The most important aspects of these mesoscopic devices is that magneti- zation **dynamics** **and** spin **transport** are coupled: a given **magnetic** configura- tion has an influence on the propagation of current (GMR effect), which itself influences the **dynamics** of the **magnetization** (spin torque). The physics of such **systems** cannot be properly captured by simple (analytically tractable) models, **and** numerical simulations that treat the **magnetic** **and** **transport** de- grees of freedom on an equal footing are needed. At present, micromagnetic simulations [37] can describe correctly the **dynamics** of the **magnetization**, whithout taking into account the effect of spin transfer on the **dynamics**. Concerning simulations that couple **magnetic** **and** **transport** degrees of free- dom, several steps have already been taken **in** that direction [49, 108, 10], which are based on two possible strategies: i) the three dimensional texture of the **magnetization** inside the sample is described correctly using micro- **magnetic** simulations, but the spatial variation of spin torque is neglected or ii) the spatial variation of the **magnetization** is neglected, using a macrospin approximation **and** spin torque is then introduced. These approximations neglect the effects related to the spatial inhomogeneity of **transport** **and** mag- netization inside the sample, which are determinant for a deep understand- ing of the physics of these **systems**. **In** particular, macrospin approximation cannot simulate the different spin wave modes excited by spin transfer ef- fect, which are essential characteristics of a STNO. The main novelty of this work is that we have developed an approach that takes into account cor- rectly both the spatial variation of spin torque **and** the three dimensional texture of the **magnetization** inside a realistic spin valve.

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de Grandmont, F-37200, Tours, FRANCE
Abstract
The Landau-Lifshitz-Gilbert (LLG) equation describes the **dynamics** of a damped **magnetization** vector that can be understood as a generalization of Larmor spin precession. The LLG equation cannot be deduced from the Hamiltonian frame- work, by introducing a coupling to a usual bath, but requires the introduction of additional constraints. It is shown that these constraints can be formulated ele- gantly **and** consistently **in** the framework of dissipative Nambu mechanics. This has many consequences for both the variational principle **and** for topological as- pects of hidden symmetries that control conserved quantities. We particularly study how the damping terms of dissipative Nambu mechanics aﬀect the con- sistent interaction of **magnetic** **systems** with stochastic reservoirs **and** derive a master equation for the **magnetization**. The proposals are supported by numer- ical studies using symplectic integrators that preserve the topological structure of Nambu equations. These results are compared to computations performed by direct sampling of the stochastic equations **and** by using closure assumptions for the moment equations, deduced from the master equation.

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[1]
Introduction
**In** 1959, the famous lecture by Richard Feynman titled “There’s plenty of room at the bottom” provided an insight of what could be achieved by making “small materials” or so called nano- materials. After six decades, our capability of producing **and** exploiting these nano-materials has evolved tremendously. The second half of the last century has witnessed a revolution of downsizing the materials to nano- **and** sub-nano scale. However, it did not take too long to realize that materials at such a small scale behaved differently than their bulk counter-parts, **and** a large number of questions emerged, particularly about the **magnetic** **and** **electronic** properties of these materials. **In** the pursuit of the answers, research on these materials received tremendous amount of attention which, at the same time, led to the discovery of several novel phenomena. One such phenomenon is Coulomb blockade, which was first observed experimentally **in** granular metallic nano-structures by Van Itterbeek et al. [1, 2] **and** then later explained by Gorter [3] on the basis of charging effect arising due to the small size of these nano-structures. Neugebauer **and** Webb proposed the theory of thermally activated charge **transport** mechanism **in** such granular materials which took into the account the charging energy; an important aspect of Coulomb blockade [4]. Later, a scaling law was proposed by Middleton **and** Wingreen to describe the current-voltage (I-V) curves **in** 1-D **and** 2-D assemblies of particles at null temperature [5], **and** was extended to finite temperatures by Parthasarathy et al. [6]. With time, the domain of Coulomb blockade expanded **and** it could be observed **in** large number of **systems** ranging from single particle device [7] to assemblies of particles [8].

