A Low Temperature Structural Transition in Canfieldite, Ag8SnS6, Single
Crystals
Tyler J. Slade1,2, Volodymyr Gvozdetskyi3, John M. Wilde1,2, Andreas
Kreyssig1,2, Elena Gati1,2, Lin-Lin Wang1,2, Yaroslav Mudryk1, Raquel A.
Ribeiro1,2, Vitalij K. Pecharsky1,4, Julia V. Zaikina3, Sergey L. Bud’ko1,2,
Paul C. Canfield1,2

1Ames Laboratory, US DOE, Iowa State University, Ames, Iowa 50011, USA
2Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA
3Department of Chemistry, Iowa State University, Ames, Iowa 50011, USA
4Department of Materials Science and Engineering, Iowa State University, Ames,
IA 50011, USA

###### Abstract

Canfieldite, Ag8SnS6, is a semiconducting mineral notable for its high ionic
conductivity, photosensitivity, and low thermal conductivity. We report the
solution growth of large single crystals of Ag8SnS6 of mass up to 1 g from a
ternary Ag-Sn-S melt. On cooling from high temperature, Ag8SnS6 undergoes a
known cubic (F$\bar{4}3$m) to orthorhombic (Pna21) phase transition at
$\approx$ 460 K. By studying the magnetization and thermal expansion between
5–300 K, we discover a second structural transition at $\approx$ 120 K. Single
crystal X-ray diffraction reveals the low temperature phase adopts a different
orthorhombic structure with space group Pmn21 (a = 7.6629(5) Å, b = 7.5396(5)
Å, c = 10.6300(5) Å Z = 2 at 90 K) that is isostructural to the room
temperature forms of the related Se-based compounds Ag8SnSe6 and Ag8GeSe6. The
120 K transition is first-order and has a large thermal hysteresis. Based on
magnetization and thermal expansion data, the room temperature polymorph can
be kinetically arrested into a metastable state by rapidly cooling to
temperatures below 40 K. We lastly compare the room and low temperature forms
of Ag8SnS6 with its argyrodite analogues, Ag8TQ6 (T = Si, Ge, Sn; Q = S, Se),
and identify a trend relating the preferred structures to the unit cell
volume, suggesting smaller phase volume favors the Pna21 arrangement. We
support this picture by showing that the transition to the Pmn21 phase is
avoided in Ge alloyed Ag8Sn1-xGexS6 samples as well as pure Ag8GeS6.

##  1 Introduction

Multinary silver chalcogenides exhibit rich structural chemistry and
electronic properties that generate both fundamental and technological
interest. A hallmark of these compounds is weak bonding between silver and
chalcogenide atoms, often resulting in diverse crystal structures, substantial
anharmonic atomic displacements, site occupancy disorder, and/or metal atoms
in unique, distorted coordination environments.[1, 2, 3, 4] Consequentially,
many silver chalcogenides exhibit high ionic conductivity and a tendency to
undergo phase transitions upon heating or cooling.[5, 6, 7, 8] The soft
bonding and structural complexity can give rise to low phonon velocities and
intrinsically glasslike thermal conductivity. As such, many silver
chalcogenides are attractive thermoelectric materials.[9, 10, 11, 12, 13, 14,
15] Compared to oxides, the bonds in metal chalcogenides are generally more
covalent, which affords smaller band gaps and higher charge carrier
mobility,[16] making some chalcogenides promising for solar energy
conversion.[17]

The past decades witnessed impressive advancement in the synthesis of
chalcogenide materials. Whereas the high vapor pressures and low boiling
points of the chalcogens, particularly sulfur, are challenges for the growth
of large single crystals of chalcogen-rich compounds, solution or flux
techniques have proven invaluable for their ability to permit chemical
reactions and crystallization at moderate temperatures compared to traditional
solid state methods.[18, 19, 20] In this regard, polychaclcogenide salts are
powerful solvents for the synthesis of alkali metal and main group
chalcogenide compounds.[20, 21] Low-melting binary metal-sulfur compositions
or eutectics can also be effective starting points for solution growth of
sulfur rich materials.[18]

Canfieldite,[22] Ag8SnS6, is a mineral belonging to the family of closely
related argyrodite compounds.[2] Like all argyrodites, Ag8SnS6 adopts a high
temperature cubic crystal structure with the F$\bar{4}3$m space group that
features a highly disordered Ag sublattice in which the Ag ions are
distributed over 3 partially occupied crystallographic sites. Previous work
indicates that at 460 K, Ag8SnS6 undergoes a transition from the high
temperature (ht) cubic phase to the room temperature (rt) orthorhombic
structure with space group Pna21.[2, 23, 24] Upon heating, the structural
transition can be visualized as the Ag sublattice “melting” into the highly
disordered state while the Sn and S framework remains essentially rigid, or
intact. As a consequence, the ht phase has high ionic conductivity supported
by mobile Ag+ ions. The heavy disorder in the “liquid like” Ag sublattice
facilitates exceptionally strong phonon scattering, and the lattice thermal
conductivity of ht-Ag8SnS6 approaches the theoretical amorphous minimum.[24]
Accordingly, recent work demonstrates that ht-Ag8SnS6 is a promising
thermoelectric material with a figure of merit zT of 0.8 near 773 K when
alloyed with Se.[24, 25] Most work on the rt, orthorhombic, variant is
concerned with nanocrystals,[26, 27, 28] which have been prepared and studied
as counter electrodes in solar cells and for photocatalytic water splitting
and dye degradation.[29, 30, 31] Notably, the rt phase has a band gap of
approximately 1.4 eV and high absorption coefficient of 104 cm-1 in the
visible range,[32] making rt Ag8SnS6 a potential light absorber for
photovoltaic applications.[33, 34]

Whereas both the ambient and high temperature forms of Ag8SnS6 exhibit
interesting properties and structural chemistry, the evolution of the crystal
structure and physical properties below room temperature are yet to be
reported. Here, we demonstrate solution growth of large ($\approx$1 g) single
crystals of Ag8SnS6 out of a Sn-S rich ternary melt, and by use of temperature
dependent magnetization, thermal expansion, and X-ray diffraction
measurements, we identify a previously unreported first-order structural phase
transition at 120 K. We find that below 120 K, Ag8SnS6 can adopt a different
low temperature (lt) orthorhombic structure with the Pmn21 space group that is
isostructural to the room temperature forms of the related Se-based compounds
Ag8SnSe6 and Ag8GeSe6. The rt to lt structural phase transition has large
thermal hysteresis, and the magnetization and thermal expansion results
indicate the room temperature phase can be kinetically arrested by rapid
cooling into a metastable state below 40 K. By comparing the room and low
temperature Ag8SnS6 polymorphs with the preferred structures of the argyrodite
analogues Ag8TQ6 (T = Si, Ge, Sn; Q = S, Se), we suggest that lower volume
unit cells (normalized per formula unit) favor the rt Pna21 form. We support
this picture by applying chemical pressure, i.e. reducing the unit cell
volume, to Ag8SnS6 by partially replacing Sn with Ge, and find the rt Pna21
phase is preserved down to 5 K in Ag8Sn1-xGexS6 and pure Ag8GeS6. Lastly, we
use density functional theory calculations to discuss the role of Ag-S bonding
in dictating the energetically preferred structure.

##  2 Experimental Details

Crystal growth: Both of the Ag-S and Sn-S binary phase diagrams have
accessible liquid regions for compositions near Ag2S and compositions between
SnS2 and SnS with eutectic temperatures as low as 740°C.[35, 36, 37] Perhaps
more importantly, studies of parts of the ternary Ag-Sn-S phase diagram
indicate the existence of an exposed liquidus surface for the primary
formation of Ag8SnS6.[38] Based on the existing phase diagram data, as well as
our experience with S-based growths, we used Ag2S (Alfa-Aesar, 99.5 $\%$), S
(Alfa-Aesar 99.99$\%$) and Sn (Alfa-Aesar, 99.99$\%$) in a molar ratio
35:38:27 respectively, giving a stoichiometry of Ag41Sn16S43 as the initial
composition of our melt.

