After a first attempt using a mechanical detection by Evans in the 60’s [41], the idea of performing **Magnetic** Resonance Force Microscopy (MRFM) was proposed in the early 90’s by John Sidles [120]. He was searching for a reliable and precise technique to determine the molecular structure of biological objects. For this purpose, he wanted a spectroscopic signature to understand the atomic structure of unknown objects. In order to achieve this goal, the atomic resolution is needed with the ability to study individual objects in the three spatial dimensions. The studied objects being fragile and sensible to external conditions, a non-destructive and non-invasive technique was also needed [121]. The development in the 80’s of the scanning surface probe techniques such as Scanning Tunnelling Microscopy (STM), Atomic and **Magnetic** Force Microscopy (AFM/MFM) gives access to a high spatial resolution. The idea is to couple a highly sensitive force sensor like a micrometre sized cantilever to the longitudinal component of **magnetic** moments, either nuclear or electronic spins. Following the principles of MRI techniques developed for medical applications, the spins precession excited by a microwave field could be localised spatially in a “resonant slice” of the sample by an appropriate field gradient. The force exerted on the cantilever being also proportional to the field gradient, the spatial resolution could be increased by keeping the same signal-to-noise ratio. The sensitivity then becomes independent of the spatial resolution.

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1 nm Ge base layer on a Si substrate with an amorphous SiO 2 top layer. The Pb film is covered by a 10 nm Ge layer, which prevents the influence of the proximity effect with the metallic dot array. In order to avoid inhomogeneities of the current, this continuous Ge/ Pb/ Ge trilayer is patterned into a transport bridge 共width w=200 m, distance between volt- age contacts d = 630 m兲 by optical lithography and chemi- cal wet etching. Measurements of the upper critical field of this transport bridge allow an estimation of the coherence length 共0兲=34 nm and of the penetration depth 共0兲 = 49 nm. The transport bridge is covered by ferromagnetic dots using electron-beam evaporation and electron-beam li- thography. The dots consist of a 3.5-nm Pd base layer and a 关Co共0.4 nm兲/Pd共1.4 nm兲兴10 multilayer with perpendicular **magnetic** anisotropy. 28 The dots are arranged in a square ar- ray with period L = 1.5 m. They have a square shape with a side length of approximately 0.8 m with slightly irregular edges.

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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 above-stated limitations of the conventional ac-susceptibility technique, namely its inability to resolve the ac response of a single **vortex** and the indirect rela- tion between the **vortex** **dynamics** and the integrated response, has provided a drive to develop alternative methods aiming to directly probe the ac properties of a supercon- ductor with single **vortex** resolution. In this chapter we discuss a recently introduced scanning probe technique, scanning ac-susceptibility microscopy (SSM), which re- veals, with unprecedented resolution, the motion and dissipation of individual units of ﬂux quanta driven by an applied ac **magnetic** ﬁeld or current [2]. The local dissi- pation can be inferred from the phase lag between the **vortex** motion and the driving force induced by an oscillatory **magnetic** ﬁeld, whereas the amplitude of the oscilla- tory **vortex** motion provides us with information about the shape of the local potential that each ﬂuxon experiences. This method has permitted us to reveal the contribution of pinning-driven (thermally activated) dissipative **vortex** motion [3], to demonstrate the nondissipative nature of the Meissner as well as the dissipative **vortex** state at microscopic scale [3] and ﬁnally, to obtain a detailed cartography of the distribution and intensity of the pinning landscape [2, 4]. This technique not only shed new light on unraveling the basic mechanisms of **vortex** dissipation with unmatched resolu- tion, but it permitted one to validate the theoretical models introduced to explain the measured integrated ac **vortex** responses in ac-susceptibility experiments [5]. We show that the technique can be readily implemented in a scanning Hall probe mi- croscopy set-up suited for low **magnetic** ﬁeld experiments [2–5] and also extended to a scanning tunneling microscopy [6] or a scanning SQUID microscopy apparatus [7] thus achieving the utmost resolution.

