity. However, the introduction of damping reduces the excitation efficiency. As the damping torque is proportional to the frequency, a higher damping inflects the curve at higher frequencies, reducing the slope and eventually decreasing the radiation resistance (not shown here). Measurements above 15GHz could allow to observe such inflection of the curve. Currently, the slope simply implies that the damping is of the order of 𝛼 = 7.5 ± 2.5 × 10 |ý . In general, the model overestimates experimental values and cannot be fitted solely using the renormalization constant c.

The SW induced RR spectrum at 10GHz is compared in Figure 58-(b) to the model predictions.

Using a low damping, the spectral features of the antenna are very detailed (dotted line). Indeed, to a given frequency corresponds a given wavevector, whose excitation efficiency by the antenna varies faster compared to the magnonic quantities (such as the group velocity). Hence the features of the antenna excitation spectrum are sharply reproduced. At higher damping (dashed line), multiple wavevectors can propagate at a given frequency since they cannot be told apart on the typical propagation length. Hence, the features of the antenna excitation spectrum are blurred. At intermediate dampings (𝛼 ~ 1 × 10 |/solid line), two typical features of the spectrum are reproduced (a shoulder in the main lobe and a well-defined secondary lobe). This damping value comes as a surprise as a low damping of the order of 2 × 10 |ý was expected (from FMR measurements 6 ). As LPE growth occurs on both faces of the sample, FMR measurements were also carried in the lab to determine the correct face. They seem to indicate a 6 × 10 |ý damping on the correct face. The damping can be more accurately extracted by analyzing the transmission spectra (see section II). Overall, the experimental spectrum is broader than in the model. We point out that a higher group velocity would not be sufficient to explain such broadening. Indeed, the uncertainty on our materials parameter does not allow a 50% to 100% increase of the group velocity. Thus, the difference must come from the way we calculate the antenna excitation spectrum. The excitation spectrum is defined by Eq. ( 142) and (143) (Bessel type of spectrum) and does not take into account the back action of the spin-wave on the current distribution. However, it is not clear whether that contribution would broaden or narrow the spectrum. Those 6 From personal communication with Jamal Ben Youssef -July 2020 considerations are left for later investigations as they're not deemed crucial to the design of the first prototype.

2) Backward Volume spin-waves in a 30µm-wide waveguide

The external magnetic field is now applied parallel to the propagation direction, and the delay line now operates with so-called Backward Volume (BW) spin-waves.

Before comparing the experimental results to the models, we remind some specificities associated with BW spin-waves. The BW spin-waves have a negative group velocity in the dipolar dominated region (small wavevectors). Thus, the propagating spin-waves have a frequency that is than lower the FMR frequency, i.e. on the BW SWs spectrum, smaller frequency means higher k-vector, contrary to DE SWs (see dispersion relation in Figure 51).

For our film thickness (𝑡 = 280 nm), their group velocity is higher than of DE SWs at frequencies larger than 5GHz (see Figure 60-b). Finally, their excitation by inductive antennas is less efficient since only the out-of-plane (z-component) of the dynamic field contributes to the excitation, as the x-component is parallel to the equilibrium magnetization direction. The maximal values of the radiation resistance are plotted in Figure 59-(a) as a function of the resonance frequency, from 4 to 14 GHz. The resistance increases linearly with frequency and is in very good agreement with the models. The experimental values (red squares) even slightly surpass the model with a normalization constant 𝑐 = 1 and a low damping (solid and dotted lines). We note that the steepness of the slope also excludes the possibility of a damping higher than 10 |/ in this configuration (dashed line). The agreement is even more striking when directly comparing the experimental spectrum to the model (see As expected, the radiation resistance is smaller than for DE (0.05 Ω. µm |-versus 0.08 Ω. µm |- at 5GHz; 0.14 Ω. µm |-versus 0.24 Ω. µm |-at 14 GHz). Furthermore, the DE values should be even higher with a lower damping (0.35 Ω. µm |-at 14GHz simply with 𝛼 = 5 × 10 |ý ).

The radiation resistance could be increased by using larger antennas. Indeed, the group velocity of BW spin-waves is larger, thus the antennas are farther apart to obtain the same delay compared to DE spin-waves. NB: For similar radiation resistance the DE configuration is still nearly two times more efficient as the spin-waves will be mainly emitted on a single side of the antenna while the BW spinwaves are emitted symmetrically. The spin-waves emitted in the wrong direction are lost.

Overall, the BW spin-waves are less efficiently excited, but their spectrum amplitude accurately follows the model expectations. Moreover, the spin-waves seem unaffected by the linewidth broadening observed in the DE configuration.

Finally, at all frequencies, the spectrum is "decorated" with periodic local maxima and minima (blue arrows in Figure 59-b). By analyzing their periodicity in different spectra, we find that their periodicity is independent of frequency and equal to ∆𝑘 = ∆g X ª ≈ 0.059 ± 0.004 rad/µm.

The pattern evokes an interference between emitted waves and waves that are reflected back after travelling a distance 𝐷 "aV . From the periodicity, we find 𝐷 "aV = ∆É ≈ 54 ± 3 µm. The distance from one antenna to the other, back and forth, being 2 × 𝐷 Dgg = 56 µm we conjecture that part of the spin-waves could be reflected back. But, this effect could also come from the phase dependent transmission (see section III.A)). Indeed, the maxima are observed when the off-diagonal impedance is purely resistive, where the insertion losses are minimized (see Figure 77). We stress that the well-defined interference pattern is another indication that the damping is low.

3) Surface Spin-waves in 4µm-wide waveguides

The auto-impedance spectrum is now measured in narrow waveguides (4µm wide). We use 20 parallel waveguides with a 2µm spacing to increase the magnetic volume. We also expect a smaller lineic RR due to the higher group velocity in such waveguides. The antennas are thus longer (120 µm) leading to resonance effects that are less pronounced due to the increase of the DC resistance (see Figure 53-b). In narrow waveguides, the quantization of the modes has an impact on the dispersion relation at the wavevectors that our antenna excites. For both BW and DE spin-waves, the quantization results in a speeding up of the spin-waves for wavevectors close to quantization wavevector (𝑘 3 -= Í for the first mode, see Figure 60-a). The confinement in narrow waveguide is thus an efficient method to control the group velocity which is at the core of the Delay Line functionality. For instance, one can reduce the dispersion by operating near the group velocity maximum, so that the delay is homogeneous in the device bandwidth (highlighted by the greyed area in Figure 60-a) at a given external field. The evolution of the group velocity with the resonance frequency is also presented in Figure 60-b) for a 4µm-wide waveguide and a full film at 𝑘 = 0.6 rad/µm. A striking feature is that the BW volume waves in waveguides have a nearly constant group velocity at all frequencies. This could be leveraged to obtain a constant delay on a wide range of frequency thanks to the tunability of the magnonic delay line. Furthermore, large group velocities can be useful to design delay lines with small delays. Indeed, the delay is limited by the size of the antennas that is constrained by the fabrication process of the chosen nominal wavevector. In all cases, the width of the waveguide must be adapted to the antenna excitation spectrum so that the effect is obtained at the desired wavevectors.

The radiation resistance for surface SWs is shown as a function of the resonance frequency in Figure 61-a). It is in close agreement with the model below 8 GHz. Above 8 GHz, the extracted RR seems to peak at 12 GHz, suggesting that our RLC model fails above the resonance for this antenna (see Figure 53-b)), probably due to the presence of a second resonance at higher frequencies that is not taken into account in our model. Thus, the effect of the damping cannot be distinguished in this case, since its impact is much greater at high frequencies. However, the values at low frequencies indicate that patterning the film has no impact on the expected radiation resistance, provided that the damping is not increased. The RR is similar to the ones obtained in the large waveguides (Figure 58 -0.1 Ω/µm at 8GHz) despite the higher group velocity and the 2/3 filling factor (2µm spacing between 4µm-wide waveguides). Bailleul's model can be adapted to work in a waveguide by rotating the susceptibility tensor in the case of DE spin-waves. We check that the result is coherent with the experiment and the Kalinikos model in Figure 61-b). The frequency, amplitude and linewidth are well reproduced which validates our approach. Only with an artificial renormalization constant 𝑐 = 2.5 does the model approach the experimental values. Getting a closer a look at the spectra (Figure 62-b) brings a partial explanation. The shape of the impedance spectrum is indeed abnormally peaked. This reminds the interference patterns observed in wide waveguides. However, it is hard to imagine this phenomenon being the sole explanation. Even with a full reflection and no damping, the impedance would only be increased by a factor of 2 and strong minima should also be observed. Besides, our attempts to adapt Bailleul's model in the BW configuration only resulted in unreasonable values (regarding both the Kalinikos model and the experimental values). Getting a better understanding of BW spin-waves in narrow waveguides is a goal for the next experimental sample. Using 50 Ohm antennas would make for an easier extraction of the experimental values. Varying the antennas' distance seems also important to exclude any interference effect. Resorting to micromagnetic simulation is also possible in the case where we would fail to improve our models.

D) Conclusion

The reflection spectra contain all the information concerning the coupling between the transducers and the magnonic degree of freedom. By correctly processing the experimental data, the measured spectra were compared to the analytical models up to a frequency of 15 GHz, despite the presence of a strong RLC resonance at 12GHz. In all the probed configurations (DE or BW SW, narrow or wide waveguides), a good qualitative agreement was found, in terms of amplitude and overall shape.

A near perfect agreement is even observed for BW spin-waves in wide waveguides and for surface spin-waves in narrow waveguides. However, the models clearly overestimate the radiation resistance for DE in a wide waveguide and underestimate BW in narrow waveguides. In both cases, the shape of the spectra is also either broader than expected (DE) or more peaked (BW), meaning that all the physical phenomena are not accounted for. Beyond those Physics considerations, the point was also to record the typical SW impedance values one can expect in real conditions. We plug these realistic values in our analytical model of the Delay Line circuit in section III to discuss the optimization of the delay line.

The information relative to the spin-waves propagation is however contained in the transmission spectrum that we describe now.

II. Spin-wave propagation: analysis of the transmission spectra

The transmission spectrum is composed of two contributions: the direct electromagnetic coupling between both antennas (or cross-talk) and the magnonic coupling via the propagation of spin-waves. As the goal is to delay the signal using the small spin-waves' group velocity, we aim at reducing the cross-talk as much as possible.

In the frequency range where the magnonic coupling dominates, information on the spin-waves propagation can be extracted from the transmission S-parameter (𝑆 "5 ; 𝑖 ≠ 𝑗). Namely the propagation losses are encoded in the amplitude and the spin-wave group velocity (or delay) is encoded in the phase. Both are crucial specifications of the delay line.

A) Cross-talk

The cross-talk is measured by the 𝑆 .-parameter. The ratio of power injected in antenna 1 and detected in antenna 2 is |𝑆 .-| . . The insertion losses are defined as:

𝐼𝐿 = -20 log -T |𝑆 .-| (155)
In the absence of spin-waves, the insertion losses should be large (no transmission). The insertion losses for different antenna designs are plotted in Figure 63. The cross-talk significantly depends on the antenna shape. The Ground-Signal-Ground (GSG) type of antenna significantly outperforms the other designs. For this reason, all the magnonic measurements showed in this part were performed on GSG antenna. The GSG spectra obtained are found to be relatively independent of the antennas length and spacing. Even GSG with meanders (such as the one used in Ref 72 ) behave similarly as straight GSG.

The GSG antennas' cross-talk is below -50dB between 0 and 10 GHz, but peaks at -30dB near 14GHz. This should not be a problem if we manage to bring the insertion losses with spinwaves down to 10dB. However, it is currently not the case. Most devices oscillate between 20dB and 30dB. Hence, above 12GHz, the cross-talk and the spin-wave transmission are of the same order of magnitude which can complexifies the analysis. In most cases, we will restrict ourselves to the 1-12 GHz frequency range to avoid spurious effects.

The reference mutual impedance, measured in the absence of spin-waves at zero external magnetic field, is plotted in Figure 64 for two antennas of length 𝐿 = 30 µm and 120 µm. The impedance peaks at 13.5 GHz. The auto-impedance due to spin-waves being of the order of 0.2 Ω. µm |-(6 Ω and 24 Ω respectively), the cross talk dominates in small antennas, while the spin-waves induced signal might be larger in long structures, when the damping is low enough. One could think that longer antennas have a higher cross-talk that short ones. However, it is found to be independent of the antenna's length. The impedances spectrum is enhanced by the RLC resonance that is stronger for short antennas. The spin-waves amplitude decays exponentially during the propagation due to the damping, resulting in propagation losses that we estimate. Generally, one can compute it as 72 :

B) Propagation losses

𝑃𝐿 = -20 log -T Γ (156)
where

Γ = |𝑍 .- 3n | + |𝑍 -. 3n | V|𝑍 -- 3n ||𝑍 .. 3n | (157)
with 𝑍 "5 3n the spin-wave induced impedance in a non-resonant circuit. Thus, to estimate the experimental propagation losses we must evaluate 𝑍 "5 3n from our measurements with RLC resonant transducers.

2) Derivation of the non-resonant spin-wave impedance 𝑍 "5 3n

Note that the physical processes generating the auto-impedance and mutual impedance are identical. The varying flux of the spin-waves under the detection antenna induces an rf-voltage that we measure. This change of voltage is then converted to an impedance. However, this impedance is not the impedance the spin-waves would induce in a non-resonant circuit.

𝑖 1 C 𝑖 1 L 𝑖 2 L 𝑖 2 C I 1 I 2 Z 11 SW Z 11 SW Z 22 SW Z 22 SW V 1 V 2 Z 21 SW /Z 12 𝑆𝑊
The RLC circuit used to model the entire delay line is shown in Figure 65. Both transducers are modeled using the parallel RLC circuit used in the previous part. The spin-waves induce mutual impedance between both antennas in the respective capacitive and inductive branches.

Let us derive the impedance induced by the spin-wave impedance 𝑍 "5 3n .

The mutual impedance measured by the VNA is given by:

𝑍 .-= 𝑉 . 𝐼 - Ú F F ÌT (158)
Applying the Kirchhoff law on both antennas:

𝐼 -= 𝑖 - z + 𝑖 - } ; 𝐼 . = 𝑖 . z + 𝑖 . } = 0 (159)
The voltage 𝑉 . is such that: Hence:

𝑉 . =
𝑉 . 𝐼 - Þ F F ÌT = 𝑍 .- 3n
which we know is not true, because the measured mutual impedance is strongly perturbed by the RLC resonance. The same assumption was made for the auto-impedance, the inductive and capacitive parts are not coupled with each other as it would be in contradiction with the experimental observations. ie, using Eq. (159):

2𝑉 . = (𝑍 } -𝑍 z )𝑖 . } + 𝑍 .- 3n 𝐼 - (162)
Let's now compute 𝑖 . } . From Eq. ( 160):

(𝑍 } + 𝑍 z + 2𝑍 .. 3n )𝑖 . } = 𝑍 .- 3n (𝑖 - z -𝑖 - } ) (163)
and

𝑖 - z = ß E _ { Ê_ EE Sà = _ EE _ { Ê_ EE Sà 𝐼 -= (_ á Ê_ EE Sà ) _ { Ê_ á Ê._ EE Sà 𝐼 -; 𝑖 - } = (_ { Ê_ EE Sà ) _ { Ê_ á Ê._ EE Sà 𝐼 -. Thus: 𝑖 . } = 𝑍 .- 3n (𝑍 } -𝑍 z ) (𝑍 z + 𝑍 } + 2𝑍 -- 3n )(𝑍 z + 𝑍 } + 2𝑍 .. 3n ) 𝐼 - (164)
And finally, plugging 𝑖 . } in Eq. ( 162):

𝑉 . 𝐼 - Þ F F ÌT = 𝑍 .- 3n 2 A1 + (𝑍 }--𝑍 z-)(𝑍 }. -𝑍 z. ) (𝑍 z-+ 𝑍 }-+ 2𝑍 -- 3n )(𝑍 z. + 𝑍 }. + 2𝑍 .. 3n ) B (165)
In the final expression, we generalize to the case where the R, L and C values depend on the antenna (not shown before for clarity).

