Alan Schuhl

Vincent Cros

Vitalii Dugaev

Robert Stamps

En primer lugar me gustaría

Gracias a todos los amigos con los que he compartido estos años en Grenoble. Tengo muchísima suerte de poder disfrutar y haber conocido a gente genial como Pablete, Emilie, Lara, Alejandro, Martí, Cristina, Diego, Elisa, Wiebke, y a las recientes "incorporaciones": Laurent, Juan, Miguel, Rubén (mi canario favorito), Miguel V. y Maite. Les agradezco inmensamente su escandalosamente genial regalo de tesis. Me mataron de la risa con su sorpresa y me hicieron sentir muy feliz.

Si alguien puede conseguir cosas como hacer que tu tesis aparezca en un número inédito del "Nationel Geographik", esa es Karla Marina. A pesar de estar temblando el mundo, se preocupó de darme ánimos cada día durante la redacción, de sorprenderme con un inmejorable final de tesis y de afrontar la difícil tarea de hacer que confíe un poquito más en mí mismo. Dichoso soy por haberme transmitido su alegría de vivir y por poder descubrir, emocionarme y resonar con alguien tan especial.

Gracias a mis grandes amigos de toda la vida. Los que uno sabe que están ahí aunque no los vea a menudo. Cuando pienso en Carmen, Pablo, Gaspar, Gonzalo y David se me dibuja una sonrisa en la cara. Mi pierna derecha también tiene mucho que agradecerle a Carmen: si no hubiera sido por nuestra conversación semanal este verano, me habría pegado un tiro en la rodilla.

Gracias a mis abuelos. Porque no me olvido de todos los días que me han cuidado siendo un enano, de los maravillosos veranos en el pueblo, de las mañanas en "los pinos" y de las tostadas de mermelada de ciruela sentado en la encimera de la cocina de la Elipa.

Gracias a mi mejor amigo, Emilio. Porque me siento inmensamente afortunado y orgulloso de nuestra amistad. A mi lado ha estado en las etapas más importantes, en las más duras y en las más felices. Una parte esencial y fuente inagotable de disfrute, alegría y humor en mi vida.

A Anni. Por todo lo que hemos compartido juntos. Por sus tiernas palabras de aliento en momentos clave. Esta tesis se la dedico a ella, como muestra del lugar tan importante que siempre ocupará en mi corazón.

A mis padres, Chon y Antonio. Más que dedicársela a ellos, siento que esta tesis es también la suya. Insignificantes serían unas palabras de dedicatoria o agradecimiento para las personas que me lo han dado todo, siempre, en todo momento. Sólo espero que este logro y todo el esfuerzo, toda la satisfacción, toda la paz y toda la alegría por conseguirlo sirvan para expresar el cariño que siento hacia ellos. In conventional electronics, information is carried by electric charges, which are stored in capacitors and driven by the application of bias potentials. This technology relays on the electronic properties of semiconductor materials, where the spin of the electron basically plays no role. In contrast, magnetic recording utilizes the local orientation of spins in ferromagnets, -i.e., the local magnetization-for data storage and magnetic elds for information reading and writing.

The discovery of giant magnetoresistance in 1988 gave birth to spintronics: an emerging technology based on the combination of both charge transport and magnetization. In turn, spintronics has given rise to a fast and not volatile memory concept: the Magnetic Random Access Memory (MRAM), and more recently to the spin transfer torque MRAM (STT-MRAM). This new class of magnetic memory presents very interesting features such as non-volatility, large read and write endurance and fast read and write operations.

Although these characteristics point the MRAMs as potential candidates to replace a number of other memory technologies, the exclusive use of ferromagnetic (F) materials for their spin-dependent transport properties presents also some drawbacks. For instance, high storage density is limited by stray elds and high current densities are still required for writing operations due to high magnetic moments inherent to Fs. Since antiferromagnetic (AF) and ferrimagnetic (FI) materials exhibit vanishing or signicantly reduced magnetic moments and stray elds with respect to Fs, this work aims to address a systematic study of spin-dependent transport properties of both materials for their potential application in non-volatile magnetic memories.

Chapter I. Introduction

In this Chapter, I present a brief introduction to spintronics as the general framework in which the subject of this thesis is involved. Some of the main underlying spin-dependent transport phenomena of this ouring technology, namely giant magnetoresistance, tunneling magnetoresistance and spin transfer torque in currently used F nanostructures are shortly explained, since they will be investigated in the following chapters in the case of AF and FI nanomaterials. I included a concise description of the essential features characterising the three dierent magnetic orderings in the rst section of this introduction.

Next, I discuss the state-of-the-art in AF and FI-based spintronics. Finally, the general outline of the thesis is presented.

I.2 Magnetic ordering

The atomic magnetic moments in solid matter can be ordered in dierent ways (see Fig. I.1) depending on the interaction energy or exchange interaction between the spins S i and S j of neighbouring atoms i and j (Heisenberg Hamiltonian for N atoms, see IV.2.1 for more details):

e ex = N i,j -J i,j S i • S j
If the exchange integrals J i,j are positive, magnetic moments will align in the same direction and in the same sense, resulting in a net macroscopic magnetization: ferromagnetic order. If they are negative, magnetic moments will align in the same direction but in alternating senses, creating two dierent sublattices with opposite magnetic orientations, giving no net magnetization: antiferromagnetic order 1 ; if in addition the opposing moments of each sublattice are unequal, then a spontaneous magnetization appears: ferrimagnetic order. For these magnetic orderings to occur, the exchange interaction energy must overcome the thermal energy: ferromagnetism and ferrimagnetism orders appear below the Curie temperature T C , and antiferromagnetism appears below the Néel temperature T N .

1. More complicated antiferromagnetic congurations can also be found for spins arranged in a noncollinear way or in the case of frustrated systems where antiferromagnetic interactions can lead to multiple ground states with non-regular distributions of localized spins.

A very general magnetic ordering: ferrimagnetism As explained above, a FI material hosts two populations of atoms with opposing and unequal magnetic moments formed from elements having an unlled d or f electron shell, which results in a spontaneous magnetization. The two dierent populations consist of ions of dierent species or similar ions occupying crystallographically inequivalent sites.

A subsystem formed by all the magnetic sites in the crystal with identical magnetic behaviour (pointing in a single direction) is called a sublattice. In the case of FIs, the magnetic moments of ions of dierent sublattices are aligned antiparallel due to a negative exchange interaction. The spontaneous magnetization is equal to the vector sum of the magnetizations of the sublattices.

Ferrimagnetism might be viewed as one of the most general cases of magnetic ordering.

As such, ferromagnetism is a particular case of ferrimagnetism in which only one sublattice is present and antiferromagnetism is a limiting case in which the two sublattices consist of identical magnetic ions and the net magnetization is zero.

As in the case of antiferromagnetism, the FI ordering notion was introduced by L.

Néel in 1948. The term "ferrimagnetism", is derived from the word ferrite, which is the name of a large class of oxides of the transition elements in which the phenomenon was rst observed. Other examples of FI materials are: magnetic garnets, transition metals compounds such as MnGa, MnCoAl, CrMnSb, and the archetypal magnet, lodestone, which is a naturally magnetized piece of the mineral magnetite (Fe 3 O 4 ), the oldest known magnetic material. The rst magnetic compasses were made out of suspended pieces of lodestone, which in Middle English means 'course stone' or 'leading stone', indicating their importance to early navigation.

Note that the ideal picture of staggered AF and FI has to be nuanced in realistic systems where magnetic interaction frustrations occur due for instance to interfacial roughness, structural defects, peculiar 3-D AF and FI spin structures, interdiusion of species and grain boundaries for polycrystals.

I.3 Conventional ferromagnetic-based spintronics

Conventional electronics has exploited until recently only the charge of the electron for technological applications. Spintronics (spin electronics) makes reference to a new technology emerged in the 1980s that takes also advantage of the electron spin to carry information. It paves the way for new revolutionary devices with spin-dependent eects arising from the interaction between itinerant spins and the magnetic properties of solid state materials. It has been used in a number of applications and has allowed for instance to strongly increase data storage capability. In comparison to conventional electronics, this new technology oers performance and additional functionalities for electronic devices.

Chapter I. Introduction

The possibility to build magnetic multilayers with individual thicknesses comparable to the electron spin diusion lengths and below, so that spin-dependent transport eects could be observed, lead in 1988 to the discovery of the Giant Magnetoresistance (GMR) eect, which is considered the beginning of the new spin-based electronics.

Giant magnetoresistance

Giant magnetoresistance (GMR) was rst observed by the groups of Albert Fert and Peter Grünberg [START_REF] Baibich | Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices[END_REF], [START_REF] Binasch | Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange[END_REF], in thin-lm structures composed of alternating F and non-magnetic (NM) conductive layers of several nanometers (thicknesses comparable to the electron mean free path). For current perpendicular to plane, the eect consists in a signicant change in the resistance for an electric current owing perpendicular to the plane of the layers when the relative magnetic conguration of the F layers switches from antiparallel to parallel (see Fig. [START_REF] Chappert | The emergence of spin electronics in data storage[END_REF] and deniton of the GMR ratio.

As depicted in Fig. I.2, the GMR ratio is dened as the dierence between the resistances in the antiparallel and the parallel states divided by the resistance of the parallel state (or divided by the resistance of the antiparallel state, depending on the convention). It strongly depends on temperature and thicknesses of the F and NM layers.

In addition to the spin-dependent reections at the interfaces, the eect can be interpreted in terms of the spin-dependent density of states (DOS) of the F electrodes for the itinerant electrons at the Fermi energy. Electronic states are spin split in the Fs, which Chapter I. Introduction leads to unequal DOS for up and down spins at the Fermi energy. In turn, the DOS at the Fermi level determines the conductance of each spin channel, i.e., up (down) spins will be scattered more likely in a F with majority down(up) electrons. For instance when no spin-ip scattering processes are considered, the bulk conductivity of spin-up and down carriers at the Fermi level is given by:

σ ↑(↓) = e 2 DOS ↑(↓) (E F )D ↑(↓)
where D ↑(↓) is the spin-dependent spin diusion constant. When an electric eld is applied to such a material, a ow of a spin-polarized current appears, with a polarization dened by the conductivities:

P = J ↑ -J ↓ J ↑ + J ↓ = σ ↑ -σ ↓ σ ↑ + σ ↓
where the J ↑(↓) denote the spin-resolved charge current densities.

Thus, when no spin-ip processes are considered, the overall resistance will depend on the magnetic orientation of the F leads.

Within three years after the discovery of GMR, the "spin valve" concept had been introduced. A spin valve is a GMR-based device with two ferromagnetic layers (alloys of nickel, iron, and cobalt) sandwiching a thin nonmagnetic metal spacer (usually copper).

The resistance of the spin valve increases typically from 5 to 10 % when the relative orientation of the magnetizations of the two layers switches from parallel to antiparallel alignment [START_REF] Wolf | Spintronics: A spin-based electronics vision for the future[END_REF]. One of the two magnetic layers is called the "free layer", since its magnetization can be easily reoriented by relatively low magnetic elds. The other magnetic layer is called the "reference layer" or "pinned" layer, since its magnetization is relatively insensitive to moderate magnetic elds. Pinning is usually accomplished by using an antiferromagnetic layer in contact with the pinned F layer, which yields to a magnetic interfacial interaction called exchange bias 2 .

Since the magnetization of the free layer can however be easily reoriented by relatively low magnetic elds, the spin-valve itself can probe the orientation of a small magnetic eld and act as a magnetic sensor through the GMR eect. This sensor can be used in a magnetic-storage read head to detect the small magnetic eld coming from an encoded bit on magnetic media (such as in a hard disk drive, HDD), as illustrated in Fig. I.3. The spin-valve sensor was introduced in 1997 by IBM to replace anisotropic magnetoresistance 2. Since the AF exhibits small or no net magnetization, the orientation of its spins is almost not aected by external magnetic elds, and it will pin the interfacial spins of the adjacent exchange-coupled F. The switching eld of the F is signicantly increased, and the hysteresis loop of the F lm is shifted away from the H=0 axis by the exchange bias eld H B , which amounts to the exchange interaction energy of spins at the F/AF interface. The exchange bias eld can be considered to be proportional to the scalar product of F and AF interfacial spins: H B ∝ S F • S AF . A change in the direction of AF spins (due e.g. to the eects of STT) will therefore show as a hysteresis loop shift. for hard-disk recording, adapted from Ref [START_REF] Chappert | The emergence of spin electronics in data storage[END_REF]. The inductive head creates a magnetic eld for writing on the recording medium.

The HDD areal recording density rapidly increased by three orders of magnitude (from 0.1 to 100 Gbit/ in 2 ) between 1991 and 2003 [START_REF] Chappert | The emergence of spin electronics in data storage[END_REF]. It gave birth to a large commercial sector within this eld of spintronics, with sales exceeding $3 billion in 2005.

Shortly after the discovery of the GMR eect, when considerable progress had been made in deposition and nanopatterning techniques, it was found that replacing the nonmagnetic metallic spacer with an insulating tunnel barrier of a few nanometers or less could result in large values of the magnetoresistance (MR) at room temperature. This eect is referred to as "tunneling MR" (TMR). The whole structure is called magnetic tunnel junction (MTJ).

Tunnelling magnetoresistance

TMR ratio is dened analogously to GMR, although the dierence in the electric resistances for parallel and antiparallel states in MTJs is signicantly enhanced. This makes MTJs particularly suitable for technological applications.

Some of the most important properties of a MTJ is that the tunnelling current depends on the applied voltage and the magnetic orientation of the two F leads, which can also be tuned by an external eld. In contrast to spin valves, the barrier plays a fundamental role as a spin lter, and TMR ratios strongly depend on the quality of the barrier and the interfaces with the leads.

Chapter I. Introduction

In the case of a MTJ, the tunnelling conductance depends not only on the DOS at the Fermi energy, but also on the tunnelling probability, which is dierent for various electronic states in the F. In other words, the tunnel barrier also has a ltering eect, i.e. electrons belonging to dierent energy bands are ltered dierently by the barrier.

However, nearly free dispersive s-bands which are hybridized with more localized d-bands in 3-d Fs can be assumed to provide all the tunnelling current [START_REF] Tsymbal | Spindependent tunnelling in magnetic tunnel junctions[END_REF].

The probability for an electron to tunnel through the barrier depends on its Fermi wave vector but the dispersive bands that dominate tunnelling are similar to free electron bands, and therefore the DOS of these bands is in turn proportional to their Fermi wave vector. Electronic bands are spin-split in F metals, which implies dierent wave vectors for up and down spin electrons and consequently a tunnelling probability that depends on spin. Hence, as in the case of GMR, TMR might also be explained in terms of the F electrodes' DOS.

For the situation depicted in Current is essentially carried by the majority channel and the electrical resistance is low. In the AP conguration, majority(minority) electrons in the left lead tunnel to the minority(majority) band in the right lead, where the DOS at the Fermi energy is low(high).

Both spin channels contribute similarly to the current. The overall conductance is however lower than in the parallel state since the DOS for both spins is low either in the left or in the right lead.

Alternatively, spin valves and MTJ can be used as a magnetic bit, with the twomemory states corresponding to antiparallel and parallel congurations. Arrays of pat-Chapter I. Introduction terned spin valves or MTJ can be used to store binary information with resistive read-out, creating a magnetic solid state memory. This is the basis for MRAMs.

The design principle of the rst MRAM generation is illustrated in Fig [ [START_REF] Prejbeanu | Thermally assisted MRAMs: ultimate scalability and logic functionalities[END_REF]. The working principle is explained in the text.

The free layer of the magnetic element to write can be reversed by the Oersted elds generated by current pulses sent through the two perpendicular lines having that magnetic element in common. There is the only point where the resulting eld is high enough for the writing process. The magnetic state of an addressed cell can be read by measuring the resistance between the same two lines. The potential main advantages with respect to electric-based semiconductor memories are non-volatility, lower energy for writing and much faster write times. The main problem is the scalability due to the use of a magnetic eld to write information. On the one hand, non-volatility and cell volume reduction impose high anisotropy constants (see IV.2.1), since the thermal excitations energy must be lower than the magnetic stability energy barrier given by KV , where K is the anisotropy constant and V is the volume of the magnetic element. In other words, the only way to increase the product KV while reducing the size of the magnet (V ) is by increasing the anisotropy constant K. On the other hand the required writing eld and consequently the required current increases proportionally with K. However, the reduced dimensions of the conducting lines restricts considerably the available current density due to electromigration [START_REF] Chappert | The emergence of spin electronics in data storage[END_REF], that consists in the diusion of metal atoms in the conductor due to momentum transfer from conducting electrons. In addition, smaller devices imply lower available power.

Last, but not least, the unavoidable spatial extension of the writing magnetic elds prevents the magnetic cells to be densely packed, since the writing process of one of them might also alter the magnetic state of its neighbouring magnetic elements, leading to undesired encoding errors.

Chapter I. Introduction

The prediction in 1996 that a spin-polarized current directly traversing each magnetic cell could be used instead of external magnetic elds for switching the orientation of the free layer represented a promising improvement to solve the mentioned scalability problems. This eect was called spin transfer torque (STT).

Spin transfer torque

Spin transfer torque in tunnel junctions and spin valves with ferromagnetic leads is one of the essential underlying phenomena of modern spintronics by which the magnetic order of a ferromagnetic (F) thin lm can be reoriented by the transfer of angular momentum from a spin-polarized current.

If the relative orientation of the magnetic layers in a spin valve or a magnetic tunnel junction has a strong impact on the current density showing as GMR or TMR eect, the spin transfer torque (STT) can be considered as the converse eect. In eect, a suciently large current density might change the relative orientation of the magnetic layers, or even reverse the magnetization direction of one of them, resulting in magnetization switching by a spin polarized current. That is, the magnetic order of a F thin lm can be reoriented by the transfer of angular momentum from a spin-polarized current coming from a not collinear F layer. This eect was theoretically predicted by Slonczewski and Berger [Slonczewski, 1995], [Berger, 1996], and has been since then the object of extensive investigations due to its applications in spintronics devices [START_REF] Wolf | Spintronics: A spin-based electronics vision for the future[END_REF]. It was experimentally shown by the groups of M. Tsoi and Ralph-Buhrman, who observed variations in the resistance of dierent F multilayers when high current densities between 10 7 and 10 8 A/cm 2 were applied [START_REF] Tsoi | Excitation of a magnetic multilayer by an electric current[END_REF]], [START_REF] Myers | Current-induced switching of domains in magnetic multilayer devices[END_REF]. The observation of STT in low-resistance MTJ was later observed using submicron-sized pillars [START_REF] Huai | Observation of spin-transfer switching in deep submicron-sized and low-resistance magnetic tunnel junctions[END_REF] for a critical current of about 8 × 10 6 A/cm 2 . When a current is applied, conducting s electrons owing through F (considered as the reference layer) become spin polarized due to the the sd exchange interaction along the direction of the magnetization M. In traversing the second F' (considered as a free layer), the injected electrons are then polarized in the direction M'. Hence, this magnetization M' exerts a torque on the spin angular momentum of the itinerant electrons and vice versa, spin angular momentum from the itinerant electrons is transferred to the localized electrons of F', responsible for the global magnetization M'. In other words, by conservation of angular momentum, the spin current exerts an equal and opposite torque on the magnetization M'. Therefore, there is a transfer of spin angular momentum from the itinerant electrons to the magnetization, and that is why it is called spin transfer.

For a suciently large current density, or critical current, the magnetization M' can completely reverse and switch from antiparallel to parallel state (and vice versa, changing the polarity of the current). Microwave precession of M' is also possible, when more complex dynamical modes are excited by current.

In such F materials, the loss of transverse spin momentum occurs over a very short distance (around 1 nm), so that STT is an interfacial eect, more ecient on a thin layer.

The amplitude of the torque per unit area is proportional to the injected current density, so that the switching current decreases proportionally to the cross-sectional area of the structure. With today's advances in nanofabrication techniques, lateral sizes of around 100 nm can be easily achieved, this represents an important advantage of spin transfer over eld-induced switching.

The spin transfer torque depends on the applied current density J and the relative orientation of the two magnetizations. It can be decomposed into one component in the plane of the layers, T , and another component out of the plane of the layers, T ⊥ . These two components read:

T = T (J, θ)M × (M × M ) T ⊥ = T ⊥ (J, θ)M × M (I.1)
The in-plane torque T with amplitude T (J, θ) is usually called "spin transfer torque", although this term will not be used here to denote only this component. The out-of-plane torque T ⊥ with amplitude T ⊥ (J, θ) is commonly called eld-like torque, since it has the same form as the torque exerted on M' due to an external magnetic eld "M". These two components are illustrated in Fig. as the cross sectional area of the magnetic cell is reduced, the current required to switch decreases proportionally, since switching occurs at a certain threshold current density.

This recent MRAM concept is called STT-MRAM.

Problematic

Despite the fact that today's nanofabrication techniques can achieve the production of magnetic nanopillars of diameter below 100 nm, the threshold current density for magnetization switching still remains very high, around 10 7 A/cm 2 [ [START_REF] Ikeda | Magnetic tunnel junctions for spintronic memories and beyond. Electron Devices[END_REF].

Although it has been signicantly reduced since the development of MTJ based on MgO barriers, and perpendicular anisotropy structures, the high magnetization together with the interfacial character of STT inherent to F materials are serious limitations for the threshold current density reduction. Moreover, the large currents necessary for STT switching are detrimental to the barrier quality, which implies a loss of endurance with respect to magnetic eld writing. In addition, the unavoidable presence of stray elds created by F thin layers represents an obstacle for high information storage density; similar to the eld-induced writing scheme, large stray elds couple in densely packed arrays of magnetic cells, leading to likely encoding errors in integrated circuits.

As discussed below, the use of antiferro and ferrimagnetic materials appears naturally as a potential solution to these issues.

I.4 Antiferromagnetic spintronics: state-of-the-art

In the eld of spintronics, the spin-dependent transport properties of Fs lie at the heart of devices working principles, hence the terminology F-spintronics. By way of contrast, AFs like IrMn, FeMn, NiMn, have been used so far mostly for their magnetic properties: as explained above, they pin the magnetization of an adjacent F via exchange bias in order to set the reference direction required for the spin of conduction electrons in spintronics devices [START_REF] Nogués | Exchange bias[END_REF], [START_REF] Baltz | Bimodal distribution of blocking temperature in exchange-biased ferromagnetic/antiferromagnetic bilayers[END_REF]. However, AF-spintronics, i.e.

spin dependent transport with AF, is now in its infancy [START_REF] Basset | Towards antiferromagnetic metal spintronics[END_REF],

[MacDonald and Tsoi, 2011] and is identied as a signicant exploratory topic in spintronics [Duine, 2011], [START_REF] Sinova | New moves of the spintronics tango[END_REF], [START_REF] Brataas | Current-induced torques in magnetic materials[END_REF]. In particular, AFs Over the last few years, a growing number of studies have considered both theoretical and device aspects of AF-spintronics [Duine, 2011], [START_REF] Sinova | New moves of the spintronics tango[END_REF], [START_REF] Brataas | Current-induced torques in magnetic materials[END_REF], [START_REF] Saidaoui | Spin transfer torque in antiferromagnetic spin valves: From clean to disordered regimes[END_REF], including current-induced AF magnetic resonance for radio-frequency components [START_REF] Gomonay | Symmetry and the macroscopic dynamics of antiferromagnetic materials in the presence of spin-polarized current[END_REF], AF domain wall motion [START_REF] Manchon | Spin-dependent diraction at ferromagnetic/spin spiral interface[END_REF], [START_REF] Logan | Antiferromagnetic domain wall engineering in chromium lms[END_REF], [START_REF] Swaving | Current induced torques in continuous antiferromagnetic textures[END_REF], [START_REF] Wieser | Indirect control of antiferromagnetic domain walls with spin current[END_REF], and tunnel anisotropic magnetoresistance with AFs for memories and logic devices [START_REF] Park | A spin-valve-like magnetoresistance of an antiferromagnet-based tunnel junction[END_REF], [START_REF] Petti | Storing magnetic information in IrMn/MgO/Ta tunnel junctions via eld-cooling[END_REF].

In particular, a rst theoretical toy model showed AF STT and GMR for metallic AF/PM/AF spin-valve-like multilayers [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF], where PM stands for a paramagnetic metallic spacer. The authors considered ideal crystalline uncompensated F monolayers with staggered AF order (i.e. two alternating F sublattices with opposite magnetizations). Owning to such alternating moment orientations, commensurate staggered torques occur generically. Furthermore, unlike the pioneering theoretical works on STT in F multilayers [Slonczewski, 1995], [Berger, 1996], which predict torques exerted by a spin polarized current close to the interface between a F and a nonmagnetic metal, STT is expected to act cooperatively through the entire volume of the AF electrodes. This feature together with the absence of shape anisotropy in AFs explain that smaller critical currents for local magnetization switching are also predicted for perfect epitaxial AFs compared to the typical values for Fs. The rst theoretical model was soon followed by experimental evidence of AF-STT with currents injected in F/AF polycrystalline bilayers [START_REF] Wei | Changing exchange bias in spin valves with an electric current[END_REF].

Ab-initio calculations based on nonequilibrium Green functions and spin-density functional theory further conrmed GMR and STT eects in AF elements for AF/NM/AF spin valve-like multilayers [START_REF] Haney | Ab initio giant magnetoresistance and current-induced torques in Cr/Au/Cr multilayers[END_REF].

Later theoretical studies have focused on the eects of disorder on STT and GMR in AF spin valves, taking into account inelastic [START_REF] Duine | Inelastic scattering in ferromagnetic and antiferromagnetic spin valves[END_REF] and elastic electron scattering [START_REF] Saidaoui | Spin transfer torque in antiferromagnetic spin valves: From clean to disordered regimes[END_REF]. While the latter shows without ambiguity that elastic Chapter I. Introduction scattering is detrimental to STT in AFs, the former predicts that in contrast to F spin valves, inelastic scattering increases the STT eciency. Authors of [START_REF] Saidaoui | Spin transfer torque in antiferromagnetic spin valves: From clean to disordered regimes[END_REF] propose the use tunnelling junctions instead of spin valves to ensure momentum conservation and get rid of the undesirable eects of elastic scattering on STT.

The diculty to pin the order parameter of an AF element along a reference direction together with the dwarfed magnetoresistance observed in AF/PM/AF multilayers [START_REF] Wang | Magnetoresistance eect in antiferromagnet/nonmagnet/antiferromagnet multilayers[END_REF] makes the experimental observation of STT in AF a technological challenge. Indirect mechanisms such as the study of exchange bias variations at F/AF interfaces due to the eect of a spin polarized current on the spin orientation of the AF interfacial layer [START_REF] Wei | Changing exchange bias in spin valves with an electric current[END_REF], [START_REF] Urazhdin | Eect of polarized current on the magnetic state of an antiferromagnet[END_REF] have been used to study STT in AFs.

Tunnel anisotropic magnetoresistance has become also a promising eect to detect the orientation of the order parameter in AF elements since its observation in exchangecoupled F/AF bilayers [START_REF] Martí | Electrical measurement of antiferromagnetic moments in exchange-coupled IrMn/NiFe stacks[END_REF] and IrMn-based tunnel junctions [START_REF] Park | A spin-valve-like magnetoresistance of an antiferromagnet-based tunnel junction[END_REF]. Tunnel anisotropic magnetoresistance stands for a strong dependence of the resistance of a M/B/NM stack on the relative orientation of magnetic moments in the magnetic layer M and the crystalline anisotropy axes via spin-orbit interactions. The amount of current tunnelling perpendicularly across the junction is dened by the density of states (DOS) of M at the Fermi energy, which depends in turn on the order parameter orientation with respect to the crystalline anisotropy axes. In particular, the authors of [START_REF] Park | A spin-valve-like magnetoresistance of an antiferromagnet-based tunnel junction[END_REF] observed more than 100% spin valve-like signal for a F/AF/B/NM stack with an AF at one side of the tunnel barrier and a non magnet at the other side.

By using the exchange spring eect [START_REF] Scholl | Creation of an antiferromagnetic exchange spring[END_REF], AF moments were rotated by the reversal of the F moments via external magnetic elds, which was detected by the measured tunnel anisotropic magnetoresistance.

Furthermore, the use of the auxiliary F layer was avoided by the authors of Ref [START_REF] Petti | Storing magnetic information in IrMn/MgO/Ta tunnel junctions via eld-cooling[END_REF], who detected distinct metastable resistance states by eld cooling NM/B/AF tunnel junctions. The work of Marti et al. [START_REF] Martí | Room-temperature antiferromagnetic memory resistor[END_REF] provides in addition demonstration of electrically readable magnetic spintronics memory devices which contains no F elements and which stores the information in the AF at room temperature.

I.5 Ferrimagnetic spintronics: state-of-the-art

Ferrimagnetic materials are now used in most permanent magnets, recording media and microwave-oriented magnetic materials. Although they are not signicantly exploited for their transport properties, they also present a good potential for spintronics devices such as MTJs due to their low magnetization and high polarization, as described below.

Chapter I. Introduction

Micro-wave applications

The most known FI material for micro-wave applications is YIG (Ytrium iron garnet).

It is a synthetic garnet FI insulator of chemical composition Y 3 Fe 2 (FeO 4 ) 3 that is used in microwave, optical and magneto-optical applications due to its low propagation loss properties [START_REF] Boona | Magnon thermal mean free path in yttrium iron garnet[END_REF], [START_REF] Odashima | Chirality-dependent magnon lifetime in a compensated halfmetallic ferrimagnet[END_REF]. In particular, YIG is highly attractive for ferrimagnetic resonance and spin pumping investigations (see IV. -4 [Hahn et al., 2014], and insulating nature [START_REF] Wang | Spin current and inverse spin hall eect in ferromagnetic metals probed by Y 3 Fe 5 O 12 -based spin pumping[END_REF]. It has been recently used to show a large inverse spin Hall eect in the AF metal Ir 20 Mn 80 [Mendes et al., 2014], which is investigated in Chapter IV.

