# All “roads” lead to rocksalt structure

Matthew Jankousky  Colorado School of Mines, Golden, CO 80401, USA  Helen Chen
Harvey Mudd College, Claremont, CA 91711, USA  Andrew Novick  Colorado School
of Mines, Golden, CO 80401, USA  Vladan Stevanović<EMAIL_ADDRESS>Colorado School of Mines, Golden, CO 80401, USA  Colorado School of Mines,
Golden, CO 80401, USA  Harvey Mudd College, Claremont, CA 91711, USA  Colorado
School of Mines, Golden, CO 80401, USA<EMAIL_ADDRESS>Colorado School of
Mines, Golden, CO 80401, USA

###### Abstract

Rocksalt is the most common crystal structure among binary compounds.
Moreover, no long-lived, metastable polymorphs are observed in compounds with
the rocksalt ground state. We investigate the absence of polymorphism and the
dominance of the rocksalt structure via first-principles random structure
sampling and transformation kinetics modeling of three rocksalt compounds:
MgO, TaC, and PbTe. Random structure sampling reveals that for all three
systems the rocksalt local minimum on the potential energy surface is far
larger than any other, making it statistically the most significant structure.
The kinetics modeling shows that virtually all other relevant structures,
including most of the 457 known A1B1 prototypes, transform rapidly to
rocksalt, making it the only option at ambient conditions. These results
explain the absence of polymorphism in binary rocksalts, answer why rocksalt
is the structure of choice for high-entropy ceramics and disordered ternary
nitrides, and help understand/predict metastable states in crystalline solids
more generally.

The rocksalt crystal structure dominates both binary compounds and entropy
stabilized multi-component systems [1, 2, 3, 4, 5, 6, 7]. Within the A1B1
chemical space, which spans tetrahedrally bonded semiconductors (GaAs, SiC,
GaN, ZnO,…), photovoltaic materials (CdTe), piezoelectrics (ZnO),
thermoelectrics (PbTe), superconductors (NbN, FeSe) and other important
functional materials, rocksalt is the most common crystal structure. There are
348 compounds with the rocksalt structure reported in the Inorganic Crystal
Structure Database (ICSD) [8] out of 704 total A1B1. Moreover, rocksalt is the
ground state structure in the Materials Project database (MP) [9] for 238 of
total 668 A1B1 compounds found in MP, as illustrated in Fig. 1. In contrast,
the next most frequent A1B1 structure, zincblende, is the ground state for
only 51 compounds. Rocksalt is also the most frequent structure among binaries
as a whole. The next most common binary structures are fluorite (also s.g.
#225) with 75 A1B2 compositions and the Uranium silicide-type structure (s.g.
#221) with 66 A1B3 compositions. Regarding the entropy stabilized systems,
disordered ternary nitrides often crystalize in the rocksalt structure [1, 2,
3]. High-entropy ceramics (oxides, nitrides, and carbides) with a 1:1 metal to
nonmetal ratio always form in the rocksalt structure even when many parent
compounds in an equimolar mixture are not rocksalts themselves [4, 5, 6, 7].

While it is commonly associated with ionic compounds, the rocksalt ground
state appears across the entire Van Arkel-Ketelaar bonding triangle [10] shown
in Fig. 1, displaying that this structure is favorable in a variety of
chemical environments. Arguments based on maximizing the packing factor for a
given combination of ionic radii [11] or Phillips’ descriptor that chemically
delineates which A1B1 systems will adopt a rocksalt structure instead of a
wurtzite structure [12] might help explain why rocksalt is the ground state
structure. However, neither of these concepts is sufficient to predict or
explain whether rocksalt compounds will have metastable polymorphs or not.

Figure 1:  (left panel) A sankey diagram showing the distribution of ground
state structures (their space group numbers) among known A1B1 compounds. Other
reported short-lived polymorphs of compounds with the rocksalt ground state
are also shown. (right panel) The Van Arkel-Ketelaar bonding triangle
populated with known A1B1 compounds. Darker hexes imply more compounds. Those
with the rocksalt ground-state are highlighted in blue with the three
compounds studied in this work depicted by red circles.

For all the abundance and diversity of compounds with the rocksalt ground
state, there are virtually no experimentally reported metastable polymorphs of
A1B1 rocksalts in MP [9], ICSD [8] or available literature. The only reported
cases are high pressure phases of the CsCl-type [84], NiAs-type [55], and KOH-
type [31], and wurtzite in extremely thin films [16, 95] (see all compounds in
Supplementary Table LABEL:RS_polymorphs). None of these 4 structures types
represent long-lived, metastable states, as they all transform quickly to the
rocksalt structure if the stabilizing pressure or strain is released
(Supplementary Figure 5).

To elucidate the physical origins behind these observations, we investigate
three specific systems with rocksalt ground state and distinct bonding
characters: ionic MgO, largely metallic TaC, and covalent PbTe. Our analysis
derives from the first-principles characterization of: (a) the sizes of the
basins of attraction of different potential energy local minima (their
“widths”), and (b) the transformation pathways and associated kinetic barriers
for polymorphic transformations (“depths” of local minima). The “widths” of
local minima correlate to experimental realizability, while the “depths” are
connected to the lifetimes of polymorphs, and, as we will show, it is the
interplay of these two features that explains the absence of polymorphism in
rocksalt binary compounds and also its dominance in entropy-stabilized
systems.

Figure 2:  (a) Networks representing the high-symmetry polymorphs identified
by random sampling in MgO, TaC, and PbTe. The rocksalt structure is the
central node of each network. All other structures are labeled according to
their space group number. The size of each node represents the frequency of
occurence of a structure in the random sample. Note that a fixed size is used
to represent the width of the rocksalt local minimum, and other points are
scaled relative to this size. Structures are arranged so that the energy
increases with the polar angle and the distance of each node from the rocksalt
center is proportional to the distance travelled by the atoms during phase
transformation of that structure to rocksalt. (b) Calculated (SSNEB) energy
profiles along the transformation from the high-symmetry structures identified
in the random sampling to the rocksalt structure.

The “widths” of local minima are evaluated by the frequency of occurrence of a
structure in the first-principles random structure sampling [18], which
involves generating a large number of initial structures with random unit cell
vectors and random atomic positions. Using density functional theory (DFT)
[19], those random structures are relaxed to the closest local minimum on the
potential energy surface. The frequency of occurrence of a given structure
after relaxations then measures the size of its attraction basin, that is, the
total volume of configuration space occupied by that local minimum.
Statistically, this frequency of occurrence represents the probability to fall
into a given local minimum [20]. It has been shown that the experimentally
realized metastable polymorphs are exclusively those with high frequencies of
occurrence in the first-principles random structure sampling [18, 21, 20, 2,
22].

The “depths” of local minima are correspondingly measured by predicting
mechanisms and associated energy barriers for diffusionless transformations
between various structures. Mechanisms are identified by mapping crystal
structures atom-to-atom in a way that minimizes the distance traveled by atoms
along the pathway [23]. Energy barriers are estimated by computing the energy
along the pathway without any relaxations. This provides an upper bound for
the actual barrier. In cases where the upper limit is high, the minimal energy
pathways are calculated with relaxations performed via solid-state nudged
elastic band (SSNEB) method [24] using the mapping result as the starting
point.

Note that this procedure assumes concerted motion of atoms during the phase
transformation, and as such does not include interface effects associated with
nucleation and growth. Because of this, when evaluating if a phase transition
is rapid or sluggish, we follow a broadly adopted semi-quantitative
distinction [25]. In short, barriers close to 100 meV/atom or lower would be
classified as allowing rapid phase transformations, while those with barriers
much above this value would be classified as characteristic of sluggish
transformations 111For example, barrier for the concerted diamond-to-graphite
transformation is estimated to be $\sim$300 meV/atom.. The transformation
mechanisms and energy profiles from this work are collected into a database of
pathways that is provided in Supplementary Information.

Exploration of the relevant crystal phases of the three rocksalt compounds
from the first-pinciples random structure sampling shows that their potential
energy surfaces share many of the same features, as summarized in Fig. 2(a).
For MgO, the rocksalt structure is found for more than half of the random
structures, or 1061 times out of 2000 total random structures needed to
achieve converged results. The frequency of the next most frequent structure
is lower by a factor of $\sim$20\. For TaC, rocksalt occurs less frequently,
making up about 4 % of the relaxed random structures, but it is more frequent
than the next one by a factor of $\sim$5\. For PbTe, the rocksalt structure
also occurs $\sim$4 % of the time (80 out of 2000), and it is about 40 times
more frequent than the next one.

