# Magnetism in the Néel skyrmion host GaV4S8 under pressure

T. J. Hicken  Current address: Department of Physics, Royal Holloway,
University of London, Egham, TW20 0EX, United Kingdom  Department of Physics,
Centre for Materials Physics, Durham University, Durham, DH1 3LE, United
Kingdom  M. N. Wilson  Department of Physics, Centre for Materials Physics,
Durham University, Durham, DH1 3LE, United Kingdom  S. J. R. Holt  Department
of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom  R.
Khassanov  Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute,
5232 Villigen PSI, Switzerland  M. R. Lees  Department of Physics, University
of Warwick, Coventry, CV4 7AL, United Kingdom  R. Gupta  Laboratory for Muon
Spin Spectroscopy, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  D.
Das  Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, 5232
Villigen PSI, Switzerland  G. Balakrishnan  Department of Physics, University
of Warwick, Coventry, CV4 7AL, United Kingdom  T. Lancaster  Department of
Physics, Centre for Materials Physics, Durham University, Durham, DH1 3LE,
United Kingdom

###### Abstract

We present magnetization and muon-spin spectroscopy measurements of Néel
skyrmion-host GaV4S8 under the application of hydrostatic pressures up to
$P=2.29$ GPa. Our results suggest that the magnetic phase diagram is altered
with pressure via a reduction in the crossover temperature from the cycloidal
(C) to ferromagnetic-like state with increasing $P$, such that, by 2.29 GPa,
the C state appears to persist down to the lowest measured temperatures. With
the aid of micromagnetic simulations, we propose that the driving mechanism
behind this change is a reduction in the magnetic anisotropy of the system,
and suggest that this could lead to an increase in stability of the skyrmion
lattice.

##  I Introduction

The lacunar spinel GaV4S8 [1] is one of very few examples of a material that
hosts a magnetic Néel skyrmion lattice (SkL) [2] throughout the bulk [3]. The
SkL, along with the cycloidal (C) phase, stabilized in different parts of the
applied field-temperature ($B$–$T$) phase diagram, occurs due to a competition
between magnetic exchange and the Dzyaloshinskii-Moriya interaction (DMI). The
latter arises owing to the the polar rhombohedral structure ($R3m$) that
GaV4S8 adopts below a structural phase transition at 42 K [4], generally
attributed to a Jahn-Teller distortion. Above this temperature the material
possesses a cubic $F\bar{4}3m$ structure. Also important in this material is
easy-axis anisotropy [5], whose direction aligns with the rhombohedral
distortion. Competition between the three interactions leads to a rich
magnetic phase diagram, with dependence on the alignment between the applied
magnetic field and crystallographic axes [3]. In zero field the ground state
comprises a complex ferromagnetic-like (FM*) configuration, where the spins on
the V atoms in each V4 cluster combine to form effective spin-1/2 units that
align ferromagnetically [6]. On increasing temperature, C ordering then takes
place in the region $5~{}\lesssim T<T_{\mathrm{c}}\simeq 13$ K. Under the
application of an applied field, in polycrystalline samples of GaV4S8, the SkL
phase is stabilized at $10\lesssim T\lesssim 13$ K,
$40\lesssim\mu_{0}H_{\text{ext}}\lesssim 100$ mT [7].

The application of hydrostatic pressure on GaV4S8 has been studied previously
at room temperature [8]. Above 35 GPa there is a pressure induced structural
phase transition that results in the high-$T$ cubic phase becoming
orthorhombic ($Imm2$), alongside a semiconductor to metallic transition and
changes in the optical properties of the material. At lower pressures, the
changes in crystal structure are more modest, with a gradual reduction in the
lattice parameter on increasing pressure. Although there has been some effort
to construct a magnetic $T$–$P$ phase diagram of GaV4S8 [9], the transitions
between the magnetically ordered states have not been thoroughly investigated.

We have previously studied the magnetism in the GaV4S8-ySey series using
magnetometry and muon-spin spectroscopy ($\mu^{+}$SR) [10, 7]. These
techniques are sensitive to subtle changes in the magnetism of this series,
revealing, for example, that the transition from FM* to C states is a
crossover, rather than an abrupt phase transition [11, 12]. This process is
likely to depend sensitively on the crystalline anisotropy in the system [13],
which has been shown to decrease as $T$ increases [5]. Here we extend our
investigation of GaV4S8 to probe the magnetic behavior under the application
of hydrostatic pressure up to $P=2.29$ GPa. With the aid of micromagnetic
simulations, we interpret the effect of pressure in terms of changes to the
relative strengths of terms appearing in an effective Hamiltonian.

##  II Experimental and Computational Details

Polycrystalline samples of GaV4S8 were synthesized and characterized as
described in Refs. [10, 14]. Measurements of the magnetization were made using
a Quantum Design Magnetic Property Measurement System. The sample was loaded
into a EasyLab Mcell pressure cell which allowed pressures of up to 1 GPa. The
pressure was measured in-situ by monitoring the superconducting transition of
a tin manometer. Temperature scan measurements were performed in an applied
magnetic field (either 10 mT or 5 T) on cooling from room temperature. Field
scan measurements were performed on decreasing field at 2 K.