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studies since one can trap a head to head or tail to tail **vortex** domain wall in a constriction with cylindrical symmetry.
V. CONCLUSIONS
We have developed an original method based upon tem- plate synthesis in a PET matrix to grow bi-conical Ni nano- wires with a constriction of a few tens of nanometers. The micromagnetic states of a single nanowire have been studied through AMR and micromagnetic modeling during magnet- ization reversal. Simulations have revealed a complex **vortex** like state whose propagation through the constriction depends on the relative angle between the applied field and the cone axis. While simulations predict no pinning of the **vortex** in the constriction, our magnetoresistance measure- ments provide evidence that small inhomogeneities allow the trapping of head-to-head or tail-to-tail domain walls in the vicinity of the constriction.

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Peter R. Wiecha, ∗ Arnaud Arbouet, Aurélien Cuche, Vincent Paillard, and Christian Girard
CEMES-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France
In this article, we propose a concise theoretical framework based on mixed field-susceptibilities to describe the decay of **magnetic** dipoles induced by non–**magnetic** **nanostructures**. This approach is first illustrated in simple cases in which analytical expressions of the decay rate can be obtained. We then show that a more refined numerical implementation of this formalism involving a volume discretization and the computation of a generalized propagator can predict the **dynamics** of mag- netic dipoles in the vicinity of **nanostructures** of arbitrary geometries. We finally demonstrate the versatility of this numerical method by coupling it to an evolutionary optimization algorithm. In this way we predict a structure geometry which maximally promotes the decay of **magnetic** transitions with respect to electric emitters.

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otherwise free of any macroscopic defects. Strongly pinning walls, introduced by heavy ion irradiation through a 30 µm thick Ni mask, define weakly pinning channels through which the **vortex** ensemble is forced to flow. An additional, essential ingredient of the experiment is the inclusion of heavy-ion irradiated contact pads directly adjacent to the channel structure, and far removed from the crystal boundaries. By injecting the transport current through these irradiated pads, one forces the transport current to flow through the bulk; shear flow of the **vortex** ensemble then takes place through the channels. The shear viscosity of the vortices in the bulk can then be probed by a standard resistivity measurement. The signatures of shear flow are clearly noticeable in the resistance curves. The nature of the various features of the resistivity curves can be unambiguously identified by comparing them to magneto-optical images of the **magnetic** field distribution created by a transport current.

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2 One-dimensional **magnetic** **nanostructures** have for the past two decades been of increased interest for the develop- ment of future spintronic devices 1–4 motivated by concepts like **Magnetic** Race Track Memory 5 and the wish to manipulate **magnetic** domain walls (DW). Cylindrical nanowires (NWs) are particularly interesting candidates to reach this goal, much due to the fast DW motion induced by an external **magnetic** field or electric current, where theoretical studies have anticipated the absence of a Walker breakdown. 6,7 In addition, curvature effects have re- cently been proved to induce effects related to topology, chirality, and symmetry, 8 and unidirectional reversal pro- cess has been reported by engineering the geometry in multi-segmented nanowires. 9 However, to further technical developments in spintronics and a better control of DW motion, a thorough understanding of the fine structures of DWs in **magnetic** NWs, in which shape and crystal structure are contributing factors to the minimization of the system’s **magnetic** energy, 10–13 is required.

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The outline of the paper is as follows. In Sec. II A we present a model of the ferromagnetic superconductor and review the result for the **magnetic** excitations spectrum in the Meissner state (see Ref. 8 ). In Sec. II B we derive the basic equations for the collective **vortex**-magnetization **dynamics**. In Secs. II C and II D the weak-binding approximation is developed and the frequency gaps between adjacent bands of the magnon spectrum are determined analytically. The numerical spectra, obtained using realistic parameters, are presented in Sec. II E . In Sec. II F the role of dissipation connected with viscous **vortex** motion is discussed. Finally, in Sec. III we consider a boundary problem for an electromag- netic wave incident at a ferromagnetic superconductor. The frequency-dependent reflectivity coefficient is examined for frequencies lying within and close to the gaps of the magnon scpectrum.

