Skyrmions as quasiparticles: free energy and entropy
Daniel Schick, Markus Wei?enhofer, Levente Rózsa, Ulrich Nowak
SSkyrmions as quasiparticles: free energy and entropy
Daniel Schick, Markus Weißenhofer, Levente Rózsa, ∗ and Ulrich Nowak Fachbereich Physik, Universität Konstanz, DE-78457 Konstanz, Germany (Dated: January 1, 2021)The free energy and the entropy of magnetic skyrmions with respect to the collinear state arecalculated for a (Pt . Ir . )/Fe bilayer on Pd(111) via atomistic spin model simulations. The simu-lations are carried out starting from very low temperatures where the skyrmion number is conservedup to the range where skyrmions are constantly created and destroyed by thermal fluctuations, high-lighting their quasiparticle nature. The higher entropy of the skyrmions at low temperature leadsto a reduced free energy, such that the skyrmions become energetically preferred over the collinearstate due to entropic stabilization as predicted by linear spin-wave theory. Going beyond the linearspin-wave approximation, a sign change is shown to occur in the free energy as well as the entropyat elevated temperature. I. INTRODUCTION
A magnetic spin configuration where the spin direc-tions span the entire unit sphere is called a magneticskyrmion [1, 2]. Magnetic skyrmions were theoretically pre-dicted to exist as solitons in the continuum two-dimensionalisotropic Heisenberg model [3], but they were demon-strated to be destabilized by the external field, magne-tocrystalline anisotropy or lattice discretization effects. Ro-bust mechanisms for the stabilization of skyrmions wereidentified later, including the Dzyaloshinsky-Moriya inter-action (DMI) [4–6], four-spin interactions [7] and the frus-tration of Heisenberg exchange interactions [8, 9]. The firstexperimental indications for the formation of a skyrmionlattice were found in MnSi via neutron scattering [10].Since then, skyrmions have been directly observed experi-mentally in other magnetic materials including Fe − x Co x Si[11, 12], Cu OSeO [13], Pd/Fe/Ir(111) [14], and GaV S [15]. Skyrmions were even observed close to or at room tem-perature in Pt/Co/MgO nanostructures [16], thin films ofFeGe [17] and Pt/Co/Ta [18], and in Co-Zn-Mn alloys [19].Skyrmions are often regarded as exceptionally stable, turn-ing them into a candidate for future use in logic and mem-ory devices [20–22]. The operation of such devices relies onthe demonstrated possibility of writing and deleting indi-vidual skyrmions [21, 23], and of moving them with electri-cal currents [24].Skyrmions are characterized by an integer topologicalcharge Q , counting the number of times the spin configura-tion wraps the unit sphere. Because the topological chargecannot be changed dynamically in a continuum model,skyrmions are often referred to as topologically protected.However, the topological charge is not a conserved quan-tity in lattice spin models as the energy barrier betweentopologically trivial states and skyrmions is finite, allowingfor the possibility of spontaneous creation and annihilationof skyrmions at finite temperature. Therefore, skyrmionsshould rather be thought of as quasiparticles, with theirlifetime following the Arrhenius law as demonstrated innumerical simulations [25, 26] and experiments [27]. Thedeciding factor for skyrmion stability can be understoodbased on linear spin-wave theory or the harmonic approx-imation of the energy functional close to the metastable ∗ [email protected] solution. In contrast to the collinear state, skyrmions arecharacterized by a number of low-frequency magnon modeswhich are easily excited by temperature, giving rise to ahigher spin-wave entropy and, consequently, a free-energypreference for skyrmions over collinear configurations as thetemperature is raised [10]. The larger entropy reduces theattempt frequency, i.e. the pre-exponential factor in the Ar-rhenius law, for skyrmion annihilation [25, 27–30], referredto as entropic stabilization. However, linear spin-wave the-ory is expected to gradually lose its validity at elevatedtemperatures, where magnon-magnon interactions becomemore prominent and the fast creation and destruction ofskyrmions makes the expansion around a well-defined equi-librium state questionable. Thermodynamic quantities ofskyrmions in this regime remain to be explored.In this paper, we calculate the free-energy and entropydifference between topologically trivial and skyrmionicstates in a wide temperature range through numerical sim-ulations for a (Pt . Ir . )/Fe bilayer on a Pd(111) sur-face. The free-energy difference is shown to decrease withtemperature, leading to a range where the free energy ofskyrmionic states with Q = 1 is lower than for topolog-ically trivial states. The dependence of this temperaturerange on the magnetic field and the system size is demon-strated. Remarkably, we also find a temperature rangewhere skyrmions possess a lower entropy than topologicallytrivial states, reversing the entropic stabilization. Theseresults highlight the thermodynamic properties and thequasiparticle character of skyrmions. II. METHODSA. Spin dynamics simulations
