Spin waves cause non-linear friction
Martin P. Magiera, Lothar Brendel, Dietrich E. Wolf, Ulrich Nowak
eepl draft
Spin waves cause non-linear friction
M. P. Magiera , L. Brendel , D. E. Wolf and U. Nowak Faculty of Physics and CeNIDE, University of Duisburg-Essen, D-47048 Duisburg, Germany, EU Department of Physics, University of Konstanz, D-78457 Konstanz, Germany, EU
PACS – Classical spin models
PACS – Atomic scale friction
PACS – Spin waves
Abstract. - Energy dissipation is studied for a hard magnetic tip that scans a soft magneticsubstrate. The dynamics of the atomic moments are simulated by solving the Landau-Lifshitz-Gilbert (LLG) equation numerically. The local energy currents are analysed for the case of aHeisenberg spin chain taken as substrate. This leads to an explanation for the velocity dependenceof the friction force: The non-linear contribution for high velocities can be attributed to a spinwave front pushed by the tip along the substrate.
Introduction. –
On the macroscopic scale the phe-nomenology of friction is well-known. However, investi-gations of energy dissipation on the micron and nanome-ter scale have led in recent years to many new insights[1]. This progress was made possible by the developmentof modern surface science methods, in particular AtomicForce Microscopy, which allows to measure energy dissi-pation caused by relative motion of a tip with respect toa substrate.Studies concerning the contribution of magnetic degreesof freedom to energy dissipation [2, 3] form a young sub-field of nanotribology, which has been attracting increas-ing interest in recent years. Two classes of models havebeen considered, which show different phenomena. Thefirst one is Ising-like spin systems with two equivalent halfspaces moving relative to each other [4–8]. In this case,friction is induced by thermal fluctuations, and hence isnot present at zero temperature. In the second class ofmodels [9–13], there is no symmetry between slider andsubstrate: The slider, representing e.g. the tip of a Mag-netic Force Microscope, interacts only locally with a pla-nar magnetic surface. While scanning the surface, the tipin general excites substrate spins and hence experiencesfriction, even at zero temperature. The present study be-longs to the second class of models.We investigate the nature of the substrate excitationscaused by the tip motion for a classical Heisenberg modelwith Landau-Lifshitz-Gilbert (LLG, [14, 15]) dynamics(precession around, and relaxation into the local field di- (a)
E-mail: [email protected] rection). As the spins are continuous variables, spin waveexcitations are possible. As we will show in the following,their properties are reflected in the velocity dependence ofthe friction force. Spin waves are increasingly attractinginterest: e.g. in the last years a new subfield of magnetism, magnonics , has been developed, where materials are stud-ied with respect to their spin wave properties [16,17]. Onemotivation is to create new devices using spin wave logicsor novel concepts of data storage.In a previous work we showed that friction in this modelis proportional to the scanning velocity v (“viscous be-haviour”), provided that the tip does not move too fast[10]. The reason can be found in continuous excitations,while the motion in the Ising-model consists of discrete ex-citations and relaxations, which yields a constant frictionforce for low v . In the present paper we focus on the lo-cal dissipation processes in order to explain, why for highvelocities deviations from the viscous behaviour exist. Simulation model. –
