Monte Carlo studies of skyrmion stabilization under geometric confinement and uniaxial strain

Abstract

Geometric confinement (GC) of skyrmions in nanodomains plays a crucial role in skyrmion stabilization. This confinement effect decreases the magnetic field necessary for skyrmion formation and is closely related to the applied mechanical stresses. However, the mechanism of GC is unclear and remains controversial. Here, we numerically study the combined effects of GC and uniaxial strain on skyrmion stabilization and find that zero Dzyaloshinskii-Moriya interaction (DMI) coupling constants imposed on the boundary surfaces of small thin plates causes a confinement effect, stabilizing skyrmions in the low-field region. Moreover, the confined skyrmions are stabilized further by tensile strains parallel to the plate, and the skyrmion phase extends to the low-temperature region. This stabilization occurs due to the anisotropic DMI coupling constant induced by the tensile strain caused by lattice deformations. Our simulation data are qualitatively consistent with experimentally observed and reported stabilization induced by tensile strains applied to the thin plate of the chiral magnet ${\rm Cu_2OSeO_3}$.

Full text

Monte Carlo studies of skyrmion stabilization under geometric
confinement and uniaxial strain
G. Diguet∗
Micro System Integration Center, Tohoku University, Sendai, Japan
B. Ducharne
INSA Lyon, Universite de Lyon, Villeurbanne Cedex, France and
ElyTMax, CNRS-Universite de Lyon-Tohoku University, Sendai, Japan
S. El Hog
Universit´e de Monastir (LMCN), Monastir, Tunisie
F. Kato and H. Koibuchi†
National Institute of Technology (KOSEN),
Ibaraki College, Hitachinaka, Japan.
T. Uchimoto
Institute of Fluid Science (IFS), Tohoku University, Sendai, Japan and
ElyTMax, CNRS-Universite de Lyon-Tohoku University, Sendai, Japan
H. T. Diep‡
CY Cergy Paris University, Cergy-Pontoise, France
1
arXiv:2208.07527v1 [cond-mat.str-el] 16 Aug 2022

Abstract
Geometric confinement (GC) of skyrmions in nanodomains plays a crucial role in skyrmion stabi-
lization. This confinement effect decreases the magnetic field necessary for skyrmion formation and
is closely related to the applied mechanical stresses. However, the mechanism of GC is unclear and
remains controversial. Here, we numerically study the combined effects of GC and uniaxial strain
on skyrmion stabilization and find that zero Dzyaloshinskii-Moriya interaction (DMI) coupling
constants imposed on the boundary surfaces of small thin plates causes a confinement effect, sta-
bilizing skyrmions in the low-field region. Moreover, the confined skyrmions are stabilized further
by tensile strains parallel to the plate, and the skyrmion phase extends to the low-temperature
region. This stabilization occurs due to the anisotropic DMI coupling constant induced by the
tensile strain caused by lattice deformations. Our simulation data are qualitatively consistent with
experimentally observed and reported stabilization induced by tensile strains applied to the thin
plate of the chiral magnet Cu2OSeO3.
∗gildas.diguet.d4@tohoku.ac.jp
†koibuchi@gm.ibaraki-ct.ac.jp; koibuchih@gmail.com
‡diep@cyu.fr
2

I. INTRODUCTION
Stabilization/destabilization of skyrmions [1–3] is a key target for future technological
applications [4–9]. The magnetic field Bplays a key role in skyrmion stabilization. Me-
chanical stresses and strains also strongly influence skyrmion stability [10, 11]. As a result,
various experimental and theoretical studies have been conducted on skyrmion stability [12–
15]. Nii et al. reported that skyrmions in MnSi are stabilized (destabilized) by compressions
perpendicular (parallel) to the magnetic field, improving the understanding of their cre-
ation/annihilation mechanism [16]. Charcon et al. reported that the area of the skyrmion
phase in the BT phase diagram increases/decreases depending on the compression direction,
where Tis the temperature [17].
On the deformation of skyrmions by mechanical strains, Shibata et al. reported that
skyrmions on thin FeGe films deform as oblong shapes along the direction of the tensile
stress [18]. Mechanical stresses are important in this phenomenon [19–22], and the shape
deformation was successfully simulated by assuming suitable magnetoelastic coupling terms
[23–25]. In addition, skyrmion deformation was numerically obtained in two-dimensional
simulations by assuming direction-dependent or anisotropic Dzyaloshinskii-Moriya interac-
tion (DMI) coefficients, as discussed in Ref. [18]. This effect was also predicted from a
quantum mechanical perspective [26]. Moreover, this shape deformation phenomenon was
simulated with the Finsler geometry (FG) modeling technique without assuming magnetoe-
lastic coupling [27].
We should note that this shape deformation of skyrmions in FeGe has the same mechanism
as skyrmion stabilization/destabilization in MnSi [16, 17]. Because tensile stress deforms
skyrmions from round to oblong shapes on FeGe perpendicular to the magnetic field, shape
deformation is considered a destabilization. Moreover, the experimental results on stabiliza-
tion/destabilization reported in [16, 17] and those on the stripe direction reported in [28],
which is compatible with skyrmion shape deformation [18], were numerically reproduced by
a single FG skyrmion model under the assumption that extensions (compressions) in one
direction correspond to compressions (extensions) in the perpendicular directions [29], where
the anisotropy in the DMI coefficient is geometrically implemented. Anisotropic coupling
constants have also been shown to stabilize skyrmions [30–33].
Another stabilization mechanism is the geometric confinement (GC) effect studied in
3

Ref.[34], which assumes magnetization anisotropy and a constant DMI coefficient. A GC
effect was experimentally observed in a nanostripe of FeGe [35], and morphological changes
in skyrmions as the nanostripe width varied were reported in [36]. Skyrmion bubbles in
centrosymmetric magnets are also influenced by GC effects [37], where the applied magnetic
field decreases with decreasing nanostripe width, indicating stabilization. Ho et al. reported
that confined skyrmions are stabilized in multilayered nanodomains [38].
Recently, Wang et al. reported experimental data on a switching mechanism for individual
skyrmions in nanodots with diameters ranging from 150(nm) to 1000(nm) [39]. In this
case, a magnetic field ~
Bwas applied perpendicular to the disk and a variable tensile strain
was electrically applied in the radial direction via a substrate. The reported data show
that skyrmions are confined in the nanodots and that the magnetic field ~
Bdecreases as
the diameter decreases. This reduction in ~
Bas the diameter changes is expected to be a
consequence of both GC and magnetoelastic effects [39].
Seki et al. reported experimental data on the dependence of the direction of the magnetic
field ~
Bon the small thin plate of the chiral magnet Cu2OSeO3, where the thickness of
the specimen is 1(µm) [40]. The reported data show remarkable stabilization when ~
Bis
perpendicular to the strain direction and parallel to the plate surface. We should note that
Cu2OSeO3is stabilized by extensions perpendicular to the magnetic field, while MnSi in Refs.
[16, 17] is stabilized by compressions perpendicular to the magnetic field. These observations
indicate that the response of Cu2OSeO3differs from those of MnSi and FeGe, at least in
the case of mechanical strains, because FeGe in Ref. [18] is expected to be destabilized by
compressions parallel to the magnetic field under the abovementioned equivalence between
compression and tension.
Interestingly, the stabilization in Cu2OSeO3by applying tensile strains is significantly
improved only when ~
Bor the skyrmion tubes (or strings) are parallel to the surface, where
the thickness 1(µm) of the plate is not large when compared with the skyrmion diameter
(∼100(nm)). Hence, skyrmions are expected to be stabilized when they are confined be-
tween narrow plates, with strong surface effects also expected. However, this stabilization
enhancement is not indicated by the combined effects of strains and GC.
In this paper, we perform Monte Carlo simulations of the GC effect for skyrmions in a
3D lattice discretized by tetrahedrons, carefully investigating the effects of DMI coefficients
on the stabilization. In the simulation model, we assume DMI coefficients of zero on the
4

