Acknowledgment

I spent more than 3 years as a PhD student in the IPCMS. I have to say that it has been a pleasure for me to study in Strasbourg and that I met exceptionnal peoples inside and outside of the lab who made these years interesting. I am more than happy to have achieved it.

First, I would like to thank my PhD advisor, Laurent Limot, for welcoming me in his team and giving me the opportunity to work in the fascinating eld of research that is the STM. He allows me great liberty while providing the necessary support when needed, even more during the writing. I thank you for the hours passing to accompain me into achieving this thesis and to have pushed me to be the best I could be. I chose the right advisor ! In second, Fabrice Scheurer was an essential element during this thesis. It is the Jack of all trade of the STM teams. When something gone wrong in the lab and noone knows why, Fabrice will. You acted like a second supervisor, a referee, a teacher and even more. I canŠt enumerate the number of time you solved my problems. I also thank you for the possibility to go to Soleil on the Deimos line. I will remember the nice moments we pass together, even when dismounting the STM.

The second member of the team is Benjamin "Big Ben" Verlhac, which enjoyed the place so much he stayed almost 5 years working in the lab. He is a skillfull scientist and I learn a lot working with him. He showed me many things about experimental work and in the treatment of datas. I also met Nicolas Bachelier, our predecessor, at the beginning of my thesis, which shared the same passion as me for baking the STM. I am still up for a race in the mud with both of you ! I'd like to thank the people from the other side of the corridor, which also use an STM. Guillaume Schull, their enlighted leader, which I thank for the casual interesting conversations and advices. Benjamin Doppagne, which shared my o ce and even my at. A nice guy with a green hand that I hope to see again. Luis Parra Lopez, the Phd student arriving at the same time as me. I hope you'll manage to attack Nimitz and whish you good luck for your manuscript. The other members of the team Anna Roslawska, Kirill Vasilev, Shuiyan Cao, Song Jang. We had a lot of good moment in and outside the lab. I would like to thanks the other members of the DSI department. Virginie Speisser, which kindly introduced me to pumps systems and help us with their maintenance and help me to become familiar with the lab when starting my Phd. And I thank you for the work you do allowing us to remove a big part of the weight due to the administrative task. A thanks to Corinne Bouillet, Hervé Bulou, Adele Carrado, Olivier Bengone, Jacques Faerber, Christine Goyhenex, Michelangelo Romeo and Celine Guillet, which shared a lot of co ee and chocolate after lunch. I had a few beamline times in Soleil synchrotron on the DEIMOS beamline and I'd like to thank their sta , especially Phillipe Ohresser who welcomed us, Loic Joly, Edwidge Otero. With their assistance, we spent days and nights acquiring and trying to extract data from our XMCD spectra.

I want to thank all the jury members who accepted to review this thesis : Jerôme Lagoute, Jörg Kröger, Christine Goyhenex and Christian Meny. I appreciated your remarks about this manuscript. I want to thank Martin Bowen for attending to my mid-term defense session. A special thank to Muriel Sicot, who followed my work through these two events and to her encouraging remarks. I want to thank the workshop members : Maxime Valentin, René Baehr, Halit Sumar, Arnaud Boulard and Nicolas Bayer, for their e ciency, whenever there was a need to build, x a piece of the machine or need nitrogen. I also thank Emmanuel Sternitzky for his help to x the electronics problems we had.

Finally, I thank all my family and friends without whom I would never had arrive at this point. You were all always incredibly supportive, which enabled me to pass every steps of my thesis, especially you mom. A thought to my dad that I hope will be proud of me.

Chapter 1

General introduction 1.1 Preamble

My thesis takes place in the research area devoted at improving how information is stored in a magnetic medium. We are interested, in particular, at pushing the storage capacity to the ultimate spatial limit, that of a single spin. This necessitates to address a spin at the atomic level even within a complex magnetic structure. This goal has stimulated the development of new spin-sensitive techniques. A nano-SQUID, for example, is capable of measuring a magnetic ux through an area as small as 75 nm in diameter [START_REF] Finkler | [END_REF]. Nitrogen-vacancy centers in diamond, which are used as atomic-sized quantum sensors in a scanning probe device, can map the vector magnetic eld with a spatial resolution at the di raction limit (500 nm) [2]. Scanning tunneling microscopy (STM) occupies a special position among these spin-sensitive techniques. To date, it is the only one that can detect a single spin while imaging a surface with atomic resolution. The rst detection of a spin in a single atom with STM goes back twenty years and consisted in the observation of zero-bias anomalies in the di erential conductance (dI/dV ) [3][4][5]. It later became possible to detect the spin by measuring the tunneling magnetoresistance through spin-polarized STM [6,7]. A breakthrough in STM spin detection came in 2004 with the development of spin excitation spectroscopy [8]. This technique uses electrically driven spin excitations to determine and manipulate the spin states of a single quantum object with knowledge of the nearby environment [9,10]. The detection of dipolar or even hyper ne interactions via the recently developed electron spin resonance STM has further paved the way towards a high-sensitivity spin mapping at the atomic-scale [11][12][13].

In this thesis, we aim at using the electrically-driven spin excitations of a molecule to study surface magnetism at the atomic-scale. The original aspect of our work is that a magnetic molecule is placed at the apex of a STM tip. The decoration of metal probe-tips by a molecule intentionally picked up from a surface has proven to be a powerful method to improve the measurement capabilities of a STM. Eigler et al. was the rst to explore tip functionalization with a single adsorbate, a xenon atom [14]. Molecules, however, are preferable to atoms as they add degrees of freedom introducing tip-surface interactions across the vacuum gap that are usually absent when using a metallic apex. These interactions give to STM an enhanced sub-molecular resolution [15][16][17], and provide new chemical insight about molecules on surfaces [18][19][20][21]. The functionalization of the tip with a CO molecule, for example, provides a spectacular contrast in the images leading to the resolution of the atomic positions and bonds within a molecule, even electrically charged [22].

To probe magnetic e ects with a molecular tip it is necessary to have a magnetic molecule at the tip apex, but its experimental realization has remained elusive until recently [23,24]. In conventional STM, the magnetic ground state of an isolated atom or molecule is inferred by collecting spin-related ngerprints in the dI/dV measured with a metallic tip using the above mentioned techniques. The ground state is in uenced by the close magnetic environment of the atom/molecule, hence single atoms and molecules can serve as spin detectors when controllably moved on the surface with the help of the tip. Exchange and surface-mediated Ruderman-Kittel-Kasuya-Yosida interactions have been spatially mapped in this way by monitoring the zerobias peak in the di erential conductance associated to the Kondo e ect [25][26][27][28], the tunneling magnetoresistance [29][30][31], the spin excitation spectra and spin relaxation times [9,[32][33][34].

Clearly, a molecule acting as a well-calibrated magnetic sensor at the tip apex can improve the detection of on-surface magnetic e ects. Firstly, this detection scheme eliminates surfacemediated interactions and bene ts from the vertical-displacement sensitivity of the STM, as the sensor-target distance is no longer imposed by the surface corrugation. Secondly, a molecule adds knowledge about the structure of the tip apex as the junction geometry may be precisely described through the comparison to rst-principles calculations. Otherwise the tip apex structure is unknown, complicating the measurement of a magnetic exchange interaction across a vacuum gap [35][36][37][38][39][40][41]. We show in this thesis that magnetic metallocene molecules can be used to decorate a tip apex and act as magnetic sensors.

The general outline of this thesis is the following. Chapter 1 brie y introduces the physical concepts at stake in the thesis. Chapter 2 is dedicated to the STM, the conductance measurements that can be done in a STM; we also give some technical details on the microscope we used. Chapter 3 presents the rst metallocene molecule of the thesis, nickelocene. We show how its spin excitations can be electrically driven even when the molecule is in a metallic environment. We will show how its spin excitations may be modi ed in the presence of a magnetic atom and of molecular vibrations. The results will be supported by density functional theory (DFT) calculations jointly carried out by the groups of M.-L. Bocquet (ENS, Paris) and N. Lorente (CFM, San Sebastián). The last two chapters present results about the functionalizing of the STM tip by two di erent metallocenes: the spin S = 1/2 cobaltocene (chapter 4) and the S = 1 nick-elocene (chapter 5). In both cases, the molecule can interact with its nearby environment and magnetic information can be gathered from the interaction with surface atoms. The detection scheme di ers slightly, however, as with cobaltocene we exploit its Kondo e ect, while with nickelocene, we will use its spin excitations. In addition, the Kondo e ect of cobaltocene also happens to be coupled to the molecular vibrational modes.

In this chapter we introduce the Kondo e ect in § 6.11; we will give the background of the discovery of the Kondo e ect and the di erent steps leading to the Kondo model. We also introduce how the Kondo e ect is detected in STM and show how molecular vibrations a ect the Kondo signature, § 1.3.1. Finally, we introduce how the spin excitations of an atom can be triggered and detected in STM, § 1.3.2.

Kondo e ect 1.2.1 Background

The metal's electrical resistance can be related to the purity of the metal and mostly to the vibrations of the lattice. High temperatures will cause higher vibrations, which result in a higher resistance. The lattice vibrations can be frozen by reducing the temperature, working with cryogenic uids for example. In this case, the temperature dependence of the resistivity can be expressed with the Bloch-Grüneisen formula:

ρ(T ) = ρ 0 + ρ el-ph (T ), (1.1)
where ρ 0 is the residual resistivity due to the metal defects and is temperature independent, while ρ el-ph (T ) ∝ T 3 at low temperature. Most metals reach the nite electrical resistance value ρ 0 at low temperatures as shown in g. 1.1(a). Materials that loose all electrical resistivity are known as superconductors.

However, in 1934, the resistance of a gold sample was observed to increase after it had reached a minimum in the resistivity-vs-temperature curve [START_REF] Haas | [END_REF], g. 1.1(b). This meant that there must be some additional scattering mechanism giving an anomalous contribution to the resistivity. The presence of minima in the resistivity-vs-temperature curve were also observed in the following thirty years in di erent systems, notably in dilute magnetic alloys [44,45], without being able to explain the phenomenon. In 1964, Knook et al. experimentally showed that the minima in the resistivity-vs-temperature curves (and the negative magneto-resistance) of the alloys were linked to the scattering by impurities having electronic con guration of partially lled d-shells [46], i.e. to impurities having a local magnetic moment. The metals reach a nite resistivity at low temperatures as expected from the Bloch-Grüneisen formula (solid curves). Reproduced from [START_REF] Allen | Boltzmann Theory and Resistivity of Metals[END_REF]. (b) Electrical resistance of Au between 1 K and 5 K. The resistivity-temperature curve reaches a minimum around 3.9 K and then starts to increase while the temperature decreases. Reproduced from [START_REF] Haas | [END_REF].

The Anderson model

In this context, Anderson proposed a model, which will be a crucial step to elucidate this anomaly [47]. His model described a magnetic impurity, i.e. a magnetic atom, embedded in a metal host. The impurity has a localized magnetic moment and a ects the electronic property of the host metal. Anderson described this localized moment as a d-orbital with an energy E with only one electron (spin up). If this state exists, an electron of the opposite spin (spin down) trying to occupy the same level will see the repulsive Coulomb interaction U of the other electron. Therefore the spin down electron will have an energy E + U , where U is the repulsive d-d interaction between the two electrons of opposite spins. This level lies above the Fermi level since it was assumed that this state is empty, g. 1.2(a). The Hamiltonian for a single localized d-orbital is:

ĤA = Ĥhost + Ĥimp + Ĥcorr + Ĥmix . (1.2)
The rst term is the unperturbed energy of the free-electron in the host metal:

Ĥhost = κ,σ κ ĉ † κσ ĉκσ , (1.3)
where κ is the unperturbed energy of the s electrons in the metal, the operator ĉ † κσ (ĉ κσ ) creates (destroys) an electron in state |κ, σ . The second term is the unperturbed energy of the d state of the magnetic impurity:

Ĥimp = E d σ d † σ dσ , (1.4)
where E d is the one-electron energy of a localized d-orbital, the operator d † σ ( dσ ) creates (destroys) an electron in state |d, σ . The third term represents the coulomb energy U between the opposite-spin electrons from the d-orbital

Ĥcorr = U nd ↑ nd ↓ , (1.5)
where ndσ is the occupation number operator of the state |d, σ de ned by ndσ = d † σ dσ . The fourth term corresponds to the s-d interaction describing the tunneling from the s band of the free electrons to a d-orbital of the impurity and vice versa:

Ĥmix = κσ (V κd d † σ ĉκσ + V * κd dσ ĉ † , κσ ). (1.6)
where V κd represents the strength of the coupling between the s-orbital of the impurity and the d-orbital of the conduction band of the metal. It results in a shift and in a broadening of the spins (up and down) energy levels, due to the hybridization of the impurity with the conduction band of the host metal. This broadening is de ned by the half width of the half maximum of the d state:

∆ = πV 2 ρ, (1.7)
where ρ is the density of states (DOS) of the host metal and |V κd | ≈ V . If the coupling is strong enough, the width of the d state crosses the Fermi level and reduces the magnetic moment of the impurity, g. 1.2(b). Additionally, an electron of the conduction band can tunnel into the impurity to form a virtual state of energy E d + U on a time scale of h/E d . This virtual state comprises two opposite spins due to the Pauli principle. Subsequently, an electron of the impurity tunnels back in the conduction band of the host. This tunnel process can be generalized to all d-orbitals of the impurity, explaining the scattering mechanism observed by Knook et al..

The Kondo model

In parallel to Anderson, Kondo used perturbation theory within the s -d interaction model to express the scattering probability of a magnetic impurity interacting with the conduction electrons of a host metal [48]. The two models are equivalent, however the Kondo model explicitly shows that: i) The scattering mechanism involves a magnetic interaction between the spin of the impurity and the spin of the conduction electrons; ii) Conduction electrons must ip their spin during the scattering process. The Kondo Hamiltonian is:

ĤK = Ĥhost + Ĥimp + Ĥex . (1.8)
It is similar to the Anderson Hamiltonian with an exchange hamiltonian Ĥex replacing Ĥcorr + Ĥmix and de ned as:

Ĥex = -J σ • Ŝ = - J 2N κ,κ [(ĉ † κ ↑ ĉκ↑ -c † κ ↓ c κ↓ ) Ŝz + ĉ † κ ↑ ĉκ↓ Ŝ-+ ĉ † κ ↓ ĉκ↑ Ŝ+ ], (1.9)
where J is negative and corresponds to the amplitude of the e ective antiferromagnetic s-d exchange interaction between the impurity and the metal host. σ is the spin density operator of the conduction electrons and Ŝ denotes the spin of the impurity (S = 1/2). ĉ † κσ (ĉ κσ ) creates (destroys) a conduction electron in state |κ (|κ ) in a host metal containing N electrons. Ŝ± are the raising or lowering operators of Ŝ.

The scattering mechanism involves an electron of the host metal that is exchange coupled to the local spin of the impurity. The Ŝoperator changes the spin of the magnetic impurity from an up to a down state, while the ĉ † κ ↑ ĉκ↓ operator changes the spin of a conduction electron from down to up. The Ŝ+ operator changes the spin of the magnetic impurity from a down to an up state, while the ĉ † κ ↓ ĉκ↑ operator changes the spin of a conduction electron from up to (c) Conduction electron after tunneling from the impurity to the conduction. The spin of the impurity switches from down to up and the spin of the conduction electron switches from up to down.

down. The total spin momentum is conserved as the electron spin and the impurity spin ip. This spin-ip mechanism is pictured g. 1.3. The total energy is conserved and the scattering process is elastic. However, the scattering involves N electrons of the host metal, leading to a logarithmic divergence of the resistivity when the temperature is close to zero and, therefore, to a minimum in the resistivity-vs-temperature curve.

By applying a many-body perturbation theory, the Kondo Hamiltonian leads to a new electron state below a characteristic temperature, also known as Kondo temperature T K :

T K E F k B exp(1/ρJ ) , (1.10)
where ρ is the DOS of the host metal, E F is the Fermi energy and k B the Boltzmann constant.

This new state comprises a correlated "cloud" of electronic spins of the host metal that magnetically interact with the impurity spin, g. with ⇑ and ⇓ are the spin orientations of the impurity and ↑ and ↓ are the spin orientation of the conduction electrons. The radius of the electron cloud screening the impurity can be de ned by:

ξ K = v F k B T K , (1.11)
where v F is the Fermi velocity. The cloud, which can easily reach an extension of a few micrometers, has been experimentally observed for the rst time only very recently [49]. In 1966, Schrie er and Wol established a relation between the Anderson model and the Kondo model [50]. They used a canonical transformation in the limit of a small s-d mixing, which is the most favorable case for the occurrence of a localized magnetic moment in a dilute alloy. In the case of low temperature, J can be expressed with the Anderson parameters U , E d and ∆, as

J = 2 π ∆ 2 1 E d - 1 E d + U < 0, (1.12)
where E F is set to zero, g. 1.5.

The Kondo e ect in STM

The spin-ip scattering associated to the Kondo e ect modi es the energy spectrum of the system. A sharp peak appears in the impurity DOS near the Fermi level, which is usually designated as a Kondo resonance. Figure 1.5 presents the spectral function of a magnetic impurity in a host metal. We can observe that the Kondo resonance is located near the Fermi level with a width proportional to k B T K . There are also two side peaks corresponding to the impurity Anderson d-level carrying the local spin (this can also be a f -level). The peaks are located at E d and E d + U and have a width of 2∆.

Li et al. presented the rst experimental observation of a Kondo e ect at the atomic scale in 1998 [4], followed shortly after by Madhavan et al. [5]. They used a cryogenic STM to spatially resolve the electronic properties of individual magnetic atoms adsorbed on a non-magnetic surface, g. 1.6(a). In both experiments, the dI/dV curves acquired above the atoms -cobalt for Madhavan et al. in g. 1.6(b) and cerium for Li et al. in g. 1.6(c) -present a zero-bias anomaly, which is absent when the dI/dV is acquired above the metal surface. Surprisingly, this feature di ers from a peak, in apparent contradiction with the predicted dI/dV curve [52]. Theory and experiment can be reconciled by remarking that the Kondo resonance in the dI/dV is subject to a Fano interference [4,5,53,54]. In 1935, in the context of ionization processes of atoms, Fano described the coupling between a discrete state and a continuum of states and showed that their interference gives rise to characteristically asymmetric peaks in the excitation spectra [55,56].

E -EF (eV) A d (
We have a similar situation in STM. The tunneling current can have two contributions at the impurity site, either from electrons tunneling into the discrete d-level of the impurity, or from electrons tunneling into the continuum of states of the conduction band of the surface. These tunneling pathways interfere and give to the Kondo resonance a Fano line shape in the dI/dV . In the measurements of Li et al. and Madhavan et al., the current is predominantly due to tunneling into the conduction band resulting in a dip in the dI/dV . In this thesis, however, we will deal with molecular systems dominated by electrons tunneling into a discrete d-level (chapters 3 and 4), which then yield a peak.

There are several measurements that can be done to experimentally assert that the zero-bias feature in the dI/dV spectrum is a Kondo resonance. The most used is to measure the temperature dependence of the resonance. The width Γ of a spin-1/2 Kondo resonance should increase with temperature as

Γ(T ) = 1.455/2 (2k B T K ) 2 + (αk B T ) 2 , (1.13)
where α = 3.83 [57]. Figure 1.7(a) presents a typical example of such a temperature dependence where it can be seen that a temperature of T > 10T K is necessary to completely wipe out the Kondo resonance [58].

Another possibility is to probe the Kondo line shape in a external magnetic eld. The line shape is predicted to split apart with increasing magnetic eld [59]. Figure 1.7(b) presents an example of such a splitting measured above a Mn atom [8]. When no external magnetic eld is applied, the dI/dV spectrum reveals a resonance at the Fermi energy, while it splits symmetrically apart when adding a magnetic eld of 4.2 or 7 T. The splitting is linear with the magnetic eld (not shown). This behavior can be easily understood by recalling that the Kondo ground state is a linear combination of |⇑, ↓ and |⇓, ↑ states ( § 1.2.3). In a magnetic eld B, one state is energetically favored with respect to the other, resulting in a Zeeman splitting of the line shape -we will detail further this line shape splitting in chapter 4. The two peaks are separated by 2gµ B B, where g stands for the Landé factor and µ B for the Bohr magneton. To observe this splitting, the magnetic eld needs to be in the order of Γ/2gµ B or higher.

Current-induced excitations in STM

To conclude this chapter, we will focus on how magnetic information may be gathered in STM through the inelastic component of the tunneling current. The important point to keep in mind The inset corresponds to the intrinsic half-width of the Kondo resonance extracted from the ts (red line of the main panel). Reproduced from [58]. (b) dI/dV spectra of a Mn atom near the boundary of an oxide patch on a NiAl (110) surface. The spectra were acquired at a temperature of T = 0.6 K at several external magnetic elds. Reproduced from [8].

is that above a threshold voltage, tunnel electrons can transfer energy to the system investigated (atom or molecule) and excite it. The threshold voltage, which corresponds to the energy of the excitation, produces an upward step in the di erential conductance (see § 2.3.2). The excitations, which can have vibrational and/or magnetic origin, are governed by selection rules and not all may be observed.

Vibrational excitations and Kondo physics

In 1968, Lambe and Jaklevic observed vibrational excitations for molecules contained in metaloxide barriers [60]. They observed a small and sharp increase in their dI/dV curves at speci c biases, which corresponded to the discrete vibrational energies of the molecules. Soon after the invention of the STM, Binnig and Persson predicted that STM should be able to probe the vibrational modes of a molecule adsorbed on a surface using the inelastic component of the tunnel current [61,62]. These theoretical studies paved the way for the experiments, which arrived in 1998 when Stipe et al. measured the vibrations and rotations of a single molecule [63][64][65].

This so-called vibrational spectroscopy will be further presented in chapter 2. Here we will speci cally address how the vibrational excitations a ect the Kondo line shape in STM, which is relevant to chapters 3 and 4. In 2005, Paaske et al. theoretically evidenced that in the presence of an electron-vibron interaction the Kondo resonance breaks up into a number of satel-lite peaks [66]. As shown in g. 6.3(a), these peaks ank the main Kondo resonance, fall at energies symmetric with bias polarity that correspond to the vibrational excitation energies of the system. Experimentally, this vibrational-assisted Kondo is exclusively observed in single molecules [66,[68][69][70][71][72]. As an example, we consider the study by Torrente et al. [70] of a TTF-TCNQ complex on Au (111), where TTF denotes tetrathiafulvalene and TCNQ denotes tetracyanoquinodimethane, g. 6. tunnels elastically (no energy loss) but ips its spin. This produces a sharp Kondo resonance at zero bias;

(2) The electron tunnels inelastically triggering a vibrational excitation in TCNQ. This produces a step in the dI/dV at ±41 meV;

(3) The electron tunnels inelastically triggering a vibrational excitation in TCNQ, but also ips its spin. This produces Kondo satellites peaks at ±41 meV.

Spin excitations

In 2004, Heinrich et al. [8] showed that spin excitations in a single-magnetic atom, i.e. excitations involving Zeeman levels, can be detected with a STM operated in an external magnetic eld. This seminal experiment by Heinrich et al. was carried out at 0.6 K on Mn atoms deposited on Al 2 O 3 islands, g. 1.9(a). The aluminum oxide preserves the spin of the Mn atom by minimizing its hybridization with the host metal, hence the spin of Mn is S = 5/2 as in vacuum. Several salient aspects can be noted in the data: i) When a magnetic eld is applied, steps appear in the dI/dV spectrum above a single Mn atom that are located at threshold biases of ±0.7 mV, g. 1.9(b); ii) The steps correspond to a 25% increase with respect to the elastic conductance -the elastic conductance corresponds to the dI/dV signal below the threshold biases; iii) The steps disappear when the eld is absent and when Mn is placed on NiAl (110); iv) The threshold biases shift linearly with the magnetic eld, gs. 1.9(c).

These ndings can be rationalized by assuming the existence of a Zeeman e ect for the spin S of the Mn atom

Ĥ = -gµ B B Ŝz , (1.14)
where B is an applied magnetic eld along an axis perpendicular to the surface (the z-axis). This axis is chosen as the quantization axis. When a magnetic eld is applied, it is possible to lift the degeneracy of the S = 5/2 state, having then |M = +5/2 as the ground state and the |M = +3/2 as the rst excited state, M is the projected spin momentum. The threshold biases of ±0.7 mV correspond then to a |M = +5/2 → |M = +3/2 excitation of the Mn atom, gs. 1.9(d). This excitation changes the projected spin momentum by δM = -1, forcing the tunneling electron to ip its spin from a |↓ to |↑ in order to conserve the total angular momentum. We will see in chapter 3 that spin excitations with δM = +1 are also possible. In this latter case the tunneling electron must ip its spin from a |↑ to |↓ to conserve the total angular momentum. This limits the detection of spin excitations to transitions with δM = ±1. The spectra are acquired with and without a 7 T magnetic eld. (c) dI/dV spectra of an isolated Mn atom on oxide using di erent magnetic elds. The spectra are centered on the step at positive bias. (d) Spin-state diagram using eq. 1.14. The red arrow designates the spin excitation observed in the experiment. Adapted from [8]. and, more recently, MgO [11,12]. In these systems, the lifetimes are typically in the range of a few 10 ns [33,73] to 1 ms [74] for 3d atoms, reaching 1 hour (at 2.5 K) for a 4f atom of Ho deposited on MgO [75]. When the magnetic atoms are directly adsorbed on a metal substrate the lifetime is greatly reduced instead. This leads to an increased broadening of spectral features and the excitations are poorly resolved in energy so that spin excitations are in general harder to observe on a metal surface [76][77][78]. Khajetoorians et al. have estimated a lifetime of 200 fs for the spin excitations of a Fe atom on Cu (111) [79] with step heights representing only 5% of the elastic conductance. The lifetime limiting process can be traced back to substrate electrons colliding onto the magnetic adsorbate and causing a de-excitation of the system. Spin excitations can be used to collect magnetic information from a variety of spin systems, e.g. phthalocyanine molecules on an oxide [80] and in molecular multi-layers [32], or magnetic impurities on semiconductors [81]. The interactions between spins in individual atomicscale magnetic structures can also be investigated as well by controllably assembling linear chains with the STM tip, one magnetic atom at a time [9]. In this thesis we will use the spin excitations of single molecules to characterize their intrinsic magnetic properties, but also to monitor their magnetic environment. As we show in the following chapters, a magnetic environment produces an e ective magnetic eld, which we can then characterize through the changes it produces on the moleculear spin excitations. These molecules may then be considered as atomic-scale "magnetometers".

