# Structures of bulk hexagonal post-transition-metal chalcogenides from
dispersion-corrected density-functional theory

S. J. Magorrian  National Graphene Institute, University of Manchester, Booth
Street East, Manchester M13 9PL, United Kingdom  V. Zólyomi  Hartree Centre,
STFC Daresbury Laboratory, Daresbury WA4 4AD, United Kingdom  N. D. Drummond
Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom

###### Abstract

We use dispersion-corrected density-functional theory to determine the
relative energies of competing polytypes of bulk layered hexagonal post-
transition-metal chalcogenides, to search for the most stable structures of
these potentially technologically important semiconductors. We show that there
is some degree of consensus among dispersion-corrected exchange-correlation
functionals regarding the energetic orderings of polytypes, but we find that
for each material there are multiple stacking orders with relative energies of
less than 1 meV per monolayer unit cell, implying that stacking faults are
expected to be abundant in all post-transition-metal chalcogenides. By fitting
a simple model to all our energy data, we predict that the most stable
hexagonal structure has P$6_{3}$/mmc space group in each case, but that the
stacking order differs between GaS, GaSe, GaTe, and InS on the one hand and
InSe and InTe on the other. At zero pressure, the relative energies obtained
with different functionals disagree by around 1–5 meV per monolayer unit cell,
which is not sufficient to identify the most stable structure unambiguously;
however, multi-GPa pressures reduce the number of competing phases
significantly. At higher pressures, an AB′-stacked structure of the most
stable monolayer polytype is found to be the most stable bulk structure; this
structure has not been reported in experiments thus far.

##  I Introduction

The hexagonal post-transition-metal chalcogenides (PTMCs) GaS, GaSe, GaTe,
InS, InSe, and InTe are layered materials with hexagonal Bravais lattices [1,
2, 3]. Due to the possibility of isolating mono- and few-layer films, in their
ultrathin form they have received considerable attention in recent years as a
new class of two-dimensional (2D) semiconductor [4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17]. The two dynamically stable structures of PTMC monolayers
are shown in Fig. 1, and are based on the honeycomb motif [6, 7]. Bulk PTMCs
have direct band gaps of $\sim 1.3$–$2.5$ eV [18, 19, 20, 21], light out-of-
plane effective masses [22, 23, 24, 25, 26, 27], and strongly nonlinear
optical properties such as second harmonic generation, optical gain, and
up-/down-conversion [28, 29, 30, 31, 32]. InSe exhibits high in-plane electron
mobility [33], which persists in the thin-film limit, and has enabled the
observation of the quantum Hall effect [13] and the demonstration of PTMCs as
candidate ultrathin transistors [4, 12]. InSe has also shown potential for
applications in photovoltaics [34, 35] and electron-beam-based data-storage
[36].

Figure 1: (a) Top and (b) side views of the $\alpha_{\rm M}$ polytype of
monolayer GaS, and (c) top and (d) side views of the $\beta_{\rm M}$ polytype
[6, 7]. Gallium and sulfur atoms are shown in red and yellow, respectively.

Thin films of PTMCs exhibit high-sensitivity broadband photoresponse [5, 8, 9,
11]. They also show a substantial increase in the band gap, from $1.3$ eV in
bulk InSe to $\sim 2.8$ eV in monolayer InSe [13], and from $2$ eV in bulk
GaSe to $\sim 3.5$ eV in monolayer GaSe [14, 16]. An offset in the location of
the valence-band maximum has been shown to develop in the thinnest films [37,
17], yielding a slightly indirect band gap, unlike the bulk. Combined with the
high density of states at the band edge, this is expected to lead to strongly
correlated phenomena in p-doped monolayer PTMCs [6, 7, 10] as well as
interesting thermoelectric properties [15].

The high tunability of the physical properties of PTMC films stems from the
strong electronic coupling between states localized on neighboring layers
[38]. For this reason, PTMCs are likely to be highly sensitive to changes
brought about by variations in stacking order. The influence of stacking and
interlayer interactions has already been shown to be important in, for
example, the metallic transition-metal dichalcogenides [39, 40, 41], which
feature multiple stacking orders very close in energy. The local stacking
order will vary continuously in the moiré superlattices formed when monolayers
are stacked with a relative rotation or lattice-constant mismatch. In twisted
bilayers of 2D materials with small misalignments and/or lattice-constant
mismatches, the constituent monolayers can adjust to maximize the size of
regions of energetically favorable stacking [42, 43]. Compared to graphene and
transition-metal dichalcogenides, PTMCs have low Young’s moduli and are highly
flexible [44, 45, 46], so in-plane relaxation can be expected to occur more
readily, starting from larger twist angles, and featuring stronger nonuniform
strain fields, than in twisted transition-metal-dichalcogenide bilayers. To
describe such reconstruction in moiré superlattices of PTMCs it is essential
first to attain a proper understanding of the energetics of the various PTMC
polytypes and structures, and the factors contributing to their formation.

In this work, we use a range of dispersion-corrected density-functional-theory
(DFT) methods to investigate systematically the energies and stabilities of
the competing polytypes of bulk layered hexagonal PTMCs. We provide an
expression for the energy per monolayer unit cell of an arbitrary bulk
hexagonal PTMC polytype. We find that each PTMC generally admits a few
polytypes that are energetically very similar, implying that crystal-growth
conditions are likely to be important. Motivated by the observation of
electronic and structural changes in PTMCs under pressure [47, 48, 49, 50], we
also investigate the pressure-dependence of the relative stability of
competing polytypes.

