Stability of Long-lived Antiskyrmions in Mn-Pt-Sn Material

TL;DR

This study calculates the room-temperature lifetime of antiskyrmions in Mn-Pt-Sn, revealing their exceptional stability due to large activation energy, with results aligning with experimental observations.

Contribution

It introduces an atomic scale modeling approach with harmonic transition state theory to accurately predict antiskyrmion stability in Mn-Pt-Sn.

Findings

Long antiskyrmion lifetime at room temperature confirmed

Stability driven by large activation energy from strong exchange coupling

Pre-exponential factor remains typical despite large system size

Abstract

The lifetime of antiskyrmions at room temperature in a Mn-Pt-Sn tetragonal Heusler material has been calculated using an atomic scale representation including nearly a million spins. The evaluation of the pre-exponential factor in the Arrhenius rate expression for this large system is made possible by an implementation of harmonic transition state theory that avoids evaluation of the eigenvalues of the Hessian matrix.The parameter values in the extended Heisenberg Hamiltonian, including anisotropic Dzyaloshinskii-Moriya interaction, are chosen to reproduce experimental observations [A. K. Nayak $\it{et\,al.}$, Nature $\textbf{548}$, 561 (2017)], in particular the 150 nm diameter. The calculated results are consistent with the long lifetime observed in the laboratory and this exceptional stability of the antiskyrmions is found to result from large activation energy for collapse due to strong exchange coupling while the pre-exponential factor in the Arrhenius expression for the lifetime is found to have a typical magnitude of 10$^{-12}$ s, despite the large number of spins. The long lifetime is, therefore, found to result from energetic effects rather than entropic effects in this system.

Equations

$$\mathcal{H}=-\sum _{⟨i,j⟩}\left[J{\mathbf{S}}_i\cdot {\mathbf{S}}_j+{\mathbf{D}}_{ij}\cdot ({\mathbf{S}}_i\times {\mathbf{S}}_j)\right]-\sum _i[\mu \text{B}\cdot {\mathbf{S}}_i+K{S}_{i,z}^2].$$

$${\tau }_0=\frac{2\pi }{\lambda {\Omega }_0}$$

$${\Omega }_0=\frac{\sqrt{\det H^{min}}}{\sqrt{\left|\det H^{sp}\right|}}.$$

$$\lambda =\sqrt{\frac{\mathbf{b}\cdot H^{sp}\mathbf{b}}{|\zeta |}},\mathrm{with}\mathbf{b}=\frac{\gamma \zeta }{\mu }{\mathbf{S}}^{sp}\times \mathbf{e},$$

Full text

I Introduction

Skyrmions and antiskyrmions are localized magnetic states that have been proposed as elements in future spintronics devices.Kiselev_2011 ; Fert_2013 ; Fert_2017 Along with interesting transport properties, such states can exhibit particle-like behavior and carry a topological charge that enhances their stability with respect to the uniform ferromagnetic or antiferromagnetic states. A key issue is the lifetime of (anti)skyrmions and its dependence on temperature and applied magnetic field. The challenge is to find or design materials where such magnetic states are sufficiently stable at ambient temperature and still small enough to be used in high density spintronic devices. Figure 1 shows the spin configuration of an antiskyrmion as well as that of a skyrmion.

So far, stability at room temperature has mainly been obtained for large skyrmions with a diameter of 50 nm or more.Everschor-Sitte_2018 ; Soumyanarayanan_2017 It is important to understand what determines the lifetime in order to guide the search for materials where smaller (anti)skyrmions are sufficiently stable at room temperature. From recent experimental studies of skyrmions in Fe1-xCoxSi, a large, destabilizing entropic contribution which reduces the lifetime of skyrmions has been reported. Wild_2017 On the other hand, theoretical studies have found that isolated skyrmions can be stabilized by entropic contributions. Varentsova_2018 ; Desplat_2018 ; Malottki_2019 ; Desplat_2020 ; Hoffmann_2020 The question we address here is whether this is also the case for the recently observed stable antiskyrmions in acentric tetragonal Heusler compounds.Nayak_2017 The diameter of these antiskyrmions is large, 150 nm, but it may be possible to modify materials parameters in some way to obtain smaller antiskyrmions that are still stable at room temperature. Antiskyrmions offer some advantage over skyrmions in that they can under some conditions move in the direction of an applied spin polarized current, while skyrmions necessarily move at an angle.Huang_2017