Chapter 2

Scanning tunneling microscopy and spectroscopy

In this chapter we brie y introduce Scanning Tunneling Microscopy (STM). We will rst focus on its working principle ( § 2.1) and imaging capacity ( § 2.2). Then we will present the spectroscopic techniques we used during this thesis ( § 2.3). Finally, we will describe our experimental setup ( § 2.4).

Working principle

Developed in 1982 by Binning and Rohrer, the STM allowed to obtain for the rst time topographic images of surfaces with an atomic resolution [82], g. 2.1(a). A few months later, using the STM, the inventors answered an experimental and theoretical problem which was the determination of the 7 × 7 reconstruction of the Si (111) surface [83], g. 2.1(b). From this, it has developed as one of the most e cient technique in surface science. The principle of measurement consists in scanning a metal tip over a conductive surface. The tip and sample act as two electrodes separated by a vacuum-tunneling barrier. When the distance between the sample and the tip is of a few ÃĆngstrÃűm, the electrons can ow through the gap between these electrodes due to the tunneling e ect. This gap constitutes a potential barrier with a height corresponding to the energy necessary to extract an electron from the metal to the vacuum, i.e. the work function. The bias voltages usually employed are much lower, which means that the passage of an electron though the junction is classically forbidden and instead occurs through quantum tunneling, g. 2.2(a). For the sake of simplicity, we describe here the tunneling process by the unidimensional SchrÃűdinger equation:

- 2 2m d 2 ψ(z) dz 2 + V (z)ψ(z) = Eψ(z), (2.1)
where is the reduced Planck constant, m and E are the mass and the energy of an electron, ψ el is the wave function at the distance z between the tip and sample and V the potential barrier. Solving this equation gives the probability to nd an electron in the sample, the tip or in the gap. The solution is in the form:

ψ ∝ exp -kz , (2.2)
where k is a decay constant with φ = V -E.

k = √ 2mφ , (2.3)
This solution shows that when a bias voltage is applied between the tip and the sample, electrons have a non-zero probability to be transmitted from one electrode to the other electrode, g. 2.2(b). This probability decreases exponentially with the distance between the two electrodes. The electron can go through the barrier by "tunneling"; a ow of such electrons is called tunneling current:

I ∝ |ψ| 2 ∝ exp -2kz .
(2.4)

The tunneling current depends exponentially on φ and on the distance z between the tip and the sample. A small variation in the distance z will induce a large change in the tunneling current. As a consequence, there is a di erence in the tunneling current even among the few last rows of atoms of the tip apex, g. 2.2(c). The current owing through the atom closest to the surface (in red) will be higher more than one order of magnitude compared to the current owing through the atoms from the second row (in blue). A STM tip is never perfectly atomically sharp and their is coexistence of di erent apices. Happily, this extreme sensitivity of the tunnel current allows the selection of only one apex for operating the STM.

Imaging

To obtain STM images of the surface, the tip needs to be displaced in order to collect the current in the area of interest. Figure 6.5(a) sketches the scanning system. The tip position is controlled using piezoelectric ceramics, which can move in the three directions of space (x, y, z). The ceramic can be contracted or expanded through the application of a well-controlled high-voltage bias, typically 100 V yields a displacement of 5 nm. It ensures the sensitivity of the tip dis- placement. There are two kind of piezoelements. Some piezoelements are assembled to form a stepper motor to ensure the so-called coarse-motion. They are driven by an electric voltage sawtooth and are used to bring the tip within tunneling distance from the surface. Once in the tunneling regime, another set of piezoelements assembled in a tube allows for scanning a surface area ranging at maximum 250×250 nm 2 , while the tip can be z-displaced up to 400 nm.

The calibration of the piezotube can be achieved using a well-known crystal. We used the step height of a Cu(100) surface to calibrate the z displacement, and images of Cu (100) with atomic resolution to calibrate the x and y displacements.

There are two main ways to record images with a STM. The simplest one consists in scanning at a constant tip height and recording the current collected in the junction. This mode is called constant-height mode and is presented in g. 6.5(b). The current will change locally in case of a change in surface corrugation. An image is then obtained from the value of the current at each position scanned. This mode may be risky when the corrugation is strong as the tip may crash into the surface. To overcome this limitation, a feedback loop is used in order to change the tip-sample distance after collecting the current. This is the most common mode, the user can then scan the sample surface with a given current and the electronics will regulate the piezo elongation to keep the current constant. This is the constant-current mode shown in g. 6.5(c). It means that the tip will retract or elongate if an obstacle is encountered, thereby avoiding a crash. The image is reconstructed from the displacement of the piezo, giving a topographic image of the surface.

The STM images can be distorted by the "creeping" of the piezoelements and by thermal drift. Generally, it a ects the topographies by adding a slope in the x, y, z components of the image and invalidates the determination of the distances and the angles from the studied surface.

Creeping occurs when a strong bias is applied to the piezo, for example when moving to a new location. The piezo will continue to move in the same direction for a certain period of time even after changing back to smaller biases. Creeping can be easily minimized with slow scanning parameters or by waiting a couple of minutes until the image becomes regular. The thermal drift originates from di erences in the coe cient of thermal expansion and in the temperature di usivity of the di erent materials in the STM, typically the scanning piezoceramics and the metal of the sample holder. Consequently, compensating thermal drift is usually required when images are intended to be used for precise determination of both, distances and angles. In our setup, the drift is of 1 nm/h at helium temperature and can be neglected in usual imaging conditions.

Tunneling spectroscopy 2.3.1 Scanning tunneling spectroscopy

To present the techniques used in this thesis in relation to tunneling spectroscopy, we recall the semi-classical Wentzel-Kramer-Brillouin (WKB) approximation to describe the current tunneling from one electrode to the other:

I(z, V ) ∝ ∞ -∞ T (z, E, V )[f (E -eV ) -f (E)]ρ t (E -eV )ρ s (E)dE, (2.5)
where f is the Fermi-Dirac distribution, ρ t and ρ s are the densities of states (DOS) of the tip and the sample, respectively. The tip and sample Fermi levels are displaced from one another by the tunneling bias V , g. 2.4. The transmission coe cient T depends on the shape of the tunneling barrier; a good approximation for the tunneling barrier is a trapezoidal shape [84][START_REF] Wiesendanger | Scanning Probe Microscopy and Spectroscopy: Methods and Applications[END_REF][START_REF] Ukraintsev | [END_REF], g. 2.4.

This shape leads to the following expression of the transmission:

T (E, z, V ) ∝ exp -kz φ e + eV 2 -(E -E ) , (2.6)
with k = 1.025 Å -1 eV -1/2 and φ e is the average work function of the sample and the tip.

The energy E refers to the energy of electrons having a momentum parallel to the junction interface; it does not play an important role in the tunneling process as electrons mainly tunnel with a momentum perpendicular to the surface. The tip is usually indented into the surface to obtain a monoatomic tip apex; hence tip and surface are made of the same metal resulting typically in φ e 4 eV. The transmission T , hence the current, therefore changes by a factor 10 for a displacement of 1 Å.

It is to notice that, by working at helium temperature, we can neglect the thermal excitations since k B T eV and

[f (E -eV ) -f (E)] 1.
If we further reasonably assume that ρ t is constant, eq. 2.5 reduces to:

I(z, V ) ∝ ∞ -∞ T (z, E, V )ρ s (E)dE, (2.7)
which shows that the tunneling current solely depends on the sample DOS and on the transmission.

As seen in eq. 2.7, one can collect information on the local DOS of the surface by measuring the tunnel current as a function of bias at a given tip-sample distance z. This technique is called Scanning Tunneling Spectroscopy (STS). A more re ned way to proceed consists in di erentiating the current relative to bias. Taking eq. 2.7, we obtain:

dI dV (z, V ) ∝ T (z, eV, V )ρ s (V ) + V 0 ρ s (E) dT (z, E, V ) dV dE. (2.8)
If we assume the transmission coe cient T constant in energy near the Fermi level, we nd:

dI dV (z, V ) ∝ ρ s (V )T (z).
(2.9)

The existence of a sample state will induce an increase in the conductance, that appears as a peak in the di erential conductance spectrum. The energy of this peak gives informations about the surface, its defects or a single object adsorbed on it.

Inelastic electron tunneling spectroscopy

Until now we have considered electrons that preserve their energy during the tunneling process, in other words electrons that tunnel elastically. Electrons, however, may loose energy during tunneling and give raise to inelastic phenomena. Jaklevic and Lambe observed that tunneling electrons can interact with the vibrational states of molecules contained in a metal-oxide barrier [60]. They found that this leads to an increase of the conductance at characteristic voltages, corresponding to the vibrational frequencies of the molecules. In STM, this energy loss can induce vibrations within a molecule as recalled in g. 2.5(a), cause transitions among Zeeman levels or excite plasmons in the tunneling junction. If we generically de ne E ex as the excitation energy, when tunnel electrons have energy |eV | < E ex they will not excite the system and the current is elastic. When instead |eV | > E ex the tunneling current is inelastic. Note that in this case the current is elastic up to eV = ±E ex , but above this value it has contributions from The inclusion of inelastic phenomena in STM is called inelastic electron tunneling spectroscopy (STM-IETS or, more commonly, IETS). The change in the slope for the current at eV = ±E ex can be viewed as the opening of a new tunneling channel for the electrons. This leads to the occurrence of steps in the corresponding dI/dV spectrum at eV = ±E ex , g. 2.5(c), and to a peak/ dip in the second derivative d 2 I/dV 2 , g. 2.5(d). The excitation energy E ex of the system Comparison between a dI/dV spectrum recorded with (a) a lock-in ampli er and (b) a numerical derivation of the I(V ) curve. The two curves were acquired in the tunneling conditions. We added the lock-in detected curve to the numerically-derived curve to highlight the change in the signal-to-noise ratio. may be measured through these curves. A rst IETS experiment was conducted by Stipe, Rezaei and Ho [64] at the scale of a single molecule. They resolved the vibrationnal C-H stretch mode of an isolated acetylene (C 2 H 2 ) on a Cu(100) surface, g. 2.6.

Lock-in detection

The dI/dV and d 2 I/dV 2 curves can be obtained numerically by di erentiating the I(V ) curve. This method presents an intrinsic source of noise degrading the quality of the signal, g. 2.7. An alternative method is to use a lock-in ampli er. The main idea is to add an ac modulation of frequency ω/2π to the voltage bias of the STM:

I = I[V + V m cos(ωt + θ)], (2.10)
where θ is an arbitrary phase imposed by the electronic circuit which is a priori unkown. If the modulation amplitude V m is small compared to the bias applied (V m |V |), eq. 2.10 can be Taylor-expanded as:

I[V + V m cos(ωt + θ)] = I(V ) + a 0 + a 1 dI(V ) dV V m cos(ωt + θ) + a 2 d 2 I(V ) dV 2 V 2 m cos(2ωt + θ) + ... (2.
11) The dI/dV and d 2 I/dV 2 curves can be measured by keeping the rst and second harmonics of the output signal, respectively. To measure the rst harmonic, the lock-in ampli er multiplies eq. 2.11 with a phase-shifted modulation signal V cos(ωt+φ) and subsequently sends the signal through a low-pass lter so that the lock-in output yields

lock-in output = a 1 V m V 2 dI(V ) dV [cos(θ -φ)].
(2.12)

The same procedure can be done for the second harmonic by using a phase-shifted modulation signal of V cos(2ωt + φ). Setting the phase φ to match the circuit phase θ allows for a better signal-to-noise ratio. In our case, the spectra of interest are located in a bias range |V | < 100 mV as the spin and vibration excitations observed in this thesis are in the range of a few mV. In order to satisfy the conditions of the Taylor expansion, we used a modulation amplitude as low as 150 µV (root mean square, rms). The choice of the modulation amplitude V m is important as it can broaden the line shape of the tunneling spectrum as we show here below. A balance has to be found between signal-to-noise ratio and broadening induced by the lock-in detection.

During this thesis, we measured all our dI/dV spectra using a lock-in ampli er. The d 2 I/dV 2 spectra were instead obtained by numerically deriving the dI/dV spectra. The signal-to-noise ratio of the tunneling current turned out in fact to be su ciently strong for both cobaltocene and nickelocene molecules.

Instrumental broadening

The line shape of a dI/dV spectrum is prone to broaden due to the temperature of the tunneling junction and/or the the lock-in detection technique used. This instrumental broadening can be present independently on temperature-related e ects of the sample DOS. We use the WKB formalism to provide an analytical description for this source of broadening.

Temperature. A lower limit to the bias resolution of the microscope is given by the Fermi-Dirac edge of the electrodes (tip and surface). When temperature increases, the Fermi-Dirac function broadens thereby smearing out the recorded spectrum. Recalling eq. 2.5 and considering T and ρ t as energy-independent constants, we have

dI dV (V ) ∝ ∞ -∞ ρ s df dV (E -eV )dE, (2.13)
that can be rewritten as a convolution product

dI dV (V ) ∝ ρ s * χ T (E -eV ), (2.14)
where

χ T (E -eV ) = df dV (E -eV ) = χ T (0) cosh[(E -eV )/(2k B T )] 2 , (2.15)
and χ T (0) = e/(k B T ). This broadening function is plotted in g. 2.8(a) and has a full width half maximum (FWHM) of δ T = 3.5 k B T . In other words, the higher the temperature, the stronger the smearing of the spectrum. At our working temperature of 2.4 K we have δ T = 700 µeV.

Lock-in. The lock-in ampli er can be a source of noise since it introduces a periodic modulation in the bias applied to the junction. This may be estimated by considering the rst harmonic obtained through the lock-in by

I ω1 = 1 τ τ /2 -τ /2 I[V + V m cos(ωt)] cos(ωt)dt (2.16)
where τ is the measurement interval time. After some math described in [88], the conductance spectrum measured with the lock-in appears as a convolution product:

dI(V ) dV lock-in = ∞ -∞ dI dV (V -V )χ m (V )dV , (2.17)
where

χ m (V ) = 2 V 2 m -V 2 πV 2 m , (2.18)
is the broadening function of the lock-in signal shown in g. 2.8(b). The modulation has then a FWHM of δ m = 1.7 V m [rms]. In our case, the measurements are performed using at best a modulation of 150 µV, which yields δ m = 250 µeV. Thermal smearing therefore dominates in our setup.

If we take into account both temperature and lock-in smearing, the line shape broadening has the general expression:

dI(V ) dV measured ∝ ρ s * χ T * χ m . (2.19)
The FWHM of the smearing function χ T * χ m depends on both δ T and δ m , and the energy resolution achieved in the tunneling spectra can be approximated as

δ eV (δ T ) 2 + (δ m ) 2 . (2.20)
This yields a resolution of δ eV = 750 µeV at 2.4 K when using a modulation of 150 µV. This value corresponds to the smallest peak width we can resolve in STS, meaning that we cannot differentiate two peaks separated by less than 750 µeV (Rayleigh criterion). The above results are for elastic electrons only. In the case of inelastic electrons, the temperature smearing changes to δ T = 5.4k B T [88], causing the resolution to decrease to δ eV = 1.1 meV. This value represents the smallest separation we can detect among inelastic steps in the dI/dV , g. 2.5(b). 6), SMA connector for current (7), sample holder (8) and tip with its piezoelectric motors (9).

Experimental setup 2.4.1 Brief description

During this thesis, we used a modi ed low-temperature microscope from Omicron operating in ultrahigh vacuum (UHV) and controlled by a Nanonis electronics, g. 2.9. It includes a STM chamber, a preparation chamber and a load-lock chamber, which are separated by gate valves.

The UHV is ensured by a primary pumping stage that consists in a scroll pump working from atmosphere pressure down to 10 -2 mbar and by a secondary stage comprising a turbo-molecular pump. To improve UHV, titanium is routinely sublimated in the chambers. The base pressure is 10 -10 mbar in the preparations chamber and 10 -11 mbar in the STM chamber. During the measurements, these pumps are stopped as they produce mechanical vibrations detrimental to STM stability, and the vacuum is maintained by ionic pumps. The base pressure in the STM results in a very low contamination rate of the sample and tip, which we can keep for several weeks without cleaning. The low temperature is reached with a cryogenic system composed of two cryostats. The inner one contains a liquid helium bath. The external cryostat contains liquid nitrogen, which cools the radiation shield surrounding the helium dewar thereby reducing He consumption. The measurement stage is also protected from thermal radiations by two rotative shields coated with gold. This enables reaching a temperature of 4.4 K for 36 hours on the sample block and 2.4 K for 15 hours when pumping on the He bath of the cryostat. During measurements, the STM head is freely suspended from the chamber through three springs, while pneumatic feet under the whole STM decouple it from ground. This minimizes the mechanical noise from the environment and allows for achieving atomic resolution.

Tip and sample preparation

The tip and sample preparation takes place in the preparation chamber. The chamber is equipped with: i) an ion gun used to sputter the sample with argon ions, ii) a local heating stage used for annealing the sample, and iii) two e-beam evaporators for thin or sub-layer metallic deposition.

During the this thesis, we used tungsten (W) tips obtained from W wires of 0.25 mm of diameter. We used a home-made system to etch the tips in a NaOH solution. We sputtered the tip for one hour in the preparation chamber to remove the contamination and oxidation. The tip was then treated in the STM chamber by indenting it into the surface (< 2 nm) while performing voltage pulses. During STM recording sequences, the operation was repeated until to obtain a monoatomic apex and a at tip DOS in the bias range of interest. Note that this procedure implies that all the tips used were covered with surface material.

We used two di erent copper surfaces during the thesis: Cu(100) and Cu (111). The samples were cleaned through argon sputtering using a base pressure of 10 -6 mbar. The ion gun creates and accelerates Ar + ions toward the sample, which remove several layers of the substrate, as well as the residual contamination and dirt from previous experiments. The argon ux also produces defects on the surface, which calls for an annealing of the sample. Annealing copper between 500 • C and 600 • C allows for the atoms to di use on the surface and eliminates surface defects.

We carried out sputter/anneal cycles until we obtained clean samples with large terraces.

We are able to deposit individual metallic atoms on the surface. At room temperature, metal atoms can di use and adsorb on stable adsorption sites (defects, step edges). In order to eliminate surface di usion, the atoms are therefore deposited on a cold substrate directly in the STM chamber. We used a home-made evaporator, which can deposit simultaneously or sequentially three di erent types of atoms. The evaporator has three magnetic wires (Fe, Co, Ni > 99.99% of purity) wound around three independent tungsten wires. A shield screens the three wires from one another. Each tungsten wire is connected to an electrode from one side and is grounded on the other side. A current is applied though the tungsten wire causing the atoms of the magnetic wire to sublimate. The current for the deposition is calibrated with a quartz balance. In order to eliminate unwanted impurities, we increased the current gradually in order to degas the wire for 30 minutes before sublimation. This method yielded reproducible coverages as low as 0.1 atom/nm.

The preparation chamber is equipped with two e-beam evaporators, one having a Co rod and the other having a Cr rod. In this thesis, we mainly prepared cobalt nanoislands on Cu (111).

To obtain these two-atomic layers high islands, we applied a current of 18 mA to the tungsten lament in the e-beam. We used a high voltage of 840 V and a current of 1.9 A to accelerate the emitted electrons and sublimate the Co atoms of the rod. With these parameters, we obtained a ux of 0.75 monolayers/min of Co. Once the evaporation achieved, we immediately transferred the sample to the STM to minimize the intermixing between Co and Cu.

Chapter 3 Vibron-assisted spin excitation in a magnetically anisotropic molecule

We have seen in chapters 1 and 2 that STM can be used to induce and detect spin excitations and vibrational excitations. To do so, it is important to decouple the molecule from the surface [33,73,75,89] since the signal may be quenched in case of a direct contact with the metal electrodes. In this chapter, we introduce the nickelocene molecule (noted Nc hereafter), which remarkably allows for spin excitations when adsorbed on a metal surface such as Cu (100). After brie y describing the adsorption of single Nc molecules ( § 3.1), we focus on self-assembled layers of nickelocene, and describe the adsorption and the spectroscopic signature of nickelocene within these layers ( § 3.2). Next, we show that by doping in-layer nickelocene with a Ni atom it is possible to give rise to a new molecular complex with magnetic anisotropy ( § 3.3). Using IETS we show that this Ni-nickelocene complex (noted NiNc hereafter) exhibits both spin and vibrational excitations. With the support of DFT, we demonstrate that a joint spin-vibration excitation is also present in NiNc resulting from the concomitant excitation of a spin and a vibration quanta. This new type of excitation should be quite common in magnetic molecules possessing magnetic anisotropy, but, as we show, can be quenched by structural changes in the molecule. This is notably the case when NiNc is placed outside of the nickelocene network ( § 3.4). These results are published in Nature Communications [90].

Nickelocene

Nickelocene is a molecule from the metallocene family. Metallocenes are composed of a single atom of one of the six elements in the middle of the 3d row (V, Cr, Mn, Fe, Co, Ni) and two cyclopentadienyl (C 5 H 5 , named Cp hereafter) ligands. The metallic atom is sandwiched between the two Cp rings as shown for Nc in g. 3.1(a). The d 8 con guration of the nickel atom results in an electronic structure of (e 2 ) 4 (a 1 ) 2 (e 1 ) 2 for the Nc molecule as shown in g. 3.1(b). It provides the molecule with a spin S = 1 as demonstrated by several studies conducted in the gas phase [91] or on powder samples [91][92][93][94][95]. As a rule of thumb, half of the magnetic moment is carried by the Ni atom, while the two Cp rings carry the other half. The adsorption and magnetic properties of nickelocene, as well as ferrocene, have been studied extensively in our group during the thesis of N. Bachellier [96] and B. Verlhac [97].

Nickelocene deposition

We deposit nickelocene by sublimating a Nc powder acquired from Sigma-Aldrich (purity 99.9%).

The powder, which is placed in a crucible, spontaneously sublimates in UHV at room temperature. We start by purifying the Nc powder by placing the crucible in the load-lock for two hours to reach a pressure in the order of 10 -6 mbar. We then transfer the crucible in the preparation chamber and wait to reach a pressure in the low 10 -6 mbar. We use the pressure, the aperture of the crucible, the sample-crucible distance and the exposure time of the sample to the molecular ux to control the molecular coverage. The deposition is carried out on a pristine surface, which needs to be cooled below 200 K -from X-ray magnetic circular dichroism measurements we know that Nc desorbs above this temperature. To this end, the surface is extracted from the cold STM for a couple of minutes and warmed up by keeping it on the wobble stick. Then, we open the valve between the preparation and the STM chambers exposing the sample to the Nc ux as shown in g. 3.2(b) for 10 to 30 seconds, depending on the pressure and the distance between the crucible and the sample. These parameters ensure the formation of Nc networks on the surface.

Adsorption of isolated nickelocene on Cu(100)

As we can see in g. 3.3(a,b), after deposition, isolated Nc molecules are found on the surface along with Nc networks. Isolated molecules have a ring-like shape, which indicates that one Cp ring lays at on the surface, while the other ring lays in vacuum, g. 3.4. The principal axis of Nc, which goes from the center of one Cp to the other, is perpendicular to the surface. We will call such a nickelocene a "vertical Nc". It has an apparent height of 3.6 ± 0.1 Å in usual tunneling conditions. To determine the adsorption site of isolated Nc, we acquire an atomically resolved image of the Cu(100) surface. Since the electronic corrugation is weak for a Cu(100) crystal, we deposit Fe atoms on the surface, which serve as "manipulated atoms". The Fe atoms are deposited though our home-made evaporator directly on the surface as described in chapter 2. When the tip is placed atop a Fe atom and is su ciently close to it (typically within 2 Å from point contact), it generates a highly localized potential-well that traps the Fe atom under the tip [98]. When we then move the tip from one point to an other, the atom follows the tip and allows for resolving the corrugation of the copper surface. When coming back to usual tunneling conditions, the atom manipulation ceases and the atom lies in a stable adsorption site of the surface, here a hollow site of Cu (100). The local atomic resolution allows for obtaining the orientation of the surface and therefore to simulate the atom positions over the whole image. We presented the Cu atoms of the surface as a circle in the inset g. 3.4(a). Since the Fe atoms are in hollow sites, we can determine that the Cp ring of isolated Nc is centered on a hollow site, g. 3.4(a). DFT calculations by M.-L. Bocquet and coworkers support the experimental observations, g. 3.4(b,c). The vertical structure is favored by 400 meV and is found to be more stable on the hollow position by 100 meV and 160 meV with respect to the bridge and top position, respectively.

Nickelocene networks on Cu(100)

There are two possible arrangements for Nc molecules when they form a layer on the Cu(100) surface. These arrangements, which we label by "compact" and "paired", have same occurrence probability in our images indicating that they are energetically equivalent. The networks are stabilized by T-shaped Van der Waals interactions between horizontal and vertical molecules [99].