$\mu^{+}$SR measurements [15, 16, 17] of polycrystalline GaV4S8 were carried
out at the $\mu$E1 beamline of the Swiss Muon Source (S$\mu$S), Paul Scherrer
Institute, Switzerland, using the GPD instrument. Both zero-field (ZF)
measurements (where no external magnetic field is applied), and transverse-
field (TF) measurements (where an external magnetic field is applied
perpendicular to the initial muon-spin polarization) were performed.
Polycrystalline samples were loaded in a double-wall piston cylinder cell made
of MP35N material [18, 19], using Daphne oil (7373) as a pressure-transmitting
medium, which was mounted in a Janis cryostat. Data analysis was carried out
using the WiMDA program [20] and made use of the MINUIT algorithm [21] via the
iminuit [22] Python interface for global refinement of parameters. We have
used the muon stopping sites calculated in Ref. [10], along with the MuESR
code [23], to perform simulations of magnetic field distributions as seen with
$\mu^{+}$SR in GaV4S8.

Micromagnetic simulations were carried out using the ubermag package [24, 25].
A $512\times 512\times 1$ grid of cells (side length 0.8 nm) with periodic
boundary conditions in all directions was simulated. The $C_{nv}$ crystal
class was used, with micromagnetic parameters based on previous experimental
and computational work [3, 5, 26]. We set exchange $\mathcal{A}=0.05975$ pJ/m,
DMI $D_{0}=0.03057$ mJ/m2, magnetocrystalline anisotropy $K_{0}=16$ kJ/m3, and
saturation magnetization $M_{\text{s}}=33.07$ kA/m. The easy axis $\bm{u}$ was
out of the plane of the simulation, aligned with the magnetic field direction.

##  III Results & Discussion

The results of magnetization measurements made at several applied pressures
are shown in Fig. 1. We start by considering the measurements performed under
the application of a small $\mu_{0}H=10$ mT external field [Fig. 1(a)] with
the intention of studying the system close to the magnetic ground state. In
these measurements, the pressure cell adds a significant background to the
measured magnetization. The data in Fig. 1(a) are therefore normalized to the
maximum magnetization $M_{\mathrm{max}}$ at each pressure. There is no
significant change in this quantity with pressure, which appears to vary
randomly by a few percent. As the temperature is decreased, the measurements
share the same features: (i) a small peak at $T~{}\simeq$ 13 K, (ii) a range
of $T$ where $M/M_{\mathrm{max}}\lesssim 0.2$, and (iii) a rapid increase in
the magnetization at low $T$.

Figure 1: Magnetization measurements of GaV4S8 under the application of
various external pressures. (a) A temperature scan in an applied field of 10
mT. Data are normalized to the maximum magnetization at each pressure. Inset
shows an enlargement of the region around $T_{\text{c}}~{}\simeq$ 13 K. (b) A
field scan at 2 K. (c) The change in magnetization associated with the Jahn-
Teller induced structural phase transition, measured in an applied field of 5
T. The 0 GPa measurements are performed without the pressure cell. As in (a),
the data are normalized. The dashed line marks $T_{\text{JT}}$ at $P=0$ GPa
according to Ref. [4].

We attribute each of the three features to different physical origins. (i) The
peak at $T~{}\simeq$ 13 K marks $T_{\text{c}}$, which increases very slightly
with pressure, although this change is small compared to other features. As
$T_{\text{c}}$ is set, predominantly, by the strength of the exchange
interaction $\mathcal{A}$, we infer that pressure does not significantly
change this parameter. (ii) The range of $T$ where $M/M_{\mathrm{max}}$ is
small corresponds to the C state, where the rotating spin structure leads to
almost zero magnetization. (iii) The increase in magnetization as $T$
decreases further is consistent with the crossover to the FM* phase. The
sample magnetization does not increase further once the FM* state is
stabilized over the entire sample. Pressure has a marked effect on the
temperature at which the magnetization increase occurs, suggesting that as the
pressure is increased, the C to FM* crossover occurs at successively lower
$T$. There are two changes to the spin Hamiltonian that might explain this
behavior: an increase in the strength of the DMI (making the twisting C state
more preferable compared to the FM* state), or a decrease in the easy-axis
anisotropy (making the spins less likely to align along a particular
direction). These two possibilities are discussed in more detail below.

Measurements were also performed at $T=2$ K as a function of applied field
[Fig. 1(b)]. These results are not normalized, and show the magnetization
saturates at effectively the same value regardless of applied pressure. There
is a slight reduction in magnetization with increasing pressure, however this
is very small ($<1.7\%/$GPa), and is therefore hard to unambiguously say
whether this is an artifact from the pressure cell. The pressure cell does
have a significant impact on the magnetization, hence the absolute values
extracted should be treated with caution; the measured magnetization is
significantly suppressed compared to measurements of the same samples measured
outside of the pressure cell (where saturation is about
$0.85~{}\mu_{\text{B}}/$f.u.).

Figure 1(c) shows the temperature dependence of the magnetization under the
application of a $\mu_{0}H=5$ T external field, where the system is in a
field-polarized state below $T_{\text{c}}$, around the temperature below which
the Jahn-Teller distortion occurs, $T_{\text{JT}}$($=42$ K at $P=0$ [4]). As
in Fig. 1(a), the data have been normalized. The effect of the pressure cell
(proportionally larger above $T_{\text{c}}$ due to the small sample signal)
results in a pressure-independent shift of features to a slightly higher $T$.
There appears to be a systematic increase in $T_{\text{JT}}$ as $P$ increases
(around 1.3 K/GPa at these pressures). Reference [8] reports that above 35
GPa, at room temperature, the system adopts the $Imm2$ structure.