Applying the formula (165) to the reference measurement (𝑍 "5 dDg = 𝑍 "5 (𝐻 = 0)) gives the nonmagnetic background that we can subtract (Figure 56-b).

3) Estimation of the experimental propagation losses

The ratio Γ is extracted from the experimental spectra and is plotted in In 4µm-wide waveguides, the experimental values have only a qualitative agreement with the analytical curves between 3GHz and 9 GHz. Above 9GHz, the limit of our RLC model prevents us from extracting the correct Γ ¿ values. Indeed, the reference impedance of the longer antenna, used to measure the 4µm-wide waveguides, deviates from the RLC resonance. Overall, the spin-waves seem more damped in the narrow waveguides for DE spin-waves, while Γ ¿ larger than 1 are recorded for BW spin-waves.

All those inconsistencies lead us to believe that the Gilbert damping might not be the main cause of damping of DE SWs. A hypothesis could be an increased sensitivity to a certain type of inhomogeneities. LPE samples can be quite inhomogeneous in thickness in some cases. This type of inhomogeneity would cause a broadening ∆𝑘 of the dispersion relation, that could be associated with an extra damping. However, we have checked that the induced ∆𝑘 is similar for DE and BW SWs. Another hypothesis could be the edge roughness of the waveguides. Indeed, during the laser lithography process the laser path was orthogonal to the waveguides, thus the edges are not as smooth as one can obtain. It probably has no impact in 30µm-waveguides and the fact that it would damp DE SWs more strongly remains to be proven. Finally, the deposition of the antenna could lead to an alteration of the YIG magnetic properties, either through chemical interactions at the interface or local stress induced by the deposition process. The DE spin-waves are strongly emitted below the antennas, while BW spin-waves excitation region is outside the antennas due to the long range out-of-plane dynamic magnetic field. The study of the auto-impedance spectrum at higher frequencies could bring evidence of an increase of the damping below the antennas. In all cases, the reduction of the RLC resonance would greatly simplify future analysis. We stress however that the best agreement so far is obtained for short antennas for which the RLC resonance is large but in strong agreement with our model.

The propagation losses can be computed from the Γ ¿ using Eq. ( 157). The propagation losses for a wide waveguide (in the unpatterned film limit) and a fixed effective propagation distance of 24 µm is shown in Figure 67-(a) using the damping values found in Figure 66-(a). The propagation losses can be as large as 20 dB for DE spin-waves due to their higher damping and to their small group velocity (see the increase in delay in Figure 67-b). The effective distance can hardly be smaller than 24 µm with the antennas that we currently use (𝑤 QaV = 12 µm). Thus, the propagation losses plotted here indicate the right order of magnitude of the minimal propagation losses that can be achieved with these antennas and this damping. For instance, one can directly estimate from those curves that at 18GHz, the propagation losses are about 16 dB for a 50 ns delay. The margin for improvement is pretty large for DE SWs, were the abnormal damping issues be resolved.

The other contributions to the insertion losses are discussed in section III where the impedance matching problem is solved.

C) Estimation of the delay

The dispersion relation in ferromagnetic films is well understood using the Kalinikos-Slavin theory [START_REF] Kalinikos | [END_REF] . Multiple experiment confirmed the theoretical expectations in thin films using either direct techniques like µ-BLS 69 or indirect techniques like propagating spin-wave spectroscopy (PSWS) measurements 72,114 . Recently, a very complete study 121 was carried out using PSWS in a 260nm-thick YIG sample. It shows that the dispersion relation and attenuation length are well modeled in wide waveguides for all in-plane magnetic field orientations and magnitudes up to 50 mT. Although we do not expect significant deviation from theory at higher fields, we will extract the delay in a wide magnetic field range, for DE and BW spin-waves, in 4µm-wide and 30µm-wide waveguides. Checking that the spin-waves group velocity is consistent with the predictions is a crucial step towards a reliable delay line prototype.

The field orientation and the patterning have significant effects on the delay dependence on wavevector and resonance frequency as shown in Figure 68 (using analytical calculations of the group velocity, also shown in Figure 60). For instance, the group velocity depends on the spin-waves wavevector. Thus, at a given external magnetic field, the delay is not uniform within the transmission band of the delay line (greyed area in Figure 68-a). One can notice that depending on the field orientation (DE or BW spin-waves) and on the level of confinement (4µm WGs or full film), the delay is more or less stable within the transmission band. It varies particularly strongly for DE spin-waves in wide waveguides (dashed green line). In practice, a stable delay means more flexibility on the external magnetic value, which would probably alleviate some later integration constraints. However, if the magnetic field can be controlled very precisely (for instance, controlling the resonance frequency by 10 MHz, requires a precision of about 0.3mT), then a varying delay could be beneficial to tune the delay around its nominal value. Furthermore, the delay depends on the external magnetic field (ie the resonance frequency), as shown in Figure 68-(b). The curves are computed for the wavevector that is most efficiently excited by the antenna (𝑘 QaVDaaQ = 0.6 rad/µm). The delay variation with frequency are small for BW spin-waves (red lines), which could be used to design a component with a fixed delay on a wide frequency range.

Before exposing the effects of patterning on the delay in the DE and BW configuration, the methods used to extract the delay from the S-parameters is discussed using the data measured in wide waveguides (𝑤 = 24 µm).

1) Extracting the delay

The delay is given by the derivative of the phase (argument) of the complex transmission spectrum 𝑆 .-(𝜔):

𝜏 cDOQ3 = - 𝜕𝜙 𝜕𝜔 (166)
with 𝜙 = arg (𝑆 .-(𝜔)). If the spin-wave signal is much larger than the cross-talk, then the phase variations are directly related to the phase accumulated during the spin-wave propagation 𝜙 ¿dW¿ 3n = -𝑘𝐷 Dgg . Thus, Eq.

(166) leads to 𝜏 cDOQ3 = 𝐷 Dgg /𝑣 ø , with 𝑣 ø = 𝜕𝑘/𝜕𝜔.

In practice, taking the derivative of a noisy data will yield even noisier data. This is why most papers 71,72,114,121 used the spin-wave induced impedance (or inductance) transmission spectrum maxima to derive the group velocity. By extracting the interval ∆𝑓 between two real part maxima (accumulated phase of 2𝜋), the group velocity is computed as 𝑣 ø = 𝐷 Dgg × ∆𝑓.

In our case, we want to follow the group velocity as a function of frequency in the transmission band. To avoid complex peak detection schemes, we use smoothing to perform the derivation of the phase. First, the unwrapped phase (arg (𝑆 .-)) is smoothed using a 10MHz moving average. Then, instead of performing the derivation on all points (0.5 MHz spacing in most measurements), only points every 10MHz are kept so that the accumulated phase is large between two points and the signal to noise ratio is low. This method allows the extraction of the delay with little noise and a 10 MHz precision.

An example of such computation is shown for DE and BW spin-waves in a 24µm-wide waveguide in Figure 69 . The difference could be explained by the short lifetime found in the DE configuration. Indeed, large wavevectors are more strongly attenuated due to their longer delay, resulting in a deformation of the 𝑆 .-spectrum compared to the 𝑆 --, with the maximum of amplitude shifted to lower wavevector value. Additionally, the delay is computed for an effective distance travelled by spin-waves 𝐷 Dgg = 28 µm that is the center to center distance between both antennas, under the assumption of point-like emitter and receiver. However, the attenuation length being 18 µm, comparable to the antenna extent ℰ = 12µm, the signal is dominated by spin-waves emitted by the most right meander of the left antenna and detected by the most left meander of the right antenna. The effective distance travelled by spin-waves is thus lower, further lowering the real delay. We note that the effect of the quantization in the 24µm-wide waveguide is clearly visible, with lower delays than in a full film (dashed lines) and a strongly varying delay for DE spin-waves in the transmission band. Let's now check if further confinement (4µm-wide waveguides) stabilizes the delay.

2) Effect of patterning in DE configuration

The delay extracted from experiments in 24µm and 4µm-wide waveguides in the DE configuration is shown in As explained in the previous section, the delay is overestimated in the wide waveguide probably due to the effect of damping. However, in narrow waveguides, the delay is very close to the predictions.

There should be no deformation of the spectrum towards lower k-vectors due to damping, because the delay is stable in the wavevector range around the antenna excitation spectrum. However, there should be a slight reduction of the effective distance due to damping, which we do not observe. Hence our measurement precision must not be sufficient to clearly capture such effect.

The delay at the transmission maximum (averaged over 3 points, i.e. 30 MHz) is plotted as a function of the resonance frequency in Figure 70-d). At low frequencies (4-10 GHz), the values are very similar for both wide and narrow waveguides. A decent agreement is found for the 24µm-wide waveguide (dotted line) using a nominal wavevector of 𝑘 QaV = 0.4 rad. µm |-. The delay is nearly 5ns higher than expected in 4µm-wide waveguides at frequencies higher than 10 GHz. This could be due to slight change of the dispersion relation due to the parallel waveguides. Indeed, the correct slope can be obtained by using an effective waveguide width of 8µm. Checking it using micromagnetic simulation can be quite challenging given the dimensions needed if one wants to keep the cell size below the exchange length. However, in this range of wavevector, the dynamic is dominated by the dipolar interaction, thus a large cell size could be used. One could also generalize the semi-analytical formalism developed for magnonic directional coupler presented in Ref 27 to an arbitrary periodic waveguide system. Finally, we point out the effect of cross-talk on the spectrum and delay in Figure 70-c). The spectrum and the delay are particularly deformed for the 24µm wide waveguide. Indeed, the spin-wave amplitude is smaller due to the lower magnetic volume (24µm instead of 80µm for the 20 narrow parallel waveguides) and is consequently more sensitive to the cross-talk. In general, the spin-wave spectrum unpredictably shifts its shape when the frequency is varied in the DE configuration, which shows that not all the physical phenomena are accounted for in our current understanding.

3) Effect of patterning in BW configuration

The delay extracted from experiments in 24µm and 4µm-wide waveguides in the BW configuration is shown in Figure 71-a-b). At 8.5 GHz, the delay increases from 14 ns to 18 ns in wide waveguides in the transmission band while it decreases from 18 ns to 15 ns in narrow waveguides. Those values are in good agreement with predictions shown in Figure 71-c). We attribute the better agreement between analytical predictions and experimental observations to the low magnetic losses detected in BW configuration. The stability region is located at a higher wavevector than for DE spin-waves, near 𝑘 = 0.8 rad. µm |-. Thus, with our current antennas, patterning 4µm waveguides does not necessarily increase the delay stability in the transmission band but rather inverse its slope at 8.5 GHz (red dots in (a)). At 12 GHz, however, there seems to be some stabilization of the delay (red dots in (b)).

The delay at the transmission maximum (averaged over 3 points) is plotted as a function of the resonance frequency in Figure 71-d). At low frequencies (4-11 GHz), the values are very similar for both width of waveguide once again (point up and point down triangles). The expected stabilization effect of quantization is not observed at low frequencies (solid red line). The slope for the narrow waveguides corresponds to a wide waveguide with effective width of 18µm, which could be related to the use of parallel waveguides again. Indeed, by changing the orientation, the waveguides interaction is strengthened compared to the DE configuration. Two magnetic dipoles interact more strongly if they are side by side rather than when they are vertically stacked. It is for this reason that the thickness modes are closer to each other in frequency in the forward volume wave direction and so are the width modes in the DE configuration. The assumption can be tested using micromagnetic simulations. Between 12GHz and 14GHz, the results are clearly perturbed by the cross-talk when plotting the spectra individually.

In general, the obtained spectra are clearer in the BW configuration. They correspond better to the "smooth main lobe" expectations. Future measurements will verify whether this smooth definition persists at higher frequencies (15-20 GHz).

D) Conclusion and perspectives

In this part, we have presented different methods to process the experimental transmission data.

In particular, we worked on its amplitude to extract the propagation losses and damping, and on its phase to extract the delay. The magnetic losses were shown to be higher for DE SWs than for BW SWs, although the origin of such disparity is not clear at the moment. Patterning also seemed to generate extra losses for DE SWs, but not for BW SWs. For future samples, an emphasis should be placed on the lithographic process (smoothening the edges and isolating the antennas from the YIG using an insulating material like HfO2). Furthermore, the analysis was complexified by the RLC resonance in our antennas. Improving the design of the antennas using rf-simulation software like Sonnet would help identify more clearly the origin and properties of the observed losses. The extracted delays were in good agreement with the theoretical expectations. Patterning was shown to decrease the variations of the delay within the transmission band for DE SWs. However, the stability of the delay versus the resonance frequency was not greatly improved, in particular for BW SWs. We have proposed ways to check the influence of the spacing of the parallel waveguides on the group velocity using micromagnetic simulation or analytical calculations.

III. Delay line rf-circuit: impedance matching

Let's now consider the full delay line circuit. This chapter's goal is to understand where the injected power goes and how to maximize the transfer of energy through the magnonic circuit.

In a first step, we assume that we can design an antenna having negligible capacitive and inductive effects in the desired frequency range. In such simplified circuit, the transducer impedance (resistance) is in series with the spin-wave induced impedance. This simplified model allows us to decompose the different sources of insertion and return losses and to solve the impedance matching problem under general assumptions. Then, we consider an RLC resonance circuit, as observed in the experiment and look to optimize the design and compare it to the result obtained in the simplified case. This allows us to determine how detrimental to the performance the RLC resonance is. In all cases, the cross-talk effects are neglected. Indeed, we aim for insertion losses lower than 20dB (22dB already obtained) while the cross-talk of the current design already exceeds the 30dB mark. Finally, we will conclude with some perspectives on the potential for improvement of the magnonic delay line prototype.

A) Delay line simplified rf-circuit

The magnonic delay line simplified equivalent circuit is represented in Figure 72. The goal is to transmit the signal from the generator to the load through the magnonic part in order to delay the signal.