The post-transition-metal (PTM) atoms that we consider are indium and gallium,
while the chalcogen atoms that we consider are sulfur, selenium, and
tellurium. The PTM atoms are strongly bonded in vertical dimers lying on a
hexagonal sublattice. Each PTM atom is strongly bonded to three chalcogen
atoms lying on a different hexagonal sublattice to the PTM dimers. There are
two different single-layer polytypes, as shown in Fig. 1: the chalcogen atoms
may all lie on the same sublattice, or the top and bottom chalcogen atoms may
lie on different sublattices [6, 7]. The former structure, referred to as the
$\alpha_{\rm M}$ monolayer polytype, is slightly more stable and has vertical
mirror symmetry $\sigma_{h}$ about the center of the layer, although it lacks
inversion symmetry (D3h point group). The latter structure, referred to as the
$\beta_{\rm M}$ monolayer polytype, does not have vertical mirror symmetry,
but it does have inversion symmetry (D3d point group). In bulk hexagonal PTMCs
there are further possibilities for polytypism due to the different ways in
which the layers can be stacked. Our reference structure is the simplest
possible bulk structure, which consists of AA-stacked $\alpha_{\rm M}$-PTMC
monolayers, with a four-atom primitive unit cell.

A range of polytypes and stacking orders have been reported for the bulk
structures of the PTMCs obtained in experiments [51]. The $\beta$ [52] and
$\varepsilon$ [53] 2H polytypes both have $\sigma_{h}$ reflection symmetry,
with the former also having an inversion center. Meanwhile, the $\gamma$ 3R
polytype [54] has a single-layer primitive unit cell, and has neither
inversion nor $\sigma_{h}$ reflection symmetry. A polytype known as $\delta$,
consisting of a four-layer unit cell with two interfaces between successive
layers stacked as in the $\beta$ polytype and the other two interfaces stacked
as in the $\gamma$ polytype, has also been reported for GaSe [55]. Note that
the $\beta_{\rm M}$ monolayer polytype should not be confused with the $\beta$
polytype of bulk PTMCs: the former refers to the inversion-symmetric monolayer
shown in Figs. 1(c) and 1(d), the latter to the bulk crystal in which non-
inversion-symmetric monolayers [the $\alpha_{\rm M}$ monolayer polytype shown
in Figs. 1(a) and 1(b)] stack into an AB-type bulk crystal that now exhibits
inversion symmetry. To avoid confusion, in Sec. II we adopt a notation for
PTMC stacking that enables unambiguous characterization of all PTMC crystals
irrespective of the monolayer polytypes or stacking order of successive
layers.

The rest of this paper is structured as follows. Our approach for enumerating
physically relevant PTMC structures is described in Sec. II. We present a
fitting function to describe the energetics of PTMC polytypes in Sec. III. We
compare the DFT energies of PTMC polytypes obtained with different
exchange–correlation functionals in Sec. IV. Our analysis of the most stable
polytypes, including the effects of pressure, is presented in Sec. V. We
examine the relationship between the electronic band gap and the energetic
stability of polytypes in Sec. VI. Finally, we draw our conclusions in Sec.
VII. Our DFT simulation parameters can be found in Appendix A.

##  II Characterization of structures

The bulk hexagonal PTMC geometries we have examined are as follows. (i) We
assume that each sublayer of chalcogen atoms and each sublayer of vertical PTM
dimers lie at the A, B, or C hexagonal sublattice sites, because energy minima
are overwhelmingly likely to occur at these high-symmetry configurations. (ii)
We assume that each chalcogen sublayer lies on a different hexagonal
sublattice to the PTM sublayer; our DFT calculations for InSe confirm that the
energy is around 2 eV per monolayer unit cell higher each time the chalcogen
atoms are on the same sublattice as the PTM dimers. In general a two-layer
structure is a 2H polytype in Ramsdell notation [56], and a three-layer
structure is a 3H polytype. However, there are exceptions; e.g., the $\gamma$
structure is a 3R polytype with a rhombohedral primitive Bravais lattice.
Nevertheless, for consistency and ease of automation, we have used a hexagonal
unit cell in all our calculations.

We have performed DFT calculations for all such two-layer and three-layer bulk
structures. We refer to each of these configurations by a character string
summarizing the 2D hexagonal sublattice sites for each sublayer of atoms in
the unit cell. Upper-case letters (A, B, and C) are used for PTM-dimer
sublayers; lower-case letters (a, b, and c) are used for chalcogen sublayers.
The 2D hexagonal sublattice sites for a single sublayer are shown in Fig. 2.
For example, the string “aBabCa” describes a two-layer bulk structure in which
the PTM dimers lie on the B and C sublattices, while the chalcogen atoms in
the first layer are all at the A sublattice sites and the chalcogen atoms in
the second layer are at the B and A sublattice sites. In this notation, the
$\varepsilon$ polytype [53] is aBabCb, the $\beta$ polytype [52] is aBabAb,
the $\gamma$ polytype [54] is aBabCbcAc, and the $\delta$ polytype [55] is
aBabAbaCacAc.