Skyrmions have been studied for several systems and recent reviews have summarized results obtained.Fert_2017 ; Everschor-Sitte_2018 The annihilation of a skyrmion can occur through various mechanisms, in particular collapse in the interior of the sampleBessarab_2015 ; Lobanov_2016 ; Stosic_2017 ; Uzdin_2018 and escape through the boundary of the magnetic domain.Stosic_2017 ; Uzdin_2018 ; Bessarab_2018 Two skyrmions can also merge to form a single skyrmion (and the reverse can also occur, i.e. a division of a skyrmion into two).Muller_2018 The calculated lifetime estimates have taken into account the influence of a magnetic fieldBessarab_2018 , point defects,Uzdin_2018 and the width of the track where the skyrmion resides.Uzdin_2018 Calculations have also been carried out for skyrmions in antiferromagnets.Bessarab_2019 ; Potkina_2020 Recent calculations have addressed how the various materials parameters, such as the Dzyaloshinskii-Moriya interaction (DMI) and anisotropy constants affect the activation energy for the collapse of a skyrmion.Varentsova_2018 The results show that the activation energy is not simply a function of the size of the skyrmion although the two tend to be correlated.

Fewer studies have been carried out on antiskyrmions. They are stabilized by anisotropic Dzyaloshinskii-Moriya interaction (DMI) while isotropic DMI stabilizes skyrmions.Meshcheriakova_2014 ; Nayak_2017 Some aspects of antiskyrmions in systems with anisotropic DMI have been studied theoretically.Gungordu_2016 ; Hoffmann_2017 ; Huang_2017 ; Kovalev_2018 Antiskyrmions can also be stabilized by the magnetostatic interactions, for example in ion-irradiated Co/Pt multilayer filmsPhatak_2016 . Antiskyrmions have, furthermore, been discussed in relation to skyrmion–antiskyrmion pair production,Koshibae_2014 ; Koshibae_2016 ; Stier_2017 in particular in frustrated ferromagnetic films.Leonov_2015 ; Malottki_2017 ; Zhang_2017 Monte Carlo simulations using parameters estimated from electronic density functional theory calculations have been used to simulate both skyrmions and antiskyrmions in the Pd/Fe/Ir(111) system and the predicted spin-polarized scanning tunneling microscopy images found to be similar.Dupe_2016 Antiskyrmions as well as skyrmions in ferromagnetic films have been simulated using the micromagnetic approximation and the effect of the dipole-dipole interaction shown to provide larger stabilization for antiskyrmions.Camosi_2018 The energy barrier for collapse cannot, however, be evaluated within the micromagnetic approximation since the collapse mechanism involves a singularity that requires a discrete lattice representation.

It is of great interest for potential spintronic applications to carry out more studies of antiskyrmions. Only a few calculations of antiskyrmions have been reported so far, and they have mainly focused on stabilization by frustrated exchange rather than the anisotropic DMI.Malottki_2017 ; Ritzmann_2018 ; Desplat_2019 Simulation studies of the factors that affect thermal stability have, furthermore, been limited so far to rather small skyrmions that are unstable at room temperature. An important challenge is to extend the simulation methodology in order to make it applicable to large enough lattices to accurately represent the large (anti)skyrmions that have been found experimentally to be stable at room temperature.

In this article, the stability of antiskyrmions is evaluated using an atomic scale representation and harmonic transition state theory. The calculations reproduce large antiskyrmions observed in the Mn–Pt–Sn inverse Heusler compound.Nayak_2017 The long lifetime at room temperature is found to be due to high energy barrier for collapse resulting mainly from strong exchange interaction. The pre-exponential factor which includes the entropic effects is, however, found to be of a typical magnitude, 10*-12* s, despite the large number of spins involved.

II Simulated system

The system is described by a Heisenberg-type Hamiltonian

[TABLE]

Here $J$ is the exchange constant for the nearest neighbor magnetic moments ($J>0$), $\mathbf{D}_{ij}$ is the Dzyaloshinskii-Moriya vector lying in the plane of the sample ($x$-$y$ plane), $K$ is the uniaxial anisotropy constant, B is the magnetic field, and $\mathbf{S}_{i}$ is the vector of unit length in the direction of the magnetic moment at site $i$ of a square lattice. The summation $\langle i,j\rangle$ runs over pairs of nearest neighbor sites. We note that the dipole-dipole interaction is not included in the Hamiltonian in the present calculations. Free boundary conditions are used resulting in an energy barrier for the escape of the antiskyrmion through the boundary and eliminating translational invariance that would lead to zero modes. Such modes require special treatment in the lifetime calculation.Ivanov_2017

Depending on the type of DMI, this Hamiltonian can give rise to skyrmions (Bloch or Néel) or antiskyrmions. If the DMI vector points along the bond connecting sites $i$ and $j$, a Bloch type skyrmion (shown in fig.1(b)) can form. The vector can be written as $\mathbf{D}_{ij}=({\hat{r}}_{ij}\cdot{\hat{x}})\mathbf{D}_{1}+({\hat{r}}_{ij}\cdot{\hat{y}})\mathbf{D}_{2}$ where ${\hat{r}}_{ij}$ is a unit vector pointing from site $i$ to site $j$. An anisotropic DMI with $\mathbf{D}_{1}=(D,0,0)$ and $\mathbf{D}_{2}=(0,-D,0)$ in Eq. (1) supports a symmetric antiskyrmion (shown in fig.1(a)). Bogdanov_1989 ; Gungordu_2016