The quadrupole composed of the two transducers are characterized by the impedance matrix

𝑍 ̅ Vd = ¨𝑍-- Vd 𝑍 -. Vd 𝑍 .- Vd 𝑍 .. Vd © = ¶ 𝑟 0 0 𝑟
• when no spin-waves are present (taking into account only ohmic losses and neglecting the cross-talk). At resonance, the spin-wave contribution is given by

𝑍 ̅ 3n = A 𝑍 -- 3n 𝑍 -. 3n 𝑍 .- 3n 𝑍 .. 3n B.
The S-parameters matrix can then be obtained using the formula (𝐼 ̅ . is the two-dimension identity matrix):

𝑆 ̅ = (𝑍 ̅ -𝑍 T 𝐼 ̅ . ) ⋅ (𝑍 ̅ + 𝑍 T 𝐼 ̅ . ) |- (167)
From the S-parameters, one can define and compute the insertion losses:

IL = -20 log -T (|𝑆 .-|) (168)
and the return losses:

RL = -20 log -T (|𝑆 --|) (169)
Our goal is to minimize insertion losses (introducing the delay line between the load and the generator should result in small additional losses) and to maximize return losses (our delay line is matched to the generator and load so that only a small portion of the signal is reflected when entering port n°1).

1) Impedance matching method

The impedance matching problem depends on the spin-wave configuration. Indeed, the DE spin-waves are non-reciprocal while the BW spin-waves amplitude is evenly split between left propagating and right propagating magnons. At the point of maximal radiation resistance, the total impedance matrix is given by:

𝑍 ̅ "z = A 𝑟 + 𝑧 3n 𝑟 3n Γ è Γ ¿ 𝛿 𝑟 3n Γ è Γ ¿ (1 -𝛿) 𝑟 + 𝑧 3n B ( 170
)
Where r is the antenna resistance, 𝑧 3n is the spin-wave auto-impedance, 𝑟 3n = ℛ𝑒(𝑧 11 𝑆𝑊 ) the spin-wave RR, Γ è = 𝑒 "è represents the phase of the 𝑧 .- 3n impedance term (depends on the kvector and the antenna distance), Γ ¿ = exp(-

" t½½ z ‹ÐÐ
) is the propagation losses ratio and 𝛿 = Let's first consider the case with no damping (Γ ¿ = 1) and perfect non-reciprocity (𝛿 = 0).

𝑍 ̅ 3n = ¨𝑟 + 𝑧 3n 0 𝑟 3n Γ è 𝑟 + 𝑧 3n © (171)
The complex spin-wave impedance 𝑧 3n has a typical shape given in Figure 73, obtained from the Bailleul's model at 10 GHz (𝐻 T = 0.28 T). The impedance 𝑧 3n is normalized by the maximal radiation resistance, so we can use it as a typical spin-wave impedance of 1 Ohm.

To artificially obtain zero reciprocity (𝛿 = 0), 𝑧 -. 3n is set to zero while 𝑧 .- 3n is normalized by its absolute value (|𝑧 .- 3n |). In the absence of damping and non-reciprocity, |𝑧 .- 3n | is similar to the radiation resistance (see Figure 73-b), which justify the use of 𝑟 3n in the off-diagonal terms in equation ( 27).

The 𝑧 .- 3n phase (Γ è ) is given by the distance between the antennas and the k-vector. Here, an arbitrary typical effective distance of 𝐷 Dgg = 24 µm is chosen. It can be modified to study the impact of the Γ è term later on.

Using this mock 𝑧 3n , we now determine the optimal value for spin-wave radiation resistance and transducer ohmic resistance in order to minimize the insertion losses and maximize the return losses.

NB:

The chosen frequency (10 GHz) is unimportant in this derivation. Indeed, the amplitude is renormalized, so the frequency only influences the signal bandwidth which plays no role in the Now that we know what we are aiming for, let's examine what can be achieved using our magnonic delay line. For a given design and frequency, both the spin-wave radiation resistance and the transducer resistance depend on the length of the antenna. Thus, changing the antenna length is equivalent to moving along a straight line passing through the origin, whose slope is given by the ratio (Λ) between the spin-wave radiation resistivity and the antenna resistivity:

Λ = 𝜌 3n 𝜌 Vd (172)
For instance, for the 120-µm long antenna, 𝜌 Vd ≈ .ý ï -.T ≈ 0.2 Ω/µm, while for a wide waveguide, at 10 GHz in the DE configuration, 𝜌 3n ≈ 0.16 Ω/µm (see Figure 58), resulting in a slope Λ = 0.8. In such conditions, the "available" insertion losses are represented in Figure 75-a) by the dotted line 𝑟 3n = Λ × 𝑟 Vd . There exists a point along the line for which the insertion losses are minimal and the return losses maximal. This point thus satisfies our impedance matching conditions. Its position is defined by the equations: with the unique solution:

𝑟 3n + 𝑟 Vd = 𝑍 T (173)
𝑟 Vd = 𝑍 T 1 + Λ ; 𝑟 3n = Λ𝑍 T 1 + Λ (175)
For instance, for a design with a ratio Λ = 0.8, the best insertion losses possible are about 12.5 dB. Only by changing the design or the operating frequency can the ratio Λ be increased and the insertion losses decreased. The evolution of the insertion losses at the matching conditions is given in Figure 75-b).

Increasing the Λ ratio to 5 allows to improve the insertion losses to roughly 7.5 dB. There are multiple ways to increase the ratio Λ. We describe them in the third sub-section (see page 140)

2) Impact of non-reciprocity

In the experiment, the non-reciprocity of DE spin-waves is not perfect (𝛿 = 0). The experiment gives 𝛿 = J1 EF Sà J J1 FE Sà Ê1 EF Sà J ≈ 0.15. Besides, the insertion losses accessible using BW spin-waves can be obtained by using 𝛿 = 1/2. Indeed, the typical BW spin-wave impedance spectra are very similar to the typical DE spin-wave spectra. We stress that the linewidth, the group velocity or As 𝛿 increases, the insertion losses get larger. Indeed, it means that a part of the spin-waves is emitted in the opposite direction and does not reach the other antenna. For 𝛿 = 0.5, the insertion losses are thus nearly 6 dB higher.

In the presence of reciprocity (𝛿 ≠ 0), the relative phase between 𝑧 --and 𝑧 .-can have a noticeable influence on the insertion losses. This effect is shown in Figure 77-b), where the change of phase creates oscillation in the insertion losses spectrum. The insertion losses are minimal when 𝑧 --and 𝑧 .-are in phase (resistive mutual impedance). Thus, the minimal insertion losses can be off-centered compared to the mutual impedance amplitude peak (see orange curve in Figure 77-b).

The impact of phase remains relatively weak (~ 1 dB) as shown in Figure 77-a

3) Improving 𝚲

The ratio between the ohmic resistivity and the spin-wave radiation resistivity Λ has a large influence over the insertion losses. Increasing Λ increases the share of power that is useful to the delay line (ie not lost in the transducer ohmic losses). The effect depicted in Figure 75-b) and the insertion losses term follows the equation:

𝐼𝐿 = 𝐼𝐿(Λ = +∞) -20 log -T ¨Λ Λ + 1 © (176)
The first obvious way to increase Λ is to operate at high frequencies. Indeed, the spin-wave radiation resistance scales with frequency due to its inductive nature, while the ohmic losses are independent on frequency. For instance, the projection using Bailleul's model with damping 𝛼 ú = 5 × 10 |ý and renormalization factor 𝑐 = 0.6, gives at 18 GHz:

𝜌 3n ≈ 0.5 Ω/µm (177)
which yields Λ = 2.5 and ideal insertion losses close to 10 dB with 𝛿 = 0.15.

Another way to increase Λ is to reduce the transducer resistivity by making it thicker or wider. Figure 78 shows the dependence of Λ on the gold thickness. Λ is increased by a factor 2 by increasing the gold thickness from 150nm (current design) to 330nm. However, the efficiency gradually decreases. In order to get another factor of 2, the thickness must be increased to 950nm, which requires specific fabrication processes. The sub-linear behavior originates from the increase of distance between the middle current plane and the ferromagnet, which reduces the spin-wave induced radiation resistivity.

The width of the transducer can also be increased. It reduces the resistivity and generally does not affect the spin-wave radiation resistance if the group velocity is comparable at the wavevectors excited by the transducer. However, the distance between antennas must be large enough to accommodate wider transducers. Thus, it is only an option when designing delay lines with large delays. The width of the transducer can also be constrained by bandwidth considerations.

Finally, one can use a meander shape (see Ref 72 ). We checked on the sample that the meander shape does not change the antenna resonance and cross-talk. Doubling the number of meanders multiplies the resistivity by 2 but increases the maximal radiation resistance by 4. The counterpart is that the transducer gets wide, so it's again only an option for long enough delays.

To conclude this part, by increasing the frequency and the gold thickness, we see that we can already obtain Λ of the order of 5 for DE spin-waves, which brings the IL within 3dB of the optimum.

4) Stability of the matching conditions

The frequency dependence of the radiation resistivity means that Λ depends on frequency, so it is not possible to be perfectly matched at all frequencies. Each design will have optimal insertion losses at a given frequency and working at another frequency will result in additional losses. We show the amplitude of such additional losses in Figure 79. The losses associated with a transducer optimized for a frequency of 15 GHz are shown versus the optimal losses that can be achieved if the transducer length was optimized for each separate frequency (dotted lines). The mismatch barely increases the insertion losses (< 1 dB). However, the effect is large on the return losses. We note that the return losses can be higher in the "rigid" case. Indeed, the return losses are associated to the optimal insertion losses but are not optimal themselves. The maximal return losses are obtained for purely resistive antennas which is not what we are interested in. Thus, being slightly mismatched in 𝑆 .-can improve the 𝑆 --matching. It can also lead to large 𝑆 --mismatch, for instance far from 15GHz. We however note that the return losses remain higher than 15dB on the 8-20GHz range. Thus, tuning our delay line over a wide frequency range is possible since the insertion losses are barely affected, and the return losses are not critically damaged.

Similarly, it is possible to misevaluate the Λ parameter of a given design. In which case, the length of the antenna may not be optimal. In Figure 80-a), the insertion losses are plotted as a function of the transducer length. For the calculation we chose an operating frequency of 15

GHz, Λ = 2.5 and ρ ÿž = 0.2 Ω/µm, leading to an optimal length 𝐿 W¿V = äT -Êð * -! Ð † ≈ 71 µm. Transducers whose length that are within a factor 2 from the optimal length only result in extra insertion losses of 1dB due to the mismatch. Thus, the insertion losses are not too sensitive on the exact length of the antenna. However, the return losses (Figure 80-b) quickly drop to 10dB if the length is a factor of 2 larger or smaller than optimal. Consequently, if one wants to maintain good return losses, the uncertainty on the optimal length should be kept in the order of few tens of percent.

5) Impact of damping

Damping leads to propagation losses that are equal to:

𝑃𝐿 = -20 log -T ¶exp ¶- 𝜏 cDOQ3 𝜏 •• = -20 log -T ¨exp ¨-𝐷 Dgg 𝐿 QVV ©© (178)
with 𝜏 the magnon lifetime, 𝜏 cDOQ3 the delay, 𝐷 Dgg the effective distance traveled by spin-waves and 𝐿 QVV the attenuation length. The propagation losses simply add up to the insertion losses. It has no impact on the return losses except by reducing the spin-wave induced resistivity.

B) Delay line with RLC resonance rf-circuit

In the current design, the transducers form a strongly resonant RLC circuit, such that:

𝑍 ̅ "z = A 𝑍 -- fz} 𝑍 -. fz} 𝑍 .- fz} 𝑍 .. fz} B (179)
with •𝑍 } + 𝑍 z + 2𝑧 "" 3n -. ?

with 𝑍 z = 𝑅 + 𝑗𝜔𝐿; 𝑍 } = 𝑅 -5 ð} , with typical values chosen close to those obtained in the experiment, defining an RLC resonance near 12GHz:

𝐿 = 1 nH ; 𝐶 = 0.2 pF (182)
Similarly to the previous part, the resistance R depends on the length of the antenna, which we seek to adjust to minimize the insertion losses and maximize the return losses. NB: Here we assume that both antennas are identical (described by the same R, L and C values and 𝑧 "" 3n = 𝑧 55 3n ).

Having brief look at the new impedance matrix elements already reveals that solving the impedance matching problem is not as straightforward in the RLC case. Due to the RLC resonance, the matching conditions now depend on frequency. Far from resonance, either 𝑍 } or 𝑍 z will be very large and we end up again in the non-resonant case (𝑍 "" fz} = 𝑅 + 𝑧 "" 3n ; 𝑍 "5 fz} = 𝑧 "5 3n ), for which the impedance matching problem has been solved. However, close to resonance, the off-diagonal term 𝑍 "5 fz} now depends on the diagonal term 𝑧 "" 3n (in the denominator of Eq. ( 181)). Thus, in the limit where 𝑧 "" 3n is large (which must be the case to minimize insertion losses), the transmission is reduced, and the insertion losses are increased where the spin-wave impedance is maximum.

The goal of this section is to give a brief overview of the impedance matching problem, at resonance (12 GHz) and "farther" from resonance (18GHz). This brief look is enough to show in what ways the RLC resonance is detrimental to the DL performance.

The section is also meant as a demonstration of all the functions implemented in the dedicated python notebook design tool. This tool can be used later, either to analyze more deeply the impact of the RLC resonance, to implement new models for the antennas or to check whether different L and C values would improve the DL performance.

1) Example of impedance matching at 12 GHz and 18 GHz

In the case of a non-resonant circuit, the ideal matching conditions are obtained for 𝑟 Vd = 0 and r %& = 50 Ω. In Figure 81, the S-parameter spectrum is plotted for the RLC and the nonresonant circuit, for 𝑅 = 0 and r %& = 50 Ω.

Near the resonance, at 12 GHz (Figure 81-a), the insertion losses are increased by 10 dB due to the large 𝑧 "" 3n amplitude at its peak at 12.08 GHz (green solid line). The return losses are also down from 40 dB to 15 dB due to the resonance (red solid line compared to the red dotted line).

The lowest insertion losses in the spectrum (8 dB) remain close to what could be obtained in a non-resonant circuit, while the return losses are still quite large (~15 dB). Hence, the main effect of the resonance, is to split the main lobe, thus reducing the bandwidth and slightly degrading the insertion and return losses. At 18 GHz, farther away from the RLC resonance, all those effects are still present but are weaker. Both insertion losses and return losses are less reduced compared to the non-resonant case.

One could wonder whether it is possible to improve the matching by changing the values of 𝑟 Vd = 0 and r %& = 50 Ω. In Figure 82, the return and insertion losses at the point of maximal impedance are plotted. Contrary to the non-resonant case, the return losses are now in contradiction with the insertion losses: maximizing the return losses at the impedance peak also maximizes the insertion losses. Thus, we end up with two different strategies to match our impedance. It is possible to maximize the insertion losses at the impedance peak, and thus preserving the bandwidth and accept the low return losses (8dB). For instance, by choosing to be near the point 𝑟 Vd = 0 and r %& = 22 Ω, for which the insertion losses are minimal at the peak impedance (see spectra in Figure 83-a).

It is also possible to maximize the insertion losses at another point of the spectrum and have high return losses, but the spectrum is cut in half. This is the case for instance at the point 𝑟 Vd = 0 and r %& = 90 Ω (see spectra in Figure 83-b).