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Email addresses: julien.tranchida@cea.fr (J.
Tranchida), pascal.thibaudeau@cea.fr (P. Thibaudeau), stam.nicolis@lmpt.univ-tours.fr (S. Nicolis)
relaxation, etc.) of the system. Under these condi- tions, numerical simulations of the stochastic effects **in** such **magnetic** **systems** are raising the question of the validity of the Markovian assumption that is commonly used, that is to have short memory **in** the sense that the correlation time is very short [8]. **In** order to test this assumption, by taking mem- ory effects explicitly into account, a colored-form for the noise has to be considered. **In** spin **systems**, the noise is represented by ~˜ ω as a random vector whose the components, ˜ ωi , are Gaussian random variables with zero mean **and** a the finite correla- tion time, which is encoded **in** its two-point function as follows:

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DOI: 10.1103/PhysRevLett.98.077203 PACS numbers: 75.75.+a, 75.47.m, 76.50.+g, 85.75.d
**In** **magnetic** multilayers, electron **transport** properties are governed by the relative **magnetization** orientation. This effect is at the origin of giant magnetoresistance or tunnel magnetoresistance **and** is widely used **in** spintronics devices. Moreover, a spin-polarized current reciprocally interacts with the **magnetization** through direct transfer of spin angular momentum. This interaction was theoretically addressed ten years ago independently by Slonczewski **and** Berger [ 1 , 2 ]. Since then, the concept of spin transfer torque attracted a considerable interest; for example, it was ex- perimentally demonstrated that a dc current could induce **magnetization** reversal **in** a ferromagnetic nanostructure, provided the current density is larger than a critical value j c [ 3 ]. The influence of spin transfer torque on the magneti- zation **dynamics** has also been widely studied **in** relation to potential applications such as rf oscillators. **In** particular, electrical measurements at microwave frequency provided evidence of steady state precession induced by a dc current of the order of the critical current [ 4 , 5 ].

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I. INTRODUCTION
Idealizations are often employed, when modeling real **magnetization** **dynamics**. Far from equilibrium, mag- netic fluctuations, modeled by white noise are exam- ples of such an idealization **and** the **magnetization** pro- cesses predicted **in** this way have been observed **in** ultra- fast reversal **magnetization** experiments. However real fluctuations always have finite auto-correlation time **and** the corresponding spin system **dynamics** is, rather, de- scribed by non-Markovian stochastic processes 1 . Despite the practical relevance of non-Markovian processes, how- ever, these have received much less attention than Marko- vian processes, chiefly, because of the simplicity of the theory **and** the familiarity of the corresponding mathe- matical tools 2 . The study of the **magnetization** dynam- ics of super-paramagnetic **systems**, however, is a suitable framework **and** provides motivation to describe **magnetic** fluctuations, induced by contact to a heat bath 3 , that are sensitive to non–Markovian effects.

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Both the XMCD data **and** the simulations show that, despite the strong ferromagnetic exchange coupling between the Ni **and** Fe sublattices **in** NiFe, they apparently lose their net **magnetization** **in** a very dissimilar manner. The same situation is also observed **in** ferrimagnetic Gd(FeCo) **and** DyCo 5 , where two antiferromagnetically coupled Gd **and** Fe or Dy **and** Co sublattices demagnetize on substan- tially di®erent time-scales. Thus, we observe the same transient decoupling of the exchange-coupled **magnetic** moments **in** two material **systems** that are strikingly di®erent, i.e., ferromagnetic versus anti- ferromagnetic coupling, itinerant versus localized type of **magnetic** ordering **and** **in**-plane versus out- of-plane **magnetic** anisotropy. Consequently this distinct **dynamics** seems to be a general property of multi-elemental **magnetic** compounds that are driven nonadiabatically to a highly nonequilibrium state by fs laser excitation. Moreover, **in** agreement with Eqs. ( 3 ) **and** ( 4 ), the observed characteristic demagnetization times scale with the **magnetic** moment of the sublattices i.e., the larger the mag- netic moment the slower the demagnetization pro- cess [see Fig. 4(b) ]. For the materials investigated here, we obtain a characteristic change of the demagnetization time per **magnetic** moment of 90 20 fs/ B . Consequently, although **in** NiFe, Fe demagnetizes slower than Ni, **in** Gd(FeCo) the de- **magnetization** of the Fe-sublattice is much faster than the demagnetization of Gd. This distinct dy- namics is even more pronounced **in** DyCo 5 given the large di®erence (a factor of 6) between the Dy **and** Co **magnetic** moments.