boundary surfaces due to the surface effects of the 3D lattice and show that this assumption
notably improves skyrmion stability. The effects of strains on the stabilization are also
investigated by assuming lattice deformation corresponding to tensile deformation without
the magnetoelastic coupling terms in the Hamiltonian. The GC and magnetoelastic effects
both are studied with a model with heterogeneous DMI coefficients that effectively become
anisotropic due to strain effects. We should note that stabilization/destabilization caused
by strains was successfully simulated by assuming anisotropic DMI constants corresponding
to higher-order derivatives in [41]. Our results on the direction dependence of DMI are
consistent with those reported in [41].
II. BACKGROUND
A. Geometric confinement of skyrmions
(a) (b)
c-sky nc-sky c-sky
FIG. 1. Illustrations of a (a) nonconfined skyrmion (nc-sky) configuration and a (b) confined
skyrmion (c-sky) configuration in a small domain. The red circles in (a) enclose skyrmions touching
the boundary, which are nc-sky and unstable. In Ref. [44], all skyrmion configurations were
reported to be confined, as in (b), and no nonconfined configurations, as in (a), were reported. The
magnetic field was applied along the zdirection.
First, we briefly introduce the GC of skyrmions using illustrations (Figs. 1(a), (b)). Y.
Wang et al. recently reported experimental results on skyrmions confined in small disks or
nanodots with diameters ranging from 150 ∼1000(nm), with the magnetic field applied per-
pendicular to the nanodots. A substrate was electrically deformed to influence the skyrmion
5

configurations in the nanodots. In this small domain, surface effects are expected to be
strong [39]. In fact, no nonconfined skyrmions were reported in Ref. [39]. Figures 1(a) and
(b) show illustrations of nonconfined and confined skyrmions in a small domain. The main
target of the study in Ref. [39] was not the GC effect but electric field-driven switching
among individual skyrmions; however, the results indicate that GC is closely connected to
this switching.
B. Dzyaloshinskii-Moriya interaction-dependent confinement
(a) (b)


FIG. 2. (a) 3D lattice of size (N, NB, NT, Ntet) = (14548,100313,167930,82164) discretized by
tetrahedrons, on which two models were simulated in Ref. [44] by using the MMC technique
[42, 43]. The symbols denoted by N, NB, NTand Ntet are the total numbers of vertices, bonds,
triangles, and tetrahedrons, which satisfy the condition N−NB+NT−Ntet =1, which is the same
condition as in the tetrahedron (N, NB, NT, Ntet ) = (4,6,4,1). (b) Boundary surfaces denoted by
Pxand Py, with the normals perpendicular to the magnetic field direction ~
B= (0,0,−B). This
lattice is also used in the following section.
Next, we show simulation results of two different types of skyrmion configurations re-
ported in Ref. [44]: confined skyrmions (c-sky) and nonconfined skyrmions (nc-sky). More-
over, we discuss the reason for the emergence of these two different configurations by clar-
ifying a difference in the two models studied in [44]. In Ref. [44], Metropolis Monte Carlo
(MMC) simulations [42, 43] were performed on a three-dimensional (3D) lattice of a cube
(Fig. 2(a)), with the magnetic field applied along the zdirection, as shown in Fig. 2(b).
Snapshots of Neel-type skyrmions, including nonconfined and confined configurations, are
shown in Figs. 3(a) and (b). The skyrmions enclosed by the red circles in Fig. 3(a) were
6

(a) (b)
c-skync-sky
Neel type
Neel type
FIG. 3. Snapshots of Neel-type skyrmion configurations with (a) nonconfined (nc-sky) and (b)
confined (c-sky) skyrmion. The skyrmions in (a) enclosed by red circles touching the boundary Px
(Fig. 2(b)) are reported to be unstable, and all skyrmions in (b) confined inside the boundary are
stable [44].
reported to be unstable, with some of the skyrmions touching the boundary disappearing
and new skyrmions emerging on the boundary after long MC simulations [44]. As a result,
the positions of the nonconfined skyrmions may fluctuate or change, while the positions of
the confined skyrmions remain constant.
The Hamiltonian corresponding to the configurations in Ref. [44] is given by
S=λSFM +DSDM −SB,
SFM =X
ij
(1 −~σi·~σj),
SB=X
i
σi·~
B, ~
B= (0,0,−B),
(1)
where the ferromagnetic interaction (FMI) and Zeeman energies SFM and SBhave the same
expression in the two models corresponding to the snapshots in Figs. 3(a) and (b). The
models differ in terms of their DMI energies SDM, which are given by
SDM =X
ij
(~eij ×~ez)·(~σi×~σj),(Fig.3(a)),
SDM =X
ij
nij(~eij ×~ez)·(~σi×~σj),(Fig.3(b)),
~eij =ex
ij, ey
ij, ez
ij,|~eij|= 1,
(2)
7

where ~eij is the unit vector from vertices ito j,~ez(= (0,0,1)) is the unit tangential vector
along the zaxis indicating the magnetic field direction, and nij corresponds to the total
number of tetrahedrons sharing the bond ij with a normalization factor. The factor nij in the
model in Fig. 3(b) appears depending on the discretization of the continuous Hamiltonian,
and more detailed information is given in Appendix A. These Hamiltonians correspond to
Neel-type skyrmions [45–48].
(a) (b)
boundary inside
periodic b.c.
boundary inside
free b.c.
   
FIG. 4. Illustrations of the position dependence of nij on the lattice in (a) periodic boundary
conditions and (b) free boundary conditions, where nij is the total number of squares sharing the
bond ij with a normalization factor. The normalization factor in nij is fixed to 1 in this case for
simplicity. In (a), nij is independent of the position of ij and is given by nij = 2, while in (b),
nij =1 on the boundary and nij = 2 inside the boundary.
We should note that nij is not uniform over the lattice and emphasize that nij has
a smaller value on the surfaces than inside the boundary. This position dependence or
inhomogeneity in nij is clear because the bonds ij on the boundary surfaces are shared
among fewer tetrahedrons than those inside the boundaries. To clarify this inhomogeneity,
Figs. 4(a) and (b) show cases of the 2-dimensional model corresponding to periodic and free
boundary conditions on a regular square lattice, with the normalization factor fixed to 1 for
simplicity. In the periodic boundary model, all bonds are shared by two neighboring squares;
hence, we have nij = 2, which is a constant and can thus be removed from the Hamiltonian
by redefining the coupling constant Das 2D→Dor fixing the normalization factor to 1/2.
Therefore, SDM in the periodic boundary model is given by the first expression in Eq. (2)
indicated by (Fig. 3(a)). In contrast, in the free boundary model, nij is not constant and
8

depends on the position of bond ij. Figure 4(b) shows that nij = 1 (nij = 2) on (inside) the
boundary. This result implies that SDM in the free boundary model is given by the second
expression in Eq. (2) indicated by (Fig. 3(b)).
In the case of the 3D lattice shown in Fig. 2 (a), nij (see Appendix A) varies depending
on the position; however, the mean values of nij on the boundary surface and inside the
boundary are considerably different. To show this difference, we calculate effective DMI
coupling constants by including the factor nij, as follows:
DS
x=1
Pij∈Py1X
ij∈Py
nij|ey
ij|,
DS
y=1
Pij∈Px1X
ij∈Px
nij|ex
ij|
DV
x,y =1
Pij∈V\S1X
ij∈V\S
nij|ey,x
ij |, S =Px∪Py∪Pz
(3)
where Pij∈Px,y 1 denotes the total number of bonds on surfaces Pxor Py(Fig. 2(b)). Note
that these constants do not depend on the spins or magnetic field and depend only on the
lattice structure. The reason why we call this “effective coupling” is that the DMI energy
responds to both lattice deformations and surface effects, and these responses are reflected
in the constants. The inclusion of |ex,y
ij |is natural because the DMI energy depends on lattice
deformations.
More specifically, the DMI energy
(~eij ×~ez)·(~σi×~σj) = ey
ij(~σi×~σj)x−ex
ij(~σi×~σj)y(4)
at the bond ij depends on the distribution of ~eij , which is expected to vary as the lattice
deforms. Therefore, in the DMI expression shown in Eq. (4), the direction or components
of ~eij effectively play a role in the direction-dependent microscopic DMI coupling constants.
The constants DS,V
µin Eq. (3) are the mean values and considered to be direction-dependent
macroscopic DMI coupling constants. The factor nij depends on whether the bond ij is
located on the boundary surfaces; therefore, nij is included in the expression of DS,V
µ. Thus,
we can observe how surface effects are reflected in the DMI energy by investigating DS,V
µ.
The superscripts Sand Vin DS
µand DV
µindicate that the constants are defined on and
inside the boundary surfaces, respectively, and Pij∈V\Sin DV
µdenotes the sum over the
bonds ij in the whole volume Vminus the boundary surfaces S=Px∪Py∪Pz. The reason
why Pzis not included in the definition of DS
x,y is that the boundaries parallel to ~
B, namely,
9