Figure 2: (a) 2D hexagonal sublattice labels A, B, and C for each sublayer,
used to construct structure label strings for bulk PTMCs. (b) Color-coded
structure of the “aBabCb”-stacked $\varepsilon$-GaSe as an example. The colors
of the atoms in panel (b) correspond to the colors of the sublattice sites in
panel (a).

PTMC structures are energetically invariant if we perform any rigid operations
(translations, rotations, or reflections). In-plane translations from one
sublattice to another correspond to even permutations of the sublattice labels
A, B, and C; thus, e.g., “aBabCa” is energetically equivalent to “bCbcAb.” In-
plane point rotations through $60^{\circ}$ or reflections in vertical planes,
together with translations, correspond to odd permutations of the sublattice
labels; thus “aBabCa” is equivalent to “aCacBa.” PTMC structures are also
equivalent under vertical displacements, which correspond to rotating the
structure strings through three characters; thus “aBabCa” is equivalent to
“bCaaBa.” Finally, structures are energetically invariant under reflections in
horizontal planes; thus “aCacBa” is equivalent to “aBcaCa.”

A program was written to loop over all valid structure strings (i.e., strings
in which each chalcogen atom is at a different sublattice site to the
neighboring PTM dimer) for multilayer bulk structures. Energetically
equivalent structure strings were eliminated and DFT input files for the
remaining structures were generated. We find that there are $2$ inequivalent
one-layer structures (these being the $\alpha_{\rm M}$ and $\beta_{\rm M}$
polytypes with AA stacking), $12$ inequivalent two-layer structures (two of
these being supercells of the one-layer structures), $62$ inequivalent three-
layer structures, $494$ inequivalent four-layer structures, $4292$
inequivalent five-layer structures, and $42158$ inequivalent six-layer
structures. The atomic positions and lattice vectors were relaxed within DFT
at zero external pressure, subject to the constraint of the initial symmetry.
The imposition of symmetry constrains the unit cell to be hexagonal and
constrains the atoms to 2D hexagonal sites, but it allows the sublayers to
relax in the out-of-plane direction and it also allows the $a$ and $c$
hexagonal lattice parameters to relax.

##  III Fit to the bulk PTMC energies

To represent the energy of each structure $S$ we fit

$\displaystyle E(S)$ $\displaystyle=$ $\displaystyle E_{\rm c}+\frac{1}{N_{\rm
l}(S)}\left[n_{\rm nc}(S)E_{\rm nc}+n_{\rm np}(S)E_{\rm np}\right.$ (1)
$\displaystyle\hskip 55.00008pt\left.{}+n_{\rm ab}(S)E_{\rm ab}+n_{\rm
snn}(S)E_{\rm snn}\right]~{}~{}~{}$

to the energy $E$ per monolayer unit cell, where $N_{\rm l}(S)$ is the number
of PTMC monolayers in structure $S$, $n_{\rm nc}(S)$ is the number of places
in the unit cell in which neighboring chalcogen atoms are on different
hexagonal sublattice sites, $n_{\rm np}(S)$ is the number of places in the
unit cell in which PTM dimers in neighboring layers are on different hexagonal
sites, $n_{\rm ab}(S)$ is the number of $\beta_{\rm M}$-polytype layers in the
unit cell, and $n_{\rm snn}(S)$ is the number of places in which the next-
nearest chalcogen atom is on the same hexagonal site as a PTM dimer. For our
aBa reference structure (AA-stacked $\alpha_{\rm M}$-PTMC), $n_{\rm
nc}(S)=n_{\rm np}(S)=n_{\rm ab}(S)=n_{\rm snn}(S)=0$. Hence the fitting
parameter $E_{\rm c}$ describes the total energy per monolayer unit cell of
the aBa structure. $E_{\rm nc}$ is the energy associated with neighboring
chalcogen atoms lying on different hexagonal sublattice sites rather than the
same sublattice site. $E_{\rm np}$ is the energy associated with PTM dimers in
neighboring layers not lying on the same sublattice. $E_{\rm ab}$ is the
energy of the $\beta_{\rm M}$ polytype of a single layer relative to the
energy of the $\alpha_{\rm M}$ polytype. Finally, $E_{\rm snn}$ is the energy
associated with second-nearest-neighbor chalcogen atoms lying on the same
hexagonal site rather than different sublattice sites. The energy of structure
$S$ relative to the aBa structure is $E_{\rm rel}(S)=E(S)-E_{\rm c}$.

The quality of the resulting fits is illustrated for the DFT-PBE-MBD* data in
Fig. 3. The fitted parameters and the root-mean-square (RMS) error in the fit
per degree of freedom are reported in Table 1. The two- and three-layer
structures are all distinct, with the sole exception of the aBa and aBc
structures. These were independently relaxed for the two- and three-layer
cases, and both the two- and three-layer versions were included in the fit.
The DFT-PBE-MBD* energy difference between the equivalent two- and three-layer
aBa and aBc structures is around $1$–$2$ meV per monolayer unit cell,
suggesting that the data suffer from a random error of this order of magnitude
due to the finite ${\bf k}$-point sampling grids and uncertainties in the
relaxed geometries. Thus the RMS errors shown in Table 1 are primarily due to
noise in the data rather than any shortcoming in the fitting function of Eq.
(1). The fit to the GaTe and InTe energy data is clearly significantly poorer
than for the other PTMCs.