Synthetic-ferrimagnets

The eects of STT in "synthetic-ferrimagnet" layers have been investigated in spin valves [START_REF] Smith | Coresonant enhancement of spin-torque critical currents in spin valves with a syntheticferrimagnet free layer[END_REF] and MTJ [START_REF] Cornelissen | Free layer versus synthetic ferrimagnet layer auto-oscillations in nanopillars processed from MgO-based magnetic tunnel junctions[END_REF], [START_REF] You | Magnetic noise spectra and spin transfer torque of a magnetic tunnel junction with an exchange biased synthetic ferrimagnetic reference layer[END_REF]. Syntheticferrimagnets are structures of the form F1/NM/F2, where F1 and F2 are two ferromagnetic lms of dierent thicknesses separated by a nonmagnetic metallic spacer promoting strong antiparallel coupling between the two F layers. Synthetic-ferrimagnets are expected to provide a high volume to withstand thermal uctuations while keeping the eective magnetic moment per area low [START_REF] Ikeda | Magnetic tunnel junctions for spintronic memories and beyond. Electron Devices[END_REF], [START_REF] Sousa | Synthetic ferrimagnet free layer tunnel junction for magnetic random access memories[END_REF], [START_REF] Inomata | Magnetic switching eld and giant magnetoresistance eect of multilayers with synthetic antiferromagnet free layers[END_REF], [START_REF] Tezuka | Single domain observation for synthetic antiferromagnetically coupled bits with low aspect ratios[END_REF], [Tezuka et al., 2003a].

Half-metallic ferrimagnets

The performance of spintronics devices is directly related to the spin polarization by the magnetic electrodes of spin valves and MTJ [START_REF] Inomata | Highly spin-polarized materials and devices for spintronics[END_REF]. A small magnetization combined with high spin polarization would considerably enhance the eects of STT for spintronics applications [START_REF] Klaer | Disentangling the Mn moments on dierent sublattices in the half-metallic ferrimagnet Mn 3-x Co x Ga[END_REF]. This advantageous combination can be achieved by the concept of half-metallic ferrimagnetism. Half-metals have a metallic resolved DOS for the half-metallic Ti 2 CoAl ferrimagnet calculated in [Bayar et al., 2011] using density functional calculations.

a main group element [START_REF] Bayar | Half-metallic ferrimagnetism in the Ti 2 CoAl heusler compound[END_REF]. In particular, Co-based Heusler alloys are promising materials for spintronics applications and have been increasingly investigated [START_REF] Inomata | Highly spin-polarized materials and devices for spintronics[END_REF]. These Heusler alloys can show both F order, such as Co 2 MnAl, Co 2 MnSi, Co 2 MnGe, Co 2 MnSe, and FI order, such as Mn 3-x Co x Ga or Ti 2 CoAl (see Fig.

I.9 (b)). The latter have been pointed out as promising materials for highly ecient STT devices in future spintronics applications [START_REF] Klaer | Disentangling the Mn moments on dierent sublattices in the half-metallic ferrimagnet Mn 3-x Co x Ga[END_REF], [START_REF] Bayar | Half-metallic ferrimagnetism in the Ti 2 CoAl heusler compound[END_REF]. Interestingly, the nearly half-metallic FI Mn 3 Ga, with a 88% spin polarization at the Fermi energy has also been explicitly identied as a suitable material for STT applications due to its high spin polarization, high Curie temperature and a low magnetic moment [START_REF] Balke | Mn 3 Ga, a compensated ferrimagnet with high curie temperature and low magnetic moment for spin torque transfer applications[END_REF].

Chapter I. Introduction

I.6 Outline of the thesis

The presence of undesirable stray elds and the need of very high current densities for current-induced magnetization switching being the main drawbacks of ferromagnets for spintronics applications, this work aims to address a comprehensive study of alternative materials with two dierent magnetic orders, namely antiferromagnets and ferrimagnets.

The vanishing magnetization of AFs implies the absence of surrounding stray elds, whereas the partially compensated magnetic structure of FIs makes these stray elds to be signicantly reduced. In addition, lower demagnetizing elds imply lower critical currents for STT-related magnetization reversal.

Current-induced magnetization switching in thin F layers is favoured by spin transfer torques acting through all the F volume to reorient homogeneously the localized magnetic moments. However, very high current densities are required to produce STT over long characteristic lengths in Fs. In contrast, as explained above, STT is expected to act on a much longer length scale in AFs. Longer length scales are also likely to be observed in FIs due to the similarities between the magnetic structure of AFs and FIs. The spatial distribution of STT is therefore of fundamental interest, and is investigated thoroughly in this thesis.

Lower critical currents for STT switching together with the absence (or reduction) of stray elds in AF and FI would eventually lead to lower device power consumption and ultimate downsize scalability.

This work includes a theoretical investigation of STT and TMR in AF and FI-based magnetic tunnel junctions and an experimental study of STT characteristic lengths in AFs.

In Chapter II the tight binding (TB) model and the nonequilibrium Keldysh formalism used here for the calculations on spin-dependent transport in magnetic tunnel junctions are described. The main outcome of this chapter is the derivation of an analytical expression for the retarded Green function for AF and FI leads, which is required for all the following calculations. Density of states (DOS) and local density of states (LDOS) for the AF and FI leads are calculated from the retarded Green function and next examined through their energy dependence. Their dispersion relations are also computed, which will be useful for the following chapter. A new method to extend the mentioned calculations to fully compensated AF is nally proposed.

Chapter III reports the theoretical results obtained using the theory developed in

Chapter II for 1-D and 3-D geometries. The rst part is focused on the spatial distribution of STT in AF tunnel junctions (AF-MTJ). The important novelty with respect to previous works is the use of a tunnelling barrier instead of a metallic spacer. STT is expected to be more robust face to disorder in MTJ, and TMR ratios are considerably higher than the equivalent GMR in spin valve structures.

The dependence of STT and TMR on the applied voltage, magnetic and electronic characteristics of the AF leads and the barrier properties are discussed.

The second part presents the rst theoretical work on STT in FIs. In particular, the inuence of the magnetic and electronic properties of the FI on the spatial distribution of torques within the FI leads is determined quantitatively. As in the rst part of the Chapter, the eects of the applied voltage on STT and TMR are also analysed. Due to the similar spatial behaviour of STT in FIs and fully compensated

AFs, a preliminary study on the later is presented at the end of the Chapter.

Since the eects of disorder were not taken into account in the theoretical calcu- The conclusions of this work as well as the perspectives for further research are included in Chapter V.

In addition to the denitions given along the text, a list of acronyms and abbreviations can be found in the glossary provided on page 137.

Appendix A recalls the computation of the nonequilibrium Keldysh function, necessary to obtain the density operator or lesser Green's function.

The calculation of the analytical expression for the retarded Green's function for AF and FI leads is described thoroughly in Appendix B, starting from the resolvent of a Bethe lattice.

Finally, Appendix C reports a preliminary experimental investigation on the eect of a spin polarized current on AF nanostructures.

Chapter II Quantum transport in magnetic tunnel junctions with tight binding models: nonequilibrium Keldysh formalism

In this chapter we will derive all the theoretical tools needed to investigate spin dependent transport properties of a magnetic tunnel junction with two dierent magnetic orders: AF and FI. We shall examine a steady state regime, but in a nonequilibrium state, since a bias voltage is considered to be applied across the junction, leading to dierent chemical potentials in each lead of the junction. Tight binding (TB) models and the nonequilibrium Keldysh formalism are particularly adapted in this case, and they will be used throughout this chapter to obtain the results presented in the next one. This latter formalism is an extension of the theory developed by Caroli et al. to calculate the tunneling current in a non magnetic tunnel junction [START_REF] Caroli | Direct calculation of the tunneling current[END_REF], and it is briey described at the end of this Chapter. Even for the case of non-interacting electrons, it has proven to be more convenient to calculate physical quantities in conventional F-based tunnel junctions, compared to Landauer or Kubo-like approaches as demonstrated in previous works [START_REF] Kalitsov | Spin-transfer torque in magnetic tunnel junctions[END_REF], [START_REF] Theodonis | Anomalous bias dependence of spin torque in magnetic tunnel junctions[END_REF], [START_REF] Kalitsov | Spin-polarized current-induced torque in magnetic tunnel junctions[END_REF], [START_REF] Edwards | Self-consistent theory of current-induced switching of magnetization[END_REF]. In addition, as discussed in the following sections, the spatial distributions of torques and spin densities are needed here, taking into account the specic local magnetic distribution of the leads, which cannot be treated by simple energy band approach to a tunneling contact.

I will thus present a method to study spin dependent transport properties in AF and FI tunnel junctions, whose magnetic order is much more complex than that of the extensively studied conventional F-based tunnel junctions. for the most general case of FI-ordered leads. F and AF leads will be also studied, but they can be considered as limiting cases of the more complex FI order. The thickness of the insulating layer (taken here of a few atomic layers) allows the electrons to tunnel through the barrier when a voltage is applied between two metallic leads. The relative orientation of the order parameters of the two magnetic layers has a strong impact on the most important properties of the magnetic tunnel junction depending on spin dependent transport, namely the TMR and STT. In the case of F and FI leads this relative orientation can be changed via an applied magnetic eld.

The current-carrying electrons responsible for the spin-dependent transport properties of the tunnel junction are assumed to be dispersive nearly free s-like electrons (characteristic of the 3d magnetic materials' electronic structures). Some properties of the tunnel junction such as the tunnelling conductance depend on the tunnelling probability of the dierent electronic states in the magnetic leads, however, as pointed out by Stearns [Stearns, 1977], most of the tunnelling current is carried by dispersive bands, since d electrons decay much faster than s electrons into the barrier region due to their large eective mass. Moreover, in the case of bulk Ni in [110] direction, the dispersive s band of majority spins is the only one that crosses the Fermi level. More realistic electronic structures of the insulator and the barrier interfaces are also disregarded here, thus, one of the essential assumptions here is that transport properties are mainly determined by dispersive s electrons. The spin of the electron is assumed to be conserved throughout the entire junction.

Therefore transport occurs through two independent spin channels. It follows that electrons coming from one lead in a given spin state are accepted by the unlled states of the same spin in the other lead. In addition, when a voltage is applied across the junction, nonequilibrium electrons from the occupied states below the Fermi level in the left lead tunnel to the empty states of energy higher than the Fermi level in the right lead (lowered by the bias-dependent chemical potential) [START_REF] Tsymbal | Spindependent tunnelling in magnetic tunnel junctions[END_REF]. These electrons stay as hot electrons in the right lead, since energy losses (or quantum dephasing) due to inelastic scattering processes are neglected here (the emission of a magnon combines for instance spin ip and energy loss of the electron).

The cross section of typical tunnel junctions used in spintronics applications is large in comparison with the longitudinal dimensions (thickness) of the magnetic layer stack.

Thus, boundary conditions on each cross section are expected to have little impact on the Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism tunnel junction properties. From a computational point of view, the calculation of these properties is greatly simplied if the quantum connement on each section is completely ignored. Cross sections are thus commonly assumed to be eectively innite, and periodic boundary conditions are applied in all the theoretical calculations.

Due to the fabrication process, actual tunnel junctions contain however a certain amount of structural disorder in both leads, in the barrier and at the barrier/leads interfaces, especially in amorphous barriers. This disorder might be present in the form of impurities, interstitial or vacancy defects, interdiusion at the interfaces, interface

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism roughness, grain boundaries, stacking faults, etc. Collective excitations such as phonons (crystal lattice atoms) and magnons (lattice spin structure) contribute also to disorder in a "dynamic" way. This disorder is responsible for the elastic and inelastic scattering processes experienced by electrons in their motion through the tunnel junction. One of the main consequences of these scattering processes is the intermixing between transport modes with dierent transverse momenta k in the plane perpendicular to transport and consequently the loss of coherence in the tunnelling process. However, since ballistic transport is considered here, all these scattering processes are neglected. Invoking the narrow layers' thickness of typical tunnel junction stacks, the electron mean free path (which is of the order of 50 monolayers in typical AF metals [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF]) is then assumed to be longer than the typical longitudinal dimensions of the junction. Calculations are carried out in the "clean limit" [START_REF] Saidaoui | Spin transfer torque in antiferromagnetic spin valves: From clean to disordered regimes[END_REF], that is in the absence of electron momentum scattering by defects or other types of disorder. Thus, the structural quality of the junction could be decisive to observe the quantum eects reported in the next chapter, which are intended to be robust in epitaxially-grown tunnel junctions. Although the spin relaxation processes as well as momentum scattering with impurities are expected to weaken STT and TMR in AF-based tunnel junctions, the transport calculation model is expected to capture the essential features of spin dependent transport in AF-based tunnel junctions since this approach was successful to predict and explain STT and TMR properties in traditional F-MTJs (see for instance, [START_REF] Theodonis | Anomalous bias dependence of spin torque in magnetic tunnel junctions[END_REF] and [START_REF] Kalitsov | Spin-transfer torque in magnetic tunnel junctions[END_REF]). The precise eect of disorder is nevertheless out of the scope of this work, although it will be discussed as future research.

II.1.2 One-particle basis

The absence of defects together with the periodic boundary conditions lead to the translational invariance of the tunnel junction in the plane parallel to the barrier/leads interfaces. It is then useful to split the Hamiltonian of the system into a longitudinal part depending exclusively on the y direction of transport and a transversal part depending on x and z directions [Datta, 2000]:

Ĥ = ĤLNG (y) + ĤTRV (x, z)
ĤTRV |k = k |k
Here k is the energy of the plane wave or Bloch state, or in other words, the dispersion relation. The choice of the particular periodic distribution on each plane will determine the expression of k , and it will be treated in a subsequent section.

Longitudinal component of the Hamiltonian: discrete atomic-like basis

For the non-periodic longitudinal part of the Hamiltonian, a typical TB discrete basis of localized orbitals is used. These localized orbitals are assumed to be spatially conned, that is, only the overlap between nearest neighbours is not negligible.

With the above considerations, the Hilbert one-particle space consists of mix states |p, k , σ , where p denotes the site index or position in real space of the electronic orbital and σ stands for the spin state. This is a useful choice, since as pointed out by Datta [Datta, 2000], the matrix elements of the entire Hamiltonian in this basis are considerably simplied:

p, k , σ| Ĥ |q, k , σ = p, k , σ| ĤLNG |q, k , σ + p, k , σ| ĤTRV |q, k , σ = ( p, σ| ĤLNG |q, σ + k )δ σ,σ δ k ,k
As described above, two important assumptions are made here:

1) No spin ip processes connecting dierent spins σ and σ are present (however ĤL is spin-dependent).

2) Elastic and inelastic scattering processes connecting the two transverse modes (or reciprocal lattice vectosr) k and k are neglected. Coherent transport is assumed.

As a consequence of assumption 2) each k Bloch state can be treated as an independent transport channel; the individual k contributions to the system transport properties are thus considered independently: each transverse mode k has an extra energy of k that is added up to the longitudinal energy whenever the total energy of the electron is used. Physical quantities will then depend on Ek , and the contributions of all the transport channels are considered through a k integration of (sum over) all the k states in the rst Brillouin zone. The momentum conservation in the plane of layers thus requires that no scattering other than the specular scattering at perfect interfaces takes place. Transport across the whole junction is therefore coherent. From an experimental point of view, this ballistic approach is only applicable to perfectly at interfaces, that is, to ideal epitaxially grown magnetic tunnel junctions. For rough electrode/barrier Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism interfaces or scattering by defects there is no reciprocal lattice vector conservation.

k + σ λ ĉ † λ,k ,σ ĉλ,k ,σ + λ,µ,k ,σ t λ,µ ĉ † λ,k ,σ ĉµ,k ,σ (II.2) ĤR = λ ,k ,σ k + σ λ ĉ † λ ,k ,σ ĉλ ,k ,σ + λ ,µ ,k ,σ t λ ,µ ĉ † λ ,k ,σ ĉµ ,k ,σ (II.3) ĤB = i,k ,σ k + i ĉ † i,k ,σ ĉi,k ,σ + i,j,k ,σ t i,j ĉ † i,k ,σ ĉj,k ,σ (II.4) Ĥint = k ,σ t a,α ĉ † a,k ,σ ĉα,k ,σ + t b,α ĉ † b,k ,σ ĉα ,k ,σ + H.c. (II.5)
Here, ĉ † p,k ,σ creates one s electron with spin σ on layer p in the Bloch state labelled by the transverse wave number k (translational invariance in the xz plane is assumed through the entire junction).

k is the in-plane kinetic energy of the Bloch state and t p,q the spin-independent hopping matrix element between sites p and q. The coupling of the left (right) lead to the barrier is considered through the hopping parameter t a,α (t b,α )

between the rst (last) layer of the barrier and last (rst) layer of the left (right) lead.

h.c. denotes the hermitian conjugate.

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

The spin-dependent on-site energies σ p within the leads are split into the s orbital energy ( 0 ) and the magnetic s -d exchange interaction between itinerant spins and localized magnetic moments (∆ σ p ): σ p = 0 + ∆ σ p We set 0 = 1.5 eV in both leads and t= -1 eV in all regions so that all the FIs analysed here are fully characterised by their on-site dependent s -d interaction ∆ σ p . Unlike the case of ordinary F-MTJ with homogeneous exchange splitting value within the electrode, ∆ σ p for FI-MTJ here not only alternates in orientation, but also varies in magnitude from one layer to the next one, which denes two dierent sublattices denoted here as A and B (see Fig. 1). Considering a right FI lead whose rst layer next to the B/FI interface is formed by fully uncompensated up spins (sublattice A), the spin splitting in layer λ writes :

∆ ↑(↓) λ =    -(+)∆ A if λ is odd -sublattice A- +(-)∆ B if λ is even -sublattice B-
As depicted in Fig. 1, majority and minority bands in the ferrimagnetic leads are split by δ = ∆ A -∆ B . When an external bias V is applied, the on-site energies inside the insulator are considered to drop linearly with the number of layers (here N B =3) from B

= 5 eV at the FI/B interface. The parameters modelling the electronic properties of the FI leads constitute a reasonable choice used previously for ordinary F-MTJ based on magnetic transition metals and their alloys [START_REF] Stamenova | Currentdriven magnetic rearrangements in spin-polarized point contacts[END_REF], [START_REF] Zhang | Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets[END_REF].

II.2 Observables to calculate

The spin-dependent transport properties of the magnetic tunnel junction are analysed here through four essential physical quantities (observables): spin density, spin current density, charge current density and spin transfer torque. Thus, the corresponding quantum operators must be calculated. Their expectation values are then computed in order to obtain measurable quantities.

As it is known from statistical mechanics, the expectation value or thermal average at time t of any operator Ô(t) can be calculated using the density matrix ρ(t) [START_REF] Bruus | Introduction to Manybody quantum theory in condensed matter physics[END_REF]:

Ô(t) = T r[ρ(t) • Ô] (II.6)
where the symbol T r denotes a trace over all many-body states (over all possible indices),

i.e. a trace in the Fock space [START_REF] Stefanucci | Nonequilibrium Many-Body Theory of Quantum Systems[END_REF]. In a nonequilibrium state, the density matrix is expressed in terms of the lesser Green's function Ĝ< , dened by the ensemble average:

G < α,β (t, t ) = i ĉ † β (t )ĉ α (t) (II.7)
Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

where α, β represent any set of quantum numbers describing the one particle Hilbert state of the system. With this denition, the density matrix is the same as the equal-time lesser

Green's function:

ρ α,β (t) = ĉ † β (t)ĉ α (t) = -iG < α,β (t, t) (II.8)
Thus, the lesser Green's function allows to calculate the time-dependent ensemble average of any one-body operator:

Ô(t) = -iT r[ Ĝ< (t) • Ô] (II.9)
The origin of the lesser Green's function in the framework of the nonequilibrium Keldysh formalism is described in II.8. The observables studied in this thesis are developed in terms of Green's functions in the following sections.

II.2.1 Spin density at site λ in the right lead

The spin's mean value at time t in a plane λ perpendicular to the direction of transport in the right lead reads:

Ŝλ (t) = T r[ρ(t) • Ŝλ ] = -iT r[ Ĝ< (t, t) • Ŝλ ] = - i 2π T r[ Ĝ< (ω) • Ŝλ ]dω = - i 2π T r[ Ĝ< (E/ ) • Ŝλ ]dE = - i 2π k ,σ,σ T r[ Ĝ< λ λ (E/ , k )] • SdE
Where Ĝ< λ λ is the lesser Green's function 2 × 2 matrix in spin space and Ŝλ is the local spin operator at site λ . The explicit time-dependence disappears because only the stationary state is considered. The k dependence of Ĝ< λ λ is only shown at the end of the equation. Taking the thermodynamic limit V = N a 2 -→ ∞, where N is the number of unit cells in the λ plane and a is the lattice parameter, the summation in k states can be replaced by an integral:

k -→ V (2π) 2 F BZ dk = N a 2 (2π) 2 F BZ dk (II.10)
Using this relation, the local spin reads:

Ŝλ = S λ = - i 2π N a 2 (2π) 2 T r[ Ĝ< λ ,λ (E/ , k ) • S]dk dE = - i N a 2 16π 3 T r[ Ĝ< λ ,λ (E, k ) • σ]dk dE function formalism
Where σ is the usual vector whose components are the spin operators (Pauli matrices).

For the sake of simplicity, the lesser Green's function's dependence on E and k will be omitted in the notation from now on. The spin density (spin by unit of surface) in the plane λ is, nally:

s λ = - i 16π 3 T r[ Ĝ< λ λ • σ]dEdk (3-D) (II.11)
In 1-D, the integration in k is not necessary and the prefactor 1/(2π) 2 of Eq. (II.10) disappears, so that Eq. (II.11) becomes:

s λ = - i 4π T r[ Ĝ< λ λ • σ]dE (1-D) (II.12)
The integration in the reciprocal space is restricted to the rst Brillouin zone, FBZ.

By projecting the lesser Green's function into the three Pauli matrices components of σ, the spin density in 3-D at site λ in the x, y and z directions are given by:

s x λ = - i 16π 3 T r σ [ Ĝ< λ λ • σx ]dEdk = - i 16π 3 [G < λ λ (1, 2) + G < λ λ (2, 1)]dEdk s y λ = - i 16π 3 T r σ [ Ĝ< λ λ • σy ]dEdk = - i 16π 3 [G < λ λ (1, 2) -G < λ λ (2, 1)]dEdk s z λ = - i 16π 3 T r σ [ Ĝ< λ λ • σz ]dEdk = - i 16π 3 [G < λ λ (1, 1) -G < λ λ (2, 2)]dEdk
where the components of the lesser Green's function operator in spin space are dened as follows:

Ĝ<

λ λ = G < λ λ (1, 1) G < λ λ (1, 2) G < λ λ (2, 1) G < λ λ (2, 2) = G <↑↑ λ λ G <↑↓ λ λ G <↓↑ λ λ G <↓↓ λ λ

II.2.2 Spin current density at site λ in the right lead

The spin current density is dened as the tensor product of the velocity operator and the spin density at site λ :

Q = ŝλ ⊗ v (II.13)
In the case of the tunnel junction considered here, transport takes place only in the y direction, only the Qxy , Qyy , Qzy components are nonzero, so that Q is not a tensor, but Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism a vector:

Q = ŝλ ⊗ (0, vy , 0) = (ŝ x , ŝy , ŝz ) t • (0, vy , 0) =    0 ŝx vy 0 0 ŝy vy 0 0 ŝz vy 0    Q = ( Qxy , Qyy , Qzy ) = (ŝ x vy , ŝy vy , ŝz vy ) (II.14)
The left index is in spin space and the right index is in real space. For the sake of clarity, the y index in real space will be omitted in the notation, so that the three components in spin space of the spin current vector in the y direction are denoted by:

Q = ( Îx , Îy , Îz ) (II.15)
The velocity operator can be extracted from the Heisenberg equation:

vy = d dt ŷ = i [ Ĥ, ŷ]
The diagonal parts of the TB Hamiltonian commute with the ŷ operator. Let's calculate the the matrix elements of the velocity operator in the Hilbert space:

p, σ| [ Ĥ, ŷ] |q, σ = p, σ| Ĥ ŷ |q, σ -p, σ| ŷ Ĥ |q, σ = q p, σ| Ĥ |q, σ -p p, σ| Ĥ |q, σ = (q -p) p, σ| Ĥ |q, σ = (q -p)t p,q
If only nearest neighbours are considered:

vy = i t p,σ (ĉ † p,σ ĉp+1,σ -ĉ † p+1,σ ĉp,σ ) (II.16)
Combining Eqs. (II.11) and (II.16), the spin current density between layers λ and λ + 1 for the 3-D case and the spin current between corresponding sites for the 1-D case can be obtained in terms of the lesser Green's function:

Îλ ,λ +1 = t 16π 3 T r σ [( Ĝ<σ,σ λ +1,λ -Ĝ<σ,σ λ ,λ +1 ) • σ]dEdk (3-D) (II.17) Îλ ,λ +1 = t 4π T r σ [( Ĝ<σ,σ λ +1,λ -Ĝ<σ,σ λ ,λ +1 ) • σ]dE (1-D) (II.18)
The x, y and z components of the spin current in spin space are thus:

Îx λ ,λ +1 = t 16π 3 T r σ [( Ĝ<σ,σ λ +1,λ -Ĝ<σ,σ λ ,λ +1 ) • σx ]dEdk (II.19)
Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

Îy λ ,λ +1 = t 16π 3 T r σ [( Ĝ<σ,σ λ +1,λ -Ĝ<σ,σ λ ,λ +1 ) • σy ]dEdk (II.20) Îz λ ,λ +1 = t 16π 3 T r σ [( Ĝ<σ,σ λ +1,λ -Ĝ<σ,σ λ ,λ +1 ) • σz ]dEdk (II.21)

II.2.3 Charge current density

The charge current density can be extracted from Eq. (II.17 , where e is the electronic charge [START_REF] Edwards | Self-consistent theory of current-induced switching of magnetization[END_REF]:

I = et 8π 3 T r σ [ Ĝ< λ +1,λ -Ĝ< λ λ +1 ]dEdk = et 2π [ Ĝ< λ +1,λ (1, 1) + Ĝ< λ +1,λ (2, 2) -Ĝ< λ λ +1 (1, 1) -Ĝ< λ λ +1 (2, 2)]dEdk (II.22)
Obviously, charge current density is conserved along the junction, and in particular at the right interface, where it writes:

I = et 2π T r σ [ Ĝ< α b -Ĝ< bα ]dEdk = et 2π [ Ĝ< α b (1, 1)+ Ĝ< α b (2, 2)-Ĝ< bα (1, 1)-Ĝ< bα (2, 2)]dEdk

II.2.4 Torque on site λ in the right lead from the spin density current

In analogy to the continuity equation for the charge or the particle number, an equivalent continuity equation can be established for the spin density [START_REF] Stiles | Anatomy of spintransfer torque[END_REF].

The characteristic dierence is that spin is not conserved due to the local exchange eld inside the leads, giving rise to an external torque T exerted by the lattice:

dŝ dt = i [ Ĥ, ŝ] = -∇ • Q + T (II.23)
Since by denition it is the right index of the spin current tensor that is in real space, the divergence of this tensor at the right hand side of Eq. (II.23) is the divergence of each of its columns (the spatial derivative is taken with respect to the real space index). As mentioned above, only the Qxy , Qyy , Qzy components are nonzero (all the spatial derivatives with respect to x and z variables are also zero due to the translational invariance in the xz plane of the tunnel junction):

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

∇ • Q = ∂ k Qik =    ∂ x Qxx + ∂ y Qxy + ∂ z Qxz ∂ x Qyx + ∂ y Qyy + ∂ z Qyz ∂ x Qzx + ∂ y Qzy + ∂ z Qzz    =    ∂ y Qxy ∂ y Qyy ∂ y Qzy    =    ∂ y Îx ∂ y Îy ∂ y Îz    = ∂ y Î
The discrete TB basis imposes in turn the discrete form of the partial derivative of the spin current in the real y direction:

∇ • Q = ∂ y Î = Îλ -1,λ -Îλ ,λ +1 = ( Îx λ -1,λ -Îx λ ,λ +1 , Îy λ -1,λ -Îy λ ,λ +1 , Îz λ -1,λ -Îz λ ,λ +1 )
Here, the subscripts λ and µ of Îλ ,µ stand for the layers/sites λ and µ correlated by the Lesser Green's function in terms of which the spin current is written (see below).

T accounts for all the external torques that act to change the direction of the spin density. Since only coherent transport is considered, no spin-ip scattering processes are taken into account, there is no transfer of angular momentum between the lattice and the spin current due to spin-ip; thus, no terms as ŝ τ ↑↓ (where τ ↑↓ is a spin ip relaxation time) [START_REF] Stiles | Anatomy of spintransfer torque[END_REF] appear in (II.23). The external torque is exerted by the exchange eld and any external eld that might be present. In the absence of external elds, the lattice is the only responsible for the torque exerted on transport electrons. In a steady state, the time derivative of the spin density is zero, and the current-induced torque exerted by transport electrons on the magnetization is in competition with the torque due to exchange, anisotropy elds and phenomenological damping (in the context of a Landau-Lifshitz Gilbert equation), and causes the spin in a given atomic plane to deviate from the anisotropy axis [START_REF] Edwards | Self-consistent theory of current-induced switching of magnetization[END_REF]. In the stationary regime, Eq.

(II.23) becomes:

T = ∇ • Q (II.24)
The torque exerted on the total magnetization is then equal to the net ux of spin current in a magnetic volume V. The torque deposited on layer λ then reads:

Tλ = Îλ -1,λ -Îλ ,λ +1 (II.25)
By using Eqs. (II.17) and (II.18), T λ writes:

T λ = t 16π 3 T r σ [( Ĝ< λ ,λ -1 -Ĝ< λ -1,λ -Ĝ< λ +1,λ + Ĝ< λ ,λ +1 ) • σ]dEdk (3-D) (II.26)
Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

T λ = t 4π T r σ [( Ĝ< λ ,λ -1 -Ĝ< λ -1,λ -Ĝ< λ +1,λ + Ĝ< λ ,λ +1 ) • σ]dE (1-D) (II.27)
The torque is then decomposed into parallel or in-plane (x direction, transversal to the order parameter) and perpendicular or out-of-plane (y direction) components.