Finally, here we showed only the ideal conditions with 𝑟 Vd = 0. It is still possible to define a Λ such that 𝜌 3n = Λ𝜌 Vd . The higher the Λ, the lower the insertion losses. Again, a compromise must be established: either low insertion losses, wide bandwidth but low return losses; or similar insertion losses, smaller bandwidth but better return losses.

2) Conclusion and perspective

We have established that the RLC resonance is detrimental to the Delay Line performance. To understand its origin and improve the design, a simulation study has been started by R. Lebrun on finite element software Sonnet. A new set of GSG antennas are also being tested on a second prototype to calibrate the simulations and find an optimum.

We have developed a python tool that predicts the insertion and return losses of the Delay Line.

The tool was used to study the impedance matching problem in the simple case of resistive antennas. The influence of key parameters such as non-reciprocity, phase and damping were exposed. We demonstrated that the performance also depends on the ratio of the SW radiation resistivity to the ohmic resistivity (Λ). For each Λ, an optimal length can be found for the antennas. We also showed the stability of those optimal conditions when the length is varied.

The tool was then used on the more complex RLC resonant circuit developed to describe the experiment. We showed that the impedance matching problem has no optimal solution and that a compromise must be found depending on the specifications. Far from the resonance, the effect is small, and keeping the same matching conditions as in the resistive case is a good compromise.

The goal is now to get closer to the identified optimal conditions, with a focus on improving the transducers. Finally, a packaged prototype must be developed in order to compare the magnonic Delay Line to the existing components.

Conclusion and outlook Conclusion

This thesis was aimed at studying the generation, propagation and manipulation of spin-waves in nanostructures of thin films of Y3Fe5O12 (YIG) and one of its doped variant (BiYIG), in order to enable the design of spin-wave based signal processing schemes and devices operating in the radiofrequency band. Two goals have been pursued in parallel:

• Provide a better understanding of radiofrequency spin-wave devices for analog applications such as filters or delay lines • Explore the physics of spin-orbit-torques applied to propagating spin-waves and understand the non-linear phenomena that have so far hindered any practical amplification scheme in the field of magnonics.

Analog devices have been fabricated using standard nano-lithography techniques. They are of two sorts:

• Using the propagating spin-wave spectroscopy technique, delay lines based on a 300 nm thick YIG film have been characterized. We have measured insertion losses of 20-30 dB at 15 GHz for delays between 15 ns and 30 ns. By modelling the radiofrequency response of our devices, we have identified the key relevant characteristics of our magnonic component: the spin-wave induced radiation impedance, the damping and the delay. We compared their magnetic field and wavevector dependence to analytical models and discussed the models' accuracy. The impedance matching problem was solved in the optimal case of a purely resistive transducer and in the case of an RLCresonant transducer, such as the one measured in the experiment. We have identified the key parameters that influence the delay line losses and demonstrated a clear path toward a realistic optimized target in terms of losses and functionalities (delay stability, bandwidth). All the analysis and modeling tools have been implemented in open source python programs to help design the future iterations of magnonic components at the CNRS/Thales Lab and beyond.

• Additionally, we have studied the efficient implementation of radiofrequency filtering in ultra-thin YIG waveguides (20 nm-thick) using periodic nanostructurations. A widthmodulated waveguide was studied using phase-resolved µ-BLS and brought a clear image of the propagating magnon modes near and inside the band gap. The experimental results were accurately reproduced using full-scale micromagnetic simulations on the GPU-based MuMax3 software, paving the way toward the reliable optimization of magnonic crystals' properties using simulations. We also unveiled the role of fabrication defects that give rise to extra losses for frequencies outside the rejection band. Using a thickness-modulated waveguide, we strongly suppressed these extra losses, demonstrating a very efficient implementation of 1D magnonic crystals in the thin film limit.

Finally, active nano-magnonic devices have been fabricated. We have demonstrated a spinwave amplification scheme based on the spin-orbit-torque phenomenon. This study illustrates the potential of the recently developed high-quality BiYIG films grown by PLD at the CNRS/Thales lab for magnonic research. By engineering the perpendicular magnetic anisotropy, we showed that we can both achieve a very high temperature stability (50 times larger than YIG) on a wide range of temperatures (260K to 400K) and a suppression of the leading detrimental non-linear coupling term in supercritical SOT systems. We also took advantage of the large magneto-optical coupling to perform precise time-resolved µ-BLS experiments on BiYIG/Pt systems. One of the main fundamental findings is the full characterization of the transient regime of the spin-current induced auto-oscillation. We showed that the auto-oscillation slowly grow in the well understood low amplitude regime. And most importantly, that as long as their saturation is not reached, the magnetic system still obeys the linear response theory. Taking advantage of this time window (few hundred nanoseconds), we propagated rf-excited spin-waves at supercritical currents and observed an absence of detrimental non-linear interactions with the auto-oscillations. Furthermore, we achieved the SOT-driven amplification of spin-waves. We also revealed a frequency or wavevector dependence of the amplification efficiency in both the subcritical and the supercritical regime.

Outlook

RF-simulations and new measurements are ongoing to optimize the delay line transducers. Their improvement will be key to demonstrate a delay line with close-to-optimal parameters and that can be reliably modelled. New functionalities can also be added to the delay line, such as frequency filtering or controllable phase shifting. The integration of the device in a packaged component is also planned later this year. The goal is to progress in the development of a technological platform that will facilitate the increase of TRL levels of magnonic devices.

Many studies are demonstrating the potential of magnonic materials with antisymmetric exchange such as the Dzyaloshinskii-Moriya interaction. In general materials driven magnonics could open new perspectives beyond the classical material platforms (YIG or Permalloy). For instance, during the course of this thesis, I have explored magnonic devices based on low loss manganites multiferroics heterostructures for voltage controlled magnonic crystals.

Going beyond the classical spin-wave physics, spin-orbit-torque driven magnons Bose-Einstein condensates are a new playground to explore the physics of quantum magnonics and its coupling with quantum optics and eventually with quantum information processing. This new physics is today in its infancy. No doubt that many discoveries will be driven by its fundamental and societal interests.

magnéto-optique, s'est imposé durant ma thèse comme le matériau idéal pour explorer les effets du SOT sur les magnons.

Deux objectifs ont été poursuivis en parallèle : -Apporter une meilleure compréhension des dispositifs magnoniques hyperfréquences pour des application analogiques telles que les filtres ou les lignes à retard.

-Explorer la physique du couple de spin orbite (SOT) appliqué aux ondes de spins progressives et comprendre les effets non-linéaires qui ont jusqu'à présent empêchés la réalisation d'un dispositif efficace d'amplification des ondes de spins. A la suite des résultats expérimentaux et analytiques, un objectif de pertes d'insertion 𝐼𝐿 = 10 dB et de perte de retour 𝑅𝐿 = 20 dB a été jugé atteignable. L'objectif est désormais de se rapprocher des conditions optimales identifiées, en se concentrant sur l'amélioration des transducteurs et la réduction des pertes de propagation en configuration DE. Enfin, un prototype entièrement intégré doit être développé afin de comparer la ligne à retard magnonique aux composants existants.

I) Dispositifs magnoniques analogiques

B) Filtres hyperfréquence dans des films ultra-fins de YIG

Lorsqu'une onde se propage dans un milieu périodique, une partie de l'onde est réfléchie à chaque période et interfère de manière destructive avec l'onde incidente pour les vecteurs d'ondes satisfaisant les conditions de Braggs : 𝑘 a = 𝑛𝜋/𝑎 où 𝑎 est la période. Aux fréquences correspondant à ces vecteurs d'ondes de Braggs s'ouvrent des bandes interdites dans la relation de dispersion à l'intérieur desquelles les ondes sont fortement atténuées. Ce phénomène physique peut donc être utilisé pour réaliser une fonction de filtrage. C'est notamment le cas en optique où des milieu périodiques, appelé « cristaux photoniques », ont été longuement étudiés [START_REF] Joannopoulos | Photonic Crystals: Molding the Flow of Light[END_REF] . Pour les ondes de spins, de tels systèmes sont nommés « cristaux magnoniques » (voir revue 33 ). Au cours des dernières années, différentes implémentations ont été proposées, basées sur des films de YIG épais (épaisseur de quelques microns) ou sur des matériaux ferromagnétiques métalliques fins qui ont un fort coefficient d'amortissement. Cependant aucune démonstration n'a été faite sur les films ultra-fin de YIG qui combinent l'avantage d'avoir de faibles pertes magnétiques tout en étant compatible avec les méthodes de nanofabrication modernes.

Dans cette thèse, j'ai démontré l'efficacité de filtres hyperfréquences basés sur la nanostructuration périodique de guides d'ondes de YIG ultra-fin (20 nm d'épaisseur). Un guide d'onde modulé en largeur a été étudié et les résultats expérimentaux, obtenus avant ma thèse, ont été compris à l'aide de simulations micromagnétiques 19 . Cette étude, mettant en avant le rôle de la nanostructuration, de la position de l'antenne dans la maille, et de l'interaction entre les différents modes de largeur, ouvre la voie à une optimisation fiable des propriétés des cristaux magnoniques de YIG grâce aux outils de simulation micromagnétique. Dans un second temps, en utilisant une modulation en épaisseur d'un guide d'ondes de YIG, nous avons fortement supprimé les pertes engendrées par la nanostructuration en dehors des bandes interdites, démontrant une réalisation efficace d'un cristal magnonique 1D.

II) Dispositifs magnoniques actifs pour l'amplification des ondes de spins

Il a été démontré que l'application d'un courant continue dans une couche de platine adjacente à une couche de YIG peut créer un couple, appelé couple de spin orbite (SOT) sur l'aimantation 6 . Sa magnitude est proportionnelle au courant appliqué et sa direction peut être opposée au couple d'amortissement et donc compenser les pertes magnétiques pour un courant appliqué suffisamment élevé (supérieur à un courant dit « critique »). L'application d'un courant surcritique mène à l'auto-oscillation de l'aimantation, démontrant la compensation des pertes magnétiques pour un mode de précession uniforme 10,52 . Lors de sa thèse au laboratoire en 2017, Martin Collet a démontré la possibilité de moduler la longueur d'atténuation des ondes de spins progressives dans un guide d'ondes de YIG/Pt par l'application d'un couple de spin orbite 12 . Cependant, pour des courants surcritiques, la cohérence de l'onde s'effondre, et une propagation sans perte ne peut être atteinte. Une hypothèse est que lors de l'application de courants surcritiques, l'amplitude de l'auto-oscillation est saturée par des couplages nonlinéaires qui redistribuent l'énergie du mode d'auto-oscillation vers d'autres modes incohérents. Une fois ces canaux de relaxation ouverts, ils redistribueraient également l'énergie des modes progressifs vers les modes incohérents. Pour éviter ces effets non-linéaires, une idée est de propager les ondes sous forme d'une impulsion avant la saturation de l'amplitude des auto-oscillations et donc avant l'apparition des effets non-linéaires. Cependant l'application du courant entraine une forte variation de la fréquence de résonance principalement en raison du chauffage par effet Joule dans la couche de Pt. Il est donc nécessaire d'utiliser un matériau dont les propriétés sont indépendantes du courant appliqué, c'est-à-dire stable en température. Une telle stabilité a été observée en 2018 pour un film de Bi1Y2F5O12 (BiYIG), cru par ablation laser pulsée (PLD) au laboratoire par Lucile Soumah. Cette observation ouvre la voie à une étude de l'effet du couple de spin orbite sur les ondes de spins progressives dans le régime pulsé. Durant cette partie de ma thèse je me suis donc intéressé à la stabilité thermique du BiYIG ainsi qu'à ses propriétés dynamiques afin de sélectionner les films les plus adapter à une étude des effets du SOT sur les ondes de spins progressives. J'ai ensuite fabriqué des guides d'ondes à partir d'une bicouche de BiYIG/Pt similaire aux guides d'ondes (YIG/Pt) employés dans la Ref 12 . Une antenne large bande permettant l'excitation des ondes de spins sur une large gamme de fréquences a été ajoutée au précédent design. La longueur des guides d'ondes a été réduite afin de limiter la résistance de la piste de platine et permettre l'application d'impulsion courte de courant. Grâce à la microscopie Brillouin, j'ai directement observé le régime transitoire des auto-oscillations et la temporalité de l'émergence des effets non-linéaires. J'ai ensuite mesuré l'effet du SOT sur les ondes de spins progressives dans le régime continue et pulsé.

A) Propriété des films de BiYIG

L'addition d'un atome de Bi dans la maille de YIG ajoute un terme d'anisotropie magnétique uniaxiale (𝐻 P ). La fréquence de résonance dépend alors de l'aimantation effective, définie par : 𝑀 Dgg = 𝑀 e -𝐻 P . En étudiant la fréquence de résonance ferromagnétique (FMR) en fonction de la température entre 260K et 400K pour différents films ultra-fins de BiYIG, j'ai pu démontrer que l'aimantation (𝑀 e ) et le champ d'anisotropie (𝐻 P ) ont la même dépendance en température. Cette propriété remarquable pourrait venir de la contribution dominante d'un mécanisme à deux ions pour l'anisotropie uniaxiale dans ces films (anisotropie de croissance). Cela signifie notamment que pour des films pour lesquels 𝐻 P ≈ 𝑀 e à température ambiante (𝑀 Dgg ≈ 0), la fréquence de résonance ne dépend pas de la température (voir Figure 85). Le terme d'anisotropie permet également de réduire les effets non-linéaires dominants liés à l'ellipticité de la précession en compensant l'anisotropie de forme 15 (𝑀 Dgg ≈ 0). Ces deux phénomènes combinés permettent d'obtenir des auto-oscillations de fortes amplitudes et dont la fréquence est quasi-indépendante du courant 14 dans les films de BiYIG. Par ailleurs, l'étude de ces films a permis de mettre en évidence l'inhomogénéité spatiale des propriétés magnétiques due à l'anisotropie sur des échelles inférieures au micron. Ces inhomogénéités peuvent affecter la propagation des ondes de spins si elles sont trop importantes. Il a été constaté que l'élargissement inhomogène de la largeur de raie est un bon indicateur de l'homogénéité du film.

B) Effet du SOT sur les ondes de spins progressive en régime continue

C) Effet du SOT sur les ondes de spins progressives en régime pulsé

En appliquant le courant sous forme d'une impulsion de 2 µs, il est possible d'étudier le régime transitoire de l'auto-oscillation. Il apparaît que l'amplitude de l'auto-oscillation croit exponentiellement jusqu'à atteindre son orbite stable, où l'amplitude sature. Avant la saturation, l'amortissement est négatif et la croissance est exponentielle, en accord avec le modèle canonique de Slavin & Tiberkevich 16 , indiquant que le système se trouve dans un régime de faible non-linéarité avant la saturation. Pour conclure, en modulant l'anisotropie du YIG via un dopage au bismuth, nous avons montré que nous pouvions obtenir une grande stabilité thermique sur une large gamme de températures et une suppression du terme de couplage non linéaire préjudiciable dans les systèmes SOT supercritiques. Le régime transitoire de l'auto-oscillation induite par le courant de spin a été entièrement caractérisé, et un dispositif d'amplification des ondes de spin basé sur le phénomène de SOT a été démontré.

