# Lifetime of skyrmions in discrete systems with infinitesimal lattice
constant

M.N.Potkina  I.S. Lobanov  H. Jónsson  V.M. Uzdin

###### Abstract

Topological protection of chiral magnetic structures is investigated by taking
a two-dimensional magnetic skyrmion as an example. The skyrmion lifetime is
calculated based on harmonic transition state theory for a discrete lattice
model using various values of the ratio of the lattice constant and the
skyrmion size. Parameters of the system corresponding to exchange, anisotropy
and Dzyaloshinsky-Moriya interaction are chosen in such a way as to keep the
energy and size of the skyrmion unchanged for small values of the lattice
constant, using scaling relations derived from continuous micromagnetic
description. The number of magnetic moments included in the calculations
reaches more than a million. The results indicate that in the limit of
infinitesimal lattice constant, the energy barrier for skyrmion collapse
approaches the Belavin-Polyakov lower bound of the energy of a topological
soliton in the $\sigma$-model, the entropy contribution to the pre-exponential
factor in the Arrhenius rate expression for collapse approaches a constant and
the skyrmion lifetime can, for large enough number of spins, correspond to
thermally stable skyrmion at room temperature even without magnetic dipole-
dipole interaction.

###### keywords:

chiral magnet , skyrmion , topological protection , lifetime

###### PACS:

75.25.+z

††journal: Journal of Magnetism and Magnetic Materials

[inst1]organization=Department of Physics, St. Petersburg State
University,city=St.Petersburg, postcode=198504, state=St.Petersburg,
country=Russia

[inst2]organization=Department of Physics, ITMO University,city=St.Petersburg,
postcode=197101, country=Russia

[inst3]organization=Science Institute and Faculty of Physical Sciences,
University of Iceland,city=Reykjavik, postcode=VR-III, 107, country=Iceland

The lifetime of skyrmions is estimated within the framework of the harmonic
transition-state theory

Collapse inside the sample and escape through the boundary are considered.

Pre-exponential factors and activation barriers are calculated for systems
with different lattice constants corresponding to the same continuous
micromagnetic model.

The manifestation of topological protection of chiral magnetic structures in
discrete systems is shown.

##  1 Introduction

Topological chiral magnetic structures, such as magnetic skyrmions, are
promising information carriers in future generations of racetrack memory,
logical and neuromorphic devices [1, 2, 3]. For such applications, thermal
stability is a fundamental issue as thermal fluctuations and external
disturbances can lead to structural breakdown and loss of the information
encoded in the magnetic configurations. The theoretically predicted stability
of topological magnetic systems is associated with the existence of a nonzero
topological charge, an integer that cannot change under continuous variation
of the magnetization. Magnetic skyrmion and homogeneous ferromagnetic states
have different topological charges and, therefore, cannot be transformed into
each other by continuous magnetization transformation [4]. In real materials,
magnetic moments are localized at discrete sites of the crystal lattice, and
the topological arguments do, strictly speaking, not apply. Instead, one might
expect to find finite energy barriers between states with different
topological charges and if they are large enough the states can be protected
against destruction by thermal fluctuations. However, estimates of the energy
barrier for the collapse of the small skyrmions in PdFe/Ir(111) [5, 6, 7] and
Co/Pt [8], which have diameter 1-10 nm, are below 150 meV. This is of similar
magnitude as energy barriers for magnetization reversal in small Fe islands on
W(110) surface, where no topological change is involved [9, 10]. The lifetime
of nanoscale skyrmions studied so far appears to be of the same order of
magnitude as that of non-topological magnetic structures of the same size.
This means that despite the apparent topological protection, small skyrmions
are quite unstable and can hardly exist at room temperature.