To further understand the crystal structures, and the effect of pressure, we
can consider related chemically-substituted systems, where the effect of
substitution is similar to that of an external pressure. In GaMo4Se8, when the
Jahn-Teller distortion is decomposed into different normal modes, it has been
shown that [27] different amplitudes of these modes results in either the
$F\bar{4}3m$ phase (stabilized in GaV4S8 at low $P$, low $T$), or $Imm2$ phase
(stabilized in GaV4S8 at high $P$, high $T$). Further, Ref. [28] reports that
GaMo4Se8 exhibits coexistence of the ground state $F\bar{4}3m$ structure and
the metastable $Imm2$ structure at low $T$. In contrast, in GeV4S8, the
structure below $T_{\text{JT}}$ is $Imm2$ [29, 30].

One possible explanation for the changes in the structure of GaV4S8 with
pressure is that the transition observed at room temperature, $P\simeq 35$
GPa, is not the same as that observed at $T_{\text{JT}}$ in the absence of
applied pressure. An alternative, perhaps simpler, explanation could be that
(i) $T_{\text{JT}}$ increases with pressure in GaV4S8 such that
$T_{\text{JT}}$ is above room temperature for pressures above 35 GPa, and that
(ii) changes to the nature of the distortion results in the realization of the
$Imm2$ phase. For this scenario there would either need to be an increase in
the rate of change of $T_{\text{JT}}$ with pressure (extrapolating the rate
from our data is not sufficient), or a discontinuous change in $T_{\text{JT}}$
due to some significant change in the behavior of the system (which could be
related to the realization of the $Imm2$ phase).

Figure 2: Parameters extracted through fitting wTF $\mu^{+}$SR measurements of
GaV4S8 under various indicated external pressures in a applied field of $3$
mT. Asymmetries are shown on the left (a–b), with $\Lambda_{\text{r}}$ and
$B_{\text{wTF}}$ on the right (c–d) describing the effects of residual fields
on muons outside the sample, or in positions of near cancellation of the
internal field. The inset of (a) shows representative data measured at 2.29
GPa, 0.4 K, with the corresponding fit.

To further probe the magnetic phase diagram of GaV4S8, we performed
$\mu^{+}$SR measurements with the application of a weak transverse field
(wTF), and in zero applied magnetic field (ZF). Previous $\mu^{+}$SR
measurements in this regime [10, 7] revealed a complicated state of affairs
with two classes of muon sites sensitive to both static and dynamic effects,
with behavior likely resulting from the evolution of the magnetic structure
and the magnetic domains with temperature. We first consider the wTF
measurements, useful in determining the nature of magnetic transitions, which
were performed in 3 mT applied perpendicular to the initial muon-spin
direction. The measured asymmetry spectra $A_{\text{wTF}}\left(t\right)$
(example spectra are shown in the inset of Fig. 2(a), and in the Supplemental
Material [17]) have two clear contributions from the precession of the muon-
spin, one from the response of muons predominantly sensitive to the external
wTF (i.e. stopped outside the sample, or in positions where the internal field
is small), and the other from muons that stop in sites dominated by the
internal field in the material. The contribution from these second class of
muon sites track the results we obtain with ZF $\mu^{+}$SR discussed later,
but with an expected larger uncertainty than the ZF measurement. We
parameterize the data with

$A_{\text{wTF}}\left(t\right)=a_{r}\cos\left(\gamma_{\mu}B_{\text{wTF}}\right)\exp\left(-\Lambda_{\text{r}}t\right)+a_{\text{b}},$
(1)

for all $T$, excluding the first 0.2 $\mu$s of data from the fits (where the
spectra is dominated by the rapid precession and relaxation from muons
stopping in the sites sensitive to the internal field). This captures
contributions from all muon sites for $T>T_{\mathrm{c}}$, but the oscillating
component loses the contribution from those muon sites experiencing the large,
internal magnetic field resulting from magnetic order below $T_{\mathrm{c}}$.
The results from this parameterization can be seen in Fig. 2.

The relaxing asymmetry, $a_{\text{r}}$ corresponds to muons stopping at sites
where the field is $B_{\text{wTF}}$. The associated relaxation rate
$\Lambda_{\text{r}}$ can arise due to both static disorder (leading to a
distribution of local magnetic fields at the muon sites, whose width
determines the relaxation rate) and dynamic fluctuations on the muon timescale
(where the amplitude and rate of the fluctuating field determines the
relaxation rate). Note that the observed exponential relaxation usually
corresponds to dynamic fluctuations. We find that $a_{\text{r}}$ decreases
below $T_{\text{c}}$ as expected for a magnetic transition [Fig. 2(a)], where
we usually see the loss of the signal from the magnetically ordered component
of the sample. In this regime the temperature-dependent $a_{\text{b}}$ [Fig.
2(b)] mainly captures the contribution from muons with their spins directed
along the large, static internal field in the ordered regime (and hence do not
precess). The temperature evolution of this component is slightly unusual, but
likely reflects dynamic fluctuations that freeze out as $T\rightarrow 0$,
leading to a recovery of $a_{\text{b}}$. The effect of pressure does not
change $a_{\text{r}}$ within the statistical accuracy of the data and we see
no evidence for a pressure-induced magnetic phase separation from these
results. Despite the gradual temperature evolution of $a_{\text{b}}$, these
results, taken with the previous $\mu^{+}$SR, suggest the entire sample shows
long-range magnetic order below $T_{\text{c}}$ at all pressures.