In-plane torque T λ

The in-plane component reads:

T λ = t 16π 3 T r σ [( Ĝ< λ ,λ -1 -Ĝ< λ -1,λ -Ĝ< λ +1,λ + Ĝ< λ ,λ +1 ) • σx ]dEdk (3-D) (II.28) T λ = t 4π T r σ [( Ĝ< λ ,λ -1 -Ĝ< λ -1,λ -Ĝ< λ +1,λ + Ĝ< λ ,λ +1 ) • σx ]dE (1-D) (II.29)
Multiplying by σx = 0 1 1 0

to the right and taking the trace amounts to sum all o-diagonal elements of these four matrices. The parallel torque then becomes:

T λ = t 16π 3 [ Ĝ< λ ,λ -1 (1, 2) + Ĝ< λ ,λ -1 (2, 1) -Ĝ< λ -1,λ (1, 2) -Ĝ< λ -1,λ (2, 1) -Ĝ< λ +1,λ (1, 2) -Ĝ< λ +1,λ (2, 1) + Ĝ< λ ,λ +1 (1, 2) + Ĝ< λ ,λ +1 (2, 1)]dEdk
The torque on the rst site of the right lead in contact with the barrier's right interface takes a particular form that includes the correlation of the last site of the barrier and the rst site of the right lead:

T α = t 16π 3 [ Ĝ< α ,b (1, 2) + Ĝ< α ,b (2, 1) -Ĝ< b,α (1, 2) -Ĝ< b,α (2, 1) -Ĝ< α +1,α (1, 2) -Ĝ< α +1,α (2, 1) + Ĝ< α ,α +1 (1, 2) + Ĝ< α ,α +1 (2, 1)]dEdk
It will be shown that the correlation function at this interface is formally dierent from the correlation function inside the lead.

Out-of-plane torque T ⊥ λ

The out-of-plane torque writes:

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

T ⊥ λ = t 16π 3 T r σ [( Ĝ< λ ,λ -1 -Ĝ< λ -1,λ -Ĝ< λ +1,λ + Ĝ< λ ,λ +1 ) • σy ]dEdk (3-D) (II.30) T ⊥ λ = t 4π T r σ [( Ĝ< λ ,λ -1 -Ĝ< λ -1,λ -Ĝ< λ +1,λ + Ĝ< λ ,λ +1 ) • σy ]dE (1-D) (II.31)
Multiplying by σy = 0 -i i 0

to the right and taking the trace amounts to sum the (1,2) components, subtracting the (2,1) components and multiplying by i, which gives:

T ⊥ λ = t 16π 3 i[ Ĝ< λ ,λ -1 (1, 2) -Ĝ< λ ,λ -1 (2, 1) -Ĝ< λ -1,λ (1, 2) + Ĝ< λ -1,λ (2, 1) -Ĝ< λ +1,λ (1, 2) + Ĝ< λ +1,λ (2, 1) + Ĝ< λ ,λ +1 (1, 2) -Ĝ< λ ,λ +1 (2, 1)]dEdk
And again the torque on the rst site of the right lead reads:

T ⊥ α = t 16π 3 i[ Ĝ< α ,b (1, 2) -Ĝ< α ,b (2, 1) -Ĝ< b,α (1, 2) + Ĝ< b,α (2, 1) -Ĝ< α +1,α (1, 2) + Ĝ< α +1,α (2, 1) + Ĝ< α ,α +1 (1, 2) -Ĝ< α ,α +1 (2, 1)]dEdk

II.2.5 On-site torque from the exchange eld

Alternatively, since ballistic transport is considered here, torques can also be extracted from the exchange eld:

T = ∆ λ ẑ × ŝλ (II.32)
The exchange eld ∆ λ is dened as an angular frequency so that the energy required to reverse one spin in layer

λ is ∆ λ = ↑ λ -↓ λ .
T = ∆ λ ẑ × ŝλ = ∆ λ (0, 0, 1) × (ŝ x λ , ŝy λ , ŝz λ ) = ∆ λ (-ŝ y λ , ŝx λ , 0)
According to the coordinates depicted in

T λ = ∆ λ • s y λ = ∆ λ • s ⊥ λ (II.33) T ⊥ λ = ∆ λ • s x λ = ∆ λ • s λ (II.34)
II.2.6 Total torque exerted in the right lead

The denition of the local torque as the (discrete) divergence of the spin current on each site is commonly used to calculate the total torque exerted on the right lead, which is the sum of local torques:

T λ = -∇ • Q = Q λ -1,λ -Q λ ,λ +1 (II.35) T = ∞ λ =1 (Q λ -1,λ -Q λ ,λ +1 ) (II.36) For the specic case of a F lead, T = Q 0,1 -Q ∞,∞ = Q 0,1 , since the components of Q λ ,λ +1
transverse to M' decay to zero as λ → ∞ [START_REF] Kalitsov | Spin-transfer torque in magnetic tunnel junctions[END_REF], [START_REF] Kalitsov | Spin-polarized current-induced torque in magnetic tunnel junctions[END_REF], [START_REF] Brataas | Non-collinear magnetoelectronics[END_REF]. The total spin torque exerted on the right F lead is therefore simply the spin current at the B/F interface [START_REF] Stiles | Anatomy of spintransfer torque[END_REF].

However, it is likely that spin density current does not vanish far away from the barrier interface for the specic case of an AF lead, i.e. Q ∞,∞ = 0, as discussed in [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF]. The last simplication to compute the total torque can no longer be used to calculate the total torque on an AF. In addition, in the case of FI and AF leads, the spin density current terms in the sum

∞ λ =1 (Q λ -1,λ -Q λ ,λ +1 ) do not cancel
since the orientation of localized spins must be taken into account through a switching sign in front of each term: a positive torque on one site tries to turn the localized spin in one sense, but a torque of the same sign acting on the next site tends to align the corresponding localized spin in the opposite sense. Heisenberg eld is strong enough to make the lattice rigid, so that both eects are counterbalanced and the torque will not be eective. This is illustrated in T

T || eective = ∞ λ =1 (-1) λ • T || λ (II.37) function formalism
⊥ eective = ∞ λ =1 (-1) λ +1 • T ⊥ λ (II.38)
That is the reason why for this thesis I studied the local distribution of torques T λ on each site/layer of the right lead instead of the single spin current density at the barrier interface, which is sucient only for F leads.

II.3 Green's and Keldysh functions to calculate

As discussed in II.2, one needs to calculate the lesser Green's function to compute the required local observables. In addition, three more Green's functions need to be introduced: Keldysh, advanced and retarded Green's functions:

F α,β (t, t ) = -i [ĉ α (t), ĉ † β (t )] - (II.39) G a α,β (t, t ) = -iθ(t -t ) [ĉ α (t), ĉ † β (t )] + (II.40) G r α,β (t, t ) = iθ(t -t) [ĉ α (t), ĉ † β (t )] + (II.41)
where θ(t) is the Heaviside step function, [A, B] -= AB -BA is the quantum commutator and [A, B] + = AB + BA is the quantum anticommutator. A more detailed description of these functions and their physical signicance is given in II.8.

When the stationary state is considered, the correlation functions only depend on the time dierence τ = t -t . It is then advantageous to take the Fourier transform of these functions with respect to this time dierence. In addition, since we consider spin

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism dependent transport, the lesser Green's function becomes a 2 × 2 matrix in spin space correlating sites/layers p and q within the tunnel junction. With these assumptions, its Fourier transform can be expressed in terms of nonequilibrium Keldysh, advanced and retarded Green's functions [START_REF] Lifshitz | Physical Kinetics, Course of Theoretical Physics[END_REF]:

Ĝ< p,q = 1 2 ( Fp,q + Ĝa p,q -Ĝr p,q )

(II.42)

The three correlation functions at the right hand side (r.h.s.) of Eq. (II.42) need to be calculated by solving a set of Dyson equations in order to obtain the lesser Green's function.

II.3.1 Retarded and advanced Green's functions

Let's start with the calculation of the retarded and advanced Green's functions.

Given that Ĝa lm = ( Ĝr ml ) H , where H denotes hermitian conjugate, i.e. transposing the 2 × 2 matrix in spin space and taking the complex conjugate of its elements, it suces to calculate the retarded Green's function, from which the advanced Green's function is straightforwardly obtained. In addition, since all the observables of interest are evaluated in this work in the right lead of the tunnel junction, only the retarded Green's function correlating sites at the right lead is considered here. In particular, three Green's functions are needed: Ĝ< α ,b , Ĝ< b,α and Ĝ< λ ,µ . They correlate atomic sites across the right interface and inside the right lead, and are given by all the diagrams or possible paths connecting the two indices. Taking into account possible excursions of the electron across the interfaces, these three Green's functions can be expressed in terms of the two surface Green's functions of the barrier, Ĝa,a and Ĝb,b [Caroli et al., 1971]: ĝλ ,µ are the retarded Green's functions for the right electrode alone, i.e., when it is not coupled with the rest of the junction through the right barrier interface. Thus, λ and µ are both contained in the right lead, otherwise ĝλ ,µ = 0. T is the hopping parameter (coupling) across the right barrier/right lead interface.

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

II.3.2 Nonequilibrium Keldysh functions

The nonequilibrium 2 × 2 Keldysh function matrix is calculated using the Dyson Quantum Kinetic equation. The detailed derivation of Fλ µ , Fb α and Fα b is shown in Appendix A for the interested reader. Their nal expressions read:

Fbα = T 2 D Bĝ ba ( Î + Σaa Âĝ aa ) fαα Ĝa aα + D B(ĝ ba Σaa Âĝ ab + ĝbb )T fα α ( Î + T Ĝa bα ) (II.46) Fα b = T L N Σbb ĝba M fαα Ĝa ab + T L N fα α Ĝa bb (II.47) Fλ µ = ĝλ α T Den -1 ĝba T ( Î + Σaa Âĝ aa ) fαα T Ĝa ab T ĝa α µ fλ µ + fλ α T Ĝa bb T ĝa α µ + ĝλ α T Den -1 (ĝ bb + ĝba Σaa Âĝ ab )T ( fα µ + fα α T Ĝa bb T ĝa α µ ) (II.48)
The denitions of the matrices Σaa , Σbb , Â, B, D, N , M , L and Den -1 are given in the same Appendix.

II.3.3 Lesser Green's functions

The lesser Green's functions correlating atomic sites in the right lead can now be easily obtained using the results of the previous sections and Eq.(II.42):

Ĝ<

α b = 1 2 ( Fα b + Ĝa α b -Ĝr α b ) = 1 2 [ Fα b + ( Ĝr bα ) H -Ĝr α b ] = 1 2 [T L N Σbb ĝba M fαα Ĝa ab + T L N fα α Ĝa bb + ( Ĝr bb T ĝr α α ) H -ĝr α α T Ĝr bb ] (II.49) Ĝ< bα = 1 2 ( Fbα + Ĝa bα -Ĝr bα ) = 1 2 [ Fbα + ( Ĝr α b ) H -Ĝr bα ] = 1 2 [T 2 D Bĝ ba ( Î + Σaa Âĝ aa ) fαα Ĝa aα + D B(ĝ bb + ĝba Σaa Âĝ ab )T fα α ( Î + T Ĝa bα ) + (ĝ r α α T Ĝr bb ) H -Ĝr bb T ĝr α α ] (II.50)
Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

Ĝ< λ µ = 1 2 ( Fλ µ + Ĝa λ µ -Ĝr λ µ ) = 1 2 [ Fλ µ + ( Ĝr µ λ ) H -Ĝr λ µ ] = 1 2 [ĝ λ α T Den -1 ĝba T ( Î + Σaa Âĝ aa ) fαα T Ĝa ab T ĝa α µ + fλ µ + fλ α T Ĝa bb T ĝa α µ + ĝλ α T Den -1 (ĝ bb + ĝba Σaa Âĝ ab )T ( fα µ + fα α T Ĝa bb T ĝa α µ ) + (ĝ r µ λ + ĝr µ α T Ĝr bb T ĝr α λ ) H -ĝr λ µ -ĝr λ α T

II.4 Calculation of the Retarded Green's function for a one dimensional ferrimagnet

A 1-D FI chain of localized moments seen by one itinerant spin is characterized in a TB model by two on-site energies 1 and 2 (alternating from one site to the next one) and a hopping parameter t. This 1-D chain corresponds to a lattice without closed loops that can be mapped into a Bethe lattice or Cayley tree which is completely characterized by its number of nearest neighbours Z = 2 or its connectivity K = Z -1 [Economou, 2006].

Splitting the 1-D TB Hamiltonian into an unperturbed site-diagonal part plus an odiagonal perturbation and using renormalized perturbation expansion [Economou, 2006] allows to perform the calculation of the system's resolvent ĝλ ,µ (z), where z is a complex variable. This resolvent is a more general denition of the Green's function in the complex plane, from which the retarded Green's function can be dened as follows:

ĝr λ .µ (E) = lim s→0 + ĝλ ,µ (z = E + is)
where E stands for the energy variable belonging to the real axis of the complex plane and s is the imaginary part of z.

The detailed derivation of the retarded Green's function from this resolvent is given in Appendix B. The retarded Green's function correlating sites l and m in the FI chain has a complex structure: it is a spin-dependent function of the energy E, and takes

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism dierent expressions depending on which interval of E it is evaluated on and what sites it correlates. In eect, the imaginary part of the retarded Green's function correlating a site l with itself is proportional to the LDOS at that site. This imaginary part does not vanish in two energy regions: the lower band and the upper band. Thus, there are three forbidden energy regions: the rst one for energies lower than the lower band edge E 2 , the gap separating the lower and the upper bands between 2 and 1 and the third one for energies higher than the upper band edge E 1 (see Fig.

II.3). g r (l, m, E) =                      g r f 1 (l, m, E) if E < E 2 g r gap (l, m, E) if 2 < E < 1 g r f 2 (l, m, E) if E 1 < E g r lb (l, m, E) if E 2 < E < 2 g r ub (l, m, E) if 1 < E < E 1
E c = 1 + 2 2 : Im[g rσ 2,2 (E) = Im[g rσ 1,1 (-E + E c )].
The real part for a given spin in one sub-lattice and the real part for the same spin in the other sub-lattice are point-symmetric with respect to the center of the gap:

Re[g rσ 2,2 (E) = -Re[g rσ 1,1 (-E + E c )].

II.4.2 Energy dependence of the retarded Green's function for an innite ferromagnetic chain

In the case of a F chain, the on-site energy for a given spin is constant regardless of the lattice site. Since all localized moments are equal and point in the same direction, there is only one type of site, so that the two sub-bands of the FI case arising from the two FI sublattices merge into a single band, as can be seen in Fig. II.5. The imaginary part (proportional to the LDOS) has square root singularities at the band edges that are characteristic of one dimensional systems [Economou, 2006].

Similar to the FI case, the imaginary part for a given spin is even-symmetric with respect to a vertical axis crossing the center of the band and the real part is odd-symmetric with function formalism respect to the center of the band.

II.5 Calculation of the retarded Green's function for a semi innite chain

Once the retarded Green's function for an innite chain is calculated, it is necessary to calculate the retarded Green's function for a semi-innite chain. Note that the uncoupled Green's functions at the r.h.s. of Eqs. (II.49), (II.50) and (II.51) refer to the right semi-innite magnetic lead in contact with the non-magnetic barrier. To calculate the correlation functions of the right lead, we cut the innite chain between sites 0 and 1 (sub-lattices B and A of the innite chain), so that site 1 will be the rst site of the right semi-innite chain. The chain cut will be considered as a perturbation term added to the previous 1-D Hamiltonian, in order to use a Dyson equation to compute the Green's function for the semi-innite chain:

Ĥ = Ĥ0 + V
where:

Ĥ: Hamiltonian for the semi innite chain Ĥ0 : Hamiltonian for the innite chain

V : Perturbation

For sites n and m on the same side of the cut: n, m ≥ 1 or n, m ≤ 0, we have Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

n| Ĥ0 |m = n| Ĥ |m -→ n| V |m = 0
However, two sites n and m on dierent sides of the chain can not be correlated:

n| Ĥ |m = 0 = n| Ĥ0 |m + n| V |m -→ n| V |m = -n| Ĥ0 |m
We shall study the matrix elements n| Ĥ0 |m to compute the perturbation V :

The Tight Binding Hamiltonian for one spin in an innite chain reads:

Ĥ0 = l l |l l| + t |l l + 1| + t |l + 1 l| (II.52)
Projecting Eq. (II.52) on sites n and m on dierent sides of the chain gives:

n| Ĥ0 |m = l t( n|l l + 1|m + n|l + 1 l|m ) = l t(δ n,l δ l+1,m + δ n,l+1 δ l,m ) =t(δ n+1,m + δ n-1,m )
For this expression to be dierent from 0, we must have n + 1 = m or n -1 = m (n and m own to dierent sides of the cut).

Case 1) n ≤ 0 on the left side of the chain and m ≥ 1 on the right side of the chain.

The only two sites giving a contribution to the perturbation are in this case: n = 0, m = 1:

n| Ĥ0 |m = tδ n,0 δ m,1
Case 2) n ≥ 1 on the right side of the chain and m ≤ 0 on the left side of the chain.

The only two sites giving a contribution to the perturbation are in this case: n = 1, m = 0:

n| Ĥ0 |m = tδ n,1 δ m,0 Therefore: n| Ĥ0 |m = t(δ n+1,m + δ n-1,m ) = t(δ n,0 δ m,1 + δ n,1 δ m,0 ) V = n,m n| V |m |n m| = -t n,m (δ n,0 δ m,1 + δ n,1 δ m,0 ) |n m| = -t(|0 1| + |1 0|) function formalism
Once the perturbation operator is known, the Green's function of the semi-innite chain (perturbed system) ĝr can be calculated applying a Dyson equation to the Green's function of the innite chain (unperturbed system) ĝr 0 :

ĝr = ĝr 0 + ĝr 0 V ĝr ĝr is the retarded Green's function corresponding to Ĥ (semi innite chain) and ĝr 0 is the retarded Green's function corresponding to Ĥ0 (innite chain, non perturbed Hamilto- nian).

Projecting Dyson equation into sites n and m gives:

g r (n, m, E) = g r 0 (n, m, E) + p,q g r 0 (n, p, E)V (p, q)g r (q, m, E) = g r 0 (n, m, E) -t p,q g r 0 (n, p, E)(δ p,0 δ q,1 + δ p,1 δ q,0 )g r (q, m, E) = g r 0 (n, m, E) -tg r 0 (n, 0, E)g r (1, m, E) -tg r 0 (n, 1, E)g r (0, m, E) (II.53)
If we are interested only on the right hand side of the chain:

n, m ≥ 1 -→ g r (0, m, E) = 0
The amplitude of probability for going from one point at the right hand side of the cut to the last site on the left hand side (site 0) is zero. We have:

g r (n, m, E) = g r 0 (n, m, E) -tg r 0 (n, 0, E)g r (1, m, E) g r (1, m, E) = g r 0 (1, m, E) -tg r 0 (1, 0, E)g r (1, m, E) -→ [1 + tg r 0 (1, 0, E)]g r (1, m, E) = g r 0 (1, m, E) -→ g r (1, m, E) = g r 0 (1, m, E) 1 + tg r 0 (1, 0, E) g r (n, m, E) = g r 0 (n, m, E) - tg r 0 (n, 0, E)g r 0 (1, m, E) 1 + tg r 0 (1, 0, E) (II.54)
Or in matrix form in spin space:

ĝr (n, m) = ĝr 0 (n, m) -tĝ r 0 (n, 0)[ Î + tg r 0 (1, 0)] -1 ĝr 0 (1, m) (II.55)
For this formula to be valid, n and m must be ≥ 1: the rst site on the right hand side

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism of the chain is site number one.

If we are interested only on the left hand side of the chain:

n, m ≤ 0 -→ g r (1, m, E) = 0 g r (n, m, E) = g r 0 (n, m, E) -tg r 0 (n, 1, E)g r (0, m, E)
In all the calculations for the tunnel barrier, the only correlation function needed in the left lead is the surface Green's function on the last site: g r αα . If the convention of the cut between sites 0 and 1 is kept: g r αα = g r 00 .

g r αα = g r (0, 0, E) = g r 0 (0, 0, E) -tg r 0 (0, 1, E)g r (0, 0, E)

g r αα = g r (0, 0, E) = g r 0 (0, 0, E) 1 + tg r 0 (0, 1, E) (II.56)
Or in matrix form in spin space: As in the innite case, the LDOS for up and down spins in a given site are just shifted, and they are symmetric with respect to the center of the single band. These LDOS approach the bulk value shown in Fig. II.5 in an oscillatory way as well.

ĝr αα = [ Î + tg r 0 (0, 1, E)] -1 ĝr 0 (0, 0, E) (II.

Change of quantization axis

To take into account the angle θ between the magnetizations in the left and right leads, the surface Green's function ĝr αα must be rotated in spin space.

The matrix of a spin rotation around an axis given by the unitary vector n is:

ÛR (n, θ) = Îcos(θ/2) -i(n • σ)sin(θ/2)
In the case where n = j: function formalism

ÛR (θ) =

cos(θ/2) -sin(θ/2) sin(θ/2) cos(θ/2)

The surface Green's function at the left lead then becomes:

ĝ αα = ÛR • ĝαα • Û H R = cos(θ/2) -sin(θ/2) sin(θ/2) cos(θ/2) • g ↑↑ αα 0 0 g ↓↓ αα • cos(θ/2) sin(θ/2) -sin(θ/2) cos(θ/2) ĝ αα =    g ↑↑ αα cos 2 (θ/2) + g ↓↓ αα sin 2 (θ/2) 1 2 sin(θ)(g ↑↑ αα -g ↓↓ αα ) 1 2 sin(θ)(g ↑↑ αα -g ↓↓ αα ) g ↑↑ αα sin 2 (θ/2) + g ↓↓ αα cos 2 (θ/2)   
Calculation of the retarded Green's function for a semi-innite chain hosting one or two magnetic impurities The Green's functions calculated above describe perfectly periodic magnetic chains.

They can thus be used to model magnetic tunnel junctions with ideal epitaxially grown crystalline structures, in the absence of defects. The specic eect of the natural disorder usually encountered in actual polycrystalline junctions is therefore neglected. A rst ap-function formalism proach to analyse the eects of disorder is considering the presence of a given number of magnetic impurities with a specic spatial distribution, which break the former periodicity. These magnetic impurities might be present as a result of conventional sputtering deposition processes. Although this analysis was eventually not pursued, the results of the calculations for one and two magnetic impurities in the right lead are described below to be considered for future work.

Once the retarded Green's function of the semi-innite right lead is calculated, it can be used to compute the retarded Green's function of a right lead hosting a substitutional magnetic impurity on a generic site l. For doing so, the Hamiltonian is split again into two parts:

Ĥ = Ĥ0 + V
where:

Ĥ: Hamiltonian for a semi innite chain with one magnetic impurity function formalism Ĥ0 : Hamiltonian for the pure semi innite chain V : Perturbation due to a magnetic impurity on site l

In spin space:

V = |l ˆ l| = |l ↑ imp 0 0 ↓ imp l| Here, ↑ imp ( ↓ imp
) is the dierence in energy of an itinerant spin up(down) at site l with and without the magnetic impurity. The Dyson equation for the perturbed Hamiltonian then reads: ĝr = ĝr 0 + ĝr 0 V ĝr . ĝr is the retarded Green's function corresponding to Ĥ

(semi-innite chain with a magnetic impurity on site l) and ĝr 0 is the retarded Green's function corresponding to Ĥ0 (semi-innite chain without magnetic impurity).

Projecting the Dyson equation on sites n and m inside the right lead (n, m ≥ 1) gives:

ĝr (n, m) = ĝr 0 (n, m) + i,j ĝr 0 (n, i)V (i, j)ĝ r (j, m)
where the matrix elements of the perturbation in the TB basis are:

V (i, j) = i| V |j = i|l ˆ l|j = δ i,l δ j,
ĝr (l, m) = ĝr 0 (l, m) + ĝr 0 (l, l)ˆ ĝr (l, m) -→ [ Î -ĝr 0 (l, l)ˆ ]ĝ r (l, m) = ĝr 0 (l, m) -→ ĝr (l, m) = [ Î -ĝr 0 (l, l)ˆ ] -1 ĝr 0 (l, m) ĝr (n, m) = ĝr 0 (n, m) + ĝr 0 (n, l)ˆ [ Î -ĝr 0 (l, l)ˆ ] -1 ĝr 0 (l, m) (II.58)
The calculation of the retarded Green's function for a semi-innite chain hosting two magnetic impurities is straight forward: considering the retarded Green's function ĝr 0 corresponding to one magnetic impurity at site l, a Dyson equation can be used to get the retarded Green's function corresponding to two magnetic impurities located at sites l and p, ĝr :

ĝr (n, m) = ĝr 0 (n, m) + ĝr 0 (n, p)ˆ [ Î -ĝr 0 (p, p)ˆ ] -1 ĝr 0 (p, m) (II.59)
where ˆ is the matrix of energy dierences for an itinerant spin at site p with and without

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism the second impurity.

II.6 Dispersion relation for a one dimensional ferrimagnet

The electron motion inside the junction depicted in Fig. II.1 can be viewed as a scattering problem of electrons going out and coming into the left and right electrodes. As long as coherent transport is considered, the conservation of the transverse wave vector allows to ignore in a rst step the dimensions perpendicular to the transport direction, reducing the calculations to a 1-D problem. In addition, the probability for an electron to travel through the barrier depends on its Fermi wave vector [Slonczewski, 1995], [START_REF] Tsymbal | Spindependent tunnelling in magnetic tunnel junctions[END_REF]]. This Fermi wave vector is dierent for up and down spins in F and FI materials, since majority and minority bands are spin split. As a consequence, their dierent tunnelling probabilities result in an imbalance of electric current carried by up and down spins, which is precisely at the origin of TMR. In particular, one of the assumptions made by Julliere [Julliere, 1975] was that the conductance of a spin channel was proportional to the product of the two DOS at the Fermi energy in the F electrodes.

It is therefore important to study the scattering states characterized by the wave vector k y parallel to the y direction through the dispersion relations and total DOS in the leads of the tunnel junction. Some important results of the next chapter are directly related to the features displayed by the dispersion relations of F, AF and FI chains. Let's start with the calculation of the dispersion relation for the general case of a FI chain.

As explained above, a FI can be considered as an alternating chain composed of two types of site per unit cell (on-site energies A and B ) which are coupled by the hopping parameter t. Within the tight-binding approximation, the 1-D Hamiltonian for a given itinerant spin-reads:

Ĥ = +∞ n=-∞ ( A |n, A n, A|+ B |n, B n, B|)+ +∞ n=-∞ (t |n, A n, B|+t |n, B n + 1, A|+H.c.)
or, in second quantized form:

Ĥ = +∞ n=-∞ ( A â † n ân + B b † n bn ) + +∞ n=-∞ (tâ † n bn + tâ † n+1 bn + H.c.)
where n is the cell index and â ( b) is the anihilation operator of one electron on a site A (B).

Since we are dealing with a periodic system, we can take the lattice Fourier transform of the Hamiltonian to pass from the site representation to the momentum representation,

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism using the usual convention: ân = k∈F BZ e ikna âk (a being the lattice constant):

n â † n ân = n,k,k e -ikna â † k e ik na âk = k,k â † k âk n e i(k -k)na = k,k â † k âk N δ k,k = N k â † k âk n â † n bn = N k â † k bk n â † n+1 bn = n,k,k e -ik(n+1)a â † k e ik na bk = N k e -ika â † k bk
The Hamiltonian then reads:

Ĥ = A k â † k âk + B k b † k bk + t k [(1 + e -iak )â † k bk + H.c.]
with k ∈ [-π/a, π/a], a being the lattice constant.

This Hamiltonian is not diagonal yet, since it mixes â and b operators; we obtain a simpler 2 × 2 eigensystem projecting it on the basis {|k, A ; |k, B }:

k, A| Ĥ |k, A k, A| Ĥ |k, B k, B| Ĥ |k, A k, B| Ĥ |k, B • |k, A |k, B = E(k) |k, A |k, B
The secular equation then reads:

A -E(k) t(1 + e ika ) t(1 + e -ika ) B -E(k) = 0
which gives the dispersion relation:

E(k) = A + B 2 ± A -B 2 2 + 4t 2 cos 2 (ka/2) (II.60)
The sign -designs the lower band and the sign + the upper band. For a linear chain with A sites hosting localised up spins, the on-site spin-dependent energies can be split as:

↑(↓) λ =    ↑(↓) A = 0 -(+)∆ A ↑(↓) B = 0 + (-)∆ B
Where 0 is the spin-independent s-orbital energy of the itinerant spins. With this con- vention, the spin-resolved dispersion relation is:

E ↑ (k) = 0 - δ 2 ± ∆ A + ∆ B 2 2 + 4t 2 cos 2 (ka/2) (II.61)
Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

E ↓ (k) = 0 + δ 2 ± ∆ A + ∆ B 2 2 + 4t 2 cos 2 (ka/2) (II.62)
Majority and minority bands are thus split by δ and the band edges for up-spins are, in ascending order: E

↑(↓) 2 , ↑ A ( ↓ B ), ↑ B ( ↓ B ), and E ↑(↓) 1
, where

E ↑(↓) 1 = ↑(↓) A + ↑(↓) B 2 + ↑(↓) A - ↑(↓) B 2 2 + 4t 2 E ↑(↓) 2 = ↑(↓) A + ↑(↓) B 2 - ↑(↓) A - ↑(↓) B 2 2 + 4t 2
The bandgap is therefore: B = ∆ A + ∆ B , which is also consistent with the results of section II.4. The density of k-states as a function of energy can be calculated directly from the dispersion relation:

DOS ↑(↓) (E) = dk πdE E =
Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

                     -(2E - ↑(↓) A - ↑(↓) B ) πa(E - ↑(↓) A )(E - ↑(↓) B ) 4t 2 (E - ↑(↓) A )(E - ↑(↓) B )
-

1 if E ↑(↓) 2 < E < ↑ A ( ↓ B ) (lower band) 2E - ↑(↓) A - ↑(↓) B πa(E - ↑(↓) A )(E - ↑(↓) B ) 4t 2 (E - ↑(↓) A )(E - ↑(↓) B )
-

1 if ↑ B ( ↓ A ) < E < E ↑(↓) 1 (upper band)
The factor 1 π comes from the normalization dEDOS(E) = 1.