To simulate a solid magneticmaterial, we consider a chain of N =320 classical, nor-malised dipole moments (“spins”, cf. fig. 1) S i = µ i /µ s ,where µ s denotes the material-dependent atomic magneticmoment. The spins represent magnetic moments of singleatoms, arranged with a lattice constant a along the x -axis.Two lattice constants above the spin chain, a magnetic tip S tip moves with constant velocity v = v e x . Its magnetisa-tion is fixed in z − direction. At the beginning of eachsimulation the tip is positioned at the centre of the chain.In order to keep boundary effects small, we use a con-veyer belt technique with anti-periodic boundary condi-p-1 a r X i v : . [ c ond - m a t . o t h e r] J un . P. Magiera et al. Fig. 1: Snapshot of a simulation. The colour encoding denotes the spins’ orientation in the yz -plane (the tip is moved alongthe spin chain axis to the right). In front of the tip a spin wave is visible, i.e. an oscillation of the spins around the x -axis. tions: When the tip has moved by one lattice constant,the boundary spin at the back end is deleted and a newspin with opposite direction is added at the front end ofthe chain. This shift puts the tip back to the centre of thesimulation cell.This paper analyses, how local excitations contributeto magnetic friction. Each substrate spin contributes itsexchange interaction, H ( i )sub , and its interaction with thetip field, H ( i )tip to the Hamiltonian H = N (cid:88) i =1 (cid:16) H ( i )sub + H ( i )tip (cid:17) = N (cid:88) i =1 H ( i ) . (1)As the substrate spins represent a ferromagnetic solid, weuse the anisotropic Heisenberg-Hamiltonian, H ( i )sub = − J S i · ( S i +1 + S i − ) − d z S i,z . (2) J> S i and its nearest neighbours i ±
1. In order to avoiddouble counting, half of the pair interaction is attributedto either spin. d z = − . J is the anisotropy constant: Hereit defines an easy plane anisotropy, thus the substratespins prefer an alignment in the xy -plane. The movingtip interacts with each substrate spin by the dipolar inter-action, H ( i )tip = − w S i · e i )( S tip · e i ) − S i · S tip R i , (3)where R i = | R i | is the length of the distance vector R i = r i − r tip , and e i its unit vector e i = R i /R i . r i and r tip denote the position vectors of the substrate spins and thetip, respectively. w quantifies the dipole-dipole couplingof the substrate and the tip, with w | S tip | = 10 Ja in thispaper.While the tip magnetisation direction is fixed in time,the substrate spins are allowed to change their orientation.To simulate their dynamics, we solve the LLG equation,˙ S i = − ˜ γ [ S i × h i + α S i × ( S i × h i )] , (4)numerically via the Heun integration scheme, where ˜ γ = γ [ µ s (1 + α )] − with the gyromagnetic ratio γ . The firstterm represents the Lamor precession of each spin in theeffective field, h i = − ∂ H ∂ S i (5) with the precession frequency ˜ γ | h i | . The precessional mo-tion preserves energy. Dissipation is introduced by thesecond term which causes an alignment towards h i . α is amaterial constant which can be obtained from ferromag-netic resonance experiments and represents the coupling ofeach spin to a reservoir of zero temperature. By adding astochastic term to the effective field, it is possible to studythe influence of finite temperatures as done in [10, 11].However, in order to analyse the non-equilibrium excita-tions it is advantageous to suppress thermal spin waves bysetting temperature equal to zero in this work.In order to discuss frictional losses occurring in the sys-tem the global energy balance was analysed in [10]: d H dt = P pump − P diss , (6) P pump = N (cid:88) i =1 P ( i )pump = N (cid:88) i =1 ∂ H ( i )tip ∂ r tip · ˙ r tip , (7) P diss = N (cid:88) i =1 P ( i )diss = N (cid:88) i =1 ˜ γα ( S i × h i ) . (8)The only explicit time-dependence of the Hamiltonian H stems from the motion of the tip. It leads to the first termin eq. (6), which is the energy pumped into the spin systemper unit time by an outside energy source that keeps thetip moving. Accordingly we call it the “pumping power”.Its local contribution, P ( i )pump , is the energy transferred perunit time from the tip to substrate spin S i . The frictionforce, F = − F e x , the substrate exerts on the