To test our ability to safely grow Ag8SnS6 out of the ternary melt in a well
controlled manner, we first placed the Ag2S-S-Sn mixture into a fritted
alumina crucible set,[39] sealed the crucible set into an amorphous silica
ampule under a 1/4 atm of high purity Ar, and gradually heated to 800°C over
10 hours. After holding at 800°C for 10 hours, the ampule was removed from the
furnace, inverted into a centrifuge, and decanted.[39, 40] All of the
materials passed through the frit with (i) no apparent attack on the alumina
crucible, (ii) no evaporative loss of material from the crucible, (iii) no
evidence for over-pressure in the growth ampule. Based on these results, we
performed a second growth, again heating to 800°C, dwelling for 10 h, and then
cooling over 200 hours to 625°C. After decanting the excess liquid, large,
mirror-faceted, single crystals of Ag8SnS6 weighing up to 1 gram with
dimensions as large as 5-10 mm were recovered (see the inset of Figure 1). Ge
alloyed samples were prepared in the same manner, substituting elemental Ge
(Alfa-Aesar, 99.99$\%$) for Sn using initial compositions of
Ag41(Sn1-xGex)16S43 to produce single crystals of Ag8Sn1-xGexS6 (x = 0.1,
0.25, 1).

Powder X-ray diffraction: To confirm the structure and phase purity of the
flux grown crystals, samples were analyzed with powder X-ray diffraction
(PXRD). Pieces of the crystals were ground by hand in a mortar and pestle to a
fine powder, sifted through a 35 micron mesh sieve, and diffraction patterns
were collected at room temperature on a Rigaku Miniflex-II instrument
operating with Cu-K$\alpha$ radiation ($\lambda$ = 1.5406 Å) at 30 kV and 15
mA. The crystal structure was refined using the Rietveld method with GSAS-II
software.[41]

Magnetization: Temperature-dependent magnetization measurements were performed
in Quantum Design Magnetic Property Measurement System (MPMS-classic and
MPMS-3) SQUID magnetometers operating in the DC measurement mode and in a
field 10 kOe. To avoid potential oxygen adsorption on the surface of the
samples, several randomly oriented crystals (total mass $\sim 80$ mg) of
Ag8SnS6 were sealed in a partial argon atmosphere between two fused silica
rods of similar length in a fused silica tube. The length of the rods and the
tube were such that the only additional contribution to the magnetic signal
from the assembly comes from the gap between the rods in which the samples are
contained. This contribution was subtracted using the size of the gap and
previous calibration. The powder samples were measured in a gel capsule which
was mounted in a straw. To allow for evacuation, pinholes were poked in the
straw and in the top of the capsule. For the powder samples, a separate
background from an empty capsule was measured and subtracted. The measurements
were conducted between 5–300 K with a heating/cooling rate of 0.5
K$\cdot$min-1 between each temperature step. To study hysteresis and low
temperature metastability, we also quenched (cooled as rapidly as possible, at
$\approx$ 12 K$\cdot$min-1) the sample to base temperature and measured on
warming. In addition, time dependent measurements of the magnetization were
performed at 40 K, 60 K and 80 K for up to 3,000 minutes (see Figure S4).
These measurements were made after rapidly cooling ($\approx$ 12
K$\cdot$min-1) the sample from 300 K to the target temperature (where the
samples were cooled with the field applied).

Thermal expansion: Measurements of the macroscopic length change of a crystal
of Ag8SnS6 were performed using strain gauges. Similar to the design of Kabeya
et al.,[42] the full setup used a Wheatstone configuration consisting of two
strain gauges and two thin-film resistors. One strain gauge (type FLG-02-23,
Tokyo Sokki Kenkyujo Co., Ltd. with R $\approx$ 120 $\Omega$) was fixed
rigidly to the sample by using Devcon 5 minute epoxy (No. 14250). The second
(identical) strain gauge was fixed to a sample of tungsten carbide, which has
a comparatively small thermal expansion coefficient over a wide temperature
range, to roughly balance out the intrinsic resistance changes of the strain
gauges. In addition, two thin-film resistors with similar and almost
temperature independent absolute resistances of $\approx$ 120 $\Omega$ were
used to complete the Wheatstone bridge. The measured resistance changes were
converted into relative length changes using the known gauge factor of the
strain gauges (that was assumed to be temperature independent). The cryogenic
environment was provided by a closed cycle cryostat (Janis SHI-950 with a base
temperature of $\approx$ 3.5 K). Temperature control was provided by a
LakeShore 336 temperature controller. Helium exchange gas was used to allow
for fast cooldown times. The measurements were not aligned along a specific
crystallographic direction.

Structure determination: The crystal structure of Ag8SnS6 was determined at
295, 160 and 90 K by single crystal X-ray diffraction. Single crystal data
were collected with a Bruker D8 VENTURE diffractometer (Photon CMOS detector,
Mo-I$\mu$S microsource and Oxford Cryosystem 800 low temperature device). Data
integration, absorption correction, and unit cell determination were performed
with APEX 3 software. The starting atomic parameters were obtained by direct
methods with the SHELXS-2017.[43] Subsequently, the structures were refined
using SHELXL-2017 (full-matrix least-squares on Fo2).

Single crystal X-ray diffraction data were first collected at 295 K.
Afterwards, the sample was slowly cooled to 90 K at a rate 1 K$\cdot$min-1,
held at 90 K for 30 minutes, and a full data set was collected at 90 K. The
next datasets were collected on heating at 160 K and 295 K. In both cases, the
crystal was warmed at 6 K$\cdot$min-1 and held at the target temperatures for
30 minutes prior to collection. Lastly, single crystal data was obtained on a
crystal cooled as quickly as possible to 90 K. To accomplish this, the crystal
was dismounted and rapidly (a few seconds) mounted back, directly into the
nitrogen gas flow, allowing for fast cooling down to 90 K.

Variable-temperature lattice parameters: The temperature dependent lattice
constants were determined by tracking the room temperature (rt) orthorhombic
$(8\,0\,4)_{\text{rt}}$, $(0\,4\,4)_{\text{rt}}$, and $(0\,0\,6)_{\text{rt}}$
Bragg peaks collected on single crystals. Measurements were performed on a
four-circle diffractometer using Cu K$\alpha_{1}$ radiation generated from a
rotating anode X-ray source using a germanium (1 1 1) monochromator. Scattered
X-rays were measured using an FMB Oxford avalanche pixel diode (APD) single
detector system. The sample was mounted on a copper pin using Crystal Bond.
Sample was mounted in an ARS 4-800 K He-closed cycle refrigerator. The sample
space is sealed with He exchange gas within the inner Be dome. Two further Be
domes were mounted to act as a heat shield and the vacuum shroud.

Because during the growth process, the crystals are cooled through a known
high temperature cubic to orthorhomic structural transition that occurs at 460
K[24], the orthorhombic samples used in our experiments are highly twinned,
i.e. cryptomorphic. The crystals form with facets perpendicular to the [1 1
1]${}_{\text{PC}}$ (PC = pseudo-cubic) direction which in the rt orthorhombic
structure split into the [2 0 1]${}_{\text{rt}}$ and [0 1 1]${}_{\text{rt}}$
directions. We aligned the crystal within the [0 1 1]${}_{\text{rt}}$-[1 1
1]${}_{\text{rt}}$ plane, which gives access to a large number of peaks from
the available twin domains. Two peaks, the $(8\,0\,4)_{\text{rt}}$ and
$(0\,4\,4)_{\text{rt}}$, were isolated from other peaks to give a measure of
the a and b lattice parameters. The out of plane $(0\,0\,6)_{\text{rt}}$ peak
was measured to determine the c lattice parameter. We first cooled from 300 K
to 10 K at 0.2 K$\cdot$min-1. The diffraction data used to determine the
lattice parameters were collected by following the evolution of the above
peaks on warming at 0.1 K$\cdot$min-1, and the lattice parameters were
calculated using Braggs’ law.

Computational Methods: Total energies and density of states (DOS) were
calculated in density functional theory[44, 45] (DFT) with PBEsol[46]
exchange-correlation functional, a plane-wave basis set and projected
augmented wave[47] (PAW) method as implemented in VASP.[48, 49] A
$\Gamma$-centered Monkhorst-Pack[50] k-point mesh of (6$\times$6$\times$4) and
(3$\times$6$\times$4) with a Gaussian smearing of 0.05 eV was used for the lt-
and rt-structures with 30 and 60 atoms in the unit cells, respectively. A
kinetic energy cutoff of 323.4 eV was used to fully relax the unit cell and
atomic positions until the absolute force on each atom is below 0.02 eV/Å. For
bonding and anti-bonding analysis, the LOBSTER[51] code was used to construct
the crystal orbital Hamiltonian population (COHP) for nearest neighbor Ag-S
pairs with the bond distance from 2.2 to 3.0 Å.