Theoretical model calculations of skyrmion lifetime have, however, shown that
it is in principle possible to identify material parameters giving stable
skyrmions at room temperature. Parameter values can be found that lead to
large enough energy barrier for skyrmion collapse and small enough pre-
exponential factor in the Arrhenius expression for the lifetime [11]. The
question remains whether materials can be found that correspond to these
parameter values. The manipulation of magnetic properties at atomic scale is a
challenging task and the possible range for parameter values in quite narrow.
On the micron scale for large skyrmions, effective parameters can be modified
more easily. For example, in layered structures containing ferromagnetic (FM)
layers and hard metal layers with large spin-orbit interaction, the effective
Dzyaloshinsky-Moriya interaction (DMI) can be varied in a controlled way by
the choice of materials and layer thickness. In such structures, skyrmion
states have been found to be stable even above room temperature [12, 13]. Such
modification of effective parameters that can lead to increased skyrmion
stability is not directly related to topological protection. Moreover, as the
number of magnetic moments in the skyrmion increases, the magnetic dipole
interaction becomes more important and some have argued that it is the dipole
interaction that is responsible for the observed stability of micron-scale
skyrmions [14]. Whether it is possible to have stability at room temperature
in thin layers with negligible dipole interaction is still an open question.
Furthermore, for skyrmions in antiferro- and ferrimagnets [15], as well as in
synthetic antiferromagnets [16], where the magnetic dipole-dipole interaction
is suppressed, the mechanism of stabilization still needs to be elucidated.

The stability of magnetic states depends not only on the activation energy of
the transitions the system can undergo from these states but also on the pre-
exponential factor in the Arrhenius expression for the rate of the magnetic
transitions which depends on the the shape of the energy surface in the
vicinity of the initial and transition states [17, 18]. The role of the pre-
exponential factor increases with increasing temperature. According to recent
theoretical studies, [11] this can be used to create small 2D skyrmions that
are stable at room temperature, although materials with the parameters
proposed in [11] have not yet been found. A change in the pre-exponential
factor by more than 30 orders of magnitude for small changes in the magnetic
field has been reported from experimental measurements [19].

When the size of a skyrmion is increased, the direction of neighboring
magnetic moments becomes more similar and the continuous magnetization model
becomes more appropriate for describing the system. How does topological
protection present in continuous models manifest itself in the limit of
infinitesimal lattice constant in discrete models? What limit does the energy
barrier for skyrmion annihilation and the skyrmion lifetime approach when the
ratio of the lattice constant and the skyrmion radius becomes infinitesimal?
These questions are addressed in the present article. We calculate the energy
of the skyrmion state, the minimum energy path between the metastable skyrmion
and the homogeneous FM state, and the activation energy of the skyrmion
collapse as well as escape through a boundary for a gradually decreasing
lattice constants while the skyrmion radius is kept constant. The pre-
exponential factor in the Arrhenius for the life time is estimated within the
harmonic approximation to transition state theory (HTST) [17] for several
values of the lattice constant. All parameters of the system, such as the
exchange and anisotropy constants as well as the DMI are chosen in such a way
as to keep the size of the skyrmion and its energy unchanged for small enough
lattice constant. The parameter values for the discrete lattice model are
scaled with the lattice constant in such a way as to be consistent with the
same continuous micromagnetic model. The calculated results can be interpreted
either in terms of trends when the lattice constant is decreased and the
skyrmion size remains the same, or in terms of increased skyrmion size if the
lattice constant is taken to be the same.

The article is organized as follows. First, in section 2. the model and the
computational methodology is described. Then, in section 3, the results are
presented along with discussions. Conclusions are given in section 4.

##  2 Model and methods

In this section the Hamiltonian of the systems studied is defined and the
computational methods used to estimate the energy barrier and pre-exponential
factor in the Arrhenius expression for the life time described.

###  2.1 Model

The energy of the magnetic system is described by a Heisenberg-type expression
for the energy of a two-dimensional (2D) square lattice

$E[{\bf S}]=-\sum_{<j,k>}(J_{jk}{\bf S}_{j}\cdot{\bf S}_{k}+{\bf
D}_{j,k}\cdot({\bf S}_{j}\times{\bf
S}_{k}))-K\sum_{j}{S^{2}_{j,z}}-\mu\sum_{j}{\bf B}\cdot{\bf S}_{j},$ (1)

where ${\bf S}_{i}$ is a unit vector giving the direction of the magnetic
moment at site i, $\mu$ is the value of magnetic moments taken to be the same
on all sites, $J$ is the exchange parameter that is non-zero only for nearest-
neighbour sites, and K is anisotropy parameter. The DMI vector ${\bf D}_{j,k}$
is taken to lie in the plane of the lattice and point in the direction along
the vector connecting atomic sites $j$ and $k$, thereby stabilizing a Bloch-
type skyrmion. Formally, the lattice constant does not enter in eqn. (1), but
when the same system is described with higher resolution, i.e. smaller lattice
constant, the parameter values corresponding to this system need to be changed
in a certain way. In order to achieve this, a connection between the
Heisenberg-type model and a micromagnetic model with continuous distribution
of magnetisation is used. There, the energy with respect to the homogeneous FM
state can be written as