The system being modeled is a (Pt . Ir . )/Fe bilayeron a Pd(111) surface. The magnetic Fe moments are de-scribed by the following classical atomistic Hamiltonian: H = 12 (cid:88) i (cid:54) = j S i J ij S j + (cid:88) i S i K S i − µ s (cid:88) i B · S i . (1)Here, i, j are site indices and µ s is the spin magnetic mo-ment. The interaction tensors J ij include Heisenberg ex-change in its diagonal terms J ij = 13 Tr J ij , DMI in its an- a r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec tisymmetric part D ij ( S i × S j ) = 12 S i (cid:0) J ij − J Tij (cid:1) S j , andtwo-site anisotropy in its traceless symmetric part. K de-notes the on-site anisotropy tensor. B is an applied exter-nal field perpendicular to the surface. The exchange coeffi-cients J ij and K were determined by ab initio calculationsusing the Korringa–Kohn–Rostoker [31, 32] multiple scat-tering formalism with the relativistic torque [33] method,and are reported in Refs. [34, 35].The dynamics of the spin system is described by thestochastic Landau–Lifshitz–Gilbert (LLG) equation, ∂ S i ∂t = − γ (1 + α ) µ s S i × ( H i + α S i × H i ) , (2)where α is the damping parameter and γ the gyromag-netic ratio. H i is the local effective field with H i = ζ i − ∂ H /∂ S i . ζ i is a Gaussian noise term with (cid:104) ζ i ( t ) (cid:105) = 0 and (cid:104) ζ i,µ ( t ) ζ j,ν ( t (cid:48) ) (cid:105) = δ i,j δ µ,ν δ ( t − t (cid:48) )2 αk B T µ s /γ , with i, j denoting different spins and µ, ν representing differentCartesian coordinate directions.The simulations were performed on a two-dimensionaltriangular lattice with periodic boundary conditions. Thedamping constant was set to α = 1 . The considered inter-action parameters allow for the stabilization of skyrmionicobjects with different topological charges at zero temper-ature [34, 36–38]. The simulations were started from acollinear field-polarized state or from a configuration witha prepared skyrmionic structure of given topological chargeand the time evolution was calculated according to thestochastic LLG equation for a time of . µs with a to-tal of 5 independent realizations for each temperature andinitial condition. B. Calculation of thermodynamic quantities
The free energy and the entropy of a skyrmion are notdefined for single microstates of the system; therefore, theycannot be calculated as time or ensemble averages. In-stead, they were determined from the time dependence ofthe topological charge and the energy of the system at var-ious values of the external parameters.The topological charge of a continuous vector field canbe calculated as a surface integral with the spin vectors S of unit length, Q = − π (cid:90) d r S · ( ∂ x S × ∂ y S ) . (3)The sign convention is chosen such that a skyrmion on anout-of-plane-oriented collinear background has a topologi-cal charge of Q = 1 , while an antiskyrmion is described by Q = − . For our spin model simulation we use a discretizedversion of Eq. (3) [39], calculating Q via Q ( S ) = − (cid:88) { i,j,k } (cid:18) S i · ( S j × S k )1 + S i · S j + S i · S k + S j · S k (cid:19) , (4)where S i ( i = 1 , , denotes the spin unit vectors on anearest-neighbor triangle on the lattice. This value is cal-culated for all simulated triangles and summed up to deter-mine the topological charge for the entire system. For the simulated system with periodic boundary conditions, Q isalways an integer value.To each value of the topological charge Q we assign theconditional free energy F Q . This can be connected to theconditional partition function Z Q , defined as the phase-space integral of the Boltzmann exponential factor over allconfigurations with topological charge Q , F Q = − k B T ln( Z Q ) = − k B T ln (cid:32)(cid:90) Q ( S )= Q exp ( − β H ( S )) d S (cid:33) , (5)with β = 1 / ( k B T ) . The direct calculation of the condi-tional partition functions is numerically not feasible. How-ever, the difference in free energies between two values ofthe topological charge, for example and , can be calcu-lated from the ratio of partition functions using the corre-sponding condition: ∆ F = F − F = − k B T ln( Z /Z ) . (6)During a numerical simulation, a total number N of spinconfigurations, or recorded events, are created. Once thesystem has reached thermal equilibrium, the configurationsare generated with the probabilities according to the Boltz-mann distribution. Therefore, we expect the number ofrecorded events N Q fulfilling condition Q divided by thetotal number of recorded events N to be the equal to thepartition function Z Q in relation the total