##  3 Results and Discussion

Figure 1: Powder X-ray diffraction pattern obtained on a powdered crystal of
Ag8SnS6. The solid lines are the fitted results of a Rietveld refinement and
are in good agreement with the room temperature Pna21 structure of Ag8SnS6
with refinement statistics R${}_{\text{wp}}$ = 4.95 and GOF = 1.39. The inset
is a picture of a flux grown crystal on a mm grid; the grid is reflected in
two of the facets.

The inset of Figure 1 displays an example of a single crystal grown from the
Ag-Sn-S melt. To determine the identity and phase purity of the crystals, we
analyzed samples with powder X-ray diffraction at room temperature. Figure 1
shows a representative powder pattern and refinement of the experimental data.
The crystals adopt an orthorhombic structure with the Pna$2_{1}$ space group
consistent with the room temperature (rt) phase of Ag8SnS6. The refined
lattice constants are a = 15.3155(9) Å, b = 7.5532(4) Å, c = 10.7000(6) Å, in
agreement with prior reports of Ag8SnS6.[27, 23, 24] The experimental pattern
has no observable peaks from secondary phases, supporting successful synthesis
of phase-pure materials within the accuracy of our PXRD analysis. Close
inspection revealed the crystals are heavily twinned, essentially
cryptomorphic, forming with facets perpendicular to the pseudo-cubic [1 1
1]${}_{\text{PC}}$ direction. The twinning likely happens when cooling through
the structural transition between the ht-cubic and rt orthorhombic phases that
occurs at 460 K in Ag8SnS6.[24]

Considering the known structural diversity among the broader argyrodite family
and the limited low temperature data on Ag8SnS6, we explored the potential for
phase transitions in Ag8SnS6 below room temperature. We first measured the
magnetization of Ag8SnS6 between 5–300 K. The temperature dependent magnetic
susceptibility (M/H) is presented in Figure 2a, which shows M/H is negative at
all temperatures, indicating diamagnetic behavior, and is consistent with our
expectation of Ag8SnS6 being a valence precise semiconductor. At high
temperatures, M/H has a weak temperature dependence and increases slowly on
cooling from -2.6$\times$10-4 emu$\cdot$mol-1 at 300 K to -2.55$\times$10-4
emu$\cdot$mol-1 at 120 K. The susceptibility data has an abrupt discontinuity
at 120 K, falling from -2.55$\times$10-4 emu$\cdot$mol-1 to -2.62 $\times$10-4
emu$\cdot$mol-1. Given Ag8SnS6 is a moderate gap semiconductor with no local
moment behavior expected or measured, the clear transition observed in M/H
suggests Ag8SnS6 undergoes a structural phase transition below $\approx$ 120
K.

Figure 2: Temperature dependent physical properties of Ag8SnS6 single
crystals. (a) Magnetic susceptibility (M/H), (b) Magnitude of the thermal
expansion. (c–e) lattice parameters, and (f) unit cell volume. All datasets
were measured on heating. The lattice parameters are shown using the basis of
the rt phase (art = 2a${}_{\text{lt}}$).

To provide more detailed characterization of the phase transition, we tracked
the thermal expansion of Ag8SnS6 between 5 and 300 K. Figure 2b displays the
measurement results, given as the magnitude of the temperature dependent
length change normalized to the initial length at room temperature,
[L(T)-L(300 K)]/L(300 K) (referred to in the text as $\Delta$L/L)). As shown
in Figure 2b, the values of $\Delta$L/L show a rapid and large drop when
warming above 120 K, implying the sample dimensions suddenly contract when
heating above this temperature. The sudden, jump-like, length change suggests
a first-order phase transition, and is characteristic of a structural
transformation. The transition temperature closely matches what is observed in
the magnetization data, and together the two measurements strongly suggest
Ag8SnS6 undergoes a first-order structural transition at 120 K. The feature
near 30 K in Figure 2b is an artifact caused by the slightly different
temperature dependencies of the resistances of the strain gauges (used to
determine $\Delta$L/L) at low temperatures.

Having found evidence for a previously unknown structural phase transition in
Ag8SnS6, we performed single crystal X-ray diffraction at 295 K, 160 K and 90
K to determine the crystal structure of the low temperature (lt) phase.
Details regarding the structural refinements are given in the Supporting
Information in Table S1, and the crystallographic information regarding the
atomic positions, thermal displacement parameters, and individual Ag-S and
Sn-S bond lengths are provided in Tables S2-S11. Figure 3 compares the room
and low temperature structures of Ag8SnS6 as determined by our single crystal
diffraction analysis conducted at 295 and 90 K.

The first dataset collected at 295 K agrees with the expected rt orthorhombic
unit cell with space group Pna21 ($\\#$33) and a = 15.2993(9) Å, b = 7.5479(4)
Å, c = 10.7045(6) Å, and phase volume V/Z = 309.03(3) Å3, where Z = 4 is the
number of formula units per unit cell in the rt structure. These results are
consistent with earlier literature.[52, 24] The rt structure is complex, as
illustrated by Figures 3a and 3b. All the atoms occupy the Wyckoff position 4a
and are labeled respectively as Ag1-Ag8, Sn, and S1-S6. The local environments
of the Ag atoms are diverse, with Ag found in distorted tetrahedra (Ag2 and
Ag6) as well as trigonal planar (Ag1, Ag3, Ag4, and Ag7) and linear (Ag5)
coordination by S. As given in Table S10 of the Supporting Information, the Ag
coordination environments are all distorted from the ideal geometries with a
wide span of Ag-S bond lengths ranging from 2.4 to 2.69 Å. Each Sn atom is
likewise coordinated by a distorted tetrahedron of S atoms with bond lengths
of approximately 2.37 Å. The Ag-S and Sn-S tetrahedra form a complicated three
dimensional corner sharing network, with some tetrahedral vertices also linked
by the linear and trigonal planar coordinated Ag atoms as shown in Figure 3a
and 3b.

Figure 3: Crystal structure of Ag8SnS6 determined at 295 K and 90 K. The 295 K
(rt) unit cell viewed along the (a) b-axis and (b) slightly titled off the
a-axis. (c) The same perspective of (b) showing only the S atoms. (d-f) show
the same for the 90 K (lt) structure with the unit cell doubled along the
a-axis for comparison with the rt phase. The perspectives in (c) and (f)
emphasize the straightening of the S columns along the a-axis in the lt
polymorph. The color code is as follows: Ag: grey, Sn: purple, S: yellow.

Analysis of the 90 K diffraction data confirms the structural phase
transition. We find that at low temperatures, Ag8SnS6 adopts a different
orthorhombic structure with the space group Pmn21 ($\\#$ 31) and cell
parameters a = 7.6629(5) Å, b = 7.5396(5) Å, c = 10.6300(5) Å, V/Z = 307.08(4)
Å3 (Z = 2 in the lt structure). The lt polymorph of Ag8SnS6 is isostructural
to the room temperature forms of the related Se-based argyrodite compounds
Ag8SnSe6 and Ag8GeSe6.[53, 54, 55] Several perspectives on the low temperature
variant of Ag8SnS6 are illustrated in Figures 3d-f. We show the lt structure
with the unit cell doubled along a for ease of comparison with the rt
structure. In the lt polymorph, Ag atoms occupy three 4b and two 2a wyckoff
sites, Sn – one 2a wyckoff site, and S – one 4b wyckoff site. As in the rt
phase, there is one Sn atom occupying a distorted tetrahedron of S atoms with
bond lengths of approximately 2.38 Å. The lt unit cell contains five unique Ag
atoms, three of which (Ag1, Ag2, and Ag5) are found in acentric trigonal
planar coordination by sulfur and two (Ag3 and Ag4) that are tetrahedrally
coordinated. The Ag–S bond lengths again cover a wide range from 2.44 to 2.79
Å (see Table S10 in the Supporting Information). Unlike the rt phase, the lt
structure does not contain Ag atoms in linear coordination.