Compact network

N. Bachellier et al. [96] found that the compact network has a rectangular unit cell, g. ognize the ring shape of the single molecule adsorbed vertically, the detection of the horizontal Nc is not trivial. Low-bias images were acquired at 1 mV and 10 pA to get a better contrast on the corrugation of the image presented g. 3.5(b). The packing density of the network is 0.030 molecules per Å 2 . The unit cell does not follow the crystal symmetry and four possible orientations of the molecular network relative to the crystal orientation underneath were found. These orientations have angles :

ψ 1 = 53 ± 3 • , ψ 2 = -ψ 1 = -53 • , ψ 3 = 90 • + ψ 1 = 143 • and ψ 4 = 90 • -ψ 1 = 27 • .
Defects regularly appear in the compact network; a defect is highlighted with a dashed circle in g. 3.5(a). As seen in its pro le, g. 3.5(c), the defect has a 0.2 Å corrugation. It is di erent from the vacancy left when a horizontal nickelocene is taken away from the network with the tip (at least 1.5 Å) and therefore we can exclude a missing nickelocene. This may be due to the horizontal molecules pushing the vertical one, further work is needed to settle this question.

Nc/Cu(100)

a 1 (Å) a 2 (Å) θ (deg)

Paired network

The paired network on Cu( 100) is similar to the ferrocene paired network [100], g. 3.6(a,b). The unit cell is a parallelogram (lattice parameters are given in tab. 3.1) with a total of 8 molecules, 4 horizontal and 4 vertical. The packing density is 0.030 molecules/Å 2 , which is the same density than for the compact network. As the compact network, it adopts 4 angles with respect the Cu(100) orientation:

ψ 1 = 53 ± 3 • , ψ 2 = -ψ 1 = -53 • , ψ 3 = 90 • + ψ 1 = 143 • and ψ 4 = 90 • -ψ 1 = 27 • .
A di erence compared to the compact network comes from the fact that the vertical Nc molecules are arranged as dimer-like pairs. The spacing among the dimer-like pairs is 9.0 ± 0.3 Å. Two di erent kinds of pairs of vertical molecules are observed. They are organized in alternating rows and are highlighted in g. 3.6(a) by solid and dashed green lines. The spacing between paired molecules is evaluated to 7.9 ± 0.3 Å from the solid line pro le and to 6.5 ± 0.3 Å from the dashed line pro le, g. 3.6(c). There is also a di erence in the corrugation between the two kinds of pairs, with a maximum of 0.6 Å for the dashed line and 0.5 Å for the solid line.

M.-L. Bocquet and coworkers performed DFT+D2 computations for both networks and simulated the corresponding STM images [99]. The optimized structure is presented in g. 3.7(a). DFT calculations show that the inclusion of horizontal molecules lowers the adsorption energy by 50%. The networks have adsorption energies close to 1.5 eV pointing to a weak chemisorption (table 3.1). The similar adsorption energies of the networks explains their equiprobable occurrence on the surface. The presence of horizontal molecules enables to closely reproduce the structural features extracted from the experimental images as shown in the simulated image of g. 3.7(b). For example, the feeble di erence in corrugation between the two kinds of pairs can be attributed to horizontal molecules. Their presence introduces a geometrical constraint that causes the vertical molecules in the paired network to tilt by 3 • in the Nc-Nc pair with smallest spacing (dashed circled pair) and by 8 • in the other pair (solid circled pair), g. 3.7(c). Also, vertical Nc are pushed up and stabilize at a position 0.1 Å higher compared to isolated Nc. The surface-molecule interaction is therefore weakened in the network compared to isolated Nc, but energetically compensated by the van der Waals interaction with the adjacent nickelocene molecules. Similar results are found for the compact network. Given the dominant contribution of van der Waals interactions to the stability of these networks, we can expect to have similar arrangements on most metal surfaces as observed experimentally for Cu(111) [99] and Pb(111) [START_REF] Mier | Molecular templates of spin textures on superconducting surfaces[END_REF]. These networks can therefore be considered as a generic property of nicke-

Magnetism and spin excitation of nickelocene

Having characterized the adsorption of nickelocene, we now focus on its magnetic properties and, in particular, on the possibility of triggering and detecting spin excitations within the molecule through IETS. As mentioned earlier, Nc in the gas phase (or free Nc), is a spin S = 1 system. Nickelocene possesses in fact a D 5h symmetry, which favors a spin S = 1 triplet compared to the S = 0 singlet state. Susceptibility measurements on a Nc powder show that the molecule also possesses an uniaxial magnetic anisotropy (noted D hereafter) with D = 4 meV [92,93]. Generally speaking, magnetic anisotropy is a consequence of spin-orbit coupling, which, due to a reduced symmetry of the magnetic center, causes the magnetic moment to align along certain favorable directions. Magnetic anisotropy introduces a small energy di erence of order a few meV between spin orientations called magnetic anisotropy energy. This phenomenon has been the focus of research for decades as the bistability it introduces in nanoscale

Ĥ = D Ŝz 2 . (3.1)
Nickelocene on copper has been proved to preserve its spin S = 1 [99,[START_REF] Ormaza | [END_REF], as well as its magnetic anisotropy, although D reduces to 3.2 ± 0.1 meV due to charge transfer (see g 5.2 in chapter 5). The relative robustness of such magnetic properties against external perturbations is related to the cyclic π* orbital of the Cp rings, although this point needs to be further clari ed. This di erentiates Nc from other atomic systems [10,58,75,89,89,103] in which, as noted in chapter 1, the local environment is a crucial factor determining the magnetic anisotropy and/or the spin.

Within a STM junction, the |S = 1, M = 0 and |S = 1, M = ±1 become accessible to IETS, even in the absence of a magnetic eld. The dynamical scattering model of M. Ternes [105] can be used to simulate the dI/dV line shape associated to a spin excitation. The model provides information on electron and spin scattering occurring between the Nc and the electronic bath of the host metal. The model uses the spin Hamiltonian of eq. 3.1, but also includes and Anderson-like Hamiltonian (see § 1.2.2)

Ĥ = D Ŝ2 z + ĤA , (3.2)
where

ĤA = -J σ • Ŝ + U Î. (3.3)
U is the scattering potential and Î is the identity operator. J represents the antiferromagnetic exchange scattering between the electronic spins of the host metal (copper here) and the localized spin of Nc. This is also known as the Kondo scattering parameter and is crucial to correctly reproduce the dI/dV line shape [106,107].

To clarify this point, g. 3.9(a) presents IETS spectra for Nc along with line shape simulations based on eq. 3.2. We use here a nickelocene attached to the copper apex of the tip, g. 3.9(b). This setup will be presented in chapter 5 and yields a IETS nearly identical to that of Nc adsorbed on copper. The interest of such a setup is that the dI/dV can be acquired at various distances z from the Cu(100) surface (z = 0 corresponds to a nickelocene in contact with the surface). As shown, the inelastic thresholds remain constant with z, hence D is constant with distance. An enhancement of the di erential conductance at voltages corresponding to the ex- citation threshold is however observed, which is increasingly pronounced as the contact point is approached. The line shape simulations nicely reproduce the experimental spectra, yielding an estimate for the dimensionless coupling constant J , g. 3.9(c). Quite clearly, when Kondo correlations are not accounted for, i.e. when the line shape is simulated using eq. 3.1, the cusps are absent, dashed line in g. 3.9(a). The changes of J with z demonstrate that the cusp depends strongly on the hybridization of nickelocene with the surface. Similar conclusions were reported for another molecular system [108].

The above IETS data is a characteristic signature of Nc, which holds in most molecular or metallic environments. Within our experimental resolution, we nd that the IETS spectrum is the same for isolated Nc, g. 6.4(a), or for Nc embedded in the networks described in § 3.2. The spectrum turns out to be robust when changing surface symmetry [START_REF] Ormaza | [END_REF] and surface material [24,[START_REF] Mier | Molecular templates of spin textures on superconducting surfaces[END_REF], or even when attaching Nc to the apex of the STM tip [23,24,109]. We found that the magnetic properties of Nc tend, however, to change in the presence of magnetic systems. When coupled to single magnetic atoms, Nc forms new and fascinating complexes that can exhibit Kondo physics and/or spin excitations that are intertwined with vibrational excitations.

In the following, we present a prototypical example of such a complex, NiNc.

3.3 Nickel-Nickelocene complexes on Cu(100)

Structure

To build Nickel-Nickelocene (NiNc) complexes, we exposed the Nc layer to single Ni atoms, which, in the following, we will call Ni adatoms for clarity. The nickel evaporation was carried out directly on the cold sample as described in chapter 2. After Ni deposition, bright spots appear in the network that we identify as NiNc complexes. Our experimental observations regarding the formation and properties of the NiNc complex showed no signi cant di erence between the two Nc networks, therefore in the following we will only focus on the paired network. As presented in g. 3.10(a,b), the molecular complexes show a ring shape, higher (5.8 ± 0.2 Å) than the rest of the layer (4.1 ± 0.2 Å) as can also be seen on the line pro le of g. 3.10(c). The ring shape indicates that a Cp is still exposed to vacuum in NiNc. We then assign the larger height of NiNc to the presence of a Ni atom directly under the Nc molecule.

M.-L. Bocquet, N. Lorente and coworkers performed spin-polarized DFT calculations to determine the adsorption geometry of the NiNc complex. Electron transmission calculations were also performed. They continued the work of previous publications [START_REF] Ormaza | [END_REF]109] treating the adsorption, chemical and physical properties of adsorbed Nc molecules on Cu(100). In the present case, a Ni atom was added to the substrate and the adsorption of Nc on this nucleation center was studied. Calculation details can be found in [90]. The optimized structure, which is presented in g. 3.11(a-c), con rms the experimental ndings. When the Ni adatom is under the Nc, the NiNc complex is 3 eV more stable than when the adatom is on top. This is similar to the results found for Co-ferrocene complexes [110]. The Ni adatom is located 2.4 Å above the copper surface, but is misaligned 0.5 Å with the Nc molecule as visible in g. 3.11(c). This results in a tilt in the principal axis of the Nc molecule, g.

Spectroscopy

Figure 3.12 presents the dI/dV and d 2 I/dV 2 spectra obtained on a NiNc complex. We have included the spectrum of Nc as a reference. The NiNc spectrum presents several step-like features symmetric in bias, which we assign to inelastic excitations. Using the d 2 I/dV 2 spectrum, we nd a peak (dip) at |E 1 | = 4.3 ± 0.4 meV (noted 1), which is close to the one of Nc, and two more peaks (dips) at |E 2 | = 33.9 ± 0.5 meV and |E 3 | = 38.1 ± 0.6 meV (noted 2 and 3, respectively). In the following, we identify the origin to each of these three excitations.

Feature 1: Spin excitation

Given the similarity to the spin excitation spectrum measured above Nc, we identify 1 to be a spin excitation. To con rm this assignment, DFT calculations were carried out. Using the relaxed structure of NiNc determined above, we nd that the d xz and d yz frontier orbitals of For energies that are accessible with the STM, the transmission is controlled by two eigenchannels that are themselves the contributions of Lorentzian-like functions as can be seen following the transmission curves of the rst and second channels. These Lorentzian-like functions energetically match the projected density of states corresponding to the two π molecular orbitals of Nc. For electron energies below -0.8 eV, the transmission involves d-electron channels of the Ni adatom.

Nc in the complex are spread out in a range of ±1 eV around the Fermi level, g. 3.13. The NiNc complex has a total magnetic moment of 1.4 µ B , corresponding to an antiferromagnetic coupling between the Ni adatom (-0.2 µ B ) and Nc (+1.6 µ B ) with a charge transfer of 0.1 electrons from the substrate. The antiferromagnetic coupling is visible in the isosurface of the spin-polarized density presented in g. 3.11(b).

The DFT calculations point therefore to an e ective spin of S = 1. The strong resemblance with the spin excitation of Nc further allows us to identify the spin spectrum with the one originating from a S = 1 system. Consequently, we model 1 via the spin Hamiltonian of The cusps observed near |E 1 | are associated to a Kondo tting parameter J ρ = -0.4 ± 0.2, which is typical for Nc, g. 3.9(c).

Feature 2: Vibrational excitation

We assign peak 2 to a vibrational excitation with energy |E 2 | = 33.9 ± 0.5 meV. A rst clue is the observation of a red shift of the peak and dip while approaching the tip towards NiNc, g. 3.14(a). We explored several tip-NiNc distances by increasing the tunneling current from 50 pA to 1 nA. While peak 1 stays at the same energy (tab. 3.2), the second peak shifts up to values of 1.5 meV. This red shift has already proved to be a marker for a vibrational excitation [111,112].

A second clue comes from the spatial dependence of the IETS signal above the NiNc complex. This can be monitored through dI/dV maps realized at a constant-height z. The resulting images indicate the spatial location were the excitation is strongest. For a scan at a bias of 4.5 mV, corresponding to peak 1, we obtain a ring shape map and observe a tilt as for the Nc molecule, g. 3.14(b). The spin excitation of peak 1 is therefore located on the Cp ring. However, for a scan at 32.5 mV, corresponding to peak 2, we obtain a di erent shape. The maximum of the intensity is found at the center of the ring g. 3.14(c), which con rms that the origin of the signal is di erent compared to 1.

To understand this di erence in the maps, N. Lorente and coworkers computed the vibrational modes for the relaxed structure of the NiNc complex. Three vibrational modes were found around |E 2 | = 33.9 ± 0.5 meV. The rst mode at ω 1 = 29.1 meV corresponds to a molecular frustrated, i.e. hindered, rotation. We discarded it due to the lack of concordance with experimental observations. The second mode at ω 2 = 31.8 meV and the third mode the simulated conductance across the molecule, we nd instead a dominant contribution for the 35.5 meV mode with a spatial dependence matching experimental observations, g. 3.14(e). Excitation 2 is then assigned to this latter mode. This excitation, which is sketched in g.3.14(f), is modeled with a molecular vibration of frequency ω = 35.5 meV using the Hamiltoninan:

Ĥvib = ω( b † b + 1 2 ) (3.4)
where the molecular vibration is created (annihilated) by b † ( b).

Feature 3: Vibron-assisted spin excitation

We now focus on excitation 3. Several observations hint for a relation between the third peak and the two other peaks, and lead to the assignation of this excitation as a vibron-assisted spin excitation. Firstly, one can notice that the energy |E 3 | is actually the sum of the energies

|E 1 | and |E 2 | |E 3 | = |E 1 | + |E 2 |. (3.5)
Secondly, there is a similarity in the shape of peaks 1 and 3 as both exhibit cusps, which are a characteristic of a spin excitation. Thirdly, the red shift observed for peak 2 when approaching the tip to the NiNc complex is also present for peak 3, g. 3.14(a) and in tab. 3.2. Another striking relation can be unraveled by acquiring dI/dV spectra at di erent positions of the complex, from one side to the other passing by its center, g. 3.15(a). While the intensity of peak 1 show at most a 10% change over the molecule, the intensity of peaks 2 and 3 become up to 3 time larger in the center of NiNc compared to the Cp rings. The dI/dV map acquired at a bias of 37 mV close to excitation 3 is in fact similar to the one of peak 2, g. 3.14(c).

In order to model the experimentally observed dI/dV spectrum, we express the Hamiltonian of NiNc by

Ĥ = D Ŝ2 z + ĤA + Ĥvib + σ W dσ † dσ ( b + b † ). (3.6)
As previously, we take a single d-level for the molecule and introduce the states d annihilated and created by dσ and dσ † with spin σ, respectively. We also assume that the electron and vibration couple with a strength W when the state d is populated. This term is estimated to be W = 20 meV from DFT and corresponds to a weak electron-vibration coupling. Equation (3.6), which does not include a spin-vibron coupling, shows that a vibron is excited when an electron tunnels into the molecule. This in turn a ects the electronic correlation included in ĤA , as well as the probability of spin excitation, which strongly depends on the occupation of the molecular orbital [107]. having enough energy to excite the spin and a vibrational excitation. This translates into a spin excitation displaced upward in energy by the vibrational excitation energy.

This excitation mechanism is similar to the coupled-spin vibrational Kondo e ect and its replicas of the Kondo resonance in the tunneling spectra at energies close to ω. As we have seen in chapter 1 ( § 1.3.1), the electron-vibron interaction can produce satellite Kondo resonances in the molecular conductance at the bias of the vibron's excitation energy [66,[68][69][70][71][72]. Consistent with this assignment, the relation between the three excitation thresholds re ects

|E 3 | D + ω.
The distortion of the NiNc molecule is negligible when the molecular vibration is active as expected for a weak electron-vibration coupling. A renormalized value of D would be observed otherwise leading to

|E 3 | = |E 1 | + |E 2 | [113, 114].
The relative amplitudes of the steps in the dI/dV spectrum of NiNc can also be explained using this framework. For a quantitative estimate of the step amplitudes, we use an extended phenomenological model to t the full spectrum, g. 3.16(c). We use a constant elastic contribution σ 0 , a vibrational step of amplitude σ 2 accounted for following [60]. The two spin excitations of amplitudes σ 1 and σ 3 are accounted for using a t based on a dynamical scattering model [105]; we use the same tting parameters for the two spin excitations, except for their amplitudes σ 1 and σ 3 . As shown in g. 3.16(c), the t to the data is satisfactory, while the energy onset of the steps are related following eq. 3.5 (tab. 3.3). We also nd that all spectra recorded obey (tab. 3.3)

1 (meV) 2 (meV) 3 (meV) 1 + 2 (meV) σ 1 /σ 0 σ 2 /σ 0 σ 3 /σ 0 (σ 1 σ 2 )/
( σ 3 σ 0 ) = ( σ 2 σ 0 )( σ 1 σ 0 ), (3.7)
where the vacuum barrier thickness is accounted for by the elastic contribution σ 0 . This relation indicates that the spin and the vibrational excitations occur independently one from another with transition rates proportional to σ 1 /σ 0 and σ 2 /σ 0 , respectively, while the transition rate of the combined excitation σ 3 /σ 0 is their product. This can lead, eventually, to a vibron-assisted spin excitation that exceeds in intensity the spin excitation, g. 3.15(a). This is similar to a mechanism already observed in an experiment using a tip functionalized with a Nc molecule above a nickelocene on the surface. In this con guration, we can excite one nickelocene giving spin excitation 1, but one can also excite the two nickelocenes at same time with the same electron, which leads to a double-spin excitation [START_REF] Ormaza | [END_REF]. In this case, the excitation intensity is up to four times the one of the single excitation.

Isolated Nickel-Nickelocene complexes

To highlight the importance of the Nc layer for observing the inelastic excitations in the NiNc complex, we engineer the NiNc complex outside the layer via a tip-assisted manipulation [110].

Using an STM, it is possible to pick a Nc molecule with the tip, g. 3.17(a), and to place it on the top of a Ni atom, g. 3.17(b). When we image this complex, we recognize the ring shape of the Nc, which means that the Nc molecule is adsorbed on the top of the Ni atom as for the complex in the layer, g. 3.17(c). One can notice that the ring is not tilted as for the NiNc complex in the network but is parallel to the surface. In the line pro le, g. 3.17(d), the complex appears higher (5.6 ± 0.2 Å) than an isolated Nc molecule (3.5 ± 0.2 Å). It is similar to the height of the complex in the layer (5.8 ± 0.2 Å).

DFT calculations give a clear picture of the new adsorption con guration adopted by NiNc, g. 3.18(a). The Ni atom is adsorbed in a hollow position on Cu (100) under the Nc molecule. The Cp ring of the Nc is centered on the Ni atom and found to be parallel to the Cu surface, as observed experimentally. The Ni atom is located at 2.47 Å from the copper surface, which is also similar to the 2.47 Å for the complex in the layer. We relate the di erence in the angle of the Cp ring to the absence of the layer, i.e. to the absence of steric constraint due to the other Nc molecules.

A typical dI/dV spectrum we acquired on a isolated NiNc complex is presented g. 3.18(b). Instead of the three excitations steps described earlier, we have now only one broad peak centered at the Fermi level. This peak is assigned to a spin S = 1/2 Kondo e ect. The lowest unoccupied molecular orbital of an isolated NiNc can in fact be qualitatively represented by the mixing of the p z orbitals of C and of the d xz and d yz orbitals of Ni placed around the Fermi level. This results in a calculated magnetic moment of 1.17

Summary and conclusion

In this chapter, we presented the Nc molecule and its characteristic spectroscopic signature. We showed that its properties can be changed by exploiting Nc layers and the addition of Ni atoms, which leads to the formation of a NiNc complex. We could identify a spin and a vibrational excitation in an in-layer NiNc, as well as a vibron-assisted spin excitation. The latter is shown to be shifted in energy compared to the spin excitation.

Spin excitations are increasingly observed in organometallic molecules coupled to metal electrodes [32,73,80,[START_REF] Ormaza | [END_REF]108,[116][117][118][119]. Along with spin excitations, electrons may also trigger molecular vibrational modes that modify [64,[120][121][122][123][124] or even suppress [125,126] molecular conductance. Even if the presence of both excitations could be evidenced in a molecular system [127,128], experimental observations regarding their interplay have remained elusive. In this sense, the concomitant excitation of a spin and a vibration quanta we observed in NiNc can be considered one of the rst observation of the kind. At the same time as us, the team of W. Ho reported similar ndings [129].

Our study suggests that the vibron-assisted spin excitation can be present in molecules with magnetic anisotropy and that it can be easier to detect than the vibrational excitation associated to it. With this in mind, we re-explored the spectrum of a in-layer Nc. As shown in g. 3.19, an inelastic step is evidenced at ±28 mV, which we then assign to a vibron-assisted spin excitation.

Compared to the spectrum of Nc presented in g. 3.12, the tip is here positioned in the center of the Cp ring to boost the vibrational excitation. Future work could consist in validating this assignment with the help of an external magnetic eld.

Chapter 4

The Kondo e ect of a molecular tip as a magnetic sensor

As we showed in chapter 1, the Kondo e ect is a quantum phenomenon occurring when a magnetic impurity -molecule or atom-couples to a non-magnetic host metal [START_REF] Hewson | The Kondo Problem to Heavy Fermions[END_REF]. The electrons of the metal coherently screen the impurity and produce a Kondo resonance in the conductance which is located at the Fermi energy and has a characteristic width of k B T K . The Kondo resonance changes when the impurity is coupled to its local magnetic environment. Magnetic information can be gathered by closely monitoring the resonance line shape, hence a Kondo impurity, whether an atom or molecule, may be regarded as an atomic scale magnetic sensor. Such an approach as been widely used in STM [25][26][27][28]31,[START_REF] Wahl | [END_REF][132][133], but the discrete interatomic separations imposed by the substrate lattice prevent a detailed investigation. In a seminal work [36], to circumvent this limitation a tip was decorated with a Kondo atom and its separation from another Kondo atom on the surface was continuously changed. This study, however, su ered from the poor structural and magnetic knowledge of the tip apex.

We show in this chapter that to improve tip control, one can exploit the Kondo e ect carried by a molecular tip. To date, the experimental realization of such a tip has remained elusive. To build a molecular Kondo tip, we resort to cobaltocene [Co(C 5 H 5 ) 2 , Cc hereafter], g. 4.1(a). The general outline of this chapter is the following. We rst present cobaltocene and its adsorption on a metal surface ( § 4.1). We will exclusively focus on the Cu(100) surface. We then show how to place a cobaltocene at the tip apex and, with the help of rst-principles calculations, characterize the tip structure ( § 4.2). We show that such a molecular tip yields reproducible Kondo physics and we demonstrate its magnetic sensitivity across the vacuum gap ( § 4.3). Finally, we show that the Kondo signature is intertwined with molecular vibrational modes that depend on how cobaltocene is contacted to its metallic environment ( § 4.4). These results are published in Nano Letters [134].

Cobaltocene

Cobaltocene is a molecule from the metallocene family that is composed of two cyclopentadienyl (Cp) rings sandwiching a Co atom, g.

Cobaltocene deposition

We deposit Cc onto a Cu(100) surface using a powder from Sigma-Aldrich (purity 99.9%). The deposition procedure is identical to the one of nickelocene (chapter 3) as the powder spontaneously sublimates in UHV at room temperature. The Cc powder is placed in a crucible and puri ed in the load-lock for two hours to reach a pressure in the order of 10 -6 mbar. The crucible is then transferred to the preparation chamber. The base pressure of the preparation chamber typically increases from 10 -10 mbar to 10 -6 mbar. To control the amount of Cc deposited onto the surface, we use the pressure, the aperture of the crucible, the sample-crucible distance and the exposure time of the sample to the ux. The deposition is done on a cold Cu(100) substrate as the molecule can degrade on copper at room temperature. The sample is therefore extracted from the cold STM stage (4.4 K) for 4 minutes in order to warm it up to a temperature estimated below 150 K. Then, we open the valve between the preparation and the STM chambers. The sample is exposed to the Cc ux as shown in g. 4.2(b) for 10 to 30 seconds, leading to the formation of self-organized networks of Cc. Isolated molecules can also be observed coexisting with the networks. In this chapter we will focus on the isolated Cc molecules.

Adsorption on Cu(100)

After the Cc deposition, isolated molecules are found on terraces and on step edges of the surface, g. 4.3(a). The Cc molecules on the terraces exhibit a ring-like pattern, inset of g. 4.3(b), that we assign to a Cp group being exposed to vacuum. The ring-like pattern observed and the apparent height of 400 ± 50 pm, g. 4.3(b), are typical for metallocenes [99,100], and point to a Cc adsorbed with its molecular axis parallel to the surface normal. The di erence in height between a Cc adsorbed at a step edge and on a terrace is 35 pm, which is smaller than the step height of copper (180 pm). This indicates that Cc adsorbs on the bottom edge of the step and is likely tilted.

To determine the adsorption site of Cc on a terrace, we enhance the spatial resolution of the STM through a tip-assisted manipulation of an adsorbed atom [137]. After sublimating single Fe atoms onto the surface, we use a tip-assisted manipulation to drag the Fe atom on Cu (100) and thereby image the corrugation of Cu (100) [top right corner in g. 4.3(c)], which, otherwise, is not observable in normal tunneling conditions. The Fe atoms serve also as markers to identify the position of the hollow sites of Cu (100). A grid representing the Cu(100) surface is simulated and positioned onto the image of g. 4.3(c) so that the Fe atoms are located in a hollow position.