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1.2 Domain wall motion **in** **magnetic** nanostructures
**In** principle, the operation of DW based devices is based on the displacement of DWs between at least two positions using either applied field or current pulses. Therefore, it is important to understand how DWs pin **and** depin **in** a **magnetic** nanostructure. It has been shown that there are several ways to pin DWs **in** **magnetic** nanostructures. For instance, the artificial pinning centers which can be properly defined using lithographies allows one to pin a DW at precise position **in** a nanowire. **In** nanowires with **in**-plane **magnetization**, there are various designs of artificial pinning sites which are mostly the notches or the constrictions with various widths **and** depths [42, 43]. For nanowires with out of plane **magnetization**, the pinning sites can be created using the local geometry of nanowires such as constrictions [44, 33], Hall crosses [45]. It can also be obtained by local change of the layer thickness [46], local decrease of the anisotropy using ion irradiation [47], pinning due to the edge roughness or lithographic defects [48]. Importantly, **in** such high anisotropic **systems**, the pinning sites due to intrinsic defects of layers are critical **and** can also efficiently pin DW [49]. The pinning can be due to a decrease of energy through the change of the DW length or by a reduction of the magnetocrystalline energy **in** the pinning potential landscape.

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Beyond these standard TRM measurements which are performed at constant temperature (after the initial quench), experiments involving temperature variations during ageing below
T g usually reveal very striking characteristic features **in** standard spin glasses [13, 14]. For example, it is now well established that ageing at any T p < T g has no apparent effect on the state at all other temperatures sufficiently different from T p . This was found, for instance, **in** ZFC experiments **in** which modified cooling protocols were applied [13, 14]. While **in** the usual ZFC protocol the system is cooled (at a constant cooling rate) **in** zero field from above T g down to the lowest temperature, **in** the modified protocols, the cooling is interrupted at some temperature(s) T p **and** resumed after some waiting time t w . **In** both cases, a small dc field is applied at the lowest temperature reached, **and** the **magnetization** is recorded while the system is heated up at a constant rate. The ZFC **magnetization**, as measured **in** the modified protocol, clearly showed marked singularities (‘dips’) centred around the temperature(s) T p . Far enough from T p , it recovered the values of the usual (reference) ZFC **magnetization** (see [13, 14]). This showed that ageing at T p during t w did not affect the system at a temperature T different from T p . It also implied that the system kept the memory of ageing at T p while it was at lower temperatures **and** was able to retrieve the aged state previously reached at T p . These results proved that **in** spin glasses different spin correlations are building up at different temperatures, **and** that the correlations built up at any temperature remain imprinted at lower temperatures.

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1. Introduction:
The discovery of new multiferroic compounds exhibiting a strong magneto-electric coupling has aroused great interest since the beginning of the century, justified both by the fundamental issues involved **and** the prospects for technological applications [1]. The interest of these compounds lies **in** the coupling between orders, **magnetic** **and** electrical, with the possibility, from a static point of view, to manipulate the **magnetization** by applying an electric field [2]. The more recent discovery of magneto-electric excitations has opened up a new field investigation [3]. **In** multiferroics, these hybrid excitations, called electromagnons can be understood as magnons excited by the electrical component of a wave electromagnetic **and** are the signature **in** the dynamic regime of magneto-electric coupling [4, 5]. Understanding the mechanisms behind these new excitations is one of the recent challenges of condensed matter physics, **and** the possibility of modulating these excitations via a field electric **and** / or **magnetic** is also an avenue explored for future applications to be defined **in** the field of information **transport**, **magnetic** refrigeration **and** spintronic devices for example. These materials **in** which the magnetism **and** the ferroelectricity are coupled have been widely studied [6, 7, 8]. Studies on RMn 2 O 5 oxides have shown an important magneto-caloric effect (MEE) that is associated with a

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Since 1940, the studies of the **magnetization** **dynamics** **in** different **magnetic** **systems** have become of large interest, **in** particular with respect to the industrial applications such as **magnetic** memories. One of the first **magnetic** recording devices based on the **magnetization** **dynamics** used ferrite heads to write **and** read the information. Because the ferrite permeability falls above 10 MHz [Doyle 1998], the read/write process was possible only with a reduced rate. An important improvement (i.e. decreased response time) was obtained using **magnetic** thin film heads. Upon reduction of the dimensions of the **magnetic** system (i.e. reduced film thickness compared to the other two dimensions) strong demagnetizing fields will be induced, **and** thus a high value of the demagnetizing factor (~1) creating a large anisotropy field perpendicular to the film surface. **In** this way, reasonable permeabilities for applied frequencies larger than 300 MHz were obtained **in** thin **magnetic** films devices [Doyle1998].