Pxand Py, are considered to be relevant to the confinement of skyrmion strings that appear
parallel to Pxand Py.
0
1
2
D:Dx
:Dy
nij=1 nij>1
S
(a)
0
1
2
D
:Dx
:Dy
nij=1 nij>1
V
(b)
FIG. 5. Effective coupling constants (a) DS
µ(µ=x, y) and (b) DV
µ(µ=x, y), which are defined in
Eq. (3). The x, y axes are shown in Fig. 2(b). The data on the left side with the notation nij = 1
in both (a) and (b) are obtained under the condition nij = 1 in Eq. (3) and correspond to the
model shown in the first of Eq. (2) denoted by (Fig. 3(a)). The data on the right side enclosed
by the red circles correspond to the model shown in the second of Eq. (2) denoted by (Fig. 3(b)),
with nij >1 indicating that DS
µis approximately 3 times smaller than DV
µ, implying that DMI
coefficients on the boundary surfaces are substantially smaller than those inside the boundaries.
Figures 5(a) and (b) show DS
µand DV
µ, which are defined in Eq. (3). The data denoted by
nij =1 correspond to the model described by the first SDM in Eq. (2). Note that DS
µ(µ=x, y)
can be calculated directly, with (1/R2π
0θdθ)R2π
0θ|cos θ|dθ = (1/R2π
0θdθ)R2π
0θ|sin θ|dθ =
2/π '0.64, which is almost the same as DS
µwhen nij =1 in Fig. 5(a). DV
µ(µ=x, y) can also
be calculated by three-dimensional integrations, such that DV
µ= 0.5 [44], and the results in
Fig. 5(b) for nij = 1 are identical to this value.
These calculations and the data plotted in Figs. 5(a) and (b) show that DS
µand DV
µ
are almost the same or that DS
µis only slightly larger than DV
µwhen nij = 1. In contrast,
the data enclosed by the red circles on the right side of the figure correspond to the model
described by the second SDM in Eq. (2) and indicate that DS
µis almost three times smaller
than DV
µ. This observation indicates that the numerically obtained confinement shown in
Fig. 3(b) is caused by a large difference in the DMI coupling constants on the boundary
surfaces and inside the boundaries.
Finally in this subsection, we comment on why DS
µand DV
µin Eq. (3) are considered to
10

be effective coupling constants for DMI. The original coupling constant Dhas a constant
value and is independent of external stimuli such as mechanical stress and strain. However,
it is reasonable to consider that DMI coupling constants are influenced by strains because
strains are generally connected to deformations of atom arrangements. If we start with the
DMI energy in Eq. (2) or Bloch type energy, which is considered in the following section, the
strains directly influence ~eij included in the energy. This strain-induced DMI modification
can also be applied to a case in which the DMI depends on the position and varies with the
lattice structure itself. Indeed, it is possible to include nij in DS
µand DV
µbecause nij depends
on the position in the lattice, namely, whether the position is on or inside the boundaries,
although this value remains unchanged by strains. Interestingly, this position dependence
of DMI is naturally implemented as a surface effect by a suitable discretization technique
without any assumptions, as mentioned above (Appendix A).
III. SIMULATION RESULTS
A. Geometric confinement model
This section presents the main results of this paper. In the previous section, we confirmed
that skyrmions are confined in small domains if the surface DMI coefficient is substantially
smaller than the bulk DMI coefficient. If the DMI coefficient is small on surfaces such as Px
and Pyin Fig. 2(b), skyrmions cannot appear on Pxand Pyand are thus confined inside the
domain boundary. Based on this observation, in this paper, we propose a model in which
the DMI coefficient is set to zero on the boundary surfaces parallel to the magnetic field ~
B,
which is applied along the ydirection, as shown in Fig. 6(a).
To observe the effect of a zero DMI coefficient on the boundary surface, we also study a
standard model, with SDM defined on the bonds and no DMI position dependence. Thus,
two different models, model A and model B, are studied in this paper. The total Hamiltonian
11

(a) (b)








  









   



tensile stress
FIG. 6. (a) DMI coefficients are fixed at zero on the boundary surfaces Pyand Pzparallel to the
magnetic field ~
Balong the ydirection, and (b) a lattice deformation characterized by εxby a
tensile stress along the xdirection, where εx=0 and εx= 0.02 are assumed in the simulations, and
the volume remains unchanged such that LxLyLz(εx= 0.02) =L0
xL0
yL0
z(εx=0) for simplicity.
S, which is the same as that in Eq. (1), and the energy terms are defined as follows:
S=λSFM +DSDM −SB,
SFM =X
ij
(1 −~σi·~σj), SB=X
i
σi·~
B, ~
B= (0,−B, 0),
SDM =X
ij
~e 0
ij ·(~σi×~σj),(model A),
SDM =X
ij
Γij~e 0
ij ·(~σi×~σj),Γij =


0 (ij ∈Py∪Pz)
1 (otherwise) ,(model B),
~e 0
ij =(1 + εx)ex
ij,ey
ij
√1 + εx
,ez
ij
√1 + εx.
(5)
The DMI energy SDM of model A differs from that of model B, while SF M has the same
expression in both models. The SDM expressions in models A and B are of Bloch type,
in contrast to the cases defined in Eq. (2) for Figs. 3(a) and (b). The ~
Bdirection also
differs from that in the case defined in Eq. (1). Moreover, the unit vector ~eij along bond
ij is replaced by ~e 0
ij, which is generally not of unit length when εx6= 0, to implement
strain effects. The reason why ~eij is replaced by ~e 0
ij is that ~eij originally corresponds to a
tangential vector ∂~r/∂x along bond ij, and ∂~r/∂x is not always of unit length in general
scenarios (Appendix A).
The partition function Z is given by
Z=X
~σ
exp(−S(~σ)/T ),(6)
12