Based on this grid, cobaltocene preferentially adsorbs with the bottom Cp ring centered on a hollow site of Cu(100).

To con rm these ndings, we use density functional theory (DFT) calculations to perform an en- ergetic analysis of the molecular adsorption. These calculations were performed by N. Lorente and coworkers; details of the calculation are given in [134]. The Cc was relaxed in three highlysymmetric sites, corresponding to a Cp ring centered in a top, bridge and hollow site of Cu (100). Cobaltocene preferentially adsorbs on a hollow site, g. 4.3(d), in agreement with the experimental data. In this con guration, the bottom Cp ring is positioned at a distance of 259 pm from the surface, g. netic moment of Cc is carried by the Co atom and is 0.75 µ B , which corresponds to an e ective spin of S = 1/2; the Cp rings have zero magnetic moment. The π orbitals are nearly degenerate and spatially add up to yield the ring-like shape of Cc, g. 4.4(b). The π orbitals of cobaltocene are similar to those observed in nickelocene, either when it is adsorbed on a surface (chapter 3) [99] or on a tip (chapter 5) [109].

The cobaltocene tip 4.2.1 Tip preparation

To prepare a Cc-tip with atomic control, we transfer a Cc from Cu(100) to the tip apex of the STM. The transfer process is analogous to other metallocene molecules [START_REF] Ormaza | [END_REF]109,110]. We start by covering the W tip with surface material through soft indentations into the pristine surface. It is essential to ensure a mono-atomically sharp metallic apex. A single Cc can then be attached to the tip by vertically moving the tip towards the Cc molecule within 100 pm from point contact at a bias of -20 mV or lower, g. 4.5(a,b,c). Similarly, one may scan the tip across the molecule close to the contact regime. This procedure, which yields a better control than a vertical displacement of the tip toward the molecule, is depicted in g. 4.5(d,e) for a cobaltocene molecule on Cu(100). We rst start by imaging the molecule with a typical tunnel conductance of G = 1 nS, g. 4.5(d). When scanning across the molecule, we then increase the conductance to G = 300 nS by approaching the tip to the surface, the bias polarity being negative, g. 4.5(e). Finally when we image again the area with G = 1 nS, the molecule is no longer on the surface, g. 4.5(f). The presence of Cc on the tip apex produces a molecular pattern in the adatom image, g. 4.5(f), which otherwise appears featureless if the tip apex is metallic, g. 4.5(d). The molecular pattern is attributed to a Cc that is adsorbed on the tip apex with its molecular axis DFT calculations for a Cc-tip in con gurations 1 to 3: Energy di erence δE among tip con gurations, distances between the Cu adatom and the nearby C atoms of the Cp ring, tilt angle of Cc relative to the surface normal, magnetizations of Cc and Co. The chemical accuracy of DFT was tested by using the Grimme DFT+D2 and DFT+D3 corrections [139,140], and by using the optPBE van der Waals density functional [141]. The results are robust to the method employed. The table presents the results for the Grimme DFT+D2.

Determining the tip-sample distance

The tip-surface distance can be deduced through the point contact of the Cc-tip with a copper atom of the Cu(100) surface, similar to what is done with metal tips [143]. Contact formation is monitored by recording changes in the tunneling conductance G = I/V as a function of tip-surface distance z. A conductance-versus-displacement measurement acquired at -5 mV is presented in g. 4.8(a). The contact distance (z = 0) is identi ed trough a change in slope of the curve [143]. Upon retracting the tip, cobaltocene remains attached to the tip apex so that the procedure can be repeated several times. A direct contact of the Cc-tip to a magnetic atom was however not possible as cobaltocene is then transferred atop the atom at distances z ≤ 90 pm. To circumvent this problem, we subtract the apparent height of the adsorbed atom, e.g. h = 115 pm for an iron atom as shown in g. 4.8(b), from the contact distance to obtain the absolute distance to the atom: z(Fe) = z -h, g. 4.8(c). 100) terrace; all dI/dV measurements were acquired at 2.4 K. The spectrum exhibits a sharp peak near zero-bias and is anked by peaks nearly symmetric with bias polarity, solid black lines in g. 4.9(a). The dominant zero-bias peak is assigned to a Kondo resonance resulting from the spin-ip scattering of tip conduction electrons by the S = 1/2 spin of Cc. The nearly symmetric shape of the peak indicates that the quantum interference with the continuum of the tip electrons is negligible (see § 1.2.4). The spatial distribution of the Kondo resonance in the molecule, inset of g. 4.9(a), follows the two-lobe pattern of the π yz orbital, revealing that the Kondo e ect is carried by this orbital.

The Kondo e ect of a cobaltocene tip

The side peaks are assigned to a Kondo e ect where tunneling electrons have their spin ipped when elastically scattering o the Cc spin, but with su cient energy to activate a vibrational mode in the molecule [66]. As we mentioned in chapter 1 ( § 1.3.1), this vibrational-assisted Kondo produces peaks that fall at biases corresponding to the vibrational excitation energies of the molecule [68][69][70][71][72]. While the Kondo resonance showed little dependence on tip con guration, g. 4.10, the intensity of the side peaks changed from tip to tip. For simplicity, in the following we restrain the presentation to con guration 2 without loss of generality. To enrich the discussion of the Kondo line shape, we include in our analysis the Cc molecules that are adsorbed at the bottom of the edge of Cu(100) steps. Remarkably, these Cc have same experimental markers as the Cc-tip. They have a two-lobe pattern when imaged and their dI/dV spectrum is nearly mirror symmetric in bias to the spectrum acquired with a Cc-tip, g. 4.6(b). Such a resemblance to a Cc-tip hints to a cobaltocene interacting with a single step-edge atom of Cu(100). In the following, we focus onto the Kondo resonance and postpone the discussion of the side peaks to § 4.4. 100) using Cc-tips in con guration 1, and 2 and 3 (-60 mV, 100 pA). A Frota t (not shown) using eq. 4.1 yields a Γ of 0.76, 0.88 and 0.68 meV, respectively, which emphasizes that the Kondo resonance is similar for the three tip con gurations. The vibrational Kondo peaks show, instead, tip dependency, in particular their amplitude.

Kondo line width

To validate the Kondo nature of the zero-bias peak, we probe the temperature dependence of the dI/dV spectrum of the Cc-tip and that of the Cc at the step edge. A set of spectra is presented g. 4.11(a). The peak broadens progressively with temperature, which is a signature of a Kondo system. The dI/dV spectra are simulated using a Frota function [144,145], solid red line in g. 4.11(a):

σ(eV, Γ) = h iΓ/(iΓ + eV -K ) + σ 0 , (4.1)
where indicates the real part, while σ 0 , h and K are constants. The thermal smearing introduced by the tunneling mechanism and the smearing of the lock-in detection (see chapter 2) is also accounted for in the simulation [146]. As shown in g. 4.11(b), the evolution of Γ(T ) with temperature is well reproduced by eq. 1.13 ( § 1.2.4). We obtain T K = 4.3 ± 1.3 K and α = 5.0 ± 0.3, solid line in g. 4.11(b). The relatively small value of T K indicates a weak coupling of Cc to the metal tip apex (or the step edge). The asymptotic slope α is slightly higher compared to the predicted value for a spin-1/2 Kondo system in the strong coupling regime where α = 3.83 when T T K [57]. This might indicate that the system is not completely in the strong coupling regime, which is consistent with our working temperature being close to T K . Similar ndings have been reported for other Kondo systems [58,128].

The Kondo nature of the zero-bias peak can be further validated by monitoring its evolution when the Cc-tip is brought into contact with the surface, gs. 4.12(a,b,c). The line width Γ, therefore T K , increases monotonically with z once the tip is in contact with the surface, g. 4.12(d), similar to other metallic junctions comprising a single Kondo impurity [39,147,148]. This reects a reinforcement of the Kondo e ect, which we attribute to the additional hybridization with the surface. The presence of a Kondo e ect in the contact regime also indicates that cobaltocene still has an e ective spin close to S = 1/2, possibly owing this behavior to the nonmagnetic Cp rings sandwiching the Co atom.

Magnetic sensing across the vacuum gap

We now turn to the major result of this chapter, i.e. the possibility of using a Cc-tip to gather magnetic information from the sample. We focus on a prototypical system consisting of a single Fe atom on Cu(100). The iron atoms, which are deposited on the cold Cu(100) surface as described in chapter 2, adsorb on the hollow sites of Cu(100), g. 4.3(c). One of the interests of using a Fe atom is that its dI/dV signal at the Fermi level is negligible compared to that of the Cc-tip. In other words, the Fe atom is spectroscopically dark as

|T Fe | 2 |T Cc | 2 , where |T Cc | 2
and |T Fe | 2 denote the excitation intensities of Nc and of Fe, respectively. This aspect, which we further detail in the following chapter ( § 5.2.1), is due to strong screening and broadening e ects induced by the metal surface on the Fe atom [79]. Hence, the presence of iron can only be indirectly inferred by the changes observed in the dI/dV of the cobaltocene tip. 100). (c) dI/dV spectra acquired above Cu (100). Solid red lines: Tunneling spectra acquired when decreasing z from 60 to 0 pm; z = 0 corresponds to point contact. Solid blue lines: Contact spectra acquired when decreasing z from 0 to -90 pm. Inset: Zoom on the tunneling regime. For clarity, the spectra are shifted by 0.5 nS from one another. (d) Γ extracted from the dI/dV spectra using eq. 4.1.

A typical set of dI/dV spectra recorded above the Fe atom at di erent tip-atom distances z is presented in g. 4.13(a). The spectra are acquired with one of the lobes of the Cc-tip centered on the atom. While the z = 150 pm spectrum is indistinguishable from the one acquired above the bare Cu(100), for smaller distances a splitting of the peak is apparent. The splitting becomes increasingly stronger as z is decreased, the average energy position of the two peaks remaining unchanged within experimental resolution. The intensity of the two peaks is asymmetric for opposite voltage polarities. The magnetic coupling of a Kondo atom to another magnetic atom [25,36], or a magnetic electrode [28,39], is known to split the Kondo resonance into two peaks. The spin-split spectrum of Cc then indicates that the Cc-tip to is magnetically coupled to the Fe atom across the vacuum gap. In the following, we show that a double magnetic information can be extracted from the spectra: The spin-split Kondo peak allows for probing the exchange interaction between Cc and Fe, while the amplitudes of the spin-split peaks provide a measure of an adsorbate-related spin polarization.

Model Hamiltonian

Simpli ed Hamiltonian

To rationalize these observations, we follow our previous work on the nickelocene-terminated tip [23], which we will present in chapter 5. We use a simpli ed spin Hamiltonian to describe the magnetically coupled Cc-Fe system:

Ĥ = -gµ B B Ŝz , (4.2)
where the z-axis corresponds to the out-of-surface direction -for simplicity, we neglect the small tilt angle of Cc. Even though Kondo correlations are not explicitly accounted for in eq. 4.2, similar spin Hamiltonians have been shown to quantitatively describe the energies of spin-split Kondo peaks [25,58,132,149]. This e ective Zeeman Hamiltonian consists of the gyromagnetic factor (g = 2), the Bohr magneton (µ B = 57.9 µeV/T), and the exchange eld (B) produced by the Fe atom and acting on Cc along the z-axis. Within mean-eld theory, B = |J| S z,Fe /gµ B , where S z,Fe ≈ 3/2 is the e ective spin of Fe on copper [23,79] and J is the magnetic coupling between Cc and Fe. We will see in chapter 5 that for nickelocene the coupling is antiferromagnetic (J < 0) and Ising-like ( § 5.2.3); we reasonably assume that it is also the case here for cobaltocene. The spin-split zero-bias peak is then ascribed to a pair of conductance steps placed symmetrically about zero bias, g. 4.13(b). We have seen in chapter 1 that in the presence of an external magnetic eld, inelastic electrons can trigger spin excitations among Zeeman levels. Here, it is the exchange eld B exerted by the iron atom that lifts the degeneracy among the |M = ±1/2 states of cobaltocene, with M as the magnetic quantum number. The steps in gs. 4.13(a,b) are then associated to inelastic tunneling events in which tunneling electrons excite Cc from its magnetic ground state |M = +1/2 to the excited state |M = -1/2 , inset of g. 4.13(c). They are located at a bias |eV 0 | = gµ B B corresponding to the energy separating the two Zeeman levels.

General Hamiltonian

The simpli ed Hamiltonian displayed in eq. 4.2 can be deduced from the more general Hamiltonian:

H = D Fe Ŝ2 z,Fe -J Ŝz,Cc • Ŝz,Fe . (4.3)
The rst term accounts for any possible anisotropy of the Fe adatom on Cu (100). The Fe atom can possess only axial out-of-plane anisotropy since it is adsorbed on the hollow site of a fourfold symmetric Cu(100) surface. The sign and strength of D Fe is, however, still unknown, but we note that the magnetic anisotropy for Fe on a Cu( 111) is D Fe = -0.85 ± 0.2 meV [79,150].

The second term of the equation accounts for the coupling between the two spin systems. The spin-state diagram and the eigenfunctions of the Cc-Fe system are sketched in g. The ground state |ψ 0 is a doublet state and corresponds to an antiferromagnetic con guration where the Fe spin is anti-aligned with the Cc spin along the z-axis. The excited state |ψ 1 corresponds to the ferromagnetic con guration. One can therefore limit the analysis to a |M Fe = -3/2 spin without loss of generality. This enables to simplify eq. 4.3 by neglecting the anisotropy term of Fe and reformulating the interaction term as J Ŝz,Fe Ŝz,Cc = gµ B B Ŝz,Cc ; please note that with this convention the exchange eld B points in the opposite direction of the Fe spin. This then leads to the simpli ed Hamiltonian of eq. 4.2.

-2D Fe -J -2D Fe -J/2 Ψ 0 Ψ 1 Ψ 2 Ψ 3 -3J/2 0 Ψ 0 Ψ 2 M Cc = 1/2,M Fe =± 1/2 Ψ 1 Ψ 3 M Cc = ±1/2,M Fe =± 1/2 M Cc = 1/2,M Fe =± 3/2 M Cc = ± 1/2,M Fe =± 3/2 ± ± |T | Cc 2 |T |

Exchange interaction

A satisfying line shape simulation based on eq. 4.2 is presented as a solid red line in g. 4.13(a). The simulation is based on the dynamical scattering model developed by M. Ternes [105,149], which we introduced in chapter 3. As highlighted in g. 4.13(b), we have included Kondo scattering in our simulations. The spin-split Kondo peaks correspond then to inelastic steps having a cusp at their threshold, similar to what we presented for nickelocene in g. 3.9(b) of chapter 3. Figure 4.13(c) presents quantitative estimates for B as extracted from the simulations carried out at various distances z. The exchange eld is a function of z of the form exp (-z/λ) with a decay length of λ = 23 ± 1 pm, which is characteristic of an exchange interaction across a vacuum gap [23,24,37,38,41]. The coupling is |J| ≈ 1.3 meV at the shortest probed Fe-Cc distance, similar to other exchange coupled systems [39,40].

In the discussion above, we neglected the tilt angle of Cc relative to the z-axis (θ hereafter). The strength of the magnetic interaction is determined by the hybridization between the Cc and Fe orbitals, which depends on the Fe-Cc distance (z) and on the angle between orbitals, hence on θ. The general expression of the exchange eld is then a priori B = |J(z, θ)| S z,Fe /gµ B . Since experimentally we show that the exchange eld, i.e. J, follows a simple exponential law with z, which is typical for an exchange interaction, we may then assume that θ does not change with z. The tilt angle is therefore a xed parameter and can be neglected in our analysis. In practice, it is di cult to anticipate how J will change with θ as this depends on the nature of the magnetic coupling, which in the present case is a mixture of a direct and a superexchange interaction. A comparative study among the di erent Cc-tip con gurations could eventually clarify the role of θ.

Spin polarization

The simulations also capture the line shape asymmetry observed experimentally. This asymmetry η = (h

+ -h -)/(h + + h -) (4.4)
stems from a selection rule for spin excitations [33,105], and re ects changes in the inelastic step height at negative and positive voltage (h -and h + ), g. 4.13(b). This mechanism, which is similar to conventional spin-polarized STM, is sketched in g. 4.15. The excitation of Cc from the ground state |M = +1/2 to |M = -1/2 requires a change in spin angular momentum of δM = -1. Because the total angular momentum must be conserved, it can only be induced by electrons that compensate for this moment by ipping their spin direction during the tunneling process from |↓ to |↑ . At negative bias, the tunneling process may be viewed as a |↓ electron hopping from the substrate into the molecular orbital with transmission T ↓ . At positive bias, the tunneling direction reverses and the transmission from tip to sample is T ↑ . The height of the steps corresponds thus to h -∝ T ↓ and h + ∝ T ↑ , which yields:

η = (T ↑ -T ↓ )/(T ↑ + T ↓ ). (4.5)
From our simulations, we nd a constant value of η = -20 ± 5% for z ≤ 100 pm, while at higher distances we cannot conclude given the large error in the estimates of η. The asymmetry provides a measure for the adsorbate-related spin polarization owing to the spin S = 1/2 nature of the Cc-tip [151]. Indeed, following Loth et al., when the tip and the sample are magnetic, the net spin polarization of the inelastic tunnel junction (η) reduces to the product of the sample (η s ) and tip (η t ) spin polarizations: η = η s η t . If this behavior is similar to conventional spinpolarized tunneling, an important di erence is to be noticed. In conventional spin-polarized tunneling, the net polarization in the tunnel junction results from the spin-polarized density of states in the tip and the sample. Here, instead the tip's spin is treated quantum-mechanically.

For instance, if we turn back to the general Hamiltonian, g. 4.14, the tip is a superposition of up and down states. The description in terms of a spin-polarized density of states is therefore inaccurate and a quantum mechanical description of the form is preferable

η t (i, f ) = | f | Ŝ+ |i | 2 -| f | Ŝ-|i | 2 2| f | Ŝ |i | 2 , ( 4.6)
where |i and |f are the initial and nal states, respectively, while Ŝ+ and Ŝare the raising and lowering operators, respectively. With this de nition, the spin polarization arises from how the tip's spin interacts with the tunneling electron. For cobaltocene the spin excitation involves the transition M = +1/2 → M = -1/2 and the tip asymmetry is η t = -1. The junction spin polarization reduces to that of the sample η = -η s and we have η s = 20 ± 5%.

It is well known that the hybridization across the vacuum barrier between a molecular orbital and a metal electrode can "weight" the spin-dependent electron transmission (T ↑ and T ↓ ) into the s, p and d states of the sample [152,153]. For instance, DFT calculations show that when electrons tunnel between a nickelocene tip and a Fe atom ( § 5.2.4) [23], there is a strong contribution from the π molecular orbitals to the electron transmission. This is also probably the case for cobaltocene. The spin polarization η s therefore cannot be taken as a measure of the "absolute" spin-polarization of the sample, but remains a reliable quantity for measuring a magnetic contrast.

Vibrational excitations and Kondo side peaks

To conclude this chapter, we turn back to the Kondo side peaks brie y mentioned in § 4.3. These peaks arise because of the coupling of the singly-occupied π yz orbital to molecular vibrations. Figure 4.16(a) presents the d 2 I/dV 2 spectrum acquired above Cu( 100) with a Cc-tip in con guration 2; g. 4.16(b,c) present spectra acquired above two locations of a cobaltocene adsorbed at a step edge of Cu( 100). The peak positions, which, as expected [63], are symmetric with bias polarity, are marked by tics in the spectra. Figure 4.17 summarizes all the peak positions we could observe experimentally in an energy range up to 50 meV; the gure also includes some peak positions collected with other Cc-tip con gurations. The peak positions, hence the energies of the vibrational modes, show negligible dependence on Cc adsorption or intramolecular position. Their intensity, instead, may vary substantially, which hints to modes having di erent electron-vibration couplings. This is apparent in the simulations of the Kondo spectral function done by N. Lorente and coworkers, which is presented in Fig. 4. 16(d). The position of the simulated side peaks are xed to the DFT-computed vibrational energies of conguration 2 . A satisfying agreement with experiment is found, but clearly some modes are favored in the experimental line shape due to a stronger electron-vibration coupling. Further work will be needed to understand this behavior [154]. Please note that the present Kondo calculation accounts for the dynamical Jahn-Teller e ect of cobaltocene by treating the electronic The DFT-computed energies correspond to all relaxed tip con gurations. Please note that the Kondo resonance impairs the observation of modes having energy < 10 meV. Some modes may also not be observable due to a weak electron-vibration coupling, which depends on molecular con guration on the surface/tip and how it is addressed.

and vibrational degrees of freedom self-consistently to all orders in the electron-vibration coupling [155].

For a complete picture of the Kondo e ect of Cc, we present in g. 4.18(a) a typical dI/dV spectrum of single-Cc on a Cu(100) terrace. Compared to previous Cc-tip spectra, the line shape has a more complex pattern. It is much broader with a dominant peak located at 25 mV and a second peak at 125 mV, with a reproducible shoulder centered at zero-bias. The Kondo spectral function can be again computed assuming that molecular vibrations are still present when Cc is adsorbed on the terrace, g. 4.18(b). The calculation was performed with the computed vibrational frequencies, using the Kondo temperature and the electron-vibration coupling as tting parameters. The Kondo temperature was xed to that of the Cc-tip, implicitly assuming that the Kondo e ect is still carried by a single molecular orbital. The spectrum is broad showing two characteristic peaks at 30 mV and at 120 mV. Despite the large number of parameters, the calculation conclusively shows the prevalence of vibrational signatures in the spectrum. Unlike the Cc-tip, all possible vibrational modes of Cc are excited in the range from 2 meV to 395 meV. This leads to a quasi-continuum of excitations that are packed in two regions, one at 30 meV and the other at 120 meV, corresponding to the distribution of modes in the molecule. The low-energy features in the dI/dV can then be unambiguously assigned to a Kondo e ect in the presence of strong electron-vibration coupling in the Cc molecule.

Summary and conclusion

We have shown that with a cobaltocene molecule we can produce a Kondo resonance at the tip apex, which is anked by side peaks resulting from vibrational excitations. The Kondo resonance can be used to probe a magnetic exchange interaction across the vacuum gap and to determine a sample-related spin-polarization. These results open the exciting perspective of obtaining a magnetic contrast in the STM images by monitoring the changes in the Kondo resonance. A variety of magnetic systems can be investigated in this way, ranging from simple ones, e.g. atoms and molecules, to magnetic surfaces. With the help of an external magnetic eld, the information gathered at the atomic-scale with a Cc-tip, either through the exchange interaction or the spin polarization, should prove su cient for obtaining a magnetic contrast in STM images, thereby o ering an alternative to other spin-polarized scanning probe techniques [35,156]. The physics of two Kondo impurities may also be thoroughly explored with a Cc-tip, adding control to previous work by Bork et al. [36] and extending it to single Kondo molecules.

Chapter 5

Molecular spin excitations at the tip apex as a magnetic sensor

In this chapter we study the functionalization of a STM metallic tip by a spin S = 1 molecule possessing magnetic anisotropy (D), nickelocene (Nc). This project was already started when I arrived and was the main topic of Benjamin Verlhac's thesis [97]. I helped him to nalize it, while learning how to use the STM. As we showed in chapter 4, a well-calibrated sensor attached to the tip apex allows for freely addressing a surface target. This detection scheme eliminates surface-mediated interactions and bene ts from the vertical-displacement sensitivity of the STM, since the sensor-target distance is no longer imposed by the surface corrugation. As in the previous a chapter, we we will exploit electrically-driven spin excitations. The main di erence with cobaltocene is that the magnetic anisotropy of nickelocene eliminates the Kondo ground state [58,97], and xes a spin orientation for the molecular tip in its ground state, simplifying and adding control to the measurement.

We start the chapter by presenting how we characterize a nickelocene tip ( § 5.1). We then show how the nickelocene tip (Nc-tip), through changes in its characteristic spectroscopic signature, can be used to investigate a magnetic atom on a surface. Here we probe a Fe atom on a Cu (100) surface and show that magnetic information can be extracted from the conductance spectra, more speci cally the d 2 I/dV 2 spectra ( § 5.2). Finally, we show that the same information can be obtained for a magnetic surface such as a cobalt nanoisland ( § 5.3). We show, in particular, that with the inelastic component of the tunnel current we can visualize the magnetic corrugation of the nanoisland with atomic-scale precision. These results are published in Science [23].

The nickelocene tip

As we showed in chapter 4, it is possible to attach a cobaltocene molecule to the apex of the STM tip which is covered with copper. Since the procedure is identical for nickelocene, we refer the reader to § 4.2.1 for details concerning the tip preparation and to § 4.2.3 for the tip-surface distance calibration. Hence, we directly jump to the presentation of the spectroscopic signature of a Nc-tip. The d 2 I/dV 2 spectrum acquired with a Nc-tip positioned above a Cu(100) surface is presented in g. 5.1. We also present a d 2 I/dV 2 spectrum acquired above a Nc on Cu(100) using a metallic tip. With the Nc-tip, we observe a peak at positive bias and a dip at negative bias with energies symmetric to zero. Evidently, Nc retains its gas-phase spin-triplet ground state |S = 1, M = 0 , with the peak (dip) corresponding to the |M = 0 → |M = ±1 spin excitation.