Figures 3b and 3e illustrate how the room and low temperature structures have
similar connectivity of atoms, manifested most clearly in the chains/columns
of S atoms running down the a axis. We emphasize this in Figures 3c and 3f by
showing only the positions of the S atoms in each polymorph. In the rt phase,
the S columns are slightly nonlinear, but in the lt structure the chains
straighten such that the S atoms overlay when viewed down the a-axis. In this
light, the rt structure can be viewed as a distortion of the lt phase, in
which the new periodicity resulting from the bending of the S chains doubles
the unit cell along the a direction. Notably, Pna21 ($\\#$33) is the maximal
non-isomorphic “klassengleich” (class-equivalent) subgroup with index of 2
(k2) of the space group Pmn21 ($\\#$31).[56] Therefore, upon transitioning
from lt to rt phase on warming, the primitive-orthorhombic cell is retained,
but the volume is doubled due to the doubling of the a-lattice parameter.
After the transition, each 2a site in the lt structure transforms into a
single 4a site in the rt structure, while the 4b site in lt structure becomes
two 4a sites in rt structure. This relationship is somewhat unusual, differing
from the more common reduction of symmetry on cooling. Comparing the three
polymorphs of Ag8SnS6, the crystal structure becomes increasingly distorted as
the temperature rises. Starting from the lt Pmn21 arrangement, the linear S
chains become bent on transitioning into the rt Pna21 structure, and this is
accompanied by an increase in the diversity of local coordination environments
adopted by the Ag atoms. Finally, the Ag sublattice enters a highly disordered
state in the ht cubic phase with high ionic conductivity, where the Ag atoms
are distributed over three partially occupied sites.

After determining the lt crystal structure, we next measured the temperature
dependence of the lattice parameters by following the room temperature (rt)
orthorhombic (8 0 4)${}_{\text{rt}}$, (0 4 4)${}_{\text{rt}}$, and (0 0
6)${}_{\text{rt}}$ diffraction peaks collected on single crystals. Figure 4
shows the evolution of the Bragg peaks on warming in the vicinity of the
transition, between $\approx$ 100-140 K. Each peak shows sudden changes as the
temperature rises above $\approx$ 120 K, indicative of the phase change. The
(0 0 6)${}_{\text{rt}}$ peak shifts to lower 2$\theta$ by $\approx$ 0.1°. The
(8 0 4)${}_{\text{rt}}$ reflection moves to higher 2$\theta$ and is
significantly reduced in intensity. The (0 4 4)${}_{\text{rt}}$ peak moves to
higher 2$\theta$ and has strongly enhanced intensity. Close inspection of the
data collected at $\approx$ 120 K shows reflections corresponding to each
structure, implying coexistence of both phases across the transition region,
which is characteristic of a first-order transformation (the peaks
corresponding to each phase are marked by arrows in Figure 4).

Figure 4: Temperature dependence of the (a) (0 0 6)${}_{\text{rt}}$, (b) (8 0
4)${}_{\text{rt}}$, (c) and (0 4 4)${}_{\text{rt}}$ Bragg peaks of a Ag8SnS6
single crystal measured upon warming between $\approx$100–140 K on 4-circle
diffractometer.

Figures 2c-e display the temperature dependent lattice parameters extracted
from the data. The lattice parameters all show abrupt discontinuities at 120
K, consistent with the magnetization and thermal expansion measurements. On
warming from the base temperature, the a-, b-, and c-lattice parameters all
increase weakly up to approximately 120 K, after which the a\- and b-lattice
parameters rapidly contract and the c-lattice parameter expands. The unit cell
dimensions and phase volume all increase gradually with further heating above
120 K. The values at room temperature and 90 K agree with those determined by
the single crystal structural refinements discussed above. We emphasize here
that below the structural transition, a${}_{\text{rt}}$ $\approx$
2a${}_{\text{lt}}$ and Zrt/Zlt = 2 in the lt phase. The contraction along a
shown in Figure 2c implies a shrinking of the d-spacing along the a-axis on
heating above 120 K. The net result is a decrease in V/Z from 308 Å3 to 306 Å3
when warming across the transition at 120 K (see Figure 2f). We also measured
variable temperature powder XRD; however, the structural transition becomes
either completely arrested or broadened in the powdered samples. The variable
temperature powder XRD data are presented and discussed in the Supporting
Information (Figure S1).

Given that the transition is structural and appears to be first-order in
nature, we also collected magnetization and thermal expansion data upon
heating and cooling. The magnetic data is shown in Figure 5a, demonstrating
large hysteresis in M/H between heating and cooling at 0.5 K$\cdot$min-1.
Comparable hysteresis is also detected in the thermal expansion data, as
displayed in Figure 5b. Since the extent of the hysterisis can depend on the
rate of temperature change, we quenched our samples (i.e. cooled as rapidly as
possible to 5 K) and then measured upon slowly warming. For both magnetization
and thermal expansion measurements, the trend from 5-60 K (after quenching)
appears to match a rough extrapolation of the 130-300 K data to lower
temperatures. As the sample is heated above 60 K, both magnetization and
thermal expansion datasets show a clear drop followed by a rapid increase on
warming at 120 K. These results imply that fast cooling traps the crystal in
the rt phase. The metastable state is maintained upon slow warming until about
60-70 K, where both the M/H and $\Delta$L/L data return to the slow
heating/cooling manifold.

Figure 5: Thermal hysteresis of the structural transition in Ag8SnS6. (a)
Temperature dependent magnetic susceptibility and (b) thermal expansion. See
text for details about quenching.

We furthermore measured the thermal expansion with different heating and
cooling rates. The results are presented in Figure S3 in the Supporting
Information, and indicate cooling at a rate faster than 4-5 K/min is
sufficient to arrest the crystallographic transformation. On warming, a clear
transition to the lt polymorph begins at $\approx$ 60 K, and this
transformation is relatively insensitive to the heating rate. Given that the
metastable state appears to be preserved below 60 K, we lastly tracked the
magnetic susceptibility over time at T = 40 K, 60 K, and 80 K to determine how
quickly the thermodymically stable lt variant is recovered after quenching to
each temperature. Figure S4 in the supporting information shows the results.
We find that the transition into the lt polymorph begins almost immediately at
80 K, takes 100’s of minutes to become noticeable at 60 K, and remains in the
metastable rt phase for the duration of our 1000 minute measurement at 40 K.

The lt-rt phase transition can be considered as a reconstructive phase
transition,[57, 58] e.g. chemical bonds are broken and re-formed. Such
transitions involve considerable atomic motions and are always first-order
with hysteresis. The hysteresis can be understood as a case of kinetic
arrest,[59] caused by the competition between the Gibbs free energy (the
transformation driving force), the strain energy (transformation opposing
force), and the thermal energy that allows for atomic rearrangement. Our
thermal expansion and magnetization data suggest the thermal energy below at
least 40 K is insufficient to drive the structural rearrangement to the lt
polymorph, allowing for the sample to be arrested in the rt structure by rapid
cooling. On warming the quenched sample, the structure transforms to the
thermodynamically favored lt structure when the thermal energy overcomes the
activation barrier. These results suggest that Ag8SnS6 may serve as a model
system for the study of time-temperature-phase relations associated with a
quenchable, structural phase transition. The advantages of Ag8SnS6 are (i) it
is a stoichiometric phase that can be grown in high purity form and (ii) the
transition from one phase to another can be monitored by a contact-free
measurement such as magnetization.

Although we were not able to perform single crystal X-ray diffraction studies
below 90 K, i.e we could not reach the quenched, metastable rt structure that
exists for long periods of time below 40 K, we could still directly compare
the two orthorhombic structures (rt and lt versions) at the same temperature
by taking advantage of the hysteresis of the phase transition (i.e. in the
90–125 K region in Figure 5). Given that the width the hysteresis depends on
cooling rate (Figure S3), we cooled a sample as rapidly as possible to 90 K
and collected single crystal diffraction data (see the Experimental Section
for more details). After fast cooling, the diffraction data revealed an
orthorhombic Pna21 unit cell (a = 15.2724(9) Å, b = 7.5197(5) Å, c =
10.6569(7) Å, V/Z = 305.98(3) Å3), consistent with rt Ag8SnS6. The quickly
cooled, 90 K, results are in qualitative agreement with the behavior of the
lattice parameters shown in Figure 2c-f. Table S10 in the Supporting
Information compares Sn-S and Ag-S bond lengths determined by the single
crystal data at each temperature, and we find that at 90 K, most bonds are
slightly longer in the lt variant than in the 90 K rt Pna21 arrangement,
consistent with the larger V/Z of the lt structure.