where P~σ denotes the sum over all possible spin configurations ~σ ={~σ1, ~σ2,··· , ~σN},~σi∈
S2(unit sphere). The simulation unit is given by kB= 1 and a= 1, where kBand aare the
Boltzmann constant and lattice spacing, respectively. The lattice spacing acorresponds to
the mean bond length in the lattice. The mean bond length corresponds to the distance
between two neighboring atoms in a coarse-grained approach; hence, the vertices do not
always represent real atoms but instead may represent lumps of atoms, as in other lattice
models [49].
A tensile strain is applied along the xaxis of the lattice in model B with the parameter
εx(≥0) such that the original side lengths (L0
x, L0
y, L0
z) change to
(L0
x, L0
y, L0
z)→(Lx, Ly, Lz) = (1 + εx)L0
x,L0
y
√1 + εx
,L0
z
√1 + εx,(εx≥0).(7)
This deformation condition ensures that the lattice volume remains unchanged, as shown in
Fig. 6(b). Under this condition, a tensile stress along the xaxis is equivalent to compressive
stresses along the yand zaxes, as discussed in Ref. [29]. We should note that magnetoelastic
terms are not included in S; instead, the DMI coefficients effectively become direction- and
position-dependent due surface effects caused by Γij and strain effects caused by the lattice
deformation. Detailed information is provided in the following subsections.
Here, we comment on the MC simulations. For the initial states of the MC simulations,
the ground states are generated by the technique proposed in Ref. [50]. To generate equilib-
rium configurations of the spin variable, the Metropolis MC technique [42, 43] is used, as in
Ref. [44]. The total number of MC sweeps (MCSs) is set to 1 ×108or 2 ×108, with 2 ×108
MCSs performed in the skyrmion phase and phase boundaries and 1 ×108MCSs performed
in the other phases.
B. Effect of zero DMI on the boundary surfaces
First, we show how the zero DMI condition stabilizes skyrmions by investigating snapshots
of models A and B in Figs. 7(a)-(c) and Figs. 7(d)-(f), respectively. The parameters are
noted in the figures and captions. The snapshots in the upper row show that stable skyrmions
do not occur in model A in those parameter regions, and the snapshots in the lower row
show that these unstable skyrmions change to clearly separated stable skyrmions.
To better understand the difference between models A and B, we plot a phase diagram
13

(d) (e) (f)
model A
model B
𝑇 = 1, 𝐵 = 0.4 𝑇 = 1, 𝐵 = 0.6 𝑇 = 1.4, 𝐵 = 0.8
model A model A
model B model B
(a) (b) (c)
Bloch Bloch
Bloch
FIG. 7. Snapshots of Bloch-type skyrmions in model A obtained with the parameters (a) (T , B)=
(1,0.4), (b) (T, B) = (1,0.6), and (c) (T , B) = (1.4,0.8). (d), (e) and (f) show snapshots of model
B obtained with the same parameters. The parameters (λ, D) are fixed to (λ, D)=(1,0.9) in both
models. The only difference between the models is the DMI coefficient on the boundary surfaces Py
and Pz, as shown in Eq. (5). Skyrmions are visualized by showing only spins ~σ with σy≥0, where
the magnetic field ~
B= (0,−B, 0) is in the negative ydirection. The yaxis is shown in Fig. 6(a).
Skyrmions are considerably stabilized in model B by surface effects, with εx= 0. The diameter of
the skyrmion strings decreases with increasing |B|, while the distance between the strings remains
unchanged, as expected.
of Bvs. Tand the corresponding snapshots of both models in the ranges −0.2≥B≥ −1.0
and 0.2≤T≤3.4. The parameters (λ, D) in Figs. 8(a)–(h) are fixed to (λ, D) = (1,0.9),
similar to those in Figs. 7(a)–(f). The BT phase diagram of model A is plotted in Fig. 8(a),
and snapshots of this model are shown in Figs. 8(b)–(h). In model A, the skyrmion phase
can be divided into confined skyrmion (c-sky) and nonconfined skyrmion (nc-sky) phases,
which correspond to Fig. 8(g) and Fig. 8(h), respectively. The nonconfined skyrmions in
Fig. 8(h) have oblong shapes along the zdirection, which is consistent with the experimental
data presented in Refs. [35, 36]. The stripe phase is also observed and can be divided into
confined and nonconfined phases. A nonconfined stripe configuration is shown in Fig. 8(e),
which is denoted as “stripe”. For a sufficiently large |B|, we observe a forced ferromagnetic
phase, which is denoted as “ferro”, as well as a paramagnetic phase, denoted as “para”, for
14

0 1 2 3
-1
-0.5
:c-sky
:nc-sky
:c-skfe
:nc-skfe
:stripe
B
T
model A
:ferro
:para
:c-skpa
:nc-skpa
:st-para
(a)
(b)
(c)
(e)
(f)
(g)
(h)
(d)
FIG. 8. (a) BT phase diagram of model A, with (b)–(h) showing snapshots of the upper view
(upper part) and side view (lower part). The upper view in (b) shows that the skyrmions are
incomplete, which is denoted as the “confined skyrmion ferromagnetic” (c-skfe) phase. (c) “c-
skfe”: only one skyrmion string is present; (d) “nc-skfe”: no complete skyrmion string is present
and some of incomplete skyrmions touch the upper wall; (e) “stripe”: three stripes touch the upper
and lower walls; (f ) “para”: the spin directions are essentially or almost random; (g) “c-sky”: three
confined skyrmion strings are present; and (h) “nc-sky”: three nonconfined skyrmion strings are
present.
sufficiently high T. In these ferro and para phases, no differences are observed between the
confined and nonconfined phases. We also observe intermediate phases between these four
pure phases, which can also be divided into confined and nonconfined phases when one of the
pure phases is sky or stripe. Thus, the BT phase diagram includes many symbols; however,
several configurations are shown as snapshots, clarifying interesting behaviors corresponding
to the confined and nonconfined phases. Remarkably, the BT phase diagram in Fig. 8(a)
shows that the c-sky phase appears only in a region with a small area in the central part of
the diagram.
Next, we show the results obtained by model B with εx= 0 in Figs. 9(a)–(h). The BT
15

0123
-1
-0.5
:c-sky
:nc-sky
:c-skfe
:nc-skfe
:c-stripe
B
T
model B
:ferro
:para
:c-skst
:nc-skpa
:st-para
=0
(a)
(b)
(c)
(e)
(f)
(g)
(h)
(d)
FIG. 9. (a) BT phase diagram of model B, with (b)–(h) showing snapshots of the model at several
points. The area of the confined skyrmion phase (c-sky) in (a) is significantly larger than that in
Fig. 8(a). Skyrmion strings are not always complete in the confined sk-fe (c-skfe) phase in (b),
(c), and (f), and three skyrmion strings that do not touch the walls are found in the c-sky phase in
(g) and (h). A confined skyrmion stripe (c-skst) and confined stripe (c-stripe) appear in (d) and
(e), respectively. The stripes in (d) and (e) are both in the xdirection, in contrast to the stripe
configuration in Fig. 8(e), which is in the zdirection.
phase diagram in Fig. 9(a) shows that the area of the c-sky phase is significantly larger than
that in model A in Fig. 8(a) on both sides of the higher and lower regions on the Taxis
and both directions on the Baxis. Moreover, no nonconfined phase, denoted by “nc-∗∗”,
is observed, at least in the region of T≤3, in this case. If we compare the snapshots in
Figs. 8(d) and 9(d), which are both obtained at (B , T )=(−0.4,0.6), it is clear that the spin
configuration in model A is considerably stabilized in model B due to surface effects caused
by Γij in Eq. (5). The nc-sky states in Fig. 8(h) also change to c-sky states in Fig. 9(h), and
the shape of the skyrmions changes from oblong to round. This shape change occurs due
to the surface effect introduced by Γij. One additional point that should be noted is that
the stripe configurations in Fig. 8(e) at B=−0.2 change to confined stripe configurations,
16

denoted by “c-stripe”, in Fig. 9(e). This change occurs due to the same surface effect.
C. Stabilization by tensile strain
0 1 2 3
-1
-0.5
:c-sky
:nc-sky
:c-skfe
:nc-skst
:stripe
B
T
model B
:ferro
:para
:c-skpa
:nc-skpa
:st-para
=2%
(a)
(b)
(c)
(e)
(f)
(g)
(h)
(d)
FIG. 10. (a) BT phase diagram of model B under a tensile strain of εx=0.02, with (b)–(h) showing
snapshots of the model at several points. The area of the confined skyrmion phase (c-sky) in (a)
is larger than that in Fig. 9(a). Skyrmion strings are complete in (c) and (d) in contrast to those
in Figs. 9(c) and (d). Some of the stripes in (e) do not touch the upper and lower boundaries and
are confined, and the stripe direction also changes to vertical from horizontal in Fig. 9(e). These
changes are caused by the combined effect of strains and GC.
In this subsection, we show the results of model B obtained under a small strain of
εx=Lx/L0
x−1 = 0.02 along the xaxis (see Fig. 6(b) and Eq. (7)). The BT phase diagram
and snapshots are shown in Figs. 10(a)–(h). In this case, a nonconfined skyrmion phase
does not appear. Moreover, the confined skyrmion phases at B=−0.4 and B=−0.6 extend
to the low Tregion, including the lowest temperature of T= 0.2. The stripe direction in Fig.
10(e) is along the zdirection, which differs from the ydirection in Fig. 9(e), and returns to
the same direction as in model A, as shown in Fig. 8(e). We should note that the stripes in
17