The spectrum can be accurately reproduced using the the dynamical scattering model of M. Ternes [105], yielding a magnetic anisotropy of D = 3.7 ± 0.3 meV which mean that the spin hard axis is aligned with the molecular (z) axis. The error bar on D re ects a statistical average over several tips. On the surface, the magnetic anisotropy lowers to D = 3.2 ± 0.1 meV. To elucidate this behavior, M.-L. Bocquet and coworkers used DFT to calculate the relaxed electronic structure of a Nc in the gas phase, on a Cu(100) and on a metallic tip apex -the tip Figure 5.1: Inelastic spectrum of a Nc-tip. d 2 I/dV 2 spectra acquired with the Nc-tip above the Cu(100) surface (solid blue line) and with the metal tip positioned above a Nc on Cu(100) (solid grey line). The feedback was opened at a sample bias of V = -20 mV and at a tunnel current of I = 0.14 nA. The solid red lines are simulated spectra using a dynamical scattering model [105]. The insets sketch the junction con gurations.

Gas Phase

Nc on Cu (100) Nc on tip was mimicked by a single Cu atom on Cu(100). They then estimated the magnetocrystalline anisotropy energy (MAE) and showed that it changes depending on the metallic environment of Nc. Details of the calculation are given in [START_REF] Ormaza | [END_REF]. They nd a MAE of -1.54 meV for Nc in the gas phase, g 5.2(a), -1.27 meV for Nc on the Cu(100), g 5.2(b), and -1.49 meV for Nc adsorbed on the tip, g 5.2(c). The 17% di erence in the MAE is consistent with the experimental changes found for D. As in other molecular systems [157], the changes in D result from the elongation or contraction of Nc along its molecular axis. When Nc is adsorbed on the surface, a charge transfer of 0.1e -from the surface to Nc causes a contraction of nickelocene compared to the gas phase, g. 5.2(b). On the tip, instead, the charge transfer reverts and nickelocene gives 0.14e -to the tip, but the internal distances remain similar to those of the gas phase, hence D is close to the anisotropy found for the free Nc. The charge transfer, whether to the surface or to the tip, remains limited as the molecule retains an e ective spin S = 1 as proven by our spectroscopic data.

When the molecule is attached to the apex of the tip, we can observe a molecular pattern when imaging a single atom on the surface. In chapter 4, we used this approach to characterize the adsorption of cobaltocene on the tip apex ( § 4.2.1). Figure 5.3(a) shows that single Fe atoms present a ring-like pattern similar to the one of the Nc on the surface with the exception of a tilt angle, characterized by the fact that one side of the atom appears brighter than the other. This tilt angle can be determined through the height di erence in the line scan for both sides of the imaged atom, g. 5.3(b), yielding θ = 10 ± 5 • . The Nc molecule is strongly bound to the tip as proven by point contact measurements we carried out [109]. N. Lorente and coworkers used DFT to model the structure of the Nc-tip; details of the calculation are given in [109]. The calculated angle of θ = 13 • and image nicely agree with experiments, g. 5.3(c). The tilt angle results from Nc being bound to the Cu tip apex through two C atoms, inset of g. 5.3(c). Please note that in the previous DFT calculation shown in g. 5.2(c) nickelocene was forced in the center of the Cu atom to ease the calculation of the MAE. The projected density of states (PDOS), g. 5. 2). This behavior is di erent from a cobaltocene tip as recalled in g. 5.3(e). As we showed in ( § 4.2.2), a Cc-tip has only a contribution from the π yz orbital at the Fermi level, resulting in a characteristic two-lobe pattern.

Nc-tip (d) -2 -1 0 1 2 E-E F (eV) -2 -1 0 1 2 PDOS (1/eV) Co(dyz) Co(dxz) Cc-tip (e) -2 -1 0 1 2 E-E F (eV)

Exchange coupling across the vacuum gap

Probing a magnetic exchange interaction across a vacuum gap is experimentally demanding [35][36][37][38][39][40][41] because scanning probe techniques su er from a poor structural and magnetic characterization of the tip apex. We have shown that here instead we have a well-calibrated sensor attached to the tip apex. In this part, we will see how we can use the Nc-tip to probe the sample magnetism. We will rst focus on single Fe atoms and then turn to a cobalt nanoisland. These are well-known magnetic systems and constitute a model playground for testing the Nc-tip.

Spin excitations in the presence of an iron atom

We deposited the Fe atoms onto the Cu(100) surface using the protocol described in § 2.4.2. As anticipated in chapter 4, at the Fermi level the Fe atoms have no spectroscopical signature in our conductance spectra, i.e. they are spectroscopically dark. For instance, g. 5.4 compares a spectrum acquired above a Fe atom with a metal tip and with a Nc-tip. It can be seen that with the metal tip, the intensity associated to the iron atom is well below that of nickelocene. Consistently when using the Nc-tip we notice no di erence between the d 2 I/dV 2 acquired above a Fe atom and above copper ( g. 5.1).

To explore the magnetic coupling between the Nc-tip and the Fe atom, we used several tips having tilt angles θ between 5 • and 10 • . A typical set of spectra in a range between z = 133 and 80 pm is presented in g. 5.5(a). For large distances (z ≥ 130 pm), the spectrum is identical to the one measured with the Nc-tip above Cu (100). At shorter distances, the peak and dip in the d 2 I/dV 2 progressively split apart. The maximum splitting is measured for the shortest distance, 80 pm. At smaller distances the nickelocene is transferred to the Fe atom to form a FeNc complex, similar to the NiNc complex (see § 3.4). The peak and dip positions can be precisely determined by simulating the line shape via the model developed by M. Ternes [105].

The peak (dip) positions, which are presented in g. 5.5(b), are nearly symmetric with respect to their average value [dashed line in g. 5.5(b)]. This average position corresponds to the magnetic anisotropy of nickelocene and we nd D = 3.5 ± 0.2 meV. Along with the splitting, an asymmetry in the peak (dip) intensity is observed with bias polarity, g. 5.5(a,b). At positive bias, the peak occurring at the lowest sample bias has higher intensity compared to the one with the highest sample bias. At negative bias, we observe the exact opposite for the dips. When measuring in the same z-range above the copper surface, no change is observed in the spectra, g. 5.5(c). The splitting and asymmetry observed have magnetic origin.

Model Hamiltonian

To rationalize these ndings, we use the general Hamiltonian introduced in chapter 4 (eq. 4.3), adding to it the magnetic anisotropy of the Nc-tip. We neglect the tilt angle of Nc on the tip without loss of generality, and choose the z-axis perpendicular to the surface, which is then parallel to the molecular axis. The Hamiltonian reads,

Ĥ = D Ŝ2 z,N c + D Fe Ŝ2 z,F e -J Ŝz,Nc • Ŝz,Fe , (5.1)
where the rst two terms account for the magnetic anisotropy of Nc and Fe (D Fe < 0 and D Nc > 0), respectively, while the second term accounts for the coupling between Fe and Nc. As we show hereafter the coupling J is Ising-like and antiferromagnetic (J < 0) at the distances explored in our measurements. The eigenfunctions of the system, |ψ i (i = 0, 1, i.e. because the Fe atom is spectroscopically dark. Finally, ψ 0 → ψ 4 is forbidden since it does not conserve total spin momentum. The change in spin momentum for Fe and Nc amounts to δM = ±2 and cannot be compensated by a spin ip of the tunneling electron. The ground state |ψ 0 is a doublet state and corresponds to an antiferromagnetic con guration where the Fe spin is anti-aligned with the Nc spin along the z-axis. The excited state |ψ 2 corresponds to the ferromagnetic con guration. One can therefore limit the analysis to a |M Fe = -3/2 spin without loss of generality. This enables to simplify eq. 5.1 by neglecting the anisotropy term of Fe and reformulating the interaction term as J Ŝz,Fe Ŝz,Nc = gµ B B Ŝz,Nc ; please note that with this convention the exchange eld B points in the opposite direction of the Fe spin. This then leads to the simpli ed Hamiltonian:

9/4D Ψ 0 Ψ 5 Ψ 4 Ψ 3 Ψ 1 Ψ 2 D Nc +9/4D +3J/2 0 2 0 3 0 1 0 4 0 5 Ψ 5 M Nc = ± 1, M Fe = ± 1/2 Ψ 4 Ψ 1 M Nc = ∓ 1, M Fe = ± 1/2 M Nc = 0, M Fe = ± 1/2 Ψ 3 M Nc = ± 1, M Fe = ± 3/2 Ψ 2 M Nc = ∓ 1, M Fe = ± 3/2 M Nc = 0, M Fe = ± 3/2 Ψ 0 D+gμBB D-gμBB 0 Fe
Ĥ = D Ŝ2 z,N c -gµ B B Ŝz . (5.2)
This e ective Zeeman Hamiltonian consists of the gyromagnetic factor, the Bohr magneton (µ B = 57.9 µeV/T), and the exchange eld (B) produced by the Fe atom and acting on Nc along the z-axis. Within mean-eld theory, B = |J| S z,Fe /gµ B , where S z,Fe ≈ 3/2 is the e ective spin of Fe on copper [23,79], and g = 1.89 [24]. When using eq. 5.2, the spin excitations observed in the spin split-spectrum of Nc correspond then to |M = 0 → |M = 1 for the lowest bias and to |M = 0 → |M = -1 for the highest bias with M as the magnetic quantum number of nickelocene, g. 5.6(b).

Exchange coupling

Based on eq. 5.2, the peaks (dips) in the spectra split-apart by 2gµ B B and therefore their positions yield a measurement for the exchange eld. In g. 5.7(a), we plot the eld B extracted from the spectra using the ts presented in g. 5.5(a), as a function of z, the distance between extracted from the spectra of g. 5.5(a) using eq. 5.2. Inset: DFT calculated con guration of the tunnel junction for a distance of 100 pm with isosurface of the spin density (antiferromagnetic coupling). (b) Computed transition energies using [105] while increasing the coupling (bottom x-axis) or the exchange eld (top x-axis).

the Fe atom and the Nc-tip. The exchange eld can be tted with an exponential function of the form exp (-z/λ), demonstrating that Fe and Nc are exchange coupled. We nd a decay length of 34 ± 2 pm, which is in the same order as the cobaltocene tip ( § 4.3.3) and other exchange interactions explored with STM [24,37,38,41]. We nd |J| ≈ 0.9 meV at z = 80 pm, which is close to values obtained with other tip-adsorbate interactions [39,40].

Before going further, we wish to clarify some assumptions made earlier concerning the coupling J between the Nc-tip and the Fe atom. The Ising coupling along the z-axis has instead symmetric spin-split peaks where their average position is constant and equal to the magnetic anisotropy D. The latter situation matches closely our experimental observations, g. 5.5(b). We can therefore conclude that the coupling is Ising-like and along the z-axis.

To determine the sign of the coupling, M.-L. Bocquet and coworkers used DFT to compute the fully relaxed structure of the Nc-tip above an Fe atom on a Cu (100). Calculation details are given in [23]. Several tip-atom distances are considered closely matching the experimental setup. At each distance, the exchange E ex = E P -E AP is calculated, where E P (E AP ) corresponds to the energy of the junction when the spins of Nc and of Fe are parallel (antiparallel), inset of g. 5.7(a). They nd that the AP con guration is always favored and obtain an energy di erence of 13 meV between the P and AP con guration for a distance z = 120 pm, while no di erence is observed above 300 pm. This short-range antiferromagnetic coupling is attributed to the hybridization of the d-orbitals of Fe with the molecular orbitals of the Nc-tip.

Spin polarization

As presented in chapter 4 for the cobaltocene tip ( § 4.3.3), along with the exchange eld, we may also extract a spin polarization from the d 2 I/dV 2 spectra acquired with the Nc-tip. This is possible by measuring the amplitudes of the peak and dip associated to the each spin excitation. When Nc is excited from the |M = 0 ground state to the |M = +1 excited state, g. 5.8(a), it requires that the tunnel electron ips its spin from |↑ to |↓ in order to conserve the total spin angular momentum. At negative bias, an electron |↑ is emitted from the spin-up density of states (DOS) of Fe atom to the molecular orbital of the Nc-tip with transmission T ↑ , g. 5.8(a). This leads to a dip in the d 2 I/dV 2 spectrum of amplitude h -∝ T ↑ . At positive bias, an electron |↑ tunnels from the molecular orbital of the Nc-tip to the spin-down DOS of the Fe atom with transmission T ↓ . This leads to a peak in the d 2 I/dV 2 spectrum of amplitude h + ∝ T ↓ . The dip amplitude is weaker than the peak amplitude, which gives a non-zero spin asymmetry η = (h + -h -)/(h + + h -). When considering |M = 0 → |M = -1 , g. 5.8(b), the excitation of Nc requires an electron starting in a |↓ state and ending in a |↑ state, resulting in a spin asymmetry of -η.

The spin asymmetry versus B, which is extracted from the simulations of g. 5.5(a), is plotted g. 5.8(c). We discarded the values below 5 T as the spin-split peaks (dips) can no longer be resolved. We nd a constant value of η = 32%, while a statistical average over several Nc-tips yields a lower value of η = 23%. As for the cobaltocene tip, the asymmetry provides a measure for the adsorbate-related spin polarization [151], and we treat it quantum mechanically using eq. 4.6 of chapter 4. For spin excitation |M = 0 → |M = +1 , the tip asymmetry is η t = 1 and the junction spin polarization reduces to that of the sample, η = η s , hence we have η s = 32%. The sample asymmetry is the same for spin excitation |M = 0 → |M = -1 .

To improve our understanding of the spin asymmetry, N. Lorente and coworkers computed the elastic transmission for a junction composed of a Fe atom on a Cu(100) surface and a Nc-tip at a distance of 100 pm from the contact regime. The spin of the Fe atom is xed to |M Fe = -3/2 and an antiparallel alignment is imposed between the Fe and the Nc-tip spins. The computed transmission is presented in g. 5.9(a), while the spin-polarized PDOS of Nc is presented in g. 5.9(b). Even if the spins are probably non-collinear in the experiment, information can still be gathered regarding the orbital contributions. The transmission has its strongest contribution at the energies where the PDOS of the π orbital of Nc is highest, which underlines the strong molecular character of the transmission. The spin polarization η s probably then re ects the polarization of the adsorbate DOS, but is weighted by the spin-dependent Fe hybridization with the π orbitals of Nc.

Detecting surface magnetism

As we showed above, the Nc-tip can be used as a magnetic probe to interact with a magnetic atom on the surface. In the second part of this chapter, we extend the study to a magnetic surface in order to demonstrate how we may acquire an image with a magnetic corrugation. We will work with cobalt nanoislands on a Cu (111) surface as they are a well-known ferromagnetic system and therefore represent a model playground for our new magnetic probe.

Cobalt nanoislands

We deposited cobalt islands on a Cu (111) surface as presented in chapter 2 ( § 2.4.2). We obtain islands of various sizes but we will focus only on islands larger than 10 nm to avoid edge e ects [158,159]. At low temperature, the islands possess a magnetization perpendicular to the surface, g. 5.10(a). Contrary to the ground state of a Fe atom, which is a linear combination of up |M Fe = +3/2 and down orientations |M Fe = -3/2 , the island magnetization is locked along the z axis at our working temperature and we assume it to be upward without loss of generality.

The Co nanoislands have a triangular shape, g. 5.10(b), and are two atomic rows high, g. 5.10(c). The bright dots in the image corresponds to Nc molecules, which were dosed onto the sample for tip functionalization. Nickelocene adsorbs either on the nanoislands or on their bottom edges, g. 5.11(a). Since there are less Nc molecule on the Cu (111) surface, the molecule preferably adsorbs on the nanoislands than on copper as for other molecules [160]. We noticed that it was easier to transfer Nc from the sample to the tip by performing a point contact to edge molecules rather to island molecules. This might indicate that Nc is more strongly bound to the island than to its edge.

We use the Nc-tip to scan at constant-height the Co island with a bias close to the Fermi level (V = -1 mV), g. 5.11(b). The image is acquired in the small area depicted by a white box in g. 5.11(a). We observe a pattern which is similar to the one obtained by dragging a Cu atom on the Co island, g 5.11(c). This pattern corresponds to the atomic resolution of the Co island, where we can distinguish the top position of the Co atoms from the hollow position. Such a high-resolution image cannot be obtained with the Nc-tip on the Cu(111) surface, which is non magnetic. This gives a rst hint on the magnetic origin of this image.

Exchange coupling and magnetic corrugation

We investigated the in uence of the Co nanoislands on the characteristic signature of the Nctip. We start by a rst d 2 I/dV 2 spectrum with the Nc-tip above the Co island "far away" from the surface, 150 pm, with the tip positioned above a surface atom. The spectrum is identical to the one acquired above copper, g. 5.1, hence the island can be considered as spectroscopically dark. We then vertically approach the Nc-tip towards the island and acquire d 2 I/dV 2 spectra at di erent distances z, from 150 pm to 70 pm as shown g. 5.12(a). The behavior is similar to the one observed for the Fe atom: Once in the range of the magnetic coupling, the inelastic peak (dip) of the Nc-tip splits apart.

To rationalize this behavior, we use eq. 5.2 where B represents now the exchange eld produced by the cobalt atom onto the Nc-tip. The eld lifts the degeneracy between the |M = +1 and |M = -1 states of the Nc-tip, and two spin excitations become accessible, a low-bias one |M = 0 → |M = +1 and a high-bias one |M = 0 → |M = -1 , g. 5.6(b). We t our spectra with the dynamical scattering model of M. Ternes [105], including an e ective magnetic eld along the z axis to simulate the exchange eld and a spin polarization to take into account the spin asymmetry. The evolution of the exchange eld versus z is presented g. 5.12(b). As for the Fe atom, it follows an exponential behavior of the form exp (-z/λ), with a decay length λ = 50 ± 5 pm. For the shortest distance z = 70 pm, we nd B = 26 T. This corresponds to a Co-Nc coupling of J ≈ 3.2 meV using B = J S Co /gµ B and the DFT-computed e ective spin S Co ≈ 0.9. The low and high-bias amplitudes change little with bias polarity, which indicates that the spin asymmetry is weak. We nd a value of η < 10% for the island spin-polarization at the Fermi level.

Having determined the exchange eld, we now focus on the corrugation observed in the constantheight image of g. 5.11(b). Firstly, it is important to remark that there is a correlation between corrugation and the existence of spin-split inelastic peaks (dips). When the tip is placed at z = 150 pm from the island, for instance, the corrugation is lost, while it is present when the tip is placed at z = 80 pm. Second, the corrugation depends on the junction bias. As the applied bias is moved away from the Fermi level, the corrugation is increasingly lost. Figures 5.13 present images of a same cobalt area acquired at z = 80 pm using a negative bias of -1 and -5 mV, respectively.

To link the bias dependence of the corrugation to the spin-split spectra, we present in g. 5.13(c) d 2 I/dV 2 spectra acquired with the Nc-tip positioned in top and hollow positions of the island as depicted in g. 5.13(a). For the sake of a proper comparison, the Nc-Co distance for both spectra is xed to z = 80 pm by positioning the tip above a top site. As shown, the average position of the peaks (dips) is the same in the two spectra, this value corresponding to the magnetic anisotropy D of nickelocene. However, the splitting is stronger above the top site compared to the hollow site, which means that the exchange eld is stronger at the top site. Since at low bias (-2 < V < 2 mV) the main contribution to the signal comes from the inelastic peak (dip) associated to |M = 0 → |M = +1 , minute changes in the position of this peak (dip) are then su cient to produce a contrast in the image. To further clarify this point, in g. 5.13(d) we present the ratio between the tunnel current recorded above a top position and above a hollow position versus sample bias (z = 80 pm). For a sample bias of -1 mV, the I top is 3 times more intense than I hollow . While increasing the voltage bias, we can see this ratio decreasing rapidly. For a sample bias of -5 mV, I top is 1.2 time more intense than I hollow , which results in the loss of the contrast. Since the position of the inelastic peak (dip) is determined by the exchange eld, the contrast re ects spatial changes in the exchange eld. Using g. 5.13(c), the highest current (top position) can then be associated to an exchange eld of 21 T; the lowest current (hollow position) can be instead associated to an exchange eld of 15 T.

To con rm the magnetic origin of the contrast in the images, N. Lorente and coworkers computed the exchange energy using DFT (E ex = E P -E AP ) for di erent tip-surface distances with the Nc-tip above top, hollow and bridge sites in the center of a Co island, g. 5.14(a). For distances below 100 pm, which correspond to the z-range explored experimentally, the exchange energy changes depending on the computed site, g. 5.14(b). The top site is shown to possess a stronger exchange energy compared to the bridge and hollow sites, in agreement with our ndings. This is due to the local character of the d orbital, making the probe very sensitive to a displacement above the surface.

Summary and conclusion

We showed that the spin excitations of a Nc-tip can be used as a marker to probe the magnetism of a single object adsorbed on a metallic surface or a magnetic surface, even without external magnetic eld. Information may be gathered on the exchange eld and the spin polarization of the sample, which can be further exploited to image the magnetic texture of surface. In future experiments, we intend to work at lower temperatures to improve our energy resolution in the spectra, which will improve the detectability of the exchange elds. We hope to detect exchange elds as low as 2 T . We also wish to use an external magnetic eld to obtain experimentally the sign of the magnetic coupling J [24,40] and improve the magnetic contrast in the images.

Chapter 6 Conclusion and perspectives

In this thesis we studied the magnetic properties of two metallocene molecules: Cobaltocene (Cc) and nickelocene (Nc). We showed that both molecules can be attached to the tip apex of a STM, to be freely displaced over the surface. These molecular tips can undergo spin excitations, which change when the tip interacts magnetically with the sample. Through these spin excitations it is then possible to monitor the surface magnetism with atomic-scale resolution. We also showed in the thesis that metallocene molecules present strong vibrational excitations, which can be intertwined with spin excitations.

In chapter 3 we provided an overview on the structural and electronic aspects of nickelocene and Ni-nickelocene complexes (NiNc). The Nc molecule retains its vacuum spin (S = 1) and magnetic anisotropy even upon adsorption onto a copper surface. This then leads to characteristic spin excitations, which are visible as peaks and dips in the inelastic tunneling spectra (IETS). We then showed that Nc can be doped with single Ni atoms to form a NiNc complex on copper. Under speci c conditions, the NiNc complex gives rise to spin and vibrational excitations also visible in IETS. Interestingly, a new vibrational-assisted spin excitation is evidenced resulting from the concomitant excitation of a spin and a vibration quanta. Our study suggests that these excitations may be rather common in molecules possessing magnetic anisotropy.

Chapter 4 was dedicated to the thorough study of the Cc molecule, where the focus was set on the functionalization of the STM tip with the molecule. We demonstrated to which extent the functionalized probe can be characterized, structurally and electronically. The cobaltoceneterminated tip (Cc-tip) gives rise to a Kondo resonance with vibrational side bands. The most salient result consisted in the modi cation of the Kondo resonance upon approaching an adsorbed Fe atom. The single resonance at large tip-atom distances gradually evolves into split resonances upon reducing the tip-Fe separation. This observation was traced to the Cc-Fe exchange interaction, the resonances separation providing a measure of the exchange eld (B) exerted by the Fe atom on Cc. Moreover, from the asymmetry of the IETS spectra with respect top zero bias voltage, an adsorbate-related spin-polarization was inferred. These results open the perspective of obtaining a magnetic contrast in the STM images by monitoring the changes in the Kondo resonance. They also o er to explore the physics of two Kondo impurities separated by a vacuum gap, adding control to previous work by Bork et al. [36] and extending it to single Kondo molecules. During our experiments we also obtained a spectra quite di erent for the Cc adsorbed on a at terrace. Unlike the Cc-tip, in this case, several vibrational modes can be excited leading to a quasi-continuum of excitations.

Chapter 5 extended the study to a nickelocene-decorated tip (Nc-tip), which exhibits a S=1 as its spin ground state and, in contrast to Cc (S=1/2), no Kondo e ect. We showed that the magnetic exchange interaction between a Nc-tip and an adsorbed Fe atom likewise leaves its ngerprint in IETS when vertically approaching the tip towards Fe. With decreasing separation the inelastic excitation signals gradually split apart. The peak (dip) separation is a measure of the exchange eld exerted by the Fe atom on Nc. The bias voltage asymmetry of the inelastic peak (dip) heights can be used to extract an adsorbate-related spin polarization. We also demonstrated that an atomic-scale magnetic contrast can be obtained for Co islands on the basis of the exchangesplit spin excitations.

In the last decade, the decoration of metal probe-tips by a molecule intentionally picked up from a surface has proven to be a powerful method to improve the measurement capabilities of a STM. Metallocene-tips represent the rst example of molecular tips providing access to surface magnetism. The technique is new and requires to be consolidated through the investigation of model magnetic systems if we wish to generalize its use for magnetic imaging. Simple questions need still to be addressed. For example, can we probe a surface with a complex spin texture? In the general case where the spin texture investigated is arbitrary, the spin Hamiltonian of metallocenes will read Ĥ = H 0 -gµ B (B x S x + B y S y + B z S z ). When the exchange eld is out-of-plane (B z ), the IETS spectrum is spin-split as we showed. We speculate that an in-plane exchange eld (B x and/or B y ) should shift the peak/dip of the IETS spectrum. To experimentally validate this, well-known surfaces with in-plane magnetization could be investigated, where the magnetic corrugation in the images will be carried by changes in B x and/or B y . Surfaces with cycloidal spin order are also of interest as both a splitting and a shift of the inelastic peaks (dips) should be observed.

Another important point to address is whether it is possible to probe spintronic properties related to single molecules. Organometallic molecules such as phthalocyanines play a central role in molecular spintronics and represent the obvious extension of our investigation on single atoms. It will be interesting to check whether the exchange eld and spin polarization can be spatially mapped within single phthalocyanine molecules. We expect the spin polarization to vary and change in sign across the molecule as observed in spin-polarized STM measurements [163,164]. Other possible extensions could consist in using more sophisticated met-allocene complexes at the tip apex to increase tip functionality and possibly also exploit the vibrational excitations for an increased image resolution as demonstrated for CO-terminated tips [START_REF] Chiang | Atomic-scale magnetometry with a mobile molecular quantum sensor JSTM 2019[END_REF].