tip is givenby F = (cid:104) P pump (cid:105) v , (9)where the angular brackets denote a time average over atleast one period a/v . P ( i )diss represents the energy current from spin S i into theheat bath. In other words, this is the energy dissipated atsite i per unit time. Dissipation always occurs when thesystem relaxes towards the ground-state, in which the spinat site i is aligned with the local field-direction h i . With-out tip movement, P pump is zero and P diss leads the systemquickly to its ground state. For a tip moving at constantvelocity, a steady non-equilibrium state is reached, wherethe time averaged derivative (cid:104) d H /dt (cid:105) vanishes, becauseall energy pumped into the system is dissipated. Thenthe two power terms in eq. (6) cancel.p-2pin waves cause non-linear friction j E (cid:31) i (cid:30) j E (cid:31) i (cid:31) (cid:30) P pump (cid:31) i (cid:30) (cid:30) P pump (cid:31) i (cid:30) P pump (cid:31) i (cid:31) (cid:30) d (cid:31) (cid:31) i (cid:31) (cid:30) dtd (cid:31) (cid:31) i (cid:30) (cid:30) dt d (cid:31) (cid:31) i (cid:30) dtTip MovementHeat Bath P diss (cid:31) i (cid:31) (cid:30) P diss (cid:31) i (cid:30) P diss (cid:31) i (cid:30) (cid:30) …… Fig. 2: Illustration of the local energy balance. The arrows rep-resent the directions in which the energy transfers are countedpositive: a positive j ( i ) E rises the energy at site ( i ), but lowersthe one at site ( i − When evaluating the local energy balance instead ofeq. (6), energy currents j E within the substrate have to betaken into account, which transport energy from one spinto its neighbour (cf. fig. 2). By taking the time derivativeof the local Hamiltonian one obtains: d H ( i ) dt = P ( i )pump − a (div j E ) ( i ) − P ( i )diss , (10) j ( i )E = − J ( S i − S i − ) · ˙ S i − + ˙ S i − J S i · ˙ S i − − S i − · ˙ S i ) . (11) Simulation Results. –
Let us consider the steadystate in a co-moving frame: the local quantities do notdepend on spin index i and time t separately, but onlyon the (continuous) coordinate x i = R i ( t ) · e x . For con-siderations, where all spins are equivalent, we can dropthe index i , e.g. the tip position is always at x = 0.In its vicinity, fig. 3(a) shows the local pumping power,as well as the discrete divergence of the energy current,(div j E ) ( i ) = (cid:0) j ( i +1)E − j ( i )E (cid:1) /a , as functions of x .The physical interpretation of fig. 3(a) is the following:When the tip approaches, a substrate spin lowers its en-ergy by adjusting to the inhomogeneous tip field at thecost of the exchange interaction. When the tip has passedby, it returns asymptotically to its higher energy in theabsence of the tip field. This means that the tip injectsenergy P pump ( x ) ∝ v per unit time at x < x >
0. With respect to origin and curve shape,this is very similar to an electrical charge passing by acharge of opposite sign on a straight line. The apparentcentral symmetry holds only up to first order in v , though.The small asymmetry, not noticeable in fig.3(a), is due todissipation, which will be discussed below. But first wederive the steady state current within the chain.In the steady state, we have˙ S i = ˙ S ( x i ) = v ∂ x S ( x i ) = v S i +1 − S i a , (12)where the third equality, due to using the difference quo- P pump v (cid:45) av (cid:33) j E (cid:72) a (cid:76) v (cid:61) (cid:61) (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) x (cid:64) a (cid:68) po w e r (cid:144) v e l o c it y P (cid:144) v (cid:64) J (cid:144) a (cid:68) (cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) 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10 0 10 2010 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) position on the chain x (cid:64) a (cid:68) j E (cid:72) i (cid:76) (cid:64) Γ J (cid:144) Μ s (cid:68) , P (cid:64) Γ J (cid:144) Μ s (cid:68) Fig. 3: (a) Pumping power and transported power, rescaledby velocity, at α = 0 .