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I - Introduction.
The ability to tailor the potential of electrons on the scale of their de Broglie wavelength has opened the new horizons in nanoelectronics. **Dynamics** of quantum particles in these heterogeneous fields, shaped by continuous spatial variations of potential as well as its gradient, attracts a growing attention in several fields of atomic, optical and solid state physics. Namely, engineering of complicated potential barriers for controlled transport of electrons in semiconductor superlattices and heterostructures /1- 3/, is widely used in microelectronic systems. This approach, generalized for traveling and tunneling regimes in motion of quasiparticles, proves to be the useful tool for analysis of the **dynamics** of polaritons in molecular crystals /4/ as well as quantum defects /5/ and **magnetic** moments /6/ in solids. A special attention was brought to periodical potentials, particularly to the **dynamics** of atom wavepackets in **magnetic** potentials, supported by current – carrying wires /7/ and, in particular, to the control of atomic ensembles and matter waves in optical lattices, arising from a set of interfering laser beams /8-10/. A wealth of literature has been devoted to transport and trapping of quantum objects in the double-well (DW) potentials of both natural and technological origin /11-13/.

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DOI: 10.1103/PhysRevLett.115.067202 PACS numbers: 75.78.-n, 75.60.Jk, 77.55.Nv, 85.80.Jm
Understanding and controlling magnetism in laterally confined **nanostructures** has been an area of high interest in the past decade. In planar structures the lateral confinement often leads to a ground state where a **vortex** core is present, the Landau flux closure and **vortex** domain configurations being the most obvious examples. Apart from fundamental interest in these topological entities, **magnetic** structures containing **vortex** cores have been recognized for applica- tions in **magnetic** random access memory [1] and micro- wave oscillators including spin torque **vortex** oscillators [2] . Excitation of **vortex** core motion [3 –5] is not only essential for microwave emission in spin torque **vortex** oscillators, but has recently been shown to provide a means to flip the direction of the **vortex** core itself [3] , allowing for data writing in **magnetic** memory applications.

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The capacity to identify chaotic behavior from the time series data from the nanocontact **vortex** oscillator opens up a number of perspectives for both fundamental and applied studies. The magnetoresistance signal repre- sents an indirect measurement of the **vortex** core polarity, whose **dynamics** is challenging to probe electrically. Our study may provide a way of studying the inertial effects and transient **dynamics** related to core reversal in nan- odevices. The chaotic **dynamics** measured in the mag- netoresistance signal is also associated with the erratic generation of regular patterns (as shown in the insert of Fig. 3 ), which could lead to the determination of sym- bolic **dynamics** for the system and hence open the way towards controlling the chaotic properties of the oscillator at the nanoscale. Finally, the use of chaotic **dynamics** in spintronics could lead to the development of novel appli- cations in information processing, such as physical-layer encryption and random number generation [ 41 ].

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Here, we demonstrate experimentally that the chaotic regime of the NCVO involves simple aperiodic waveform patterns. These can be encoded into bit sequences, which are correlated with the core-polarity state of the **magnetic** **vortex**. First, we describe time-resolved signals from the NCVO at 77 K and validate their chaotic characteristics from sensitivity to initial conditions and correlation dimension analysis. Then, we show that the time traces are in fact only composed by a few waveform patterns which are ordered aperiodically in the chaotic regime. By reconstructing attractor geometries from the measured time ser- ies, we reveal the symbolic **dynamics** of chaotic NCVOs, which is in good agreement with the patterns observed in simulation. We extract bit sequences based on this symbolic analysis and show that the generated bits can achieve maximal values of the Shan- non block entropy and Lempel–Ziv complexity.