Fig. 10(e) are partly or almost entirely confined, and this behavior differs from that in Fig.
8(e), where the stripes are not confined. When compared with Figs. 9(e) and 8(e), these
changes in Fig. 10(e) are caused by the combined effect of strains and GC.
To see the origin of these morphological changes or strain-induced stabilization, we cal-
culate the effective DMI coupling constants DS
µand DV
µ, which have similar definitions to
those shown in Eq. (3). Because the direction of the magnetic field ~
Bchanges in this case,
the definition of DS
µalso changes. Hence, we formulate the new definitions as follows. First,
DS
µin model A is given by
DS
x=1
Pij∈Pz1X
ij∈Pz|e0x
ij |,(model A)
DS
y=1
Pij∈Py∪Pz1X
ij∈Py∪Pz|e0y
ij |,(model A)
DS
z=1
Pij∈Py1X
ij∈Py|e0z
ij |,(model A).
(8)
In model B, all DS
µ(µ=x, y, z) are defined to be zero on Py∪Pz, while DS
xand DS
zare
nonzero on Px. Therefore, we have
DS
x=


0 on Py∪Pz,
1
Pij∈Px1Pij∈Px|e0x
ij |on Px
,(model B)
DS
y= 0 (model B)
DS
z=


0 on Py∪Pz
1
Pij∈Px1Pij∈Px|e0x
ij |on Px
,(model B).
(9)
We should note that e0y
ij =0 on Px; hence, DS
y=0 on all boundary surfaces. Finally, DV
µhas
the same definition in both models, which is given by
DV
µ=1
Pij∈V\S1X
ij∈V\S|e0µ
ij |(µ=x, y, z), S =Px∪Py∪Pz,(model A,model B) (10)
The main difference between these DS,V
µand those in Eq. (3) is changes in the boundary
surfaces due to a change in the direction of ~
B. The factor nij , which was introduced in Eq.
(3), is assumed to be nij = 1 in models A and B. In these models, the zero DMI condition
imposed by Γij in Eq. (5) on the boundary surfaces leads to surface effects, in contrast to the
model described in the preceding section [44], where nij automatically appears for a specific
discretization of the continuous Hamiltonian. We should note that the unit vector ~eij from
18

vertex ito vertex jalso changes due to lattice deformations when εx6=0. This change from
~eij to ~e 0
ij when εx6=0 effectively makes DS,V
µdirection-dependent, thus playing a role in the
magnetoelastic coupling. Therefore, neither uniaxial anisotropy, such as −KxPi(σx
i)2, nor
more general magnetoelastic coupling terms are included in the Hamiltonian.
0 0.02 0.04
0.62
0.64
0.66
D
Dx
S
(a)
Dy
Dz
surface
x
model A
0 0.02 0.04
0.48
0.5
0.52
D
Dx
V
(b)
Dy
Dz
bulk
A B
x
FIG. 11. (a) DS
µ(µ=x, y, z) vs. εxin model A and (b) DV
µ(µ=x, y, z) vs. εxin models A and B.
The dashed line in (a) indicates that the simulations of model A are performed only at εx= 0, and
the dashed lines in (b) with symbols A and B indicate that the simulations of models A and B are
performed at strains of εx= 0 and εx= 0.02, respectively.
DS
µvs. εxand DV
µvs. εxare plotted in Figs. 11(a) and (b), where the lattice deformation
defined by εxis given in Eq. (7). The simulations of model A are performed only for εx= 0,
as indicated by the dashed line in Fig. 11(a); however, we plot DS
µvs. εxto observe how
DS
µvaries with εx.
Along the tensile strain direction x,DS
xand DV
xboth increase with increasing strain
εx. In Ref. [39], the corresponding coupling constant Dxdecreases with increasing tensile
strain, and the response of Dxto the tensile strain is opposite to that shown in the plotted
data of DS
xand DV
xin Figs. 11(a) and (b). However, these two different behaviors, namely,
increasing and decreasing, are consistent because the sign of the DMI energy SDM in Eqs.
(5) and (2) is opposite to that in Ref. [39] and always negative because the continuous form
of the DMI energy R~σ ·(∇×~σ)d3xis replaced by the discrete expression −Pij ~e 0
ij ·(~σi×~σj).
As a result, changes in the two coupling constants DS,V
xand Dxhave the same effect on
SDM (<0) in model B and SD M (>0) in the model in Ref. [39]. Thus, the increase in DS,V
xin
−DS,V
x|SDM |(>0) in model B is equivalent to a decrease in Dxin DxSDM (>0) in Ref. [39].
Note that DV
z(6=0.5) at zero strain (⇔εx= 0) slightly deviates from 0.5 in Fig. 11(b). This
19

result occurs because the distribution of ~eij deviates slightly from isotropic to nonisotropic
in the zdirection. This type of anisotropy is expected in the case of tetrahedral lattices with
flat boundary surfaces, where one side of each tetrahedron is forced to be on the same flat
surface, and the area of Pzis relatively large (Fig. 6(a)). However, this deviation in DV
zin
model B is constant, independent of the strain, and relatively small (1.4%); hence, it does
not have a substantial influence on the results.
The decrease in DV
y, where the ydirection is the direction of the ~
B(= (0,−B, 0)) axis,
is consistent with the result of model 2, which is a 3D Finsler geometry model aiming
to reproduce the results of MnSi, in Ref. [29], where Dzalong the ~
B(= (0,0,−B)) axis
decreases as skyrmions stabilize. Moreover, the increase in DS,V
xwith increasing εxalong
the tensile strain direction is consistent with that of Dxvs. εxin the 2D FG model proposed
in Ref. [27] to reproduce experimental results of FeGe in Ref. [18]. In this 2D model, the
variable εx<0 corresponds to tension and is opposite to the definition in this paper, where
εx>0 corresponds to tension. Note that model B in this paper aims to reproduce one of the
results reported in Ref. [40] for Cu2OSeO3. Thus, the stabilization mechanism depending on
anisotropic DMI is expected to be common among MnSi, FeGe and Cu2OSeO3. In contrast,
a tensile stress was applied along the ~
Baxis in Ref. [29] to stabilize the skyrmions, and this
stabilization is inconsistent with the result in this paper. In fact, skyrmions are stabilized
in model B if tension is applied perpendicular to the ~
Baxis, while in model 2 in Ref. [29],
stabilization occurs when tension is applied along the ~
Baxis.
The variation in DS,V
µas εxvaries, which is plotted in Figs. 11(a) and (b), is expected for
regular cubic lattices if suitable discrete Hamiltonians are used. These expected responses
in DS,V
µare discussed in the latter part of the following subsection.
D. Effect of DMI anisotropy caused by strains
Increases/decreases in DV
xand DV
zare caused by tensile strains with εx(>0), and these
variations in DV
xand DV
zinfluence the skyrmion configurations in model B. The reason why
skyrmion strings along the ydirection are influenced by DV
xand DV
zis that the diameter
ratio of the string depends on the characteristic lengths 2λ/DV
xand 2λ/DV
z[11]. Note that
DS
µ=0 on Pyand Pzin model B. The snapshots in Figs. 12(a) and (b) correspond to those
in Fig. 9(d) for εx= 0 and Fig. 10(d) for εx= 0.02, respectively, where (B, T )=(−0.4,0.6).
20