1. Negative pumping power represents anenergy return from the spin chain to the tip, positive an energyinjection from the tip into the chain. (b) Energy current forfive different time steps (points, for α = 0 . αv . tient, holds up to first order in a . Plugging that into eq.(11), the current j ( i )E reads j ( i ) E = Jv a (1 − S i − · S i +1 ) . (13)This shows that j E transports the exchange energy to bepaid for orienting the spins according to the inhomoge-neous tip field. Correspondingly, the source of this current(div j E >
0) is behind, and its sink (div j E <
0) is in frontof the tip, as seen in fig. 3(a).The terms discussed so far are reversible and hence in-dependent of the damping constant α : To first order in v they add up to zero in eq. (6). The origin of dissipationis that the spin pattern does not follow the tip instanta-neously, but with a delay, which corresponds in the co-moving frame to a lag ∆ x ∝ αv [10]. It is a manifestationof the driving out of equilibrium, which the spin relaxationmust counteract. Fig. 3(b) shows that P diss ( x ) /j E ( x ) isindeed a constant ∝ αv , which we may call the drivingforce . As pointed out in fig. 3(a), j E is proportional to v . This only holds true for velocities not much larger than v ≈ . γJa/µ s (see eq. (16) below). This implies thatfor small velocities P diss ∝ αv , which gives rise to a fric-tion force, eq. (9), proportional to αv . This is analogous to electrical power P = UI , where the voltage U provides the driving force for the current I . p-3. P. Magiera et al. Fig. 3(b) shows an important qualitative difference be-tween energy currents for tip velocities below, respectivelyabove v . For low velocities, the energy current is concen-trated around the tip position in an essentially symmetricway: Whatever exchange energy is released behind the tip,is reabsorbed in front of it. For high velocities, however,an additional shoulder in front of the tip appears. Thisshoulder represents a part of the energy current, whichcan leave the tip’s immediate neighbourhood and propa-gates further along the spin chain, until it is damped out.We call this contribution non-confined . The propagationrange depends on the damping constant α , as can be seenin fig. 4. The lower the damping constant, the farther thecurrent extends.In order to evaluate this quantitatively, we define thenon-confined energy current as j nc ( x ) = j E ( x ) − j E ( − x ) for x> . (14)It is plotted for several α -values in fig. 4(b). The axesare rescaled in order to show that the range shrinkswith increasing damping approximately like α − . , andthat the amplitude of the non-confined current also de-creases roughly like α − . . The range and the ampli-tude of the non-confined current combine in such a way,that the integral over j nc ( x ) is nearly proportional to α − .Hence, when multiplied by the driving force ∝ αv , the α -dependence nearly cancels. The contribution of the non-confined excitations to friction is therefore approximatelyindependent of α , in contrast to the confined contribu-tion discussed above. As will be explained below, the twocontributions also depend differently on velocity.The non-confined excitations can be regarded as spinwaves. An excitation means a deflection of a spin S i fromthe local field h i , leading to dissipation at the correspond-ing site according to eq. (8) and a precession around h i ina plane perpendicular to h i . Accordingly, e i,a = h i × e y | h i × e y | and e i,b = h i × e i,a | h i × e i,a | (15)form an appropriate local basis to illustrate the excita-tions. Spins far in front of the tip experience a field whichpoints in x -direction, thus the e i,a -component points in z -direction. Near the tip the basis changes as sketched inthe inset of fig. 5.In the low velocity regime, a deflection from the localfields is present solely in the vicinity of the tip, accordingto the purely confined contribution to friction discussedabove. For velocities, where non-confined currents canbe observed, additional oscillations are present in front ofthe tip. A Fourier analysis of our simulation data showsthat their wavenumber has approximately a linear velocitydependence: k ∝ v − v with v ≈ . γJa/µ s . (16)The resulting empirical coefficient 0.31 must be expectedto depend on system parameters like the tip field’s shape (cid:72) a (cid:76) Α (cid:61)
Α (cid:61)
Α (cid:61)
Α (cid:61)
Α (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) position on the chain x (cid:64) a (cid:68) c u rr e n t j E (cid:64) Γ J (cid:144) Μ s (cid:68) (cid:72) b (cid:76) Α (cid:61)
Α (cid:61)
Α (cid:61)