Title: Magnonic circuits based on nanostructured ultra-thin YIG for radiofrequency applications Keywords: spin waves, YIG, magnonic, spin-orbit torque, SOT, spintronic Abstract: This thesis was aimed at studying the generation, propagation and manipulation of spinwaves in nanostructures of thin films of Y3Fe5O12 (YIG) and one of its doped variants (BiYIG), in order to enable the design of spin-wave based signal processing schemes and devices operating at radiofrequencies. Two goals have been pursued in parallel:

-Provide a better understanding of spin-wave radiofrequency devices for analog applications such as filters or delay lines.

-Explore the physics of spin-orbit-torque (SOT) applied to propagating spin-waves and understand the non-linear phenomena that have so far hindered any practical magnonic amplification scheme.

Prior to the present work, spin-orbit effects in YIG|Pt microstructures have been demonstrated to be a powerful new paradigm to control the propagation and emission of coherent spin-waves. An intense research effort has been put in the lab to grow epitaxial nanometer thick YIG films by pulsed laser deposition (PLD). The film thickness was reduced down to a few nanometers while preserving its excellent magnetic properties. More recently, ultra-thin Bi-doped YIG films have been developed, introducing a large and controllable perpendicular magnetic anisotropy term (PMA) while preserving the excellent damping properties of YIG. Additionally, the Bi atoms induce a very large Faraday rotation angle which increases the sensitivity of direct imaging methods like Brillouin Light Scattering microscopy (BLS) by nearly two orders of magnitude. This important step forward on the material aspect opened up new possibilities for the realization of magnonic devices that can have a large impact on the ICT industry. Indeed, microfabrication of YIG is now possible thanks to the advent of high-quality nanometer thick YIG films. In the meantime, BiYIG, with its controllable PMA and large magneto-optical coupling, quickly imposed itself as a material whose properties are particularly well suited to the study of SOT effects.

During this thesis, analog devices have been fabricated using standard nano-lithography techniques. They are of two sorts:

-A delay lines based on a 300 nm thick YIG film have been characterized. The impedance matching problem was analytically solved, and we demonstrated a clear path toward realistic optimized targets in terms of losses and functionalities. All the analysis and modeling tools have been implemented in open source python programs.

-Additionally, we have studied the efficient implementation of radiofrequency filtering using periodic nanostructurations thanks to the advent of 20 nm thick YIG films with low damping. A width-modulated waveguide was studied, and the experimental results were accurately reproduced using full-scale micromagnetic simulations, paving the way toward the reliable optimization of magnonic crystals' properties. Using a thicknessmodulated waveguide, we strongly suppressed the extra losses induced by defects, demonstrating a very efficient implementation of a 1D magnonic crystal.

Finally, active nano-magnonic devices have been fabricated. By engineering the perpendicular magnetic anisotropy of Bi-doped YIG, we showed that we can both achieve a very high temperature stability (50 times larger than YIG) on a wide range of temperatures (260K to 400K) and a suppression of the leading detrimental non-linear coupling term in supercritical SOT systems. The transient regime of the spin-current induced auto-oscillation was fully characterized, and a spin-wave amplification scheme based on the spin-orbittorque phenomenon was demonstrated. We also revealed a frequency or wavevector dependence of the amplification efficiency in both the subcritical and the supercritical regime. Titre: Circuits magnoniques à base de YIG ultra-mince nanostructuré pour les applications hyperfréquences Mots clés: Ondes de spins, YIG, magnonique, couple de spin orbite, SOT, électronique de spin Résumé: Cette thèse porte sur l'étude de la génération, la propagation et la détection d'ondes de spins dans des nanostructures élaborées à partir de couches ultra-minces (quelques nanomètres d'épaisseur) de Y3Fe5O12 (YIG) et un de ses variant dopé (BiYIG). Le but est de permettre le développement de dispositifs pour le traitement des signaux hyperfréquences, basés sur les ondes de spins. Deux objectifs ont été poursuivis en parallèle :

-Apporter une meilleure compréhension des dispositifs magnoniques hyperfréquence pour des application analogiques tel que les filtres ou les lignes à retard.

-Explorer la physique du couple de spin orbite (SOT) appliqué aux ondes de spins progressives et comprendre les effets non-linéaires qui ont jusqu'à présent empêchés la réalisation d'un dispositif efficace d'amplification des ondes de spins.

Ce travail repose sur les études démontrant que les effets spin-orbite dans les microstructures YIG|Pt constitue un puissant paradigme pour le control de la propagation et de l'émission cohérente d'ondes de spins. Il s'appuie également sur l'expertise développée au laboratoire dans la croissance de couches ultra fines épitaxiées de YIG, élaborées par ablation laser pulsée (PLD). Plus récemment, des couches ultra-minces de YIG dopé au Bismuth (BiYIG) ont été développées. L'introduction du bismuth induit une forte anisotropie magnétique perpendiculaire (PMA), tout en préservant les excellentes propriétés dynamiques du YIG. De plus, le bismuth augmente fortement l'angle de rotation Faraday et améliore donc la sensibilité des méthodes d'imagerie directe, telle que la microscopie à diffusion Brillouin (BLS). Ces avancées importantes sur l'aspect matériaux ont ouvert de nouvelles possibilités pour la réalisation de dispositifs magnoniques. En effet, la micro-fabrication du YIG est désormais possible. Par ailleurs, le BiYIG, avec sa PMA accordable et son fort couplage magnéto-optique, s'est imposé durant ma thèse comme le matériau idéal à l'exploration des effets SOT.

Au cours de cette thèse, des dispositifs analogiques ont été fabriqués à l'aide de techniques standard de nano-lithographie. Ils sont de deux sortes :

-Des lignes à retard basées sur un film de YIG de 300 nm d'épaisseur ont été caractérisées. Le problème de l'adaptation d'impédance a été résolu de façon analytique, nous permettant d'établir des objectifs ciblés, basés sur nos paramètres expérimentaux, dans le but d'optimiser les caractéristiques techniques des lignes à retard magnoniques. Les outils d'analyse et de modélisation développés au cours de cette thèse ont été implémentés dans des programmes python et mis en libre accès.

-Par ailleurs, nous avons démontré l'efficacité de filtres hyperfréquence basés sur la nano-structuration périodique de guides d'ondes de YIG fin (20 nm). Un guide d'ondes modulé en largeur a été étudié et les résultats expérimentaux reproduits avec précision à l'aide de simulations micromagnétiques à l'échelle, ouvrant la voie à une optimisation fiable des propriétés des structures périodiques de YIG (cristaux magnoniques). En utilisant un guide d'ondes modulé en épaisseur, nous avons fortement supprimé les pertes engendrées par les défauts de fabrication, démontrant une réalisation efficace d'un cristal magnonique 1D.

Enfin, des dispositifs nano-magnoniques actifs, dont les propriétés sont contrôlées par l'application d'un courant continue, ont été fabriqués. En modulant l'anisotropie du BiYIG, nous avons montré que nous pouvions obtenir une grande stabilité thermique sur une large gamme de températures et une suppression du terme de couplage non linéaire préjudiciable dans les systèmes SOT supercritiques. Le régime transitoire de l'auto-oscillation, induite par le courant de spin, a été entièrement caractérisé, et un dispositif d'amplification des ondes de spin, basé sur le phénomène de SOT, a été démontré.

F 10 -

 10 is the exchange length in SI units 48 NB: All formulas will be given in SI units -𝐴 DC in [J/m] 𝑀 e in [A/m] -𝜇 T = 4 × 10 |} [H/m] -𝜇 T 𝑀 e in [T] -

Case n°1 :

 n°1 Isotropic thin film ferromagnet of normal 𝑧̂ (𝑁 I Qa = 0, 𝑁 I 11 =

  𝜃 « = 0: (out-of-plane magnetic field) o if 𝐻 T > 𝑀 e : 𝜃 = 𝜃 « = 0 o if 𝐻 T < 𝑀 e : 𝜃 = arcos( « w xe ) -If 𝐻 T ≫ 𝑀 e (the sample is saturated): o 𝜃 ≈ 𝜃 « at order zero o 𝜃 ≈ 𝜃 « + x y .« w sin(2𝜃 « ) at first order Case n°2: Thin film ferromagnet of normal 𝑧̂ (𝑁 I 11 = 1) and uniaxial perpendicular anisotropy (𝑁 I Qa = 𝑁 I P with 𝜃 P = 0

Figure 2 .Figure 2 :

 22 Figure 2: a) Schematic of the magnons modes corresponding to the quantization conditions 𝑚 = 0 and 𝑛 = 1, 𝑛 = 2 or 𝑛 = 3 for a waveguide of width 𝑤 b) Dispersion relation 𝑓(𝑘 C ) for these three magnon modes with the external magnetic orthogonal to the waveguide axis (Damon-Eshbach configuration) and using typical thin YIG film parameters: 𝛾 = 28 ØÙÚ Û , 𝜇 T 𝑀 e = 0.176 T, 𝐴 DC = 3.6 pJ. m |-, H T = 100 mT, 𝑡 = 20 nm, 𝑤 = 0.5 µm, 𝐻 c = -𝑁 // D 𝑀 e = -0.45 mT.

𝑯Figure 3 :

 3 Figure 3: Schematics of the precession of the magnetization vector 𝑀 SS⃗ induced by the conservative torque T S S⃗ ë = -𝛾𝑀 SS⃗ × 𝐻 S S⃗ Dgg . The magnetization is brought back toward its equilibrium position by the damping torque T S S⃗ Ê = -ì x y 𝛾(𝑀 SS⃗ × (𝑀 SS⃗ × 𝐻 S S⃗ Dgg ))

Figure 4 :

 4 Figure 4: Real part (blue curve) and imaginary part (orange) of the Kittel susceptibility tensor component 𝜒 -(a) as a function of frequency for an in-plane field 𝐻 T = 0.2 𝑚𝑇 and (b) as a function of in-plane magnetic field for frequency 𝑓 = 7.68 𝐺𝐻𝑧 Typical YIG-film parameter: 𝜇 T 𝑀 e = 176 𝑚𝑇 ; 𝛼 = 2 × 10 |ý ; 𝐴 DC = 3.6 × 10 |-. 𝐽/𝑚

Figure 5 :

 5 Figure 5: Schematics of the precession of the magnetization vector 𝑀 SS⃗ induced by the conservative torque T S S⃗ ë = -𝛾𝑀 SS⃗ × 𝐻 S S⃗ Dgg . The magnetization is brought back toward its equilibrium position by the damping torque T S S⃗ Ê , due to the natural relaxation of the material and to the spin pumping phenomena. The SOT is applied in the anti-damping direction T S S⃗ | . Its amplitude is proportional to the charge current applied in the Pt layer.

Figure 6 :

 6 Figure 6: PLD setup. Picture (a) and scheme (b) (Courtesy of L. Soumah 73 )

Figure 7 :

 7 Figure 7: Microscope images after each lithography step for a typical magnonic device (studied in Part 1) a) Alignment marks and numbering of the devices are defined using laser lithography and evaporation. b) BiYIG/Pt waveguides are defined with e-beam lithography and etching over a hard mask. The waveguides end with a large area to contact the Pt layer. c) A large insulating square is deposited on top of the waveguide using laser lithography and Chemical Vapor Deposition or Atomic Layer Deposition d) A rf stripe antenna is defined using e-beam lithography.

Figure 9 :

 9 Figure 9: Sketch of the operating principle of PSWS for transmission measurements. The spin-waves are excited by the dynamic rf-field (ℎ f/ ) created by the flow of an rf-current (𝐼 f/ ) in a conductive antenna. The spin-waves propagate and generate a varying flux under a second antenna, inducing an rfvoltage (Δ𝑉 "acPˆDc ). The quantity relating the rf-current to the induced voltage is the spin-wave induced inductance. In reflection measurements, the excitation and detection are performed by the same antenna.

Figure 10 :

 10 Figure 10: Light intensity through the FPI1 when scanning the mirror distance 𝑑 -= 𝑑 - T + ∆𝑑 around the distance 𝑑 - T = i . λ.

Figure 11 :

 11 Figure 11: Schematic of the TFPI with its Fabry Pérot interferometer (FPI 1 and FPI 2) with mirror distance d1 and d2, mounted with an angle 𝛼 to suppress other order transmission peaks. Frequencies are scanned by moving the stage thanks to piezo actuators.

Figure 12 :5

 12 Figure 12: Transmission of the light scattered by a magnon of frequency 𝑓 `= 5 GHz, through two FPI with distances 𝑑 - T = 𝑛 -𝜆 " and 𝑑 . T = 𝑛 . 𝜆 u , such that 𝑑 - T ≈ 10 mm and 𝑑 . T ≈ 𝑑 -cos(𝛼) with 𝛼, the angle between the two FPI. The stage is scanned, so that 𝑑 -= 𝑑 - T + ∆𝑑 and 𝑑 . = 𝑑 .T + ∆𝑑 cos(𝛼). This ensures that the main order is similar for both FPI, while other orders correspond to different scan distances ∆𝑑 for FPI1 compared to FPI1, ie they appear shifted in frequency. The transmission through both FPI is shown at the bottom, the other transmission orders are thus suppressed.

Figure 14 :

 14 Figure 14: (a) FMR curve measured at 10 GHz on a BiYIG film (blue) and the corresponding anti-Lorentzian fit (dotted red). Inset: measured resonances as a function of fields. (b) Resonance field as a function of frequency for 3 BiYIG films: in-plane (blue triangles), out-of-plane (red) and compensated (green). The Kittel law fit is indicated with the dotted lines and its result is logged in the inset table.

Figure 15 :

 15 Figure 15 : Effective magnetization as a function of temperature for seven BiYIG films grown by PLD.

Figure 17 :

 17 Figure17: Effective magnetization as a function of temperature for films near compensation, PLD-grown 21 nm thick (green curves, both as in fig.13) and LPE grown 80 nm thick (black).

Figure 16 :

 16 Figure16: Ferromagnetic resonance at 8 GHz at 260K (dotted) and 400 K (continuous line) for an inplane BiYIG film (blue) and a compensated BiYIG (green, both also in fig.12 and 13).

Figure 18 -

 18 a). A histogram of all recorded resonance frequencies is displayed in Figure18-b). While most points are within 15 MHz from the average FMR, some local "hotspots" can be up to 150 MHz above the average FMR.

Figure 18 :

 18 Figure 18: (a) 2D map of the local ferromagnetic resonance (b) Histogram of the resonance linewidths (5MHz bins). Dotted lines indicate the half-height linewidth. The frequency channel width is 50MHz and the pixel size 500𝑛𝑚 × 500𝑛𝑚.

Figure 19 ,

 19 the experimental decay length measured in an 0.5 µm wide BiYIG(21nm)/Pt(7nm) waveguide is shown for six different frequencies. The attenuation length follows the expected frequency dependence. It increases due to the increase of group velocity with the wavevector for DE spin-waves. The damping is determined at 𝛼 = 2.2 ± 0.3 × 10 |/ . This high value is consistent with the presence of Pt on this waveguide. The results are similar to the values obtained in 20nm-thick YIG/Pt waveguides12 .