The majority of other structures found in the random sampling have no symmetry
at all (s.g. #1). They represent a “sea” of local minima, the number of which
increases exponentially with the system size [27]. Extrapolation to an
infinite number of initial random structures would return negligible
probabilities of occurrence for all of them. There are also more symmetric
structures that exhibit stable frequencies of occurrence as a function of the
number of random structures. These occur much less frequently than rocksalt
and are consequently less likely to be realized. They share some common
structural features with the rocksalt structure. Most are built from different
arrangements of 6-fold coordination polyhedra including face-sharing and/or
edge-sharing octahedra and trigonal prisms. Structures consisting of these
motifs include those with s.g. # 160, 166, 187, and 194 in MgO, s.g. #
160,166,187, and 194 in TaC, and s.g. # 160 and 186 in PbTe. MgO is the only
chemistry which exhibits symmetric structures that do not all fall into the
above categories. These other structures include the tetrahedrally coordinated
zincblende (ZB, s.g. #216) and wurtzite (WZ, s.g. #186) structures, a
hexagonal boron nitride (hBN, s.g. # 194), and a buckled hBN (s.g. #136)
structure, as well as a structure consisting of MgO6 octahedra and MgO5
trigonal bipyramids with long channels running in the c-direction (s.g. #
140).

We now turn to the question of the lifetimes of higher energy, symmetric
structures found by the random sampling. If any of them could potentially be
realized in spite of their narrow local minima it is important to ask whether
they would be long-lived (metastable). Interestingly, for all three
chemistries, structure mapping combined with SSNEB calculations finds that all
symmetric structures identified by the random sampling have low energy
barriers ($\lesssim 100$ meV/atom) for transformations to the rocksalt ground
state. The results are summarized in Fig. 2(b) and the mechanisms are provided
in Supplementary Information as part of the pathways database. For the 6-fold
coordinated structures, conversion to rocksalt generally involves a slipping
process that transforms trigonal prisms to octahedra and/or changes the
orientation of octahedra from face- to edge-sharing. For the structures with
MgOx polyhedra with $x<6$, mechanisms also include an increase in coordination
from 4 or 5 to 6.

Among structures found in random sampling, the highest barrier for
transformations to rocksalt is 127 meV/atom. This is for the TaC structures
with s.g. #194 and 160. Their predicted transformation mechanisms have the
same rate limiting steps because the structure with s.g. #160 is an
intermediate local minimum along the pathway from s.g. #194 to the rocksalt
structure. As a result of the relatively low energy barriers for
transformation to rocksalt, all symmetric structures from the random sampling
are expected to have short lifetimes at ambient conditions.

These results explain the lack of metastable polymorphs in binary rocksalts.
They imply that for all three studied chemistries the local minimum
corresponding to the rocksalt structure is 5 to 40 times larger than any other
and is hence, statistically much more significant. Additionally, all other,
statistically non-negligible local minima would be short-lived as the kinetic
barriers are not high enough to ensure kinetic trapping.

Figure 3:  Sankey diagrams showing the kinetic flow of 457 prototype
structures found in the ICSD spanning the A1B1 chemical space for each of MgO,
TaC, and PbTe. The bars on the left represent the distribution of the A1B1
prototypes across 7 crystal systems. Bars in the center display the
distribution of space groups that these prototypes relax into. Structures
which link to the s.g. #225 bar at the right end have kinetic barriers for
transformation to rocksalt less than 150 meV/atom.

What about other structures? It is also interesting to investigate whether
these conclusions hold for other structures not found by the random sampling.
According to ICSD, A1B1 compounds crystallize in 457 different structures
types (see Methods). To test which of these would constitute local minima for
the three studied chemistries and how shallow they might be, we substituted
the MgO, PbTe and TaC chemistries in all A1B1 prototype structures from ICSD
and relaxed them using DFT. For the resulting relaxed structures, we estimated
their kinetics for transformation to the rocksalt using our structure mapping
method. The results are summarized in Fig. 3.

Figure 4:  Energies of reported polymorphs for A1B1 chemistries where rocksalt
is reported as a high energy polymorph. Bars represent the energy of the
rocksalt relative to the convex hull, experimentally realized rocksalts are
blue, while theoretical ones are grey. Other experimentally realized
polymorphs with energy below rocksalt are filled blue dots, while polymorphs
with energy above rocksalt are open blue circles, and high pressure structures
with energy above rocksalt are additionally marked with a black dot. The only
reported polymorphs with energy above rocksalt are those in the exception
categories specified in the text.

For MgO, TaC, and PbTe, respectively, 245, 174, and 283 of the 457 prototypes
relax spontaneously to rocksalt structure implying zero or nearly zero
barriers. For the remaining structures, pathways that have static energy
profiles with maximum energy below 150 meV/atom are considered characteristic
of a rapid transformation to rocksalt. We use 150 meV/atom because our
structure mapping result is an estimate of the upper bound for the barriers.
The ssNEB calculations will, in most cases, relax to a minimal energy pathway
with a lower barrier. Adding this criterion, a total of 412, 251, and 369 of
the 457 prototypes are estimated to transform quickly to the rocksalt
structure in MgO, TaC, and PbTe.

These are relatively large fractions of common A1B1 structures, in particular
for MgO and PbTe. TaC seems a little different, but it is important to note
that the 108 prototypes that relax directly to structures with s.g. #194
include 38 that end in the nickeline (NC) structure type, which has a low
barrier for transformation into rocksalt according to our SSNEB calculations
(see Fig. 2(b)). These are marked by the blue rectangle in Fig. 3. The same is
true for s.g. #160 in TaC, s.g. #216 for MgO, and s.g. #186 for PbTe. In
summary we can say that 291 out of 457 prototypes (64%) transform quickly to
rocksalt in TaC. While not all prototypes transform rapidly into rocksalt,
those that do not are, according to the random structure sampling,
statistically insignificant and based on the collected evidence should be very
hard if not impossible to realize.

What if rocksalt is not the ground state? For many A1B1 compounds the ground
state is not rocksalt, but it does appear as either a high pressure phase or a
phase that can be epitaxially stabilized. Our results are relevant for these
situations too, as only the structures that have energies in between the
ground state and the rocksalt structure are plausible candidates for the
metastable states. Most of those with energies above rocksalt will tend to
transform rapidly into the rocksalt which will then likely transform quickly
into the ground state, making them short-lived. In this way, the energy of the
rocksalt structure then represents an energy limit for the likely metastable
states. This is of course true for the larger fraction of prototypes that do
transform rapidly. To what extent the rocksalt structure a limit of
metastability for the A1B1 chemistry can be seen in Fig. 4.

We find that for most A1B1 compositions which have rocksalt as a high energy
structure according to MP [9], other experimentally reported polymorphs have
their total energy equal or below the rocksalt structure. The only exceptions
to this trend (open circles) are (i) high pressure phases that transform
quickly to rocksalt and are not metastable (open circles with black dot), (ii)
misfit layered compounds, which are formed by thin layers of two different
compounds (see SI) [28], and (iii) instances with misassigned structure type.
This is all evident if we take SiC as an example. It has a relatively
corrugated potential energy surface with polytypes in the family of
zincblende, wurtzite, and a tetrahedrally coordinated rhombohedral structure
all forming as metastable polymorphs and all having energies below rocksalt as
confirmed by the random structure sampling [22]. A polymorph network and a
Sankey diagram for SiC are provided in Supplementary Figures 7 and 8.

Metastable polymorph discovery beyond A1B1. The realization that in A1B1
chemistries most structures with energies above rocksalt will transform
quickly to rocksalt and then to the respective ground-state may offer a
general insight for the discovery of metastable polymorphs in other
compositions. Stoichiometries other than A1B1 may behave the same way provided
that a structure playing the role of the rocksalt exists. Some evidence for
the A1B2 chemistry can be found in the literature. Namely, the fluorite
structure (also s.g. # 225) is the most commonly observed A1B2 structure and
it is well documented (Ref. [23] and the references therein) that the
transformations from the highest-pressure fluorite structure to other observed
lower pressure phases of SnO2 including the ground-state rutile are all rapid,
indicating fluorite may play a role similar to that of rocksalt in A1B1 (See
Supplementary Information Table 7). Indeed, A1B2 compositions with the
fluorite ground state have only cotunnite as a possible metastable polymorph,
and there are reports of it also transforming rapidly into fluorite [157].
Additionally, all known polymorphs of SiO2 have their total energies below
that of fluorite (see Supplementary Information Table 8).