$E=\int\omega({\bf r})d^{2}{\bf r},$ (2)

where

$\omega({\bf r})=\mathcal{A}{\|\nabla\bf S(r)\|}^{2}-{\mathcal{D}}({\bf
S(r)}\cdot\mathrm{rot}\bf S(r))-\mathcal{K}({\bf K^{0}}\cdot\bf
S(r))-\mathcal{M}({\bf B}\cdot\bf S(r))$ (3)

The integration is taken over the whole $\mathbb{R}^{2}$ space. The exchange
stiffness $\mathcal{A}$, DMI density $\mathcal{D}$, anisotropy density
$\mathcal{K}$ and magnetization $\mathcal{M}$ are proportional to the exchange
parameter $\it J_{jk}$, modulus of the DMI vector ${\bf D}_{j,k}$, anisotropy
constant $K$ and $\mu$ in (1), respectively. The coefficients of
proportionality depend on the type of crystal lattice, number of exchange
parameters used in lattice model, etc. While the correspondence between
continuous and lattice models is ambiguous in that different lattice
parameters can correspond to a given micromagnetic model, the scaling of
parameters in the discrete model is determined uniquely and depends only on
the dimensionality of the system. For a 2D system, micromagnetic parameter
values can be converted into parameter values for a discrete model of a square
lattice with lattice constant $h$ by

$J=2\mathcal{A},\quad D=h\mathcal{D},\quad
K=h^{2}\mathcal{K},\quad\mu=h^{2}\mathcal{M}.$

This gives a scaling law for the values of the parameters as the lattice
constant is changed in such a way that correspondence is maintained to the
same micromagnetic model

$D\left(h/N\right)=\frac{D(h)}{N},\quad
K\left(h/N\right)=\frac{K(h)}{N^{2}},\quad\mu\left(h/N\right)=\frac{\mu(h)}{N^{2}}$
(4)

while $J$ is independent of $N$. Within this framework, calculations were
performed for the following set of dimensionless parameters and $N=1$:
$D/J=0.35$, $K/J=0.16$, $\mu B/J=0.02$. The cell size, $l=30$ ($30\times 30$
spins subject to periodic boundary conditions) was chosen to be sufficiently
large compared to the skyrmion size to prevent the influence of boundary
conditions on the state of the skyrmion. In what follows, we will refer to $N$
as the scaling parameter, which denotes how the lattice constant is scaled.

In addition to the extended 2D system, calculations are also presented for a
magnetic track with finite width in one dimension. Periodic boundary
conditions are then only applied in the direction of the track while free
boundaries are then used in the perpendicular direction, along the edges of
the track. The escape of the skyrmion through the edge of the track is studied
also as a function of $N$.

###  2.2 Methodology

The lifetime of the skyrmion state is estimated within the framework of HTST
for magnetic degrees of freedom [17]. Within this approximation, the lifetime,
$\tau$, has an Arrhenius dependence on temperature

$\tau=\tau_{0}e^{\Delta E/k_{B}T}$ (5)

where $\Delta E$ is the activation energy given by the energy difference
between the saddle point and the initial state, $T$ is the absolute
temperature and $\tau_{0}$ is the pre-exponential factor related to the
entropy difference between the transition state and the initial state, as well
as the flux through the transition state. In a continuum description of the
system, a smooth transformation of the magnetization connecting the skyrmion
and FM states does not exist since these states have different topological
charge [4]. However, for a discrete lattice description, such a magnetic
transition is possible for any lattice constant, but there can be a
significant energy barrier for annihilation making the skyrmion state stable
for practical purposes. The question addressed here is how the activation
energy and the pre-exponential factor depend on the scaling factor, especially
in the limit of infinitesimal lattice constant which is the closest a discrete
model comes to the continuum model.