Table 1: Parameters in the fit of Eq. (1) to our two- and three-layer DFT-PBE-MBD* PTMC energy data, together with the RMS error per degree of freedom. The parameters and the RMS error are in units of meV per monolayer unit cell. Parameter $E_{\rm c}$ in Eq. (1) (the total energy of the aBa structure) contains an arbitrary, pseudopotential-dependent offset, and is therefore not reported here.  PTMC | $E_{\rm nc}$ | $E_{\rm np}$ | $E_{\rm ab}$ | $E_{\rm snn}$ | RMS error
---|---|---|---|---|---
GaS |  $-47$. | $529$ |  $-1$. | $709$ |  $24$. | $028$ |  $-1$. | $202$ |  $1$. | $11$
GaSe |  $-56$. | $010$ |  $-1$. | $692$ |  $18$. | $742$ |  $-0$. | $857$ |  $1$. | $02$
GaTe |  $-79$. | $411$ |  $-0$. | $662$ |  $17$. | $002$ |  $-0$. | $804$ |  $2$. | $04$
InS |  $-69$. | $786$ |  $-1$. | $793$ |  $17$. | $499$ |  $-0$. | $621$ |  $0$. | $842$
InSe |  $-76$. | $943$ |  $-1$. | $347$ |  $15$. | $917$ |  $1$. | $513$ |  $1$. | $09$
InTe |  $-98$. | $382$ |  $0$. | $041$ |  $15$. | $794$ |  $2$. | $931$ |  $3$. | $11$

Figure 3: Scatter plots showing the fit of Eq. (1) to the DFT-PBE-MBD* energy
data for (a)–(c) gallium chalcogenides and (d)–(f) indium chalcogenides.
$E_{\rm rel}$ is the energy relative to the AA-stacked $\alpha_{\rm M}$-PTMC
structure [aBaaBaaBa ($={}$aBaaBa${}={}$aBa)].

##  IV Comparison of DFT functionals

We have computed the DFT energies within the local density approximation (LDA)
and the Perdew-Burke-Ernzerhof (PBE) variant of the generalized gradient
approximation [57]. We compare a representative set of semiempirical
dispersion-correction schemes: Grimme 2006 (G06) [58]; Ortmann, Bechstedt, and
Schmidt (OBS) [59]; and the many-body dispersion (MBD*) method [60, 61]. We
have also investigated the optB86b (MK) and optB88 (BO) nonlocal van der Waals
density functionals [62]. DFT simulation parameters such as the plane-wave
cutoff energy are summarized in Appendix A.

We obtained a complete set of DFT-PBE-G06 and DFT-PBE-MBD* total-energy
results for all two- and three-layer structures and fitted Eq. (1) to the
data. We also obtained DFT-LDA, DFT-PBE, DFT-LDA-OBS, and vdW-DF data for all
two-layer structures, to assess the performance of these functionals. The DFT
results for two-layer structures are shown in Fig. 4. The corresponding
results for three-layer structures are shown in Fig. 5. The disagreements
between different dispersion corrections indicate the limitations of DFT in
studies of layered structures. Alternative methods such as quantum Monte Carlo
approaches are required to provide independent benchmarks [63]. We regard the
DFT-PBE-MBD* method as somewhat more reliable than the other dispersion
corrections because it describes many-body interactions and screening effects
beyond a description by pairwise interatomic potentials, and because it has
been extensively benchmarked against diffusion quantum Monte Carlo data [64].

Figure 4: DFT energy $E_{\rm rel}$ of two-layer structures relative to the AA-
stacked $\alpha_{\rm M}$ polytype [aBaaBa ($={}$aBa)], for (a) GaS, (b) GaSe,
(c) GaTe, (d) InS, (e) InSe, and (f) InTe. Different exchange-correlation
functionals and dispersion-correction methods have been used. The “PBE-vdW”
results were obtained using the PAW method with the optB86b vdW-DF [62]. The
other vdW-DF data [65] are similar to the optB86b results shown.

Figure 5: DFT energy $E_{\rm rel}$ of three-layer structures relative to the
AA-stacked $\alpha_{\rm M}$ polytype [aBaaBaaBa ($={}$aBa)], for (a) GaS, (b)
GaSe, (c) GaTe, (d) InS, (e) InSe, and (f) InTe. Different dispersion-
correction methods are used.

##  V Structural stability

###  V.1 Zero external pressure

Using Eq. (1) together with the parameters shown in Table 1, we find that for
GaS, GaSe, GaTe, and InS the most stable hexagonal structure is aBabAb, which
corresponds to the $\beta$ polytype described in experiments [52]. This
structure consists of an AA′ stacking of $\alpha_{\rm M}$-polytype monolayers
and has D6h point group and P$6_{3}$/mmc space group. For GaSe, this structure
is more stable than the $\varepsilon$ and $\gamma$ polytypes by $0.86$ meV per
monolayer unit cell, and it is more stable than the $\delta$ polytype by a
mere $0.43$ meV per monolayer unit cell. The most energetically stable
structures of GaSe with unit cells of up to six layers are shown in Table 2.