Relevance to high entropy systems. The arguments connecting “widths” of the
local minima on the potential energy surface to experimental realizability
apply also to atomic disorder and entropy stabilized systems. An intriguing
fact is that virtually all equimolar mixtures of A1B1 binaries are stabilized
in the rocksalt structure even when a number of components are not rocksalts
themselves. The exceptions are high-entropy borides, which form in a hexagonal
structure with layers of edge sharing octahedra which share faces with the
octahedra in adjacent layers, likely due to the tendency of boron to form
rings. While simplistic entropy arguments based on counting configurations of
atoms on a given lattice offer an explaination for the formation of the
mixture as opposed to the phase separation [6, 7], there is no explanation as
to why the rocksalt structure is observed and not some other. Namely,
contributions to configurational entropy from different arrangements of atoms
are the same irrespective of the type of the crystal structure.

Based on the previous discussion we can say that in a given mixture each
individual arrangement of atoms having the rocksalt structure as the
underlying (parent) structure could be considered a relatively “wide” local
minimum. The width of the rocksalt local minimum is also relevant for mixtures
as it allows relatively large distortions away from the ideal rocksalt
structure while still remaining inside the rocksalt basin of attraction. Then
all those arrangements of atoms (cations and/or anions) will collectively form
a very wide “valley” on the potential energy surface corresponding to the
disordered rocksalt phase. Because of all this it could be hypothesized that
rocksalt will remain statistically the most significant structure even when a
large fraction of components are not rocksalts themselves.

We put this hypothesis to test by performing the random structure sampling for
a hypothetical (MgO)(ZnO)(CuO) mixture. Out of the three oxides, only MgO
crystalizes in the rocksalt structure, while both ZnO and CuO strongly favor
4-fold coordination in their ground states. Random structure sampling results
show disordered rocksalt as the highest occurring structure-type with
different atomic configurations spanning a relatively narrow energy range as
evidenced by the thermodynamic density of states (see Fig. 9 in Supplementary
Information). Hence, in spite of only 1/3 of cations favoring rocksalt
structure, it remains statistically the most significant crystal structure of
all.

Another interesting example supporting our discussion is ZnZrN2 ternary
nitride, which is found in thin-film synthesis to always form in the
disordered rocksalt structure in spite of having a well-defined non-rocksalt
(layered) ground state [2]. The case of ZnZrN2 is particularly interesting as
many other structure-types are calculated to have lower energies than the
cation-disordered rocksalt phase [2]. The random structure sampling reported
in Ref. [2] once again shows that the cumulative frequency of occurrence of
the collection of the disordered rocksalt configurations is $\sim$100 times
higher than the ground state, which appears as a very narrow local minimum in
comparison to the rocksalt. Hence, our findings are also consistent with the
difficult experimental realization of the ZnZrN2 in its ordered ground state
structure and the realization of the disordered rocksalt instead, as a
consequence of the high configurational entropy of the disordered rocksalt
originating from the large fraction of the configurational space that can be
assigned to the rocksalt structure.

##  I Conclusion

In this paper we examine the lack of metastable polymorphs among compounds
with A1B1 stoichiometry and the rocksalt ground state. Using first-principles
random structure sampling and transformation kinetics modeling we show that a
combination of probabilistic and kinetic factors is responsible for the lack
of polymorphism in this important group of materials. Namely, we demonstrate
using ionic MgO, metallic TaC and covalent PbTe as case studies, that
metastable polymorphs are not found among A1B1 rocksalts because (a) the
rocksalt structure represents a very large local minimum on the potential
energy surface in comparison to other structures, which makes it statistically
much more significant than any other, and (b) that virtually all relevant
crystal structures including those found by the random structure sampling as
well as the majority of the A1B1 structure prototypes transform rapidly to the
rocksalt structure. Moreover, the minority of the prototypes that do not
transform rapidly all represent a very narrow local minima according to the
random sampling, and, are therefore statistically insignificant. In other
words, while they may be candidate metastable structures their experimental
realization is predicted to be very difficult if not entirely unlikely. In
addition, we showed that same arguments can be extended to explain observed
polymorphs in A1B1 compounds that do not have rocksalt ground state, and
possibly also in other chemistries. Probabilistic factors (“width” of local
minima) are also found to be responsible for the realization of the disordered
rocksalt structure in high-entropy ceramics and disordered ternary nitrides.
Lastly, all transformation pathways from A1B1 prototypes to the rocksalt
structure are collected in a database provided in Supplementary Information.

###### Acknowledgements.

This work is supported by the National Science Foundation, Grant No.
DMR-1945010 and was performed using computational resources provided by
Colorado School of Mines and using services provided by the PATh Facility,
which is supported by the National Science Foundation award #1836650. The
support of the NSF-funded REU program at Colorado School of Mines, Grant No.
DMR-1950924, is also acknowledged.

##  II Methods

###  II.1 First-principles random structure sampling

We evaluate which local minima are statistically relevant via the first-
principles random structure sampling [18]. In this sampling, lattice vectors
with a random choice of parameters $a,b,c,\alpha,\beta,\gamma$ are generated.
A given number of atoms are then distributed over two interpenetrating grids
of points, one for cations and the other for anions, to promote cation-anion
coordination. Grids are constructed by discretizing a family of planes defined
by the reciprocal lattice vector ${\bf G}=n_{1}{\bf g}_{1}+n_{2}{\bf
g}_{2}+n_{3}{\bf g}_{3}$ with a random choice of $n_{1},n_{2},n_{3}$. Cations
and anions are distributed along the minima and maxima of a plane wave defined
by $cos({\bf Gr})$, respectively. To prevent atoms from being placed too close
to one another, a probability distribution is constructed of Gaussians
centered at the location of each atom, and new atoms are placed in regions
where the sum of Gaussians is low. In this work supercells with 24 atoms are
generated, and the number of random structures is chosen based on the
convergence of the appearance of different structures within the samples.
Plots demonstrating this convergence can be found in the Supplementary
Information Figure 10. This results in approximately 2000 structures for MgO,
1000 structures for TaC, and 2000 structures for PbTe. Once a structure is
generated, it is then relaxed using DFT to its closest local minimum on the
potential energy surface. After relaxations are finished the relaxed
structures are then grouped into classes of equivalent structures if (a) their
space group assignments are the same, (b) their energies are within 10
meV/atom, (c) their volumes are within 5 %, and (d) and the coordination
numbers of atoms up to 3rd coordination shell are the same.

###  II.2 Density functional theory relaxations

Relaxations of the random structures include all degrees of freedom (cell
shape, volume, atomic positions) and are performed using the conjugate
gradient algorithm such that each structure relaxes to its closest local
minimum on the potential energy surface with high probability. Forces for the
conjugate gradient algorithm are evaluated with density functional theory [19]
using the Vienna ab initio Simulation Package (VASP) [30]. The relaxtions are
performed with the generalized gradient approximation, specifically, the
Perdew-Burke-Ernzerhof exchange-correlation functional [31]. Electronic
degrees of freedom are described within the projector-augmented wave (PAW)
formalism [32]. The plane-wave cutoff used in all calculations is 540 eV, and
the Monkhorst-Pack k-point grids [33] are automatically generated with a
length parameter of 20 to achieve uniform subdivisions along the reciprocal
lattice vectors in all structures. The electronic structure is considered
converged when the change in total energy between steps of the Davidson
algorithm is less than $10^{-6}$ eV. The ionic positions are considered
converged when the change in energy between steps of the conjugate gradient
algorithm for ionic positions is less than $10^{-5}$ eV. For numerical reasons
cell shape and volume relaxations are restarted at least 3 times followed by a
self-consistent run. All structures with final pressures above 3 kbar
calculated in the final, self-consistent run are re-relaxed until final
pressures are all below 3 kbar.

###  II.3 Barrier approximation

Once high symmetry structures are identified via random sampling, their
barrier for a concerted transformation to the rocksalt structure is
approximated. Optimal atom-to-atom mapping is performed between each relevant
local minimum and the rocksalt structure as in [23]. In this mapping,
different representations of the unit cell of each structure are compared to
identify those combinations with maximum volumetric overlap, to minimize the
distortion of the unit cell along the pathway. For cells with maximum
volumetric overlap, atoms are matched between the two structures by way of the
Munkres algorithm [34], an iterative solution of the assignment problem.
Iterating over unit cell combinations with the Munkres algorithm returns
mappings with minimized distance traveled by the atoms during the concerted
transformation. For each mapping, the first shell coordination is computed for
ten images along the transformation. We find that mappings with monotonic
changes in coordination have smaller barriers, so these are used as starting
points where possible. If there is no path with monotonic change in
coordination, the path chosen is that with the smallest change in
coordination, if multiple paths have the same change in coordination along the
path, the path with the shortest distance is used.