Résumé 6.1 Préambule

Ma thèse porte sur le stockage de l'information numérique sous forme magnétique, en particulier le but est d'atteindre la limite ultime, celle du stockage sur un seul spin et ce même si l spin se trouve dans une structure magnétique complexe. Pour se faire, nous utilisons un microscope à e et tunnel (STM), sous ultra-vide et à 2.4 K, qui permet la détection d'un spin unique tout en imageant avec une résolution atomique. Nous allons également nous servir de molécules, que nous placerons sur la pointe, a n de générer électriquement une excitation de spin et étudier le magnétisme de surface à une échelle atomique. Cette approche permet déplacer librement la molécule au dessus de la cible en s'a ranchissant des interactions due à la surface tout en béné ciant de la sensibilité verticale du STM. Un autre avantage est de bien dé nir la partie active de la pointe. Les molécules utilisées, le nickelocène et le cobaltocène, dans cette thèse font partie de la famille des métallocènes qui sont magnétiques et possèdent des marqueurs spectroscopiques permettant de sonder leur environnement.

Chapitre 1 : Introduction

Dans ce chapitre, nous introduisons l'e et Kondo en donnant le contexte de sa découverte et les di érentes étapes ayant conduit à ce modèle. Nous allons présenter comment l'e et Kondo est détecté par STM et comment les vibrations moléculaires peuvent a ecter la signature Kondo. En n, nous allons introduire l'excitation de spin d'un atome unique et sa détection par STM.

L'e et Kondo

Excitation induite par courant en STM

Nous allons maintenant voir comment des informations magnétiques peuvent être obtenues avec un STM en utilisant le courant tunnel inélastique.

Excitation vibrationnelle et Kondo

I(z, V ) ∝ ∞ -∞ T (z, E, V )[f (E -eV ) -f (E)]ρ t (E -eV )ρ s (E)dE. (6.1)
Cette expression fait intervenir la distribution de Fermi-Dirac f et les densités d'état de la pointe (ρ t ) et de la surface ((ρ s ) ainsi que la transmission de la jonction T :

T (E, z, V ) ∝ exp[-kz φ ef f + eV 2 -(E -E ) avec k = 1.025 Å -1
eV -1/2 et φ ef f le travail de sortie moyen du substrat et de la pointe.

Cette dépendance exponentielle du courant par rapport à la distance pointe-échantillon permet de faire des images dŠun échantillon, en scannant la surface ligne par ligne, avec une précision de lŠordre du dixième dŠangström. Une boucle de rétro-action permet dŠajuster la hauteur de la pointe a n de garder un courant tunnel constant. Les déplacements verticaux de la pointe lors de ces scans sont alors enregistrés et directement liés à la topographie de la surface de lŠéchantillon, g. 6.5.

Spectroscopie à e et tunnel

Le courant tunnel dépendant de la densité d'états électroniques (DOS) locale de la surface, il est possible dŠobtenir des informations sur la surface à partir du courant tunnel. En travaillant à la température de l'hélium, nous pouvons négliger l'excitation thermique et en contrôlant que ρ t soit constant, le courant tunnel ne dépend alors plus que de la DOS de l'échantillon et de la transmission :

I(z, V ) ∝ ∞ -∞ T (z, E, V )ρ s (E)dE.
Ainsi, en e ectuant un balayage de tension, il est possible de sonder spectroscopiquement la surface et même d'objets déposés dans la jonction, c'est la spectroscopie à e et tunnel. La présence de structures dans le spectre dépend de la con guration électronique de la jonction. Ces structures sont d'autant plus facile à observer dans les spectres de conductance di érentielle (dI/dV), qui apparaît comme un pic dans la conductance di érentielle, où l'énergie de ces pics nous donne des informations sur la surface (gap HOMO-LUMO, e et Kondo...).

dI dV (z, V ) ∝ ρ s (V )T (z). (6.2)
Il existe également des électrons qui tunnellent de manière inélastique, c'est à dire qu'ils cèdent de l'énergie lors du processus de tunneling, qui est donnée au système et permet de l'excité. Par exemple, un électron peut exciter une transition entre un plasmon, un état vibronique [64] ou deux niveaux Zeeman, comme c'est le cas pour l'excitation de spin [8]. Le courant inélastique apparaît comme une augmentation du courant dans les spectres de conductance, sous forme d'une marche symétrique centrée autour du niveau de Fermi, et sŠexplique par lŠouverture d'un nouveau canal inélastique dans lequel tunnellent les électrons inélastiques.

Le chapitre se termine par la présentation des techniques de préparation de nos échantillons, ainsi qu'une description de notre STM. Grâce à cet outil, nous atteignons une résolution en énergie de l'ordre de 750 ţeV pour les structures élastiques et 1.15 meV pour les structures inélastiques. Lorsque le Nc est déposé par évaporation sur la surface, il peut s'adsorber soit en molécule isolée, g. 6.6(c), soit sous forme d'un réseau de molécules, en jouant sur la température et le ux du dépôt, g. 6.6(d). En ce qui concerne les molécules isolées, l'adsorption se fait sur un site "hollow" par un ligand en contact avec la surface; l'autre faisant face à la pointe, donne sa forme annulaire à la molécule en imagerie STM, g. 6.6(c). Cette étude a été complétée par une simulation de l'adsorption de la molécule sur une surface de Cu(100) au moyen de la Density Functional Theory (DFT) qui donne le même site d'adsorption. Quand les molécules sont adsorbées en réseaux, il existe deux formations possibles le réseau "compact" et le réseau "appareillé". C'est sur ce dernier que nous allons réaliser nos expériences. Une image STM du réseau "appareillé", g. 6.6(d), permet de mettre en évidence les paramètres de mailles. Ces réseaux ont été simulés en DFT et comparés aux résultats expérimentaux. Ces réseaux ressemblent structurellement à ceux obtenus pour le ferrocène [100], un autre métallocène. Après avoir étudié l'adsorption du Nc sur le Cu(100), nous allons étudier ses propriétés magnétiques. Des mesures ont montré que le spin de la molécule était conservé lors de lŠadsorption sur la surface, ainsi que la présence d'une anisotropie magnétique (D) estimée à 3.2 meV, ce qui a été con rmé par des simulations DFT [START_REF] Ormaza | [END_REF]. Cette anisotropie magnétique xe l'état de moment de spin fondamental à |S = 1, M = 0 et |S = 1, M = ±1 pour l'état excité, g. 6.7(a). La DFT montre un faible transfert de charge entre la molécule et la surface, ce qui s'explique par les ligands qui "isolent" le métal de la molécule du métal de la surface. De plus, la DFT indique que le moment magnétique du Nc est conservé lors de l'adsorption, ainsi que son spin S = 1 [99,[START_REF] Ormaza | [END_REF]. Le spectre dI/dV de Nc donne une structure ressemblant à une marche et donne un pic dans le spectre d 2 I/dV 2 à 3.2 mV, g. 6.7(b,c), nous amenant à considérer l'ouverture d'un canal inélastique correspondant à l'excitation de l'anisotropie magnétique de Nc. Des résultats identiques ont été obtenus pour des molécules intégrées dans un réseau. Les électrons inélastiques passant au travers de la jonction doivent e ectuer un "spin ip" a n de conserver le moment de spin total et respecter les règles de sélection. Le programme de M. Ternes [105] permet de prendre en compte les paramètres de couplages d'échange entre le Nc et les électrons du métal, responsables des épaulements dans le spectre en dI/dV, g. 6.7(c). Il est important de souligner que des résultats identiques ont été obtenus pour les molécules formant le réseau de Nc.

Complexes de Ni-Nc sur Cu(100)

Nous allons maintenant présenter les résultats du papiers de Bachellier et al.. [90] dans lequel nous avons dopé un réseau de Nc avec des atomes de Ni a n de créer un nouveau complexe Le cobaltocène (Cc) est une molécule de la famille des metallocènes composé d'un atome de Co et de deux ligands cyclopentadienyl encadrant l'atome métallique, ce qui en fait une molécule de spin 1/2, g. 6.9(a,b). Après avoir déposé les molécules de Cc sur une surface de Cu(100), les images STM montrent la présence de molécules de Cc isolées sur les terrasses et les marches de la surface sur trois sites d'adsorbtions di érents. Sur la terrasse, les molécules de Cc présentent une forme d'anneau, g. 6.9(c) comme pour les autres métallocènes [99,100]. Cette molécule est adsorbée avec son axe moléculaire parallèle à la surface, g. 6.9(d), avec l'anneau centré sur un site "hollow". On retrouve ce résultat en simulant la structure relaxée par DFT, g. 6.9(c,e). La simulation du moment magnétique de la molécule de Cc sur Cu(100), montre qu'il est principalement porté par le Co (0.75 µ B ), correspondant à un spin e ectif S = 1/2, alors que le moment magnétique du Nc était principalement porté par l'anneau. Ce sont les molécules de Cc sur les terrasses que nous utiliserons pour fonctionaliser la pointe STM. Nous observons les pics satellitaires sur le spectre d 2 I/dV 2 acquis avec une pointe Cc en con-guration 2, g. 6.12(a), que nous avons simulé, g. 6.12(b), et pour la molécule de Cc à la marche, g. 6.12(c). Dans les deux cas, la position des pics est symétrique par rapport à la tension. De plus, que la molécule soit sur la pointe ou à la marche, la position des pics est similaire, mais l'intensité des pics change, les di érents modes de vibrations ayant des couplages électron-vibration di érents. La g. 6.12(d) liste la position des modes observés expérimentalement et les compare à ceux simulés. On constate qu'il manque certains modes de vibration. En n, a n d'avoir une étude complète sur le couplage électron-vibration, nous présentons le spectre de la molécule de Cc adsorbée sur la terrasse, g. 6.12(e). Dans ce cas, le spectre est di érent, plus large et dominé par un premier pic à 25 mV et un second pic à 125 mV. Dans ce cas et contrairement à la pointe de Cc, tous les modes de vibrations sont excités et regroupés en deux régions qui correspondent aux pics dominants du spectre.

6.6 Excitation de spin d'une pointe moléculaire comme sonde magnétique.

Ce dernier chapitre est une étude de la pointe STM fonctionalisée avec un Nc, réalisée en collaboration avec mon prédécesseur Benjamin Verlhac [23]. La pointe peut être utilisée comme une sonde magnétique en étudiant le changement de signature spectroscopique lors de l'approche d'objets magnétiques. Dans un premier temps, nous allons réaliser l'approche sur un atome de Fe adsorbé sur la surface, qui n'a pas de signature spectroscopique. Nous présentons une série de spectres de l'approche de la pointe de Nc au dessus du Fe, g. 6.14(a); pour une large distance z = 133pm, le spectre est identique à celui de la pointe sur la surface, alors que lorsque l'on se rapproche, le pic se divise en deux pics symétriques due à l'e et Zeeman, qui lève la dégénérescence entre les deux états excités du nickelocène. Le plus gros écart entre les pics est observé à 80 pm; en dessous de cette distance, la molécule tend à se désorber de la pointe pour s'adsorber sur le Fe pour former un complexe FeNc. La position moyenne des pics correspond à la valeur de l'anisotropie magnétique D = 3.5 ± 0. Dans cette seconde partie, nous verrons l'application de la pointe de Nc comme sonde magnétique pour l'étude d'îlots de Co sur Cu (111), une surface ferromagnétique présentée g. 6.15(a). Lorsque nous approchons la pointe de l'îlot de Co, nous assistons, comme pour Fe, à une levée de dégénérescence de l'état excité de Nc, g. 6.15(b), dont nous pouvons obtenir le champ d'échange, g. 6.15(c). Nous trouvons également η < 10%. Ce qui est remarquable dans cet exemple c'est la possibilité de combiner la résolution atomique du STM ainsi que le couplage d'échange a n de réaliser des topographies magnétiques de l'îlot. Lorsque l'on image un îlot de Co à 80 pm et à -1 mV, g. 6.15.(d), on obtient une résolution atomique que l'on perd sur le Cu (111) ou en changeant la tension à des valeurs plus élevées comme -5 mV, g. 6.15(e). Il est possible de lier cette dépendance en tension à la corrugation visible sur les images en observant la di érence de séparation des pics pour les sites "hollow" et "top, g. 6.15(f). À bas voltage (-2 < V < 2 mV), la contribution principale venant du signal inélastique, un léger changement dans la position est alors su sant pour produire le contraste observé, g. 6.15(d). Cela se traduit explicitement dans le ratio du courant entre le site "top" et "hollow", g. 6.15(g). Dans ce cadre, le plus haut courant ("top") peut être associé à un champ d'échange de 21 T et 15T pour le courant le plus faible ("hollow")

Spin and vibrational excitations of single metallocene molecules in the junction of a scanning tunneling microscope

Résumé La fonctionnalisation d'une pointe-sonde métallique par une molécule a grandement amélioré la façon de sonder une surface avec un microscope à effet tunnel (STM). Les degrés de liberté de la molécule introduisent des interactions pointe-surface dans le gap tunnel qui sont généralement absentes avec une pointe métallique conventionnelle. Ce travail de thèse vise à utiliser une pointe moléculaire afin de sonder le magnétisme de surface à l'échelle atomique. Nous montrons qu'il est possible de décorer une pointe STM par une molécule magnétique de métallocène et d'en contrôler électriquement ses états de spin en la transformant ainsi en un capteur magnétique. Deux types de métallocènes sont utilisés: le cobaltocène de spin S=1/2 et le nickelocène de spin S=1. Nous montrons que les deux molécules permettent de caractériser le magnétisme d'un atome en surface au travers d'une interaction d'échange avec ce dernier. Les excitations de spin dans ces molécules peuvent, cependant, être entremêlés à des vibrations moléculaires, qui dépendent de la structure métallocène-pointe.

Mots clés : Cobaltocène, nickelocène, STM, spectroscopie à excitation de spin, vibrations moléculaires, effet Kondo

Résumé en anglais

The decoration of a metal probe-tip by a molecule intentionally picked up from a surface has proven to be a powerful method to improve the measurement capabilities of a scanning tunneling microscope (STM). The degrees of freedom of the molecule introduce tip-surface interactions across the vacuum gap that are usually absent when using a metallic apex. This thesis aims at using a molecular tip to probe surface magnetism at the atomic-scale. We show that magnetic metallocene molecules can be used to decorate a tip apex and act as magnetic sensors through electrically driven spin excitations among their spin states. Two different metallocenes are used: The spin S=1/2 cobaltocene and the spin S=1 nickelocene. In both cases, the molecule can interact with its nearby environment and magnetic information can be gathered from an exchange interaction with surface atoms. Spin excitations can however be intertwined with molecular vibrations that depend on how the metallocenes molecules are contacted to their metallic environment.

Keywords: Cobaltocene, nickelocene, STM, spin excitation spectroscopy, molecular vibrations, Kondo effect

6. 6

 6 Excitation de spin d'une pointe moléculaire comme sonde magnétique. . . . . . 6.6.1 Pointe de nickelocène. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Couplage d'échange à travers le vide et polarisation. . . . . . . . . . . . 6.6.3 Détection de magnétisme de surface. . . . . . . . . . . . . . . . . . . . . 6.12 Étude des pics satellitaires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Présentation de la pointe de Nc. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Cas de la pointe de Nc au dessus d'un atome de Fe. . . . . . . . . . . . . . . . . 6.15 Cas de la pointe de Nc au dessus d'îlot de Co. . . . . . . . . . . . . . . . . . . . . List of Tables 3.1 Parameters for the compact and paired networks of nickelocene . . . . . . . . . 3.2 Evolution of the red-shift versus tip-complex distance. . . . . . . . . . . . . . . 3.3 Energy onsets and transition rates for di erent spectra of Ni-nickelocene . . . . 4.1 Structural properties and magnetism for a cobaltocene tip . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Anomaly in the electrical resistance at low temperature. (a) Resistivity of Ti, Zr and Hf versus temperature.The metals reach a nite resistivity at low temperatures as expected from the Bloch-Grüneisen formula (solid curves). Reproduced from[START_REF] Allen | Boltzmann Theory and Resistivity of Metals[END_REF]. (b) Electrical resistance of Au between 1 K and 5 K. The resistivity-temperature curve reaches a minimum around 3.9 K and then starts to increase while the temperature decreases. Reproduced from[START_REF] Haas | [END_REF].

Figure 1 . 2 :

 12 Figure 1.2: The Anderson model. (a) Unperturbed energy levels in the absence of s-d admixture. (b) Density of states with the s-d hybridization for a coupling V κd . The localized spin-up and down magnetic levels have width 2∆ and are shifted. Reproduced from [47].

Figure 1 . 3 :

 13 Figure 1.3: Elastic spin ip scattering mechanism. (a) Conduction electron before tunneling from the conduction band to the impurity. (b) Creation of a virtual state at E d + U through the combination of two spins, which point in opposite directions due to the Pauli principle. (c) Conduction electron after tunneling from the impurity to the conduction. The spin of the impurity switches from down to up and the spin of the conduction electron switches from up to down.

Figure 1 . 4 :

 14 Figure 1.4: Kondo ground state. (a) Electron scattering by a magnetic impurity without screening above T K and (b) with screening below T K .

2 Figure 1 . 5 :

 215 Figure 1.5: Spectral function of a Kondo impurity. Local d-orbital spectral function A d (E) for a temperature T = 4 K. The singly occupied state lies below the Fermi energy at E d and is broadened by 2∆ due to its hybridization with the conduction electron of the host metal. The Coulomb repulsion U (in red) separates the unoccupied states from the singly occupied state. The Kondo resonance occurs at approximately E F and has a width proportional to k B T K (in orange). Parameters used for the calculation: E d = 0.84 eV and U = 2.84 eV. Inspired from [51].

Figure 1 . 6 :

 16 Figure 1.6: First Kondo resonance detected with a STM. (a) Constant-current image (400 × 400 Å 2 ) of Co atoms on Au(111) at 4 K. (b) dI/dV spectra taken with the STM tip above a single Co atom and above the nearby Au(111) surface. The dashed line is a simulated Fano line shape. (c) dI/dV spectrum acquired above a Ce impurity on Ag(111). The solid line is a simulated Fano line shape. Panels (a) and (b) are reproduced from [5], panel (c) is reproduced from [4].

Figure 1 . 7 :

 17 Figure 1.7: Temperature and magnetic-eld dependence of the Kondo resonance. (a) dI/dV spectrum of a Co atom on Cu 2 N/Cu(100) measured at di erent sample temperatures.The inset corresponds to the intrinsic half-width of the Kondo resonance extracted from the ts (red line of the main panel). Reproduced from[58]. (b) dI/dV spectra of a Mn atom near the boundary of an oxide patch on a NiAl(110) surface. The spectra were acquired at a temperature of T = 0.6 K at several external magnetic elds. Reproduced from[8].

  3(b). Within this complex, TCNQ possesses an unpaired electron making it a spin S = 1/2 organic molecule. The single electron resides in a molecular orbital, which is more extended than d orbitals and therefore easily subject to a coupling with molecular vibrations. At the center of TCNQ, the dI/dV spectrum shows a Kondo resonance accompanied by two side peaks at ±41 meV. These are assigned to satellite Kondo peaks arising from a vibration of TCNQ, which DFT calculations place at 41.6 meV, inset of g. 6.3(b). The three processes involved in the dI/dV spectrum are summarized in g. 6.3(c): (1) The electron

Figure 1 . 8 :

 18 Figure 1.8: Kondo e ect and vibrational excitations. (a) Conductance vs bias voltage for three vibrational modes di ering in energy. The modes are noted by a, b and c. (b) dI/dV spectrum obtained on the center of a TCNQ molecule revealing various elastic and inelastic processes during electron tunneling, noted (1) to (3). The inset shows the motion of TCNQ carbon atoms associated to the vibrational mode involved in the inelastic process. The vibration consists in an in-plane ring deformation, accompanied by the stretching of the C=C(CN 2 ) double bonds. The arrows are proportional to the amplitude of motion. (c) Energy scheme of Kondo (1), vibrational excitation (2) and vibrational-assisted Kondo (3). Panel (a) is adapted from [66], panels (b) and (c) from [67].

Figures. 1 .

 1 Figures.1.9(b) demonstrates that the presence of the insulting layer is essential for observing the spin excitation in the dI/dV spectrum. It also ensures a partial decoupling of Mn from the metal, thus extending the electronic lifetimes for the spin states of the Mn atom. Nowadays, spin excitations are mostly observed on atoms adsorbed on thin insulating layers such as Cu 2 N

Figure 1 . 9 :

 19 Figures.1.9(b) demonstrates that the presence of the insulting layer is essential for observing the spin excitation in the dI/dV spectrum. It also ensures a partial decoupling of Mn from the metal, thus extending the electronic lifetimes for the spin states of the Mn atom. Nowadays, spin excitations are mostly observed on atoms adsorbed on thin insulating layers such as Cu 2 N

Figure 2 . 1 :

 21 Figure 2.1: First images published with STM. (a) Topography of a CaIrSN 4 (100) surface. (b) Two individual line scans exhibiting triple, double and mono-atomic steps. (c) Pseudo 3D image of two complete 7 × 7 unit cells. (d) Top view of (c) clearly exhibiting the six-fold rotational symmetry around the rhombohedron corners. (a) and (b) reproduced from [82], (c) and (d) reproduced from [83].

Figure 2 . 2 :

 22 Figure 2.2: Tunneling current. (a) Sketch of the STM junction in the classical case. The electron cannot cross the energy barrier. (b) Sketch of the STM junction in the quantum case. The electron wave function spreads across the energy barrier. (c) Representation of a STM junction.

Figure 2 . 3 :

 23 Figure 2.3: Acquiring a STM image. (a) Sketch of the acquisition of an image. (b) Constantheight mode. (c) Constant-current mode.

Figure 2 . 4 :

 24 Figure 2.4: Scheme of a STM junction. (a) For a positive sample bias, the electrons tunnel from the occupied state of the tip to the unoccupied state of the sample, (b) while for the negative sample bias, the electrons tunnel from the occupied state of the sample to the unoccupied state of the tip. φ e stands for the average work function and represents the e ective tunneling barrier. The energy is expressed with respect to the Fermi level of the system, E F .

Figure 2 . 5 :

 25 Figure 2.5: Inelastic process in STM. (a) Sketch of an inelastic process in STM. The energy E ex is lost by the electrons during the excitation. We can detect this excitation through : (b) a change in the slope of the current versus bias curve at eV = ±E ex , (c) a step at eV = ±E ex in the dI/dV curve, (d) a dip (peak) at eV = -E ex (eV = E ex ) in the d 2 I/dV 2 curve.

Figure 2 . 6 :

 26 Figure 2.6: First detection of a molecular vibration in STM. In their IETS experiment, Stipe, Rezaei and Ho acquired the current-versus-bias, dI/dV , d 2 I/dV 2 curves above a C 2 H 2 molecule and a Cu(100) substrate, noted 1-2 on the spectrum. (a) They observed a change of slope in the current-versus-bias curve, highlighted by a blue arrow. (b) This change in the current-versusbias curve leads to a step in the dI/dV spectrum, highlighted by a blue arrow. (c) It becomes a peak at 358 mV in the second derivative d 2 I/dV 2 , highlighted by a blue arrow. Reproduced from [64].

Figure 2 . 7 :

 27 Figure 2.7: Lock-in detection and spectroscopy. Comparison between a dI/dV spectrum recorded with (a) a lock-in ampli er and (b) a numerical derivation of the I(V ) curve. The two curves were acquired in the tunneling conditions. We added the lock-in detected curve to the numerically-derived curve to highlight the change in the signal-to-noise ratio.

Figure 2 . 8 :

 28 Figure 2.8: Temperature and the lock-in broadening in tunneling spectroscopy. (a) Thermal broadening function resulting from the temperature dependence of the occupation of electron states. (b) Broadening function related to the use of a lock-in ampli er [87].

Figure 2 . 9 :

 29 Figure 2.9: Overview of our STM. (a) Image of the STM chamber (1) with ltered input and output connectors (2), home-made evaporator for atom deposition (3) pneumatic feet (4). (b) Image of the three decoupling springs (5). (c) Image of the head of the STM with the SMA connector for the sample bias (6), SMA connector for current(7), sample holder (8) and tip with its piezoelectric motors(9).

Figure 3 . 1 :

 31 Figure 3.1: Structure and electronic con guration of nickelocene. (a) Structure of nickelocene (Ni: green, C: grey, H: white). (b) Electronic d-manifold con guration of nickelocene yielding a spin S = 1.

Figure 3 . 2 :

 32 Figure 3.2: Scheme showing nickelocene deposition. (a) Chamber before opening the valve, while we have Nc in the preparation chamber at a pressure around 10 -6 mbar. (b) The right panel shows the Nc ux coming from the crucible to the sample, after opening the valve.

Figure 3 . 3 :

 33 Figure 3.3: Image of adsorbed nickelocene. (a) Layer of Nc molecules on Cu(100) in a paired con guration. This image of 19 × 19 nm 2 was taken using a sample bias V = 20 mV and a tunneling current I = 20 pA. (b) Layer of Nc molecules on Cu(100) in a compact con guration. On both panels we can see isolated Nc molecules. This image of 23 × 23 nm 2 was taken using a sample bias V = 20 mV and a tunneling current I = 20 pA.

Figure 3 . 4 :

 34 Figure 3.4: Adsorption of isolated nickelocene. (a) STM image of isolated Nc (V = 20 mV, I = 50 pA, 40 × 40 Å 2 ) and two iron adatoms on the Cu(100) surface. The inset corresponds to a STM image of the Cu(100) surface obtained by a tip-assisted manipulation of a Fe atom on the surface (V = 10 mV, I = 75 pA, 16.5 × 16.5 Å 2 ). (b) Top view of the DFT simulation for various adsorption sites of a vertical nickelocene on Cu(100); the top (red), bridge (green) and hollow (blue) adsorption sites have been highlight. (c) Side view of the DFT simulation to determine the adsorption height of a vertical nickelocene on Cu(100).