partition func-tion Z : N Q /N = Z Q /Z. (7)Using Eqs. (6) and (7), a formula to calculate free energydifferences can be derived, only based on the number ofrecorded events during simulation time that fulfill a certaincondition [40], ∆ F , count ( T ) = − k B T ln( N /N ) = − ∆ F , count ( T ) , (8)which we apply to spin configurations with different topo-logical charges Q . Equation (8) requires that the simulationexplores a large part of the phase space, such that a suffi-cient amount of changes in the topological charge take placeduring simulation time, with both states being present ata considerable number of time steps.At lower temperatures, switching events between con-figurations with different topological charges become ex-ceedingly rare, as is expected from the Arrhenius law τ ∝ exp ( − ∆ E/ ( k B T )) , where τ is the lifetime of thestate and ∆ E is the energy barrier separating it from aconfiguration with a different topological charge. In theregime where no changes in topological charge take placeduring the simulation, another method must be employedto calculate the free energy. Using the internal energy U Q = (cid:104)H ( S ) (cid:105) Q ( S )= Q and a single value of ∆ F at a hightemperature T calculated from Eq. (8), ∆ F can be de-termined at all temperatures based on the internal energydifference ∆ U = U − U via [40, 41], ∆ F , integ ( T ) = ∆ F , count ( T ) TT − T (cid:90) TT ∆ U ( T (cid:48) ) d T (cid:48) T (cid:48) . (9) ∆ U is easily accessible as the time average of the Hamil-tonian, which may be determined from independent simula-tions initialized in states with different topological charges,rather than requiring a high number of switching eventsduring a single run. Note that Eq. (9) may still becomenumerically inaccurate close to T = 0 where the denomi-nator goes to zero.The entropy difference between skyrmionic and topologi-cally trivial states can be calculated by the negative deriva-tive of the free energy difference, ∆ S = − ∂ ∆ F ∂T . (10)which is done numerically by using the central finite differ-ence.The average value of the topological charge can be calcu-lated as a time average of Q using the number of recordedevents with corresponding topological charge: (cid:104) Q (cid:105) time = (cid:80) Q QN Q (cid:80) Q N Q . (11)This method of calculation is appropriate at high temper-atures. However, at low temperatures there are no changesin topological charge during the simulation time, makingthe calculated value dependent on the initial condition.For the low-temperature calculations, we use the formula (cid:104) Q (cid:105) ensemble = (cid:80) Q Q exp( − β ∆ F Q ) (cid:80) Q exp( − β ∆ F Q ) , (12)where ∆ F Q is determined from Eq. (9). Note that herethe free-energy difference of the entire system has to beused and not an average per spin. Since this method re-quires performing simulations with different initial valuesof Q , we refer to it as an ensemble average. For simplic-ity, the summation in Eq. (12) were restricted to the values Q = − , , , which is possible since skyrmions and anti-skyrmions are both stable at zero temperature in the sys-tem [34]. Calculating the free-energy difference for othervalues of the topological charge is numerically demanding,and their relative Boltzmann weight is significantly lower.When the number of states with other topological chargesincreases, this restriction of the sum to Q = − , , is nolonger a good approximation, and Eq. (11) can be useddirectly instead. III. RESULTS
The zero-temperature energies of the different states ofthe considered system are displayed in Fig. 1. For lowerexternal fields the ground state is a spin spiral which trans-forms into a collinear state at around B s ≈ . T [42]. Theskyrmion lattice is not a ground state for any value of theexternal field, since its energy already exceeds that of thefield-polarized state at B s . A single isolated skyrmion leadsto a positive energy contribution to the field-polarized statefor fields above B m ≈ . T. Due to the short-range attrac-tive interaction between the skyrmions, this field is slightlylower than where the energy curves for the skyrmion lattice