We now consider $\Lambda_{\text{r}}$ and $B_{\text{wTF}}$ [Figs. 2(c–d)],
whose behavior below $T_{\mathrm{c}}$ reflects muons stopping outside the
sample or in positions where the internal field is small. Considering first
the low-temperature FM∗ to C transition, at 0.74 GPa, both
$\Lambda_{\text{r}}$ and $B_{\text{wTF}}$ show a local maximum coincident with
the crossover, consistent with magnetometry results and a similar peak in the
relaxation rate observed at ambient pressure [7]. As $P$ is increased, these
peaks are pushed to lower $T$ (consistent with the magnetization results),
before the peak becomes impossible to resolve at 2.29 GPa, suggesting the C
state persists down to lower $T$ than measured. A peak in $\Lambda_{\text{r}}$
is not unexpected for a phase transition or crossover, however, the peak in
$B_{\text{wTF}}$ is more unusual. We suggest that since, as $T$ increases, the
anisotropy in GaV4S8 decreases [5], this will lead to a changing orientational
preference of the magnetic domains in the sample, altering the magnetic field
experienced by the muons outside the samples due to the macroscopic fields
from the different grains. The field experienced by the muons will then peak
in the FM∗ state when domains are most aligned, which should occur at the
highest temperature where the FM∗ state is realized. The peak in
$B_{\text{wTF}}$, therefore, coincides with the transition between the FM∗ and
C states.

Figure 3: (a) Example ZF $\mu^{+}$SR asymmetry spectra of GaV4S8 at various
pressures, all at 2 K. The internal field extracted from fitting the 1.68 GPa
data is shown in (b), and compared to the zero-pressure result reported in
Ref. [7]. All other measured pressures show the same shape [17]. The average
internal field for $8$ K $<T<T_{\text{c}}$ (i.e. in the C phase at all
pressures) at each pressure is shown in (c). (d) Simulations of the internal
field distribution as seen with ZF $\mu^{+}$SR for the cycloidal state in
GaV4S8 under various different possible conditions.

ZF $\mu^{+}$SR measurements were also performed as a function of pressure,
with example spectra shown in Fig. 3(a). Below $T_{\text{c}}$ the asymmetry
spectra $A\left(t\right)$ are well described by

$A\left(t\right)=a_{1}\cos\left(\gamma_{\mu}B_{\text{ZF}}+\phi\right)\exp\left(-\sigma_{1}^{2}t^{2}\right)\\\
+a_{2}\exp\left(-\Lambda_{2}t\right)+a_{3}\exp\left(-\sigma_{3}^{2}t^{2}\right)+a_{4}.$
(2)

Above $T_{\text{c}}$ only the $a_{3}$ and $a_{4}$ terms are needed. The first
term accounts for muons that stop in sites with a component of $B_{\text{ZF}}$
perpendicular to the initial muon-spin, leading to coherent precession,
whereas the second term accounts for muons that stop with the initial muon-
spin parallel to the local field and relax due to dynamics in the local field
distribution. In these measurements, there is a significant contribution from
the pressure cell, which is captured by the third and fourth terms, leading to
several parameters that are temperature-independent:
$a_{1}/(a_{1}+a_{2}+a_{3})$, $a_{2}/(a_{1}+a_{2}+a_{3})$,
$a_{3}/(a_{1}+a_{2}+a_{3})$. We find that $a_{1}+a_{2}+a_{3}$ increases with
$T$, and $\sigma_{3}$, and $a_{4}$ predominantly capture the temperature-
dependence of the cell. We are unable to extract any useful information from
$\Lambda_{2}$ (which is determined by dynamic fluctuations). We also find that
$\sigma_{1}$ (likely to be determined by a mixture of static disorder and
residual dynamics) is temperature-independent, and hence was refined
simultaneously at all $T$. The most interesting parameter is therefore the
internal field $B_{\text{ZF}}$ [Fig. 3(b)]. (The behavior we observe here
contrasts with our previous work [7] where two internal fields were detected,
one higher field, broadly matching the low-pressure $B_{\text{ZF}}$ found
here, and one lower field, which is not observable in this configuration,
likely due to the competing signal from the pressure cell.) At all pressures,
$B_{\text{ZF}}$ increases slightly with increasing $T$, as previously observed
[7] (shown for comparison), before collapsing to zero above $T_{\text{c}}$ as
expected.

Table 1: Method of extracting transition temperatures shown in Fig. 4 from different measurement types. Measurement | Parameter | Transition | Extracted feature
---|---|---|---
Magnetization | $M$ | $T_{\text{c}}$ | Local maximum
| $M$ | FM${}^{*}\rightarrow$ C | $M=M_{\text{max}}/2$
wTF $\mu^{+}$SR | $\Lambda_{\text{r}}$ | $T_{\text{c}}$ | Local maximum
| $\Lambda_{\text{r}}$ | FM${}^{*}\rightarrow$ C | Local maximum
| $B_{\text{wTF}}$ | $T_{\text{c}}$ | Local minimum
| $B_{\text{wTF}}$ | FM${}^{*}\rightarrow$ C | Local maximum
ZF $\mu^{+}$SR | $B_{\text{ZF}}$ | $T_{\text{c}}$ | $B_{\text{ZF}}=0$

Despite no notable changes in the saturation magnetization with pressure [Fig.
1(b)], the local internal field significantly increases as the pressure is
increased, as shown in Fig. 3(c). This suggests a change to the structure of
the C state that does not lead to a change in the net magnetization of the FM∗
state. To explore possibilities that explain this, we consider the changes to
the Hamiltonian previously suggested: an increase in the DMI, or a decrease in
the anisotropy. An increase in DMI with pressure would be expected to lead to
a decrease in the cycloidal period (as the wavelength
$\lambda_{\text{C}}\propto\mathcal{A}/D$). We can simulate the effect this
change in the period would have on the distribution of fields seen at the muon
site following the approach in Ref. [7], which was shown to describe the ZF
$\mu^{+}$SR data. The values of $\sigma_{1}$ and $\Lambda_{2}$ are the same
order of magnitude as the comparable parameters $\Lambda_{i}$ in Ref. [7],
suggesting the width of the distribution of magnetic fields at the muon site
remains similar upon application of pressure. We therefore conclude that the
simulations are likely still good descriptors of the ZF $\mu^{+}$SR data.