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5.3.7 FOREBODY/WING/TAIL RESULTS
As expected, the addition of a forebody has an important impact on the flow, specifically, the presence of forebody vortices and their interaction with the leading-edge ones result in very substantial changes of the airloads at moderate to high angles of attack. As no attempt was made to fix transition on the forebody, an erratic behavior of the static loads was observed which is contrary to the case of the delta wing that separation is well defined at the sharp leading edge. Fig. 46 depicts the loads observed at several runs for σ = 30°. Most of the scatter occurs at small roll angles although it is present everywhere to a lesser extent. The loads are not symmetrical (or anti-symmetrical) about the origin due to the asymmetric forebody **vortex** shedding, which as suggested by the load discontinuities switch position at φ ~ -8°. Tests could not be performed near the above roll angle as potentially damaging, rapidly diverging lateral oscillations of the sting were induced. The large oscillation amplitude suggests that a positive feedback results from the coupling between the forebody **vortex** switching and model motion.

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vortices. However, at a slightly lower temperature, T /T c = 0.93, the interstitial vortices freeze up
leading to a strong reduction of the ac screening length. We propose a simple model for the **vortex** response in this system which fits well to the experimental data. Our analysis suggests that the observed switching to the high mobility regime stems from a resonant effect, where the period of the ac excitation is just large enough to allow interstitial vortices to thermally hop through the weak pinning landscape produced by random material defects. This argument is further supported by the observation of a pronounced enhancement of the out-of-phase response at the crossover between both dynamical regimes.

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From the Bohr-van Leeuwen theorem and without internal degrees of freedom, the magnetization of a classical system is always null. 7 Because magnetism is a quantum effect, the **magnetic** molecular **dynamics** (MMD) cannot mimic the magnetization **dynamics** without introducing the spins as supplemental dynamical variables. In a classical fluid theory, the first formulation of the **dynamics** of classical particles with spin degrees of freedom was given early by Turski. 8 More recently, another coupling of the molecular **dynamics** (MD) and atomic spin **dynamics** was presented by Antropov et al. 9 , 10 This approach is based simultaneously on a quantum mechanical derivation of localized moment and atomic equations of motion but is a colossal consumption of computational resources. However, in simplifying the full set of first-principles equations of motion, Akbar et al. 11 have demonstrated a few-hundred-atom simulation in which com- plex antiferromagnetic order and helical structures are found for γ -Fe. Moreover, explicit calculations of spin damping have been reported on bulk, monolayers, and atomic wires of Fe, Co, and Ni. 12

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KEYWORDS: Magnetoresistance, platinum oxide, nanocontact, spin-dependent tunneling
C alculations suggest that platinum atomic chains of 1 nm or longer should exhibit spontaneous Hund ’s rule superparamagnetism at low temperature. 1 −4 This would imply that the spins can be frozen in a direction determined by an external **magnetic** ﬁeld. Moreover, the conductance of a single Pt atomic contact is dominated by conducting channels de ﬁned by the 8 valence electrons, and it is expected to vary as a function of the chain length and interatomic distance. 1,3,5 In ideal **magnetic** atomic structures, only one spin channel may be available, so that the conductance can vary in multiples of e 2 /h

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Our simulation scheme differs from previous works @32,33# in that the initial state corresponds to a set of N particles connected to each other in a rigid-rod-like configu- ration and are subsequently subjected to a **magnetic** field of desired strength, which allows the chain to relax and fluctu- ate. Previous works @32,33# focused on the kinetics of chain formation for which the appropriate initial state is a random configuration of the set of N particles. We simulate the ther- mal fluctuations of a single chain of variable length ~N532 particles to N5108 particles!, where all particles of the chain are moved in each MC step; the motion of one particle at a time is not sufficient to generate independent configurations and therefore calculate the ensemble averages. The dimen- sionless parameter l ~which is the ratio between the induced dipole-dipole interactions and the thermal energy! and the number of particles in the chain are the only parameters in the MC simulation. The fluctuation spectra of the chains were simulated using different **magnetic** field strengths and chain lengths. The configurations generated by the MC simu- lations were studied by calculating the same correlation functions that were used to treat the experimental data @Eqs. ~5! and ~6!#. The results are presented in Figs. 8–11.

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