Figure 3 . 5 :

 35 Figure 3.5: Compact network of nickelocene. (a) STM image of the compact network (V = 20 mV, I = 20 pA, 70 × 70 Å 2 ). The unit cell is shown by solid white lines. a 1 ,a 2 represent the lattice vectors and are apart by an angle θ = 90 • . The orientation of the layer relative to the [011] and [01 1] directions of Cu(100) is indicated by the angle ψ and was determined using the atomically-resolved image of the Cu(100) in the inset. (b) STM low-bias image of the compact layer (V = 1 mV, I = 10 pA, 28 × 46 Å 2 ) revealing the presence of the horizontal molecules. The molecular arrangement is sketched in orange with horizontal Nc depicted as hourglasses and vertical Nc as circles. (c) Line pro les along the solid and dashed red lines in panel (a).

Figure 3 . 6 :

 36 Figure 3.6: Paired network of nickelocene. (a) STM image of the paired network (V = 20 mV, I = 20 pA, 70 × 70 Å 2 ). The unit cell is shown by solid white lines. a 1 ,a 2 represent the lattice vectors and are apart by an angle θ = 41 • . The orientation of the network relative to the [011] and [01 1] directions of Cu(100) is indicated by the angle ψ. It was determined using the atomically-resolved image of the Cu(100) in the inset. (b) Low-bias STM image of the paired network (V = 1 mV, I = 10 pA, 28 × 46 Å 2 ) revealing the presence of horizontal molecules. The molecular arrangement is sketched in orange with horizontal Nc depicted as hourglasses and vertical Nc as circles. (c) Line pro les along the solid and dashed green lines in panel (a).

Figure 3 . 7 :

 37 Figure 3.7: Simulation of the nickelocene paired network. (a) Top view of the optimized paired network with only one layer of copper presented. (b) Side view of the two di erent paired structures. The horizontal molecules have been removed to highlight the tilt of the vertical molecule. (c) Simulated STM image of the nickelocene paired network on Cu(100).

1 Figure 3 . 8 :

 138 Figure 3.8: Spin excitation in nickelocene. (a) Energy diagram of a nickelocene with a scheme of the spin excitation process. (b) Comparison of the spin excitation signature between the Nc layer in red and the isolated molecule in green with the feedback loop opened at I = 50 pA and V = -15 mV. We used a vertical o set for clarity. (c) Sketch of the spin excitation in Nc before and after tunneling.

  Figure 6.4(b) presents a d 2 I/dV 2 spectrum acquired above a Nc in a molecular network on Cu(100). Since the vibrational excitations of nickelocene adsorbed on the surface can be discarded in this energy range[104], the peak (dip) corresponds to the spin excitation |S = 1, M = 0 → |S = 1, M = ±1 occurring when inelastic tunnel electrons have a threshold energy |eV | ≥ D = 3.2 meV. The spin excitation obeys a selection rule as the spin angular momentum of the system (molecule plus tunneling electron) needs to be conserved during the tunneling process. The selection rule imposes to the spin of the electron to ip during the tunneling process. For instance, if the electron enters in the junction in the |↑ state, Nc is excited to the |S = 1, M = +1 , g. 6.4(c); the electron will then exit the junction in the |↓ state. Conversely, if the electron enters |↓ , Nc is excited to the |S = 1, M = -1 and the electron will exit the junction in the |↑ state.

Figure 3 . 9 :

 39 Figure 3.9: Kondo scattering in nickelocene. (a) dI/dV spectra at various z (indicated on the gure), along with their simulation based on a dynamical scattering model where we take into account only the Kondo scattering (solid red lines), i.e. U = 0. The blue dashed line corresponds to a simulation without Kondo scattering. z = 0 corresponds to a nickelocene in contact with the Cu(100) surface. (b) Sketch of a metal tip functionalized by a Nc molecule. (c) Kondo scattering parameters J ρ extracted from the simulations in panel (b). ρ is the electron density of the substrate near the Fermi level.

Figure 3 . 10 :

 310 Figure 3.10: Ni adsorption on a nickelocene paired network. (a) STM image of a NiNc complex in a Nc paired network on Cu(100) (V = 20 mV, I = 20 pA, 16 × 16 nm 2 ). Inset: Close-up view of a NiNc complex with the molecular arrangement sketched in orange with horizontal Nc depicted as hourglasses and vertical Nc as circles (V = 20 mV, I = 100 pA, 4 × 4 nm 2 ). (b) STM image acquired at the edge of a paired network showing a NiNc complex and an isolated Nc on Cu(100) (V = 20 mV, I = 100 pA, 10 × 10 nm 2 ); (c) Line pro les along the green, red and blue dashed lines of panel (b).

Figure 3 . 11 :

 311 Figure 3.11: DFT calculations of a NiNc complex in a paired network. (a) Top view and (b) side view of the DFT optimized structure of the NiNc complex in a Nc paired layer; some molecules have been removed for clarity in (b). The orange rectangle marks the NiNc complex. The red arrow indicates the position of the Ni adatom. (c) Side view of the spin density map of the paired structure containing two Ni adatoms. The black dashed line represents the principal axis of the Nc molecule and highlights the misalignment between the Nc molecule and the Ni adatom. The orange rectangle marks the NiNc complex. Yellow: Isosurface of the spin-up density; red: spin down.

Figure 3 . 12 :

 312 Figure 3.12: Spectroscopy of NiNc. (a) dI/dV spectrum and (b) d 2 I/dV 2 spectrum acquired above the Cp ring of a NiNc complex in the paired layer (V = -80 mV and I = 200 pA). The solid red line is a t based on a dynamical scattering model [105]. The spectrum shows three features, better seen in the d 2 I/dV 2 spectrum and labeled 1, 2 and 3. The NiNc complex is compared to the dI/dV spectrum acquired above the Cp ring of a Nc (dashed curve) in the paired network (V = -70 mV and I = 100 pA) in (a).The arrows indicate the position where the spectra were acquired.

Figure 3 . 13 :

 313 Figure 3.13: PDOS and elastic transmission for Ni-nickelocene. (a) Full elastic transmission (black dots) and the transmission of each eigenchannel mainly originating in the π orbitals of the Nc molecule (solid line: spin up, dashed line: spin down). The third eigenchannel (blue lines) negligibly contributes to the total transmission. (b) Projected density of states onto Ni(3d xz ), Ni(3d yz ), C(2p z ) atomic orbitals (the dashed lines correspond to the Ni-adatom d-states). The frontier π molecular orbitals of Nc have contributions from Ni(3d xz ), Ni(3d yz ) and C(2p z ) atomic orbitals.The calculations are performed for a tip-molecule distance of 4.9 Å as measured from the tip apex atom to the topmost molecular C. For energies that are accessible with the STM, the transmission is controlled by two eigenchannels that are themselves the contributions of Lorentzian-like functions as can be seen following the transmission curves of the rst and second channels. These Lorentzian-like functions energetically match the projected density of states corresponding to the two π molecular orbitals of Nc. For electron energies below -0.8 eV, the transmission involves d-electron channels of the Ni adatom.

eq. 3 .

 3 3 that includes an axial magnetic anisotropy D . The z-axis is chosen along the molecular axis running through the center of the Cp rings. Within this viewpoint, 1 is assigned to a spin excitation occurring between the ground state |S = 1, M = 0 and the doubly degenerate |S = 1, M = ±1 excited states of NiNc, see Fig.3.16(b). The onset |E 1 | corresponds then to D . Using the t of § 3.2.3 -solid red line in Fig.3.12(a)[105], we obtain D = 4.6 ± 0.2 meV.

Figure 3 . 14 :

 314 Figure 3.14: Vibrational excitation in a NiNc complex. (a) d 2 I/dV 2 spectra acquired above the center of a NiNc complex for di erent tip-molecule distances. The feedback loop was opened with a constant tunnel bias set to -20 mV. The current was successively set to 50 pA (solid black line), 500 pA (solid blue line) and 1 nA (solid red line) in order to decrease the tip-molecule distance. The spectra were renormalized by the opening current. The vertical lines indicate the peak position of excitation 2. Panels (b) and (c) present dI/dV maps acquired at a constantheight with a bias of 4.5 mV (1) and 32.5 mV (2) respectively. The feedback loop was opened at -60 mV and 500 pA above a Nc molecule of the layer and we used a modulation of 3 mV rms. (d) Computed vibrational modes energetically close to peak 2. (e) Computed ratio between the inelastic and elastic tunneling conductance using the 35.5 meV mode. (f) Sketch of a vibrational excitation by the tunneling current in a NiNc complex.

Figure 3 . 15 :

 315 Figure 3.15: Vibron-assisted spin excitation in a NiNc complex. (a) dI/dV spectra acquired at four locations (indicated by the tips in the inset) above NiNc. The feedback loop was opened at -30 mV and 300 pA above all locations in order to have same amplitude for the step associated to 1. (b) dI/dV map acquired at a constant-height with a bias of 37 mV corresponding to the peak 2. The feedback loop was opened at -60 mV and 500 pA above a Nc molecule of the layer and we used a modulation of 3 mV rms. (c) Sketch of the vibron-assisted spin excitation in a NiNc complex.

Figure 3 .

 3 16(a) presents the computed spectrum based on eq. 3.6, where D = 4.6 meV and ω = 35.5 meV. The computed spectrum showed little dependence on the parameters used for the molecular orbital. The electron-phonon interaction in the presence of spin excitations is su cient to reproduce the experimental spectrum, but the inclusion of dynamical electron correlations in the calculation is essential to correctly grasp the amplitude of the vibrational step.

Figure 3 .Figure 3 . 16 :

 3316 Figure 3.16(b) sketches the corresponding eigenenergies and allowed excitations. The spin excitation goes from the |S = 1, M = 0; n = 0 state to the |1, ±1; 0 state, where S stands for the spin of the NiNc, M for its magnetic quantum number and n for the vibrational quantum number. The vibrational excitation goes from |1, 0; 0 state to the |1, 0; 1 state. The third excitation goes from the |1, 0; 0 state to the |1, ±1; 1 state and is associated to a tunneling electron

Figure 3 . 17 :

 317 Figure 3.17: Creating an isolated NiNc complex. Panels (a) and (b) sketch the transfer of Nc from the surface to the tip and from the tip to the Ni atom, respectively. The transfer occurs when the nickelocene tip is 1 Å from point contact to the Ni atom. (c) Pseudo-3D images before/after creating an isolated NiNc complex (V = -20 mV, I = 20 pA, 5 × 5 nm 2 ). (d) Line pro les of an isolated NiNc complex and of an isolated Nc.

Figure 3 . 18 :

 318 Figure 3.18: Spectroscopy and DFT calculations of isolated NiNc. (a) DFT optimized structure of the isolated NiNc complex on Cu(100), (b) side view of the spin density map of the isolated NiNc complex on Cu(100) showing than the spin is mostly located on the Nc. Red: spin up, Yellow: spin down. (c) dI/dV spectrum measured above the Cp ring of an isolated NiNc on Cu(100) with the feedback opened at -50 mV and 20 nA. The solid red line is a Frota-Fano t.

Figure 3 . 19 :

 319 Figure 3.19: Spectroscopy in the center of a layer-integrated Nc. dI/dV spectrum acquired above the center of a layer-integrated Nc (solid black line). The solid red line corresponds to high-resolution data acquired between -35 mV and -20 mV, and between 20 mV and 35 mV. The feedback loop was opened at -30 mV and 300 pA for all spectra. Inset: Arrow indicates the position where the spectra were acquired.

Figure 4 . 1 :

 41 Figure 4.1: Structural conformation and electronic con guration of cobaltocene. (a) Structural conformation. (b) Electronic d-manifold resulting in a spin S = 1/2.

Figure 4 . 2 :

 42 Figure 4.2: Cobaltocene deposition. (a) Chamber before opening the valve, while we have Cc in the preparation chamber at a pressure around 10 -6 mbar. (b) The right panel shows the Cc ux coming from the crucible to the sample after opening the valve.

  4.1(a). It is slightly smaller than nickelocene, the Cp-Cp distance being 3.5 Å instead of 3.7 Å. Cobaltocene has 19 valence electrons, i.e. one electron more compared to the closed shell con guration of ferrocene, endowing Cc with a spin S = 1/2. The electrons occupy the d xz and d yz orbitals, resulting in a 2E 1g ground state as con rmed by experiments[135,136]. The degeneracy of the singly occupied doublet d xz and d yz is removed by a Jahn-Teller distortion as sketched in g. 4.1(b).

Figure 4 . 3 :

 43 Figure 4.3: Cobaltocene adsorption on Cu(100). (a) Cobaltocene molecules on Cu(100) (V = 20 mV, I = 30 pA, 12 × 8 nm 2 ). Inset: ×1.6 zoom of Cc molecules at a step edge. (b) Line pro les across two Cc molecules at the step (red and blue solid lines), of the pristine Cu step (solid orange line) and across a cobaltocene on the copper terrace (solid green line). Inset: Closeup view of Cc on a terrace (-70 mV, 20 pA, 2 × 2 nm 2 ). (c) Cobaltocene on a Cu(100) terrace co-adsorbed with a Fe atom (-70 mV, 20 pA, 4 × 4 nm 2 ). Inset: Cu(100) surface imaged through a tip-assisted manipulation of the Fe atom (inset: 7 mV, 280 nA, 1 × 1 nm 2 ). A grid representing the Cu(100) surface is simulated and positioned onto the image using the Fe atoms as markers. (d) Cobaltocene adsorption site as determined by DFT. (e) Side view of the relaxed structure of Cc on Cu(100) as computed by DFT.

Figure 4 . 4 :

 44 Figure 4.4: PDOS and wave function of cobaltocene on Cu(100). (a) Projected density of states (PDOS) for Cc when adsorbed on Cu(100) in the relaxed con guration of gs. 4.3(d,e). The solid red and green lines correspond to the projection onto the Co(d xz ) and the Co(d yz ) atomic orbitals, respectively, while the total PDOS of Cc is in light grey. Top panel: spin-up PDOS, bottom panel: spin-down PDOS. (b) Calculated wave function density of the π xz and π yz orbitals of Cc in vacuum and their combination.

Figure 4 . 5 :

 45 Figure 4.5: Preparing a cobaltocene tip. Panels (a,b,c) sketch the Cc transfer from the surface to the tip apex through a vertical displacement of the tip. Panels (d,e,f) present the Cc attachment to the tip through a line scan performed close to contact regime. (d) Image of Cc and of a Fe atom (-20 mV, 20 pA, 5 × 5 nm 2 ). (e) Image acquired by starting the scan at -20 mV and 20 pA, and by subsequently changing to -1 mV and 300 pA when scanning above Cc. (f) Image of the same area after picking up the molecule (-20 mV, 20 pA, 5 × 5 nm 2 ).

Figure 4 . 7 :

 47 Figure 4.7: PDOS of the cobaltocene tip. Projected density of states (PDOS) for the Cc-tip in con guration 1. The solid red and green lines correspond to the projection onto the Co(d xz ) and the Co(d yz ) atomic orbitals, respectively, while the total PDOS of Cc is in light grey. Top panel: spin-up PDOS, bottom panel: spin-down PDOS.

Figure 4 .

 4 Figure 4.9(a) presents a typical dI/dV spectrum acquired with a Cc-tip positioned above a Cu(100) terrace; all dI/dV measurements were acquired at 2.4 K. The spectrum exhibits a sharp peak near zero-bias and is anked by peaks nearly symmetric with bias polarity, solid black lines in g. 4.9(a). The dominant zero-bias peak is assigned to a Kondo resonance resulting from the spin-ip scattering of tip conduction electrons by the S = 1/2 spin of Cc. The nearly symmetric shape of the peak indicates that the quantum interference with the continuum of the tip electrons is negligible (see § 1.2.4). The spatial distribution of the Kondo resonance in the molecule, inset of g. 4.9(a), follows the two-lobe pattern of the π yz orbital, revealing that the Kondo e ect is carried by this orbital.

Figure 4 . 8 :

 48 Figure 4.8: Tip-sample distance. (a) Conductance-versus-tip-displacement curve acquired at a sample bias of -5 mV with a Cc-tip above the Cu(100) surface. The change of slope in the curve indicates the contact point. (b) Line pro les of Cc and of a Fe atom on Cu(100). (c) Sketch showing how we determine the distance between a Cc-tip and a magnetic atom.

Figure 4 . 9 :

 49 Figure 4.9: Di erential conductance of cobaltocene: Tip versus step edge. (a) dI/dV spectrum taken with a Cc-tip above the bare Cu(100) surface (feedback loop opened at -70 mV, 200 pA). Inset: Constant-height dI/dV image acquired with the Cc-tip above a Fe atom (-2 mV, 1.6 × 1.6 nm 2 ). The feedback loop was opened at distance of z = 0.4 nm above the center of the Fe atom. (b) dI/dV spectrum of a Cc adsorbed at a step edge of Cu(100) surface acquired with a Cu-terminated tip (-75 mV and 200 pA). Inset: Constant-height dI/dV image acquired above a Cc adsorbed at a step edge (4 mV, 2.5 × 2.5 nm 2 ). The feedback loop was opened at distance of z = 0.4 nm above one of the Cc lobes. The black tics in panels (a) and (b) mark the position of Kondo side peaks due to molecular vibrations.

Figure 4 . 10 :

 410 Figure 4.10: Di erential conductance of cobaltocene: Tip con gurations. dI/dV spectra acquired above Cu(100) using Cc-tips in con guration 1, and 2 and 3 (-60 mV, 100 pA). A Frota t (not shown) using eq. 4.1 yields a Γ of 0.76, 0.88 and 0.68 meV, respectively, which emphasizes that the Kondo resonance is similar for the three tip con gurations. The vibrational Kondo peaks show, instead, tip dependency, in particular their amplitude.

Figure 4 . 11 :

 411 Figure 4.11: Di erential conductance of cobaltocene: Temperature. (a) dI/dV spectra of Cc adsorbed on a step-edge of Cu(100) for di erent temperatures (feedback loop opened at -70 mV, 300 pA). For clarity, the middle and top spectra are shifted vertically by 4 and 8 nS, respectively. The solid red lines are simulated line shapes using eq. 4.1 with K = 0.5 ± 0.1 meV. (b) Temperature-dependence of Γ extracted from the dI/dV spectra. The solid line correspond to the curve calculated using eq. 1.13.

Figure 4 . 12 :

 412 Figure 4.12: Kondo e ect versus tip-sample distance. Panels (a) and (b) sketch the contact of a Cc-tip to Cu(100). (c) dI/dV spectra acquired above Cu(100). Solid red lines: Tunneling spectra acquired when decreasing z from 60 to 0 pm; z = 0 corresponds to point contact. Solid blue lines: Contact spectra acquired when decreasing z from 0 to -90 pm. Inset: Zoom on the tunneling regime. For clarity, the spectra are shifted by 0.5 nS from one another. (d) Γ extracted from the dI/dV spectra using eq. 4.1.

Figure 4 . 13 :

 413 Figure 4.13: Spin-split Kondo resonance. (a) Set of dI/dV spectra acquired with the Cc-tip positioned above a Fe for various tip-atom distances; for clarity, the spectra are shifted vertically from one another by 1 µS. The solid red lines are a line shape simulation based on a dynamical scattering model [105, 149]. Inset: Sketch of the experimental setup. (b) Line shape simulation with (solid grey line) and without (dashed black line) Kondo scattering. In the latter case, no cusps are present at the inelastic steps. The asymmetry is η = 24%. (c) Distance dependence of the excitation threshold |eV 0 | (y-axis: left) and of the exchange eld (y-axis: right). The exponential t described in the text is plotted as a solid red line. The inset represents the spinstate diagram of the system and the spin excitation based on eq. 4.2.

  4.14. The eigenfunctions of the system, |ψ i (i = 0, 1, 2, 3), are linear combinations of |M Cc ; M Fe . The only spin excitation detected is ψ 0 → ψ 1 and involves a spin-ip of the Cc molecule. Excitation ψ 0 → ψ 2 has instead intensity |T Fe | 2 since it involves a change of spin momentum for Fe, while ψ 0 → ψ 3 has intensity |T Fe | 2 |T Cc | 2 since it involves a change of spin momentum for both Fe and Cc. As Fe is spectroscopically dark, i.e. as |T Fe | 2 |T Cc | 2 , these excitations do not lead to distinguishable spectroscopic features in the dI/dV .

2 Figure 4 . 14 :

 2414 Figure 4.14: Excitations in the spin-coupled iron-cobaltocene system. Spin-state diagram for a cobaltocene exchange coupled to a Fe atom on Cu(100) with D Fe = -0.85 meV and J = -1.3 meV (antiferromagnetic coupling). We have taken S Cc = 1/2 and S Fe = 3/2. The blue arrow indicates the spin excitation that is observed experimentally. The red arrows indicate the excitations that are not observed. The intensities of each excitation are noted aside the arrows (|T Cc | 2 : Excitation intensity of Nc, |T Fe | 2 : Excitation intensity of Fe).

Figure 4 . 15 :

 415 Figure 4.15:Inelastic tunneling above a Fe atom. At negative bias, inelastic electrons tunnel from the spin-down states of the Fe atom to the spin-up states of the Cc-tip with transmission proportional to T ↓ , leading to an inelastic step of intensity h -. At positive bias, the junction polarity is reversed and inelastic electrons tunnel from the spin-down states of the Cc-tip to the spin-up states of the Fe atom with transmission proportional to T ↑ , leading to an inelastic step of intensity h + . During the tunneling process, the electrons excite Cc from |M = +1/2 to |M = -1/2 . The weaker amplitude for h + compared to h -re ects the di erence in the transmission (T ↑ = T ↓ ).

Figure 4 . 16 :Figure 4 . 17 :

 416417 Figure 4.16: Kondo side peaks of cobaltocene. (a) d 2 I/dV 2 spectrum of a Cc-tip in con guration 2 (feedback loop opened at -70 mV and 2 nA). (b-c) d 2 I/dV 2 spectra for a Cc at a step edge (-75 mV and 200 pA). The spectrum in (b) and (c) were acquired by positioning the Cu-terminated tip at two locations of Cc, which are marked, respectively, by a red and yellow dot in the image of panel (b). (d) Derivative of the computed spectral function for a Cc-tip in con guration 2. The black tics indicate the vibrational peaks.

Figure 4 . 18 :

 418 Figure 4.18: Cobaltocene on a Cu(100) terrace. (a) dI/dV spectrum of Cc adsorbed on a terrace of Cu(100). The feedback loop was opened at -175 mV, 200 pA. (b) Computed spectral function of Cc with a quasi-continuum of vibrational excitations (see text).

Figure 5 . 2 :

 52 Figure 5.2: Calculated electron density and MAE for nickelocene. (a) Nc in the gas phase, (b) Nc molecule adsorbed on Cu(100), (c) Nc on a tip, which is modeled as a Cu atom on a Cu(100) surface. An electron transfer occurs between the molecule and the surface as sketched by the blue arrow.

Figure 5 . 3 :

 53 Figure 5.3: Nickelocene adsorption on the tip apex. (a) STM image of a molecule and of two co-adsorbed Fe atoms acquired with a Nc-tip (V = -15 mV, I = 30 pA, 4.9 × 4.9 nm 2 ). (b) Line scan of the Fe atom along the red dashed line in panel (a). The line scan yields the tilt angle θ. (c) Simulated STM image using the DFT relaxed structure of a Nc molecule. Inset: DFT relaxed structure of the Nc-tip used in (c). Panel (d) and (e) present the PDOS for the Nc-tip and the Cc-tip, respectively. The solid red and black lines correspond to the projection onto the d xz and the d yz atomic orbitals. Top panel: spin-up PDOS, bottom panel: spin-down PDOS.

  3(d), indicates that the mixing of C(sp), Ni(d xz ) and Ni(d yz ) orbitals results in partially-populated π xz and π yz molecular orbitals, giving a magnetic character to Nc on the tip apex. The calculated magnetic moment for Nc is 1.7µ B (e ective spin S ≈ 1), the Ni atom carrying 1µ B while the remaining 0.7µ B are located on the two Cp rings. The ring-like pattern of nickelocene is attributed to the contribution of the nearly degenerate π xz and π yz molecular orbitals, similar to what we presented for cobaltocene adsorbed on Cu(100) ( § 4.1.

Figure 5 . 4 :

 54 Figure 5.4: Di erential conductance of a Fe atom on Cu(100). The solid red (black) line presents a d 2 I/dV 2 spectrum acquired above a Fe atom on Cu(100) at a distance of z = 130 pm using a Nc-tip (a metal tip). The insets sketch the junction con gurations. The feedback was opened at a sample bias of V = -15 mV and at a tunnel current of I = 0.4 nA. The Nc-tip is positioned su ciently far above Fe to avoid a magnetic coupling.

Figure 5 . 5 :

 55 Figure 5.5: Exchange coupling with an iron atom. (a) d 2 I/dV 2 spectra acquired above the Fe atom on Cu(100) at distances ranging from z = 133 to z = 80 pm. (b) Peak and dip energy positions extracted from the spectra of panel (a). The color scale anking the panel corresponds to the d 2 I/dV 2 amplitudes extracted from (a) and is given in units of G 0 /V (G 0 = 2e 2 /h: quantum of conductance). (c) d 2 I/dV 2 spectra acquired above Cu(100) surface at distances ranging from z = 133 to z = 80 pm. For clarity, the spectra in (a) and (c) were shifted vertically from one another by 5×G 0 /V. The solid red lines correspond to simulations using the dynamical scattering model of [105].

Figure 5 . 6 :

 56 Figure 5.6: Excitations in the spin-coupled Fe-Nc system. (a) Spin-state diagram using the general Hamiltonian of eq. 5.1 (S Nc = 1, S Fe = 3/2 and J < 0). The blue arrows indicate the observed spin excitations of the system. The red arrows indicate the excitations that are not observed. (b) Spin-state diagram using the simpli ed Hamiltonian of eq. 5.2.