The parameters in the extended Heisenberg Hamiltonian are chosen here to mimic the Mn–Pt–Sn inverse Heusler compound where antiskyrmions have been observed over long time scale at room temperature.Nayak_2017 The diameter of an antiskyrmion in this material has been measured to be approximately 150 nm, corresponding to 230 lattice constants, and accurate atomic scale modeling therefore requires a square lattice containing at least $900\times 900$ lattice points, i.e. nearly a million spins. In order to evaluate the activation energy and estimate the lifetime of the antiskyrmion, it is necessary to use a discrete atomic lattice, rather than the continuum approximation invoked in micromagnetic simulations.

The parameters in the Hamiltonian used here are chosen to be consistent with the previously determined micromagnetic model parameters for this system.Nayak_2017 There, the antiskyrmions at $T=300$ K were modeled using the following parameter values: exchange stiffness $A=1.2\times 10^{-10}$ J/m, Dzyaloshinskii-Moriya parameter $d=6\times 10^{-3}$ J/m2, saturation magnetization $M_{s}=445$ kA/m, external field $B=0.29$ T, zero anisotropy, and an in-plane lattice constant of $a=$ 0.63 nm and out-of-plane lattice constant of $c=$ 1.22 nm. These micromagnetic parameter values are converted to parameters for the atomic scale lattice Hamiltonian as $J=2cA=1830$ meV, $D=acd=29$ meV, $K=0$, and $\mu=a^{2}cM_{s}=1.37$ meV/T. We note, however, that the micromagnetic simulations can only be used to determine these quantities relative to the exchange parameter, $J$, so our calculations are carried out in terms of scaled parameters $D/J=0.016$ and $\mu/J=7.5\cdot$10*-4* T*-1*.

III Calculations of the lifetime

The lifetime of a magnetic state can generally be described by an Arrhenius rate law, $\tau=\tau_{0}\exp(E_{a}/k_{B}T)$, where $E_{a}$ is the activation energy for the annihilation event and $\tau_{0}$ is the so-called pre-exponential factor. The two parameters, $\tau_{0}$ and $E_{a}$, can be estimated using the harmonic approximation to transition state theory (HTST) for magnetic degrees of freedom.Bessarab_2012 ; Bessarab_2013 It is based on an analysis of the multidimensional energy surface of the system, describing how the energy depends on the angular variables specifying the direction of all the magnetic moments in the system. In the case of interest, the antiskyrmion corresponds to a local minimum on the energy surface, whereas the homogeneous ferromagnetic phase corresponds to the global minimum. The activation energy for annihilation can be estimated as the highest rise in energy along the minimum energy path (MEP) connecting the antiskyrmion minimum to the ferromagnetic minimum. The point of highest energy on the MEP corresponds to a first order saddle point on the energy surface. The MEP can be found using the geodesic nudged elastic band method.Bessarab_2015 The computational effort is reduced here by making use of prior knowledge about the shape of the MEP and by focusing only on a small part of the MEP near the maximum. Lobanov_2017

In HTST, the pre-exponential factor, $\tau_{0}$, is related to the relative vibrational entropy of the initial and transition states as well as the flux through the transition state. The vibrational entropy is estimated by approximating the energy surface in the vicinity of the local minimum and the saddle point as quadratic functions to obtain the frequencies of the vibrational modes. The vibrational entropy of a state is then given by the product of the vibrational frequencies for that state. Normally this involves evaluating the eigenvalues of the Hessian matrixBessarab_2012 but for large systems this becomes a challenging calculation. Below, we briefly describe a more efficient method for evaluating $\tau_{0}$ that does not require the evaluation of the eigenvalues and makes it possible to carry out calculations for the large system studied here. Additional information about the method can be found in Ref. Lobanov_2020 The lowest couple of eigenvalues of the Hessian are still calculated explicitly using the Lanczos method to ensure that only one negative eigenvalue is found at the obtained saddle point and to test for the possible presence of zero modes.

An atomic scale simulation of such a large system is computationally challenging because of the large number of variables. In order to reduce computational effort we use the following scaling method. A sequence of calculations is carried out for systems with decreasing in-plane lattice constant, $a$, and increasing number of spins in such a way as to keep the values of the parameters corresponding to the micromagnetic model constant. Letting $a_{N}=a_{1}/N$ denote the in-plane lattice constant after $N$ iterations and referring to the relationship between micromagnetic model parameters, which are kept constant, and atomic lattice parameters, which depend on the lattice constant, the following scaling relationships are obtained: $D_{N}=D_{1}/N$ and $\mu_{N}=\mu_{1}/N^{2}$ while the exchange constant $J$ is unchanged as it is independent of $a$. The number of spins is 45$N$ x 45$N$, starting from $N=1$, and eventually reaching the 900 x 900 when $N=20$ giving spacing between sites that corresponds to the lattice constant of the Mn–Pt–Sn inverse Heusler compound. The calculation for an MEP for a given $N$ starts from an initial guess obtained from the calculation with $N-1$. Fig. 3 shows the spin configurations corresponding to the first three values of $N$.