As can be seen in In contrast to the LDOS, the total spin-resolved DOS for the scattering k states is symmetric with respect to the center of the gap.

II.6.1 Limiting case: antiferromagnetic chain

In the case of an AF chain, the opposite magnetic moments are equal, so that:

∆ A = ∆ B = ∆ ↑ A = ↓ B , ↓ A = ↑ B , B = 2
• ∆ and δ = 0 (majority and minority bands are no longer split because there is no macroscopic magnetization). The dispersion relation and DOS become:

E ↑ (k) = E ↓ (k) = 0 ± ∆ 2 + 4t 2 cos 2 (ka/2) (II.63)
Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

DOS σ (E) =                    -(2E -σ A -σ B ) πa(E -σ A )(E -σ B ) 4t 2 (E -σ A )(E -σ B )
-

1 if E σ 2 < E < ↑ A ( ↓ B ) (lower band) 2E -σ A -σ B πa(E -σ A )(E -σ B ) 4t 2 (E -σ A )(E -σ B )
- The Fermi wave vectors for up and down spins coincide in this case.

1 if ↑ B ( ↓ A ) < E < E σ 1 (upper band)

II.6.2 Limiting case: ferromagnetic chain

If all the localised moments are equal and point in the same direction, a ferromagnetic order is established. There is only one sublattice, and the energy for a given spin is constant and site-independent:

↑(↓) = 0 -(+)∆
The dispersion relation and DOS become:

E ↑ (k) = 0 -∆ + 2tcos(ka) (II.64) E ↓ (k) = 0 + ∆ + 2tcos(ka) (II.65)
Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism Underlying the ballistic approach considered here, that is, momentum conservation in the plane of the layers (k ), is the assumption that these planes are translational invariant. function formalism

DOS ↑(↓) (E) = - 1 2πat 1 1 - E -( 0 -(+)∆) 2t 2
The specic conguration of localized atoms distributed periodically on each of the planes determines the 2-D dispersion relation that must be used in coherent-based theoretical calculations, as explained in section II.1.1. A number of works on spin dependent transport in spin valves and magnetic tunnel junctions [START_REF] Kalitsov | Spin-transfer torque in magnetic tunnel junctions[END_REF], [START_REF] Theodonis | Anomalous bias dependence of spin torque in magnetic tunnel junctions[END_REF], [START_REF] Kalitsov | Spin-polarized current-induced torque in magnetic tunnel junctions[END_REF], [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF] (II.66

)
where 0 is the on-site energy on the plane. A successive parallel disposition of these planes constitutes the model for a ferromagnetically ordered lead (spins might be rotated by an angle θ, but the on-site energy 0 is the same in every plane for a given itinerant spin). In turn, the antiferromagnetic order can be modelled in dierent ways. One of them is considering an alternating disposition of uncompensated planes where each of them lies between two planes rotated 180 o . The macroscopic magnetization is zero since the magnetization of a layer of uncompensated spins is compensated by the next one. An itinerant spin will successively encounter alternating uncompensated planes, so that its on-site energy will be layer-dependent. Another way to model AF order is considering a lead built up with parallel planes of compensated spins, as the one depicted in The lattice parameter is a in both cases

This conguration would be closer to reality for instance in IrMn 3 FCC crystal lat- tice structures, which are deposited in the (1, 1, 1) direction. In these structures, the close-packed plane is in turn the (1, 1, 1) plane, which is formed of compensated spins [START_REF] Srinivasan | Antiferromagnetic iridium manganese based intermediate layers for perpendicular magnetic recording media[END_REF].

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

Consequently, the dispersion relation in Eq. (II.66) must be recalculated for these compensated planes, as described in the next section. In addition to the replacement of Eq.

(II.66) by the new dispersion relation in all the calculations, the integration in k space will now take place in a dierent Brillouin zone, according to the new reciprocal vectors of the direct lattice shown in a 1 = a(e 1 + e 3 ), a 2 = a(e 1 -e 3 )

They give rise to the following new reciprocal vectors:

b 1 = π a (e 1 + e 3 ), b 2 = π a (e 1 -e 3 )
Two linear combinations of atomic-like orbitals centred on each of the two sub-lattices Φ(r -R i ) are used as basis of the in-plane Bloch states:

|Ψ 1 k = 1 √ N R 1 e ik • R 1 |Φ(r -R 1 ) |Ψ 2 k = 1 √ N R 2 e ik • R 2 |Φ(r -R 2 )
where, N is the number of unit cells of each of the sub-lattices.

Any Bloch state in the 2-D periodic lattice can be expressed as a linear combination of these basis Bloch functions:

|Ψ k = α k |Ψ 1 k + β k |Ψ 2 k
By denition, the Bloch states are eigenstates of the periodic 2-D Hamiltonian:

Ĥ |Ψ k = k |Ψ k
This equation can be arranged in a matrix form considering a 2-D space spanned by the two basis Bloch functions above:

H 11 H 12 H 21 H 22 • α k β k = k • α k β k
where the matrix elements are dened as the projections of the Hamiltonian into the Bloch basis:

H i,j = Ψ i k | Ĥ |Ψ j k .
For this eigenvalue problem to have a solution the secular equation must be satised:

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

H 11 -k H 12 H 21 H 22 -k = 0
Considering an atomic function overlap with nearest neighbours only, the following relations hold:

Φ(R 1 )| Ĥ |Φ(R 1 ) = 1 δ R 1 ,R 1 Φ(R 2 )| Ĥ |Φ(R 2 ) = 1 δ R 2 ,R 2 Φ(R 1 )| Ĥ |Φ(R 2 ) = t = 0 only if R 2 = R 1 + ae 1 ; R 2 = R 1 -ae 3 ; R 2 = R 1 -ae 1 ; R 2 = R 1 + ae 3
The matrix elements are thus greatly simplied:

H 11 = Ψ 1 k | Ĥ |Ψ 1 k = 1 N R 1 ,R 1 e -ik • R 1 e ik • R 1 Φ(R 1 )| Ĥ |Φ(R 1 ) = 1 N R 1 ,R 1 e -ik • (R 1 -R 1 ) δ R 1 ,R 1 = 1 H 22 = 2 H 12 = Ψ 1 k | Ĥ |Ψ 2 k = 1 N R 1 ,R 2 e -ik • R 1 e ik • R 2 Φ(R 1 )| Ĥ |Φ(R 2 ) Ψ 1 k | Ĥ |Ψ 2 k = t N R 1 e -ik • R 1 [e ik • (R 1 +ae 1 ) + e ik • (R 1 -ae 3 ) + e ik • (R 1 -ae 1 ) + e ik • (R 1 +ae 3 ) ] = t N R 1 (e ikxa + e -ikza + e -ikxa + e ikza ) = 2t[cos(k x a) + cos(k z a)] = γ(k) = γ * (k) = Ψ 2 k | Ĥ |Ψ 1 k
With these simplications, the secular equation reads:

1 -k γ(k) γ(k) 2 -(k) = 0 = [ 1 -k ][ 2 -k ] -γ 2 (k) = 2 k -( 1 + 2 ) k + 1 2 -γ 2 (k)
which gives the following dispersion relation for a square 2-D lattice of compensated spins:

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

k = 1 + 2 ± ( 1 + 2 ) 2 -4[ 1 2 -γ 2 (k)] 2 = 1 + 2 ± ( 1 -2 ) 2 + 16t 2 [cos(k x a) + cos(k z a)] 2 2 (II.

II.8 Nonequilibrium Keldysh formalism

This section intends to give a very brief description of the Keldysh formalism and the origin of the Green's functions needed for physical systems out of equilibrium. It is freely taken mainly from [START_REF] Caroli | Direct calculation of the tunneling current[END_REF]], [Roermund, 2010], [START_REF] Rammer | Quantum eld-theoretical methods in transport theory of metals[END_REF]] and [START_REF] Haug | Quantum Kinetics in Transport and Optics of Semiconductors[END_REF].

The usual equilibrium diagram technique at zero temperature involves the calculation of the time-ordered (also called casual) zero-temperature single particle Green's function.

This correlation function is mainly dened by a chronological product of the form:

Φ 0 | T { Â(t) B(t )} |Φ 0
where T is the usual chronological ordering operator which orders observables from t = -∞ to t = +∞, Â(t) and B(t ) are eld or creation and annihilation operators in the Heisenberg picture acting at times t and t and |Φ 0 is the ground state of the system (space variables are omitted for the sake of clarity). When the system is aected by a perturbation -interactions between particles are turned on for instance-it is useful to employ the interaction representation, in order to get rid of the unknown exact eigenstate of the system, and express the correlation function in terms of the ground eigenstate |Φ 0 of the unperturbed Hamiltonian instead, generally much easier to calculate. The connection between the exact and the non-perturbed ground state is accomplished by using the Gell-Mann and Low theorem, which yields to the interaction representation of the last expression:

Φ 0 | S(-∞, +∞) T { Ã(t) B(t )S(+∞, -∞)} |Φ 0
where the evolution operator in the interaction representation or matrix S is dened as:

S(t , t) = T {exp[-ı t t Ṽ (t 1 )dt 1 ]}
Here, the total Hamiltonian has been split into an unperturbed part H0 and a perturbation Ṽ . This perturbation is supposed to be adiabatically turned on from t = -∞ and turned o at t = +∞. The tilde symbol identies operators in the interaction representation.

In the presence of irreversible eects, the evolution of the system from t = -∞ to t = +∞ amounts no longer to a simple phase shift of its state, since the system absorbs or emits energy during the evolution. As a consequence, the exact state of the system at t = +∞ is in general not known (actually this is one of the things one wants to compute with the Green's functions); thus, all correlation functions must also refer only to the state |Φ 0 at t = -∞, when the perturbation is o and the Hamiltonian is easier to solve.

The diagram technique is thus generalized by the following statement:

Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

Φ 0 | S(-∞, +∞) T { Ã(t) B(t )S(+∞, -∞)} |Φ 0 = Φ 0 | TC {S C Ã(τ ) B(τ )} |Φ 0 (II.68)
where TC is an operator ordering times from right to left, not as usual from t = -∞ to t = +∞, but along a contour C made of a "positive" branch going from τ = -∞ to τ = +∞ and with a "negative" branch going from τ = +∞ to τ = -∞ (contour-ordering operator), as shown in Fig. II.17.

Figure II.17 Representation of the Keldysh contour C, where T { Ã(t 1 ) B(t 2 )} = B(t 2 ) Ã(t 1 ) but TC { Ã(τ 1 ) B(τ 2 )} = Ã(τ 1 ) B(τ 2 ). The time variable τ on the Keldysh contour must specify on which branch it is located.

In the preceding expression, the contour-evolution operator ŜC corresponds to an evolution forward in time from τ = -∞ to τ = +∞ and then backward in time from τ = +∞ to τ = -∞:

S C = S(-∞, +∞)S(+∞, -∞) = TC {exp[ C Ṽ (τ )dτ ]}
where the times of S(+∞, -∞) are on the positive branch and the times of S(-∞, +∞) on the negative branch. It is clear that the presence of this term will introduce times on the negative branch of the contour in (II.68).

Times on the positive branch are ordered from τ = -∞ to τ = +∞ and times on the negative branch are ordered from τ = +∞ to τ = -∞. Any time of the negative branch is considered as posterior to any time of the positive branch. Now, each operator must specify both its time and on which side of the contour it lays, the positive or the negative branch. Hence, the Green's functions will take additional indices and the usual perturbation expansion of the Dyson equation takes a matrix form, as described below. Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism

G C p,q (t, t ) = -ı Φ 0 | TC {ĉ p (t + )ĉ † q (t + )} |Φ 0 = -ı Φ 0 | T {ĉ p (t)ĉ † q (t )} |Φ 0 GC p,q (t, t ) = -ı Φ 0 | TC {ĉ p (t -)ĉ † q (t -)} |Φ 0 = -ı Φ 0 | T {ĉ p (t)ĉ † q (t )} |Φ 0 G > p,q (t, t ) = -ı Φ 0 | TC {ĉ p (t -)ĉ † q (t + )} |Φ 0 = -ı Φ 0 | ĉp (t)ĉ † q (t ) |Φ 0 G < p,q (t, t ) = -ı Φ 0 | TC {ĉ p (t + )ĉ † q (t -)} |Φ 0 = -ı Φ 0 | ĉ † q (t )ĉ p (t) |Φ 0
Here, the operator T orders times from +∞ to -∞ (anti-chronological order) and the subscripts of the creation and annihilation operators indicate site indices of a TB Hamiltonian. In a diagram calculation, integrals are thus performed along the contour starting and ending at τ = -∞, and not from t = -∞ to t = +∞; this is equivalent to perform the integration from -∞ to +∞ and to sum over the subscripts + and -corresponding to the branches of the contour. Because the negative branch of the contour goes from τ = +∞ to τ = -∞, any point of this branch corresponds to a (-) sign due to the dierential in time, or equivalently any interaction on this branch corresponds to a (-) sign. Summing over subscripts + and -is equivalent to using 2 × 2 matrices. Therefore the usual diagram technique still holds, if one denes a Green's function matrix:

G = G ++ G +- G -+ G -- = G C G < G > GC
The Dyson equation then reads:

G p,q (t, t ) = G 0 p,q (t, t ) + G 0 p,k (t, t 1 )Σ k,l (t 1 , t 2 )G l,q (t 2 , t )dt 1 dt 2 (II.69)
where

Σ = Σ C Σ < Σ > ΣC function formalism
is the self-energy matrix and G 0 is the Green's function matrix in the absence of the perturbation. Obviously (II.69) represents 4 equations, but only 2 are independent. In our case we are concerned with a system in the steady state, i.e., all Green's functions depend only on the time dierence t -t , and it is useful to Fourier transform (II.69) with respect to time and work in the energy domain. Eq. (II.69) then writes:

G = G 0 + G 0 ΣG (II.70)
where all quantities are 2 × 2 matrices with respect to the + and -indices. The fact that Eq. (II.69) represents only two independent equations becomes clearer if one makes the following canonical transformation (rotation

): G → G = R -1 GR, with R = (1 + ıσ y )/ √ 2,
where σ y is one of the Pauli matrices. Then, the transformed Green's function matrix becomes:

G = 0 G a G r F
where we nally get the retarded, advanced and Keldysh functions: The same transformation for the self-energy yields:

G r = G C -G < = -GC + G > G a = G C -G > = -GC + G < F = G C + GC = G > + G <
Σ = Ω Σ r Σ a 0 with Σ r = Σ C + Σ < = -( ΣC + Σ > ) Σ a = Σ C + Σ > = -( ΣC + Σ < ) Ω = Σ C + ΣC = -(Σ > + Σ < )
Thus with the transformed (invariant) Dyson equation we obtain three equations and the Fourier transforms of the equations for G a and G r are obviously complex conjugates:

G r = G r 0 + G r 0 Σ r G r G a = G a 0 + G a 0 Σ a G a F = F 0 + G r 0 Σ r F + F 0 Σ a G a + G r 0 ΩG a
The rst two equations are the Dyson equations for the retarded and advanced Green's functions and the third one is the Quantum Kinetic equation.They provide a complete description of the system out of equilibrium.

In the case considered in this work, the perturbation consists in the connection of the leads with the isolating barrier through the couplings or hopping parameters at the interfaces. It is instantaneous and can be written as:

V (t) = {t α,a ĉ † α (t)ĉ a (t) + t α ,b ĉ † α (t)ĉ b (t)} + H.c.
where the time t is either on the positive or on the negative branch of the contour so that the interaction cannot connect the two branches. Consequently, Σ has the following form in the transformed Dyson equation:

Σ = 0 t α,a t α,a 0 (δ p,α δ q,a + δ p,a δ q,α ) + 0 t α ,b t α ,b 0 (δ p,α δ q,b + δ p,b δ q,α )
The unperturbed system at t = -∞ consists of the leads and barrier disconnected and maintained at dierent chemical potentials. Then the hopping at the interfaces is turned on adiabatically and non-equilibrium quantities are evaluated when the steady state is established. function formalism

Numerical implementation

The complexity and size of the systems considered in this work are unaordable for analytical calculations to give quantitative theoretical predictions. Instead of using a simplied model analytically solvable, numerical simulations are performed here to take into account a more detailed description of the system. The numerical calculations are carried out by a quantum transport code built in Fortran 90. I coded the new spatially-resolved retarded, advanced, Keldysh and lesser Green's functions developed previously, together with all the observables described in II.2 for AF (uncompensated and compensated) and FI systems in 1-D and 3-D. A pre-existing code that was intended to calculate the interfacial spin current density in a AF/B/AF junction was used. The computation was performed in the CEA computer cluster "Summer", with a total of 288 Intel cores. The calculation of the local torque deposited in a single layer for 3-D systems took approximately one month. The bottleneck was the k numerical integration in the FBZ, which was performed by the multidimensional adaptive integration algorithm Cuhre, included in the Fortran 90 interface. A number of convergence tests was performed all along this work to check the validity of the results presented in the next chapter.

Chapter III

Spin dependent transport in antiferro and ferrimagnetic tunnel junctions

This chapter is devoted to the investigation of the amplitude and characteristic lengths of STT and tunnelling magnetoresistance in epitaxial AF-and FI-based tunnel junctions using the theoretical tools developed in Chapter II.

First, a comprehensive analysis of the essential features characterising the spatial distribution of STT in AF materials is presented. This analysis is addressed from two dierent but physically equivalent points of view, namely the spatial variation of the transverse components of spin currents and spin accumulation, as described in II.2.5.

The STT out-of-plane component exhibits a staggered spatial distribution similar to its inplane component. This behaviour is specic to the use of a tunnel barrier and signicantly diers from the out-of-plane torques reported in previous works using a metallic spacer, as outlined in Chapter I.

Next, the dependence of the charge current on the intrinsic magnetic properties of the leads as well as the magnetic state of the tunnel junction is analysed in order to examine the TMR ratio in purely AF tunnel junctions, which is of great technological interest.

The TMR ratio is shown to be very sensitive to the specic magnetic properties of the AF leads, reaching values comparable to typical magnetoresistances found for usual spin valves.

FI materials merge characteristic features of both Fs and AFs, namely a spontaneous macroscopic magnetization together with a partially compensated magnetic structure. As a consequence, combined STT features of Fs and AFs may also occur in FIs, giving rise to high local STT values and long range STT spatial distributions. In fact, due to their complex magnetic structure, FI-based tunnel junctions exhibit a more interesting STT spatial behaviour, and are examined in a subsequent section. The spatial distribution of STT is shown to have non-trivial wave patterns which are strongly aected by the intrinsic magnetic and electronic properties of the FI material as well as by the applied bias across the junction. The strong sensitivity of TMR ratios to the electronic structure and magnetic properties shown for AF leads is also observed in FI-based tunnel junctions.

III.1 Spin dependent transport in antiferromagneticbased tunnel junctions

The STT picture proposed in [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF] for AFs arises from changes in the exchange eld experienced by localized moments due to their magnetic interaction with nonequilibrium spin densities originated by conduction electrons. This is in contrast with the widely used theoretical framework for F systems, where the basic idea is that the precession of an electron about the magnetization of a F yields to a change in this magnetization by conservation of angular momentum, which is equal to the imbalance of inward and outward spin uxes. Due to the vanishing magnetization in AF materials, this formalism can only be applied locally, unless the alternative mechanism for STT proposed in [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF] is used.

Together with the reection-based polarizing mechanisms previously reported for AFs [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF], [START_REF] Saidaoui | Spin transfer torque in antiferromagnetic spin valves: From clean to disordered regimes[END_REF], we emphasize that despite the fact that the Apart from spin-valve structures using metallic spacers, usual F-based spintronics takes advantage since long of tunnel barriers (B) [START_REF] Moodera | Large magnetoresistance at room temperature in ferromagnetic thin lm tunnel junctions[END_REF]. Whereas GMR relies on spin dependent scattering at interfaces, tunnel magnetoresistance (TMR) rather relates to spin dependent densities of states. In addition, tunnel barriers lter the wave vectors' angles of incidence of incoming electrons, which lowers the eect of dephasing in three dimensional systems (3-D).

In eect, the tunnelling current distribution over the two-dimensional Brillouin zone (corresponding to the wave vector component parallel to the barrier interface) is strongly localized in certain regions. For instance, STT is mostly determined by electrons with perpendicular incidence and decreases quickly with |k | in F tunnel junctions [Manchon et al., 2008a]. Similar behaviour is characteristic for spin dependent tunneling and TMR as demonstrated both using rst principles calculations [START_REF] Butler | Spin-dependent tunneling conductance of Fe|MgO|Fe sandwiches[END_REF],

tight-binding [START_REF] Mathon | Theory of tunneling magnetoresistance of an epitaxial Fe/MgO/Fe(001) junction[END_REF], [START_REF] Kalitsov | Spin-transfer torque in magnetic tunnel junctions[END_REF] and free electron approaches [Manchon et al., 2008a]. Such "a tunnelling cone" ltering mechanism which is absent in metallic structures weakens the eect of momentum dephasing processes [START_REF] Ralph | Spin transfer torques[END_REF]. Therefore, we investigate the specic behaviour of STT and TMR in AF-based tunnel junctions using tight binding calculations in the framework of Chapter III. Spin dependent transport in antiferro and ferrimagnetic tunnel junctions the Keldysh formalism both for 1-D and 3-D geometries. Due to the use of tunnel barriers, we anticipate unusual behaviours compared to AF-based spin-valves with metallic spacers, as will be detailed below.

I recall in Fig. III.1 the tunnel junction model thoroughly described in the previous chapter, for the particular case of AF leads. These semi-innite leads are modelled as a chain of uncompensated localized spins with translational invariance in the plane perpendicular to the electron ow (y axis).

Following the method explained in Chapter II, a single band tight binding Hamiltonian for the itinerant electrons with a hopping parameter t= -1 eV in all regions is used. The insulating spacer here is a barrier of N B sites (resp. layers) for 1-D (resp. 3-D) with a spinindependent on-site energy B = 5 eV. The magnetic properties of each AF are embedded in the spin-splitting ∆, which accounts for the s-d magnetic interaction between itinerant spins and localized moments. The absolute value of the spin splitting ∆ is constant, but alternates in sign for a given spin from one site(layer) to the next one due to the alternating orientation of the localized magnetic moments within the AF leads. In particular, localized up (resp. down) spins build up a magnetic sublattice in which the spin-dependent on-site energy is given by The term "voltage-induced" is used here in order to distinguish the torques present even at equilibrium (zero voltage), from the torques originated by the electrons owning to the narrow transport energy window opened by bias voltages (these bias voltages separate the Fermi levels in both electrodes of the tunnel junction by varying the chemical potential at the right electrode). In addition, it is generally accepted that the term voltage-induced is more appropriate to describe tunneling transport while the term current induced is more suitable for metallic spin-valve systems [Slonczewski, 2005].

↑(↓) = 0 -(+)∆.
Let's rst discuss the in-plane component of the torque. This component exhibits the perfectly staggered distribution that was previously observed by Núñez et al.

[ [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF] for 3-D AF-based spin valves (see introduction chapter). The alternating sign of the torque (staggered character) is produced by the alternating localized magnetic moments of each sublattice in the AF lead. These alternating moments are responsible for the alternating exchange eld seen by transport electrons. In turn, as described in Eq. (II.33), the constant out-of-plane spin density originated by these transport electrons couple to the alternating exchange eld, which leads to the alternating sign of the torque from one site to the adjacent site.

Unlike the random distribution of the out-of-plane torque shown with such metallic spacer, in the case of an insulating spacer analysed here, the non-equilibrium, i.e. the voltage induced part of this torque component is also staggered. This is valid regardless of the left lead's magnetic order since we observed similar behaviour using AF and FI leads as polarisers (results with AF polarisers are shown in subsequent sections for 3-D junctions).

The red and blue thick lines in This RKKY conduction-electron-mediated interaction, often referred to as interlayer exchange coupling (IEC) between localized spins in the left and right leads [START_REF] Ruderman | Indirect exchange coupling of nuclear magnetic moments by conduction electrons[END_REF], [Kasuya, 1956], [Yosida, 1957] is comparable in magnitude to the transport-induced out-of-plane torque near the B/AF interface (even higher at certain points). It decays so gradually that the total out-of-plane torque is strongly af- deviations from this dependence were observed for torques in AF spin valves, and were ascribed to the multiple spin-dependent reections in the metallic spacer (similar to F spin valves) [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF], [START_REF] Saidaoui | Spin transfer torque in antiferromagnetic spin valves: From clean to disordered regimes[END_REF], which are expected to be reduced here in the case of an insulating spacer. Consequently, the value θ = π/2 expected to maximise in a rst approximation the torque amplitude was used in all the numerical simulation performed in this work concerning STT in AFs.

Spatial distribution of torques and spin densities

The local distribution of the the two torque components is illustrated in

Comparison of antiferromagnetic tunnel junctions and spin valves

An important point to discuss is the dierence between the random spatial distribution of the out-of-plane torque reported in ref. [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF] for AF-based spin valves and the staggered and ordered character of this torque component shown here for tunnel junctions. This might be ascribed to the multiple spin-dependent reections that take place in the metallic spacer but not in the insulating barrier. In eect, in the case of tunnel junctions, the evanescent waves decaying exponentially do not stay in the barrier, which reduces considerably the quantum interference between spin-dependent leftward and rightward electrons' wave functions. In addition, the spatial prole of a given spin density component generated by itinerant electrons reected o an AF displays a complex spatial pattern [START_REF] Saidaoui | Spin transfer torque in antiferromagnetic spin valves: From clean to disordered regimes[END_REF]. Thus, in the particular case of the in plane spin component, quantum interference in a metallic spacer may lead to a non-coherent spatial distribution of the in-plane spin density (and consequently out-of-plane torque) both in the spacer and in the right lead. In contrast, as analogous staggered spatial distributions of the in-plane torque are present in both systems, the k -ltering eect of tunnel junctions is unlikely to be at the origin of the dierent out-of-plane torque behaviour, since the spin precession dephasing of the dierent k states would aect similarly both torque components.

Relation between local spin densities and torques

The As can be seen in the inset, the amplitude increases slightly with the spin splitting ∆ for a constant value of 0 . However, the orbital energy 0 has a stronger impact. The open symbol at ∆ = 1 eV represents an AF with an orbital energy 0 = 2 eV, to be compared to 0 = 1.5

eV for the rest of the solid points. This increase of 0.5 eV in the orbital energy leads to a dramatic increase in the torque amplitude. An increase of the orbital energy keeping the exchange splitting constant amounts to a shift of the DOS to higher energies, maintaining the gap constant as well. This approaches the singularity of the DOS' lower edge to the Fermi energy, providing more states available for the charge current, and producing more torque. This demonstrates the importance of the electronic structure and consequently the position of the Fermi energy for the STT in AF-based tunnel junctions.

Eects of roughness at the barrier interfaces

As is already known, the tunnelling current is very much inuenced by the electronic structure, interdiusion and roughness at the barrier interfaces [START_REF] Tsymbal | Spindependent tunnelling in magnetic tunnel junctions[END_REF].

Chapter III. Spin dependent transport in antiferro and ferrimagnetic tunnel junctions

The interface sensitivity is often explored experimentally by the insertion of ultrathin layers called dusting layers at the barrier-electrode interface. In particular, interface roughness leads to uctuations in the barrier thickness and its coupling to the leads, strongly aecting tunnelling conductance.

In an attempt to examine the eects of the barrier interfaces, the STT behaviour is next analysed as a function of the hopping parameters t α,a and t b,α , which quantify the coupling at the barrier interfaces in the tunnel junction model used here (see II. strong distortion of the in-plane component near the interface for lower coupling, i.e., the torque at the rst layer of the lead becomes radically higher than the value that would correspond to the sublattice it owns to. Although this eect is harmful for the eective torque since the in-plane torque delivered at the rst layer is subtracted to the torques deposited in the next layers (it tends to rotate the order parameter in the opposite sense of the torque deposited in the rest of the layers), the staggered character of both torques Chapter III. Spin dependent transport in antiferro and ferrimagnetic tunnel junctions is preserved in the bulk of the lead. In contrast, the distortions of the out-of-plane torque near the interface are more important for high coupling. As mentioned above, these interfacial eects are crucial to be minimised in order to have high values of the eective torque; they are analysed experimentally in Chapter IV for spin valve-like systems, and will be the object of theoretical future work, together with the eect of impurities using the method proposed in II.5 and most sophisticated techniques to treat the disorder such as coherent potential approximation (CPA). In any case, the staggered pattern of the STT spatial distribution seems to be robust against low currents/low coupling at the interface.