FIG. 1. Phase diagram of the (Pt . Ir . )/Fe/Pd(111) system.The energy per spin is shown in the spin spiral (SS), skyrmionlattice (SkL) and field-polarized (FP) states as a function ofmagnetic field. B m and B s denote the metastability field for anisolated skyrmion and the transition field from the spin spiralto the field-polarized state, respectively. and the field-polarized states cross in Fig. 1. In the follow-ing, we discuss the thermal properties of skyrmions at fieldvalues higher than B m .As a first step, it has to established whether configura-tions with topological charge Q = 1 may indeed be iden-tified as skyrmions in our simulations, especially at highertemperatures. Besides the localized, cylindrically symmet-ric equilibrium skyrmion known at zero temperature, a Q = 1 spin configuration may denote a combination oftwo skyrmions plus an antiskyrmion or a completely dis-ordered state with various signs of the local topologicalcharge density. However, the latter configurations turn outto be significantly higher in energy and are consequently ex-pected to occour very rarely in our simulations for a widetemperature range. In Fig. 2(c), the topological charge isshown over a short timescale for a sample run with visiblechanges in the topological charge over time. The frequencyof changes in the topological charge depends on the tem-perature of the system, with skyrmion lifetimes quickly de-creasing with temperature as expected from the Arrheniuslaw. We calculate a time average of each spin’s Carte-sian coordinates over the time intervals denoted by thicklines in Fig. 2(c) in order to demonstrate that the recordedspin structures with Q = 1 are actually skyrmions. Notethat the individual spins do not have unit length after tak-ing the average. Figure 2(a) demonstrates that the aver-age structure still consists of a downwards-pointing core inan upwards-pointing background, with the spin directionsspanning the whole unit sphere as indicated by the color-coding. The time average over configurations with Q = 0 ,shown in Fig. 2(b), still resembles the collinear state evenat this elevated temperature.The difference in internal and free energy between the Q = 0 and Q = 1 states can be seen in Fig. 3(a). For us-ing Eq. (9), ∆ F , count was determined at a temperature of T = 190 K by comparing the number of states with differ-ent topological charges. Because the free-energy differencesagree between the counting and the integral methods for (a)(b) 50 55 60 65-101 (a) (b)(c)
FIG. 2. Time-averaged spin configurations for topologicalcharges (a) Q = 1 and (b) Q = 0 . (c) Time evolution of thetopological charge for a sample run with external field strength B = 1 T, T = 80 K and number of spins N S = 25 × . The spinconfigurations in (a) and (b) result from a time average for eachspin’s Cartesian coordinates over the indicated time intervals. temperatures T > K , the choice of T is not criticalfor our results. The deviations at lower temperature canbe attributed to the limitations of the simulation lengthdiscussed after Eq. (8).Since the metastability field B m for skyrmions in Fig. 1 issignificantly lower than the value of B = 1 T used in Fig. 3,at low temperatures the field-polarized state is strongly pre-ferred. ∆ F has a minimum at around T ≈ K with avalue below , showing that the skyrmion quasiparticleswith the short lifetimes shown in Fig. 2 are energeticallypreferred for a certain temperature range in this systemeven for such a high value of the external field. FromEq. (8), it is clear that skyrmions occur more often at thesetemperatures than topologically trivial states. For highertemperatures, ∆ F is slightly positive, but the ratio of theenergy barrier between topological charges to the temper-ature is small, resulting in rapid changes in Q .The low-temperature behavior of the internal and thefree energy may be understood based on linear spin-waveexpansion [43]. Using classical statistics for the spin waves,which is appropriate for modeling spin dynamics simula-tions based on the stochastic LLG equation, it is found that ∆ U is a constant as a function of temperature, while ∆ F changes linearly due to the entropy difference. These pre-dictions agree with the simulation results up to T ≈ K, FIG. 3. (a) Internal- and free-energy difference per spin betweenskyrmion ( Q = 1 ) and topologically trivial ( Q = 0 ) states asa function of temperature, for B = 1 T and N S = 25 × .The free-energy difference ∆ F , count is calculated from Eq. (8)and ∆ F , integ from Eq. (9). (b) Difference in entropy per spinbetween skyrmionic and topologically trivial states as a functionof temperature, for the same parameters. with the negative slope of ∆ F indicating the entropic sta-bilization of skyrmions [25, 27–30]. Note that for the con-sidered external field and system size, the free-energy dif-ference only reaches negative values at higher temperaturewhere deviations from linear spin-wave theory are observed,particularly in the rapid reduction of the internal-energydifference.From ∆ F , we calculate ∆ S using Eq. (10), with theresult shown in Fig. 3(b). Between K and K, theentropy difference is approximately constant, in agreementwith linear spin-wave theory. The deviations at lower tem-perature may be caused by a logarithmic divergence of theentropy difference, caused by the translational Goldstonemode in the harmonic approximation of the energy [30] orfound in the calculation of the configurational entropy ofskyrmions [44]; unfortunately, this possible effect cannot bedisentangled