Table 2: Most energetically competitive structures of GaSe with up to six layers in the unit cell, together with some other structures of interest. The DFT-PBE-MBD* energy of each structure relative to aBa is shown.  Structure | Energy (meV/cell/layer)
---|---
aBabAb ($\beta$-GaSe) | $-59.416$
aBabAbaBabAbaBabCb | $-59.130$
aBabAbaBabAbaBacAc | $-59.130$
aBabAbaBabAbaCacAc | $-59.130$
aBabAbaBabCbcBcbCb | $-59.130$
aBabAbaCacAc ($\delta$-GaSe) | $-58.988$
aBabAbaBabCb | $-58.988$
aBabAbaBacAc | $-58.988$
aBabAbaBabCbcAc | $-58.902$
aBabAbaCacAcbCb | $-58.902$
aBabAbaBabCbaBabCb | $-58.845$
aBabAbaBabCbaBacAc | $-58.845$
aBabAbaBabCbcAcbCb | $-58.845$
aBabAbaBacAcaBacAc | $-58.845$
aBabAbaBacAcbCbcAc | $-58.845$
aBabAbaCabAbaBabCb | $-58.845$
aBabAbaCabAbaBacAc | $-58.845$
aBabAbaCabAbaCacAc | $-58.845$
aBabAbaCabAbcBcbCb | $-58.845$
aBabAbaCacAcbCbcAc | $-58.845$
aBabAbaCacBcbCbcAc | $-58.845$
aBabAbcBcbAbaBacAc | $-58.845$
$\vdots$ | $\vdots$
aBabCb ($\varepsilon$-GaSe) | $-58.559$
aBabCbcAc ($\gamma$-GaSe) | $-58.559$
$\vdots$ | $\vdots$
aBacBc | $-56.010$

On the other hand, for InSe and InTe the most stable hexagonal structure is
aBacBc. This consists of an AB′ stacking of $\alpha_{\rm M}$-polytype
monolayers. For InSe this structure is more stable than the $\varepsilon$ and
$\gamma$ polytypes (aBabCb and aBabCbcAc) by just $0.17$ meV per monolayer
unit cell. The most stable structure differs from that of the gallium
chalcogenides and indium sulfide by a horizontal translation of every second
layer. Nevertheless, this structure also has D6h point group and P$6_{3}$/mmc
space group. The most stable structures of InSe in unit cells of up to six
layers are shown in Table 3.

Table 3: Most energetically competitive structures of InSe with up to six layers in the unit cell, together with some other structures of interest. The DFT-PBE-MBD* energy of each structure relative to aBa is shown.  Structure | Energy (meV/cell/layer)
---|---
aBacBc | $-76.943$
aBabCbaBacBcaBacBc | $-76.888$
aBabCbaCabCbaBacBc | $-76.888$
aBabCbaCabCbaCabCb | $-76.888$
aBabCbaCabCbaCacBc | $-76.888$
aBabCbaBacBc | $-76.860$
aBabCbaCabCb | $-76.860$
aBabCbaCacBc | $-76.860$
aBabCbaCabAbcAc | $-76.844$
aBabCbaCabCbcAc | $-76.844$
aBabCbaBabCbaBacBc | $-76.832$
aBabCbaBabCbaCabCb | $-76.832$
aBabCbaBabCbaCacBc | $-76.832$
aBabCbaBacAcaBacBc | $-76.832$
aBabCbaBacAcbAbcAc | $-76.832$
aBabCbaBacAcbAbcBc | $-76.832$
aBabCbaBacBcaCacBc | $-76.832$
aBabCbaBacBcbAbcBc | $-76.832$
aBabCbaCabAbaCabCb | $-76.832$
aBabCbaCabAbaCacBc | $-76.832$
aBabCbaCabCbaBacAc | $-76.832$
aBabCbaCacBcbAbcAc | $-76.832$
$\vdots$ | $\vdots$
aBabCb ($\varepsilon$-InSe) | $-76.777$
aBabCbcAc ($\gamma$-InSe) | $-76.777$
$\vdots$ | $\vdots$
aBabAbaCacAc ($\delta$-InSe) | $-76.020$
$\vdots$ | $\vdots$
aBabAb ($\beta$-InSe) | $-75.264$

As a test, we have relaxed the structures of aBabAb (the $\beta$ polytype) and
aBabCb (the $\varepsilon$ polytype) GaSe without any symmetry constraints. The
initial lattice vectors and atom positions were randomly offset by a small
amount from their exact hexagonal-cell values, and the positions and lattice
vectors were relaxed within DFT-PBE-MBD* at zero pressure. This did not lead
to a lowering of the total energy relative to the hexagonal cell, thus
providing direct evidence in support of our assumption that the unit cell is
hexagonal in all cases and that the atoms lie in horizontal sublayers on
hexagonal sublattice sites. Direct confirmation that the structures that we
have found to be most energetically stable in any of the PTMCs are also
dynamically stable is provided by the DFT-PBE-MBD* phonon dispersion curves
shown in Fig. 6. On the other hand, it is known that a monoclinic structure of
GaTe is more stable than the hexagonal structures studied here [66, 67].

Figure 6: DFT-PBE-MBD* phonon dispersion curves of (a) aBabAb (the $\beta$
polytype) GaSe and (b) aBacBc GaSe. The results were obtained using the method
of finite displacements in different sizes of supercell.