To study this dependence, the minimum energy path (MEP) between the skyrmion
and the FM state is calculated. The point of highest energy along the MEP, a
first order saddle point on the multidimensional energy surface given by eq.
(1), minus the initial state energy gives the activation energy for the
transition. For $N=1$ the dimensionality of the energy surface is already
high, 1800, and it increases as the square of the scaling parameter, $N^{2}$.
For small $N$, the MEP can easily be calculated using the geodesic nudged
elastic band method [20, 21]. When $N$ becomes large, this, however, becomes a
computationally demanding task. Since the shape of the path is known it is
possible to focus only on the region around the maximum and thereby reduce the
effort significantly [22]. In this way calculations can be carried out for
large values of $N$, even larger than $N=100$.

In HTST, the energy surface near the initial state and near the saddle point
on the energy surface corresponding to the transition state is approximated by
a quadratic expansions. Zero modes corresponding to degrees of freedom for
which the energy does not change significantly need to be treated separately.
The expression for the attempt frequency $f_{0}=1/\tau_{0}$ is [17, 23]

$f_{0}=\frac{1}{2\pi}(2\pi
k_{B}T)^{\frac{(P_{sk}-P_{sp})}{2}}\frac{V_{sp}}{V_{sk}}\sqrt{\sum\limits_{j=2}^{D-P_{sp}}{b_{j}^{2}}{\epsilon_{{sp},j}}}\frac{\prod\limits_{j=1}^{D-P_{sk}}{\sqrt{\epsilon_{{sk},j}}}}{\prod\limits_{j=2}^{D-P{sp}}{\sqrt{\epsilon_{{sp},j}}}}$
(6)

Here $\epsilon_{sk,j}$ and $\epsilon_{sp,j}$ are the eigenvalues of the
Hessian matrix for the initial skyrmion state and at the saddle point,
respectively, $V_{sk}$ and $V_{sp}$ give the volume corresponding to zero
modes while $P_{sk}$ and $P_{sp}$ the number of such modes. The summation for
the saddle point does not include the negative eigenvalue $\epsilon_{sp,1}$
and starts at $j=2$. The number $D$ in the upper limit of the summations is
equal to the dimension of the energy surface $2N^{2}l^{2}$. The variables
$b_{j}$ denote the expansion coefficients for the unstable mode at the saddle
point derived from the linearized Landau-Lifshitz equation of motion. More
precisely $b_{j}=\frac{\gamma}{\mu}\left({\bf e}_{sp,1}\cdot\left[{\bf
S}_{sp}\times{\bf e}_{sp,j}\right]\right)$, where $\gamma$ is the gyromagnetic
ratio, $\mu$ the length of the magnetic moments, ${\bf e}_{sp,1}$ and ${\bf
e}_{sp,j}$ are the unit vectors along the first eigenvalue (normal to dividing
surface) and the j-th eigenvector of the Hessian at the saddle point, and
${\bf S}_{sp}$ corresponds to the spin configuration at the saddle point [24].

First, consider the entropic contribution to the expression for $f_{0}$ with
the product of square roots of the positive eigenvalues of the Hessian in the
numerator (for skyrmion state) and denominator (for transition state)(6).
These expressions can be rewritten as

$\prod\limits_{i}^{D^{\ast}}\sqrt{\epsilon_{i}}=\exp\left(\frac{1}{2}\sum\limits_{i}^{D^{\ast}}\ln{\epsilon_{i}}\right)=\left[\exp\left(\frac{1}{D^{\ast}}\sum\limits_{i}^{D^{\ast}}\frac{1}{2}\ln{\epsilon_{i}}\right)\right]^{D^{\ast}}$
(7)

The upper limit $D^{\ast}$ in eq. (7) takes into account the absence of zero
modes at the saddle point and in the skyrmion state, as well as the negative
eigenvalue at the saddle point.

Recently, another algorithm for calculating the entropic contribution to the
pre-exponential factor has been presented [24]. This contribution can be
written via determinants of the Hessian at the minimum and saddle point. An
algorithm for calculating these determinants without searching for eigenvalues
of the Hessian but using recursive relations makes use of the sparse structure
of the Hessian matrix for energy in the Heisenberg-like energy expression when
only short-range interactions are included. The method makes it possible to
calculate determinants for systems containing millions of magnetic moments.
This makes it possible to calculate the pre-exponential factor and estimate
the lifetimes of micron-scale topological structures with atomic resolution
[25].