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the resistivity ρ follows ρ ∝ T 5/3 [6, 18]. This again is at
odds with the antiferromagnetism suggested by the GGA calculations.
All this points to the possible importance of the elec- tronic correlations that are not correctly incorporated **in** the local or semi-local DFT. **In** this paper, we show that including local dynamical many-body effects sig- nificantly improves the description of iron. Within a local-density-approximation+dynamical mean-field the- ory (LDA+DMFT) framework we obtain the ground state properties **and** the equation of states (EOS) of both ferromagnetic α **and** paramagnetic ǫ phases of iron as well as the α−ǫ transition pressure **and** volume change **in** good agreement with experiment. The strength of the elec- tronic correlations is significantly enhanced at the α → ǫ transition. This leads to a reduced binding, which ex- plains the relatively low value of the measured bulk mod- ulus **in** ǫ−Fe. The calculated resistivity has a jump at the transition but the magnitude of the jump is severely underestimated compared with the experimental value, which points to additional scattering not present **in** our local approach.

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Figure 7: Open loop (blue) vs closed loop frequency response. Weighting parameters are: q 1 = 3
**and** q 2 = 12, r = 1.
is possible to observe the benefits of the proposed controller. The resonance peak that might cause undesirable oscillations have been eliminated thanks to the proposed optimal linear control. Note also that the low-frequency gain has kept unchanged.

Chapter 6
Conclusions
The goal of this thesis was to see the contribution of inertia **in** damped macrospin dy- namics. A macrospin mass (tensor of inertia) was introduced not related to a real mass dis- placement, but to magnetisation inertia. Following the work performed initially by Gilbert **in** his search of a mechanical analogue for the macrospin **dynamics**, a generalised Gilbert equation was derived within the recent theory of mesoscopic nonequilibrium thermodynam- ics (MNET), accounting for macrospin’s inertia. A new relaxation time τ depending on inertia **and** damping separated the behaviour of macrospin’s **dynamics** into two regimes: the long time scale regime t >> τ , where the equation of Gilbert was found as a long time scale limit, **and** the short time scale regime t << τ where a new phenomenon was predicted, nutation. The exact time scale of the new phenomenon couldn’t be foretold as the values of the introduced inertia tensor are unknown. However, if the nutation phenomenon exists **in** macrospin **dynamics**, it will be observed for time scales shorter than the picosecond scale, for small values of the damping coeﬃcient.

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The combination of the resonant core switching **and** the bi-stability of the gyrotropic mode under a perpendicular field led to a proposal for an efficient read/write process for this future memory.
The second main achievement of this thesis is the demonstration of the collective **dynamics** inside nano-discs coupled by the dipolar interaction. Before addressing the problem of coupled vortices, for which no complete theoretical description is available, we have studied the case of perpendicularly saturated disc pair placed nearby laterally. The f-MRFM probe was used to provide a local field gradient needed to tune differentially the frequency of the spin wave modes **in** each disc. It also provides a local detection of the **dynamics**. This experimental control of the system at the nanoscale allowed us to measure accurately the coupling strength, or dynamical splitting, which is the frequency splitting between the two coupled modes. The influence of the geometry on the dipolar interaction has also been investigated. **In** particular the influence of the pair separation as well as the asymmetry between discs was measured. Moreover, the theoretical analysis **and** the numerical simulations developed here could explained satisfactorily the collective modes frequencies as well as their relative amplitude.