While experimental results [55, 53, 52, 54, 68] support our determination of
the $\alpha_{\rm M}$ monolayer structure as the most stable form, and also
agree that the most stable structure of GaS is aBabAb (the $\beta$ polytype)
[52], for InSe and GaSe the aBacBc structure calculated to be most stable is
not one of the commonly observed structures in experiments. Specifically, the
experimental work on InSe finds most often the $\gamma$ polytype (aBabCbcAc)
[54] and occasionally the $\varepsilon$ polytype (aBabCb) [68], neither of
which has inversion symmetry. For GaSe the $\varepsilon$ polytype (aBabCb)
[53] and the $\delta$ polytype (aBabCbcBcbAb) [55] are reported, against our
result of aBabAb (the $\beta$ polytype). It should be noted that our results
show several structures for each PTMC of comparable stability on a sub-meV-
per-monolayer-unit-cell scale. This has important consequences, not only on
the theoretical side, with the structure returned as the most stable being
sensitive to the van der Waals functional chosen, but also on the experimental
side, suggesting that the polytype of a PTMC crystal must be highly sensitive
to the crystal growth conditions. Indeed, it supports the observation of
multiple stacking faults and regions of different polytypes within a single
sample [69], and suggests that the synthesis of different PTMC polytypes
should be possible with careful tuning of experimental conditions. On the
theoretical side, an important conclusion is that a computational method with
an accuracy and precision of around $0.1$ meV per monolayer unit cell is
required to determine the most stable PTMC structure reliably. The $>10$ meV
per monolayer unit cell spread of DFT results with different van der Waals
correction schemes and the $\sim 1$ meV per monolayer unit cell disagreement
between independently relaxed equivalent two- and three-layer structures,
together with the disagreements with experiment regarding the most stable
structures, demonstrate that dispersion-corrected DFT is not currently capable
of such accuracy and precision.

We compare our relaxed lattice parameters with both previous DFT results and
experimental results in Table 4. Where comparison is possible, our dispersion-
corrected DFT-PBE calculations agree with experimental results to within $0.2$
Å (often an order of magnitude better). The hexagonal $a$ lattice parameter is
almost the same for all structures of a given PTMC, reflecting the in-plane
rigidity of the individual layers. However, the $c$ lattice parameter is much
more sensitive to the structure, as shown in Fig. 7. High-energy structures
generally have larger lattice parameters $c$.

Table 4: Hexagonal lattice parameters $a$ and $c$ of $\beta$-GaS (aBabAb), $\varepsilon$-GaSe (aBabCb), and $\gamma$-InSe (aBabCbcAc) obtained using various methods. Results without citation were obtained in the present work.  |  $\beta$-GaS (aBabAb) |  $\varepsilon$-GaSe (aBabCb) |  $\gamma$-InSe (aBabCbcAc)
---|---|---|---
Method |  $a$ (Å) |  $c$ (Å) |  $a$ (Å) |  $c$ (Å) |  $a$ (Å) |  $c$ (Å)
DFT-LDA | 3.541 | 15.214 | 3.762 | 15.666 |  |
DFT-LDA-OBS | 3.517 | 14.939 | 3.695 | 15.346 |  |
DFT-PBE | 3.633 [67], 3.626 | 16.677 [67], 17.633 | 3.823 [67], 3.811 | 17.848 [67], 18.201 | 4.091 [67] | 26.982 [67]
DFT-PBE-G06 | 3.570 | 15.497 | 3.740 | 15.899 | 3.942 | 25.251
DFT-PBE-MBD* | 3.583 | 15.266 | 3.771 | 15.744 | 4.031 | 24.919
vdW-DF2-C09 | 3.575 [67] | 15.460 [67] | 3.761 [67] | 15.943 [67] | 4.028 [67] | 24.996 [67]
Experiment | 3.587 [52] | 15.492 [52] | 3.743 [70] | 15.919 [70] | 4.002 [54] | 24.946 [54]

Figure 7: Hexagonal lattice parameter $c$ divided by number of layers
$n_{\text{layers}}$ against ground-state total energy $E_{\rm rel}$ for DFT-
PBE-MBD*-optimized structures of (a) bulk GaS, GaSe, and GaTe and (b) bulk
InS, InSe, and InTe. In each case the ground-state total energy $E_{\rm rel}$
is plotted relative to that of the aBa structure. The dashed lines show linear
fits to $c/n_{\text{layers}}$ against energy for each material.

###  V.2 Nonzero pressure

At zero temperature the most thermodynamically stable polytype is the
structure with the lowest enthalpy $H$. At sufficiently low pressures $p$ we
may approximate the enthalpy of a PTMC structure as

$H\approx E_{0}+pV_{0}+O(p^{2}),$ (2)

where $E_{0}$ and $V_{0}$ are the zero-pressure energy and volume. The
enthalpy is plotted against pressure for energetically competitive structures
of gallium chalcogenides and indium chalcogenides in Fig. 8. Of the two-layer
structures, aBabCb (the $\varepsilon$ polytype), aBacBc, and (in GaS, GaSe,
and InS) aBabAb (the $\beta$ polytype), have DFT-PBE-MBD* energies within a
few meV per monolayer cell of each other at zero pressure, but at higher
pressure, aBabAb (the $\beta$ polytype) is clearly disfavored. More generally,
the application of pressure simplifies the picture by reducing the number of
competing structures and increasing the relative enthalpies of those
structures. In GaSe, the three-layer structures aBabCbcAc (the $\gamma$
polytype) and aBabAbcAc are competitive at low pressure. This is not the case
in InSe. However, at very high pressures, three-layer structures may be
favored in InSe. Below 7.1 GPa, the aBacBc structure of InSe is favored; above
7.1 GPa, the aBacBacBc structure of InSe is favored. The structures favored at
high pressures feature PTM dimers on the same sublattice and neighboring
chalcogen atoms on different sublattices, as would be expected from steric
considerations. At low pressure it is once again clear that accuracy and
precision of around $0.1$ meV per monolayer unit cell are required to identify
the most stable polytype unambiguously.