An upper bound for the energy barrier of this transformation can be computed
by calculating the energy of each image along the path. Energy is calculated
using the same parameters as the relaxations, but the structures along the
path are not allowed to relax. The energies are calculated for structures
identified from the random sample, if the barrier
($E_{max,path}-E_{polymorph}$) is found to be greater than 100 meV/atom, we
perform solid state nudged elastic band (SSNEB) calculations to find the
minimum energy pathway near the pathway predicted as above [24]. The SSNEB
calculations are considered converged when the force perpendicular to the
tangent along the path is less than 0.1 eV/Angstrom. Initial calculations use
9 intermediate images, if these calculations exhibit intermediate local
minima, the local minima are relaxed, and a new SS-NEB calculation is
performed between the local minima along the pathway in order to converge each
step along the path.

###  II.4 Database of transformation mechanisms

To determine if prototypical A1B1 structures are likely to transform quickly
to the rocksalt structure and to begin the population of a database of NEB
starting points, we used the AFLOW Xtalfinder [35] to classify each A1B1
structure in the ICSD as belonging to an AFLOW prototype [36]. For some
prototypes, all atoms occupy fixed Wyckoff positions and the only variable
parameter is the lattice constant. For these cases, only a single
representative of that prototype is needed in each chemistry. For those where
there exist structures in the ICSD with different lattice constant ratios and
variable wyckoff positions such that atom positions vary between members of
the same prototype, structures are classified as equivalent using the AFLOW
Xtalfinder [35], such that every distinct structure found in the ICSD for a
given prototype is represented. If swapping A and B sites leads to different
coordination of A and/or B atoms, the swapped structure is also considered.

For each prototypical structure identified in this way, the A and B sites are
populated with the elements of the three example compounds. The lattice
constant is then scaled such that the average bond length between A and B
atoms is equal to the bond length found in the rocksalt structure for each
chemistry. Atoms are then perturbed by 0.3 Angstroms, and the perturbed
structures are relaxed as described in the “Density Functional Theory
Calculations” section.

The resulting structures are grouped according to their AFLOW prototype
labels, and the resulting distribution of space groups is the second column of
nodes in Fig. 3. An atom-to-atom map is generated from each unique structure
to the rocksalt structure, as in the “Barrier approximation” section. Only the
static barriers are computed for these structures. Those structures with a
kinetic barrier to the rocksalt structure of 150 meV or less are considered to
have rapid kinetics, and are added to the rocksalt node in the third column of
Fig. 3. The structures with static barriers of greater than 150 meV/atom are
grouped by their space group in third column of Fig. 3.

## References

  * Sun  _et al._ [2019] W. Sun, C. J. Bartel, E. Arca, S. R. Bauers, B. Matthews, B. O. nanos, B.-R. Chen, M. F. Toney, L. T. Schelhas, W. Tumas, J. Tate, A. Zakutayev, S. Lany, A. M. Holder, and G. Ceder, Nature Materials 18, 732 (2019).
  * Woods-Robinson  _et al._ [2022] R. Woods-Robinson, V. Stevanovic, S. Lany, K. N. Heinselman, M. K. Horton, K. A. Persson, and A. Zakutayev, Phys. Rev. Mater. 6, 043804 (2022).
  * Bauers  _et al._ [2019] S. R. Bauers, A. Holder, W. Sun, C. L. Melamed, R. Woods-Robinson, J. Mangum, J. Perkins, W. Tumas, B. Gorman, A. Tamboli, G. Ceder, S. Lany, and A. Zakutayev, PNAS 116, 14829 (2019).
  * Oses  _et al._ [2020] C. Oses, C. Toher, and S. Curtarolo, Nature Review Materials 5, 295 (2020).
  * Rost  _et al._ [2015] C. M. Rost, E. Sachet, T. Borman, A. Moballegh, E. C. Dickey, D. Hou, J. L. Jones, S. Curtarolo, and J.-P. Maria, Nature Communications 6, 8485 (2015).
  * Sarker  _et al._ [2018] P. Sarker, T. Harrington, C. Toher, C. Oses, M. Samiee, J.-P. Maria, D. W. Brenner, K. S. Vecchio, and S. Curtarolo, Nature Communications 9, 4980 (2018).
  * McCormack and Navrotsky [2021] S. J. McCormack and A. Navrotsky, Acta Materialia 202, 1 (2021).
  * Belsky  _et al._ [2002] A. Belsky, M. Hellenbrandt, V. L. Karen, and P. Luksch, Acta Crystallogr., Sect. B 58, 364 (2002).
  * Jain  _et al._ [2013] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Persson, APL Materials 1, 011002 (2013).
  * Allen  _et al._ [1993] L. C. Allen, J. F. Capitani, G. A. Kolks, and G. D. Sproul, Journal of Molecular Structure 300, 647 (1993).
  * Kummerfeld  _et al._ [2008] J. K. Kummerfeld, T. S. Hudson, and P. Harrowell, The Journal of Physical Chemistry B Letters 112, 10773 (2008).
  * Phillips [1970] J. C. Phillips, Rev. Mod. Phys. 42, 317 (1970).
  * Abrahams and Bernstein [1965] S. Abrahams and J. Bernstein, Acta Crystallographica (1,1948-23,1967) 18, 926 (1965).
  * Weir  _et al._ [1986] S. Weir, Y. Vohra, and A. Ruoff, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 33, 4221 (1986).
  * Hull and Keen [1999] S. Hull and D. Keen, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 59, 750 (1999).
  * Martínez-Boubeta  _et al._ [2012] C. Martínez-Boubeta, A. S. Botana, V. Pardo, D. Baldomir, A. Antony, J. Bertomeu, J. M. Rebled, L. López-Conesa, S. Estradé, and F. Peiró, Physical Review B 86, 041407 (2012).
  * Min  _et al._ [2012] N. K. Min, K. Yong-Il, J. Younghun, L. S. Mi, K. B. G., C. Ran, C. Sang-Il, S. Hyunjoon, and P. J. T., Journal of the American Chemical Society 134, 8392 (2012).
  * Stevanović [2016] V. Stevanović, Phys. Rev. Lett. 116, 075503 (2016).
  * Hohenberg and Kohn [1964] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  * Jones and Stevanović [2017] E. B. Jones and V. Stevanović, Phys. Rev. B 96, 10.1103/PhysRevB.96.184101 (2017).
  * [21] C. M. Caskey, A. Holder, S. Shulda, S. Christensen, D. Diercks, C. Schwartz, D. Biagioni, D. Nordlund, A. Kukliansky, A. Natan, _et al._ , J. Chem. Phys. 144, 144201.
  * Jankousky  _et al._ [2023] M. Jankousky, E. M. Garrity, and V. Stevanović, Physical Review Materials 7, 053606 (2023).
  * Stevanović  _et al._ [2018] V. Stevanović, R. Trottier, C. Musgrave, F. Therrien, A. Holder, and P. Graf, Phys. Rev. Mater. 2, 10.1103/PhysRevMaterials.2.033802 (2018).
  * Sheppard  _et al._ [2012] D. Sheppard, P. Xiao, W. Chemelewski, D. D. Johnson, and G. Henkelman, J. Chem. Phys. 136, 074103 (2012).
  * Trinkle  _et al._ [2003] D. R. Trinkle, R. G. Hennig, S. Srinivasan, D. M. Hatch, D. Jones, H. T. Stokes, R. C. Albers, and J. W. Wilkins, Physical Review Letters 91, 025701 (2003).
  * Note [1] For example, barrier for the concerted diamond-to-graphite transformation is estimated to be $\sim$300 meV/atom.
  * Stillinger [1999] F. H. Stillinger, Phys. Rev. E 59, 48 (1999).
  * Wiegers [1996] G. Wiegers, Progress in Solid State Chemistry 24, 1 (1996).
  * Morris  _et al._ [2001] E. Morris, T. Groy, and K. Leinenweber, Journal of Physics and Chemistry of Solids 62, 1117 (2001).
  * Kresse and Hafner [1993] G. Kresse and J. Hafner, Phys. Rev. B 47, 10.1103/PhysRevB.47.558 (1993).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 10.1103/PhysRevLett.77.3865 (1996).
  * Kresse and Joubert [1999] G. Kresse and D. Joubert, Phys. Rev. B 59, 10.1103/PhysRevB.59.1758 (1999).
  * Monkhorst and Pack [1976] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).
  * Kuhn [1955] H. W. Kuhn, Naval research logistics quarterly 2, 83 (1955).
  * Hicks  _et al._ [2021] D. Hicks, C. Toher, D. C. Ford, F. Rose, C. D. Santo, O. Levy, M. J. Mehl, and S. Curtarolo, npj Comput. Mater. 7, 30 (2021).
  * Mehl  _et al._ [2017] M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. Hart, and S. Curtarolo, Computational Materials Science 136, S1 (2017).