strain
effect


shrink
extend
FIG. 12. Illustration of a tensile strain effect caused by lattice deformation along the xaxis in
model B, with snapshots of (a) εx= 0 and (b) εx= 0.02, where the surface DMI is assumed to
be DS
µ= 0 on the boundaries Pyand Pz(Fig. 6(a)). An increase in DS,V
x, denoted by DS,V
x%,
enlarges |SDM |, causing the diameter of the skyrmion to decrease along the xaxis, while a decrease
in DS,V
z, denoted by DS,V
z&, causes the diameter of the skyrmion to extend along the zaxis.
These snapshots show the effect of the nonzero strain εx(=0.02). We should emphasize that
confined skyrmions can exist only in the central region between the plates Pzbecause the
zero DMI (⇔DS
µ= 0) on Pyand Pzremoves skyrmion configurations from these surfaces.
Thus, the confined skyrmions effectively feel repulsion from Pz, the width of which is not
quite larger than or comparable to the skyrmion size. As a result, the stripe configurations
become parallel to Pz, as shown in Fig. 12(a), in this region of Band T, and such anisotropic
stripe configurations change to skyrmion configurations and are stabilized by the variations
in DV
xand DV
z, as shown in Fig. 12(b). If the width of the plate becomes sufficiently large,
the stripe direction tends to be isotropic and not always parallel to Pz; therefore, the tensile
strain along the xdirection is not always effective for stabilization.
It should also be noted that the oblong shape of the skyrmions along the xdirection
(Fig. 12(a)) in the low magnetic field region (⇔|B|= 0.4) looks different from that in the
FeGe nanostripe in Ref. [35], where shape deformation is observed in the zdirection as the
stripe width increases. However, this oblong shape along the xdirection is also observed
21

in the same material, namely, FeGe, when the width is sufficiently narrow [36]. Therefore,
the result shown in Fig. 12(a) is consistent with the results in Refs. [35, 36]. On the
other hand, the response of the skyrmion shape to stresses with respect to the stability in
Cu2OSeO3differs from that in FeGe and MnSi, as described above, and our model B results
are consistent with the response in Cu2OSeO3in Ref. [40]. Since strain and GC effects are
both implemented in model B, these findings indicate that the same GC effects in sufficiently
narrow nanostripes occur in different materials, with the strain effects depending on the
material. Here, both effects modify the effective couplings DV
yand DV
zin model B, implying
that the strain-induced variations in DV
yand DV
zdepend on the materials, although the
changes in the skyrmion shape due to direction-dependent DMI coefficients are independent
of the material. The changes in the direction-dependent DMI coefficients in response to the
strain and in the skyrmion morphology due to variations in the nanostripe width are both
interesting. However, the numerical data presented in this paper are insufficient for studying
these problems, and further numerical studies are necessary.
0 1 2 3
-0.3
-0.2
-0.1
model B
SDM/NB
T
B=-0.6
(a)
skyrmion
x=2%
V V
0 1 2 3
0
0.2
0.4
0.6
model B
SFM/NB
T
:x=2%
:x=0 B=-0.6
(b)
skyrmion
x=0
V V
FIG. 13. (a) SV
DM /NV
Bvs. Tand (b) SV
F M /NV
Bvs. Tin model B, where NV
Bis the total number
of internal bonds and is defined as NV
B=Pij∈V\S1, where S=Px∪Py∪Pz. The regions between
the dashed lines are the skyrmion phase for ε= 0 and ε= 2%. The results show that SV
DM and
SV
F M are not influenced by the strain.
Before presenting the effective DMI energies, we plot the bulk DMI energy SV
DM /NV
Band
bulk FMI energy SV
F M /NV
Bin Figs. 13(a) and (b), where NV
Bis the total number of internal
bonds and is defined as NV
B=Pij∈V\S1. SV
DM and SV
F M are calculated only inside the
22

boundary, such that
SV
DM =X
ij∈V\S
~e 0
ij ·(~σi×~σj),
SV
F M =X
ij∈V\S
(1 −~σi·~σj),
NV
B=X
ij∈V\S
1, S =Px∪Py∪Pz.
(11)
The plots show that SV
DM and SV
F M are both independent of the strain. For the total FMI
energy SF M , which is not plotted and is given by SV
F M plus the surface FMI energy on S,
no dependence of SF M on strain is observed, as expected.
0 1 2 3
-0.1
-0.05
model B
SDM/NB
T
:x=2%
:x=0 B=-0.6
(c)
skyrmion
x=2%
xV
0 1 2 3
-0.2
-0.15
-0.1
model B
SDM/NB
T
:x=2%
:x=0 B=-0.6
(d)
skyrmion
x=0
zV
0 1 2 3
-0.12
-0.1
-0.08
-0.06
model B
SDM/NB
T
:x=2%
:x=0 B=-0.6
(a)
skyrmion
x=2%
xV
0 1 2 3
-0.12
-0.1
-0.08
-0.06
model B
SDM/NB
T
:x=2%
:x=0 B=-0.6
(b)
skyrmion
x=0
zV
FIG. 14. (a) Sx
DM /NV
Bvs. Tand (b) Sz
F M /NV
Bvs. Tin model B, where Sµ
DM is given by Eq.
(12). (c) ¯
Sx
DM /NV
Bvs. Tand (d) ¯
Sz
F M /NV
Bvs. Tin model B, where Sµ
DM is given by Eq. (13).
NV
Bis the total number of internal bonds and is defined as NV
B=Pij∈V\S1, S=Px∪Py∪Pz.
The regions between the dashed lines are the skyrmion phase. The direction dependencies of Sµ
DM
in (a), (b) such that |Sx
DM |(εx= 0.02) >|Sx
DM |(εx= 0) and |Sz
DM |(εx= 0.02) <|Sz
DM |(εx= 0) are
almost the same in ¯
Sµ
DM in (c), (d), at least in the skyrmion phase 1 ≤T≤2.6.
On the other hand, an increase in DV
x(decrease in DV
z) due to the positive strain εx=0.02,
as shown in Fig. 11(b), increases (decreases) the direction-dependent DMI energy |Sx,z
DM |
23

along the x(z) direction, where Sx,z
DM (<0) is defined as
Sx,z
DM =X
ij∈V\S~e 0
ij ·(~σi×~σj)|e0x,z
ij |
|~e 0
ij|2,
~e 0
ij =(1 + εx)ex
ij,ey
ij
√1 + εx
,ez
ij
√1 + εx, S =Px∪Py∪Pz.
(12)
The quantities Sx
DM and Sz
DM are effectively considered to be the xand zcomponents of
SDM (Appendix B), where “effective” means that Sµ
DM is slightly modified from the original
SDM and not used in the Hamiltonian, similar to the effective coupling constants in Eqs.
(8), (9) and (10); however, these values can be extracted from the numerical results and are
useful for qualitative arguments.
Figures 14(a) and (b) show that the absolute value of |Sx
DM |(|Sz
DM |) when ε= 2% is larger
(smaller) from that when εx= 0. This result is consistent with the case of a regular cubic
lattice. Indeed, in the case of a regular cubic lattice, we have |e0x
ij |= 1+εx,|e0y
ij |=|e0z
ij |=
1/√1 + εx, and |e0x
ij |≥|~e 0
ij|,|e0z
ij |≤|~e 0
ij|, where ~e 0
ij is given by ~e 0
ij =(e0x
ij ,0,0) or ~e 0
ij =(0, e0y
ij ,0)
or ~e 0
ij = (0,0, e0z
ij ). Therefore, it is easy to verify that |e0x
ij |/|~e 0
ij|2<|e0x
ij |/|e0x
ij |2= (1 + εx)−1<
1 = |ex
ij|/|~eij|2, and hence, we obtain |Sx
DM |(εx= 0.02) >|Sx
DM |(εx= 0). Note that Sx
DM and
Sz
DM are both negative in Figs. 14(a) and (b). The relation |Sz
DM |(εx= 0.02) <|Sz
DM |(εz= 0)
is also obtained by the relations |e0z
ij |/|~e 0
ij|2>|e0z
ij |/|e0z
ij |2= (1 + εx)1/2>1 = |ez
ij|/|~eij|2.
These relations |Sx
DM |(εx= 0.02) >|Sx
DM |(εx= 0) and |Sz
DM |(εx= 0.02) <|Sz
DM |(εx= 0)
are compatible with the results shown in Figs. 14(a) and (b). Note that the value of
~e 0
ij ·(~σi×~σj)(<0) is negative at low temperature region including the skyrmion phase;
hence, the decrease (increase) in Sx,z
DM changes to an increase (decrease) in |Sx,z
DM |. The
increase (decrease) in |Sx
DM |(|Sz
DM |) implies that the period of helical order or the diameter
of the skyrmions along the x(z) axis shrinks (extends), as shown in Figs. 12(a) and (b). In
other words, the zero DMI on Pzcauses the skyrmions to be oblong along the xdirection
in the relatively low |B|region, while the positive strains εx>0 cause the skyrmions to be
oblong along the zdirection in the relatively low Tregion, where thermal fluctuations are
suppressed. Thus, we consider that these competing effects stabilize the skyrmions in the
low Tand low |B|regions, as shown in Fig. 10(a).
The direction-dependent or anisotropic DMI energy is not always uniquely determined,
24