from the inaccuracies caused by the numericalintegration in Eq. (9) and differentiation in Eq. (10).As the temperature is increased, the entropy differencebetween the considered states is reduced. Remarkably,skyrmions actually have a lower entropy for certain tem-peratures than Q = 0 states. The temperature range where ∆ S is negative is confined between the extrema of ∆ F ,as expected from the derivative expression (10), and shiftedtowards higher temperatures as compared to the tempera-ture range with negative ∆ F . It is established that in thelow-temperature limit, the thermodynamic stabilization ofskyrmions can be explained by a competition between pos-itive internal-energy difference and negative entropy differ-ence contributions. However, these results indicate that forskyrmion quasiparticles with reduced lifetimes at elevatedtemperature, the stabilization mechanism is more complex,and the role of the internal-energy and the entropy termsmay become reversed as both of them change sign. In theskyrmion lifetime, the pre-exponential factor of the Arrhe-nius law is affected by the entropy differences of skyrmionscompared to the topologically trivial states [25, 27–30],meaning that the decrease in the entropy difference mayalso influence the lifetime in this regime. However, notethat the exponential factor of the Arrhenius law also losesits validity where the temperature becomes comparable tothe energy barrier.In Fig. 4, ∆ F is calculated for different external mag-netic fields applied perpendicular to the surface and differ-ent sizes of the simulated system. It is visible in Fig. 4(a)that strong external fields increase the internal- and free-energy difference at zero temperature, which can suppressthe minimum for ∆ F , meaning that skyrmions cannotbecome favored even at elevated temperatures. At lowermagnetic fields, the temperature range where skyrmionsare energetically preferred is larger and it extends to lowertemperatures. Also the minimum value of ∆ F is evenlower than for higher field values. The lower limit of thetemperature range where skyrmions are stable is expectedto reach K at B m ≈ . T, where isolated skyrmionson an infinite collinear background become energeticallypreferable (cf. Fig. 1).Simulations with different system sizes are compared inFig. 4(b). In our finite-size system, the skyrmions inter-act with themselves via the periodic boundary conditions,thereby raising the internal-energy difference and withthat also the free-energy difference. Since even isolatedskyrmions have a higher internal energy at zero tempera-ture than the collinear state, the free-energy per spin alsodecreases with increasing the system size for a fixed num-ber of skyrmions as the relative size of the field-polarizedareas increase. It is obvious that simulating with too smallsystems can cause the minimum in ∆ F to only have pos-itive values, which means that topologically trivial statesare always preferred over Q = 1 states. On the other hand,increasing the number of simulated spins lowers the free-energy difference per spin, as can be seen by comparingthe case N S = 25 × to N S = 28 × , making Q = 1 states also being preferred over a wider temperature range.Note that further increasing the system size may cause theformation of multiple skyrmions in the system at elevatedtemperature with a high probability, which effect was to beavoided in our simulations, similarly to Ref. [25].At high temperatures, the topological charge can take FIG. 4. (a) Free-energy difference per spin as a function oftemperature for different magnetic fields at a simulated systemsize of N S = 25 × spins. (b) Free-energy difference per spin asa function of temperature for different simulated system sizes atan external magnetic field of B = 1 T. All curves are calculatedusing Eq. (9). many different values during a single simulation, and thetime average in Eq. (11) was used for calculating the aver-age topological charge. The resulting average of the topo-logical charge (cid:104) Q (cid:105) as a function of temperature T is shownin Figure 5. At low temperature where the transition timesbetween different topological charges exceeds the simula-tion times, the ensemble average in Eq. (12) was approx-imated by using the free energies of topological charges Q = − , , and . For the data with topological charge Q = − we performed simulations with an antiskyrmion asthe initial condition. Since the antiskyrmion has consider-ably higher energy than the skyrmion in the system [34],we found no minimum either in the free-energy difference ∆ F − or in the entropy difference ∆ S − .In Fig. 5 we notice that at around T ≈ K, there isa regime with average topological charge above . for thetime average. This further indicates that skyrmions areenergetically preferred at elevated temperatures in the con- Time avg.Ensemble avg.