The simulated distribution [7] produced for the experimentally observed C
state ($\lambda_{\text{C}}=17$ nm, and the $\bm{q}$-vector perpendicular to
the [001] direction [31]) is shown in Fig. 3(d). We first consider the effect
on the distribution that the small change in lattice parameters arising due to
the pressure would have. Taking the change in cell volume from Ref. [8], and
assuming the muon sites remain unchanged (a reasonable assumption given the
minimal change to the lattice parameters), we can see that the spectrum is
almost unchanged. Reducing the cycloidal period to $\lambda_{\text{C}}=8$ nm
(equivalent to approximately doubling the relative strength of the DMI)
changes the form, but not the position of the features (i.e. $B_{\text{ZF}}$)
in the spectrum. It is therefore unlikely that this can explain the observed
increase in internal field. Further, as changes to $\mathcal{A}$ would also
result in changes to $\lambda_{\text{C}}$, these results also support the
conclusion from the magnetization measurements that $\mathcal{A}$ does not
significantly change with pressure.

One possible effect of a reduction in the anisotropy as the pressure is
increased could be a changed preference for the direction of the
$\bm{q}$-vector of the cycloidal state. In Fig. 3(d) we show that, if the
$\bm{q}$-vector were to point along the [001] direction, the larger-field peak
is shifted to higher fields, as seen experimentally. We note that changing the
$\bm{q}$-vector was the only way we could find to simulate the observed
increase in internal field without changing the net magnetization. Whilst we
cannot say for certain that this is the only orientation of the
$\bm{q}$-vector that would reproduce this result, we found no other feasible
direction when randomly sampling, or testing specific high symmetry
directions. We, therefore, suggest that a change in the anisotropy is a
possible explanation for both the reduction in temperature of the C to FM*
crossover, and the increase in internal field with increasing pressure.

We have summarized our magnetization and $\mu^{+}$SR results with a suggested
$P$–$T$ phase diagram in the absence of an applied field, Fig. 4. We have
extracted the transition temperatures from the different types of measurement
as shown in Table 1. We find that $T_{\text{c}}$ slightly increases with
applied pressure, whereas the C to FM* crossover decreases in $T$ as the
pressure increases. The error bar on the highest pressure, low temperature
point represents the minimum temperature we were able to measure.

Figure 4: The $P$–$T$ dependence of GaV4S8 based on magnetization (green) and
$\mu^{+}$SR (ZF: red; wTF: blue) measurements. Closed symbols indicate
$T_{\text{c}}$, above which the system is paramagnetic (PM), and open symbols
represent the ferromagnetic-like (FM∗) to cycloidal (C) transition. Circles
indicate the point comes from the extracted value of the internal field [Fig.
2(d), Fig. 3(b)], squares represent points from a feature in
$\Lambda_{\text{r}}$ [Figs. 2(c)], and triangles represent points from
magnetization [Fig. 1(a)]. The points are extracted as detailed in Table 1.
Figure 5: Results of micromagnetic simulations using the parameters in Sec.
II. (a) An example cycloidal (C) state. (b) Difference in energies between the
C and ferromagnetic (FM) states as a function of $D$ and $K$, with the
preferred cycloidal period shown in (c). (d) Energies of the skyrmion lattice
(SkL) state compared to the ground state (either FM, solid line, or C, dashed
line) as a function of applied magnetic field for different combinations of
$D$ and $K$.

To further study the effects on the magnetic states of these possible changes
in the DMI and/or anisotropy, we have performed micromagnetic simulations. We
use the Hamiltonian,

$\hat{\mathcal{H}}=-\mathcal{A}\bm{m}\cdot\nabla^{2}\bm{m}+D\left(\bm{m}\cdot\nabla
m_{z}-m_{z}\nabla\cdot\bm{m}\right)\\\
-K\left(\bm{m}\cdot\bm{u}\right)^{2}-\mu_{0}M_{\text{s}}\bm{m}\cdot\bm{H},$
(3)

which is appropriate for a $C_{nv}$ crystal system, with the micromagnetic
parameters defined in Sec. II. The exchange $\mathcal{A}$ term attempts to
locally align $\bm{m}$ whereas the DMI $D$ term prefers periodic rotation of
$\bm{m}$. The anisotropy $K$ term gives a preference for alignment along the
easy axis $\bm{u}$, and the final term lowers the energy when $\bm{m}$ is
aligned along the external field $\bm{H}$. We initialize different magnetic
configurations before allowing the micromagnetic solver to find a local energy
minimum, allowing us to probe the energy of many possible states.

In the absence of an applied field we find that the FM state is the most
energetically favorable, with the C state only slightly higher in energy.
(Note that, on the length scale of this simulation, each micromagnetic cell
contains around four V4 clusters, hence the FM∗ and FM state are
indistinguishable.) An example of the C state simulated can be seen in Fig.
5(a). We study the effect of changing the $D$ and $K$ parameters, keeping the
other parameters fixed, and identify the difference in energy between the
lowest energy C state (i.e. that which has the preferred C period) and the FM
state. This is shown in Fig. 5(b), where the lowest energy state is marked. As
expected, both increasing $D$ and decreasing $K$ change the preferred state
from FM to C. The minimum energy cycloidal period is shown in Fig. 5(c), and
shows that changes in $\lambda_{\text{C}}$ are mainly correlated with changes
in $D$.