All “roads” lead to rocksalt structure - Supplementary Information

Matthew Jankousky

Helen Chen

Andrew Novick

Vladan Stevanović

Table 1:  All compounds in the Materials Project Database with a rocksalt ground state and an experimentally reported polymorph. Inspection of references shows that there are no metastable polymorphs. compound | polymorph space groups | structure types | realization condition | citations
---|---|---|---|---
INa | 141 |  | mislabeled | [1, 2]
PrSb | 123 | CsCl | high pressure | [3, 4]
CeSb | 123 | CsCl | high pressure | [5, 6]
PuSb | 123, 221 | CsCl | high pressure | [7, 7, 8]
IRb | 221 | CsCl | high pressure | [9, 10]
LaAs | 123 | CsCl | high pressure | [11, 12]
PbTe | 221, 62 | CsCl, Distorted RS | high pressure | [13, 14, 15, 16]
KCl | 221 | CsCl | high pressure | [17, 10]
TlI | 221, 63 | CsCl, Distorted RS | high pressure | [18, 19, 20]
CTl | 99, 221 | CsCl | high pressure | [21, 22, 23]
MgO | 186, 187 | Wurtzite | thin film | [24, 25, 26]
CaTe | 221 | CsCl | high pressure | [27, 28]
TlCl | 221, 63 | CsCl, Distorted RS | high pressure | [19, 29, 30]
ClAg | 11, 63 | Distorted RS, KOH | high pressure | [31, 31, 31]
HCr | 186 | Wurtzite | H interstitial in metal, theoretical | [32, 32]
TeSr | 221 | CsCl | high pressure | [27, 27]
SMn | 216, 186, 221 | CsCl, Wurtzite | high pressure, alloying | [33, 33, 34, 35]
SmTe | 221 | CsCl | high pressure | [36, 37]
NpAs | 221 | CsCl | high pressure | [38, 39]
PbSe | 42, 221, 62, 63 | CsCl, Distorted RS | high pressure | [40, 40, 41, 42, 43]
TePr | 221 | CsCl | high pressure | [36, 36]
SnTe | 216, 221 | CsCl | high pressure | [44, 45, 46]
BrCs | 221 | CsCl | high pressure | [19, 9]
BaTe | 221 | CsCl | high pressure | [47, 48]
SeBa | 221 | CsCl | high pressure | [49, 50]
KN | 129 |  | mislabeled | [51, 52]
OBa | 129, 194 | NiAs | high pressure | [53, 54, 55]
ThAs | 221 | CsCl | high pressure | [56, 39]
NdSb | 123 | CsCl | high pressure | [57, 4]
TbSe | 186 | Wurtzite | thin film | [58, 59]
SmSe | 186 | Wurtzite | thin film | [60, 59]
PrAs | 123 | CsCl | high pressure | [61, 62]
SSr | 221 | CsCl | high pressure | [63, 64]
BrRb | 221 | CsCl | high pressure | [10, 65]
NAg | 64 |  | mislabeled | [66, 67]
PuAs | 221 | CsCl | high pressure | [68, 69]
PCe | 221 | CsCl | high pressure | [70, 71]
LaN | 129 | CsCl | high pressure | [72, 73]
HCs | 221 | CsCl | high pressure | [74, 75]
TeIn | 140, 221 | CsCl | high pressure | [76, 76, 77]
DySe | 186 | Wurtzite | alloyed thin film | [59, 78]
LaS | 64, 129 | CsCl | misfit layered compound, high pressure | [79, 80, 81]
SnSb | 216, 221 | CsCl | high pressure | [44, 82, 83]
ClNa | 221 | CsCl | high pressure | [84, 85]
SPb | 5, 39, 12, 221, 62, 63 | CsCl, Distorted RS | high pressure | [86, 87, 88, 89, 41, 13, 90, 91]
EuTe | 221 | CsCl | high pressure | [36, 36]
LaSe | 12 |  | misfit layered compound | [92, 81]
SmS | 139 | Distorted RS | misfit layered compound | [93]
NpTe | 221 | CsCl | high pressure | [94, 7]
MnO | 186 | Wurtzite | thin film | [95]
AsU | 221 | CsCl | high pressure | [96, 39]
NPr | 129 | CsCl | high pressure | [97, 98]
CaO | 64 |  | composite crystal | [99, 100]
LaSb | 123 | CsCl | high pressure | [6, 5]
FCs | 221 | CsCl | high pressure | [10, 9]
BrAg | 11 | KOH | high pressure | [31, 31]
KI | 221 | CsCl | high pressure | [101, 102]
BrTl | 221, 63 | CsCl, Distorted RS | high pressure | [103, 104, 19]
NdAs | 123 | CsCl | high pressure | [61, 105]
ClCs | 221 | CsCl | high pressure | [106, 107]
FRb | 221 | CsCl | high pressure | [102, 108]
PuTe | 221 | CsCl | high pressure | [7, 109]
SmAs | 123 | CsCl | high pressure | [61, 110]
NpSb | 123 | CsCl | high pressure | [7, 111]
KBr | 221 | CsCl | high pressure | [112, 113]
ClRb | 221 | CsCl | high pressure | [114, 10]
ThSe | 221 | CsCl | high pressure | [115, 116]
ICs | 221, 62 | CsCl, Distorted RS | high pressure | [117, 9, 19]
BiCe | 123, 221 | CsCl | high pressure | [118, 118, 119]
FK | 221 | CsCl | high pressure | [10, 120]
ThSb | 221 | CsCl | high pressure | [56, 56]
EuS | 139 | Distorted RS | misfit layered compound | [121, 122]
Figure 5:  Computed pathways and barriers for experimentally reported polymorphs of rocksalt compounds: CsCl-type in NaCl, NiAs-type in BaO, KOH-type in AgCl, and wurtzite-type in MgO. space group | frequency | $\Delta E_{f}$ | NEB barrier | description | name
---|---|---|---|---|---
number |  | (eV/atom) | (eV/atom) |  |
225 | 1061 | 0 | 0 | rocksalt (RS) | MgO_24_1960
194 | 42 | 0.035 | 0.057 | hexagonal boron nitride (hBN) | MgO_24_1015
72 | 1 | 0.060 |  | hBN with fault between rings | MgO_24_648
186 | 8 | 0.068 | 0.088 | wurtzite (WZ) | MgO_24_1977
166 | 2 | 0.076 | 0.099 | RS with stacking fault (face sharing octahedra) | MgO_24_1988
194 | 1 | 0.076 | 0.030 | RS with stacking fault (face sharing octahedra) | MgO_24_345
14 | 1 | 0.078 |  | hBN with fault | MgO_24_265
13 | 1 | 0.079 |  | distorted trigonal bipyramids | MgO_24_202
160 | 26 | 0.081 | 0.101 | rhombohedral WZ polytype | MgO_24_1296
140 | 1 | 0.082 | 0.001 | trigonal bipyramids and octahedra with pores in the c-direction | MgO_24_141
36 | 3 | 0.087 |  | buckled hBN | MgO_24_828
136 | 1 | 0.088 | 0.092 | buckled hBN | MgO_24_607
160 | 1 | 0.098 | 0.053 | face and edge sharing octahedra, trigonal prisms | MgO_24_1011
166 | 58 | 0.101 | 0.041 | RS with stacking fault (face sharing octahedra) | MgO_24_266
12 | 1 | 0.102 |  | RS with channels in c-direction | MgO_24_527
216 | 13 | 0.102 | 0.044 | Zincblende (ZB) | MgO_24_775
12 | 1 | 0.105 |  | distorted hBN | MgO_24_1806
36 | 1 | 0.113 |  | buckled hBN | MgO_24_1699
187 | 8 | 0.130 | 0.30 | face sharing octahedra and trigonal prisms | MgO_24_1220
12 | 1 | 0.130 |  | RS with channels in c-direction | MgO_24_1971
194 | 13 | 0.150 | 0.062 | RS stacking fault - all octahedra face sharing | MgO_24_728
36 | 21 | 0.155 |  | square pyramids | MgO_24_1080
12 | 1 | 0.161 |  | see-saws and tetrahedra with channels | MgO_24_1181
36 | 1 | 0.164 |  | square pyramids | MgO_24_1817
38 | 1 | 0.168 |  | square pyramids and tetrahedra | MgO_24_1246
161 | 1 | 0.218 | 0.009 | disordered 7-fold coordination | MgO_24_1609
43 | 4 | 0.220 |  | disordered 5-, 6-, and 7-fold | MgO_24_304
15 | 2 | 0.221 |  | disordered see-saw, tetrahedra, square pyramids | MgO_24_937
194 | 1 | 0.244 | 0.019 | nickeline (NC) | MgO_24_1249
Table 2:  Random sampling structures identified for MgO space group | frequency | $\Delta E_{f}$ | NEB barrier | description | name
---|---|---|---|---|---
number |  | (eV/atom) | (eV/atom) |  |
225 | 45 | 0 | 0 | rocksalt (RS) | C1Ta1_24_18
166 | 4 | 0.066 | 0.89 | trigonal prisms and octahedra | C1Ta1_24_990
160 | 10 | 0.072 | 0.127 | trigonal prisms and octahedra | C1Ta1_24_849
187 | 2 | 0.092 | 0.067 | trigonal prisms and octahedra | C1Ta1_24_719
194 | 2 | 0.123 | 0.127 | nickeline (NC) | C1Ta1_24_770
166 | 1 | 0.158 | 0.0636 | RS with stacking fault | C1Ta1_24_909
63 | 1 | 0.206 |  | distorted hBN | C1Ta1_24_451
12 | 1 | 0.216 |  | layerd trigonal prisms, distorted trigonal bipyramids, octahedra | C1Ta1_24_345
44 | 1 | 0.244 |  | trigonal prisms and octahedra | C1Ta1_24_416
12 | 1 | 0.264 |  | distorted trigonal prisms and 7-fold coordination | C1Ta1_24_581
Table 3:  Random sampling structures identified for TaC space group | frequency | $\Delta E_{f}$ | NEB barrier | description | name
---|---|---|---|---|---
number |  | (eV/atom) | (eV/atom) |  |
225 | 84 | 0 |  | rocksalt (RS) | PbTe_24_761
12 | 2 | 0.051 |  | RS with channels in c-direction | PbTe_24_46
186 | 1 | 0.078 | 0.028 | layered face-sharing octahedra | PbTe_24_1279
160 | 2 | 0.084 | 0.007 | layered trigonal prisms and octahedra | PbTe_24_1527
12 | 1 | 0.108 |  | disordered octahedra and square pyramids | PbTe_24_1532
12 | 1 | 0.117 |  | disordered octahedra and square pyramids | PbTe_24_1716
38 | 1 | 0.131 |  | RS with site swap | PbTe_24_417
38 | 1 | 0.174 |  | RS with site swap | PbTe_24_1226
Table 4:  Random sampling structures identified for PbTe space group | frequency | $\Delta E_{f}$ | NEB barrier | description | name
---|---|---|---|---|---
number |  | (eV/atom) | (eV/atom) |  |
216 | 36 | 0 | 0 | zincblende (ZB) | SiC_12_1523
160 | 4 | 0 | 0.237 | tetrahedral rhombohedral WZ polytype | SiC_12_723