III.1.3 Charge current density and antiferromagnetic tunneling magnetoresistance

Given the importance of the TMR for non-volatile memory applications, we next examine the voltage dependence of the charge current density (responsible for the voltageinduced part of the torque) for dierent angles θ of the left lead order parameter (see Fig. current can be considered as proportional to the applied bias in linear response theory [START_REF] Bruus | Introduction to Manybody quantum theory in condensed matter physics[END_REF], dierences between the three magnetic states are sensible only in the F case. In contrast, for the AF case, the three congurations can only be distinguished for voltages higher than a threshold of around 0.4 V. Higher voltages produce charge current saturation and decrease at very dierent rates for the three magnetic congurations. This is clearly illustrated by the TMR voltage dependence depicted in which amounts to a steep change in the number of states available in the right lead. Further investigations are however necessary in order to clarify this point. For bias below 0.4 V, the TMR is around -1% (negative TMR is also found for instance in the case of FI leads due to negative spin polarization [START_REF] Kaiser | High negative tunneling magnetoresistance in magnetic tunnel junctions with a ferrimagnetic cofeâgd electrode and a CoFe interface layer[END_REF]), but interestingly it presents a dramatic increase with voltage from the threshold, reaching values as high as 90% for a bias of 0.9 V, to compare with the GMR ratio of around 10% shown in [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF]. This bias dependence of the TMR in this AF-based tunnel junction is opposite to the usual case of ordinary F-based tunnel junctions, where TMR reaches very high values (a TMR ratio in excess of 1000% for an epitaxial Fe/MgO/Fe junction has already been reported using tight-binding bands tted to an ab initio band structure of iron and MgO [START_REF] Mathon | Theory of tunneling magnetoresistance of an epitaxial Fe/MgO/Fe(001) junction[END_REF]), but decreases signicantly with applied voltage, as shown in Fig. . From the experimental point of view, TMR ratios of 604% at room temperature and 1144% at 5 K approaching the theoretically predicted value [START_REF] Mathon | Theory of tunneling magnetoresistance of an epitaxial Fe/MgO/Fe(001) junction[END_REF] were found in sputtered CoFeB/MgO/ CoFeB F-based tunnel junctions [START_REF] Ikeda | Tunnel magnetoresistance of 604% at 300k by suppression of ta diusion in CoFeB/MgO/CoFeB pseudospin-valves annealed at high temperature[END_REF], while only a MR ratio of 0.5% was observed in the AF spin valve structure IrMn/Cu/IrMn [START_REF] Wang | Magnetoresistance eect in antiferromagnet/nonmagnet/antiferromagnet multilayers[END_REF]. This low MR ratio indicates the high sensitivity of AF STT and MR to momentum scattering due to disorder, and points out AF-based tunnel junctions as an attractive system for spintronics applications due to its momentum conservation properties [START_REF] Saidaoui | Spin transfer torque in antiferromagnetic spin valves: From clean to disordered regimes[END_REF].

Chapter III. Spin dependent transport in antiferro and ferrimagnetic tunnel junctions plane torque (as well as the total out-of plane torque in the bulk) deposited in an AF lead is found to be perfectly staggered, in contrast with the random behaviour previously reported for a metallic spacer.

The STT spatial pattern is independent of the magnetic nature of the left lead, i.e. F, AF or FI, which points out that the main role of polariser is played by the closest spins next to the barrier's left interface.

In addition, it is demonstrated that unlike conventional F-based tunnel junctions, AFbased tunnel junctions can show monotonically increasing TMR with voltage, reaching values as high as 90% for a bias of 0.9 V. However, the particular voltage dependence of the TMR is shown to be strongly aected by the electronic structure of the AF leads [START_REF] Merodio | Spin-dependent transport in antiferromagnetic tunnel junctions[END_REF]].

In the following section, the analysis carried out for AF-based tunnel junctions will be extended to the case of FIs, for which AFs constitute simpler limiting cases. As will be seen, the additional magnetic complexity inherent to FI materials yields to a richer physics concerning the STT spatial behaviour in FI-based tunnel junctions.

III.2 Spin dependent transport in ferrimagnetic-based tunnel junctions

In this section, we present a theoretical study of STT in ferrimagnet based tunnel junctions. We show that electronic structure parameters such as band widths and exchange splittings of the FI leads strongly inuence STT. In particular, the STT spatial distribution within the leads shows a striking spin-modulated wave-like behaviour resulting from the interplay between the exchange splittings of the two FI sublattices. This wave-like behaviour can also be tuned via the applied voltage across the junction. Additionally, the fundamental parameter for quantifying STT characteristic lengths in FI metals is identied here, which shall also be accessible to experiments for instance by ferromagnetic resonance and spin pumping measurements. As will be discussed in the next chapter, I used these methods to measure experimentally characteristic penetration depths of a spin current into an AF (limiting case of a FI).

All the calculations here concerning STT in FIs are performed at θ = π/2, since this angle is expected to maximise STT, as in the case of conventional F-MTJ [Manchon et al., 2008a]. While the in-plane torque is zero at equilibrium (V= 0V), the out-of-plane torque is not since it accounts for the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, as mentioned in the previous section. Interestingly, both in-plane and out-of-plane torques are staggered and commensurate with the lattice moment orientations, similar to the case of AF leads (see Fig. III.2). Here, in addition, and due to the specic FI order of the lead, the torques exhibit a striking wave-like behaviour: the FI sublattices A and B host two torque waves dephased by half of a period one with respect to the other. Note also that the higher the exchange eld, the higher the amplitude of the corresponding torque wave.

Chapter III. Spin dependent transport in antiferro and ferrimagnetic tunnel junctions Indeed, as explained in II.2.1, spin accumulation is calculated independently from Eq. II.12, its spatial distribution oscillates exactly at the same period as that of the STT and the ratios T /S y and T ⊥ /S x are constant. Using spin accumulation thus provides an alternative method for calculating torques, not only in ordinary F-MTJ but also more generally in FI-and AF-MTJs.

The variables controlling the modulation of the STT are next examined.

III.2.2 Period and amplitude of the spin-modulated torque waves

The STT waves presented here can be characterized by their amplitude A (similar to the AF case) and their period P. Both parameters are depicted in In turn, the period P of the waves increases monotonically with the applied bias, despite the two plateaus at 0.15 and 0.4 V caused by the discrete character of the period and the discrete voltages considered in the calculations, so likely not ascribed to physical reasons. It is worth noting that while the rst half period of the torque wave tries to turn the magnetization in one direction, the next half period tries to do the opposite, since the wave changes sign for each sublattice, and the exchange energy is high enough to maintain the lattice rigid. In order to have a high eective torque, it is therefore important to apply a high voltage, but not far from the bias at which saturation of the torque waves amplitude is reached.

Inuence of spin splittings on the torque modulation

The intrinsic magnetic properties of the FI leads have also a very strong inuence in the torque wave shape, and are next examined. It is apparent that the spatial prole of the in-plane component is much smoother than that of the out-of plane component. This is due to the strong inuence of the slowdecaying RKKY interaction in 1-D geometries, whose behaviour is similar to the AF case.

The subtraction of this zero voltage interaction gives the uniform prole shown in Fig.

III.13.

Apart from the higher amplitude of the in-plane torque for the FI shown in The linear t in Fig. III.16 shows that for a given voltage, the torque wave's period (P) is inversely proportional to the dierence (δ) of magnetic s -d exchange interactions between itinerant spins and localized magnetic moments in each of the two FI sublattices:

P ∝ 1 δ (III.1)
Given the above relation, the limiting case characterized by δ = 0, i.e. when the FI becomes a fully compensated AF, results in a diverging period. This agrees with the perfectly staggered and non oscillating T spatial distributions described in III.1.1 and reported previously in AF-based spin valves [START_REF] Núñez | Theory of spin torques and giant magnetoresistance in antiferromagnetic metals[END_REF]. Interestingly, a Taylor expansion at the Fermi energy of the dispersion relation for a 1-D FI up to second order in the wave vector recasts relation (III.1) in terms of the Fermi wave vectors for up and down spins for the case of low band lling: (1/δ = 10 eV -1 ). Dashed line is a linear t to the data.

P ∝ 1 δ ∝ 1 (k ↑ f ) 2 -(k ↓ f ) 2 = 1 (k ↑ f + k ↓ f )(k ↑ f -k ↓ f ) (III.2)
This expression is related to the length scales at which torque oscillations decay in conventional F-MTJ from the B/F interface. The two factors 1/

(k ↑ f -k ↓ f ) and 1/(k ↑ f + k ↓ f )
are respectively proportional to the voltage-induced and RKKY torque oscillations period in F-based tunnel junctions [START_REF] Manchon | Spin-dependent diraction at ferromagnetic/spin spiral interface[END_REF]. Moreover, as pointed out by Berger, the following relation between the exchange eld ∆ and the spin dependent Fermi wave vectors applies in Fs [Berger, 1996]:

( 2 /2m)((k ↑ f ) 2 -(k ↓ f ) 2 ) = -2µ B ∆
where ∆ is dened here as an actual magnetic eld. Consequently, the δ parameter plays the role of a global eective exchange eld in FIs, similar to the F case. This is consistent, since the net magnetization of the FI is higher for higher values of δ. In addition, for the range of energies considered here, the parameter δ and k ↑ f -k ↓ f are proportional to each other with great accuracy; the period P can thus be viewed as inversely proportional to the spatial frequency ∆k = k ↑ f -k ↓ f . Similar to the spatial precession of up and down spin components of the scattering state in Fs, the period of the STT waves in FIs results in a rst approximation from the precession around the net magnetization.

Inuence of the bandwidth on the torque modulation

Here it is also shown that the bandwidth of the electronic band structure strongly inuences both the torque waves' period and the amplitude of the oscillations. As a result of the TB model used, both upper and lower bandwidths can be parametrized by the hopping parameter t, which determines electron mobility inside the whole structure.

Although the eect of the variation of t on STT will be examined, its particular value is the same in all regions of the FI-MTJ, i.e. only homogeneous variations are considered.

The eect of local changes of the hopping/coupling at the B/FI interface are analysed in shows the wave period dependence on t for dierent FI parameters.

The two FI represented by crosses and open circles display the same period regardless of t since they share the same value of δ despite unequal spin splittings. The period is however proportional to t in every case and results from the electron mobility increase for higher values of the hopping.

The amplitude of the oscillations A scales with t 5 , as can be seen in Fig. III.17-(b); the power factor accounts for the lesser Green's function dependence on the hopping parameter in three regions of the FI-MTJ and the hoppings/couplings at to interfaces.

It is noteworthy that the hopping parameter does not inuence the commensurability between the torques and the lattice moment staggered character. As can be seen, the period slightly increases with bias for 1/δ ≥ 4, but for a xed value of this applied bias, the period is inversely proportional to δ. Moreover, the slope of the three linear ts corresponding to dierent applied bias is very similar, with a value of around 8 sites • eV. Therefore, an increase of the bias roughly amounts to a vertical shift of the period linear t. The voltage dependence of this vertical shift is illustrated in Fig.

III.14 for a 1-D FI/B/FI junction with 0 = 1.5 eV, ∆ A = 0.5 eV and ∆ B = 0.25 eV.

Barrier eects on the torque wave's period and amplitude

In order to test the robustness of the wavy STT proles shown above, the inuence of the barrier properties was next investigated through the variation of barrier height U B , the number of sites N B and the hopping elements t α,a = t b,α at the interfaces.

For all the set of barrier parameters considered here, the staggered and wavy spatial distribution of torques is always present (not shown). While these three parameters aect the amplitude of the torques (see Fig. III.19), only a slight variation of the oscillations period was detected for dierent hoppings at the interfaces (a variation of 4 sites for a period of 34 sites (roughly 10%) was observed when the hopping at the interfaces t α,a = t b,α was multiplied by 4). Manchon et al., 2008a], [START_REF] Wang | First-principles study of spintransfer torques in layered systems with noncollinear magnetization[END_REF]), the torque waves are found to be still present. The damping is introduced by the dephased torque contributions due to k integration.

The out-of-plane component of the torque seems to be as smooth as the in-plane component, except at the rst layers next to the interface, where strong distortions appear. This is explained by the short range action of the RKKY interaction in 3-D geometries.

Similarly to the AF case, the zero voltage contribution of the STT decays very fast from the interface, not aecting the STT in the bulk, as can be appreciated in the FI with δ= 0.17 eV, but only two appear for the FI characterised by δ= 0.10 eV.

Since damping spoils the periodic character of these oscillations, the concept of period P previously discussed no longer stands. Of note, these damped oscillations cannot be dened as pseudo-oscillations, and a new parameter ξ is therefore introduced to keep track of the wavy length scale. We dene ξ as the number of layers between the B/FI interface and the rst layer at which the torque wave gets zero, as illustrated in As can be seen both components are reduced when the coupling becomes weaker, but the staggered and wavy prole do not disappear. TMR ratios are shown to be strongly sensitive to the magnetic properties of the FI. This is similar to the AF case, where TMR is shown to be strongly inueced the The same numerical instability problem encountered for AF leads is found here, i.e. the J-V curves are not symmetric for positive and negative voltages. A more detailed study of this issue should be carried out in future works since it must be taken into account to chose the better FI material for technological applications.

III.2.6 Spatial distributions of spin density and spin transfer torque in 3-D antiferromagnetic-based tunnel junctions with compensated layers: out-of-plane torque modulations

Although this section is devoted to AF-based tunnel junctions, it has been included here since the spatial behaviour of the out-of-plane torque component exhibit striking similarities with that of FI. As explained in chapter 2, AF order can be modelled by alternating layers of compensated spins (called G-type AF in ref [START_REF] Saidaoui | Spin transfer torque in antiferromagnetic spin valves: From clean to disordered regimes[END_REF]).

Actual crystal lattices such as IrMn 3 are deposited in the (1,1,1) direction, perpendicular to a family of compensated planes. It is therefore interesting to investigate whether the STT is still present in these congurations, where there is no net magnetization in the xz plane of the system, perpendicular to transport. The spatial distribution of STT for a The in-plane and out-of-plane torques components are still present but they are reduced by 3 and 2 orders of magnitude respectively with respect to the uncompensated case. In addition, it is impossible to know whether these torques are ecient, since their local distribution on each atomic site of the fully compensated planes cannot be calculated with the formalism used here. In other words, if the torque deposited in each compensated layer has the same sign in every atomic site, then it will not be ecient, since opposite magnetic moments on the compensated plane are intended to rotate in opposite senses.

However, rst principles calculations applied to AF spin valves show that this spatial distribution within each compensated plane of a FeMn AF can be polarised dierently in inequivalent sublattices, being thus ecient [START_REF] Xu | Spin-transfer torques in antiferromagnetic metals from rst principles[END_REF].

Here, the same staggered behaviour reported for uncompensated AFs is found for the in-plane component, but the out-of-plane component appears to be modulated analogously to the FI case. The oscillations are damped too (which is not the case for uncompensated AF), and the sign of the torque deposited in each sublattice is conserved, which is advantageous for the eective torque. Again, these modulations can be characterized by As can be seen in the top panel of the Figure, the dependence of ξ on the spin splitting ∆ in fully compensated AFs is analogous to that of ξ on 1/δ. The pseudoperiod ξ does not follow a well-dened law, but it increases monotonically with the spin splitting.

Higher values of ∆ would be favourable for the eective torque, since the amplitude of the modulation is higher for the rst beating, however the left panel of The energy dependence of the LDOS is depicted in Figs. II.4 and II.5 for a FI and a F chain respectively. Thus, the LDOS at the Fermi level takes only one value in a F chain and two values in a FI or AF chain, for a given spin. This is clearly illustrated in Fig.

III.28. However, when the chain is cut, the translational symmetry is broken and not all sites are equivalent any more, as can be observed in Figs. II.6 and II.8. For a F lead, the LDOS corresponding to the semi-innite chain exhibits a double wavy pattern that oscillates around the constant value for the innite chain. One wave is located in even sites and the other in odd sites, they have the same amplitude, but are spatially shifted.

For the FI case, there exist multiple waves oscillating around the two constant values for the innite chain. The amplitude of the waves hosted in sublattice A (where the spin splitting is higher) are also higher. This could explain the oscillatory behaviour of the spin density, but the dependence of the period on the parameter δ is opposite. Note that the spatial period of the LDOS waves in Fig. III.28 (d) is much longer than that of (c), although they correspond to FIs with δ= 0.25 eV, and δ= 0.1 eV respectively. In other words, similar oscillatory behaviour is observed in STT and LDOS, however, their variations with δ are opposite in certain cases, and they cannot be directly linked.

In turn, the LDOS pattern encountered for AF leads is far more complex than the

III.2.8 Summary

In conclusion, a detailed analysis of the STT spatial distribution in FI-based tunnel junctions has been carried out using a TB Hamiltonian that captures its essential electric and magnetic qualitative characteristics, in the framework of the nonequilibrium Keldysh formalism. The in-plane and out-of-plane torques are found to be spatially staggered, and they exhibit a striking wave-like behaviour perfectly modulated by spin accumulation.

Thus, this work has addressed the equivalence of torques computed from the exchange eld along the z direction in FI-MTJ and through the divergence of the spin current.

The characteristic lengths of the torque oscillations are shown to be strongly dependent on the electronic and magnetic features of the FI, namely the spin splittings of the two FI sublattices and the bandwidth parametrized here by the hopping matrix element of the TB Hamiltonian. Furthermore, the fundamental parameter that governs the torque waves period in 1-D FI-MTJ and the spatial extension of the damped torque oscillations in 3-D FI-MTJ is identied. This fundamental parameter can be considered as an eective exchange eld in FIs, similar to the homogeneous exchange eld in the F case. The mentioned characteristic lengths can also be tuned via the applied bias across the junction [Merodio et al., 2014c].

These theoretical results may be of importance to choose the best suited FI material for STT-based spintronics devices and will serve as a guideline for experiments on spin penetration length in these materials using, for example, ferromagnetic resonance and spin pumping eect [START_REF] Ghosh | Penetration depth of transverse spin current in ultrathin ferromagnets[END_REF], [Merodio et al., 2014a]. In addition, experiments such as spin-transfer driven ferromagnetic resonance and spin-torque diode eect Chapter IV

Experimental determination of STT characteristic lengths in antiferromagnets

IV.1 Introduction

The property that STT acts throughout the entire volume of an AF rely on phasecoherent interferences and will surely apply strictly only in idealized AFs with perfect epitaxially grown crystal lattices. However, as explained in II.1.1, realistic polycrystalline AFs contain spin disorder that will produce noncoherent spin scattering both in the AF bulk and at interfaces: impurities, interstitial or vacancy defects, interdiusion at the interfaces, interface roughness, grain boundaries,stacking faults, complex spin structures, e.g. 3Q for IrMn and FeMn, etc. Therefore, we decided to experimentally investigate the eect of disorder on the STT characteristic lengths in two dierent AFs: IrMn and FeMn. However, as explained in IV.3.1 the required lateral size of the samples used here to study spin-dependent characteristic lengths in AFs is too large to deposit a thin and continuous insulating layer of good quality. Therefore, spin-valve-like structures were used here instead of AF tunnel junctions, replacing the insulating layer with a thicker metallic spacer. The averaging over the incidence angles of the incoming electrons wave vectors will thus come into play. This wave vectors averaging might aect the STT eciency through the AF thickness, in contrast to the 3-D theoretical predictions performed in III for tunnel barriers, which lter the wave vectors incidence angles.

Some of the critical parameters for spin dependent transport in general are: 1) the spin penetration depth originating dierent mechanisms like spin ip related to the terminology: spin diusion length and spin precession dephasing with the associated spin dephasing length terminology, and 2) the spin mixing conductances, since they both control current perpendicular to plane GMR [START_REF] Bass | Spin-diusion lengths in metals and alloys, and spin-ipping at metal/metal interfaces: an experimentalist's critical review[END_REF]. Magnetoresistive and dynamic

Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets experiments are the most common tools to study spin length scales and mixing conductances in thin lms [START_REF] Bass | Spin-diusion lengths in metals and alloys, and spin-ipping at metal/metal interfaces: an experimentalist's critical review[END_REF]. However, these two types of studies are not ideal for AF materials since the former shows very low magnetoresistive signals and the latter requires very high (THz) frequencies for dynamics excitation. Earlier attempts to characterize characteristic lengths in IrMn and FeMn using NiFe/Cu/AF/Cu/NiFe spin valves and cryogenic-temperature are reported in in Refs. [START_REF] Park | Measurement of resistance and spin-memory loss (spin relaxation) at interfaces using sputtered current perpendicular-to-plane exchange-biased spin valves[END_REF], [START_REF] Acharyya | Spin-ipping associated with the antiferromagnet IrMn[END_REF], [START_REF] Acharyya | A study of spin-ipping in sputtered IrMn using Py-based exchange-biased spin-valves[END_REF]. Although the authors could not extract denite values they concluded on signicant spin ipping at (IrMn,FeMn)/Cu interfaces and pointed towards nanometric spin penetration depths in IrMn and FeMn. An alternative way to determine spin absorption and spin mixing conductances in thin lms was recently implemented. It is based on the spin pumping phenomenon, the STT reciprocal eect and is best suited

for AFs [Berger, 1996]- [START_REF] Ghosh | Penetration depth of transverse spin current in ultrathin ferromagnets[END_REF]. The method indirectly monitors spin absorption in materials surrounding a F res layer excited at resonance by ferromagnetic resonance (FMR). The oscillating F res magnetization transfers spin angular momentum to conduction electrons of the adjacent layers. Loss of spin angular momentum by the conduction electrons results in F res resonance linewidth broadening. This latter is related to the attenuation (i.e. to the Gilbert damping α) of the F res excitations. One can distinguish between local, i.e. intrinsic losses, that is inside the precessing F res (α 0 ), and non-local, i.e. extrinsic damping where spins are lost outside the F res (α pump ), i.e. absorbed by the surrounding materials/interfaces under study. For various materials/interfaces, this method compares the spin absorption eciency (related to spin mixing conductances) since depending on the materials/interfaces properties the spins can be entirely absorbed or backscattered into the F res . Additionally, this technique determines spin absorption length scales by investigating the F res damping variations with the thickness of the neighbouring spin sink (i.e. absorbing material under investigation) [START_REF] Mizukami | The study on ferromagnetic resonance linewidth for NM/80NiFe/NM (NM=Cu, Ta, Pd and Pt) lms[END_REF].

In this chapter, spin dependent transport in two typical AF polycrystalline thin lms, IrMn and FeMn is studied using the spin pumping technique, in order to measure the characteristic length scales and thereby to determine spin absorption mechanisms in these two materials. The rst part is devoted to a more detailed description of the underlying FMR and spin pumping working principles necessary to understand how characteristic lengths in the two AFs are measured. The sample preparation process, measurement setup and experimental results are addressed in the second part.

IV.2 Ferromagnetic resonance and spin pumping

The spin pumping eect and the FMR technique are directly related to the magnetization behaviour of a F element when a time-varying magnetic eld is applied. The time evolution of the magnetization is captured by the Landau-Lifshitz-Gilbert equation, which describes the inuence of external magnetic elds and the presence of adjacent materials Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets next to the F on its magnetization dynamics. A brief discussion of this equation is thus essential to understand FMR and spin pumping eect underlying features.

IV.2.1 Magnetization dynamics: Landau-Lifshitz-Gilbert equation

In the absence of STT, the time dependence of the total or spontaneous magnetization M for a magnetic system obeys the following dynamical equation known as Landau-Lifschitz-Gilbert (LLG) equation [Lakshmanan, 2011]:

dM dt = -γM × H e + α |M| (M × dM dt ) (IV.1)
where the modulus |M| is assumed to remain constant, so that the magnetization vector can be written as M = |M| • m, with m being a unit vector pointing in the magnetization direction. The two terms at the right hand side of the LLG equation are described in detail below.

First term of the Landau-Lifschitz-Gilbert equation: magnetization precession

The rst term at the right-hand side of equation (IV.1) corresponds to the magnetization precession around an eective eld H e whose origin is described bellow. γ = -g e 2m e is the gyromagnetic ratio (g is the Landé factor, e and m e are the charge and the electron mass respectively) and µ 0 is the magnetic permeability in vacuum.

Exchange energy

Exchange interaction occurs between the spins of neighbouring atoms. This short range interaction tends to align nearby spins in the same direction and has a purely quantum mechanical origin. A change in the direction of the spins leads to a change in the electrostatic repulsion between neighbouring electrons, since the spatial overlap of the electronic wave functions permitted by the exclusion principle is dierent for dierent spin congurations. The exchange energy for a system of N atoms with spins S i reads (Heisenberg Hamiltonian):

e ex = N i,j -J i,j S i • S j
where J i,j are the exchange integrals related to the spatial overlap of neighbouring electronic orbitals. A positive(negative) value of these exchange integrals leads to a ferromagnetic(antiferromagnetic) order. Thus, the exchange interaction is responsible for the spontaneous ordering of atomic magnetic moments occurring in magnetic solids.

In a classical approach, a suitable expression of the exchange energy must be available at a continuous scale, by averaging the atomic description. For instance, the exchange energy for a simple cubic lattice of lattice parameter a reads:

e ex = JS 2 a dV ((∇m x ) 2 + (∇m y ) 2 + (∇m z ) 2 )
where S = |S i |. The exchange coupling constant J depends on temperature because of the temperature variation of the interatomic distances.

Magnetocrystalline anisotropy energy

In a classical picture, exchange interactions makes all spins to be parallel, but the energy remains invariant whatever the direction the spins are aligned with. Exchange interaction determines only the orientation of the sublattice magnetizations relative to each other. However, the total magnetic energy of a ferromagnet does depend on the particular orientation of the spontaneous magnetization with respect to the crystal lattice or crystallographic axes. The main energy contribution which depends on the direction of the magnetization is called magnetocrystalline anisotropy energy e a and is several orders of magnitude weaker than the exchange interaction. It is minimum when the magnetization is aligned along certain equivalent directions related with the symmetry Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets operations living invariant the environment of each magnetic atom. These energetically favourable directions are called "easy axis".

The origin of the magnetocrystalline anisotropy is the spin-orbit interaction Ĥ = λ L • Ŝ, which couples spin angular moments with orbital angular moments (in contrast with exchange interaction which couples only spin angular moment). Thus, the orientation of the spins is indirectly related with the direction of the orbital moment. This orbital moment created by the orbital motion of electrons couples in turn with the crystal eld created by neighbouring charges. Consequently, the energy of these orbitals depends on their orientation with respect to the surrounding crystal eld. The spin orientation will then align with the preferential direction that minimises the spin-orbit interaction. For cubic crystals, the anisotropy energy up to sixth order in the magnetization components reads:

e a (m) = K 1 (m 2 x m 2 y + m 2 y m 2 z + m 2 z m 2 x ) + K 2 m 2 x m 2 y m 2 z
where the anisotropy constants K 1 and K 2 depend on the material both in magnitude and sign, and strongly vary with temperature.

In small samples and ultrathin magnetic lms other anisotropy contributions such as surface, interface, and exchange anisotropies might come into play. This eect is attributed to the reduced symmetry of the atoms close to a surface with respect to those in the bulk. Hence the form of the magnetocrystalline anisotropy is dierent at the surface.

Zeeman energy

The Zeeman energy is simply the energy of the interaction between the magnetization and an external magnetic eld:

e Zeem = -µ 0 M • H ext
Obviously, the magnetization tends to align with the external eld in order to minimise this energy term.

Magnetostatic (stray eld) energy

The magnetostatic energy arises from the coupling of the magnetization with the eld created by matter, i.e. the eld created by the magnetization itself, called "stray eld"

or "demagnetizing eld":

e dem = - 1 2 µ 0 M • H d
The internal demagnetizing eld H d is opposite to the magnetization, and reduces the external applied eld. The magnetostatic energy is also small in comparison with the Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets exchange energy contribution, but is a long range contribution responsible for the domain formation in ferromagnetic materials, which reduces the magnetic energy stored in the eld. The stray eld or "demagnetizing eld" is proportional to the magnetization; the proportionality is given by the demagnetizing factors N i,j of the demagnetizing tensor N : H d = -N • M. N depends on the shape of the sample. Indeed, the demagnetizing eld is high when the magnetization is in a direction of small dimension. It will induce the magnetization to orient towards the largest dimensions of the sample, for instance, in the plane of a thin lm or along the axis of a nanowire.

Second term of the Landau-Lifschitz-Gilbert equation: intrinsic damping of the magnetization precession

The second term at the right-hand side of Eq. IV.1 is a phenomenological term which accounts for damping processes (similar to a viscous force in an harmonic oscillator). The phenomenological damping factor α is a positive number in the range of 0.01-0.04 for longitudinal magnetic recording media [START_REF] Inaba | Damping constants of Co-Cr-Ta and Co-Cr-Pt thin lms. Magnetics[END_REF] This dissipative term is the responsible for the magnetization to be eventually aligned with the eective eld, otherwise the precession movement would continue forever and the equilibrium would never be reached. The smaller the value of α the longer the magnetization precesses before equilibrium is achieved. It acts perpendicular both to the magnetization M and the direction of motion dM dt

, and just like the precession term con-Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets serves the magnitude of the magnetization. As will be described bellow, it explains the linewidth of the absorption signal in resonance experiments.

IV.2.2 Ferromagnetic resonance

Detecting the precessional motion of the magnetization in a ferromagnetic sample is the underlying principle of ferromagnetic resonance (FMR) to measure its magnetic properties. This method utilizes an applied static magnetic eld H DC that exerts a torque on the sample magnetization and makes the magnetic moments to precess around the total eective eld H e ; in order to counterbalance the damping term in Eq. (IV.1), a transverse RF eld H RF (microwaves ranging typically 1-35 GHz) is applied perpendicular to the static eld H DC . As explained below, when the RF eld frequency ω matches the resonance frequency extracted from Eq. (IV.1) and determined by the applied H DC eld, the microwave power is absorbed by the sample. The most important magnetic properties of the sample are extracted from the power absorption Lorentz-like resonance signal. 1. The sample is measured in a region where the applied static eld H DC aligned along the x direction is homogeneous.

2. The macrospin approximation is made, i.e., the magnetization of the sample is spatially uniform. The spins in the small volume of the sample remain parallel by virtue of the exchange interaction.

3. The magnetization lies in-plane and the surface dimensions are much larger than the thickness. The demagnetizing factors therefore reduce to: N x = N y = 0 and N z = 1.

Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets 4. The two surfaces originate a uniaxial perpendicular anisotropy which is small compared to the demagnetizing eld. This surface anisotropy (not negligible only in ultrathin magnetic layers) is ascribed to the reduced symmetry of atoms near the interface, and can be described as a surface eld contribution to the total eective eld: H S = 2K S a µ 0 M s t , where t is the thickness of the F layer and K S a is a perpendicular anisotropy constant in units of J/m 2 . Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets

χ yy = χ yy + iχ yy = M s (A + iα ω γ )[AB - ω γ 2 (α 2 + 1) -iα ω γ (A + B)] [AB - ω γ 2 (α 2 + 1)] 2 + [α ω γ (A + B)] 2 (IV.3)
where A = M e + H DC + H uni and B = H DC + H uni .