Having established micromagnetics can reproduce the observed states, we can
examine the effect of applying a magnetic field. We consider the energy of the
skyrmion lattice (SkL) state compared to the ground state (either FM or C,
depending on the micromagnetic parameters chosen). We find that, for $D_{0}$,
$K_{0}$, (i.e. the zero-temperature parameters we expect in the absence of any
applied pressure) the SkL state is hard to stabilize, with very few
initialized SkL states retaining this character. The SkL state is never found
to be lower in energy than the C or FM state, as expected from the measured
phase diagram, where the SkL is only stabilized via thermal fluctuations above
$T=0$ in an applied field, typically over a range of 2–3 K below
$T_{\text{c}}$. Changing the values of $D$ and $K$ makes stabilizing the SkL
much more probable, with many more initialized SkL states staying in that
configuration, and the energy of the skyrmion state above the ground state
(normally C, apart from at high field in the $D=D_{0}$, $K=0$ case where the
ground state becomes FM) is much reduced. These micromagnetic simulations
therefore suggest that, if the effect of changing pressure is a change in the
$D$ or $K$ parameters, the SkL should still be stabilized at elevated
temperatures, as it is in the absence of an applied pressure, but with it
likely to persist down to lower $T$. However, since the SkL is still never
found to be the lowest energy state it is unlikely to ever be stabilized at
zero temperature.

TF $\mu^{+}$SR has previously been shown to be able to identify the SkL phase
in GaV4S8 via an increase in the internal field and the associated relaxation
rate [7]. We have performed TF $\mu^{+}$SR measurements of GaV4S8 at 50 mT and
100 mT (fields which stabilize a SkL in the absence of applied pressure) under
the application of pressure. The asymmetry spectra
$A_{\text{TF}}\left(t\right)$ are well described by

$A_{\text{TF}}\left(t\right)=\sum_{i=1}^{2}a_{i}\cos\left(\gamma_{\mu}B_{\text{TF},i}+\phi_{i}\right)\exp\left(-\sigma_{i}^{2}t^{2}\right)+a_{\text{b}}.$
(4)

The $i=2$ term captures the effects of muons stopping in the pressure cell
when the spins subsequently precess, hence we employed simultaneous refinement
of the associated temperature-independent parameters: $B_{\text{TF},2}$,
$\phi_{2}$, and $\sigma_{2}$. The total asymmetry $a_{1}+a_{2}$ increases at
$T_{\text{c}}$ as expected, with $a_{\text{b}}$ capturing temperature
dependence that appears to be coming from the cell (presumably from muons that
stop at sites in the cell where the total magnetic field is zero). The first
term, sensitive to the magnetism in GaV4S8, has a temperature-independent
$\phi_{1}$ which was simultaneously refined at all temperatures. The remaining
parameters $B_{\text{TF},1}$ and $\sigma_{1}$ are shown in Fig. 6.

At all applied pressures, the behavior is broadly similar in both parameters.
We first consider the measured internal field, Figs. 6(a,c). At 50 mT, there
is an enhancement of $B_{\text{TF}}$ compared to the applied field, as
previously seen in the absence of applied pressure [7]. There is no large
increase in this parameter over a limited temperature range that would signify
the stabilization of the SkL in that region, as has previously been observed
at ambient pressure. Instead, the internal field seems enhanced for all
$T<T_{\text{c}}$. Comparatively, at 100 mT the enhancement from the applied
field is smaller.

Figure 6: Parameters extracted from fitting of TF $\mu^{+}$SR measurements of
GaV4S8 at different applied pressures and magnetic fields. Dashed lines in (b)
and (d) are guides to the eye.

Considering now the relaxation rate $\sigma$ [Figs. 6(b,d)], we resolve two
clear regimes of behavior at 0.81 GPa and 1.68 GPa in both the 50 mT and 100
mT measurements, as marked by dashed lines. The crossover $T$ between these
two regimes decreases as the pressure is increased, following a similar trend
seen earlier for the FM∗ to C transition. By 2.23 GPa it is difficult to split
the behavior into separate regimes, again as seen in the zero-field data. We
conclude, therefore, that pressure acts to reduce the characteristic
temperature scales of the magnetic field fluctuations in this part of the
phase diagram. We stress that we are unable to resolve those features seen at
ambient pressure [7] that unambiguously mark the onset of the SkL. However,
since fluctuations in the field distribution experienced by the muon have
previously been shown to track the onset of the skyrmion phase, we might
speculate that these data are consistent with the SkL phase extending to lower
$T$, as anticipated from the micromagnetic results. The phase diagram would
therefore look more like that of GaV4Se8, which has a SkL stable down to low
$T$ [32].

##  IV Conclusion

Our magnetization and $\mu^{+}$SR measurements of GaV4S8 reveal that the
application of pressure lowers the crossover temperature of the C to FM*
transition. By 2.29 GPa, we cannot observe this crossover, and suggest that
GaV4S8 remains in the C state down to the lowest measured temperatures. On
increasing the applied pressure, the internal field as measured by $\mu^{+}$SR
increases without any corresponding increase in the magnetization. Through
simulations of the distribution of magnetic fields at the muon stopping sites,
we suggest this occurs due to a reorientation of the direction of the
$\bm{q}$-vectors due to a reduction in the magnetic anisotropy of the system.
Investigation of this phenomena with a $\bm{q}$-resolved experimental probe
would be highly beneficial in a future study. Micromagnetic simulations
suggest further that a decrease in the magnetic anisotropy should lead to an
increase in stability of the SkL state in GaV4S8.