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Reviews 2016, 308, Part 2, 346-380. (b) Gómez-Coca, S.; Aravena, D.; Morales, R.; Ruiz, E. Large
**magnetic** anisotropy **in** mononuclear metal complexes. Coordination Chemistry Reviews 2015, 289–
290, 379-392. (c) Frost, J. M.; Harriman, K. L. M.; Murugesu, M. The rise of 3-d single-ion magnets **in**
molecular magnetism: towards materials from molecules? Chemical Science 2016, 7, 2470-2491. 4. (a) Zadrozny, J. M.; Greer, S. M.; Hill, S.; Freedman, D. E. A flexible iron(ii) complex **in** which zero-field splitting is resistant to structural variation. Chemical Science 2016, 7, 416-423. (b) Lee, W.- T.; Jeon, I.-R.; Xu, S.; Dickie, D. A.; Smith, J. M. Low-Coordinate Iron(II) Complexes of a Bulky Bis(carbene)borate Ligand. Organometallics 2014, 33, 5654-5659. (c) Liu, Y.-Z.; Wang, J.; Zhao, Y.; Chen, L.; Chen, X.-T.; Xue, Z.-L. Four-coordinate Co(ii) **and** Fe(ii) complexes with bis(N-heterocyclic carbene)borate **and** their **magnetic** properties. Dalton Transactions 2015, 44, 908-911. (d) Lin, P.-H.; Smythe, N. C.; Gorelsky, S. I.; Maguire, S.; Henson, N. J.; Korobkov, I.; Scott, B. L.; Gordon, J. C.; Baker, R. T.; Murugesu, M. Importance of Out-of-State Spin–Orbit Coupling for Slow **Magnetic** Relaxation **in** Mononuclear FeII Complexes. Journal of the American Chemical Society 2011, 133, 15806-15809. (e) Scepaniak, J. J.; Harris, T. D.; Vogel, C. S.; Sutter, J.; Meyer, K.; Smith, J. M. Spin Crossover **in** a Four- Coordinate Iron(II) Complex. Journal of the American Chemical Society 2011, 133, 3824-3827.

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(Pt 5 , Pt 6 ) as well as (Fe 2 , Fe 3 ) **and** (Fe 5 , Fe 6 ). **In** the FM configuration the central **and**
outermost Fe atoms have a slightly larger spin moments than the other atoms **and** not far from the bulk one. This is **in** very good agreement with DFT calculations performed by Ebert et al [18] (see Fig. 1 of their paper). The same behavior is observed **in** the AFM ordering. It is also seen that the spin moment of the Pt atoms increases with d **in** the FM case **and** **in** the AFM case the spin moments of the Pt atoms between AFM Fe layers nearly vanish. However **in** the N = 135 cluster as **in** the N = 141 cluster the Pt atoms which cover completely the two (001) surfaces have a non negligible spin moment of the same sign as the neighboring Fe layer **in** accordance with the positive exchange coupling between Pt **and** Fe atoms found **in** first-principles calculations [15]. This effect certainly favors the AFM configuration of this cluster as well as for the N = 147 cluster. On the other hand the presence of Fe atoms on the first or last (001) layer of Pt atoms leads to a FM order as **in** the N = 19, 43, 55, 141 clusters.

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Theorem 1.3. For a generic element of P, the control problem ( 1.3 ) is approximately controllable.
The proofs of Theorems 1.2 **and** 1.3 can be found **in** Sections 2.4 **and** 2.5 , respectively. They are based on a general sufficient condition for controllability proved **in** [ 10 ] **and** recalled **in** Section 2.1 below. **In** a nutshell, such a condition is based, on the one hand, on a nonresonance property of the spectrum of the Schr¨odinger operator **and**, on the other hand, on a coupling property for the interaction term (see the notion of connectedness chain introduced **in** Definition 2.1 ). These properties are expressed as a countable number of open conditions. Their density is proved through a global analytic propagation argument. **In** Section 3 , we present two generalizations of these results, motivated by the applica- tions. First, **in** Subsection 3.1 , we consider a situation where the gate only partially covers the upper side of the rectangle domain. Then, **in** Subsection 3.2 , we take into account **in** our model the self-consistent electrostatic Poisson potential, as a perturbation of the applied potential V .

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[ 6 , 7 ].But informations on the carrier **dynamics** **and** energy relaxation processes **in** such InAs/InP QDs are still lacking. Recently, the growth of self-organized InAs QDs on misoriented InP(113)B substrates has been proposed to get QDs with both quantum sizes **and** a high surface density [8]. **In** the optimized structures, the control of the maximum QD height of the sample yields the control of their wavelength emission [9]. These structures are named double-cap quantum dots (DC-QDs) **and** emit at 1.55 µm at room temperature [10]. The optical properties of a single DC-QDs layer have been analyzed [10,11,12], **and** lasing structures were obtained with such nanostructures with low threshold current densities [13,14]. **In** a previous study, we managed to dissociate the two capture (or relaxation) mechanisms described **in** literature: phonon **and** Auger assisted relaxations [15]. Nevertheless, a complete dynamic analysis, necessary to give us useful information for the description of the properties of a InAs/InP laser emitting at 1,55 µm, was still lacking.

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