Figure 19 :

 19 Figure 19: Attenuation length measured experimentally (red dots) and computed from Kalinikos-Slavin theory 26 (solid line) in a 0.5µm-wide BiYIG/Pt waveguide. The shaded area represents the error on the Gilbert damping value obtained from a fit. Parameters: 𝜇 T 𝑀 e = 0.176 T; 𝛾 = 28.1 GHz/T; 𝐻 P = 0.2112 T; 𝐻 T = 0.2 T; 𝐴 DC = 3.6 pJ/m; 𝑤 = 0.5 µm; 𝑡 = 21 nm; 𝐻 cD`Qø = -4.7 mT; 𝛼 = 2.2 ± 0.3 × 10 |/ .

Figure 20 :

 20 Figure 20: (a) Optical image of the 0.5µm-wide BiYIG/Pt waveguide. The DC contacts are insulated from the rf-antenna by an insulating SiO2 square. (b) Schematic representation of the device. The BLS laser spot is focused from the back of the sample.

Figure 22 :

 22 Figure 22: (a) Current dependence of the integral BLS intensity normalized to the value at I=0 mA (red) and of the inverse of the integral BLS intensity (blue) (b) Peak frequency of the spectrum (red) as a function of DC current. Solid line is a linear fit of the inverse BLS intensity. Vertical dotted line marks the critical current at which the damping is compensated. The data are obtained at 𝐻 T = 0.2 𝑇.

Figure 21 :

 21 Figure 21: Taken from Fig. 2 in Ref 8 (a) Current dependence of the spectrally integrated BLS intensity and of its inverse value. Vertical dashed line marks the current Ic, at which the magnetic damping is completely compensated by the spin-orbit torque. Straight solid line is the linear fit of the experimental data (b) Current dependence of the frequency of the quasi-uniform ferromagnetic resonance and of the effective magnetization. Solid line is the parabolic fit of the experimental data at I < IC. The data were obtained at H = 0.1 T.

Figure 23 :

 23 Figure 23: Integral BLS intensity (4.4GHz-5.4GHz spectrum) as a function of time for different DC current values with a 20ns sliding average. Time t=0 corresponds to the start of the 2µs DC current pulse. The transient regime of auto-oscillations can be decomposed in two phases: 1. The increase of the incoherent thermal magnon fluctuations dominates (0-50ns); 2. The thermal fluctuations saturate and the auto-oscillation amplitude grows exponentially until it reaches its stable orbital. (100ns-Xns)

Figure 24 :

 24 Figure 24: For 9 DC current values, the exponential growth rate is extracted by a linear fit of the logarithm of the BLS amplitude. (a) During the first phase (0-50ns), the thermal fluctuations growth rate seems independent on the current. The mean (blue curve) gives 𝛤 gOPˆV ≈ 21 ± 3 𝑟𝑎𝑑. µ𝑠 |- (variance indicated by the shaded region, the errors bars are obtained from the fit). (b) During the second phase, the auto-oscillations growth rate scales linearly with the supercriticality factor 𝜁 -1 = (𝐼 "} -𝐼 ˆ)/𝐼 ˆ (see upper abscise) The linear fit (blue line in (b)) x-axis intercept gives an evaluation of the critical current 𝐼 ˆ= 0.845 ± 0.05 mA (dashed black line) and the slope gives the natural relaxation 𝛤 d = 𝛼 Dgg ¶𝜔 « + 𝜔 x t½½ 2 ⁄ • = 21.2 ± 1 rad. µ𝑠 |-.

  ), Γ . = 2Γ d × (𝜁 -1) and the slope of Γ . versus (𝜁 -1) directly gives the relaxation rate of the BiYIG: Γ d ≈ 21.2 ± 1 rad. µs |-. The spinwave lifetime related to the relaxation rate is: 𝜏 = -… † ≈ 47 ns. It is similar, within the error bars, to the typical growth time of the incoherent thermal magnons found during phase 1.

Figure 25 :

 25 Figure 25: a) Colormap of the normalized BLS spectrum as a function of time (4ns integration -50MHz channels) for a 2µs DC current pulse of 1 mA starting at t=0ns. b) Zoom on the end of the DC pulse (t=2000ns). The auto-oscillation frequency is at the bottom of the thermal spectrum (near 4.9GHz)

Figure 26 -

 26 a gives us the temperature and Oersted field induced frequency shift (constant term 𝜔(𝑝 = 0)) and the non-linear frequency shift (linear term 𝑁𝑝). As the first-order non-linear coefficient 𝑁 is known, 𝑝 and the precession angle can be deduced. At maximal amplitude at 1.1 mA, the non-linear frequency shift is roughly ∆𝑓 `QC ≈ 110 ± 10 MHz, leading to a spin-wave power 𝑝 `QC = .¸∆g h‹u @ = 0.096 ± 0.01. It gives us an estimate for the maximal cone angle achieved in the experiment: 𝜃 `QC ≈ arcos(1 -2𝑝 `QC ) ≈ 0.63 ± 0.03 rad ≈ 36 ± 2 °. Thus a much larger precession angle is obtained than in YIG/Pt disks at threshold currents in Ref 10 (𝜃 = 1.3°

Figure 26 :

 26 Figure 26: Auto-oscillation frequency versus BLS intensity during a 2µs DC current pulse (0.9mA in red, 0.95 in green, 1mA in blue, 1.1 mA yellow) during their growth (a) and decay (b).(a) The frequency depends linearly on the auto-oscillation power during the growth phase (up triangles). The dotted lines show a linear fit allowing to extract the non-linear frequency shift that is independent on current. The auto-oscillation reaches a stable orbital (dots) till the end of the pulse. (b) The auto-oscillation amplitude quickly decays when the pulse ends (down triangles). As the temperature also decays, the frequency shifts down more steeply than during the growth phase (dotted lines from fit in (a)) NB: Both the frequency and intensity are extracted using a Lorentzian fit of the BLS spectrum. The integration constant used is 4ns. A 40 ns moving average is used for the growth phase (a) and 8 ns for the decay phase (b). Points shown are spaced by 20ns in (a) and 4 ns in (b). NB2: The measurement at 1.1 mA was carried out in a preliminary run, with 7.7 times less repetitions. Hence, the low intensity spectra are noisier.

Figure 27 :

 27 Figure 27: (a)-(b) Rf-excited spin-wave intensity as a function of the distance to the antenna (solid line) for frequencies 50±10MHz and 100±10MHz above FMR respectively. The decay lengths are extracted by linearly fitting the logarithm of the intensity and are indicated in the insets.

Parameters:

  𝜇 T 𝑀 e = 0.176 T; 𝛾 = 28.1 GHz/T; 𝐻 P = 0.212 T; 𝐻 T = 0.2 T; 𝐴 DC = 3.6 pJ/m; 𝑤 = 0.5 µm; 𝑡 = 21 nm; 𝐻 cD`Qø = -4.7 mT; 𝛼 = 2.2 ± 0.3 × 10 |/ ; 𝐼 ˆ= 0.85 ± 0.02 mA.

Figure 28 :

 28 Dispersion relation of the first three quantized width modes for the BiYIG waveguide under study. The first two modes nearly overlap, while the 3 rd mode frequency is nearly 150 MHz higher. Parameters: 𝜇 T 𝑀 e = 0.176 T; 𝛾 = 28.1 GHz/ T; 𝐻 P = 0.212 T; 𝐻 T = 0.2 T; 𝐴 DC = 3.6 pJ/m; 𝑤 = 0.5 µm; 𝑡 = 21 nm; 𝐻 cD`Qø = -4.7 mT

  3 and 2.4 V, corresponding to one subcritical current (𝐼 -= 0.75 ± 0.01 mA) and three supercritical currents (𝐼 . = 0.94 ± 0.02 mA, 𝐼 / = 0.99 ± 0.025 mA, 𝐼 ý = 1.04 ± 0.025 mA). The current values are estimated from the resistance at room temperature, i.e. without Joule heating (upper bound of the current estimated value) and the Pt resistance when the voltage is applied continuously, ie with maximal Joule heating (lower bound of the current value).The BLS intensity, measured for a DC voltage of 1.7 V (~ 0.75 mA) and an rf frequency 𝑓 = 4.925 GHz (𝑃 dg = -30 dBm), is presented in Figure30-(a)as a function of time and distance to the antenna. The DC voltage is applied at 𝑡 = 0 ns and the RF pulse at 𝑡 = 50 ns. The spinwaves propagate away from the antenna, while the background signal from the auto-oscillation grows with time. Both pulses are switched off at 𝑡 = 250 ns and the BLS intensity quickly decays.

Figure 30 :

 30 Figure 30: Color maps of the BLS intensity as a function of time and distance to the antenna. The DC and rf pulse timing are schematically represented on the right side of each map.In a first run (a), the DC pulse is applied at 𝑡 = 0 𝑛s and the RF pulse at 𝑡 = 50 ns with power 𝑃 = -30 dBm. Both the contribution from the SOT (thermal magnons and auto-oscillation) and the rf-excited spin-waves are probed. The SOT contribution is uniform over the waveguide, while the rf-excited spin-waves propagate away from the antenna. At 𝑡 = 250 𝑛𝑠, both pulses are switched off and the auto-oscillation and spin-waves quickly decay. In a second run (b), the pulses are applied with similar timing, but the rf-power is now 𝑃 = -90 dBm. Hence, only the SOT contribution is measured. By subtracting both maps, we can isolate the rf spin-waves signal in the presence of SOT.

Figure 29 :

 29 Figure29: Colormap of the rf-excited spin-waves intensity as a function of time and distance to the antenna, obtained by the subtracting of the SOT-enhanced thermal background and auto-oscillation signal using the method described in Figure30. The probed area of the waveguide is schematically represented above. The resolution is 0.2 µm × 0.8 ns, with a 4ns moving average in time.

Figure 31 :

 31 Figure 31: Colormaps of the rf-excited spin-waves intensity as a function of time and distance to the antenna for frequencies 𝑓 = 4.925, 4.95, 5.0 GHz (lines) for different applied DC voltages 𝑈 = 1.7, 2.2, 2.4 V (columns).The DC voltage pulse starts at 𝑡 = 0 ns, while the rf-voltage is applied at 𝑡 = 50 ns. The spin-waves propagate away from the antenna and the slope of the intensity step gives the group velocity (white solid line). Spin-waves are excited with a constant rf power 𝑃 = -30 dBm but their intensity heavily depends on the frequency (excitation efficiency of the antenna) and DC voltage (see scale on the color bars).

Figure 33 .Figure 32 :

 3332 Figure 32: Group velocity computed from the dispersion relation (solid line -same parameters as Figure 27) and experimentally measured (red dots). The experiment matches the theory for FMR ~ 4.885 GHz.

Figure 33 :

 33 Figure 33: The highlighted areas represent the BLS intensity of the magnons emitted at the antenna between time 𝑡 = 60 ns and 𝑡 = 64 ns (a.1), 𝑡 = 110 ns and 𝑡 = 104 ns (a.2) and 𝑡 = 50 ns and 𝑡 = 130 ns (b). The group velocity used to draw up these areas, 𝑣 ø = 105 m/s, is determined by the rise of intensity of the initial magnon propagation.

Figure 33 -

 33 b is the widest than can be taken since the magnons must have enough time to propagate over a distance of 12 µm before the end of the DC pulse (t = 250 ns). The experimental intensity curves of the magnons emitted between the beginning of the rf-pulse at

Figure 34 :

 34 Figure 34: Spin-wave intensity as function of the distance to the antenna obtained by taking a crosssection parallel to the intensity edge and averaged over 80ns. At 𝑥 = 0, the BLS intensity measured from 𝑡 " = 50 ns and 𝑡 g = 130 ns is averaged. At 𝑥 = 𝑥 T , the BLS intensity is averaged between 𝑡 " = 50 + 𝑥 T /𝑣 ø and 𝑡 g = 130 + 𝑥 T /𝑣 ø . The experimental curves (solid lines) are linearly fitted (dotted lines) in a logarithmic scale for 4 frequencies 4.925 GHz (a), 4.95 GHz (b), 4.975 GHz (c), 5.0 GHz (d) and 4 voltages 1.7 V (blue), 2.2 V (orange), 2.3 V (green), 2.4 V (red).

2 NB:

 2 Horizontal propagation leads to large uncertainties on the attenuation lengths due to the divergence of the inverse function at zero.

t = 50 ns and t = 130 ns are plotted in Figure 34 .

 34 They are fitted to extract the attenuation length shown in the associated Table (Figure34).

Figure 35 :

 35 Figure 35: Propagation of magnons emitted at the antenna 10 ns (a) and 50 ns (b) after the start of the pulse (t=60ns and t=110ns) for frequency 𝑓 = 4.925 GHz and voltages 1.7 V (blue) and 2.4 V (red). The shaded areas represent the uncertainty coming from the precision on the group velocity ±5 m/s at 10ns and ±15 m/s at 50 ns. The exponential fits (dotted lines) show that the slope is positive in all cases at 2.4 V and the rf-excited spin-waves are thus amplified at this frequency.

Figure 36 :

 36 Figure 36: a) Schematic of the experiment. Inset shows the distribution of the internal static magnetic field in the MC waveguide calculated by using the MuMax3 software (b) Solid curves -calculated dispersion curves for the 0.8 and 1 µm wide YIG stripes constituting the MC. Symbols -experimentally determined dispersion for a straight 1 µm wide reference waveguide fabricated from the same YIG film

  f=4.942 GHz, l=2 µm f=4.942 GHz f=4.890 GHz

Figure 37 :

 37 Figure 37: (a) Representative examples of twodimensional SW intensity maps recorded for SW frequencies 𝑓 =4.890 and 4.942 GHz corresponding to the wavelength 𝜆 = 3 and 2 µm, respectively. (b) Propagation-coordinate dependence of the SW intensity integrated across the width of the 2D maps in the log-linear scale. Symbols -experimental data. Dashed lines -fit of the data by the exponential function. (c) 2D SW phase map recorded at frequency 4.942 GHz.

Figure 38 :

 38 Figure 38: (a) Symbols -experimentally determined dispersion curve for SWs in the MC. Solid curves -calculated dispersion curves for the 0.8 and 1 µm wide YIG stripes. The shaded area marks the frequency region Df of the magnonic band gap. (b) Symbols -measured frequency dependence of the imaginary part of the wave vector 𝑘´´ (decay) Solid curvereference dependence 𝑘´´(𝑓) calculated for a 0.8 µm wide straight waveguide.

  Dfeye to follow the position of the local maxima as a function of frequency. Below the gap, the maxima are located in the middle of the narrow parts and shifts to the middle of the wider part.

Figure 39 :

 39 Figure 39: (a) and (b) Dispersion curves for SWs at frequencies of the band gap determined from high-frequencyresolution BLS measurements. The vertical dashed line marks the Bragg wave vector kB. The shaded area marks the frequency region Df of the magnonic band gap. (c) Color-coded plot of the spin-wave intensity in the x-frequency coordinates. The dashed vertical lines mark the x position of the maximum SW amplitude that shifts in position when the frequency is increased across the band gap. (d) x-profiles of spin waves at frequencies 𝑓 dDc = 4.886 GHz and 𝑓 ¡OPD = 4.894 GHz, showing the frequency asymmetric response below and above 𝑓 ˆ.