The pre-exponential factor is evaluated separately for each $N$ but without having to determine the eigenvalues of the Hessian as has been done in previous calculations.Bessarab_2012 ; Bessarab_2013 It is difficult to obtain high enough accuracy for the eigenvalues when the number of spins is so large. Since the dipole-dipole interaction is not included here explicitly, only the nearest neighbors interact making it possible to write the Hessian matrix in a block tri-diagonal form and thereby evaluate the determinant of the Hessian. The pre-exponential factor is then evaluated from the determinant directly without having to determine the eigenvalues. It can be written as

[TABLE]

where the factor $\Omega_{0}$ is a ratio of determinants of the Hessian at the initial state minimum, $H^{min}$, and at the saddle point, $H^{sp}$, i.e.

[TABLE]

It is connected with the ratio of the entropy of the initial state and the entropy of the transition state, both evaluated within the harmonic approximation. The factor $\lambda$ is connected with the dynamics of the system through the transition state and is evaluated from a basis invariant expression as

[TABLE]

where ${\bf S}^{sp}$ is the spin configuration at the saddle point, $\zeta$ the negative eigenvalue of $H^{sp}$, $\bf e$ the corresponding eigenvector (a unit tangent vector to the MEP at the saddle point), and $\gamma$ the gyromagnetic ratio. This expression for $\lambda$ can be evaluated numerically even for large systems since it can be computed in spin-related basis without the evaluation of the whole set of eigenvectors and eigenvalues of $H^{sp}$.Potkina_2020

Figure 4 shows the energy of the antiskyrmion with respect to the uniform ferromagnetic state and the energy of the saddle point as a function of the scaling parameter $N$. The energy at the local minimum corresponding to the antiskyrmion almost reaches a constant as the parameter $N$ is increased, while the energy of the saddle point is still increasing at $N=20$. The lattice effects are still strong at $N=20$ and the Belavin-Polyakov limitBelavin_1975 of $4\pi J$ has not yet been reached. For the full 900 x 900 lattice corresponding to $N=20$, the antiskyrmion energy is found to be $E_{m}/J=1.27$ and the saddle point energy $E_{sp}/J=11.38$. Thus, the energy barrier for the collapse of the antiskyrmion is $E_{a}/J=10.11$. (See Ref. AnimationOfMEP for an animation of the MEP for antiskyrmion collapse when $N=1$).

The largest uncertainty lies in the value of the exchange parameter, $J$. The value of exchange stiffness used in the micromagnetic modeling of Nayak et al.Nayak_2017 corresponds to $J$=1830 meV for the lattice representation. Even for a much smaller value of $J$=110 meV the activation energy for collapse is large, ca. 1 eV. However, the pre-exponential factor turns out to have a typical value of 10*-12* s (see inset in figure 4). Together, these values give a lifetime of a month at room temperature. A larger value of $J$ would give even longer lifetime. This high stability is in good agreement with the reported experimental observations.Nayak_2017 The reason for the long lifetime is the large activation energy for the collapse of the antiskyrmion in this material while the pre-exponential factor, which is related to entropic effects, has a value that is similar to what has been calculated previously for the collapse of small skyrmions.Bessarab_2018

IV Conclusions

Calculations of the lifetime of large but submicron scale antiskyrmions in Mn–Pt–Sn tetragonal Heusler material are presented. The results are consistent with the observed stability at room temperature in recent experimentsNayak_2017 and show that the long lifetime is due to large activation energy for collapse rather than entropic effects. Since the atomic scale representation of this system requires roughly a million spins, the calculations are challenging and are made possible by using a scaling approach to evaluate the activation energy and an improved method for evaluating the pre-exponential factor in the Arrhenius rate expression. The latter is possible because the dipole-dipole interaction is not included here. Previous studies using the micromagnetic approach have shown that the dipole-dipole interaction makes antiskyrmions more stable with respect to skyrmions.Camosi_2018 The rate of collapse cannot, however, be evaluated from micromagnetic simulations but we expect that the inclusion of dipole-dipole interaction would further increase the activation energy while not affecting the value of the pre-exponential factor significantly.

Acknowledgments

This work was supported by the Icelandic Research Fund, the Research Fund of the University of Iceland, and the Russian Science Foundation (Grant 19-42-06302). The calculations were carried out at the Icelandic Research High Performance Computing facility.