186 | 25 | 0.002 | 0.345 | wurtzite (WZ) | SiC_12_1769
36 | 2 | 0.065 |  | disordered tetrahedra | SiC_12_949
36 | 1 | 0.131 |  | disordered tetrahedra | SiC_12_743
44 | 1 | 0.190 |  | tetrahedra mixed with C-C and Si-Si bonds | SiC_12_1296
15 | 3 | 0.324 |  | tetrahedra mixed with trigonal planar SiC3 | SiC_12_1149
12 | 1 | 0.364 |  | disordered tetrahedra, C-C bonds | SiC_12_1882
152 | 1 | 0.366 |  | disordered tetrahedra | SiC_12_690
20 | 1 | 0.403 |  | disordered tetrahedra, C-C and Si-Si bonds | SiC_12_1245
12 | 1 | 0.481 |  | disordered tetrahedra, C-C and Si-Si bonds | SiC_12_885
38 | 1 | 0.497 |  | edge-sharing tetrahedra | SiC_12_128
25 | 1 | 0.524 |  | disordered tetrahedra, C-C and Si-Si bonds | SiC_12_61
44 | 1 | 0.566 |  | disordered tetrahedra, trigonal planar SiC3, C-C and Si-Si bonds | SiC_12_1859
43 | 1 | 0.566 |  | disordered, distorted tetrahedra | SiC_12_364
107 | 1 | 0.709 |  | 5-fold, intermediate between ZB an RS | SiC_12_1932
225 | 2 | 0.740 | 0.001 | rocskalt (RS) | SiC_12_6
Table 5:  Random sampling structures identified for SiC Figure 6:  The frequency of occurence vs energy for each of MgO, TaC, and PbTe. compound | polymorph space group | structure types | realization condition | citations
---|---|---|---|---
MoN | 191 | layered material | space group misassigned | [123]
MoN | 74 | 1-D chains | space group misassigned | [124, 125]
NbN | 194 | Face sharing octahedra | degenerate energy | [126]
SnSe | 129 | tetragonal layers | misfit layered compound | [127]
SnSe | 63 | 1-D chains | space group misassigned | [41]
SnSe | 59 | 2-D ribbons | space group misassigned | [41]
SnS | 216 | zincblende | Evaporation (citation only) | [128]
SnS | 39 | layered material | misfit layered compound | [129]
SnS | 39 | layered material | misfit layered compound | [130]
SnS | 2 | layered material | misfit layered compound | [131]
CdTe | 63 | distorted RS | high pressure | [132]
CdTe | 63 | distorted RS | high pressure | [132]
GeTe | 62 | Pnma layered | citation only | [133]
GeTe | 60 | buckled square ribbons | high pressure | [134]
LiI | 194 | face sharing octahedra | degenerate | [135]
TaN | 191 | close packed planes with trigonal prisms between | space group misassigned | [136]
TaN | 191 | close packed planes with octahedra between | space group misassigned | [137]
AgI | 11 | distorted rocksalt | high pressure | [31]
AgI | 221 | CsCl-type | high pressure | [138]
HgTe | 63 | distorted rocksalt | high pressure | [139]
BiSe | 12 |  | misfit layered compound | [140]
BiSe | 12 |  | misfit layered compound | [140]
BiSe | 14 |  | nonstoichiometric | [141]
MnTe | 62 | distorted NiAs | high pressure | [142]
InSb | 59 | distorted Rocksalt | high pressure | [143]
InSb | 196 | high pressure cadmium telluride | high pressure | [143]
InSb | 221 | CsCl-type | high pressure | [144]
Table 6:  All compounds in the Materials Project Database with rocksalt reported as a polymorph. Inspection of references shows that there are no metastable polymorphs with energy above that of rocksalt. Figure 7:  A polymorph network representing the polymorphism of SiC. Zincblende is taken to be the origin. All other structures are labeled according to their space group number. The size of each node represents the frequency of occurence of a structure in the random sample. Note that a fixed size is used to represent the width of the zincblende local minimum, and other points are scaled relative to this size. Structures are arranged so that the energy increases with the polar angle and the distance of each node from the zincblende center is proportional to the distance travelled by the atoms during phase transformation of that structure to zincblende. Dashed lines represent slow transformations from wurtzite and rhombohedral to zincblende. Solid lines represent fast transformations between rocksalt and the tetrahedral polymorphs. Figure 8: Sankey diagrams showing the kinetic flow of 457 prototype structures found in the ICSD spanning the A1B1 chemical space for SiC. The bars on the left represent the distribution of the A1B1 prototypes across 7 crystal systems. Bars in the center display the distribution of space groups that these prototypes relax into. Structures which link to the s.g. #216 bar at the right end have kinetic barriers for transformation to zincblende less than 150 meV/atom. For this diagram, we assume that all strucutres with energy greater than the energy of rocksalt will transform quickly to rocksalt, and will therefore transform quickly to the zincblende ground state. Even under this condition, many fewer structures transform to the zincblende ground state in SiC than those that transform quickly to rocksalt in each of the figures in main body.  compound | polymorph space groups | structure types | realization condition | citations
---|---|---|---|---
Mg2Sn | 227,194 | Laves, cotunnite, hexagonal | high pressure, high temperature | [145, 146]
K2S | 194 |  | mislabeled (refers to K2SO4 compound) | [147]
Rb2Te | 227, 194 | cotunnite, Ni2In | high pressure, high temperature | [148]
Mg2Pb | 62 |  | alloying | [149]
Mg2Si | 227,194 | cotunnite, hexagonal | high pressure, high temperature | [146]
Cs2Se | 62, 43 | cotunnite, distorted fluorite | cotunnite degenerate, high pressure | [150]
CdF2 | 227 | cotunnite | high pressure (reversible) | [151]
Rh2As | 62 | cotunnite | high temperature | [152]
Mg2Ge | 227 | Cubic Laves phase | high pressure | [153]
Li2S | 62 | anticotunnite | high pressure (reversible) | [154]
ThO2 | 62 | cotunnite | high pressure (reversible, sluggish) | [155]
CaF2 | 62 | cotunnite | high pressure (partially reverts) | [156, 157]
SrF2 | 62,194 | cotunnite, Ni2In | high pressure (partially reverts) | [158]
Rb2S | 62,194 | anticotunnite, Ni2In | high pressure | [159]
Na2S | 62, 194 | anticotunnite, Ni2In | high pressure (reversible) | [160]
Ba2F | 62, 194 | cotunnite, Ni2In | high pressure (partially reverts) | [161]
Ba2Cl | 62 | cotunnite, hydrated structure | high temperature (reversible), hydration | [162, 163]
PbF2 | 62 | cotunnite | approximately degenerate | [164]
Table 7:  All compounds in the Materials Project Database with a fluorite ground state and an experimentally reported polymorph. Inspection of references shows that while cotunnite is reported to be metastable for some chemistries, in those cases transformation back to fluorite is at least partially observed at room temperature. ICSD label | polymorph space groups | name | energy (eV/atom)
---|---|---|---
091736 | 14 | Cristobalite | -7.908
647410 | 154 | Quartz low | -7.887
190370 | 136 | Stishovite | -7.730
075486 | 92 | Cristobalite low - alpha | -7.904
067121 | 152 | Quartz low | -7.897
077459 | 227 | Cristobalite beta high | -7.905
040900 | 20 | Tridymite | -7.909
089277 | 154 | Quartz low | -7.899
089289 | 180 | Quartz high | -7.899
056473 | 4 | Tridymite | -7.909
162660 | 122 | Cristobalite beta | -7.909
172295 | 15 | Coesite | -7.869
094091 | 19 | Tridymite | -7.908
030795 | 40 | Tridymite O | -7.910
162616 | 198 | Cristobalite beta | -7.905
200478 | 194 | Tridymite 2H high (Gibbs Model) | -7.906
162618 | 122 | Cristobalite beta | -7.910
040137 | 15 |  | -7.902
029343 | 182 | Tridymite 2H low (subcell) | -7.907
192626 | 58 | Ferrierite, siliceous | -7.900
056320 | 227 | Dodecasil 3C | -7.902
067669 | 15 | Moganite | -7.899
064980 |  | Quartz high | -6.860
085586 | 166 | Chabazite | -7.897
048153 | 191 | Dodecasil 1H | -7.905
048154 | 227 | Dodecasil 3C | -7.902
183701 | 229 | Sodalite | -7.904
065497 | 71 |  | -7.900
093975 | 181 | Quartz high | -7.899
| 225 | Fluorite | -6.661
|  | Glass ensemble | -7.529
Table 8:  Energies of named polymorphs of SiO2. All polymorphs have energy
below that of SiO2 in the fluorite structure. Figure 9:  The thermodynamic
density of states for a hypothetical compound MgZnCuO3. Despite the fact that
both Zn and Cu prefer 4-fold coordination in oxides, rocksalt is by far the
most abundant structure in random sampling of this compound. This displays the
statistical relevance of the rocksalt structure for high-entropy systems.
Figure 10:  Convergence of the high symmetry structures with respect to total
number of structures for random sampling on each of MgO, TaC, and PbTe.