Like rt Ag8SnS6, the related argyrodite compounds Ag8GeS6 and Ag8SiS6 both
adopt the Pna21 structure,[60, 61] whereas Ag8SnSe6 and Ag8GeSe6 take the
Pnm21 form.[53, 54, 55] From this information, we observe a trend relating the
unit cell dimensions (normalized to Z) to the preferred structure of each
material. Owing to the smaller size of Ge and Si compared to Sn, Ag8GeS6 and
Ag8SiS6 have smaller respective phase volumes of 299 and 296 Å3 compared to
309 Å3 for Ag8SnS6. Similarly, Se is larger than S, and Ag8GeSe6 and Ag8SnSe6
both have larger V/Z of 328 and 342 Å3 respectively. The preferred structures
indicate that increasing V/Z favors the Pmn21 arrangement. This reasoning is
consistent with the intermediate V/Z of Ag8SnS6 and the observation of both
polymorphs in this compound.

To support this argument based on crystallography, we used DFT to compare the
calculated formation energies of Ag8SnS6 and Ag8GeS6 in both Pna21 and Pmn21
structures at zero temperature. With the PBEsol exchange-correlation
functional, the fully-relaxed lattice parameters for Ag8SnS6 are a = 15.03 Å,
b = 7.40 Å, and c = 10.58 Å for the rt structure, and a = 7.63 Å, b = 7.42 Å,
and c = 10.49 Å for the lt structure. Both relaxed lattice parameters are
within 2$\%$ (smaller) than the experimental data, which is typically
acceptable for PBEsol. The corresponding total energy difference ($\Delta$E)
favors the lt structure by 0.14 eV$\cdot$f.u.-1 (or 9.5 meV$\cdot$atom-1),
agreeing with the experimental observation of the Pnm21 variant being the
ground state structure. For Ag8GeS6, the calculations indicate the Pnm21
structure is also more stable, but by only 2.9 meV$\cdot$atom-1. We note that
the calculated $\Delta$E is only weakly affected by the inclusion of spin-
orbit coupling (SOC), changing to 9.7 meV$\cdot$atom-1 for Ag8SnS6. Thus, the
following results are all from DFT calculations without SOC.

The calculations are qualitatively consistent with our suggestion that the
Pmn21 structure is favored to a greater degree for larger phase volume. This
implies that applying pressure to Ag8SnS6, or contracting the lattice, should
suppress the transition temperature. We tested this by applying chemical
pressure through isovalent alloying with the smaller Ge in place of Sn. We
prepared single crystals of Ag8Sn1-xGexS6 (nominal x = 0.10, 0.25) and pure
Ag8GeS6 following the same growth procedure used for the Ag8SnS6. Powder X-ray
diffraction analysis indicate successful Ge incorporation (Figures S4 and S5
in the Suporting Information), with a monotonic decrease in the lattice
parameters and unit cell volume moving from pure Ag8SnS6 to Ag8GeS6.
Magnetization data measured from 5 to 300 K is given in the Supporting
Information in Figure S7, and we find no evidence for a phase transition in
any of the Ge containing samples. Although we cannot definitively rule out a
disorder broadened transition, the magnetization data supports the arguments
presented above, and indicates the lattice contraction caused by even
relatively small ($\approx$10$\%$) addition of Ge is sufficient to eliminate
the transition. Suppression of the transition after lattice contraction is
likewise consistent with our X-ray analysis, which showed an increase in V/Z
when cooling through the transition. In principle, our arguments could be
further tested by expanding the lattice with “negative pressure” or alloying
with larger atoms, which we anticipate would raise the transition temperature
of Ag8SnS6. Likewise, we predict that applying pressure to Ag8GeSe6 may
stabilize a currently unreported Pna21 polymorph.

Our calculations and experimental results indicate the preferred structure of
the agryrodite members is closely related to the normalized unit cell volume.
This observation is likely grounded in the different strength of the chemical
bonding in each material, which will naturally be reflected in the bond
lengths. The structures adopted by the various argyrodite materials, along
with the experiments and calculations presented here, suggest larger volume
(longer bonds) favors the Pmn21 variant. Because the local tetrahedral
coordination environment around Sn remains essentially unchanged in each
phase, whereas the changes to the Ag-S sublattice are more significant, we
anticipate the Ag-S interactions are most important in determining the
preferred structure.

To provide more insight into the role of chemical bonding in determining the
relative stability for the two Ag8SnS6 structures, we also used DFT to
calculate the density of states (DOS) and crystal orbital Hamilton population
(COHP) for each polymorph. Figure 6a shows the calculated total DOS for each
structure of Ag8SnS6, and Figure 6b displays the atomic orbital projected
density of states (PDOS) for the lt structure to show the positions of the
relevant bands. The overall DOS for each structure are very similar. The
states from -14 to -12 eV are mostly from the S 3s orbitals and those from -8
to -7 eV are primarily from Sn 5s orbitals, respectively. The Sn 5p bands
(plotted together with Sn 5s) spread out in a broad energy range above -6 eV
and have low magnitude due to the small number of Sn atoms. The Ag 4d bands
range from -6 and 0 eV, with the main peak centered around -3.5 eV. The Ag 5s
bands spread from -6 eV to the conduction band, with greatest contribution
from -6 to -5.5 eV, where they overlap with the lower edge of the Ag 4d bands.
The S 3p bands range from -8 eV to the conduction band with two regions of
greatest contribution, one from -6 to -4.5 eV and the other from -2 to 0 eV.

Figure 6: (a) DFT calculated density of states for Ag8SnS6 in the rt (Pna21)
and lt (Pmn21) structure. (b) Orbital projected density of states for Ag8SnS6
in the lt structure. (c) Crystal orbital Hamiltonian population (COHP) plots
for nearest neighbor Ag-S interactions in Ag8SnS6 for rt and lt structures.
(d) Integrated COHP for each structure. The arrows in (c) and (d) mark the
positions of relevant antibonding COHP peaks that shift to higher energy in
the lt structure.

The COHP for nearest-neighbor Ag-S pairs is presented in Figure 6c. COHP
decomposes the density of states into nearest neighbor pair interactions and
gives information on bonding and antibonding contributions to the band
structure energy,[62] respectively denoted by negative (stabilizing) and
positive (destabilizing) COHP values. The COHP plot clearly shows the Ag-S
band hybridization from -6 to -3.4 eV gives bonding states and that the states
above -3.4 eV are anti-bonding. The relative stability of the two structures
depends on the details of how the states in these bonding and anti-bonding
regions distribute differently, which reflects the rearrangements of Ag-S
local coordination and the different bond lengths in each structure. In the
bonding region, the primary COHP peaks reach more negative values in the lt
structure. Likewise, in the antibonding region, the most notable change is the
shift of the COHP peak from -0.6 to -0.2 eV as indicated by the two vertical
arrows in Figure 6c. The effects of these changes on the overall stability are
captured by integrating the COHP (ICOHP) to the Fermi level, which gives the
contribution from the nearest neighbor Ag-S pairs to the total energy. As
shown in Figure 6d and zoomed just below the Fermi energy EF, after
integrating up to EF, we ultimately find the lt arrangement to be more stable
than the rt-structure by 2 meV/pair, in qualitative agreement with our overall
DFT energy calculations discussed above. The calculations therefore indicate
differences in the Ag-S bonding significantly impact the energetically favored
structure.

##  4 Summary and Conclusions

We grew single crystals of the semiconducting mineral Ag8SnS6 from a Ag-Sn-S
melt. Temperature dependent magnetization, thermal expansion, and X-ray
diffraction measurements demonstrate Ag8SnS6 crystals undergo a structural
transition from the room temperature (Pna21) form to a different orthorhombic
(Pmn21) structure below 120 K (on warming). The low temperature polymorph is
isostructural to the related argyrodites Ag8SnSe6 and Ag8GeSe6. We find the
transition temperature to be strongly dependent on the cooling rate, and
samples rapidly quenched at speeds faster than $\approx$ 4-5 K$\cdot$min-1 can
be trapped in a metastable form of the room temperature phase for temperature
up to at least 40 K. As such, Ag8SnS6 may serve as a model system for the
study of time-temperature-phase relations associated with a quenchable,
structural phase transition. Based on the structures preferred by each family
of argyrodite materials Ag8TQ6 (T = Si, Ge, Sn; Q = S, Se), we suggest lower
volumes (per formula unit) increasingly favor the Pna21 variant. We support
this picture by applying chemical pressure to Ag8SnS6 by alloying with Ge to
form Ag8Sn1-xGexS6, and find that a small lattice contraction from 10 $\%$
incorporation of Ge is sufficient to avoid the low temperature phase change in
Ag8Sn1-xGexS6. We lastly use density function theory calculations to suggest
that differences in Ag-S bonding determines the energetically preferred
structure.