##  3 Results and discussion

Results are presented from calculations of both activation energy and pre-
exponential factor in the Arrhenius expression for the life time of the
skyrmion. Two annihilation mechanisms are considered, radial collapse within
an extended 2D system and escape through the edge of a track with finite
width.

###  3.1 Activation energy

Figure 1: Minimum energy path for radial skyrmion collapse calculated for
several values of the scaling parameter. The activation energy for collapse
increases as the lattice constant is decreased over this range of small values
of $N$ but reaches a large $N$ saturation limit at $N\approx 100$ where the
energy maximum is $4\pi J$ with respect to the ferromagnetic state which is
taken to give the zero of energy in each case.

Figure 1 shows the computed MEPs for radial skyrmion collapse for several
values of $N$. The path length represents the total change in orientation of
all the spins and therefore increases with $N$ since a larger number of spins
then forms the skyrmion structure. The energy of the initial skyrmion state
first decreases with $N$, but already at $N>2$ the change becomes
insignificant. The energy maximum along the MEP increases with scaling
parameter and is still increasing for $N=4$. But, calculations for much larger
values of $N$ show that at $N\approx 100$ and larger the maximum energy
changes insignificantly and approaches the lower bound of the energy of a
topological soliton given by the $\sigma$-model [26]. This can be understood
from the fact that, according to the scaling relations in eq. (4), all the
parameters of the expression for the energy in eq. (1), with the exception of
the exchange parameter $J$, tend to zero with increasing $N$. Nevertheless,
the contribution of various terms in eq. (1) to the energy of the skyrmion
state does not change since the number of magnetic moments that form the
skyrmion also increases with increasing $N$. However, the number of lattice
sites corresponding to noncollinear ordering at the saddle point is much
smaller than that of the skyrmion and it increases relatively slowly with
increasing $N$. Therefore, the transition state for large $N$ is a topological
soliton that can be described within the $\sigma$-model [27] which only
includes exchange interaction. In the continuous $\sigma$-model the energy $E$
of any structure with topological charge $q$ satisfies the inequality
$E\geqslant 4\pi Jq$ [26]. Consequently, the energy barrier for the nucleation
of a structure with $q=1$ cannot be lower than $4\pi J$. Our calculations show
that the energy barrier is nearly equal to this value for large $N$ and that
the barrier for the nucleation of a skyrmion from the FM state in the
continuous case can be estimated from the soliton energy in the
$\sigma$-model. The collapse of a skyrmion passes through a state
corresponding to this solution with a small radius (infinitesimal in the
continuous limit).

For a track of finite width, a skyrmion can also be eliminated by escape
through the boundary [6, 28, 29]. This mechanism corresponds to another path
on the energy surface.

Figure 2: Minimum energy path for skyrmion escape through the boundary of a
track with finite width for several values of the scaling parameter, $N$. The
activation energy reaches a limiting, constant value for small values of $N$.
The zero of energy is taken to be that of the final ferromagnetic state.

Figure 2 shows the MEP for skyrmion escape through the sample boundary for
several values of the scaling parameter. The width of the track for $N=1$
corresponds to 30 atomic rows, about 3 times larger than the size of the
skyrmion, wide enough for the boundaries not to have significant effect on the
skyrmion state. The energy barrier for escape first decreases slightly with
$N$ but then remains constant. Thus, the activation energy for this skyrmion
annihilation mechanism is nearly independent of scaling for $N>2$. Unlike the
collapse mechanism, escape can be described with a micromagnetic model [30].

###  3.2 Pre-exponential factor

Figure 3 shows calculated eigenvalues of the Hessian arranged in increasing
order. The horizontal axis gives the number of the eigenvalue in this sequence
divided by the total number of eigenvalues, $i/D^{\ast}$. This is a monotone
function, $\epsilon_{i}(i/D^{\ast})$, on the interval $[0,1]$, defined for a
discrete set of points.