## References

  * Swanson and Davey [1955] H. Swanson and W. Davey, National Bureau of Standards (U.S.), Circular 539, 31 (1955).
  * Kirkpatrick and Dickinson [1927] L. Kirkpatrick and R. Dickinson, Journal of the American Chemical Society 49, 2327 (1927).
  * Samsonov  _et al._ [1974] G. Samsonov, M. Abdusalyamova, and K. Shokirov, Inorganic Materials 10, 790 (1974).
  * Hayashi  _et al._ [2000] J. Hayashi, I. Shirotani, Y. Tanaka, T. Adachi, O. Shimomura, and T. Kikegawa, Solid State Communications 114, 561 (2000).
  * Holbourn and Woodhams [1981] P. Holbourn and F. Woodhams, Journal of Solid State Chemistry 39, 186 (1981).
  * Leger  _et al._ [1984] J. Leger, D. Ravot, and J. Rossat Mignod, Journal of Physics C 17, 4935 (1984).
  * Dabos-Seignon  _et al._ [1990] S. Dabos-Seignon, U. Benedict, S. Heathman, J. Spirlet, and M. Pages, Journal of the Less-Common Metals 160, 35 (1990).
  * Makode  _et al._ [2013] C. Makode, J. Pataiya, M. Aynyas, and S. P. Sanyal, AIP Conference Proceedings 1536, 925 (2013).
  * Posnjak and Wyckoff [1922] E. Posnjak and R. Wyckoff, Journal of the Washington Academy of Sciences 12, 248 (1922).
  * Weir and Piermarini [1964] C. Weir and G. Piermarini, Journal of Research of the National Bureau of Standards, Section A. Physics and Chemistry 68, 105 (1964).
  * Ugur  _et al._ [2008] G. Ugur, S. Ugur, A. Erkisi, and F. Soyalp, Int. J. Mod. Phys. 22, 5027 (2008).
  * Shirotani  _et al._ [2003] I. Shirotani, K. Yamanashi, J. Hayashi, N. Ishimatsu, O. Shimomura, and T. Kikegawa, Solid State Communications 127, 573 (2003).
  * Kabalkina  _et al._ [1968] S. Kabalkina, N. Serebryanaya, and L. Vereshchagin, Soviet Physics - Solid State (New York) 10, 574 (1968).
  * Li  _et al._ [2013] Y. Li, C. Lin, H. Li, X. Li, and J. Liu, High pressure research 33, 713 (2013).
  * Korkosz  _et al._ [2014] R. J. Korkosz, T. C. Chasapis, S.-h. Lo, J. W. Doak, Y. J. Kim, C.-I. Wu, E. Hatzikraniotis, T. P. Hogan, D. N. Seidman, C. Wolverton, V. P. Dravid, and M. G. Kanatzidis, Journal of the American Chemical Society 136, 3225 (2014).
  * Fujii  _et al._ [1984] Y. Fujii, K. Kitamura, A. Onodera, and Y. Yamada, Solid State Communications 49, 135 (1984).
  * Walker  _et al._ [2004] D. Walker, P. Verma, L. Cranswick, R. Jones, S. Clark, and S. Buhre, American Mineralogist 89, 204 (2004).
  * Shamovskii and Shushkanov [1968] L. Shamovskii and A. Shushkanov, Doklady Akademii Nauk SSSR 181, 862 (1968).
  * Blackman and Khan [1961] M. Blackman and I. Khan, Proceedings of the Physical Society, London 77, 471 (1961).
  * Becker and Beck [2004] D. Becker and H. Beck, Zeitschrift fuer Kristallographie (1979-2010) 219, 348 (2004).
  * Penneman and Staritzky [1958] R. Penneman and E. Staritzky, Journal of Inorganic and Nuclear Chemistry 6, 112 (1958).
  * Strada [1934] M. Strada, Gazzetta Chimica Italiana 64, 400 (1934).
  * Wilhelm and Chiotti [1950] H. Wilhelm and P. Chiotti, Transactions of the American Society for Metals 42, 1295 (1950).
  * Johnsen and Norby [2009] R. Johnsen and P. Norby, Journal of Physical Chemistry C 113, 19061 (2009).
  * Tsirel’son  _et al._ [1998] V. Tsirel’son, A. Avilov, Y. Abramov, E. Belokoneva, R. Kitaneh, and D. Feil, Acta Crystallographica, Section B: Structural Science 54, 8 (1998).
  * Zwijnenburg and Bromley [2011] M. Zwijnenburg and S. Bromley, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 83, 024104 (2011).
  * Zimmer  _et al._ [1985] H. Zimmer, H. Winzen, and K. Syassen, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 32, 4066 (1985).
  * Sifi  _et al._ [2012] C. Sifi, M. Slimani, and H. Meradji, Materials Science and Engineering, IOP conference series 28, 012031 (2012).
  * Ungelenk [1962] J. Ungelenk, Naturwissenschaften 49, 252 (1962).
  * Roberts  _et al._ [2006] A. Roberts, K. Venance, T. Seward, J. Grice, and W. Paar, Mineralogical Record 37, 165 (2006).
  * Hull and Keen [1999] S. Hull and D. Keen, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 59, 750 (1999).
  * Snavely and Vaughan [1949] C. Snavely and D. Vaughan, Journal of the American Chemical Society 71, 313 (1949).
  * Corliss  _et al._ [1956] L. Corliss, N. Elliott, and J. Hastings, Physical Review (1,1893-132,1963/141,1966-188,1969) 104, 924 (1956).
  * Wiedemeier and Gary Sigai [1969] H. Wiedemeier and A. Gary Sigai, High Temperature Science 1, 18 (1969).
  * Mehmed and Haraldsen [1938] F. Mehmed and H. Haraldsen, Zeitschrift fuer Anorganische und Allgemeine Chemie (1950) (DE) 235, 193 (1938).
  * Chatterjee  _et al._ [1972] A. Chatterjee, A. Singh, and A. Jayaraman, Physical Review, Serie 3. B - Solid State (1,1970-17,1977) 6, 2285 (1972).
  * Yarembash  _et al._ [1971] E. Yarembash, E. Tyurin, A. Reshchikova, A. Karabekov, and N. Klinaeva, Inorganic Materials 7, 661 (1971).
  * Langridge  _et al._ [1994] S. Langridge, W. Stirling, G. Lander, J. Rebizant, J. Spirlet, D. Gibbs, and O. Vogt, Europhysics Letters 25, 137 (1994).
  * Dabos  _et al._ [1986] S. Dabos, C. Dufour, U. Benedict, J. Spirlet, and M. Pages, Physica B and C (Netherland) (79,1975-) 144, 79 (1986).
  * Wang  _et al._ [2015] S. Wang, C. Zang, Y. Wang, L. Wang, J. Zhang, C. Childs, H. Ge, H. Xu, H. Chen, D. He, and Y. Zhao, Inorganic Chemistry 54, 4981 (2015).
  * Chattopadhyay  _et al._ [1986] T. Chattopadhyay, H. von Schnering, W. Grosshans, and W. Holzapfel, Physica B and C (Netherland) (79,1975-) 139, 356 (1986).
  * Mariano and Chopra [1967] A. Mariano and K. Chopra, Applied Physics Letters 10, 282 (1967).