Figure 40 (Figure 40 :

 4040 Figure 40 (a) shows a typical intensity plot, where the amplitude of the SWs is plotted at frequencies 𝑓 = 4.900 GHz and 𝑓 = 4.875 GHz corresponding to 𝜆 = 2.25 µm and 𝜆 = 3 µm respectively. We notice a 15 𝑀𝐻𝑧 frequency shift between experimental results and the simulated one (𝑓 ˆ= 4.875 GHz in the simulation against 𝑓 ˆ= 4.890 GHz in the experiment) which is well within the error bounds caused by uncertainties in material parameters and external magnetic field. Similarly, to the experiment, the amplitude for the SWs at 𝜆 = 3 µm display a rapid decay and a large spatial beating arising from Bragg reflection. For each frequency, the imaginary part of the wave vector (k") is calculated by linearly fitting the log of the amplitude. The real part is obtained by Fourier analysis of the phase maps. The obtained dispersion relation is shown in Figure 40 (b) and defines a clear 12 MHz band gap for wave vectors k ≈ k £ . In this gap, an increase of the imaginary part of the wavevector by up to a factor of 3 is observed (see Figure 40 (c)). The 𝑘" plot also features a smaller peak near 𝑘 ≈ 1.3 𝜇𝑚 |-which is also observed on the BLS data (Figure 38 (b)). We attribute this feature to mode crossing between the 𝑛 = 1 and 𝑛 = 3 quantized transverse modes in the wider part, an expected mechanism well described in Ref 112 . A focus on the gap region is given in Figure 40 (d) and Figure 40 (e). An unambiguous vertical degeneracy and a decrease of group velocity at the bounds of the gap are observed. Contrary to

Figure 41 :

 41 Figure 41: Color-coded plot of the simulated SW intensity in the x-frequency coordinates. The dashed vertical line marks the x position of the maximum standing SW amplitude that shifts in position when the frequency is increased across the band gap. (b) Simulated intensity map of the SW amplitude‖𝑚 1 SSSSS⃗‖ in log scale as a function of x at 3 different frequencies, below the gap (f1) where the mode 𝑛 = 1 dominates, at the maximum attenuation frequency (fc), and above the gap (f2) where the onset of mode 𝑛 = 3 as well as edge modes is evidenced

Figure 42 :

 42 Figure 42: Schematic of the thickness modulated magnonic crystal.

Figure 43 :

 43 Figure 43: Imaginary part of the spin-wave inductance spectra: in reflection ∆L11 (dotted linesamplitude ×0.1) and transmission ∆L21 (solid lines) at external fields 𝐻 T = 6, 12, 20 𝑎𝑛𝑑 27 𝑚𝑇.

Figure 44 :

 44 Figure 44: (a) Normalized excitation and detection efficiency of the antenna as function of the k-vector. Calculated from the squared Fourier transform of the current density distribution. (b) Expected transmission inductance spectrum, amplitude (in black) and imaginary part (in blue), for 𝐻 T = 6 𝑚𝑇, computed analytically.

-Figure 45 :

 45 Figure 45: Dispersion relation, for an external field 𝐻 T = 11.3 𝑚𝑇, extracted from the unwrapped phase of ℐ𝑚(𝛥𝐿 .-) (blue) and from the Kalinikos-Slavin theory 12 with no fitting parameters (dotted red) and with 𝑤 Dgg = 1.6 𝜇𝑚 (dashed green) Parameters: 𝐷 Dgg = 31.5 𝜇𝑚; 𝑤 = 2.5 𝜇𝑚; 𝑡 = 23.5 𝑛𝑚; 𝜇 T 𝑀 e = 176 𝑚𝑇; 𝐻 P = -37 𝑚𝑇; 𝐴 DC = 3.6 × 10 |-. 𝐽. 𝑚 |-; 𝐻 cD`Qø = -1.1 𝑚𝑇.

  Fig 5 for an external field H T = 11.3 𝑚𝑇.Unexpectedly, the transmission amplitude is increased at almost all frequencies except around 𝑓 = 1.4 GHz where the amplitude decreases. This frequency corresponds to the Bragg k-vector (see Figure45). There is no strong indication of the presence of a second order gap. The increase of the transmission amplitude is observed at all fields. In the previous part, we showed that the opening of a gap can strongly modify the way antennas are coupled to the spin-waves due to standing waves. But this phenomenon, concentrated at the gap edges, fails to explain the signal increase over the whole spectrum. Increasing the grooves depth to 5 nm and 8 nm only strengthens the attenuation in the region of the gap (Figure46(b) and (c) respectively) with almost no extra-losses compared to 2nm. The phase remains similar outside the gap, confirming the small impact of thickness modulation on the dispersion relation and on the attenuation outside the gap.

Figure 46 :

 46 Figure 46: ℐ𝑚(∆L21) for 𝐻 T = 11.3 𝑚𝑇 for unetched waveguides (in blue), 2 nm grooves (orange), 5 nm (green) and 8 nm (red). Strong changes, indicated by an arrow, occur near the Bragg k-vector.

Figure 47 :Figure 48 :

 4748 Figure 47: (a) Dispersion relation extracted from the ∆L21 phase. (b) Attenuation computed as the amplitude ratio (in dB) between the etched and the unetched waveguides. Each point on the graphs is an experimental data point. Reference without etching in blue, 2 nm grooves in orange, 5 nm in green and 8nm in red.

Figure 49 :

 49 Figure 49: (a) ℐ𝑚(∆L21) for 𝐻 T = 11.3 𝑚𝑇 for 2 nm and 22nm grooves (b) AFM image of YIG waveguide (horizontal) with 3 grooves (vertical) (c) Height profile across a groove taken by averaging along the waveguide center (white rectangle)

Figure 50 :

 50 Figure 50: Microscope image of a typical delay line sample. The 4µmwide YIG waveguides (light brown) array is defined using laser lithography. Two gold GSG rf-antennas are deposited on top. The 4µm wide signal line is separated from the 2µm signal lines by a 2µm gap.

Figure 51 :

 51 Figure 51: Dispersion relation of the first three thickness quantized modes in the Damon-Eshbach configuration (solid lines) and in the Backward Volume configuration (dotted lines) for an unpatterned 300nm-thick YIG film. In both orientations, the modes do not overlap, and the delay line can be operated in a single thickness mode regime. Parameters: 𝑡 = 300 nm; 𝜇 T 𝑀 e = 0.176 T; 𝛾 = 28 GHz/T; 𝐻 T = 0.45 T; 𝐴 DC = 3.6 pJ/m; ;

Part 3 :𝐽

 3 Delay line demonstrator -a step toward integrated magnonics 107

Figure 52 :

 52 Figure 52: Excitation and detection efficiency spectrum of our antenna for 𝐽 T = sinc (in blue) and 𝐽 T = ℬ T (in green). The spin-waves are most efficiently excited at 𝑘 = ä rad/µm for which the dynamic magnetization 𝑚 C is in phase with the dynamical field ℎ C created by the antenna, as schematically represented in the inset. On the contrary, near 𝑘 = .ä , spinwaves are in opposite phase and cannot be excited by our antenna.

Figure 53 :

 53 Figure 53: Transducer auto-impedance spectrum measured at zero magnetic field (solid lines) and model using a distributed RLC model (dashed lines) for antennas of length 30µm (a) and 120µm (b). The resonance amplitude is 2.5 times smaller for the 120µm-long antenna. Parameters: (a) 𝑅 = 8.2 Ω; 𝐿 = 0.87 nH; C = 0.2 pF; (b) 𝑅 = 21 Ω; 𝐿 = 0.97 nH; C = 0.22 pF

Figure 54 :Figure 55 :

 5455 Figure 54: Impedance spectra obtained by subtraction (Eq. (148)) of the reference impedance, measured at zero magnetic field, to the spectra measured at different magnetic fields (H1,H2,…,Hk). The clear increase of the absolute value (a) and the 180° phase shift (real value (b)) at the 12GHz resonance demonstrates that the circuit is not in series and that the SW induced impedance spectra should be extracted in a way that takes into account the RLC resonance.

Figure 56 :

 56 Figure 56: Example of a spin-wave induced impedance spectrum 𝑍 "5 3n (𝐻) and associated background spectrum 𝑍 "5 3n (𝐻 = 0) for the reflection spectrum (a) and transmission spectrum (b).The result after subtracting the background is shown in (c). NB: Curves obtained for DE SW in a 30µm waveguide.

Figure 58 :Figure 57 :

 5857 Figure 58: a) Maximal radiation resistance measured at different resonance frequencies (symbols) for DE spin-waves. The curves are obtained from the Bailleul's model with a renormalization constant 𝑐 = 0.65 and Gilbert damping 5 × 10 |ý (solid line), 1 × 10 |/ (dashed line) and Kalinikos model (no damping). b) Example of an experimentally obtained spectrum (green line) with a resonance near 10 GHz. The red arrows indicate features (shoulder and secondary lobe) that are best reproduced by the model with damping 1 × 10 |/ (solid line). Higher damping completely blurs out the features, while a lower damping exhibits sharper ones.

4 - 9 Model

 49 𝛼 = 1 × 10 |/ ; c = 0.65 ---𝛼 = 3 × 10 |/ ; c = 0.

Figure 59 -

 59 b at 10GHz), both in terms of amplitude and linewidth.

Figure 59 :

 59 Figure 59: (a) Maximal radiation resistance measured at different resonance frequencies for BW spin-waves (symbols). The curves are obtained from the Bailleul's model with a renormalization constant 𝑐 = 1 and Gilbert damping 2 × 10 |ý (line), 1.5 × 10 |/ (dashed line). The dotted line is obtained from the Kalinikos model with zero damping. b) Example of an experimentally obtained spectrum (red line) with a resonance near 10 GHz. The green arrows indicate features (secondary lobes) that are best reproduce by the model with a uniform current distribution (solid line). The blue arrows indicate periodic local maxima that are not expected by the model.

Figure 60 : 4 )

 604 Figure 60: (a) Absolute value of the group velocity as a function of the k-vector (with an external field such that the FMR is 10GHz) in a 4µm-wide waveguide (full line) and in unbounded film (dotted). For both field orientation, the first width quantized mode is sped up near k Š -= 0.75 rad/µm. The spectrum antenna (grey area) coincides with the highest group velocity region. (b) Evolution of the group velocity at the most efficiently excited wavevector 𝑘 QaVDaaQ = 0.6 rad/µm as a function of the resonance frequency. It is in the DE configuration that the spinwaves are the most sped up, especially at high frequencies. Note that the BW spin-waves have a nearly constant group velocity in the waveguide.

Figure 61 :

 61 Figure 61: a) Maximal radiation resistance measured at different resonance frequencies for DE spin-waves in a 4µm-wide waveguide (symbols). The curves are obtained from the Bailleul's model with a renormalization constant 𝑐 = 1 and different Gilbert damping. The values shown take into account the filling factor of 2/3 (4µm waveguides with 2µm spacing). The RLC model used to extract the experimental values fails above 8 GHz for this antenna b) Example of an experimentally obtained spectrum (green line) with a resonance near 7 GHz. The experimental spectrum is well reproduced by the Kalinikos model (dotted line) and our adaptation of the Bailleul's model.

Figure 62 :

 62 Figure 62: a) Maximal radiation resistance measured at different resonance frequencies for BW spinwaves in a 4µm-wide waveguide (symbols). Contrary to the other configurations, the experimental values are significantly higher than the Kalinikos' model expectations (solid line 𝑐 = 1 ). A renormalization constant 𝑐 = 2.5 is necessary to approach the experimental values (dotted line). b) Example of an experimentally obtained spectrum (red line) with a resonance near 6.5 GHz. The Kalinikos model significantly underestimate the maximal radiation resistance.

Figure 64 :

 64 Figure 64: a) Transducer mutual-impedance spectrum measured at zero magnetic field (solid lines) for antennas of length 30µm (a) and 120µm (b). A resonant peak is observed at 13.5 GHz.The peak height is about 2.5 times smaller for the longer antennas. Note that these impedance spectra result in similar S-parameters spectra for short and long antennas.

Figure 63 :

 63 Figure 63: Insertion losses spectra for different antenna designs (schematically represented on the right) measured in the absence of spin-waves. Low insertion losses indicate direct coupling between antennas (cross-talk) which we aim to avoid. GSG are at least 10dB better than the other designs. Antennas' length and spacing does not seem to impact the cross-talk.

Figure 65 :

 65 Figure 65: Lumped RLC model of the delay line. Both antennas are parallel RLC circuits. The capacitive and inductive branches are coupled through the spin-waves propagating in the waveguide resulting in the 𝑍 "5 3n impedance.

Figure 66 .

 66 In both the 30µm-wide waveguide (a) and the 4µm-wide waveguides (b), Γ is close to one for BW spinwaves (green symbols) while it steadily decreases to 0.2 in the DE configuration (red symbols). The curves are fitted using the Kalinikos & Slavin formula to extract the damping. In wide waveguides (Figure66-a), the experimental values are well fitted by different damping values 𝛼 = 1 × 10 |ý for BW SWs and 𝛼 = 3 × 10 |ý ; ∆𝐻 T = 0.1 mT for DE SWs. Both values are lower than the damping found by FMR measurement performed on the unpatterned sample (𝛼 = 6 × 10 |ý ).

Figure 66 :

 66 Figure 66: Ratio between the SW induced mutual impedance and auto-impedance, Γ ¿ for a 30µm waveguide (a) and a 4µm waveguide (b). In both cases, the ratio for BW SWs is close to 1 while it decreases for DE SWs. The Gilbert damping is extracted by fitting Γ ¿ with the Kalinikos & Slavin formulas.

𝛼 = 3 ×Figure 67 :

 367 Figure 67: (a) Propagation losses computed with the damping values found in the experiment for wide waveguides with an effective propagation distance 𝐷 = 24 µm. The higher losses for DE SWs (green) are due to an increased damping and longer delays (b) compared to BW SWs (red).

Figure 68 :

 68 Figure 68: Analytical calculation of the delay (τ ‡ˆ›‰Š = D ˆ´´/ v ç ) using the group velocity from the Kalinikos-Slavin formula and the effective distance travelled by spin-waves 𝐷 Dgg = 28 µm. The delay is computed for a 4µm waveguide (line) and in unbounded film (dashed) for BW (red) and DE (green) spin-waves as a function of the wavector (a) (with an external field such that the FMR is 10GHz); and as a function of the resonance frequency (b) at 𝑘 QaVDaaQ = 0.6 rad/µm. In both configurations, the delay is more stable in waveguides.

  near 9.5 GHz. The BW spin-waves induce a delay that is stable within the transmission band with value 𝜏 cDOQ3 = 16 ± 3 ns (orange connected dots in (a)). From the Kalinikos-Slavin formula, the delay is expected between 15 to 17 ns (orange solid line in (b)) for wavevectors between 0.4 and 0.8 rad. µm |-(corresponding to the k-linewidth of the main lobe of the antenna excitation spectrum). The DE spin-waves delay varies from 18 ns at the lower edge of the transmission band to 25 ns at the upper edge (blue connected dots in (a)). It is lower than the expected 24 ns to 32 ns delay at 0.4 and 0.8 rad. µm |-respectively (blue solid line in (b))

Figure 70 -

 70 (a)-(b). At 8.5 GHz, the delay increases from 16 ns to 22 ns in the transmission band in the wide waveguide (blue dots). Meanwhile, the delay seems monotonous in narrow waveguides, equal to 17 ± 2 ns (green). The theoretical expectations are shown in Figure70-c) using the center to center distance between antennas.