Extraction of the resonance frequency from Kittel law

The maximum magnetization in the y direction (response) for a given H RF eld (excitation) is reached when the denominator of Eq. (IV.3) is minimum. Thus the resonance condition can be expressed as:

AB -ω res γ 2 (α 2 + 1) = 0

Here α 2 + 1 ≈ 1, since α << 1, and the resonance condition reduces to ω res γ 2 = AB. This condition gives the Kittel resonance formula, which gives the sought resonance frequency:

ω res γ 2 = (M e + H DC + H uni )(H DC + H uni ) (IV.4)
Extraction of the damping parameter α from the linewidth of the power absorption resonance signal

The time dependence of the power absorbed by the sample from the total eld applied is given by the time derivative of the Zeeman energy:

P (t) = d Zeem dt = d dt [-µ 0 M • (H DC + H RF )] = -2µ 0 m y dH RF dt = -2µ 0 Re(χ yy H RF ) dH RF dt
Using the mean value theorem, the average absorbed power absorbed during one cycle of duration T = 2π ω can be calculated straightforwardly:

P = ω 2π 2π/ω 0 P (t)dt = µ 0 ωχ yy (H RF ) 2 (IV.5)
The absorbed power depends only on the imaginary part of the susceptibility function, which is associated with irreversible processes that lead to energy dissipation in the magnetic system. FMR measures the power absorption characteristics, which can be used to extract information about the relaxation processes using Eq. (IV.5). For the FMR mea-Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets surements performed in this work, it is dicult to vary the microwave frequency over large ranges; instead, the static magnetic eld H DC is varied to reach the resonance conditions, since there is a correspondence between ω res and H DC given in Eq. IV.4. The imaginary part of the susceptibility function in Eq (IV.5) is a Lorentzian function of the applied H DC eld centred at the resonance eld H res . The linewidth of this Lorentzian is given by its full width half maxima ∆H 1/2 . However, experimentally it is easier to measure the ∆H

pp (ω) = ∆H 0 + 2 √ 3|γ| αω res (IV.6)
Where γ the gyromagnetic ratio and ∆H 0 the inhomogeneous broadening associated with spatial variations in the magnitude of the out-of-plane magnetic anisotropy. The linewidth is therefore a linear function of the frequency, with a constant term ∆H 0 that takes the sample inhomogeneity into account. If several ∆H pp measurements are performed at dierent microwave frequencies ω, then the damping factor ω can be easily obtained from the slope given in Eq. (IV.6).

IV.2.3 Spin pumping

As discussed in the previous chapters, a spin-polarized current can exert a torque on the order parameter of a magnetic element, leading to current-induced magnetic dynamics.

The reciprocal of this phenomenon is the interfacial 'pumping' of spins by a precessing antiferromagnets magnetization of a ferromagnet F res into adjacent nonmagnetic layers. In other words, the precessing magnetization vector of a ferromagnet emits a pure spin current when in contact with a normal (paramagnetic) metal (see Fig. It can be monitored using the FMR methods described previously, through the resonance signal linewidth broadening.

Using a scattering matrix approach it was demonstrated that the spin current pumped out of the ferromagnet into the NM layer is given by:

I pump s = 4π (g ↑↓ r (m × dm dt ) + g ↑↓ i dm dt ) (IV.7)
m is the magnetization and g ↑↓ dm dt

= -γm × H e + α 0 m × dm dt + γ 4πM s V (g ↑↓ r (m × dm dt ) + g ↑↓ i dm dt ) (IV.8)
where V is the volume of the F. The imaginary component g ↑↓ i in Eq. (IV.8) is negligible with respect to the real component g ↑↓ r , so that the additional extrinsic damping term in Eq. (IV.8) reads:

γ 4πM s V g ↑↓ r m× dm dt
. It has a form similar to that of the intrinsic Gilbert damping (m × dm dt

), so that both terms can be merged in the same damping term. The most general additional damping coecient due to non-local losses arising from the spin pumping eect is:

∆α = α pump= |γ| 4πM s V g ↑↓ e = |γ| 4πM s ( g ↑↓ e S ) 1 t F (IV.9)
where the eective spin mixing conductance g ↑↓ Note that the additional damping term depends on the inverse of the ferromagnetic layer thickness t F , which points out an interface eect.

The spins transmitted through the F/NM interface can continue to precess in the eective eld while they diuse away from the surface. However, since these spins take a random walk through the conductor, at any distance larger than the electron mean free path away from the interface the transverse components of the spin accumulation average out (spin dephasing), and only the component along the static magnetization remains. Besides, for a metal with a fast spin-ip relaxation time, the spin injection can be balanced by spin-ip relaxation, i.e., spin currents are absorbed quickly in the metal and exert a torque. This kind of metal is denoted a good spin sink.

To summarize, spin-pumping can be thought of as the transfer of angular momentum out of a ferromagnet with precessing magnetization into the adjacent layers due to a pure spin current. This is the source of enhancement of resonance linewidths experimentally observed in FMR measurements, which accurately probe the total (intrinsic + non-local)

Gilbert damping factor. Transport-related interface eects described by the spin mixing conductances can be analysed using FMR for various materials and interfaces. In addition, this technique investigates the spin absorption length scales in F/NM/M (where M represents a material with any magnetic order) nanostructures examining the total damping variations with the thickness of the material M under investigation. In particular, it will be used in the present thesis to determine spin characteristic length scales and spin absorption mechanisms in two typical AF polycrystalline thin lms, IrMn and FeMn. The cases where M=F and PM were thoroughly studied [START_REF] Ghosh | Penetration depth of transverse spin current in ultrathin ferromagnets[END_REF].

Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets Deposited samples: summary NiFe(8)/Cu(3)/AF(t AF )/Al(2) (nm) heterostructures have been deposited at room temperature by dc-magnetron sputtering onto thermally oxidised 3 × 2 mm 2 silicon substrates [START_REF] Baltz | Bimodal distribution of blocking temperature in exchange-biased ferromagnetic/antiferromagnetic bilayers[END_REF] , [START_REF] Ghosh | Penetration depth of transverse spin current in ultrathin ferromagnets[END_REF], in order to study spin dependent transport in two usual AF materials. The AFs with varying thicknesses t AF are made from Ir 20 Mn 80 and Fe 50 Mn 50 , respectively. The F res NiFe(8) layer is deposited from a Ni 20 Fe 80 target. The Al(2) cap oxidises in air and consequently forms an AlO x protecting lm with a low spin current absorption [START_REF] Ghosh | Penetration depth of transverse spin current in ultrathin ferromagnets[END_REF] . The Cu(3) breaks the F/AF direct magnetic exchange interactions, evidenced through negligible hysteresis loop shifts measured by vibrating sample magnetometry (VSM) at 300 and 4 K after eld cooling proce- In addition, given that Cu(3) is a light element and since its spin diusion length is much longer than its 3 nm thickness [START_REF] Bass | Spin-diusion lengths in metals and alloys, and spin-ipping at metal/metal interfaces: an experimentalist's critical review[END_REF], it does not alter spin propagation between the F and the AF.

IV.3.2 Experimental setup

A scheme of the FMR measurement set-up used is represented in Fig. IV.9. This measurement facility was developed by Abhijit Ghosh and William Bailey during Abhijit's thesis work [Ghosh, 2013]. The microwave signal is provided by a vector network analyser that generates the range of frequencies 2-20 GHz with an input power of 15 dBm to Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets remain in the linear regime. The microwave signal is transmitted through a double ground coplanar wave guide CPW situated between the magnetic poles (separated 20 mm). The sample is placed up-side down on this CPW, so that the DC (homogeneous at the sample emplacement) and RF elds are applied in-plane, one perpendicular to the other, as shown in Fig. IV.9. A phase sensitive lock-in detection was used in order to improve the signal to noise ratio. The signal processed by the lock-in amplier is a derivative of the absorbed power (which is assumed to be Lorentzian) vs applied eld. The peak-to peak linewidth ∆H pp is then extracted from this derivative, as described above. H ef f originates from two main sources: the intrinsic damping inherent to the NiFe layer (α 0 ) and the damping due to the spin current generated by the F excitations then diused through the multilayer and nally pumped/reected by the AF (α pump ). Beyond a critical length (λ AF ) characteristic of the spin dependent transport in the AF, the coherence of the spin current in the AF is lost and the damping saturates.

In the gure, λ AF denotes the critical absorption depth over which the coherence of the spin current within the AF is lost. Consequently the extra F damping due to the AF levels out above λ AF [START_REF] Ghosh | Penetration depth of transverse spin current in ultrathin ferromagnets[END_REF]. The NiFe total Gilbert damping (α) is extracted from series of F resonance spectra obtained with a broadband coplanar waveguide. Inplane DC bias magnetic eld (H DC ) is employed. A small 2 to 20 Oe modulation eld of 201 Hz is applied along the DC eld for lock-in detection of the transmitted signal to improve sensitivity and excitation frequencies (ω/2π) ranging between 2 and 20 GHz are used. For each frequency the resonance linewidth is determined by tting the resonance spectra (dierential power absorption vs H DC ) to a Lorentzian derivative. The total Gilbert damping α is extracted from Eq. IV. For the various NiFe(8)/Cu(3)/AF(t AF )/Al(2) (nm) heterostructures, the non-local damping α pump ascribed to the absorption of spin angular momentum by the AF only is straightforwardly obtained by subtracting the Gilbert damping obtained for t AF = 0 from the total Gilbert damping.

Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets Two dierent behaviours are observed depending on the AF type. In the case of IrMn, it is observed that α pump increases linearly with t IrMn and cuts o to a maximum at an empirical critical thickness λ IrMn /2 of around 1.4 nm. Like in Ref. [START_REF] Foros | Scattering of spin current injected in Pd(001)[END_REF], it was considered that the spins relax on the way forward in the IrMn depth, reect and return backward through the IrMn to the NiFe layer, thus traversing and relaxing linearly twice in the IrMn depth. The linear behaviour is similar to F spin sinks [START_REF] Ghosh | Penetration depth of transverse spin current in ultrathin ferromagnets[END_REF],

indicating that the nature of the absorption for IrMn mainly relates to dephasing of the spin current transverse component as well: the spins undergo Larmor precession as they go into the material because the majority and minority Fermi wave vectors are dierent. Spins with dierent initial conditions precess at dierent rates leading to classical dephasing.

Given that, λ IrMn is mostly related to spin dephasing proportional to the integration over the Fermi wave vectors of π/(k [START_REF] Lombard | IrMn and FeMn blocking temperature dependence on heating pulse width[END_REF] . Because of that, λ IrMn was expected to be even larger than the critical lengths of usual F: CoFeB, Co or NiFe for which λ F /2 ∼ 1.2 nm [START_REF] Ghosh | Penetration depth of transverse spin current in ultrathin ferromagnets[END_REF] , [START_REF] Bailey | Pd magnetism induced by indirect interlayer exchange coupling[END_REF] . Although the linear behaviour of α pump vs t IrMn clearly points out spin dephasing mechanism as the main source of absorption, the fact that Ir is a heavy element and carries d electrons in the conduction band, probably introduces to a lesser extent some additional spin ip mechanisms balancing the eect of lower ∆ ex and contributing to slightly reducing the characteristic penetration length to a value similar to that of usual F.

From Fig.

IV.12, it can be observed that the α pump vs t FeMn follows a totally dierent trend with an exponential [1 -exp(-2t FeMn /λ FeMn )] thickness dependence. Such a trend is typical of a paramagnetic spin sink for which the absorption of the spin current is mainly due to spin ipping [START_REF] Foros | Scattering of spin current injected in Pd(001)[END_REF] , [START_REF] Mizukami | The study on ferromagnetic resonance linewidth for NM/80NiFe/NM (NM=Cu, Ta, Pd and Pt) lms[END_REF] . For consistency, the empirical factor 2 was also considered here [START_REF] Foros | Scattering of spin current injected in Pd(001)[END_REF], taking into account the forward and backward path of spins within the FeMn layer. Fits of the raw data with the above exponential law give a λ FeMn /2 of around 1.9 nm. In this case, λ FeMn relates to spin diusion in contrast to λ IrMn originating from spin dephasing, although practically both give values proportional to critical spin penetration depths. Given that, antiferromagnets 2) (nm) stacks with various IrMn thicknesses (t IrMn ) and for various directions of magnetic eld applied during deposition (H dep ). The inset is a zoom of the low thicknesses region. level, all three α pump vs t IrMn curves superimpose: same linear increase before levelling out, same saturation damping and same saturation threshold λ (better visible from the zoom shown as inset in Fig. IV.13). This nding does not support the initial assumption that the absorption of the transverse component of the spin current pumped by the F depends on the orientation of the localized spins in the rst IrMn layers. It could be argued that the reason why this absorption is apparently isotropic is that localized spins in the IrMn are not necessarily oriented by the external eld applied during deposition due to other magnetic energies that might come into play. Thus, the isotropic character of the absorption via measurements of α for NiFe(8)/Cu(3)/NiFe(3)/IrMn(7)/Al(2) (nm) stacks with a NiFe(3) spin sink pinned by usual exchange bias eld cooling along the transversal (parallel and antiparallel to H DC ) and longitudinal directions was further tested (see Fig. Excitation frequencies only ranging between 2 and 4 GHz are used so that the DC bias eld does not exceed the pinned layer reversal eld of around 600 Oe. The dierences in the damping parameter for the three orientations of the second ferromagnet NiFe(3) are within the experimental error. These results are however not conclusive, since the eect on linewidth is so small at these low frequencies that it would be probably not within the Chapter V

Summary and perspectives

In the theoretical part of this thesis, the spin-dependent transport properties of epitaxial magnetic tunnel junctions with AF and FI leads have been investigated using a tight-binding Hamiltonian modelling their essential electric an magnetic properties, in the framework of the nonequilibrium Keldysh formalism. Special attention has been paid to the spatial distribution of STT within the metallic leads, in order to elucidate the associated characteristic lengths. Indeed, longer STT characteristic lengths are of fundamental importance for lower magnetization switching current densities in STT-MRAM applications. The STT spatial distributions has been analysed in terms of spin accumulation and divergences of the spin current transverse components. Both schemes are found to be equivalent.

Concerning the AF leads, both in-plane and out-of-plane torque components are shown to be staggered. This is a new result to be compared with the random behaviour previously reported for the out-of-plane component using a metallic spacer instead of an insulating barrier. Due to this staggered character and the non-decaying torque amplitudes in the bulk, torques in AFs are found to be very ecient and even overcome those observed in conventional F, where STT represents an interfacial phenomenon. The amplitude of the torques increases linearly with the applied bias for 1-D geometries. Increasing amplitudes are also observed for 3-D systems, but reaching saturation at voltages around 0.9 V.

We demonstrate that unlike conventional F-based tunnel junctions, AF-based tunnel junctions can show monotonically increasing TMR with voltage, reaching values as high as 90% for a bias of 0.9 V. The particular voltage dependence of the TMR is however shown to be strongly aected by the electronic structure of the AF leads.

The magnetic complexity inherent to FI structures is expected to combine STT characteristics of F and AF materials. Compared to AF, similar staggered character of both torque components was observed. Torques and localized moments in the FI structure are also commensurate, which implies again high eciency for current-induced magnetization switching. In addition, and more interesting, both torque components exhibit a striking wave-like behaviour perfectly modulated by spin accumulation. In 3-D FI-MTJ, these As mentioned above, the two site correlation functions can be expressed in terms of the already calculated locators using the expressions (5.59a) and (5.59b) and the intermediate functions (5.60) and (5.61) of [Economou, 2006]. The mathematical expression for these correlation functions not only changes depending on the energy region through the locators, but also depending on the type of sites they correlate.

The intermediate functions read:

g(l + 1, l + 1[l], E) = 2 E -l+1 + [g(l + 1, l + 1, E)] -1 g(l, l[l + 1], E) = 2 El + [g(l, l, E)] -1 Forbidden states: (E < E 2 ) ∪ ( 2 < E < 1 ) ∪ (E > E 1 )

• l = 1 , |m -l| even: rst site 1 , second site 1 g r (l, m, E) = g r (l, l, E)t |m-l| [g(l + 1, l + 1[l], E)] |m-l|/2 [g(l, l[l + 1], E)] |m-l|/2 = g r (l, l, E)t |m-l| 2 E -l+1 + [g(l + 1, l + 1, E)] -1 |m-l|/2 2 El + [g(l, l, E)] -1 |m-l|/2

g f (l 1 , m 1 , E) = g f (l 1 , l 1 , E)t |m 1 -l 1 | 2 E -2 + [g f (l 2 , l 2 , E)] -1 |m 1 -l 1 |/2 • 2 E -1 + [g f (l 1 , l 1 , E)] -1 |m 1 -l 1 |/2 (B.21) • l = 1 , |m -l| odd: rst site 1 , second site 2 g f (l 1 , m 2 , E) = g f (l 1 , l 1 , E)t |m 2 -l 1 | 2 E -2 + [g f (l 2 , l 2 , E)] -1 |m 2 -l 1 | + 1 2 • 2 E -1 + [g f (l 1 , l 1 , E)] -1 |m 2 -l 1 | -1 2 (B.22)
• l = 2 , |m -l| even: rst site 2 , second site 2 g r (l, m, E) = g r (l, l, E)t |m-l| 2 E -l+1 + [g(l + 1, l + 1,

E)] -1 |m-l|/2 2 E -l + [g(l, l, E)] -1 |m-l|/2 g f (l 2 , m 2 , E) = g f (l 2 , l 2 , E)t |m 2 -l 2 | 2 E -1 + [g f (l 1 , l 1 , E)] -1 |m 2 -l 2 |/2 • 2 E -2 + [g f (l 2 , l 2 , E)] -1 |m 2 -l 2 |/2 (B.23)
• l = 2 , |m -l| odd: rst site 2 , second site 1

g f (l 2 , m 1 , E) = g f (l 2 , l 2 , E)t |m 1 -l 2 | 2 E -1 + [g f (l 1 , l 1 , E)] -1 |m 1 -l 2 | + 1 2 • 2 E -2 + [g f (l 2 , l 2 , E)] -1 |m 1 -l 2 | -1 2 (B.24)
Lower band: (E 2 < E < 2 )

The correlation functions in the lower band are extracted from the expressions above, replacing the on-site correlation functions (locators) g f (l i , l i , E) by g lb (l i , l i , E).

Upper band:( 1 < E < E 1 )

The correlation functions in the upper band are also extracted from the expressions above, replacing in this case the on-site correlation functions (locators) g f (l i , l i , E) by g ub (l i , l i , E).

Note that all the Green functions correlating sites l and m depend only on the absolute value |l -m|, and they are therefore symetric in real space for this TB Hamiltonian.

  Soy libre. III Spin dependent transport in antiferro and ferrimagnetic tunnel junctions III.1 Spin dependent transport in antiferromagnetic-based tunnel junctions . . . III.1.1 Spatial distributions of spin density and spin transfer torque in 1-D antiferromagnetic-based tunnel junctions: staggered torques . . . . III.1.2 Spin density and spin transfer torque in 3-D antiferromagnetic-based tunnel junctions with uncompensated layers III.1.3 Charge current density and antiferromagnetic tunneling magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.2 Spin dependent transport in ferrimagnetic-based tunnel junctions . . . . . III.2.1 Characteristic lengths of spin density and spin transfer torque in 1-D ferrimagnetic-based tunnel junctions: spin-modulated torque waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.2.2 Period and amplitude of the spin-modulated torque waves . . . . . III.2.3 Spatial distributions of spin density and spin transfer torque in 3-D ferrimagnetic-based tunnel junctions: spin-modulated torque waves with damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.2.4 Pseudoperiod of the spin-modulated torque waves . . . . . . . . . . III.2.5 Charge current density and ferrimagnetic tunneling magnetoresistance103 III.2.6 Spatial distributions of spin density and spin transfer torque in 3-D antiferromagnetic-based tunnel junctions with compensated layers: out-of-plane torque modulations . . . . . . . . . . . . . . . . . . . . III.2.7 Investigation on the origin of the spatial distribution of spin transfer torque in ferrimagnetic-and antiferromagnetic-based tunnel junc-Information storage represents one of the main pillars of our modern technology-based society. Current technological needs for telecommunications, data processing, informatics or digital imaging require reliable memory devices with high access speed, low power consumption, non volatility and high memory storage density.

Figure I. 1

 1 Figure I.1 Three fundamental magnetic orderings of matter.

  I.2). The overall resistance is relatively low for parallel alignment and relatively high for antiparallel alignment. The ferromagnetic layers are spontaneously coupled antiferromagnetically through the NM metallic spacer (high resistance), but the magnetic conguration can be tuned via an external magnetic eld.

Figure I. 2

 2 Figure I.2 Schematic illustration of GMR adapted from Ref. [Chappert et al., 2007] and

(

  AMR) sensors in commercial magnetoresistive HDD read heads.

Figure I. 3

 3 Figure I.3 Spin valve magnetic sensor introduced by IBM as a magnetoresistive head

  Fig. I.4, when the magnetizations of the two F leads are parallel, the DOS at the Fermi energy for majority(minority) electrons is high(low) in both leads. Majority electrons in the left lead are transmitted through the insulating barrier and occupy the majority band in the right lead.

Figure I. 4

 4 Figure I.4 Schematic explanation of the TMR eect in terms of the spin-split DOS of two identical F leads for parallel (left) and antiparallel (right) states. The resistance in the parallel state is lower since the DOS at the Fermi level for majority down electrons at both leads is high. The Fermi levels at the left and right leads are shifted with respect to each other by an energy eU when a voltage U is applied across the structure.

  . I.5. The magnetic elements to store binary information are integrated at the crossing points of two perpendicular arrays of conducting lines, working as a static semiconductor RAM chip but with the advantage of non-volatility with power o.

Figure I. 5

 5 Figure I.5 Basic cross point architecture of MRAM adapted from Ref.

Figure I. 6

 6 Figure I.6 Conducting electrons owing through the rst thin F layer get spin polarized along the magnetization M. If magnetizations M and M' are not collinear, the spin orientation of the polarized electrons is modied again when traversing the second F' thin layer. This last change of spin angular momentum of conducting electrons implies a torque exerted on the magnetization M hence the term spin transfer torque.

  show no stray eld which is benecial for ultimate downsize scalability. Although synthetic antiferromagnets (SAF, i.e. two Fs coupled antiparallel usually by RKKY interactions) are used to overcome device malfunction at reduced lateral dimensions associated with F stray elds (e.g. crosstalk in MRAM: mutual inuence of neighbouring cells supposed to be isolated one from another, dened above), SAFs never entirely compensate, and small, Chapter I. Introduction but non-zero stray elds persist. With AFs, the net compensation is intrinsic except for a very small part at the interface (see Fig. I.7).

Figure I. 7

 7 Figure I.7 Reduction of the stray eld for AF-like magnetic structures.

  2.3) due to its extremely low damping constant α ≈ 4 • 10

  Figure I.8 Schematic illustration of the spin-resolved DOS for a half-metal showingexplicitly the gap at the Fermi level for the spin-down channel[START_REF] Inomata | Highly spin-polarized materials and devices for spintronics[END_REF].

Figure

  Figure I.9 (a) Spin-polarized DOS for the half-metallic AF V 7 MnFe 8 Sb 7 In predicted in [van Leuken and de Groot, 1995] using the local density approximation. (b) Spin-

  Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism II.1 Magnetic tunnel junction model II.1.1 Coherent transport A magnetic tunnel junction consists of a thin insulating layer of a few angstroms [Serrano-Guisan et al., 2011] separating two magnetic metal layers. The schematic geometry modelling the magnetic tunnel junctions analysed in this work is shown in Fig. II.1

Figure II. 1

 1 Figure II.1 Schemes adapted from[START_REF] Kalitsov | Spin-transfer torque in magnetic tunnel junctions[END_REF] for the specic case of FI leads. (Top) Schematic picture of the FI-MTJ consisting of left and right semi-innite FI leads separated by a nonmagnetic barrier of N B atomic layers. The magnetization M' of the right lead points along the z direction. The non equilibrium on-site torques are represented by the in-plane T and out-of-plane T ⊥ components and the spin current densities are also indicated (Q ij ). (Middle) Schematic representation of sublattices A and B in the FI leads, with dierent spin splittings ∆ A and ∆ B corresponding to up and down localized spins respectively. The magnetization M of the left lead is parallel to the FI/B interface (i.e. in the xz plane) and is rotated by an angle θ around the y axis. The Greek primed and unprimed letters denote atomic sites in the right and left FI leads, respectively, and the Latin letters denote the sites in the barrier. (Bottom) Schematic illustration of the spinresolved bands for itinerant electrons and the potential prole, where the 1-D densities of states for up and down itinerant spins are split by δ = ∆ A -∆ B and the lower and upper bands for a given spin are separated by a gap of ∆ A + ∆ B . B is the spin-independent on-site energy in the barrier and V is the potential applied through the junction. The lower dashed line indicates the Fermi level at equilibrium, set at 0 eV in all regions, when no voltage is applied.

A 1 -

 1 D and 3-D problem: real space and momentum representations As stated above, the calculation of all physical quantities of interest for the 3-D xztranlationally invariant tunnel junction depicted on Fig. II.1 is reduced to a 1-D scattering problem plus a k integration. The spatial dependence of the TB Hamiltonian for the whole junction is condensed in the quantum number specifying the layer in which an itinerant electron can be found. The strategy to study the properties of the 3-D junction is to solve rstly the 1-D scattering problem and then perform a k integration after the consideration of a specic dispersion relation modelling the planes of the layer. II.1.3 The magnetic tunnel junction Hamiltonian Given the layer structure sketched in Fig. II.1, the FI-MTJ is described using a single orbital simple cubic tight binding (TB) Hamiltonian which sums the Hamiltonian terms accounting for the isolated left (L) and right (R) electrodes, the barrier (B), and the leads-barrier interactions:

  Note the relation between the spin splitting ∆ λ and the exchange eld ∆ λ (bold sign): ∆ λ = 2∆ λ . ∆ λ is assumed to match the direction of local magnetization in the local spin density approximation (LSDA). The z component of torque is zero and the in-plane and out-of-plane components of the local torques dened in Fig. II.1 are given by the so called x and y components of spin accumulation, ŝx λ and ŝy λ respectively:

  Fig. II.1, the out-of-plane spin accumulation (perpendicular to the B/FI interface) originates the in-plane torque T λ , and the in-plane Chapter II. Quantum transport with tight binding models: nonequilibrium Green's function formalism spin accumulation (parallel to the B/FI interface) originates the out-of-plane torque T ⊥ λ :

  Fig. II.2. Thus, local torques must be summed up taking into account the orientation of localized spins on each site: local torques that tend to rotate the localized spin clockwise(counter clockwise), both in plane and out of plane, are considered positive(negative). With this convention:

Figure

  Figure II.2 (a) Eective in-plane torque in an AF lead resulting from local torques of alternating sign from layer A (up spins) to the adjacent layer B (down spins). (b) Ineective in-plane torque due to local torques of the same sign in adjacent layers; Heisenberg eld maintains the lattice rigid and the order parameter does not change orientation.

Figure II. 3

 3 Figure II.3 Schematic illustration of the dierent energy regions in which the retarded Green's function takes dierent expressions. The exact values of the band edges depending on the TB parameters of the FI are detailed in Appendix B.

  Figure II.4 Real and imaginary parts of the (on-site) retarded Green's function's diagonal elements in real space for an innite FI chain (Bethe lattice) with ∆ A = 0.5eV , ∆ B = 0.25eV and t = -1eV . Left(right) panels: up(down) spins. Top(bottom) panels: on-site retarded Green's function on the A(B) sublattice (odd sites)(even sites). The vertical lines denote divergences.

Figure

  Figure II.5 Real and imaginary parts of the (on-site) retarded Green's function's diagonal elements in real space for an innite F chain with ∆ = 0.5eV and t = -1eV . Left(right) panel: up(down) spins. Vertical lines denote divergences.

  Fig. II.6 shows the spin-resolved energy dependence of the right FI lead LDOS (proportional to the imaginary part of the retarded Green's function) for the ve sites next to the right barrier interface. An AF lead exhibit the same LDOS, but majority and minority bands are not split. However, the dierent proles of majority and minority LDOS at the B/AF interface might be the origin of spin polarization of AF leads; it will be discussed in the next chapter. The imaginary part of the surface Green's function at the rst site of the right lead is plotted on the top panel of Fig. II.6. The LDOS for up spins in a given site is symmetric to the LDOS for down spins when it is shifted to lower energies by δ = ∆ A -∆ B . Note that the LDOS in Fig. II.6 approaches the bulk value (corresponding to the innite chain) shown in Fig. II.4 as the site considered gets further away from the right barrier interface. The LDOS converges to the bulk value in an oscillatory way. This is more clearly illustrated in Fig. II.7, where the LDOS of the rst site in an innite chain (bulk) and sites 15 and 101 in a semi-innite chain are plotted as a function of energy. The behaviour of these oscillations around the Fermi level is used as a rst attempt to explain the theoretical results that will be presented in the next chapter. By way of comparison, the LDOS for the limiting case of a F semi-innite chain is shown in Fig. II.8.

Figure

  Figure II.7 Imaginary part of the on-site retarded Green's function (LDOS) for down spin in an innite FI chain at site 1 (top panel). Idem for a semi-innite chain at sites 15 and 101 (middle and bottom panels).

Figure II. 8

 8 Figure II.8 Imaginary part of the on-site retarded Green's function (LDOS) for a semiinnite F chain with ∆ = 0.5eV and t = -1eV . Blue(red) lines are for up(down) spins. α corresponds to the rst site of the right lead in contact with the barrier.

  l ˆ . The Dyson equation then reads: ĝr (n, m) = ĝr 0 (n, m) + i,j ĝr 0 (n, i)δ i,l δ j,l ˆ ĝr (j, m) = ĝr 0 (n, m) + ĝr 0 (n, l)ˆ ĝr (l, m)

Fig

  Fig. II.9 shows the dispersion relations (II.61) and (II.62), for up and down spins respectively split by the parameter δ. The Fermi wave vectors for up and down spins are thus dierent.