There are multiple reasons that application of pressure may change the
anisotropy of the system. The effect of pressure might be to make the
underlying electronic structure more isotropic, hence explaining the above
results. Further, the appearance of an easy-axis in the system is set by the
direction of the Jahn-Teller distortion; given our results on the increase of
the temperature of the Jahn-Teller distortion with pressure, and the
possibility of the changing nature of this distortion, it seems highly likely
that the anisotropy would also have corresponding changes with pressure. Given
the possible different scenarios we set out to explain the changes to the
Jahn-Teller distortion with pressure, experiments to determine the crystal
structure of GaV4S8 as a function of both temperature and pressure are an area
of research that should be explored.

Our work shows the sensitivity of the magnetism in GaV4S8 to applied
pressures, and demonstrates that changes in both the magnetic phase diagram
and underlying magnetic interaction strengths can be expected. This provides
the opportunity to tailor the magnetic interactions of this system, and
perhaps related systems in the GaV4S8-ySey series, and investigate the effects
on the magnetism. Understanding the changes these parameters make is of great
importance for understanding the behavior of these complex magnetic states,
which is essential if they are to be used for applications.

## Acknowledgments

Part of this work was carried out at the Swiss Muon Source, Paul Scherrer
Institute, Switzerland; we are grateful for the provision of beamtime. We are
grateful for computational support from Durham Hamilton HPC. The project was
funded by EPSRC (UK) (Grant Nos.: EP/N032128/1 and EP/N024028/1). M. N. Wilson
acknowledges the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC). Research data from this paper will be made
available via Durham Collections at XXX.