Figure 3: Eigenvalues of the Hessian matrix for the initial skyrmion state
(left) and at the saddle point corresponding to the transition state (right)
as a function of $i/D^{\ast}$, where $i$ is a number of the eigenvalue when
they are arranged in increasing order. $D^{\ast}$ is the total number of
positive eigenvalues. The value of the scaling parameter $N$ for each curve is
shown.

As $N$ is increased, additional eigenvalues are introduced and the scaled
index $i/D^{\ast}$ becomes more densely distributed over the interval $[0,1]$.
As $N$ goes to infinity, the eigenvalues approach a limiting function
$\epsilon(x)$. In figure 3 the left and right panels correspond to the initial
skyrmion state and transition state respectively. In both cases, convergence
has been achieved for practical purposes already at $N=2$.

In the limit of large $N$, the expression in parentheses in the exponent in
eq. (7) can be regarded as a Riemann sum for a definite integral over the
function $\ln{\epsilon(x)}$ and, therefore,

$\frac{1}{D^{\ast}}\sum\limits_{i}^{D^{\ast}}\frac{1}{2}\ln{\epsilon_{i}}\stackrel{{\scriptstyle{D^{\ast}}\to\infty}}{{\longrightarrow}}\frac{1}{2}\int\limits_{0}^{1}\ln{\epsilon(x)}dx.$
(8)

An expression of the form (8) enters both in the numerator (6) at the point
corresponding to the skyrmion state, as well as into the denominator of eq.
(6) at the saddle point.

If the integrals over $\ln\epsilon(x)$ at the minimum and at the saddle point
are different, an exponential dependence of the entropic contribution to
$f_{0}$ on the scaling parameter $N$ is obtained. For relatively small value
of $D^{\ast}$, the density of the eigenvalues of the Hessian has been analyzed
previously [31], where the importance of accurate consideration of close to
zero eigenvalues was emphasized. The case of large $D^{\ast}$ is
computationally challenging since the calculation of the function
$\epsilon(x)$ presupposes the knowledge of all eigenvalues of a matrix with a
rank of several million with high accuracy already for $N\approx 100$.

The results presented here show that the entropy term in the pre-exponential
factor for the lifetime, $\tau_{0}$, first increases as the scaling parameter
$N$ increases, but then approaches a constant value for $N>30$. The increase
in the entropy factor corresponds to five orders of magnitude while the
dynamic term, containing the $b_{j}$ in eq. (6), changes by less than one
order of magnitude. A strong increase in the pre-exponential factor with
increasing skyrmion size explains the possible stability of micrometer
skyrmions at room temperature even without taking into account the magnetic
dipole interaction and can be considered as a manifestation of topological
stabilization in the limit of an infinitesimal lattice constant.

The pre-exponential factor corresponding to skyrmion escape through the
boundary also changes with the scaling parameter $N$, but only by two orders
of magnitude, and already for $N>4$ it is practically constant. In contrast to
the case of collapse in an extended system, the escape involves unequal number
of zero modes at the minimum and at the saddle point (two for the skyrmion but
only one for the transition state). The pre-exponential factor is, therefore,
temperature dependent. The variation of the lifetime due to escape with $N$ is
much smaller than for skyrmion collapse inside the sample.

##  4 Conclusions

The activation energy and pre-exponential factor in the Arrhenius expression
for the life time of a skyrmion is estimated within HTST for a set of
decreasing values of the lattice constant while keeping the size of the
skyrmion fixed. In the limit of infinitesimal lattice constant, the activation
energy reaches a finite value, unlike the topological protection implied by a
continuum model. The energy of the transition state approaches the Belavin-
Polyakov limit of $4\pi J$ with respect to the ferromagnetic state. An
efficient method for calculating the pre-exponential factor for transitions in
large systems is used where the eigenvalues of the Hessian need not be
evaluated. The calculated results demonstrate that the entropy term in the
pre-exponential factor decreases as the lattice constant is decreased but also
approaches a constant value. The results can also be interpreted in terms of
changing skyrmion size for a fixed lattice constant and then demonstrate how
the decrease in the pre-exponential factor with increased skyrmion size makes
an important contribution to the stability of micrometer skyrmions even in the
absence of magnetic dipole interaction.

Acknowledgments

This work was supported by Russian Science Foundation (Grant 19-42-06302), the
Icelandic Research Fund, and the Research Fund of the University of Iceland.

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