Figure II. 9

 9 Figure II.9 Spin-resolved dispersion relation within the FBZ for a 1-D FI with 0 = 1.5 eV, ∆ A = 0.5 eV, ∆ B = 0.25 eV and t= -1 eV. The wave vector k is in 1/a units.

  Fig. II.10, the divergences of the 1-D FI DOS at the band edges are due to the vanishing derivative of the dispersion relation (II.60) at the center and at the edges of the FBZ. This DOS is the one shown at the bottom of Fig. II.1.

Figure

  Figure II.10 Spin-resolved DOS for a 1-D FI with 0 = 1.5 eV, ∆ A = 0.5 eV, ∆ B = 0.25 eV and t= -1 eV.

Figs

  Figs.II.11 and II.12 show that the dispersion relation and DOS for an AF chain are the same as those of a FI, however, majority and minority bands are no longer split since there is no macroscopic magnetization.

Figure

  Figure II.11 Dispersion relation within the FBZ for a 1-D AF with 0 = 1.5 eV, ∆= 0.5 eV and t= -1 eV. The wave vector k is in 1/a units.

Figure

  Figure II.12 DOS for a 1-D AF with 0 = 1.5 eV, ∆= 0.5 eV and t= -1 eV.

  As shown in Figs. II.13 and II.14, lower and upper FI/AF sub-bands for a given spin merge into a symmetric single band. Majority and minority bands are also split.

Figure

  Figure II.13 Spin-resolved dispersion relation within the FBZ for a 1-D F with 0 = 1.5 eV, ∆= 0.5 eV and t= -1 eV. The wave vector k is in 1/a units.

Figure

  Figure II.14 Spin-resolved DOS for a 1-D F with 0 = 1.5 eV, ∆= 0.5 eV and t= -1 eV.

  are based in the simple in-plane spin conguration shown in Fig. II.15 a). This spin conguration constitutes a square Bravais lattice of uncompensated spins that gives rise to the commonly used TB 2-D dispersion relation depicted in Fig. II.16-a) [Economou, 2006]: k = 0 + 2t[cos(k x a) + cos(k z a)]

  Fig. II.15b). In this case, there exist two dierent on-site energies for a given spin on the same plane, 1 and 2 , corresponding to the sites hosting up and down localized spins in the square Bravais lattice.

Figure

  Figure II.15 a) Bravais square lattice of uncompensated spins on a plane perpendicular to transport. b) Bravais lattice of compensated spins on a plane perpendicular to transport.

  Fig. II.15 b). The two sub-lattices of up and down localized spins in Fig. II.15 b) are generated by the two families of direct lattice vectors {R 1 } and {R 2 }. Any vector owning to one of these families can be expanded as a linear combination of the following primitive vectors:

  67) which is plotted in Fig. II.16 b).

Figure

  Figure II.16 a) 2-D dispersion relation for an electron on the plane of uncompensated spins shown in Fig. II.15 a) corresponding to Eq. (II.66). 0 = 1 eV and t = -1 eV. b) 2-D dispersion relation for an electron on the plane of compensated spins shown in Fig. II.15 b) corresponding to Eq. (II.67). 1 = 1 eV, 2 = 2 eV and t = -1 eV. The wave vectors k x and k z are in 1 a units.

  It is now possible to calculate expression (II.68) with the usual Feynman-Dyson diagram technique; since the times belonging to the negative branch (noted with subscript -) are posterior to the times belonging to the negative branch (noted with subscript +), one needs to use the four Green's functions called casual, anti-casual, greater and lesser, whose expressions are respectively (see Fig. II.18):

Figure

  Figure II.18 Illustration of the casual G C , anti-casual GC , greater G > and lesser G < Green's functions via their correlated times on the Keldysh contour.

  Retarded and advanced Green's functions describe the propagation of particles or excitations forward and backward in time respectively and contain all the spectral information on the local density of states. The Keldysh function satises a Quantum Kinetic equation (described below) which is analogous to the Boltzmann equation in the case of quantum systems where coherence of states is of fundamental importance. They are use to compute the lesser Green's function, which provides the average occupation number of each quantum state.

  overall DOS in AFs remains unpolarized at the Fermi level, the local interfacial DOS becomes spin polarized giving rise to TMR and STT. This is illustrated in Fig. II.6 for the FI case, the only dierence compared to AFs being the absence of split of majority and minority bands. Thus STT and TMR behavior is strongly inuenced by the two uncompensated layers next to the insulating barrier.

Figure III. 1

 1 Figure III.1 Layouts of the tunnel junction modied from [Kalitsov et al., 2009] for AF leads. (Middle) Schematic illustration of the two AF sublattices of up and down localized spins whose magnetic interaction with itinerant electrons is modeled by the staggered spin splitting ∆. The lower and upper bands for a given itinerant spin are separated by a gap of 2∆.

  Fig. III.2. These torques and spin densities were calculated using Eqs. II.29, II.31 II.12.

  Figure III.2 Computed spatial distribution of the in-plane and out-of-plane components of the voltage-induced torque (T , top, and T ⊥ , bottom) and spin density (S ⊥ , top, and S , bottom) for a 1-D F/B/AF junction with θ = π/2, V= -0.1V and ∆ = 0.5 eV for both F and AF leads. The local out-of-plane torque T ⊥ (0V) and in-plane spin density S (0V) at zero bias were subtracted to obtain the out-of-plane torque and the in-plane spin accumulation respectively (see Fig. 3 for more details). Note that the equilibrium inplane torque T (0V) and related out-of-plane spin density S ⊥ (0V) are zero. The red and blue thick lines representing the out-of-plane (top) and in-plane (bottom) spin densities respectively refer to the right-hand ordinate.

Figure III. 3

 3 Figure III.3 Computed spatial distribution of the total out-of-plane torque T ⊥ and the total in-plane spin density S for a 1-D F/B/AF junction with θ = π/2 at -0.1V and 0V (RKKY interaction) with ∆ = 0.5 eV for both F and AF leads. The blue and green thick lines representing respectively the in-plane spin densities at -0.1V (top) and 0V (bottom) refer to the right-hand ordinate.

  fected fairly deep into the electrode: subtracting the IEC to the total out-of-plane torque Chapter III. Spin dependent transport in antiferro and ferrimagnetic tunnel junctions smoothes out the oscillations (which are highlighted by the blue curve representing the in-plane spin accumulation in the upper plot of Fig. III.3), leading to the atter staggered torque shown in the lower plot of Fig. III.2. The result is that the two branches in each sublattice converge much faster to the bulk value. Unlike the nonequilibrium out-of-plane torque, the torque exerted at zero bias in each individual sublattice of the right AF lead as well as the in-plane spin accumulation are staggered. Consequently, the global out-ofplane torque spatial distribution at zero bias is not, since the sign of the local torque and spin density alternates every two sites (owning to each of the sublattices). The spatial distribution of the staggered in-plane torque is much simple, and can be characterized by the amplitude A, dened as the dierence between the torques delivered in two sites owning to each of the sublattices, as shown in Fig. III.4 (a). This gure illustrates the spatial distribution of the in-plane torque in a symmetric AF/B/AF junction, which demonstrates that similar STT spatial distributions were found using AFs instead of Fs as polarisers.

Figure

  Figure III.4 (a) Denition of the amplitude A (in µeV) of the in-plane torque in a 1-D AF/B/AF junction with θ = π/2, V= -0.1V, 0 = 1.5 eV and ∆= 0.5eV. (b) Voltage dependence of A in the same system. The line is a linear t to the data.

Figure

  Figure III.5 (Left) Angular dependence of the in-plane spin current density component for three dierent 3-D AF/B/AF MTJ with a bias of 0.5V and a barrier height U B of 9 eV. (Right) Idem for the out-of-plane component.

  Fig. III.6. It is clear that the staggered spatial distribution of the in-plane torque as well as the outof-plane torque in the bulk survives the k integration for 3-D geometries. The spatial behaviour of the in-plane component is similar in 1-D and 3-D systems.

Figure III. 6

 6 Figure III.6 Computed spatial distributions of the total in-plane (T ) and out-of-plane (T ⊥ ) components of the torque in µeV per unit of surface for a 3-D AF/B/AF junction with θ = π/2, V= -0.1V, 0 = 1.5eV and ∆= 0.5 eV in both AF leads. The inset is a zoom of the out-of-plane component from the sixth layer to stress its staggered character.

Figure III. 7

 7 Figure III.7 Spatial distribution of the out-of-plane torque in µeV per unit of surface at zero bias (RKKY interaction) for a 3-D AF/B/AF junction with θ = π/2, V= -0.1V, 0 = 1.5eV and ∆= 0.5 eV in both AF leads. The inset is a zoom of the torque from the third layer.

  local out-of-plane and in-plane spin densities responsible for the local STT are depicted in Fig. III.8. A constant out-of-plane spin density originates a perfectly staggered in-plane torque in the whole right lead. A constant in-plane spin density originates a staggered out-of-plane torque in the bulk.

Figure

  Figure III.8 (a) (Top) Computed spatial distribution of the in-plane torque (T ) and out-of-plane spin density (S ⊥ ) for a 3-D AF/B/AF junction with θ = π/2, V= -0.1V 0 = 1.5 eV and ∆ = 0.5 eV in both AF leads. (Bottom) Local ratio T /S ⊥ showing explicitly the exchange eld ∆ = 2∆ (b) (Top) Computed spatial distribution of the out-of-plane torque (T ⊥ ) and in-plane spin density S ) for the same system. (Bottom) Local ratio T ⊥ /S showing explicitly the exchange eld ∆ = 2∆. The red and blue thick lines representing the out-of-plane and in-plane spin densities in (a) and (b) respectively refer to the right-hand ordinate.

Figure

  Figure III.9 (a) Dependence of the in-plane torque amplitude A in µeV per unit of surface

  1.3). The spatial distribution of the two torque components for two dierent values of the mentioned hopping parameters at the interfaces is plotted in Fig. III.10 (a) and (b), corresponding to t b,α = 0.4 eV and 0.7 eV respectively. The results reported previously for t b,α = 1 eV are plotted in Fig. III.10 (c) by way of comparison. The hopping parameter inside the leads is kept at 1 eV.

Figure

  Figure III.10 Computed spatial distributions of the in-plane (T ) and out-of-plane (T ⊥ ) components of the torque in µeV per unit of surface for a 3-D AF/B/AF junction with θ = π/2, V= -0.1V, 0 = 1.5eV and ∆= 0.5 eV in both AF leads. The hopping parameter at the barrier interfaces (coupling) is t b,α = t a,α = (a) 0.4 eV. (b) 0.7 eV. (c) 1 eV. As in Fig. III.6, the inset is a zoom of the out-of-plane component from the sixth layer.

  Fig. III.11-(b) shows the charge current density as a function of applied bias for antiparallel, perpendicular and parallel magnetic states of left and right AF leads (the magnetic states are dened here as the relative orientation of the two layers of uncompensated spins at the AF/B and B/AF interfaces, see Fig. III.1). By way of comparison, the same features are shown in Fig. III.11-(a) for a conventional F-based tunnel junction. In the low bias region, where the charge density

Fig

  Fig. III.11-(d). The position of the threshold might be ascribed to the curvature change of the AF DOS at the Fermi level (see Fig. III.1 and II.12) when a voltage is applied,

III. 2 . 1

 21 Characteristic lengths of spin density and spin transfer torque in 1-D ferrimagnetic-based tunnel junctions: spinmodulated torque waves In order to illustrate the essential features of the STT spatial distribution in the right FI lead, a 1-D FI-MTJ for which the calculation of local torques does not require the k integration is rst examined. Fig. III.13 shows the corresponding voltage-induced part of local on-site torques and spin accumulations. The equilibrium zero voltage torques were subtracted.

Figure

  Figure III.13 Spatial distribution of the in-plane (T ) and out-of-plane (T ⊥ ) nonequilib- rium torque components and local spin density components (S ⊥ and S ) for a 1-D system; θ = π/2, ∆ A = 0.5 eV, ∆ B = 0.25 eV and V= -0.1V. The equilibrium torques and local spin densities were subtracted: T (0V) and S ⊥ (0V) equal zero in contrast to T ⊥ (0V) and S (0V) that result from interlayer equilibrium RKKY interactions. The thick red and blue curves(symbols) refer to the right-(left-)hand ordinate.

Fig

  Fig.III.13 demonstrates that the spin transfer torques calculated using Eqs. II.29 and II.31 are perfectly modulated by spin accumulation, similar to the AF case. In particular, the torque deposited in each sublattice scales with the spin density according to the spin splitting/exchange eld of that sublattice. This is shown for sublattice B in Fig.III.13 (top), (lower spin splitting/lower torques) and for sublattice A in Fig.III.13 (bottom), (higher spin splitting/higher torque).

  Fig. III.14 for clarity. In the same Figure, A and P are plotted as a function of voltage in the right panel. In contrast to the AF case (see Fig. III.4), the amplitude A does not increase monotonically with voltage, but reaches a maximum for a bias of 0.35 V. This might be related with the saturation of the current at the same voltage, which is examined in the next section for 3-D junctions.

Figure

  Figure III.14 (a) Denition of the amplitude A (in µeV) and the period P (in number of layers) of the in-plane torque in a 1-D FI/B/FI junction with θ = π/2, V= -0.1V, 0 = 1.5 eV, ∆ A = 0.5 eV and ∆ B = 0.25 eV. (b) Voltage dependence of P (open circles referred to the left hand ordinate) and A (solid circles referred to the right hand ordinate) in the same system.

  Fig. III.15 represents the local distribution of the in-plane and out-of plane torque components for two dierent FI leads, characterized by the spin splittings in each of the sublattices.

  Fig. III.15 (b) with respect to the FI plotted in Fig. III.15 (a), the most striking dierence isthe dramatic change of the period. This period depends on the interplay between the spin splittings ∆ A and ∆ B of each sublattice; in particular, the period P depends on the dierence δ = ∆ A -∆ B . A high δ leads to a short spatial period P and vice versa, a low δ leads to a longer P, as illustrated in Fig.III.16.

Figure

  Figure III.15 Computed spatial distributions of the in-plane (T ) and out-of-plane (T ⊥ ) components of the torque in µeV for a 1-D FI/B/FI junction with θ = π/2, V= -0.1V, 0 = 1.5eV and (a) ∆ A = 0.5 eV; ∆ B = 0.1 eV (δ= 0.4 eV). (b) ∆ A = 0.5 eV; ∆ B = 0.4 eV (δ= 0.1 eV).

Figure

  Figure III.16 For six dierent 1-D FI/B/FI MTJ, 1/δ dependence of the in-plane torque component period (P) for θ = π/2 and V= -0.1V. The double symbols at 4 and 10 eV -1 represent FI electrodes with unequal spin splittings ∆ A , ∆ B but the same δ = ∆ A -∆ B . These symbols are horizontally shifted to show the presence of two points/FIs. Upper(Lower) inset: spatial distribution of the parallel torque wave for 1/δ = 4 eV -1(1/δ = 10 eV -1 ). Dashed line is a linear t to the data.

Figure

  Figure III.17 (a) Dependence on the hopping parameter t of the in-plane torque component waves period P for θ = π/2 and V= -0.1V. (b) Corresponding dependence with t 5 of the torque waves amplitude. The three FI leads of (a) and (b) have ∆ A and ∆ B given in eV in the caption of (a). The lines are linear t to the data.

Figure

  Figure III.18 For dierent 1-D FI/B/FI MTJ, 1/δ dependence of the in-plane torque component period P for θ = π/2 and three dierent applied bias. The dash black, solid black and dot blue lines are linear ts to the data corresponding to -0.1V, -0.25V and -0.4V respectively.

  Fig. III.19 represents the amplitude variations with the barrier height, the number of sites and the hopping parameter at the interfaces. As can be seen in Fig. III.19 (c), a good contact at the barrier interfaces, i.e. high hopping parameter t b,α is crucial to have high torque amplitudes. The dependence on the barrier height and the number of layers from Fig. III.19 (a) and (b) is quite intuitive: the amplitude decreases with the barrier height and exponentially decays with the number of sites, due to the specic dependence of the conductance on the transmission coecients of the junction.

Figure

  Figure III.19 Dependence of the torque wave's amplitude on (a) the barrier height U B . (b) the barrier's number of sites N B . (c) the hopping parameter t b,α at the interfaces.

  Figure III.20 Computed spatial distributions of the in-plane (T ) and out-of-plane (T ⊥ ) components of the torque in µeV per unit of surface for a 3-D FI/B/FI junction with θ = π/2, V= -0.1V, 0 = 1.5eV and (a) ∆ A = 0.6 eV; ∆ B = 0.43 eV (δ= 0.17 eV). (b) ∆ A = 0.5 eV; ∆ B = 0.4 eV (δ= 0.1 eV).

Figure

  Figure III.21 Spatial distribution of the out-of-plane torque in µeV per unit of surface at zero bias (RKKY interaction) for a 3-D FI/B/FI junction with θ = π/2, V= -0.1V, 0 = 1.5eV, ∆ A = 0.6 eV and ∆ B = 0.43 eV (δ= 0.17 eV).

  Fig. III.20. It would correspond to half a period in the 1-D case. III.2.4 Pseudoperiod of the spin-modulated torque waves The two insets in Fig. III.22 illustrate the in-plane torque deposited in each layer of the right lead in two 3-D FI-MTJ with dierent δ. ξ is plotted in Fig. III.22 as a function of 1 δ which governs the oscillations period in 1-D FI-MTJ.

Figure

  Figure III.22 For four 3-D FI leads, 1/δ dependence of the torque characteristic length (ξ) for θ = π/2 and V= -0.1V. Upper(Lower) inset: spatial distribution (in µeV per unit area) of the in-plane torque wave for 1/δ = 4 eV -1 (1/δ = 16.7 eV -1 ). The dashed line is a linear t to the data.

  emphasized here the importance of the staggered character of the in-plane torque over the penetration depth given by ξ. In eect, similar to the AF case examined in III.1.1, the torque deposited in sublattice A adds up to the torque delivered in sublattice B, since localized spins owning to dierent sublattices are aligned in opposite directions (seeFig. II.2). This is in contrast to the case of ordinary F-MTJ, where positive and negative local torques are counterbalanced to give the total torque delivered, which is found to be comparable in amplitude to the torques discussed in this letter. Therefore, in-plane torques in FI leads with low values of δ are expected to be very ecient thus driving current induced order parameter dynamics.Chapter III. Spin dependent transport in antiferro and ferrimagnetic tunnel junctions eV in the case of the FI with ∆ A = 0.5 eV and ∆ B = 0.25 eV examined in the Figure.Eects of roughness at the barrier interfacesThe robustness of the STT wave patterns face to barrier eects is next tested by reducing the hopping parameters t α,a and t b,α at the interfaces, similarly to what we did for the 3-D AF case in section III.1.2, and in 1-D FI systems in section III.2.2. Fig. III.24 (a) and (b), show the STT spatial distribution corresponding to t b,α = 0.4 eV and 0.7 eV respectively. The hopping parameter inside the leads is kept at 1 eV.

Figure

  Figure III.24 Computed spatial distributions of the in-plane (T ) and out-of-plane (T ⊥ ) components of the torque in µeV per unit of surface for a 3-D FI/B/FI junction with θ = π/2, V= -0.1V, 0 = 1.5eV, ∆ A = 0.5 eV and ∆ B = 0.25 eV in both FI leads. The hopping parameter at the barrier interfaces (coupling) is t b,α = t a,α = (a) 0.4 eV. (b) 0.7 eV. (c) 1 eV.

Figure

  Figure III.25 Calculated voltage dependence of the charge current density I in nA by unit of surface

  3-D symmetric AF/B/AF tunnel junction constituted of two leads of compensated layers is depicted in Fig. III.26.

Figure

  Figure III.26 Computed spatial distributions of the STT components for a 3-D symmetric AF/B/AF junction of compensated layers with θ = π/2, 0 = 1.5eV and ∆= 0.45 eV in both AF leads. (a) In-plane torque (T ) at V= -0.1 V. (b) Out-of-plane torque (T ⊥ ) at V= -0.1 V. (c) Out-of-plane torque at V= 0V (RKKY interaction). (d) Voltageinduced-out-of-plane torque (the out-of-plane torque at zero bias was subtracted to the out-of-plane torque at -0.1V).

  their amplitude and a parameter ξ analogous to ξ for FI, dening the length of the rst modulation beating (depicted in Fig. III.26 (b)). The dependence of both parameters on the spin splitting ∆ is plotted in Fig. III.27.

Figure

  Figure III.27 Variation of the wave length scale ξ and the amplitude A with the spin splitting ∆ in a 3-D AF/B/AF junction of compensated layers with θ = π/2, 0 = 1.5eV and V= -0.1 V.

  Fig. III.28 (a), (b), (c) and (d) respectively.

Figure

  Figure III.28 Spatial variation of the local density of states (LDOS) at the Fermi level for an itinerant spin in a 1-D innite chain (blue solid circles) and a 1-D semi-innite chain (open circles) with: (a) Ferromagnetic order characterized by 0 = 1.5 eV and ∆= 0.5 eV. (b) Antiferromagnetic order, 0 = 2 eV and ∆= 1 eV. (c) Ferrimagnetic order, 0 = 1.5 eV, ∆ A = 0.5 eV and ∆ B = 0.4 eV (δ= 0.1 eV). (d) Ferrimagnetic order, 0 = 1.5 eV, ∆ A = 0.5 eV and ∆ B = 0.25 eV (δ= 0.25 eV).

  simple at distribution of spin accumulation shown in Fig. III.8. Further investigations are thus required to explain the fundamental origin of the rich spatial distributions of torques and spin densities.

Figure IV. 1

 1 Figure IV.1 Precession movement of the magnetization M(t) around the eective eld H ef f described by the rst term at the right-hand side of Eq. (IV.1).

  , usually found from comparison with experiments. It represents all relaxation mechanisms arising from the coupling of the magnetization with the environment, namely crystal lattice vibrations, magnons, conduction electrons and other external sources. Thus, the magnetization relaxes in a timescale of the order of nanoseconds.

Figure IV. 2

 2 Figure IV.2 Illustration of the precession movement of the magnetization M around the eective eld H ef f described by both terms at the right-hand side of Eq. IV.1, adapted from Ref [Rezende et al., 2013].

Figure IV. 3

 3 Figure IV.3 Scheme of the FMR measurement setup

Figure IV. 4

 4 Figure IV.4 Schematic illustration of the FMR measurement conguration: the static DC eld is applied in the plane of the sample along the x direction; the RF eld is applied in the plane of the sample perpendicular to the DC eld.

Figure IV. 5

 5 Figure IV.5 Dierential absorption ∂χ ∂H as a function of the applied static eld. The inset corresponds to the Lorentzian susceptibility χ .

  IV.6).

Figure IV. 6

 6 Figure IV.6 Spin pumping eect in a F/NM structure. The precessing magnetization of the F pumps a pure spin current into the adjacent NM material, where it is absorbed.

r

  and g ↑↓ i are the real and imaginary part of the transverse spin mixing conductance, which describes spin transport at the interface between two metals. It determines how much spin current crosses the interface. The pumped spin current, ows perpendicular to the F/NM interface, and delivers some of its angular momentum to the NM lattice, for instance by spin-ip relaxation processes or spin dephasing mechanism. The delivered angular momentum at the NM causes a loss of angular momentum to the ferromagnet F, which manifests itself as an additional damping (torque) term. This additional term α pump , or extrinsic damping has a non-local nature in the sense that angular momentum is lost outside the precessing F, that is, absorbed by the surrounding materials/interfaces under study. It must be distinguished from the usual local intrinsic damping α 0 discussed above. The extrinsic damping term must be included in the LLG equation taking into account the expression of the pumped spin current given Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets in Eq. (IV.7):

  e takes into account all additional layers and interfaces encountered by the spin current (other than the simple F/NM structure, such as the one depicted in Fig. IV.10), and is a function of the layer conguration.

  dure. Fig. (IV.8) shows the hysteresis loops for two Si/SiO 2 /NiFe(8)/Cu(3)/IrMn(t IrMn ) /Al(2) stacks with 0 and 8 nm of IrMn. They exhibit the same negligible shift of around 2.5 Oe, which is therefore likely ascribed to vortex trapped in the superconducting coil of the VSM set up. This was further conrmed by rotating the sample 180 o , which kept the eld shift invariant.

Figure IV. 8

 8 Figure IV.8 Hysteresis loops at 300K for two Si/SiO 2 /NiFe(8)/Cu(3)/IrMn(t IrMn )/Al(2) stacks with t IrMn = 0 (open circles) and 8 nm (solid circles).

Figure

  Figure IV.9 FMR measurement setup scheme

Fig

  Fig. IV.10 illustrates the FMR measurement conguration and principle. The local magnetization (m(t)) of the NiFe is excited to resonance by a small RF pumping magnetic eld (H RF ). As a consequence, the NiFe generates a pure spin current (I pump s ) when oscillating around the local eective eld (H ef f ) due to the spin pumping eect. This spin current diuses through the NiFe/Cu/AF trilayers and concurrently transfers spin angular momentum to the conduction electrons. It is so to say absorbed or reected at interfaces and within layers due to spin dependent scattering and in return it aects the

Figure

  Figure IV.10 Scheme illustrating the phenomenon for our Si/SiO 2 //NiFe/Cu/AF/Al stacks with AF = IrMn or FeMn and deposited either without or with a magnetic eld (H dep ) applied along dierent directions. A precession of the NiFe magnetization around the eective magnetic eld (H ef f ) is initiated by application of an external RF eld (H RF ) under a given static DC eld (H DC ). Relaxation of the NiFe magnetization along

  6. A linewidth versus ω/2π plots and a representative spectrum are shown in Fig. IV.11. The inhomogeneity component ∆H 0 was always controlled in order to have acceptable values ensuring a correct homogeneity of the sample. Typical values of ∆H 0 were of the order of 0.6 Oe.

Figure

  Figure IV.11 Dependence of the resonance linewidth (∆H pp ) with the excitation fre- quencies (ω/2π) for Si/SiO 2 //NiFe(8)/Cu(3)/IrMn(t IrMn )/Al(2) (nm) stacks with t IrMn = 0; 1 and 3 nm. The lines are linear t to the data. Inset: typical resonance spectrum, i.e. dierential power absorption ∂χ ∂H VS DC bias eld (H DC ) for t IrMn = 1 nm and (ω/2π) = 10 GHz; the peak-to-peak linewidth gives ∆H pp and is indicated by the arrow.

Fig

  Fig. IV.12 shows α pump vs t AF for IrMn and FeMn. For some t AF , either the same sample is measured twice or two samples of the same composition are deposited and measured. The maximum dierence is observed for t FeMn = 15 nm and denes the error bars of ∼ 1.4 × 10 -4 .

Figure

  Figure IV.12 Dependence with t AF of the AF contribution, via spin pumping, to the NiFe magnetization damping (α pump ) for Si/SiO 2 //NiFe(8)/Cu(3)/AF(t AF )/Al(2) (nm) stackswith various AF thicknesses (t AF ) and AF = IrMn and FeMn. For IrMn, the straight line is a linear t proportional to: 2t IrMn /λ IrMn for t IrMn < 1.4 nm and a guide to the eye above. For FeMn, the dashed line is an exponential t of the form: A • [1 -exp(-2t FeMn /λ FeMn )]. The spin dependent transport characteristic length in the AF is λ AF .

Figure

  Figure IV.13 Dependence with t IrMn of the IrMn contribution, via spin pumping, to the NiFe magnetization damping (α pump ) for Si/SiO 2 //NiFe(8)/Cu(3)/IrMn(t IrMn )/Al(2) (nm) stacks with various IrMn thicknesses (t IrMn ) and for various directions of magnetic eld applied during deposition (H dep ). The inset is a zoom of the low thicknesses region.

IV

Fig

  Fig. IV.15 shows the hysteresis loops for these stacks performed at 300 K to measure the pinned layer reversal eld. They conrm the longitudinal and transversal orientations of the pinned ferromagnet.

  1 )(E -2 )[4t 2 -(E -1 )(E -2 )] 1 )(E -2 )[4t 2 -(E -1 )(E -2 )]

Résumé

  En électronique de spin, le couple de transfert de spin (STT) et la magnétorésistance tunnel (TMR) dans les jonctions tunnel magnétiques à électrodes ferromagnétiques (F) sont deux phénomènes physiques essentiels. Dans cette thèse, nous présentons une étude théorique du STT dans des jonctions tunnel antiferromagnétiques (AF), où deux électrodes non plus F mais AF sont séparées par une barrière isolante non-magnétique. Plus concrètement, les comportements du STT et de la TMR sont étudiés dans des jonctions tunnel AF cristallines, et ce, à l'aide de calculs de liaisons fortes dans le cadre du formalisme de Keldysh. Nous avons observé une distribution spatiale uniforme et de signe alternatif de la composante perpendiculaire du STT, ce qui est similaire au comportement de la composante parallèle. Ces variations spatiales de la composante perpendiculaire sont cependant spéciques à l'utilisation d'une barrière tunnel et contrastent avec les eets observés par le passé pour le cas de couches séparatrices métalliques. De plus, contrairement aux jonctions tunnel F conventionnelles, nous avons montré que la TMR peut augmenter avec la tension appliquée et atteindre des valeurs du même ordre de grandeur que pour des vannes de spin usuelles : tout-métallique et à électrodes F. L'analyse eectuée pour des AF est ensuite étendue aux matériaux ferrimagnétiques (FI), pour lesquels les AF constituent, somme toute, des cas limites. La complexité magnétique additionnelle inhérente aux FI se traduit par un comportement spatial du STT beaucoup plus riche dans les jonctions tunnel FI. Nous observons notamment que les paramètres électroniques tels que les largeurs et les décalages de bandes ont une très forte inuence sur le STT. Plus particulièrement, la diérence entre les couplages d'échange inter-spin locaux des deux sous-réseaux du FI donne lieu à une distribution spatiale du STT ondulatoire qui est modulée par la densité locale de spin. Il est possible d'ajuster cet eet en jouant sur la tension appliquée aux bornes de la jonction tunnel FI. Nous trouvons de plus que la diérence entre les couplages d'échange inter-spin locaux constitue un paramètre fondamental pour la quantication des longueurs caractéristiques du STT dans les FIs. Ce paramètre peut être considéré comme un champ d'échange eectif, par similitude avec le cas usuel des Fs qui présentent un champ d'échange homogène. Pour nir, nous avons sondé expérimentalement les longueurs caractéristiques du STT dans des AFs polycristallins. Pour de l'Ir 20 Mn 80 et du Fe 50 Mn 50 , nous avons déterminé les longueurs de pénétration de spin et les mécanismes d'absorption de courants de spin à température ambiante en utilisant la résonance F et le pompage de spin. Plus précisément, nous avons associé les profondeurs de pénétration critiques à deux mécanismes d'absorption distincts: du déphasage pour l'Ir 20 Mn 80 et du retournement de spin pour le Fe 50 Mn 50 .

  through the STT eect (current-induced writing) is scalable. It turns out that the amplitude of the torque per unit area is proportional to the injected current density;

II.1.