Figure 8: Enthalpy against pressure [using Eq. (2)] for energetically
competitive structures of (a) GaS, (b) GaSe, (c) GaTe, (d) InS, (e) InSe, and
(f) InTe. The zero-pressure energy $E_{0}$ and volume $V_{0}$ data were
obtained from DFT-PBE-MBD* calculations. The enthalpies are plotted relative
to the enthalpy of the aBacBc structure (not the aBa structure). At any given
pressure, the structure with the lowest enthalpy is thermodynamically favored
at zero temperature. In panel (b) we also show DFT-PBE-MBD* enthalpies
obtained directly by relaxing the structure and lattice parameters at fixed
external pressure. The inset shows the low-pressure region in greater detail.
Note that the zero-pressure results shown here are obtained directly from the
DFT calculations and do not make use of the fit of Eq. (1); thus the relative
enthalpies shown here differ from the results shown in Tables 2 and 3 by
around a meV per monolayer unit cell.

In Fig. 8(b) we compare the linear approximation to the enthalpy [Eq. 2)] with
DFT enthalpies obtained by directly relaxing the lattice vectors at a given
external pressure. We find that the linear approximation is of quantitative
accuracy on a meV-per-monolayer-unit-cell scale for relative enthalpies up to
around $1$ GPa. Beyond this, the linear approximation provides a qualitative
picture that generally preserves the ordering of the structures, at least up
to $\sim 10$ GPa.

In all bulk PTMCs at multi-GPa pressures, the aBacBc structure is found to be
the most stable structure over a broad range of pressures in DFT-PBE-MBD*
calculations. This is the inversion-symmetric structure that is predicted to
be most stable for InSe and InTe at zero pressure, and consists of AB′-stacked
$\alpha_{\rm M}$ monolayers. Despite its ubiquitous presence in the
theoretical calculations, we are not aware that this structure has previously
been reported.

##  VI Electronic band structure

In Fig. 9 we plot the DFT-PBE electronic band structures of the theoretically
most stable polytypes of GaSe and InSe (aBabAb and aBacBc, respectively) and
the experimentally observed [53] $\varepsilon$ polytype (aBabCb) of GaSe. The
structures were relaxed using DFT-PBE-MBD*. In each case the polytypes exhibit
a direct band gap at the $\Gamma$ point of the two-layer hexagonal Brillouin
zone. The DFT-PBE band gaps, which are expected to be significant
underestimates of the true gaps [71], are $0.804$ and $0.742$ eV for the
aBabAb and aBabCb structures of GaSe, respectively. The low-energy band
structure is qualitatively similar for these two energetically competitive
structures of GaSe. The DFT-PBE band gap of the most stable structure of InSe
(aBacBc) is much smaller than the gap of GaSe, at $0.183$ eV.

Figure 9: Electronic band structures of low-energy structures of bulk PTMCs:
(a) aBabAb GaSe (the $\beta$ polytype and the lowest-energy structure in
theory), (b) aBabCb GaSe (the $\varepsilon$ polytype), and (c) aBacBc InSe
(the lowest-energy structure in theory). The horizontal dashed line shows the
Fermi energy in each case. The inset to panel (a) shows the hexagonal
Brillouin zone.

We have examined the band gap of a range of two-layer structures for each
material, finding that the vertical band gap at $\Gamma$ and the ground-state
energy of each structure are positively correlated, although with significant
noise: see Fig. 10. PTMC structures with smaller band gaps tend to be more
stable. In fact, the most stable two-layer structure of InTe has a direct gap
at $\Gamma$ of just $0.2$ meV. In most cases the vertical gap at $\Gamma$ is
the fundamental gap, especially for low-energy structures. A notable exception
is GaTe, where the vertical gap at $\Gamma$ is nonfundamental for all the two-
layer structures. In the most stable two-layer GaTe structure, the valence-
band maximum is at $\Gamma$ but the conduction-band minimum is on the
$\Gamma$–$M$ line. Previous work using DFT and many-body perturbation theory
has shown that $\gamma$-InSe changes from a direct-gap material to an
indirect-gap material under high pressure [72].

Figure 10: Vertical band gap at $\Gamma$ against ground-state total energy
$E_{\rm rel}$ relative to that of the aBa structure for DFT-PBE-MBD*-optimized
two-layer structures of (a) bulk GaS, GaSe, and GaTe and (b) bulk InS, InSe,
and InTe. The band gaps were calculated using DFT-PBE and the total energies
were evaluated using DFT-PBE-MBD*. The dashed lines show linear fits to the
gap against energy for each of the three materials. Where the symbols are
filled, the vertical gap at $\Gamma$ is equal to the fundamental band gap.