## References

  * Brasen  _et al._ [1975] D. Brasen, J. M. Vandenberg, M. Robbins, R. H. Willens, W. A. Reed, R. C. Sherwood, and X. J. Pinder, Magnetic and crystallographic properties of spinels of the type AxB2S4 where A = Al, Ga, and B = Mo, V, Cr, Journal of Solid State Chemistry 13, 298 (1975).
  * Lancaster [2019] T. Lancaster, Skyrmions in magnetic materials, Contemporary Physics 60, 246 (2019).
  * Kézsmárki  _et al._ [2015] I. Kézsmárki, S. Bordács, P. Milde, E. Neuber, L. M. Eng, J. S. White, H. M. Rønnow, C. D. Dewhurst, M. Mochizuki, K. Yanai, _et al._ , Néel-type skyrmion lattice with confined orientation in the polar magnetic semiconductor GaV4S8, Nature Materials 14, 1116 (2015).
  * Holt  _et al._ [2020] S. J. R. Holt, A. Štefančič, C. Ritter, A. E. Hall, M. R. Lees, and G. Balakrishnan, Structure and magnetism of the skyrmion hosting family GaV4S8-ySey with low levels of substitutions between $0\leq y\leq 0.5$ and $7.5\leq y\leq 8$, Physical Review Materials 4, 114413 (2020).
  * Ehlers  _et al._ [2016] D. Ehlers, I. Stasinopoulos, I. Kézsmárki, T. Fehér, V. Tsurkan, H. K. von Nidda, D. Grundler, and A. Loidl, Exchange anisotropy in the skyrmion host GaV4S8, Journal of Physics: Condensed Matter 29, 065803 (2016).
  * Holt  _et al._ [2021] S. J. R. Holt, C. Ritter, M. R. Lees, and G. Balakrishnan, Investigation of the magnetic ground state of GaV4S8 using powder neutron diffraction, Journal of Physics: Condensed Matter 33, 255802 (2021).
  * Hicken  _et al._ [2020] T. J. Hicken, S. J. R. Holt, K. J. A. Franke, Z. Hawkhead, A. Štefančič, M. N. Wilson, M. Gomilšek, B. M. Huddart, S. J. Clark, M. R. Lees, _et al._ , Magnetism and Néel skyrmion dynamics in GaV4S8-ySey, Physical Review Research 2, 032001(R) (2020).
  * Wang  _et al._ [2021] Y. Wang, S. Rahman, E. Sun, C. Knill, D. Zhang, L. Wang, V. Tsurkan, and I. Kézsmárki, From semiconducting to metallic: Jahn–Teller-induced phase transformation in skyrmion host GaV4S8, Journal of Physical Chemistry C 125, 5771 (2021).
  * Mokdad  _et al._ [2019] J. Mokdad, G. Knebel, C. Marin, J.-P. Brison, V. T. Phuoc, R. Sopracase, C. Colin, and D. Braithwaite, Structural, magnetic, and insulator-to-metal transitions under pressure in the GaV4S8 Mott insulator: A rich phase diagram up to 14.7 GPa, Physical Review B 100, 245101 (2019).
  * Franke  _et al._ [2018] K. J. A. Franke, B. M. Huddart, T. J. Hicken, F. Xiao, S. J. Blundell, F. L. Pratt, M. Crisanti, J. A. T. Barker, S. J. Clark, A. Štefančič, _et al._ , Magnetic phases of skyrmion-hosting GaV4S8-ySey ($y$ = 0, 2, 4, 8) probed with muon spectroscopy, Physical Review B 98, 054428 (2018).
  * White  _et al._ [2018] J. S. White, A. Butykai, R. Cubitt, D. Honecker, C. D. Dewhurst, L. F. Kiss, V. Tsurkan, and S. Bordács, Direct evidence for cycloidal modulations in the thermal-fluctuation-stabilized spin spiral and skyrmion states of GaV4S8, Physical Review B 97, 020401(R) (2018).
  * Clements  _et al._ [2020] E. M. Clements, R. Das, G. Pokharel, M. H. Phan, A. D. Christianson, D. Mandrus, J. C. Prestigiacomo, M. S. Osofsky, and H. Srikanth, Robust cycloid crossover driven by anisotropy in the skyrmion host GaV4S8, Physical Review B 101, 094425 (2020).
  * Izyumov [1984] Y. A. Izyumov, Modulated, or long-periodic, magnetic structures of crystals, Soviet Physics Uspekhi 27, 845 (1984).
  * Štefančič  _et al._ [2020] A. Štefančič, S. J. Holt, M. R. Lees, C. Ritter, M. J. Gutmann, T. Lancaster, and G. Balakrishnan, Establishing magneto-structural relationships in the solid solutions of the skyrmion hosting family of materials: GaV4S8-ySey, Scientific Reports 10, 9813 (2020).
  * Blundell [1999] S. Blundell, Spin-polarized muons in condensed matter physics, Contemporary Physics 40, 175 (1999).
  * Blundell  _et al._ [2021] S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt,  _Muon Spectroscopy: An Introduction_ (Oxford University Press, Oxford, 2021).
  * [17] See supplemental material for further information on the $\mu^{+}$SR technique, and some additional data.
  * Khasanov  _et al._ [2016] R. Khasanov, Z. Guguchia, A. Maisuradze, D. Andreica, M. Elender, A. Raselli, Z. Shermadini, T. Goko, F. Knecht, E. Morenzoni, _et al._ , High pressure research using muons at the Paul Scherrer Institute, High Pressure Research 36, 140 (2016).
  * Shermadini  _et al._ [2017] Z. Shermadini, R. Khasanov, M. Elender, G. Simutis, Z. Guguchia, K. V. Kamenev, and A. Amato, A low-background piston–cylinder-type hybrid high pressure cell for muon-spin rotation/relaxation experiments, High Pressure Research 37, 449 (2017).
  * Pratt [2000] F. L. Pratt, WiMDA: a muon data analysis program for the Windows PC, Physica B: Condensed Matter 289, 710 (2000).
  * James and Roos [1975] F. James and M. Roos, MINUIT: a system for function minimization and analysis of the parameter errors and corrections, Computer Physiscs Commununications 10, 343 (1975).
  * iminuit team [2021] iminuit team, iminuit – a python interface to minuit, https://github.com/scikit-hep/iminuit (Accessed: 16-09-2021).
  * Bonfà  _et al._ [2018] P. Bonfà, I. J. Onuorah, and R. De Renzi, Introduction and a quick look at MUESR, the Magnetic structure and mUon Embedding Site Refinement suite, in  _Proceedings of the 14th International Conference on Muon Spin Rotation, Relaxation and Resonance ( $\mu$SR2017)_ (2018) p. 011052.
  * Beg  _et al._ [2017] M. Beg, R. A. Pepper, and H. Fangohr, User interfaces for computational science: A domain specific language for OOMMF embedded in Python, AIP Advances 7, 056025 (2017).
  * Beg  _et al._ [2021] M. Beg, R. A. Pepper, T. Kluyver, J. Mulkers, J. Leliaert, and H. Fangohr, ubermag: Meta package for ubermag project. (Accessed: 16-09-2021).
  * Padmanabhan  _et al._ [2019] P. Padmanabhan, F. Sekiguchi, R. B. Versteeg, E. Slivina, V. Tsurkan, S. Bordács, I. Kézsmárki, and P. H. M. Van Loosdrecht, Optically driven collective spin excitations and magnetization dynamics in the Néel-type skyrmion host GaV4S8, Physical Review Letters 122, 107203 (2019).
  * Routledge  _et al._ [2021] K. Routledge, P. Vir, N. Cook, P. A. E. Murgatroyd, S. J. Ahmed, S. N. Savvin, J. B. Claridge, and J. Alaria, Mode crystallography analysis through the structural phase transition and magnetic critical behavior of the lacunar spinel GaMo4Se8, Chemistry of Materials 33, 5718 (2021).
  * Schueller  _et al._ [2020] E. C. Schueller, D. A. Kitchaev, J. L. Zuo, J. D. Bocarsly, J. A. Cooley, A. Van der Ven, S. D. Wilson, and R. Seshadri, Structural evolution and skyrmionic phase diagram of the lacunar spinel GaMo4Se8, Physical Review Materials 4, 064402 (2020).
  * Müller  _et al._ [2006] H. Müller, W. Kockelmann, and D. Johrendt, The magnetic structure and electronic ground states of Mott insulators GeV4S8 and GaV4S8, Chemistry of Materials 18, 2174 (2006).
  * Bichler  _et al._ [2008] D. Bichler, V. Zinth, D. Johrendt, O. Heyer, M. K. Forthaus, T. Lorenz, and M. M. Abd-Elmeguid, Structural and magnetic phase transitions of the v 4-cluster compound GeV4S8, Physical Review B 77, 212102 (2008).
  * Ruff  _et al._ [2015] E. Ruff, S. Widmann, P. Lunkenheimer, V. Tsurkan, S. Bordács, I. Kézsmárki, and A. Loidl, Multiferroicity and skyrmions carrying electric polarization in GaV4S8, Science advances 1, e1500916 (2015).
  * Fujima  _et al._ [2017] Y. Fujima, N. Abe, Y. Tokunaga, and T. Arima, Thermodynamically stable skyrmion lattice at low temperatures in a bulk crystal of lacunar spinel GaV4Se8, Physical Review B 95, 180410 (2017).