Α (cid:61) Α x (cid:64) a (cid:68) Α . j n c (cid:64) Γ J (cid:144) Μ s (cid:68) Fig. 4: (a) Energy currents for v =0 . α values. Whenthe damping is lowered, the current can proceed further in thesubstrate. (b) Rescaled non-confined current vs. a rescaledposition for v =0 . and amplitude, which determine the spin waves’ confine-ment. For small enough damping, the oscillations presum-ably extend arbitrarily far in front of the tip. Asymptoti-cally we can neglect the tip field and consider an isolatedspin chain with exchange interaction only. In its groundstate, all spins point into the same direction, say e x . Forsmall perturbations of S i from this direction, the LLG-equation with h i = J ( S i − + S i +1 ) (17)can be linearised ( e.g. [18]). The solution is a spin wavewith an oscillating part of δ i = e y δ cos( ika − ωt ) + e z δ sin( ika − ωt ) (18)to first order in its small amplitude δ . Its dispersion rela-tion ω ( k ) = 4 Jγµ s (1 − cos( ka )) (19)yields a group velocity v ∝ k in the long wavelength limit.The finding eq. (16) indicates that this holds true even inthe more complicated system with the inhomogeneous tipfield.Inserting S i = e x + δ i into eqs. (17) and (8) yields forthe wave of wave number k a dissipation of P diss ( k ) ∝ sin (cid:18) ka (cid:19) . (20)Using eq. (16) and assuming that the amplitude of the spinwave excitations δ does not depend on v and that ka (cid:28) v Α (cid:61) (cid:61)
Α (cid:61) (cid:61) (cid:45)
10 0 10 20 30 40 (cid:45) (cid:45) x (cid:64) a (cid:68) s p i n c o m pon e n t S i a e x e y e z h h S S e a e b Fig. 5: Spin component perpendicular to the correspondingfield. While for low velocities one precession around the localfield is observable, for higher velocities more precession cyclesin front of the tip are present. For lower damping excitationsreach further. Sketch: Definition of the field dependent basis.For two sites spin and field values are sketched. For S thelocal field points in x − direction (as the situation in the studiedsystem in front of the tip is), and the spin precesses on the bluedisk. The shift of the field for S to the bottom (as it may beinduced by the tip-field) yields also a change of the disk andthe appropriate basis. this predicts a non-linear velocity dependence of the spinwave contribution to friction like ( v − v ) v .The total magnetic friction force is thus predicted to be F ≈ Aαv + B ( α )Θ( v − v ) ( v − v ) v , (21)where Θ( x ) is the Heaviside step function, and the coeffi-cients A , B as well as v may depend on system parameterslike the tip field. In fig. 6 this total force is plotted, andthe simulation results are in good agreement with eq. (21). Conclusion and outlook. –
In this work, we couldseparate two distinct contributions to magnetic friction byexamining the energy current in a spin chain. The con-fined current results in a friction force F = Aαv , which isin accord with our earlier results for 2 d [10] and 3 d [12]substrates. Above a threshold velocity v , spin wave exci-tations may leave the tip’s immediate neighbourhood andform a damped wave packet in front of the tip, propagat-ing along with it. These excitations are the stronger, theweaker the damping. They lead to an additional contri-bution to friction with a non-linear velocity dependence.The dependence of the non-confined contribution on α isnot trivial, because the range as well as the amplitude ofthe energy current are influenced in a non-linear way., cf.fig 4(b).Important extensions of the present investigation in-clude the influence of dimensionality on the non-confinedspin waves. Here the propagation is not confined to thetip’s motion direction. The influence of thermal spin wavesand their interaction with the free spin waves is another Α (cid:61)
Α (cid:61)
Α (cid:61) Α v non (cid:45) confinedcontributions (cid:72) a (cid:76) v (cid:64) Γ Ja (cid:144) Μ s (cid:68) fr i c ti on f o r ce F (cid:64) J (cid:144) a (cid:68) (cid:231)(cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243)(cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) Α (cid:61)
Α (cid:61)
Α (cid:61) B (cid:72) v (cid:45) v (cid:76) with v (cid:61) (cid:177) (cid:231)(cid:225)(cid:243) (cid:72) b (cid:76) (cid:45) (cid:45) (cid:45) (cid:72) v (cid:45) v (cid:76) (cid:180) (cid:64) Γ Ja (cid:144) Μ s (cid:68) F n c (cid:180) v (cid:64) Γ J (cid:144) Μ s (cid:68) Fig. 6: (a) Friction force for several velocities, with the non-confined part marked by the shading. (b) Non-confined con-tributions to the friction force times velocity as well as thefunction B ( v − v ) , where v has been taken from eq. (16). open question. Studies dealing with this are already inprogress and will be reported in a future work. ∗ ∗ ∗ This work was supported by the German ResearchFoundation (DFG) via SFB 616 and the German Aca-demic Exchange Service (DAAD) through the PROBRALprogramme.
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