and the following quantities influence the effective DMI energy (Eq. (B2) in Appendix B):
¯
Sx,z
DM =X
ij∈V\S
sgn(e0x,z
ij )(~σi×~σj)x,z ,
sgn(f) = 


1 (f > 0)
−1 (f < 0) ,sgn(0) = 0.
(13)
Figures 14(c) and (d) demonstrate the expected behavior, namely, that |¯
Sx
DM |(|¯
Sz
DM |) is
larger (smaller) when εx=2% than when εx= 0. A small deviation between |Sx
DM |and |¯
Sx
DM |
when εx=0 is observed in the low Tregion. However, this deviation is not contradictory to
the expected inequality, and moreover, the T(<1) region is not the skyrmion phase when
εx= 0, and hence, this deviation does not influence the result that |¯
Sx
DM |(|¯
Sz
DM |) is larger
(smaller) when εx=2% than when εx= 0. We should note that ¯
Sx,z
DM in Eq. (13) is the same
as Sx,z
DM in Eq. (12) in the case of a regular cubic lattice (Eq. (B3) in Appendix B).
IV. SUMMARY AND CONCLUSION
In this paper, we numerically study skyrmion stabilization using a plate-shaped 3D lattice
discretized by tetrahedrons by assuming zero DMI coefficients on the boundary surfaces
parallel to the magnetic field as a geometric confinement (GC) effect. The Hamiltonian is
given by a linear combination of the standard ferromagnetic interaction (FMI) energy, the
Dzyaroshinskii-Moriya interaction (DMI) energy for Bloch-type skyrmions and the Zeeman
energy. Zero DMI is assumed on the boundary surfaces as a GC effect due to an observation
in the reported numerical data, namely, that skyrmion confinement and stability are observed
in a model in which the DMI coefficients on the boundary surfaces are approximately 3 times
smaller than those inside the boundary.
Compared with the nonzero DMI model, the stabilization effect is significantly improved
in the zero DMI model, with an increase in the area of the skyrmion phase in the BT
phase diagram. Moreover, a tensile strain implemented by lattice deformation enhances
stabilization, extending the skyrmion phase in the low-temperature region. This strain-
induced enhancement is observed only in the zero DMI model, where zero DMI conditions
are implemented as a GC effect. These numerically obtained data indicate that the zero
DMI condition and tensile strain compete with each other, thereby enhancing the stability.
25

The models in this paper are applicable to the skyrmions observed in Cu2OSeO3, in
which tensile strains perpendicular to the magnetic field stabilize skyrmions. Moreover,
the mechanism by which anisotropic DMI coefficients stabilize skyrmions in Cu2OSeO3is
expected to be similar to that in MnSi and FeGe in the sense that the variation in the
skyrmion shape according to the anisotropic DMI coefficients should be the same. However,
detailed information on the shape morphology of confined skyrmions and the dependence on
the domain size has not yet been obtained in the framework of effective interaction theories
implementing the effects of external stimuli and GC. Therefore, additional theoretical and
numerical studies are necessary to develop a unified understanding of the stability leading
to skyrmion control.
ACKNOWLEDGMENTS
This work was supported in part by JSPS Grant-in-Aid for Scientific Research (19KK0095).
The numerical simulations were performed on the supercomputer system AFI-NITY at the
Advanced Fluid Information Research Center, Institute of Fluid Science, Tohoku University.
Appendix A: Discretization of the DMI energy in a tetrahedral lattice with free
boundary conditions
In this Appendix, we show the reason for the inclusion of the factor nij in SDM in Eq. (2).
To summarize, the factor nij appears as a consequence of the discretization from a continuous
3D integral to a sum over the tetrahedron volumes in the Hamiltonian. However, we should
emphasize that the discretization is not uniquely determined and generally depends on the
lattice structure; therefore, the appearance of nij should be understood as one possibility
for modeling skyrmions.
We start with the continuous Hamiltonian given by
SDM =Z√gd3xgab ∂~r
∂xa·~σ ×∂~σ
∂xb,(A1)
where xa(a= 1,2,3) denotes the local coordinate axes, gab = (gab)−1is the inverse of the
3×3 matrix of metric functions, and gis the determinant. ~r (∈R3) and ~σ (∈S2:
unit sphere) denote a three-dimensional position vector and the spin variable. The interfacial
26

Hamiltonian for Neel-type skyrmions is obtained by replacing ∂~r/∂xawith ∂~r/∂xa
×~e z. Here,
we use the bulk-type Hamiltonian for simplicity, eliminating the magnetic field direction ~e z.
To obtain a discrete Hamiltonian on a tetrahedral lattice, we replace the volume integral
with a sum over the tetrahedrons ∆, such that R√gd3x→P∆under the assumption of
the Euclidean metric gab =δab. The tangential vector ∂~r/∂xais not always of unit length;
however, for simplicity, we replace this vector with a unit vector, such that ∂~r/∂xa→~ea.
Thus, using the unit tangential vectors ~e12 ,~e13 , and ~e14 along the local coordinate axes,
we obtain the replacements ∂~r/∂x1→~e12,∂~r/∂x2→~e13, and ∂~r/∂x3→~e14 (Fig. 15(a)).
Thus, we have a discrete SDM such that SDM =P∆(~e12 ·~σ1×~σ2+~e13 ·~σ1×~σ3+~e14 ·~σ1×~σ4). In










FIG. 15. (a) A local coordinate origin denoted by the three-dimensional position vector ~r1in a
tetrahedral lattice and the unit tangential vector ~e12 along the x1axis, and (b) a local coordinate
origin denoted by the three-dimensional position vector ~r1in a regular cubic lattice and the unit
tangential vector ~e12 along the x1axis.
this expression, the local coordinate origin is assumed to be at vertex 1; however, vertices
2, 3, and 4 can also be the coordinate origin. Thus, including the other 9 terms obtained by
replacing vertices 1→2,2→3,3→4,4→1 three times, we have SDM =P∆SDM(∆), with
SDM(∆) = (~e12 ·~σ1×~σ2+~e13 ·~σ1×~σ3+~e14 ·~σ1×~σ4
+~e23 ·~σ2×~σ3+~e24 ·~σ2×~σ4+~e34 ·~σ3×~σ4).
(A2)
The six terms in SDM(∆) have one-to-one correspondence with the bonds in tetrahedron ∆,
and therefore, SDM(∆) can be expressed by SDM(∆)= Pij(∆)~eij ·~σi×~σj, where Pij(∆)is the
sum over the bonds ij of tetrahedron ∆. Thus, we have an expression for SDM such that
SDM =P∆Pij(∆)~eij ·~σi×~σj. Here, we use the fact that the order of the sums is exchangeable,
such that P∆Pij(∆)1= Pij P∆(ij)1, where P∆(ij )1 denotes the sum over the tetrahedrons
27