Acknowledgements

Work at the Ames Laboratory (T.J.S., J.M.W., A.K., E.G., L.W., R.A.R., Y.M.,
V.K.P., S.L.B., P.C.C.) was supported by the U.S. Department of Energy, Office
of Science, Basic Energy Sciences, Materials Sciences and Engineering
Division. The Ames Laboratory is operated for the U.S. Department of Energy by
Iowa State University under Contract No. DE-AC02-07CH11358. T.J.S. was
supported by the Center for Advancement of Topological Semimetals (CATS), an
Energy Frontier Research Center funded by the U.S. Department of Energy Office
of Science, Office of Basic Energy Sciences, through the Ames Laboratory under
its Contract No. DE-AC02-07CH11358 with Iowa State University. T.J.S and E.G.
also acknowledge funding from the Gordon and Betty Moore Foundation’s EPiQS
Initiative through Grant No. GBMF4411. J.V.Z. acknowledges financial support
from the National Science Foundation (DMR 1944551). Temperature dependent
powder diffraction experiments (Y.M., V.K.P.) were supported by the Division
of Materials Science and Engineering of the Office of Basic Energy Sciences.
Earlier attempts to grow and study single crystal Ag8SnS6 samples were made by
X. Lin, W. Meier and G. Drachuck. P.C.C. acknowledges D. Argyriou for geologic
reasons. The authors thank Tom Lagrasso and Matt Kramer for useful
discussions.

Conflicts of Interest

The authors have no conflicts of interest to declare.

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#  Role of isotropic and anisotropic Dzyaloshinskii-Moriya interaction on
skyrmions, merons and antiskyrmions in the $C_{nv}$ symmetric system

Sandip Bera  Department of Physics, University of Toronto, 60 St. George
Street, Toronto, Ontario, Canada M5S 1A7

###### Abstract

The lattice Hamiltonian with the presence of a chiral magnetic isotropic
Dzyaloshinskii-Moriya interaction (DMI) in a square and hexagonal lattice is
numerically solved to give the full phase diagram consisting of skyrmions and
merons in different parameter planes. The phase diagram provides the actual
regions of analytically unresolved asymmetric skyrmions and merons, and it is
found that these regions are substantially larger than those of symmetric
skyrmions and merons. With magnetic field, a change from meron or spin spiral
to skyrmion is seen. The complete phase diagram for the $C_{nv}$ symmetric
system with anisotropic DMI is drawn and it is shown that this DMI helps to
change the spin spiral propagation direction. Finally, the well-defined region
of a thermodynamically stable antiskyrmion phase in the $C_{nv}$ symmetric
system is shown.

The Dzyaloshinskii-Moriya interaction [1, 2, 3], an asymmetric exchange
interaction, plays an important role in stabilizing the noncollinear magnetic
structure in broken inversion symmetric systems. The spin spiral (SS), whose
length scale [4, 5, 6, 7] is mostly created by DMI and symmetric exchange
interactions, is a common noncollinear ground state of a chiral magnet. This
ground state transforms into a number of interesting topologically protected
small-scale spin textures, including skyrmions [8, 9, 10, 11, 12, 13, 14] and
merons [15, 16], with the aid of magnetic fields, magnetic anisotropy, and
higher order frustrated exchange interactions. These spin textures are
classified into two types based on crystal symmetry: Neel type for $C_{nv}$
symmetric and Block type for $D_{n}$ symmetric systems [4, 5]. Whereas, the
$D_{2d}$ symmetric system [6] supports antiskyrmions [17] and antimerons [18].
Due to their tiny size [19, 20], topological stability[4, 5], and stability
over a broad range of parameter spaces [7, 21, 6, 22, 23] including magnetic
field, anisotropy, and DMI, all of these topological non-trivial phases have
had a positive impact on the field of spintronics[24, 25, 26, 27, 28, 29].

We have a semi-analytic theory [5, 7, 6] for singly symmetric skyrmions in a
ferromagnetic background. However, we still have no analytical solution for
experimentally observed asymmetric skyrmions [30, 31, 32, 33] in the presence
of easy plane anisotropy. Since the topological number remains the same [34,
7], asymmetric skyrmions have the same topological properties as symmetric
skyrmions, such as the Hall effect [35]. Asymmetric merons such as asymmetric
skyrmions have been observed experimentally [15, 36, 37, 38] in regions of
easy plane anisotropy. Even merons always appear in lattice form [15, 16, 7].
Continuum model-based analytical solutions, however, are more suitable for
single symmetric skyrmions and merons. As a result, we are unable to construct
precise phase diagrams for the lattice skyrmions, asymmetric skyrmions,
symmetric merons and asymmetric merons using the continuum model. In this
article, I describe the full phase diagram by minimizing the lattice
Hamiltonian in presence of isotropic DMI using Monte Carlo (MC) simulations
with periodic boundary conditions on both square and hexagonal lattices. These
phase diagrams depict a transition from SS $\rightarrow$ skyrmions (lattice
skyrmion, isolated skyrmion) $\rightarrow$ polar ferromagnet in the easy-axis
anisotropy limit and from merons $\rightarrow$ skyrmions in the easy-plane
anisotropy limit, all of which have already been observed experimentally [12,
13, 25, 14]. These phase diagrams also demonstrate that asymmetric skyrmions
and asymmetric merons have significantly wider regions in the parameter space
than symmetric skyrmions and symmetric merons.

The orthogonal coefficients of an anisotropic DMI are different. Heide et al.
[39] predicted such DMI in 2Fe/W(100) films using first-principles
calculations. It has been observed experimentally in an ultrathin epitaxial
Au/Co/W(110) films [40], CoFeB(211)/Pt(110) films [41]. Shibata et al. [42]
showed that the anisotropy of bulk DMI causes mechanical strain, which
produces an anisotropic skyrmion. We already know that anisotropic DMI is
direction dependant, and this strain aids in changing the spin configuration.
This DMI can also alter the spin orientation of the skyrmions in a certain
manner, leading to the eventual emergence of an antiskyrmions in $C_{nv}$
symmetric system. Hoffmann et al. [43] have first observed antiskyrmions in
(110)-oriented films with $C_{2v}$ symmetric systems. Although, we generally
know skyrmions appears in $C_{2v}$ symmetric systems. We have a theoretical
prediction [7] of antiskyrmion stabilization in $C_{nv}$ symmetric systems.
However, neither this skyrmion nor the antiskyrmion is symmetric about their
center. Consequently, there are no analytical solutions and phase diagrams for
these topological phases. Here, I first demonstrate through MC simulations how
the anisotropic DMI promotes the skyrmion to antiskyrmion transition in the
$C_{nv}$ symmetric system before illuminating the full phase diagram for this
DMI.