FIG. 5. Average value of Q calculated with ensemble average us-ing free-energy values of topological charges Q = − , , (green)and time average over all topological charges (blue). Systemsimulated with N S = 25 × spins and B = 1 T external field. sidered system. Note that the ensemble average is close tothe time average between and K, but it stays be-low . for all temperature values. Although the Q = 1 state has lower free energy than the collinear state closeto the maximum of the average topological charge, takingthe presence of antiskyrmions into account reduces the av-erage value below . . However, a lot of changes of thetopological charge are recorded at this temperature range,as can be seen in Fig. 2, and considering higher Q valuesin the time average increases the average above . . Atlow temperature, the average topological charge vanishessince skyrmions are energetically unfavorable compared tothe topologically trivial state. At very high temperature, (cid:104) Q (cid:105) again converges to zero since all microstates, includ-ing those with positive and negative topological charges,contribute with similar weights to the total partition func-tion in this limit. Figure 5 demonstrates that combiningEq. (11) accurate at high temperature with Eq. (12) whichcan be applied at low temperature enables the calculationof the average topological charge ranging from the com-pletely ordered to the completely disordered states, withreasonable agreement between the two methods in the in-termediate temperature range where both of them are valid. IV. CONCLUSION
We calculated the free-energy and entropy difference be-tween a skyrmion and the topologically trivial state in a (Pt . Ir . )/Fe bilayer on a Pd(111) surface by means ofnumerical simulations. We found that the free-energy dif-ference turns from positive to negative as the temperatureis increased, meaning structures with Q = 1 are thermo-dynamically preferred over topologically trivial states ina certain temperature range. We demonstrated that thisrange vanishes at higher magnetic fields or smaller systemsizes, where the internal energy of skyrmions with respectto the collinear state becomes higher. We showed that Q = 1 configurations in a time average can still be identi-fied with skyrmion-like spin structures in this temperaturerange, although they are frequently created and destroyedby thermal fluctuations. The preference for the formationof skyrmions at elevated temperature agrees with the pre-diction of entropic stabilization based on linear spin-wavetheory, but qualitative deviations from this approximationhave been observed in the thermodynamic quantities. Inparticular, we found that while skyrmions have higher en-tropy at low temperature, their presence reduces the en-tropy at elevated temperatures. We calculated a compositeaverage of the topological charge via combining an approx-imate average in the canonical ensemble based on the free-energy calculations at low temperature with a time averageat higher temperatures. We found the time average of thetopological charge to reach values over . in the tempera-ture range where we found skyrmions to be thermodynam-ically preferred.Although skyrmions are preferred by the free energy ina certain parameter regime, this does not mean that thesetopologically non-trivial states are stable at this tempera-ture. The deviations from linear spin-wave theory based onstable equilibrium structures are pronounced in this range,and the lifetime of skyrmions is considerably reduced asconfirmed by our simulations. This shows that skyrmionscannot be interpreted as particles with a conserved topo-logical charge, but should rather be seen as quasiparticleswith a finite chemical potential. The non-integer averagetopological number in this regime corresponds to the prob-ability of finding a skyrmion in the system, if the contri-butions from higher or opposite topological charges can beneglected. These results should stimulate further studies onthe properties of topologically non-trivial spin structures inthe presence of strong thermal fluctuations. ACKNOWLEDGMENTS
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