∆(ij) sharing bond ij. Using the symbol nij = (1/¯
N)P∆(ij)1 to represent the total number
of tetrahedrons sharing bond ij and including the factor 1/¯
N, where ¯
N=Pij P∆(ij)1/Pij 1
is the mean value of P∆(ij)1, we obtain the following expression for SDM:
SDM =X
ij
nij~eij ·~σi×~σj,
nij = (1/¯
N)X
∆(ij)
1,¯
N=X
ij X
∆(ij)
1/X
ij
1.
(A3)
Note that every tetrahedron has six bonds; thus, we have P∆Pij(∆)1 = 6 P∆= 6Ntet,
where Ntet is the total number of tetrahedrons. Combining this relation with the definition
of ¯
N=Pij P∆(ij)1/Pij 1, we obtain ¯
N= 6Ntet/NB, where NB=Pij 1 is the total number
of bonds. Using the data in the caption of Fig. 2(a), we determine that ¯
N'4.91 is the mean
value of the total number of tetrahedrons sharing a bond. Because of the factor ¯
N, the mean
value of nij is hniji=Pij nij /Pij 1 = 1. Moreover, the value nij depends on the position
of bond ij. For example, hnijion the boundary surfaces is considerably different from that
inside the boundary. This difference in the factor nij characterizes the two different models
defined in Eq. (2) and motivates us to study model B, which is introduced in Section III A.
Here, we comment on the response of the discrete Hamiltonian in cubic lattices under
lattice deformations. In the cubic lattice shown in Fig. 15(b), ∂~r/∂x1can be replaced by
~e12 along the x1axis, the same as in the case of a tetrahedral lattice, if the lattice is not
deformed. If ~e12 is replaced by ~e12 →~e 0
12 = (1 + εx)~e12 for a tensile deformation corresponding
to Eq. (7), the effective coupling constants DS,V
xdefined in Eq. (3) increase with increasing
εx, similar to those shown in Figs. 11(a) and (b) for model B on a tetrahedral lattice. The
other values of DS,V
y,z in the other directions also vary with increasing εx, which is consistent
with the behaviors shown in Figs. 11(a) and (b), because an extension in ~e 0
12 and reduction
in ~e 0
13 and ~e 0
14 are expected under tensile deformation along the x1axis. Thus, we expect
that the response of the skyrmion configuration under lattice deformation is independent of
whether the lattice is cubic or tetrahedral.
Appendix B: Canonical coordinate axis component of the DMI energy
In this Appendix, we explain why Sx,z
DM in Eq. (12) is considered the xor zcomponent
of SV
DM =Pij∈V\SΓij Eij . In SV
DM , the sum over bond ij is limited to that in Eq. (12).
28

Moreover, the factor Γij is typically included, and Eij corresponds to ~e 0
ij ·(~σi×~σj) in model
B. This Eij is direction-dependent because of ~e 0
ij, and this dependence is implemented in
DV
µin Eq. (10). Now, SDM can be written as
SV
DM =X
ij∈V\S
ΓijEij =X
ij∈V\S
ΓijEij |e0x
ij |2+|e0y
ij |2+|e0z
ij |2
|~e 0
ij|2,
=X
ij∈V\S
Γij|e0x
ij |Eij|e0x
ij |+ Γij|e0y
ij |Eij|e0y
ij |+ Γij|e0z
ij |Eij|e0z
ij |
|~e 0
ij|2
→X
ij∈V\S
DV
xEij|e0x
ij |+DV
yEij|e0y
ij |+DV
zEij|e0z
ij |
|~e 0
ij|2
=DV
xX
ij∈V\S
Eij|e0x
ij |
|~e 0
ij|2+DV
yX
ij∈V\S
Eij|e0y
ij |
|~e 0
ij|2+DV
zX
ij∈V\S
Eij|e0z
ij |
|~e 0
ij|2
=DV
xSx
DM +DV
ySy
DM +DV
zSz
DM ,
Sµ
DM =X
ij∈V\S~e 0
ij ·(~σi×~σj)|e0µ
ij |
|~e 0
ij|2,(µ=x, y, z),
Eij =~e 0
ij ·(~σi×~σj), S =Px∪Py∪Pz.
(B1)
where Γij|e0µ
ij |in the second line is replaced by DV
µin the third line. This DV
µincludes
the mean values hΓij|e0µ
ij |i and is defined in Eq. (10), where Γij = 1. In further detail, the
replacement is formulated as follows: Γij|e0µ
ij |→hΓij |e0µ
ij |i=Pij ∈V\SΓij |e0µ
ij |/Pij∈V\S1 =
DV
µ. Thus, SV
DM can be approximated by using the direction-dependent DMI energy Sµ
DM .
Note that this effective energy Sµ
DM is not used in the simulations.
Another possibility for defining the direction-dependent DMI energy, as introduced in
29

Eq. (13), is formulated as follows:
SV
DM =X
ij∈V\S
ΓijEij =X
ij∈V\S
Γij~e 0
ij ·(~σi×~σj)
=X
ij∈V\SΓij e0x
ij (~σi×~σj)x+ Γij e0y
ij (~σi×~σj)y+ Γij e0z
ij (~σi×~σj)z
=X
ij∈V\SΓij |e0x
ij |sgn(e0x
ij )(~σi×~σj)x+ Γij |e0y
ij |sgn(e0y
ij )(~σi×~σj)y
+Γij|e0z
ij |sgn(e0z
ij )(~σi×~σj)z
→X
ij∈V\SDV
xsgn(e0x
ij )(~σi×~σj)x+DV
ysgn(e0y
ij )(~σi×~σj)y+DV
zsgn(e0z
ij )(~σi×~σj)z
=DV
x¯
Sx
DM +DV
y¯
Sy
DM +DV
z¯
Sz
DM ,
¯
Sµ
DM =X
ij∈V\S
sgn(e0µ
ij )(~σi×~σj)µ,(µ=x, y, z ),
sgn(f) = 


1 (f > 0)
−1 (f < 0) ,sgn(0) = 0.
(B2)
In the fourth line, Γij|e0µ
ij |,(µ=x, y, z) are replaced by the mean values hΓij |e0µ
ij |i, as in Eq.
(B1), where DV
µis defined in Eq. (10), where Γij = 1. The final expression for ¯
Sµ
DM is the
same as that in Eq. (13).
On a regular cubic lattice, ~e 0
ij is given by ~e 0
ij =(e0x
ij ,0,0) or ~e 0
ij =(0, e0y
ij ,0) or ~e 0
ij =(0,0, e0z
ij );
therefore, we have |e0µ
ij |=|~e 0
ij|,(µ=x, y, z). Thus, Sµ
DM in Eq. (12) can be written as
Sx,z
DM =X
ij∈V\S
~e 0
ij ·(~σi×~σj)|e0x,z
ij |
|~e 0
ij|2
=X
ij∈V\S
e0x,z
ij (~σi×~σj)x,z |e0x,z
ij |
|e0x,z
ij |2
=X
ij∈V\S|e0x,z
ij |sgn(e0x,z
ij )(~σi×~σj)x,z 1
|e0x,z
ij |
=X
ij∈V\S
sgn(e0x,z
ij )(~σi×~σj)x,z .
(B3)
In the first line, Sx,z
DM (⇔Sx
DM,Sz
DM) means that ~e 0
ij =e0x,z
ij (⇔~e 0
ij =e0x
ij ,~e 0
ij =e0z
ij ) in the sum
Pij∈V\Sin the right hand side. The final expression is the same as that of ¯
Sx,z
DM in Eq. (13),
proving that Sx,z
DM =¯
Sx,z
DM (⇔Sx
DM =¯
Sx
DM,Sz
DM =¯
Sz
DM) in a regular cubic lattice.
30

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