##  I  Hamiltonian and Methods

The ferromagnetic Heisenberg spin Hamiltonian in the presence of an easy axis
magnetic field can be written as

${\cal H}=-J\sum_{<ij>}{\bf m}_{i}\cdot{\bf
m}_{j}+H_{DMI}+A\sum_{i}(m^{z}_{i})^{2}-H\sum_{i}{m}^{z}_{i},$ (1)

where ${\bf m}_{i}$ is the unit vector of the magnetization at $i^{th}$ sites.
The first term is the isotropic exchange interaction with nearest-neighbor
coupling constant, $J$. Second term, $H_{DMI}$ is an antisymmetric spin orbit
interaction, known as Dzyaloshinskii-Moriya interaction [1]. The direction of
the DMI vectoer depends on the crystal symmetries. $H_{DMI}$ is proportional
to $D\sum_{<ij>}(\hat{z}\times\hat{r}_{ij})\cdot({\bf m}_{i}\times{\bf
m}_{j})$ for $C_{nv}$, and $D\sum_{<ij>}\hat{r}_{ij}\cdot({\bf
m}_{i}\times{\bf m}_{j})$ for $D_{n}$ symmetries systems respectively. Here,
$\hat{r}_{ij}$ is unit vector from $i$ to $j$ sites. The amplitude of DMI
vector ($D$) is proportional to interaction strength between spin and orbit.
Here, $H_{DMI}$ is isotropic. $A$ is the coefficient of the magnetic
anisotropy. It is positive for easy-plane anisotropy and negative for easy-
axis anisotropy. $H$ is the effective magnetic field perpendicular to the
plane of the systems. The last two terms in Eq. (1) determine the direction of
the ferromagnet [22, 7]. When $A<0$, all spins align parallel to $H$ and form
a polar ferromagnet phase. For $A>0$ , the direction of magnetization depends
on the anisotropy and the magnetic field strength. Easy-plane anisotropy
forces all spins to lie in the plane, and the magnetic field forces the spins
to lie perpendicular to the plane. Consequently all spins make an angle with
the direction of $H$. This configuration is known as tilted ferromagnet.

The Hamiltonian given in Eq. (1) is minimised on both the square lattice and
the hexagonal lattice using a MC simulation based on the conventional
Metropolis algorithm [44, 45] with periodic boundary conditions. Hamiltonian
has many local minima or metastable state for a given parameters. To avoid
these metastable states and locate the actual thermodynamic ground state,
conventional simulated annealing techniques with MC are employed. The MC
simulation begins with a random initial spin configuration at a high
temperature. The temperature is then steadily lowered using the formula
$T_{n+1}=\alpha T_{n}$, with an MC simulation being run at each stage. The
initial spin configuration in the MC simulation is derived from the outcome of
the preceding step after the first annealing loop has been completed. Repeat
the same procedure until my final temperature reaches a very low value ( close
to zero). The simulation parameters used in my numerical calculations are:
total number of steps in annealing method $n=10^{3}$, initial temperature
$T_{0}=11J$, final temperature $\approx 5\times 10^{-4}J$, $\alpha=0.99$, and
MC steps $4\times 10^{6}$. $J$ is utilized to measure all energy-related
parameters such as $D$, $A$ and $H$, while $a$, the lattice constant, is used
to scale length. $D=\sqrt{6}J$ is chosen to construct the phase diagram in the
$H-A$ plane. This value of DMI supports a SS with periodicity close to $8a$.
We know that the periodicity of SS in $MnGe$ is about $8.3a$ [46]. So, the
chosen value of DMI belongs to real materials. All these MCs are performed
$32\times 32$ mesh, but the mesh size is increased to $64\times 64$ for some
important results (mostly at phase boundaries). My simulation always treats
$J$ as 1.

###  I.1 Numerical Results and Phase Diagrams in $C_{nv}$ Symmetric System

The Hamiltonian (1) has only three parameters, namely $D$, $H$ and $A$ because
$J$ is used to scale all energy parameters. We only have the $D-H$, $D-A$, and
$H-A$ planes for the phase diagram. The article is organized as follows: i
begin by showing the whole phase diagram for the $C_{nv}$ symmetric system
with isotropic DMI on a square and a hexagonal lattice of different parameter
planes. The numerical outcomes for the $D_{2d}$ symmetric system in $D-H$
plane are later explained. Finally, in the presence of anisotropic DMI,
numerical results and an entire phase diagram are displayed for the $C_{nv}$
symmetric system.

The final output of the MC gives a spin configuration of the system for a
given parameter. To identify the phase that this spin configuration supports,
we need to use some distinct properties: local magnetization $m^{i}_{z}$,
average magnetization $M=1/N\sum_{i}m^{i}_{z}$, local chirality
$\rho_{i}=(1/4\pi)\sum_{\delta=\pm 1}{\bf m}_{i}\cdot({\bf
m}_{i+\delta\hat{x}}\times{\bf m}_{i+\delta\hat{y}})$, and total chirality
$\mathcal{R}=(1/N)\sum_{i}\rho_{i}$, local spin asymmetric parameter
$\delta\theta_{i}=|\theta_{i+\hat{y}}-\theta_{i-\hat{y}}|-|\theta_{i+\hat{x}}-\theta_{i-\hat{x}}|$,
where N is the number of lattice sites. $\delta\theta_{i}$ gives information
about symmetry with respect to its center. Chirality helps distinguish between
topologically trivial and nontrivial phases. A detailed description of these
above parameters for different phases is given in Ref. [21]. These
characteristic variables are used to draw phase boundaries in all phase
diagrams.

The complete phase diagram for a square lattice in the $H-A$ plane at
$D=\sqrt{6}$ is shown in Fig. (1a). The large magnetic field transforms the
system into the ferromagnetic (FM) phase (Phase-1). In opposite limit, a weak
magnetic field causes the spins to form a spin spiral (SS) phase (phase-II).
Here, SS propagates along the $x+y$ direction because the sign of the DMI
vector is taken positive. These two phases, SS and FM, are topologically
trivial and emerge at extreme magnetic field values. Now increasing the
magnetic field in the SS phase in the easy-plane anisotropy region. At this
stage, all spins begin to experience an additional force in the presence of a
magnetic field. The magnetic field seeks to align all of the spins in the
field’s direction. DMI, however, aims to provide a finite rotation between the
spins. These interactions work together to stabilize a skyrmion. At low field
ranges, the skyrmion and spin spiral appear together in what is known as a
mixed phase (Phase-III). As the magnetic field increases, the number of spin
spirals decreases. Because of this, we’ve reached a point where there are only
skyrmions in the system. In this case, the separation between any two
skyrmions remains the same and forms a lattice known as skyrmion lattice
(phase-IV). The magnetic field increases the spacing between the skyrmions. As
a result, fewer skyrmions exist in a given lattice shape and they are also
smaller in size. These small-sized skyrmions are randomly distributed in what
is called the isolated skyrmion phase (phase-V). The decrease in size of the
skyrmions with magnetic field has already been predicted theoretically [7] and
observed experimentally [47]. The area of phase-V is much smaller than the
phase-IV. Finally for a slight increase in the magnetic field we reach the FM
phase. The spin configurations of skyrmion lattice phase and isolated skyrmion
phase are shown in figures (2a) and (2b), respectively. We can see that there
is a transition from SS to skyrmion to FM with magnetic field which has
already been observed experimentally [12, 13, 25, 14].

(a)

(b)

(c)

(d)

Figure 1:  Phase diagrams: (1a) $H-A$ plane at $D=\sqrt{6},~{}J=1$ and (1b)
$D-A$ plane at $H=0,~{}J=1$. The important phases in the phase diagrams are:
Ferromagnet with magnetization along $\hat{z}$ direction (I), spin spiral
(II), mixed phase of broken spin spiral and skyrmion (III), skyrmion lattice
(IV), isolated skyrmion (V), meron (VI), Planar FM (IX) and one unknown phase
(VIII). It is challenging to discern between the phase-VIII and the tilted FM
phase in my numerical simulations. So, in the phase diagrams (Figures 1a and
1c, I have highlighted them in white without suitable borders. The complete
phase diagram for a hexagonal lattice at $D=\sqrt{6},~{}J=1$ is displayed in
panel (1c) in the $H-A$ plane. Panel (d) shows the phase diagram on a square
lattice in the $D_{2d}$ system at $A=0$. Here, the antiskyrmion (VII) phase
region grew with $D$ and $H$. The spin configurations of all these phases are
shown in figure (2). Figure 2: Magnetic structures obtained from MC
simulations for different phases are shown in panels: (a) skyrmion (Sk)
lattice, (b) Isolated skyrmion, (c) meron lattice, (d) antiskyrmion (ASk)
lattice, (e) asymmetric skyrmion (A-Sk) lattice, (f) Asymmetric meron
(A-meron) lattice, (g) spin spiral (SS), (h) Ploar ferromagnet (FM), (i)
unknown phase (VIII), (j) asymmetric antiskyrmion (A-ASk) in presence of
anisotropic DMI ($H_{aDMI}$), (k) asymmetric skyrmion (A-Sk) in presence of
$H_{aDMI}$, (l) Spin spiral in presence of $H_{aDMI}$. The out-of-plane
component of magnetization is color coded, while the in-plane component is
shown by arrows.