The experimentally measured gaps of $\beta$-InSe, $\gamma$-InSe and
$\epsilon$-InSe are $1.28$ eV [73], $1.25$–$1.29$ eV [74, 75], and $1.4$ eV
[76], respectively, which are (as expected) very much larger than the DFT-PBE
InSe gaps of energetically stable polytypes shown in Fig. 10. Nevertheless, we
would expect the qualitative conclusion that the band gap of a PTMC polytype
is positively correlated with its energy to continue to hold.

We note that the dispersion of the band-edge states in the out-of-plane
direction along $\Gamma$-A is substantial. The electronic structure is very
much three-dimensional, despite the layered crystalline structure of the
PTMCs. This dispersion arises due to strong interlayer hybridization of
$p_{z}$ orbital states on chalcogen atoms in the band-edge wave functions
[38]. It is the restriction of out-of-plane momentum in ultrathin PTMC films
that gives rise to their strong thickness-dependent electronic and optical
properties, with an increase in band gap for a reduced number of layers [13,
14].

##  VII Conclusions

We have used dispersion-corrected DFT methods to examine the relative
stability of a large number of candidate bulk hexagonal PTMC polytypes. For
all PTMCs there is a clear consensus among DFT functionals that the
$\alpha_{\rm M}$ monolayer polytype, in which the chalcogen atoms lie on the
same hexagonal sublattice, is $E_{\rm ab}=16$–$24$ meV per monolayer unit cell
more stable than the $\beta_{\rm M}$ polytype, in which the chalcogen atoms
lie on different hexagonal sublattices; indeed, all experimentally observed
bulk polytypes only feature $\alpha_{\rm M}$ monolayers [55, 53, 52, 54, 68].
Our DFT-PBE-MBD* calculations show that there is an energy gain of $-E_{\rm
nc}=50$–$100$ meV per monolayer unit cell from having neighboring chalcogen
atoms on different hexagonal sublattices; again, all experimentally observed
polytypes only have neighboring chalcogen atoms on different hexagonal
sublattices [55, 53, 52, 54, 68]. The DFT-PBE-MBD* energy gain associated with
PTM dimers in neighboring layers lying on different hexagonal sublattices is
$-E_{\rm np}=0.04$–$2.7$ meV per monolayer unit cell. This leads to a tendency
to avoid AA-stacked structures at zero pressure. However, in InSe and InTe
this is offset by an energy penalty of $E_{\rm snn}=1.5$–$2.9$ meV per
monolayer unit cell associated with PTM dimers and next-nearest chalcogen
atoms lying the same hexagonal sublattices; it is geometrically impossible to
have an AB- or ABC-stacked $\alpha_{\rm M}$ structure in which PTM dimers and
next-nearest chalcogen atoms all lie on different hexagonal sublattices. The
interplay between these effects leads to a subtle, sub-meV competition between
polytypes. Disagreements between dispersion-corrected DFT total energies are
of order $10$ meV per monolayer unit cell. Disagreements between the relative
energies of the lowest-energy polytypes are of order $1$–$5$ meV per monolayer
unit cell. Only for GaS is the observed stable polytype ($\beta$) predicted by
DFT-PBE-MBD* to have the lowest energy; however, in GaSe and InSe the observed
polytypes are very close in energy to the theoretically most stable structure.
We conclude that dispersion-corrected DFT methods are not yet able to predict
the relative stability of bulk PTMC polymorphs reliably; however, they can
provide insights into the energy scales involved and the types of structures
that are favored. The small energy differences between competing polytypes
imply that a wide variety of different polytypes are likely to be found in
experiments, and that stacking faults must be common in PTMC samples.

We find that application of pressure tends to favor an aBacBc PTMC structure
that has not previously been reported. In fact this polytype is found to be
most stable within DFT-PBE-MBD* at zero pressure for InSe and InTe. We also
find that there is a positive correlation between the ground-state total
energy and the electronic band gap; energetically stable PTMC polytypes tend
to have smaller band gaps.

###### Acknowledgements.

We acknowledge useful conversations with V. I. Fal’ko. All relevant data
present in this publication can be accessed at URL???. Computational resources
were provided by Lancaster University’s high-end computing facility, and by
the University of Manchester Computational Shared Facility. SJM acknowledges
support from EC Quantum Technology Flagship Project No. 2D-SIPC.

##  Appendix A Methodology

We used DFT as implemented in the castep [77] plane-wave-basis code to compute
the relative energies of PTMC crystals in a variety of bulk hexagonal
structures. We used ultrasoft pseudopotentials to represent atomic cores and
we used plane-wave cutoff energies of at least $566$ eV. The maximum distance
between ${\bf k}$ points in the Monkhorst-Pack grid was less than $0.0189$ Å-1
in each case. The force tolerance for geometry optimization was $0.514$ meV
Å-1. We verified that near-identical relative energies for PTMC structures
were obtained using the vasp [78] DFT code with projector augmented-wave (PAW)
pseudopotentials instead of castep. In the vasp calculations the basis
consisted of plane waves with a cutoff energy of $680$ eV and the Brillouin
zone was sampled by a Monkhorst-Pack grid of $18\times 18\times 4$ points. The
crystals were fully optimized with a force tolerance of $0.005$ eV Å-1. We
also verified that castep DFT relative energies obtained using norm-conserving
pseudopotentials were in agreement (on a meV-per-monolayer-unit-cell scale)
with our results obtained using ultrasoft pseudopotentials. Full data sets can
be found in the Supplemental Information [65].

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