Figure 69 :

 69 Figure 69: a) Delay as a function of frequency for two experimental spectra measured near 9.5 GHz for BW (orange) and DE (blue) spin-waves. The delay is highlighted (connected dots) where the spinwave impedance signal is large (continuous line). b) Analytically expected delay for an effective distance of 𝐷 Dgg = 28 µm corresponding to the center to center distance between both antennas. Despite the large width of the waveguide, the effect of quantization on the delay (continuous line) is clearly visible compared to an unbounded film (dotted line)

Figure 70 :

 70 Figure 70: Delay extracted from experimental spectra for devices with a wide waveguide (𝑤 = 24 µm in blue) and with narrow waveguides (𝑤 = 4 µm in green) near 8.5 GHz (a) and 11.5 GHz (b) in the DE configuration. c) Expected delay at 8.5 GHz from analytical calculations as a function of the spin-wave wavevector. d) Delay at the maximum of amplitude of 𝑍 .--𝑍 .- dDg as a function of the resonance frequency from experiment (triangles) and analytical calculations using a nominal wavevector of 𝑘 QaV = 0.6 rad. µm |-(solid line) and 𝑘 QaV = 0.4 rad. µm |-(dotted). The error bars are computed as the standard deviation of the delay within the transmission band. NB: The effective distance used in the calculations is the center to center distance between antennas 𝐷 Dgg = 28 µm for the wide waveguide and 𝐷 Dgg = 26.5 µm for the parallel narrow waveguides.

Figure 71 :

 71 Figure 71: Delay extracted from experimental spectra for devices with a wide waveguide (𝑤 = 24 µm in blue) and with narrow waveguides (𝑤 = 4 µm in green) near 8.5 GHz (a) and 11.5 GHz (b). c) Expected delay at 8.5 GHz from analytical calculations as a function of the spin-wave wavevector. d) Delay at the maximum of amplitude of 𝑍 .--𝑍 .- dDg as a function of the resonance frequency from experiment (triangles) and analytical calculations using a nominal wavevector of 𝑘 QaV = 0.6 rad. µm |-(solid line); The error bars are computed from the standard deviation of the delay within the transmission band.

Figure 72 :

 72 Figure 72: Simplified equivalent circuit of the magnonic delay line.

Figure 73 :

 73 Figure 73: (a) Normalized spin-wave induced auto-impedance spectrum at 10 GHz (𝐻 T = 0.28 T) (b) Normalized spin-wave induced mutual-impedance in the artificial case where 𝛿 = 0. NB: The spin-wave radiation resistance (ℛ𝑒(𝑧 -- 3n ) dotted black line in b) has the exact same shape as the absolute value of the mutual impedance (J𝑧 .- 3n J) in the absence of damping and zero reciprocity (J𝑧 -. 3n J = 0)

Figure 74 :

 74 Figure 74: Insertion losses (a) and return losses (b) as a function of the transducer and spin-wave radiation resistance. The minimal insertion losses are obtained for a 50 Ω radiation resistance and a zero-resistance transducer. The return losses are maximal if the condition 𝑟 Vd + 𝑟 3n = 50 Ω is satisfied.

Figure 75 :

 75 Figure 75: (a) Insertion losses color map as a function of the transducer and spin-wave radiation resistance. The dark lines represent the accessible insertion losses for a given design at a function frequency, given by 𝑟 3n = Λ × 𝑟 Vd . The higher the slope, the higher the share of the power goes into the spin-waves and the lower the insertion losses. All such curves intersect (red dots) the impedance matching line (white) given by 𝑟 Vd + 𝑟 3n = 50 Ω where the return losses are maximal, and the insertion losses are minimal. The insertion losses at the impedance matching intersection is shown as a function of Λ in (b). The IL are higher because part of the power is absorbed in the ohmic losses. The transmitted part being proportional to ð ðÊ-, the insertion losses are such that 𝐼𝐿 = 𝐼𝐿(Λ = +∞) -20 log -T ð ðÊ-(see solid black line).

Figure 76 :

 76 Figure 76: Insertion losses at the impedance matching conditions as a function of Λ for different reciprocity ratio 𝛿.

Figure 77 :

 77 Figure 77: a) Minimal insertion losses as a function of 𝛿 for two off-diagonal phase angles 𝜓 = 0 (solid line) and 𝜓 = . (dotted). The phase has no influence in the case where 𝛿 = 0. b) Insertion losses spectrum corresponding for 𝛿 = 0 (green and blue). The minimal value corresponds to the impedance matrices indicated. For 𝛿 = 0.5, the insertion losses are minimal when the phase is . (ie resistive off-diagonal term, as in the red matrix), which occurs at the impedance peak for 𝜓 = . but not for 𝜓 = 0. The phase thus create oscillation in the insertion losses spectrum.

Figure 78 :

 78 Figure 78: Evolution of the ratio Λ as a function of the transducer thickness, normalized by its value in the experiment (150nm gold with 20nm Ti).

Figure 79 :

 79 Figure 79: At each operating frequency, the Λ is different and optimal transducer length ensures minimal insertion losses (green dashed line) and good return losses (red line). In a real delay line, the transducer length is fixed and chosen to optimize the IL at 15 GHz in this figure. Using the delay line at other frequencies than 15 GHz results in the IL and RL displayed in blue and orange. NB: The calculation is performed using Λ = 2.5 × 𝑓/𝑓 T with 𝑓 T = 15 GHz.

Figure 80 :

 80 Figure 80: For a transducer with Λ = 2.5 and operating frequency of 15GHz, the optimal length is 𝐿 W¿V = 71µm. Here is plotted the insertion losses (a) and the return losses (b) for different lengths of the transducer, from 7µm (𝐿 W¿V /10) to 700µm (10 × 𝐿 W¿V ).

Figure 81 :

 81 Figure 81: Reflection (red) and transmission (green) spectra of the Delay Line S-parameters at 12 GHz (a) and 18 GHz (b) for 𝑅 = 0 Ω and r %& = 50 Ω. The S-parameters obtained considering RLC resonance (solid lines) are compared to the non-resonant case (dotted lines). The RLC resonance worsen both the return and insertion losses. The effect is stronger at 12 GHz, near the RLC resonance.

Figure 82 :

 82 Figure 82: (a) Insertion and (b) Return losses at the peak of the spin-wave induced impedance 𝑧 -- 3n at 12GHz with parameters 𝐿 = 1 nH ; 𝐶 = 0.2 pF defining an RLC resonance at12GHz. The red line (𝑟 3n + 𝑟 Vd = 70 Ω) marks the condition that maximizes the return losses. It also maximizes the insertion losses.

Figure 83 :

 83 Figure 83: Spectra of the S-parameters for two impedance matching strategies. (a) Using a low spin-wave impedance 𝑟 3n = 22 Ω, the large bandwidth (40MHz) is preserved with low insertion losses (9dB) but also low return losses (4-8 dB) (b) Using a high spin-wave impedance 𝑟 3n = 90 Ω, the main lobe is split in two bands (20 and 15 MHz), with low insertion losses (7-10dB) and high return losses (10-30 dB).

A)

  Vers un démonstrateur de ligne à retard magnoniqueLorsque les premiers dispositifs magnoniques de traitement du signal ont été développés dans les années 1980125,129 , ils furent rapidement dépassés par les dispositifs analogiques basés sur les ondes acoustiques et les dispositifs digitaux basés sur le CMOS. Cependant, ces deux types de dispositifs deviennent très inefficaces aux hautes fréquences 120 (au-dessus de 10 GHz), fréquences actuellement exploitées par certains systèmes clefs chez Thales (radars, communications satellite) et prochainement exploités par de nombreux systèmes (réseaux 5G à large bande passante, dispositifs IOT). En particulier, le retardement d'un signal à ces fréquences résultent soit en de fortes pertes d'insertion (50dB), soit requiert des volumes importants. Par ailleurs, une ligne à retard magnonique est constituée des briques de bases de tout composant magnonique : la conversion d'un signal RF en onde de spins (excitation et détection) et leur propagation (guide d'onde magnonique) (voir Figure84-a). Dans ce contexte, le développement d'une ligne à retard magnonique est pertinent comme première étape vers l'intégration de composants magnoniques dans les systèmes Thales.

Figure 84 :

 84 Figure 84: a) Schéma de principe d'une ligne à retard magnonique contenant les trois briques de bases de tout composant magnonique : l'excitation (conversion du signal rf en onde de spins), la propagation et la détection (conversion de l'onde de spins en signal rf) b) Image microscope d'une ligne à retard fabriquée par lithographie laser. En brun clair : vingt-quatre guides d'ondes de YIG (épaisseur 300 nm, largeur 4 µm, séparation 2 µm), deux antennes rf Ground-Signal-Ground (Au(200nm) /Ti(20nm), largeur de la ligne signal 4 µm, largeur des lignes grounds 2 µm). Distance entre les antennes (centre à centre) : 26.5 µm

  Un guide d'onde de BiYIG/Pt d'une largeur de 0.5 µm et muni d'une antenne large bande (voir Figure86) a été étudié par microscopie BLS. En appliquant continument un courant souscritique, la longueur d'atténuation des ondes de spins peut être modulée pour atteindre 𝐿 QVV = 87 ± 25 µm pour une fréquence proche de la FMR, une valeur 50 fois plus grande qu'en l'absence de SOT (𝐿 QVV = 1.6 ± 0.5 µm). C'est mieux que la modulation d'un facteur 5 obtenue pour le YIG/Pt 12 en 2017. Cela pourrait être lié à la réduction des effets non-linéaires dipolaires liés à une aimantation effective proche de zéro dans le BiYIG. Étonnamment, la compensation de l'amortissement par le SOT devient graduellement moins efficace lorsque la fréquence est légèrement augmentée. À ces fréquences, la longueur d'atténuation atteint seulement 8 µm malgré l'application d'un courant proche du courant critique (modulation d'un facteur 3 par rapport à la propagation sans courant). L'amortissement effectif des modes progressifs de ce guide d'ondes est donc non nul au courant critique, et son degré de modulation semble dépendre de la fréquence ou de la longueur d'ondes du mode. De surcroit, de forts effets non-linéaires émergent toujours lors de l'application continue de courants surcritiques.

Figure 85 :

 85 Figure 85: Aimantation effective (𝜇 T 𝑀 Dgg ) en fonction de la température pour sept films de BiYIG (20 nm d'épaisseur), mesurés par résonance ferromagnétique. Les films sont crus par PLD avec différents paramètres de croissance de manière à varier le champ d'anisotropie. Quand l'aimantation effective est proche de zéro, on constate que sa valeur est indépendante de la température.

Figure 86 :Figure 87 :

 8687 Figure 86: Image microscope du dispositif à l'étude : un guide d'onde de BiYIG(20nm)/ Pt(7nm) de 500nm de large. Les ondes de spins sont excitées via une antenne rf large bande de 300 nm de large. Un champ magnétique 𝐻 = 0.2 T est appliqué orthogonalement à l'axe du guide d'onde, permettant l'application d'un couple de spin orbite en injectant un courant continue dans le Pt.

mA) 0 0.6 0.75 0.8 0.85

	FMR (GHz)	4.850	4.880	4.900	4.910	4.920

Table: FMR frequency for the applied DC currents I (

  𝑍 } 𝑖 . } + 𝑍 .. The fact that the capacitive (resp. inductive) branch of the first antenna is only coupled to the capacitive (resp. inductive) branch of the second antenna is crucial (dashed line in Figure64). Indeed, if we modeled it with a coupling between the inductive and capacitive branches, Eq.

	3n 𝑖 . } + 𝑍 .-3n 𝑖 -} = 𝑍 z 𝑖 . z + 𝑍 .. 3n 𝑖 . z + 𝑍 .-3n 𝑖 -z	(160)
	NB: (160) would write:
	𝑉 . = 𝑍 } 𝑖 . } + 𝑍 .. 3n 𝑖 . } + 𝑍 .-3n 𝑰 𝟏 = 𝑍 z 𝑖 . z + 𝑍 .. 3n 𝑖 . z + 𝑍 .-3n 𝑰 𝟏
	ie, simplifying the above equation:
	(𝑍 } + 𝑍 .. 3n )𝑖 . } = (𝑍 z + 𝑍 .. 3n )𝑖 . z ⇒ 𝑖 . z = 𝑖 . } = 0 (using Eq. (159))
	Let's now derive the expression of 𝑉 . | F F ÌT :
	2𝑉 . = 𝑍 } 𝑖 . } + 𝑍 z 𝑖 . z + 𝑍 .. 3n (𝑖 . } + 𝑖 . z ) + 𝑍 .-3n 𝐼 -	(161)

  𝑳 𝒐𝒑𝒕 𝟏𝟎 × 𝑳 𝒐𝒑𝒕 𝟐 × 𝑳 𝒐𝒑𝒕 𝑳 𝒐𝒑𝒕 /𝟐 𝑳 𝒐𝒑𝒕 /𝟏𝟎 𝑳 𝒐𝒑𝒕 𝟏𝟎 × 𝑳 𝒐𝒑𝒕 𝟐 × 𝑳 𝒐𝒑𝒕 𝑳 𝒐𝒑𝒕 /𝟐 𝑳 𝒐𝒑𝒕 /𝟏𝟎

	𝑍 "" fz} =	1 3n + 𝑍 z + 𝑧 "" 1	𝑍 } + 𝑧 "" 3n 1	(180)
	a)				b)
	𝑍 "5 fz} =	𝑧 "5 3n 2	61 +	(𝑍 } -𝑍 z ) .

Tandem Interferometer TFP-1 by JRS Scientific Instrument

Single photon counting module, COUNT BLUE-10B by Laser ComponentsFigure 13: Schematic of the full µ-BLS setup.

from personal communications with B. Diviniskiy and V. Demidov (Dec 2019)

inhomogeneities. This suggests that modeling the PMA distribution at the edges is impossible in non-0D structures.

Durant cette thèse, j'ai fabriqué des lignes à retard basées sur un film de YIG de 300 nm d'épaisseur et les ai caractérisées par des mesures électriques. La fréquence de fonctionnement du prototype a été choisie entre 10 et 20 GHz avec des retards compris entre 10ns et 50ns. J'ai également développé des outils d'analyse et de modélisation sous forme de programmes python qui ont été transmis aux équipes CNRS et Thales.

Les dispositifs ont été caractérisés en connectant les antennes à un analyseur de réseau vectoriel qui mesure la variation d'amplitude et de phase entre le signal RF injecté et les signaux réfléchi et transmis d'une antenne à l'autre. Nous montrons que l'analyse de l'amplitude permet d'extraire les pertes de propagation, tandis que la phase contient les informations sur le retard de propagation. Les conclusions de l'analyse des différents échantillons sont les suivantes :

-Les antennes GSG (Figure 84