The unwanted inuence of the writing process of a magnetic element on neighbouring cells is therefore signicantly reduced if the STT eect is used instead of non-local external switching magnetic elds (eld-induced writing). The writing current is directly sent into the magnetic element to write. Another important advantage is that the writing Chapter I. Introduction process

  or equivalently hv g /(2∆ ex ), with v g the spin-averaged group velocity and ∆ ex the exchange splitting. ∆ ex is smaller for IrMn compared with the usual F due to a Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets

↑ F -k ↓ F )

where k ↑(↓) F are the majority (minority) Fermi wave vectors,

Acknowledgments

Estas líneas de agradecimiento cierran definitivamente este manuscrito, tres años de doctorado, muchos años de duro trabajo y una etapa importantísima de mi vida. He de confesar que me ha costado más escribirlas que cualquier capítulo de la tesis. Les debo tanto a tantas de las personas que vienen a continuación, que me parecía que unas simples palabras se quedarían más que cortas. Gracias a ellas he podido llegar hasta aquí.

inmejorable. Gracias a todos por su fantástico regalo de fin de tesis: tocar el piano cómodamente sentadito de la FMR. Y a Nikolai por saber arreglar y conocer mucho mejor que yo el sistema segundos después de que yo mismo le haya explicado cómo se utiliza. Mogollón de electrones han sido bombeados gracias a ellos (y a varias muestras depositadas por el amabilísimo Stéphane). . Muchísimas gracias a Christophe (de verdad) por de la sala limpia habría acabado conmigo (y con la sala limpia).

Contento estoy de haber compartido despacho con Alexandru, Abhijit, Bertrand, no lo quiera), le recomendaría encarecidamente tener a su lado a alguien como Eldar. La tarde que pasamos juntos solucionando los errores de compilación me salvaron del desastre global.

Les debo innumerables risas (sobre todo durante las comidas) a Marina, Murat, Eldar, Léa, Andrey y Hongxin. Una vía de escape inestimable, especialmente después de mi época de "Lobo Solitario".

Gracias a Emilie y a Nahuel por haberme dejado sus transparencias para preparar la audición de la Escuela Doctoral, una de las razones por las que pude conseguir la beca. Gracias también a Christophe y Emilie por el glorioso template de Latex en el que escribí todas estas cositas científicas sobre funciones de Green y tal y tal.

Merci beaucoup a Claudine Chaffy por haber sido mi tutora durante estos dos últimos años como profesor en la universidad. Por la tranquilidad que me dio desde el principio, por todo el material que puso a mi disposición, por sus inestimables consejos y por sus fantásticos informes sobre mi trabajo.

Grazie mille a Simone, mi director de stage de Master 2. Un tipo genial (y un músico apasionado) que me escribió una carta de recomendación para conseguir la beca de doctorado y trabajar en un proyecto de otro laboratorio. Sin comentarios. Creo que deberíamos haber compartido más cervezas, más conversaciones y más conciertos aquí en Grenoble.

Merci a Matthieu, mi codirector de proyecto de ingeniería, una gran persona que siempre se ha portado estupendamente conmigo.

Due to numerical instability, the J-V curves are not symmetric in this particular case for positive and negative voltages (as it should be due to the symmetry of the junction), which is amplied in the calculation of the TMR ratio (see Fig. III.12 (bottom) ). The bumped behaviour for positive voltages suggests that the negative voltage (corresponding to right-going electrons) is the more reliable branch. TMR ratios over 10% are found in all this voltage range. In conclusion, the particular choice of the AF strongly aects not only the value of the TMR but also its voltage dependence. Chapter III. Spin dependent transport in antiferro and ferrimagnetic tunnel junctions [START_REF] Tulapurkar | Spintorque diode eect in magnetic tunnel junctions[END_REF], [START_REF] Sankey | Spin-transfer-driven ferromagnetic resonance of individual nanomagnets[END_REF], [START_REF] Sankey | Measurement of the spin-transfer-torque vector in magnetic tunnel junctions[END_REF], [START_REF] Kubota | Quantitative measurement of voltage dependence of spin-transfer torque in MgO-based magnetic tunnel junctions[END_REF],

III.1.4 Summary

where the magnitude and direction of the STT are extracted in magnetic tunnel junctions, could also demonstrate the theoretical results predicted in this work.

Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets IV.3 Experimental determination of characteristic lengths for spin dependent transport in NiFe/Cu/AF trilayers

In order to study spin absorption in AF layers with the FMR technique, F/NM/AF heterostructures were used. The F material was chosen to be NiFe since it gave the better signal-to-noise ratio in comparison with Co, the second F tested. The thickness of 8 nm was chosen to have a high α pump value (α pump is inversely proportional to the thickness of the F layer [Ghosh, 2013]) and a good layer quality (continuous layer). Cu(3 nm) was used to decouple the F from the AF. The two AF analysed here are IrMn and FeMn; their thicknesses were varied to study the evolution of α pump .

IV.3.1 Samples preparation

Sample deposition

Samples were deposited using a classical Plassys dc-magnetron sputtering equipment.

Magnetron sputtering is a thin lm deposition process in which atoms from a solid "target" fabricated from materials that one wants to deposit are ejected due to bombardment by energetic particles onto a substrate (here Si/SiO 2 (500nm)). These energetic particles are the positive ions of the inert gas Argon introduced in the vacuum chamber (Ar + ), which are part of a plasma constituted of Argon atoms, Argon ions and electrons. The plasma is created by means of electric discharges applied to the target that ionise the Ar atoms through collisions with electrons. Ar ions are accelerated by an additional DC electric eld and strike the target (bombardment) with sucient force to eject atoms from it. A magnetron placed beneath the target creates a closed magnetic eld that traps the plasma near the surface of the target, enhancing the eciency of the initial ionization process.

The atoms ejected from the target (source) reach the substrate placed in its vicinity and condense on the surface producing the desired thin lm. A shutter installed between the substrate and the target controls the deposition time. Here, the base pressure was 10 -3 mbar and typical sputtering rates range between 0.5 and 1 Angstrom/s.

Calibration of the deposited layer thickness

We checked the accuracy of the deposited thicknesses by calibration of the Plassys equipment using x-ray reectivity measurements on a control sample. The reectivity measurements consist in the analysis of the X-ray interference pattern between the Xrays reected at the surface of the sample and those reected at the interface between the deposited layer and the substrate.

Sample cut

The optimal sample size for the FMR measurements was determined to be 3 × 2 mm 2 [Ghosh, 2013]. This size gives a good FMR absorption signal but is small enough to cover an approximately homogeneous DC eld region. In addition, the macrospin approximation is expected to be applicable for this geometry. At a rst stage, the thin layers were deposited on typical thermally oxidised silicon substrates of 3 × 26 mm 2 . I hand-cut them in samples of approximately 3 × 2 mm 2 . However, high dispersion in the damping measurements was found and ascribed to the geometrical variations from sample to sample due to the imprecision of the cut. A high-precision disc saw machine was then used to eliminate uncertainty due to the sample geometry, but it was detected that the liquid used to cool down the sample during the cutting process removed partially the materials deposited. The most adequate preparation method was found to be saw cut of the substrates and then deposition of the thin layers. This sequence eliminates uncertainty in the sample sizes and problems of material removal during the cut. It gave the most reproducible results.

Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets λ IrMn concerns the spin transverse component absorption when λ FeMn is associated to both longitudinal and transverse components absorptions. Such paramagnetic α pump vs t FeMn variations at room-T are the consequence of the low exchange splitting ∆ ex of FeMn [START_REF] Nogués | Exchange bias[END_REF] , [START_REF] Bailey | Pd magnetism induced by indirect interlayer exchange coupling[END_REF] . It is not unlikely that, at room-T FeMn lms thinner than λ FeMn are paramagnetic rather than antiferromagnetic given the reduced bulk T N of FeMn compared to IrMn [START_REF] Nogués | Exchange bias[END_REF] to which addsnite size eects additionally reducing T N [START_REF] Lombard | IrMn and FeMn blocking temperature dependence on heating pulse width[END_REF], [START_REF] Zhang | Thickness-dependent curie temperatures of ultrathin magnetic lms: Eect of the range of spin-spin interactions[END_REF] . Accurately measuring and estimating nite size eects on T N is not simple and very few corresponding literature is available for AF materials. A toy model in Ref.

[ [START_REF] Zhang | Thickness-dependent curie temperatures of ultrathin magnetic lms: Eect of the range of spin-spin interactions[END_REF] reproduces nite size eects on F layers critical temperature.

The model is transferable to AFs [START_REF] Ambrose | Finite-size eects and uncompensated magnetization in thin antiferromagnetic CoO layers[END_REF] and gives the following general power law:

where n is the number of AF monolayers (ML), N 0 the AF exchange length and k an integer. For 3D Ising models, k is close to 1.6. Conversely, accurate values of N 0 are not straightforwardly accessible to experiments and models. Alternatively, T N is accessible to experiments via ultrafast measurements of F/AF exchange bias bilayers. The blocking temperature (T B ) is the temperature above which the F is no longer pinned in a xed direction by the AF. It depends on various parameters among which the F/AF interfacial coupling, the AF bulk properties (AF-AF exchange stiness and grain volume) and time. In particular, T B increases with the F magnetization sweep-rate and reaches the AF intrinsic critical Néel-T (T N ) in the nanosecond regime [START_REF] Lombard | IrMn and FeMn blocking temperature dependence on heating pulse width[END_REF] . Ref. [START_REF] Lombard | IrMn and FeMn blocking temperature dependence on heating pulse width[END_REF] is to our knowledge the only paper dealing with that: for 30 ns pulses, the critical T for IrMn reduces from 350-400 • C (i.e. bulk value) for 6.5 nm to 200 • C for 4.5 nm. In the case of FeMn, the same authors measure a reduction from 200 • C (i.e. bulk value) for 7 nm to 100 • C for 5 nm. Such measurements are compatible with T N lower than room-temperature for few nm thick FeMn.

Fig. IV

.12 also shows that for both IrMn and FeMn layers, α pump levels out for thick AF. The α pump saturation value (α pump sat ), i.e. after maximum spin absorption, seems to be slightly larger for IrMn. Given the above mentioned distinct behaviours for IrMn and FeMn, α pump sat originates from the corresponding distinct mechanisms. For the paramagnetic-like FeMn, α pump sat is mostly related to spin ipping that is bulk-like.

In contrast, for the F-like IrMn, α pump sat mostly relates to the eective spin mixing conductance (g ↑ e ↓) that mostly depends on the Cu/IrMn interface (g ↑ Cu/IrMn ↓) since the values of α pump reported in this study result from the dierence between the damping for NiFe(8)/Cu(3)/AF(t AF )/Al(2) and NiFe(8)/Cu(3)/Al(2) (nm). As described in Ref.

[ [START_REF] Ghosh | Penetration depth of transverse spin current in ultrathin ferromagnets[END_REF] , the measured values of eective spin mixing conductance from the addition of the AF layer do not depend on the spin mixing conductance of NiFe/Cu (g ↑ NiFe/Cu ↓ /S ∼ 14.4 ± 1.4nm -2 , which cancels due to the Cu Sharvin conductance correction of the same order of magnitude. In addition, to a rst approximation, given

Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets that the AF randomizes spins over short distances, the Cu/AF interface mainly drives spin mixing. If the uncompensated spins at the AF interface were fully oriented toward the same direction, Cu/IrMn spin pumping conductivity (g ↑ Cu/IrMn ↓ /S) would be expected to be similar to Cu/F, typically around 14 to 16 nm -2 . However AF interfaces are known to be highly frustrated [START_REF] Takano | Interfacial uncompensated antiferromagnetic spins: Role in unidirectional anisotropy in polycrystalline Ni 81 Fe 19 /CoO bilayers[END_REF] , [START_REF] Akmaldinov | Benet of inserting a (Cu/Pt) intermixing dual barrier for the blocking temperature distribution of exchange biased Co/(Cu/Pt)/IrMn stacks[END_REF] and the resulting overall picture gives few uncompensated spins (e.g. tiny F regions) at the AF interface, also responsible for exchange bias. While an AF spin surface in contact with a F is tuned by the interfacial F spin conguration that orients the AF uncompensated spins in a preferential direction after eld cooling, in the present case of Cu/AF the AF interfacial uncompensated spins are rather randomly oriented positively and negatively.

Therefore, the inuence of the uncompensated AF interfacial spins on the Cu/AF spin mixing conductance is hard to anticipate here. Finally, note that the Cu/IrMn interface is surely more complex due to the formation of CuMn spin-glasses [START_REF] Akmaldinov | Benet of inserting a (Cu/Pt) intermixing dual barrier for the blocking temperature distribution of exchange biased Co/(Cu/Pt)/IrMn stacks[END_REF] , [START_REF] Chouhan | Magnetic phase diagram for cumn[END_REF].

IV.3.5 Isotropic character of the spin current absorption: eects of magnetic eld during sample deposition

For the F-like spin absorbing IrMn, it might be inferred that spin pumping, i.e.

the reciprocal eect to STT should also show a directional character. An advantage of F res /N/AF trilayers with respect to F res /N/F for investigating this issue is that a F spin sink is oriented by the high resonance DC bias magnetic elds necessary for the experiment at high H RF frequencies. In contrast, once oriented, the AF is not aected by such DC magnetic elds and stays still during the experiment. As will be discussed below, this attempt to evidence the anisotropy character of spin pumping either failed within the noise level or unexpectedly do show no directional character.

For the F-like spin absorbing IrMn, Fig. IV.13 shows α pump vs t IrMn for samples fabricated with three dierent magnetic eld congurations during the deposition process:

no eld (same data as in Fig.

IV.12 reproduced here as a reference and meaning a random orientation of the IrMn interfacial uncompensated spins) and static magnetic elds (H dep )

along the transversal and longitudinal in-plane directions in order to orient the rst few IrMn layers when still paramagnetic at the rst stage of the growth. We performed atomic force microscopy measurements in order to detect surface inhomogeneities due to preferential growing of the sample during the deposition process under applied eld. No signicant anisotropies were found in the surface prole.

What matters in the following is the orientation of the IrMn interface relatively to the DC magnetic eld that governs the F res precession axis, which is along the transversal direction of the sample.

Despite the fact that the noise level precludes drawing ne conclusion on the damping Chapter IV. Experimental determination of spin transfer torque characteristic lengths in antiferromagnets depth [Merodio et al., 2014a]. Although not entirely understood, results on the directional character of spin current absorption were discussed as a basis for comparisons with potential future works.

Further research could involve: 1) Other AFs (e.g. with a variable content of heavy spin scatterers). 2) Various degrees of crystallinity up to single-crystal AF. 3) Various interfacial qualities determined by roughness, stacking faults, species intermixing, etc.; 4)

Variable temperature for studies of the para-to antiferro-magnetic transition temperature that is dicult to determine by many other techniques. 5) Nanostructured elements since AFs can be inuenced by size eects. 6) FI materials. 7) The use of an insulating barrier instead of a metallic spacer.

Note that we also tried to experimentally investigate the eect of a spin-polarized current on an AF in F/NM/F/AF structures with controlled geometries in order to complement the work of [START_REF] Wei | Changing exchange bias in spin valves with an electric current[END_REF]. Although we obtained very disperse results, this thorough attempt is described in Appendix C, along with suggestions for future studies.

Summary and perspectives modulated torques are damped over longer length scales in comparison with F-MTJ. We demonstrate that the electronic and magnetic characteristics of the FI are responsible for the specic features of the torque waves, and consequently for the STT characteristic lengths in FIs. In particular, we identify a fundamental parameter predicted to be inversely proportional to the torque waves spatial period in 1-D FI-MTJ and the spatial extension of the damped torque oscillations in 3-D FI-MTJ. This parameter corresponds to the dierence between the spin splittings of the two FI sublattices and can be considered as an eective exchange eld in FIs, similar to the homogeneous exchange eld characterising Fs. In addition, the bandwidth parametrized here by the hopping matrix element of the TB Hamiltonian is found to be proportional to the mentioned spatial periods. Regarding extrinsic variables, the STT characteristic lengths can also be tuned via the applied bias across the junction; in particular, the spatial period increases monotonically with the applied voltage. Interestingly, its aforementioned dependence on the FI eective exchange eld holds for each given voltage. Hence, STT in FI is predicted to be highly ecient due to its long decaying characteristic lengths and its staggered character commensurate with the magnetic lattice.

Since ideal coherent transport is considered throughout this theoretical part, the eects of disorder represent the obvious object of investigation for future work. Although elastic and inelastic scattering have been predicted by previous authors to have a strong impact on STT and GMR in AF-based spin valves, the use of a tunnelling barrier instead of a metallic spacer is expected to be a suitable mean to ensure momentum conservation and thus overcome at least partially the negative eects of disorder; in eect, tunnelling process is likely less sensitive to elastic scattering. In particular, the characteristic lengths over which the STT does not vanish are expected to be still longer than in conventional F-MTJs. We believe that disorder does not represent an insurmountable constraint for STT and TMR in AF-MTJs. The similarities between Fs and FIs in terms of magnetization and band splitting (which is absent for AFs) suggest that STT in FI leads might be as robust face to disorder as it is for conventional Fs.

The eects of disorder in the STT characteristic lengths in actual AF thin layers are addressed in the experimental part of this thesis. In particular, critical penetration depths and absorption mechanisms of spin currents in Ir 20 Mn 80 and Fe 50 Mn 50 spin sinks at room temperature by means of F-resonance and spin pumping were determined. Dierent room temperature absorption mechanisms of spins were evidenced: dephasing for IrMn and spin ipping for FeMn.

Further research using similar AF thin layers could involve a comprehensive study of the directional character of spin current absorption, other AFs (e.g. with a variable content of heavy spin scatterers) and dierent degrees of crystallinity and interfacial qualities.

Ideally, similar investigations could be carried out for AF and FI nanostructured elements, using in particular tunnel junctions to compare with the theoretical results of this work.

Appendix A

Calculation of the nonequilibrium Keldysh function

The Dyson or Quantum Kinetic equation for the nonequilibrium 2×2 Keldysh function matrix reads:

where the self-energy Σ pq is dened as [START_REF] Caroli | Direct calculation of the tunneling current[END_REF]:

Σ pq = T (δ pα δ aq + δ pa δ αq ) + T (δ pb δ α q + δ pα δ bq )

T is the hopping parameter across the left barrier interface and the non-equilibrium Green function matrices for the uncoupled left and right leads at local equilibrium are:

where f L and f R are the Fermi-Dirac distribution functions in the isolated left and right leads. In particular, at the left and right interfaces, we have:

Using the denition (A.2) in Eq. (A.1) gives:

Fpq = fpq + ĝpα T Faq + ĝpa T Fαq + ĝpb T Fα q + ĝpα T Fbq + fpα T Ĝa aq + fpa T Ĝa αq + fpb T Ĝa α q + fpα T Ĝa bq As explained above, we are interested in evaluating the observables at the right lead, so that the three Keldysh functions Fbα , Fα b and Fλ ,µ are needed. These Keldysh functions are coupled via the Quantum Kinetic Equation. For instance, the Quantum Kinetic Equation for Fbα involves three more coupled equations:

As previously discussed by Caroli et al [START_REF] Caroli | Direct calculation of the tunneling current[END_REF] we assume that in the range of energy of interest, the density of states of the uncoupled barrier is strictly zero.

The tunnelling process of electrons occurs in the gap of the insulator. The density of states is related to the imaginary part of the retarded Green function. In particular, the local density of states at the atomic site i reads:

This means that the retarded Green functions inside the barrier are real (the vanishing of the imaginary part applies also to the retarded Green function correlating dierent atomic sites). In addition, as will be seen below, the retarded Green functions for a given spin are symmetric in real space: g r i,j = g r j,i . The following relations are thus valid inside the barrier:

Therefore, inside the insulator, f i,j = 0, so that: fba = fbb = fba = fab = 0

The Quantum Kinetic equations for the interfacial Keldysh function Fbα are then simpli- ed to give the following system of coupled equations:

1.

Fbα = ĝba T Fαα + ĝbb T Fα α we have:

We shall use T = T for simplicity, so that:

In the same way: and a hopping parameter t. This 1-D chain corresponds to a lattice without closed loops that can be mapped into a Bethe lattice or Cayley which is completely characterized by its number of nearest neighbours Z = 2 or its connectivity K = Z -1 [Economou, 2006].

Splitting the 1-D TB Hamiltonian into an unperturbed site-diagonal part plus an odiagonal perturbation and using renormalized perturbation expansion [Economou, 2006] allows to perform the calculation of the the system's resolvent ĝλ ,µ (z), where z is a complex variable. This resolvent is a more general denition of the Green function in the complex plane, from which the retarded Green function can be dened as follows:

where E stands for the energy variable belonging to the real axis of the complex plane.

The rst step to obtain the retarded Green function for a given spin in the Bethe lattice is the calculation of the diagonal elements or locators of this function in the two types of site of the lattice. As seen below, the o diagonal elements or two site correlation functions can be expressed in terms of these locators. Let's start with the expression (5.57) of [Economou, 2006] for the locator or resolvent on a site l / l = 1 owning to a Bethe lattice with Z nearest neighbours and a connectivity K = Z -1:

In order to calculate the retarded Green function from the resolvent Eq. (B.1), it must be taken into account that the local density of states on a site l is proportional to the retarded Green function:

For E owning to the energy spectrum of the system, the LDOS is = 0, so that the imaginary part of the retarded Green function Im[g r (l, l, E)] is therefore = 0 as well. On the real axis, (z = E ∈ R), the only way to full this condition is to have a square root of a negative real quantity in Eq. (B.1). Allowed states will be conned in those regions of energy where

Let's study the sign of (B.2) to nd these energy bands:

The two real roots of this second degree equation are given by:

and the vertical axis of the parabola is given by the mean value of 1 and 2 :

According to the relative position of 1 , 2 , E 1 and E 2 , we have three dierent cases (recall that 1 > 2 ):

Conditions a) and b) imply that 4Kt 2 < 0, so that the only possibility is E 2 < 2 < 1 < E 1 . In this particular conguration, the product of the three factors at the left hand side of equation (B.2) is negative only in two intervals, so that the spectrum of the extended states splits into two sub bands, the lower one extending from E 2 =

( 1 + 2 ) 2 -

( 1 -2 ) 2 /4 + 4Kt 2 to 2 and the upper one extending from

( 2 -1 ) 2 /4 + 4Kt 2 , and they are symmetric with respect to the vertical axis of the parabola:

In these sub bands, we have:

so that the denominator of Eq.(B.1) exhibit two purely imaginary square roots:

The resolvent on the real axis (depending on the real variable E) then becomes:

determining the retarded Green function on the allowed energy spectrum g r (l, l, E) consists in choosing the sign ±i in such a way that Im[g r (l, l, E)] < 0. As seen above, outside the allowed energy spectrum R > 0 and the imaginary part Im[g r (l, l, E)] vanishes.

Forbidden states: E < E 2 , 2 < E < 1 and E > E 1

The square root in the denominator corresponds to the product [START_REF] Kollar | Green functions for nearest-and next-nearest-neighbor hopping on the Bethe lattice[END_REF]]

where the square roots are given by their principal branches.

E < E 2

In this region, (E -1 )(E -2 ) > 0 and (E -1 )(E -2 ) -4Kt 2 > 0, so we take the positive square roots:

In this region, both (E -1 )(E -2 ) < 0 and (E -1 )(E -2 ) -4Kt 2 < 0, so the two square roots are purely imaginary and we take

In this region, (E -1 )(E -2 ) > 0 and (E -1 )(E -2 ) -4Kt 2 > 0, so we take the positive square roots as well:

In the three cases, the locator has no imaginary part: g(l, l, z) is analytic so that g r (l, l, E) and g a (l, l, E) coincide.

Inside the two sub band regions E 2 < E < 2 and 1 < E < E 1 :

In order to identify the real and imaginary parts of g(l, l, E) in the two sub bands, the denominator in the r.h.s. of Eq. (B.3) must be multiplied by its complex conjugate:

, by denition. The imaginary part then reads:

The sign of Im[g(l, l, E)] is determined by the sign of E -2

Here (E -2 ) < 0 -→ we take the negative square root:

Here (E -2 ) > 0 -→ we take the positive square root:

One could also have chosen the sign of the square root by imposing Im[g(l, l, z) < 0]

in the bands and use Kramers-Kronig relations:

where P denotes the principal value of the integral.

CONCLUSION

The retarded Green function at site l (locator) is a function of the energy E, and takes dierent expressions depending on which interval of E it is evaluated on:

If the locator g(l, l, z) is evaluated on a site l / l = 2 , the numerator in Eq. (B.1) becomes 2K(z -1 ), but the choice of the square root sign remains valid:

From now on, we shall deal with a 1D FI chain, which is a Cayley tree with two nearest neighbours: K = 1.

B.1.1 One-site correlation functions (locators)

Forbidden states: E < E 2 )

Forbidden states:

Eect of a spin polarized current on an antiferromagnet: nanofabrication

C.1 Introduction

One of the initial objectives of this thesis was also to experimentally investigate the eect of a spin-polarized current on an AF in a F a /NM/F p /AF exchange-biased spin valve, following the experiments shown in Ref [START_REF] Wei | Changing exchange bias in spin valves with an electric current[END_REF]. These experiments showed partial evidence of STT acting on the AF due to high density currents spin polarized in the exchange-biased F p in contact with the AF. The second F a was merely used as an analyser to probe the magnetization direction of the F p layer (namely the switching eld at which The transmitted (reected) polarized current from the F exerts a torque on the AF spins that tends to rotate the AF order parameter to a direction parallel (perpendicular) to

Appendix C. Eect of a spin polarized current on an antiferromagnet: nanofabrication the F magnetization. Since the AF uncompensated spins at the interface are exchangedcoupled to the bulk AF, torques acting on the bulk will vary their orientation. In addition, the exchange bias eld increases (decreases) when the component of the uncompensated spins along the exchange-bias direction is altered to increase (decrease) due to STT. These changes in the exchange bias are then monitored through the switching eld of the F p by GMR detection for dierent current densities to evidence the eects of STT on the AF.

The point contact used to generate a high density electrical current and the typical variations of the electrical resistance reported in [START_REF] Wei | Changing exchange bias in spin valves with an electric current[END_REF] Although the reversal of the free F a seems to be little aected by the applied current, the average exchange-bias eld of F p is signicantly altered, which is ascribed to AF-STT. to the hysteresis loop shift [START_REF] Wei | Changing exchange bias in spin valves with an electric current[END_REF].

Joule heating was ruled out as responsible for these eects since not symmetric variations of the exchange bias were observed for positive and negative current polarities. Mutual STT between F a and F p was also discarded due to the invariant reversal eld of F a , even for dierent F thicknesses.

These experimental attempts to show AF-STT only obtained qualitative results due to the uncontrolled current path inherent to the point contact technique, which makes dicult to know precisely the injected current density values. One possible solution is the use of patterned arrays of spin valve nanopillars with controlled geometries to better control the injected electric current density. In an attempt to obtain quantitative results

Appendix C. Eect of a spin polarized current on an antiferromagnet: nanofabrication following the experiments described above, I carried out the nanofabrication of these spin valve arrays. The nanofabrication process I used is the object of the next section.

C.2 Process

Around 1000 elliptical and circular nanopillars of dierent sizes were patterned from a full sheet 2 inch wafers. Resulting from a collaboration with the Universities of Texas and Michigan, Si/SiO 2 /(Ta(5)/Cu(10))×5/Ta(5)/Cu(5)/CoFe(10)/Cu(10)/CoFe(3)/FeMn(8)/ /Au(5) (nm) stacks were provided by the authors of [START_REF] Wei | Changing exchange bias in spin valves with an electric current[END_REF] In addition, very large dispersions were obtained in the R vs H vs j mappings (see Next, the analysis carried out for AFs is extended to ferrimagnets (FI), for which AFs constitute simpler limiting cases. The additional magnetic complexity inherent to FI materials yields to a richer physics concerning the STT spatial behaviour in FI based tunnel junctions. Electronic structure parameters such as band widths and exchange splittings of the FI are shown to have a strong inuence on STT. In particular, the STT spatial distribution within the leads exhibits a striking spin-modulated wave-like behaviour resulting from the interplay between the exchange splittings of the two FI sublattices. This wave-like behaviour can also be tuned via the applied voltage across the junction. Furthermore, the fundamental intrinsic parameter for quantifying STT characteristic lengths in FI metals is identied. This fundamental parameter can be considered as an eective exchange eld in FIs, similar to the homogeneous exchange eld in the F case.

Finally, the STT characteristic lengths in AF materials are investigated experimentally.

Here, room temperature critical depths and absorption mechanisms of spin currents in Ir 20 Mn 80 and Fe 50 Mn 50 are determined by F-resonance and spin pumping. In particular, room temperature critical depths are observed to be originated from dierent absorption mechanisms: dephasing for Ir 20 Mn 80 and spin ipping for Fe 50 Mn 50 .