JFigure 5 . 7 :

 57 Figure 5.7: Exchange eld of a Nc-tip coupled to a Fe atom. (a) Exchange eld B versus zextracted from the spectra of g. 5.5(a) using eq. 5.2. Inset: DFT calculated con guration of the tunnel junction for a distance of 100 pm with isosurface of the spin density (antiferromagnetic coupling). (b) Computed transition energies using[105] while increasing the coupling (bottom x-axis) or the exchange eld (top x-axis).

Figure 5 .

 5 7(b) presents simulations carried out by M. Ternes based on his dynamical scattering model[105]. Three di erent type of coupling are considered: An isotropic Heisenberg model, an Ising coupling along the x-axis and an Ising coupling along the z-axis. The energy positions of the spin-split peaks (dips) of the inelastic spectrum are then computed for varying coupling intensity or, equivalently, exchange eld B. The Heisenberg simulation as well as the Ising coupling along the x-axis present an asymmetric splitting where the average value of the spin-split peaks increases monotonically with B.

Figure 5 . 8 :

 58 Figure 5.8: Spin polarization of a Nc-tip coupled to a Fe atom. Panels (a) and (b) sketch the mechanism leading to the asymmetry in the d 2 I/dV 2 spectra acquired above Fe. Panel (a) presents the rst excited state |M = 0 → |M = +1 , while panel (b) presents the second excited state |M = 0 → |M = -1 . (c) η versus the e ective magnetic eld, B, extracted from g. 5.5. The red dotted line highlights the value of η found at the highest eld measured.

Figure 5 . 9 :Figure 5 . 10 :

 59510 Figure 5.9: Elastic transmission and PDOS of the Nc-Fe junction. (a) Elastic transmission of the Nc-Fe junction. Solid red line: Spin up, solid black line: Spin down. (b) PDOS of nickelocene for the Nc-Fe junction. Solid black lines: 3d orbitals of the Ni atom, red dashed line: 2p orbitals of the C atoms. There is a correlation between the spin-resolved transmission curves and the the PDOS of Nc, which highlights the molecular contribution to the transmission at the Fermi energy.

NcFigure 5 . 11 :

 511 Figure 5.11: Atomic resolution of the Co island. (a) Cobalt island on Cu(111) with Nc molecules (V = -50 mV, I = 20 pA, 25 × 25 nm 2 ). (b) Constant-height image of a Co island recorded with a Nc-tip when scanning the area with white borders in panel (a) (V = -1 mV, z = 80 pm, 1 × 1 nm 2 ). (c) Image of a Co island on Cu(111) obtained by dragging a Cu atom with a metal tip as described in § 4.1.2 (V = 100 mV, I = 500 pA, 1 × 1 nm 2 ).

Figure 5 . 12 :

 512 Figure 5.12: Exchange coupling of a Nc-tip with a Co island. (a) d 2 I/dV 2 spectra acquired above a Co island on Cu(111) at distances ranging from z = 150 to z = 70 pm. The solid red line is a t based on the dynamical scattering model [105]. For clarity, the spectra are displaced vertically from one another by 5 × 10 -3 G 0 /mV. G 0 = 2e 2 /h: quantum of conductance. (b) Exchange eld versus z and. The solid red line is a t using an exponential function. The inset represent the spin-state diagram of the system.

Figure 5 . 13 :

 513 Figure 5.13: Magnetic corrugation using a Nc-tip. (a) Constant height images of the Co islands at V = -1 mV and (b) at V = -5 mV (z = 80 pm, 1 × 1 nm 2 ). (c) d 2 I/dV 2 spectra acquired at z = 80 pm on the hollow (top) site presented in blue (red) in panel (a). (d) Ratio between the current of the top site and the current of the hollow site versus sample bias (z = 80), the current used is extracted from panel (b).

Figure 5 . 14 :

 514 Figure 5.14: Computed exchange interaction for a Nc-Co junction. (a) Isosurface of the spin density for a Co bilayer corresponding to the Co islands. Spin-down (↓) and spin-up (↑) densities are plotted in red (-0.003 Å -3) and blue (+0.003 Å -3 ), respectively. The (↑) d electrons from the Co atoms are more localized than the (↓) from the sp electrons of the tip, which are dispersive in nature[158,161,162]. The Co atoms has a magnetic moment of 1.76 µ B , which is mainly carried by the d orbitals (d: 1.81 µ B , p: -0.04 µ B , s: -0.01 µ B ). The spatial con nement of sp electrons within the island[158,161,162] was not accounted for. We represent the unit cell with black line. (b) Plot of the di erence in the exchange energy (|∆E ex |) between the di erent con gurations versus the distance between the Nc-tip and the Co island. The red line corresponds to the di erence between the top and the hollow positions, while the black line corresponds to the di erence between the top and bridge positions. We mark with a red dashed line the di erence between the contact and the tunnel regime, in the contact regime, red area, the Nc molecule is covalently bound to the surface. In the inset, we present the di erent junction geometries used for the DFT calculations: top, hollow, and bridge sites. Experimentally, the di erence in exchange energy between the top and hollow sites is 2 meV when z = 80 pm, g 5.13(c).

Figure 6 . 3 :

 63 Figure 6.3: E et Kondo et excitation vibrationnelle. (a) Spectre dI/dV obtenu au centre de la molécule de tetracyanoquinodimethane; les di érents phénomènes mis en jeu sont notés de (1) à (3). (b) Schéma énergétique de l'e et Kondo (1), l'excitation vibrationnelle (2) et de l'e et Kondo assistée par vibron (3).Figure adaptée de [67].

Figure 6 . 5 :

 65 Figure 6.5: Acquisition d'une image STM. (a) Schéma de l'acquisition d'une image STM. (b) Mode de scan à hauteur constante, permettant d'obtenir une topographie de la surface.

6. 4

 4 Chapitre 3 : Excitation de spin assistée par vibron dans une molécule possédant une anisotropie magnétique Ce chapitre est une étude de la pointe STM fonctionalisée avec un Nc, réalisée en collaboration avec les précédents étudiants Benjamin Verlhac et Nicolas Bachellier[90].6.4.1 Présentation de la molécule de nickelocèneDans ce chapitre, nous commencerons par présenter la molécule de nickelocène (Nc) de la famille des metallocènes. Ces molécules sont composées dŠun métal de transition entouré par deux ligands cyclopentadiényles (C 5 H 5 ), ce qui en fait une molécule de spin 1 en phase gazeuse relativement petite et compacte, g. 6.6(a,b)[91][92][93][94][95].

Figure 6 . 6 :

 66 Figure 6.6: Présentation de la molécule de nickelocène. (a) Structure de la molécule de Nc. (b) Con guration électronique de la molécule de Nc présentant un spin 1. (c) Image STM d'un Nc isolé (V = 20 mV, I = 50 pA, 40 × 40 Å 2 ) et de deux atomes de fer adsorbés sur une surface de Cu(100). L'encart corresponds à une image STM de la surface de Cu(100) obtenue avec une résolution atomique (V = 10 mV, I = 75 pA, 16.5 × 16.5 Å 2 ). (d) Image STM d'un réseau de Nc (V = 20 mV, I = 20 pA, 70 × 70 Å 2 ); les paramètres de mailles sont présentés en superposition.

Figure 6 . 7 :

 67 Figure 6.7: Excitation de spin de la molécule de nickelocène. (a) Comparaison de l'excitation de spin caractéristique entre le réseau de Nc en rouge et la molécule isolée en vert (I = 50 pA and V = -15 mV). (b) Diagramme d'énergie du Nc. (c) Spectre dI/dV à di érentes hauteurs et leurs simulations basées sur le modèle de M. Ternes [105] qui permet de prendre en compte la di usion des électrons (en rouge).

Figure 6 . 8 :

 68 Figure 6.8: Complexes NiNc. (a) Image STM de complexes NiNc dans un réseau de molécules de Nc sur une surface de Cu(100) (V = 20 mV, I = 20 pA, 16 × 16 nm 2 ). Encart : Zoom sur un complexe Ni-Nc (V = 20 mV, I = 100 pA, 4 × 4 nm 2 ). (b) Vue de coté de la simulation du réseau de Nc contenant un complexe de Ni-Nc. (c) Vue de coté de la représentation de la densité de spin. Le trait noir représente l'axe principal de la molécule de Nc et met en évidence un mésalignement entre la molécule de Nc et l'atome de Ni. Le complexe Ni-Nc est entouré d'un rectangle orange pour le panneau (b) et (c). (d) spectre dI/dV et (e) d 2 I/dV 2 acquis au dessus du Cp d'un complexe de NiNc dans le réseau (V = -80 mV and I = 200 pA) et son t en rouge [105]. Les spectres présentent 3 structures notées 1, 2 et 3. Le spectre du complexe Ni-Nc est comparé au spectre d'une molécule de Nc dans le réseau (V = -70 mV and I = 100 pA).

6. 5

 5 Chapitre 4 : L'e et Kondo d'une pointe moléculaire comme sonde magnétique. 6.5.1 Présentation de la molécule de cobaltocène

  s'accordent avec les observations expérimentales. Le moment magnétique calculé de la pointe de Cc montre peu de di érence entre les trois con gurations et reste autour de 0.89 µ B . Le spin e ectif de la pointe S = 1/2 est porté par un électron de l'orbital π yz dont la LUMO produit le motif à deux lobes observé sur les images. C'est la pointe de Cc en con guration 2 que nous utiliserons pour expliquer nos observations sans perte d'informations. Nous avons acquis un spectre dI/dV avec une pointe de Cc en con guration 2 au-dessus d'une terrasse de Cu(100). Ce spectre présente un pic n centré sur une valeur proche de 0 V, assigné à une résonnance Kondo, ainsi que deux pics satellitaires presque symétriques en fonction de la tension due à des vibrations moléculaires, g. 6.10(d). Les pics satellitaires sont le re et de la résonance Kondo où les électrons ont su samment d'énergie pour activer un mode vibrationnel de la molécule[66]. La pointe moléculaire Cc de conformation 2 a le signal spectroscopique miroir d'une molécule de Cc adsorbée sur une marche, ce qui s'explique par une inversion géométrique de la jonction, g. 6.10(e).

6. 5 . 3 Figure 6 . 11 :Figure 6 . 12 :

 53611612 Figure 6.11: Étude de l'e et Kondo. (a) Spectres dI/dV d'une molécule de Cc adsorbée à la marche pour di érentes températures (-70 mV, 300 pA). (b) Dépendance en température de Γ extraite de spectres dI/dV de la pointe de Cc et Cc adsorbé à la marche. (c) Spectres dI/dV de la pointe moléculaire au dessus d'une terrasse en réduisant la distance pointe-surface (V = -40 mV and I = 200 pA). (d) Spectres dI/dV d'une pointe de Cc au dessus d'un atome de Fe pour di érentes distances pointe-atome, où 0 correspond au point de contact. Encart : schéma de l'expérience. (e) Champ magnétique e ectif extrait des spectres de la série (d) en fonction de la distance pointe-atome. Représentations des diagrammes d'état de spin.

6. 6 . 1 Figure 6 . 13 :

 61613 Figure 6.13: Présentation de la pointe de Nc. (a) Spectres d 2 I/dV 2 acquis avec une pointe de Nc au dessus d'une surface de Cu(100) (bleu) et pour une pointe métallique au dessus d'un Nc (gris) (V = -20 mV, I = 0.14 nA). Nous avons simulé ces spectres avec le modèle de M. Ternes [105]. Les encarts schématisent les con gurations des jonctions. (b) Image STM d'une molécule et de deux atomes de Fe (V = -15 mV, I = 30 pA, 4.9 × 4.9 nm 2 ). (c) Image STM simulée de la structure relaxée de la pointe de Nc. Encart : structure relaxée de la pointe de Nc.

Figure 6 . 14 :

 614 Figure 6.14: Cas de la pointe de Nc au dessus d'un atome de Fe. (a) Schéma de con guration de la jonction. (b) Spectres d 2 I/dV 2 acquis avec la pointe de Nc au dessus d'un atome de Fe en changeant la distance pointe de Nc -Fe. (c) Champ d'échange B extrait de la série présentée en (b) en fonction de la la distance pointe de Nc -Fe. (d) et (e) schématisent le mécanisme lié l'asymétrie des spectres d 2 I/dV 2 de (b). (f) η en fonction du champ magnétique e ectif B.

  2 meV. On peut dé nir l'hamiltonien Ĥ = D Ŝ2 z,N c +D Fe Ŝ2 z,F e -J Ŝz,Nc • Ŝz,Fe , où J est un couplage antiferromagnétique de forme Ising. De plus l'anisotropie du fer étant invisible spectroscopiquement, on peut la négliger et reformuler le terme de couplage en J Ŝz,Fe Ŝz,Nc = gµ B B Ŝz,Nc et nalement d'obtenir Ĥ = D Ŝ2 z,N c -gµ B B Ŝz . Ceci nous permet d'extraire le champ d'échange B, que nous traçons en fonction de la distance, g. 6.14(b). Cet hamiltonien nous permet de visualiser l'e et du Fe sur la pointe de Nc à travers le couplage d'échange. On remarque également une asymétrie dans l'intensité des pics en fonction de la tension, comme dans le cas du cobaltocène. Le mécanisme est présenté g. 6.14(c,d) et nous pouvons extraire une valeur η = (h + -h -)/(h + + h -), g. 6.14(e).

6. 6 . 3 Figure 6 . 15 :

 63615 Figure 6.15: Cas de la pointe de Nc au dessus d'îlot de Co. (a) Îlot de Co sur une surface de Cu(111) avec des molécules de Nc (V = -50 mV, I = 20 pA, 25 × 25 nm 2 ). (b) Spectres d 2 I/dV 2 acquis au dessus d'un îlot de Co sur une surface de Cu(111) pour des distances entre 150 et 70 pm. (c) Champ d'échange en fonction de la distance pointe de Nc-îlot de Co. (d) Image à hauteur constante de l'îlot de Co à V = -1 mV et (e) V = -5 mV (z = 80 pm, 1 × 1 nm 2 ). (f) Spectres d 2 I/dV 2 acquis sur les sites "hollow" (bleu) et "top" (rouge) d'un îlot de Co à 80 pm. (g) Rapport entre les courants obtenus en (f).

Table 3 . 1 :

 31 Calculated adsorption energy (eV) Compact 8.8 ± 0.4 15.1 ± 0.5 90 ± 2 1.49 eV Paired 12.6 ± 0.3 32.3 ± 0.8 41 ± 3 1.52 eV Parameters for the compact and paired networks of nickelocene.

Table 3 . 2 :

 32 Evolution of the red-shift versus tip-NiNc distance. Energy position of the excitations peaks of g. 3.14(a).

	Tunnel current (pA) 1 (meV) 2 (meV) 3 (meV)
	50	4.6	33.4	37.6
	500	4.7	32.6	37.1
	1000	4.7	31.9	36.8

Table 3 . 3 :

 33 Energy onsets and transition rates for di erent NiNc spectra. Energy onsets and step amplitudes extracted from the dI/dV spectrum of six NiNc complexes using the t shown in Fig.3.16(c).

	σ 2 0

Table 4 .

 4 ).

	Con g. δE (eV) C-Cu distance (pm) angle ( • ) S Cc (µ B ) S Co (µ B )
	1	0	211, 230, 288, 302, 327	16	0.89	0.83
	2	0.13	251, 206, 260, 310, 306	4	0.88	0.82
	3	0.18	216, 215, 285, 287, 316	16	0.91	0.85

1: Structural properties and magnetism for a Cc-tip.

  La résistance électrique est liée aux vibrations du réseau cristallin; à haute température, les vibrations seront plus fortes, ainsi que la résistance. Ces vibrations peuvent être diminuées en réduisant la température, donnant pour une température nie une résistance minimale. En [t!] Principe de l'e et Kondo. Système (a) avant le que l'électron ne tunnelle de la matrice vers l'impureté, (b) créant un état virtuel et (c) ce qui résulte en un spin-ip.1934, Haas et al. ont mesuré un minimum dans une courbe de résistance en fonction de la température à une température non nulle[START_REF] Haas | [END_REF]. Ce phénomène est lié à la présence d'une impureté magnétique. Anderson a mis en évidence un mélange des orbitales d des impuretés avec les bandes électroniques s du métal environnant[47]. Puis, Kondo proposa un modèle pour décrire ce phénomène à travers un couplage d'échange entre l'impureté et le métal. Un état virtuel est alors crée au dessus du niveau de Fermi, du à la di usion de spin, sur le même niveau d'énergie causant une répulsion de Coulomb[48]. Ce mécanisme qui correspond à un spin-ip élastique est brièvement décrit g. 6.1. La force de ce couplage entre lŠimpureté et le métal caractérise la température Kondo (T Le spin-ip élastique associé à l'e et Kondo donne naissance à un pic au niveau de Fermi appelé la résonance Kondo, dont la largeur est proportionnelle à k B T K , g. 6.2. Lors des premières mesures, par Li et al.[4] et Madhavan et al.[5] en 1998, montrent toutefois qu'une interférence Fano doit être prise en compte pour obtenir la forme prédite. Cette interférence est issue du couplage entre un état discret et un continuum d'état et peut produire des pics asymétriques dans le spectre d'excitation. Mesurer la dépendance en température de cette résonance est un

		10
	(a)	8			(b)	Virtual state	(c)
					E d + U
	E F	Metal host d (E ) (arb. units) A	6 4	Impurity	E d E d	Metal host	∝k B T K	U	E + U d
			2
	Figure 6.1: E -EF (eV) 0 -1 0 1	2	3

K ). Les modèles d'Anderson et de Kondo ont pu être liés par la transformation canonique de Schrie er et Wol

[50]

, permettant d'exprimer l'intensité du couplage J en fonction des paramètres de l'impureté d'Anderson U , E d et ∆.

6.2.2 L'e et

Kondo en STM marqueur expérimental de l'e et Kondo. En e et, la largeur Γ du pic Kondo d'un spin 1/2 augmente avec la température. Un autre marqueur est l'in uence d'un champ magnétique externe de l'ordre de Γ/gµ B : ce champ crée une division des pics due à l'e et Zeeman.

2

Figure 6.2: Fonction spectrale de l'e et Kondo. Inspiré de [51].

  Figure adaptée de[67].En 1968, Lambe et Jaklevic ont observés des pics dans un spectre dI/dV à des énergies auxquelles correspondent les modes vibrationnels des molécules étudiées, due l'excitations par électrons tunnelant inélastiquement[60]. Binnig et Persson avaient également prédit la possibilités d'observer des modes de vibrations d'une molécule adsorbée sur une surface avec un STM en utilisant le courant tunnel inélastique[61,62]. Leurs premières observations expérimentales dans un spectre tunnel en STM arrivèrent en 1998, lorsque Stipe et al. mesurèrent des vibrations et rotations de spin change de δM = +1, et donc pour conserver le moment angulaire total, l'électron tunnel doit changer son spin de |↑ à |↓ , ce qui correspond à un ip spin inélastique. La gure. 6.4(b) montre qu'il faut isoler le métal de la surface pour observer l'excitation de spin. Ce qui augmente considérablement le temps de vie de l'état excité. Ces excitations de spins peuvent être utilisées pour collecter des informations magnétiques sur de nombreux systèmes di érents[80,81]. De plus, il est possible de sonder les interactions entre spins à l'échelle d'un seul atome ou d'une seule molécule et cela même dans un assemblage d'entité créer en utilisant la pointe d'un STM.Le second chapitre de ma thèse reprend le principe de fonctionnement du microscope à e et tunnel, outil central de ma thèse. Cet outil a été développé par Binning et Rohrer et permit pour la première fois d'obtenir des images avec une résolution atomique[82] et, ainsi, d'élucider la

	6.3 Chapitre 2 : Microscopie et spectroscopie à e et tunnel
	6.3.1 Courant tunnel et microscopie à e et tunnel

reconstruction d'une surface de Si

(111)

[83]

. Le principe de fonctionnement du STM consiste à scanner une surface conductrice avec une pointe en métal, qui s'échangent des électrons par e et tunnel, générant ainsi un courant dit "tunnel" que l'on peut mesurer. Ce courant peut être exprimé en utilisant l'approximation semi-classique de Wentzel-Kramer-Brillouin (WKB) :

  Après évaporation des atomes de Ni dans le réseau de Nc, des motifs plus lumineux apparaissent, plus hauts que le reste du réseau de Nc, correspondant aux complexes NiNc, g.6.8(a). Le fait que l'on observe un anneau sur les images STM tend à indiquer que le Nc est adsorbé sur le Ni. Les groupes de M.-L. Bocquet et N. Lorente ont calculé la structure et la polarisation de spin en DFT pour les complexes de NiNc dans le réseau, ce qui con rme l'observation du Ni sous le Nc, g. 6.8(b,c). De plus, cela a permis de mettre en évidence un mésalignement entre l'atome de Ni et la molécule de Nc, comme permet de le voir le trait noir en pointillé g. 6.8(c). Ce qui donne un "tilt" dans l'axe principal de la molécule, observée lors d'un zoom sur un complexe, g. 6.8(a).La g. 6.8(d) présente le spectre dI/dV acquis sur l'anneau de complexes NiNc, que nous comparons à la molécule de Nc dans le réseau (en pointillé). Le spectre obtenu pour le complexe de NiNc comprend 3 motifs. Le spectre d 2 I/dV 2 , g. 6.8(e), nous renseigne sur la position en énergie de ces pics. Le premier pic 1 possède la même signature spectrale que l'excitation de spin de la molécule de Nc dans le réseau, ce qui est con rmé par les calculs DFT. Nous avons réalisé des cartes de dI/dV à hauteur constante de la dépendance spatiale du signal inélastique du second pic, montrant que le signal est plus intense au centre du complexe; alors que dans la carte correspondant au premier pic, le signal est principalement porté par l'anneau. N. Lorente a simulé les modes de vibrations théoriques de la molécule, nous permettant de mettre en relation un des modes vibrationnels théoriques avec la carte dI/dV expérimentale. Le pic 2 est associé à une excitation vibrationnelle. Pour nir le pic 3 a des similitudes avec les deux premiers pics et est associé à une excitation de spin assistée par vibron. Le spectre a une forme caractéristique à l'excitation de spin, comme celui du pic 1 ainsi qu'une dépendance spatiale similaire sur sa carte dI/dV à la carte dI/dV du pic 2. Ce mécanisme d'excitation est similaire à celui de l'e et Kondo assisté par vibron, où la réplique est déplacée d'une énergie proche de ω. Son énergie correspond e ectivement à la somme de l'énergie des deux précédents pics : excitation de spin + excitation d'un mode de vibration. Nous avons également comparé le complexe NiNc dans le réseau à un complexe NiNc isolé, fabriqué en utilisant la pointe du STM. Dans ce cas, l'adsorption du Nc sur le Ni est di érente puisque le Nc se met à plat, comme montré en DFT et par l'absence du "tilt" sur les images STM. Le spectre résultant est également di érent : nous n'obtenons plus qu'une résonance Kondo (T K = 68 ± 7 K) sur la même gamme d'énergie, associée à un spin 1/2. L'atome de Ni permet l'hybridation entre les électrons du cuivre et ceux de la molécule, changeant le spin e ectif du complexe.

moléculaire, que nous appellerons NiNc. Ce nouveau complexe conserve l'anisotropie magnétique et peut donc avoir une excitation de spin; il présente également des modes vibrationnels. En utilisant l'IETS et la DFT, nous allons mettre en évidence la présence d'un couplage spinvibration due à l'excitation simultanée de l'excitation de spin et de la vibration.

List of Figures

tilted with respect to the surface normal. Three con gurations are observed, g. 4.6(a): a twolobe pattern with one lobe brighter than the other (noted 1), a two-lobe pattern with equal intensities on the lobes (2), a one-lobe pattern (3).

Tip characterization

To gain insight into the con gurations adopted by cobaltocene on the tip apex, N. Lorente and coworkers carried out DFT calculations. The molecular tip is mimicked by relaxing a Cc molecule on a Cu adatom on Cu(100), g. 4.6(b). We nd an excellent agreement between the experimental and the simulated STM images, g. The computed magnetic moment of Cc on the tip shows negligible dependence on adsorption con guration, tab 4.1. In the case of con guration 1, for instance, the magnetic moment is 0.89 µ B with a 0.12 e -charge transfer of Cc to the metal tip. The projection on the Co atom gives 0.83 µ B , which indicates that the Cp rings have nearly no magnetic moment. The spin-polarized En 2004, Heinrich et al. [8] ont détecté une excitation de spin à l'échelle d'un atome unique avec un STM, g. 6.4(a). Ils ont noté l'apparition de pics à ±0.7 mV dans les spectres dI/dV lorsqu'un champ magnétique externe est activé; ces pics se déplacent linéairement avec l'intensité du champ, g. 6.4(b,c). Ces e ets peuvent être expliqués par la présence d'un e et Zeeman pour le spin S de l'atome de Mn, conduisant à l'hamiltonien Ĥ = -gµ B B Ŝz , où B est le champ magnétique appliqué perpendiculairement à la surface (axe z qui sera l'axe de quanti cation). Le champ magnétique permet de lever la dégénérescence de l'état S = 5/2, donnant un état fondamental et un état excité séparé de ±0.7 mV, g. 6.4(d). Lors de cette excitation, le moment Nous pouvons transférer la molécule de Cc de la surface à la pointe STM, en approchant la pointe au dessus de la molécule (100 pm du point de contact) à -20 mV ou moins. La présence du Cc à l'apex de la pointe change le motif observé sur les images des atomes adsorbés, dans ce cas le fer, g. 6.10(a). Trois con gurations di érentes sont observés lors de nos expériences, g. 6.10(b,c). Nous avons simulé les structures relaxées de Cc sur la pointe et les images STM, qui

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