The phase diagram at $A>0$ is interesting because there are no analytical
solutions for the asymmetric skyrmions and asymmetric merons in this region.
Figure (1a) shows the corresponding phase diagram. We observed both skyrmions
and SS phases here, similar to negative anisotropy. Here $\delta\theta_{i}\neq
0$ for all phases, indicating that these skyrmions are not symmetric about
their center called asymmetric skyrmions. This result is consistent with
analytical predictions [7]. The phase space region of these skyrmions is
larger in this case than it is in the $A<0$ case, which is a second crucial
fact. Given the wide spectrum of asymmetric skyrmion, this is crucial in
application. Spin configuration of a asymmetric skyrmion is shown in figure
(2e).

Apart from the existing phases described in the above paragraph, there is
another interesting phase which is also topologically non-trivial known as the
meron phase (phase-VI). This phase lies below the skyrmion-lattice phase. Like
skyrmions, merons are not symmetrical about their center in regions known as
asymmetric merons. Spin configuration of asymmetric merons is shown in figure
(2f). However, at $H=0$ the merons are formed in the background of planar
ferromagnet and are symmetric about their center. Figure (1a) shows that
$2.4-4.5$ is the range of anisotropy for symmetric merons at $H=0$. Both types
of merons appear in lattice shapes that are consistent with theoretical
predictions [15, 16, 7]. This result is new and interesting not only because
it is challenging to estimate the potential meron region using continuum
models, but also because these results are crucial for practical applications.
Phases in the white area in Fig. (1a) are difficult to identify because this
region may contain some conical phases and tilted ferromagnet.

Fig. (1b) shows the full phase diagram in the $D-A$ plane for a better
understanding of a symmetric meron. The phase region of symmetric merons
increases with both DMI and magnetic anisotropy. This phase diagram also
suggests that we need a DMI value close to the exchange interaction strength
to stabilize symmetric merons in the absence of a magnetic field. Which
indicates that the system should have strong spin-orbit coupling to support
symmetric merons. The planar ferromagnetic phase (phase IX) is observed with
large anisotropy. An unknown phase (phase VIII) is noticed between merons and
planar ferromagnetic phases. The spin configuration of these phases are shown
in Fig. (2).

The full phase diagram for the hexagonal lattice is shown in figure (1c). The
order and appearance of the phases in the phase diagram are almost the same as
in the square lattice phase diagram. However, in the hexagonal case the
boundaries of each phase are slightly larger. In other words, each axis of our
phase diagram has an additional scaling factor. Here, I consider $2a$ to be
the lattice spacing for the hexagonal lattice (where $a$ is the lattice
spacing for the square lattice). This additional scaling factor of the phase
diagram is due to this lattice spacing.

###  I.2 Antiskyrmion in $D_{2d}$ Symmetric System

The same numerical method is used to minimize the lattice Hamiltonian (1) in
$D_{2d}$ symmetric systems for square lattice. Here the phase diagram is
similar to the $C_{nv}$ symmetric system, but the $D_{2d}$ symmetric system
has antiskyrmions instead of skyrmions. Since the results are the same, we
have calculated the phase diagram in a different plane ($D-H$) for the
$D_{2d}$ systems. Fig. (1d) shows that the antiskyrmions phase (phase-VII)
regions increase with increasing DMI and magnetic field. Spin configuration of
an antiskyrmion is shown in Fig. (2d).

Figure 3:  Phase diagram with an anisotropic DMI in $C_{nv}$ symmetric system
at $A=0$. Here, the spin configurations (see Fig. (2)) of spin spiral (II),
skyrmion (IV) and antiskyrmion (VII) are slightly different with the same
phases observed in the case of isotropic DMI.

##  II Antiskyrmion in $C_{nv}$ Symmetric System with Anisotropic DMI

The following form of anisotropic DMI ($H_{aDMI}$) is considered in the
lattice Hamiltonian:

$\displaystyle H_{aDMI}=(1+\lambda)D_{x}+(1-\lambda)D_{y},$ (2)

where $\lambda$ is a positive number and represents the strength of the
anisotropic DMI. The DMI becomes isotropic when $\lambda=0$. Anisotropic DMI
refers to the fact that the DMI’s amplitude in the $x$ and $y$ directions
differ. Here, $D_{x}$ and $D_{y}$ are, in the case of an isotropic DMI vector,
the $x$ and $y$ components, respectively. The introductory section has already
described how this term is expressed for several symmetric systems. In Eq.
(2), the same amount of anisotropy added to the $x$ component of $H_{aDMI}$ is
subtracted from the $y$ component.

Here, my attention is solely on $C_{nv}$ symmetric system. It is obvious that
all skyrmions and merons will be asymmetric about their centres because our
DMI is asymmetric in nature. The entire phase diagram at $A=0$ is displayed in
Fig. (3) in the $\lambda-H$ plane. The SS phase (phase II) manifests at large
anisotropic DMI or large $\lambda$, as expected. The spin configuration of SS
phase in this instance differs from the spin configuration of SS phase in
isotropic DMI (Fig. (2l) ). For $\lambda=0.72$ and $H=1.0$ SS propagates along
$x-$direction whereas SS in isotropic DMI always propagates along
$x+y$-direction (Because the isotropic DMI vector is assumed to be a positive
quantity). This result demonstrates that we can continually modify the SS
phase propagation direction by utilising $\lambda$. In the other limit, a
ferromagnetic phase (I-phase) is observed at very low $\lambda$ and large $H$.
Skyrmions (phase-IV) are observed in a small region with low magnetic field
and low $\lambda$. These skyrmions are asymmetrical and extended in the $y$
direction. For any $\lambda$, the $y$ component of $H_{aDMI}$ is smaller than
the $x$ component. It is known that the radii [7, 6] of skyrmions and merons
depend inversely on the DMI. Thus, the extension of the skyrmions border along
the $y$ direction seen in our simulations is also supported by the theoretical
realization. Fig. (2k) depicts the spin arrangement of these asymmetric
skyrmions. Finally, for medium to high values of $H$ and $\lambda$, the phase
with the greatest interest (phase-IV) is observed. It is an antiskyrmion
according to all investigations including topological number, spin
configuration and chirality. This antiskyrmion is also not symmetric, known as
asymmetric antiskyrmion, and is thermodynamically stable. This phase exists
between the ferromagnetic phase and the SS phase. Its spin configuration is
shown in Fig. (2j). The phase space region of this asymmetric antiskyrmion
expands as $H$ and $\lambda$ increase. As a result, we see experimentally [43]
confirmed thermodynamically stable antiskyrmions in $C_{nv}$ symmetric system
with anisotropic DMI. Using the same form of anisotropic DMI, one can also
stabilize the skyrmion in the $D_{2d}$ system. In this case $D_{x}$ and
$D_{y}$ will depend on the DMI form of the $D_{2d}$ symmetric system.

In conclusion, complete phase diagrams for skyrmions in both square and
hexagonal lattices are shown. The phase diagram remains the same for these two
different lattices. The phase region of merons are seperately depicted in
different plane. The numerical results for the total phase diagram of the
$D_{2d}$ symmetric system are the same as those of the $C_{nv}$ symmetric
system. Skyrmions are not seen in this instance; antiskyrmions are. These
results are consistent with theoretical predictions. Symmetric and asymmetric
skyrmions, merons and antiskyrmions have already been detected experimentally
in many systems. In comparison to symmetric skyrmions, asymmetric skyrmions
have a larger parameter space region, which is significant for applications.
Finally, it is shown that the $C_{nv}$ symmetric system has a
thermodynamically stable antiskyrmion that is in line with the findings of an
experiment[43]. This anisotropic DMI also helps to change the direction of
spiral spiral propagation. Higher order exchange interactions and frustrated
exchange interactions are absent in our calculations. We know that exchange
can stabilize the skyrmion even for very small DMI [21]. Thus, frustrated high
order exchange interaction and anisotropic DMI can give some additional
features to the phase diagram.

###### Acknowledgements.

I would like to thank Prof. Sudhansu Shekhar Mandal for initial discussions.
I’m grateful to Abinash Kumar Shaw, Sahana Das and Sudipto Das for help in
understanding Monte Carlo simulations. The initial work of the project was
done using the computer facilities provided by the Department of Physics at
IIT Kharagpur.

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