  * Auriel  _et al._ [1993] C. Auriel, R. Roesky, A. Meerschaut, and J. Rouxel, Materials Research Bulletin 28, 247 (1993).
  * Goldschmidt [1927] V. Goldschmidt, Skrifter utgitt av det Norske Videnskaps-Akademi i Oslo 1: Matematisk-Naturvidenskapelig Klasse 1927, 1 (1927).
  * Onodera  _et al._ [1986] A. Onodera, Y. Fujii, and S. Sugai, Physica B and C (Netherland) (79,1975-) 139, 240 (1986).
  * Rogacheva  _et al._ [1986] E. Rogacheva, G. Gorne, S. Laptev, A. Arinkin, and T. Vesiene, Izvestiya Akademii Nauk SSSR, Neorganicheskie Materialy 22, 41 (1986).
  * Grzybowski and Ruoff [1984] T. Grzybowski and A. Ruoff, Materials Research Society Symposia Proceedings 22, 43 (1984).
  * Grzybowski and Ruoff [1971] T. Grzybowski and A. Ruoff, Physical Review, Serie 3. B - Solid State (1,1970-17,1977) 3, 755 (1971).
  * Lyskova and Vakhobov [1975] Y. Lyskova and A. Vakhobov, Inorganic Materials 11, 1784 (1975).
  * Grzybowski and Ruoff [1983] T. Grzybowski and A. Ruoff, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 27, 6502 (1983).
  * Juza and Liedtke [1957] R. Juza and H. Liedtke, Zeitschrift fuer Anorganische und Allgemeine Chemie (1950) (DE) 290, 204 (1957).
  * Nagib  _et al._ [1977] M. Nagib, E. von Osten, and H. Jacobs, Atomkernenergie 29, 41 (1977).
  * Lingun [1971] L. Lingun, Journal of Applied Physics 42, 3702 (1971).
  * Taylor [1984] D. Taylor, Transactions and Journal of the British Ceramic Society 83, 5 (1984).
  * Weir  _et al._ [1986] S. Weir, Y. Vohra, and A. Ruoff, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 33, 4221 (1986).
  * Gerward  _et al._ [1988] L. Gerward, J. Olsen, U. Benedict, S. Dabos, H. Luo, J. Itie, and O. Vogt, High Temperatures-High Pressures 20, 545 (1988).
  * Manfrinetti  _et al._ [2009] P. Manfrinetti, A. Provino, A. Morozkin, and O. Isnard, Journal of Alloys and Compounds 487, L28 (2009).
  * Eliseev  _et al._ [1987] A. Eliseev, I. Orlova, L. Martynova, A. Pechennikov, and V. Chechernikov, Inorganic Materials 23, 1833 (1987).
  * Singh and Srivastava [1977] A. Singh and O. Srivastava, Zeitschrift fuer Metallkunde 68, 768 (1977).
  * Miller  _et al._ [1962] J. Miller, L. Matson, and R. Himes, Proceedings of the Conference of Rare Earth Research, 2nd, Colorado 1962, 233 (1962).
  * Shirotani  _et al._ [2001] I. Shirotani, K. Yamanashi, J. Hayashi, Y. Tanaka, N. Ishimatsu, O. Shimomura, and T. Kikegawa, Journal of Physics: Condensed Matter 13, 1939 (2001).
  * Iandelli and Botti [1937] A. Iandelli and E. Botti, Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti, Serie 7 (1, 1925-1945) 25, 498 (1937).
  * Flahaut  _et al._ [1962] J. Flahaut, L. Domange, and M. Patrie, Bulletin de la Societe Chimique de France (Vol=Year) 1962, 2048 (1962).
  * Ugur [2009] S. Ugur, Materials Science and Engineering B: Solid-state Materials for Advanced Technology 162, 116 (2009).
  * Cortona [1992] P. Cortona, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 46, 2008 (1992).
  * Chen and Jiang [2010] W. Chen and J.-Z. Jiang, Journal of Alloys and Compounds 499, 243 (2010).
  * Niggli [1959] A. Niggli, Zeitschrift fuer Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie (-144,1977) 111, 269 (1959).
  * Dabos-Seignon  _et al._ [1989] S. Dabos-Seignon, U. Benedict, J. Spirlet, and M. Pages, Journal of the Less-Common Metals 153, 133 (1989).
  * Charvillat and Damien [1973] J. Charvillat and D. Damien, Inorganic and Nuclear Chemistry Letters 9, 559 (1973).
  * Vedel  _et al._ [1987] I. Vedel, A. Redon, J. Rossat Mignod, O. Vogt, and J. Leger, Journal of Physics C 20, 3439 (1987).
  * Ono  _et al._ [1974] S. Ono, K. Nomura, and H. Hayakawa, Journal of the Less-Common Metals 38, 119 (1974).
  * Schneider  _et al._ [2012] S. B. Schneider, D. Baumann, A. Salamat, and W. Schnick, Journal of Applied Physics 111, 093503 (2012).
  * Olcese [1979] G. L. Olcese, Journal of Physics F 9, 569 (1979).
  * Ponyatovskii and Bashkin [1985] E. Ponyatovskii and I. Bashkin, Zeitschrift fuer Physikalische Chemie (Frankfurt Am Main) 146, 137 (1985).
  * Zintl and Harder [1931] E. Zintl and A. Harder, Zeitschrift fuer Physikalische Chemie, Abteilung B: Chemie der Elementarprozesse, Aufbau der Materie 14, 265 (1931).
  * Faita  _et al._ [2011] F. Faita, C. Campos, K. Ersching, and P. Pizani, Materials Chemistry and Physics 125, 257 (2011).
  * Chattopadhyay  _et al._ [1985] T. Chattopadhyay, R. Santandrea, and H. von Schnering, Journal of Physics and Chemistry of Solids 46, 351 (1985).
  * Olcese [1961] G. Olcese, Atti della Accademia Nazionale dei Lincei, Classe di Fisiche, Matematiche e Naturali, Rendiconti 31, 256 (1961).
  * Friese  _et al._ [1997] K. Friese, O. Jarchow, and K. Kato, Zeitschrift fuer Kristallographie (1979-2010) 212, 648 (1997).
  * de Boer  _et al._ [1991] J. de Boer, A. Meetsma, T. Zeinstra, R. Haange, and G. Wiegers, Acta Crystallographica, Section C: Crystal Structure Communications 47, 924 (1991).
  * Charifi  _et al._ [2012] Z. Charifi, H. Baaziz, Y. Saeed, A. Hussain Reshak, and F. Soltani, Physica Status Solidi B - Basic Solid State Physics 249, 18 (2012).
  * Kolobyanina  _et al._ [1972] T. Kolobyanina, S. Kabalkina, L. Vereshchagin, M. Kachan, and V. Losev, High Temperatures-High Pressures 4, 207 (1972).
  * Losev  _et al._ [1970] V. Losev, S. Kabalkina, and L. Vereshchagin, Soviet Physics - Solid State (New York) 12, 2374 (1970).
  * Abrahams and Bernstein [1965] S. Abrahams and J. Bernstein, Acta Crystallographica (1,1948-23,1967) 18, 926 (1965).
  * Zhang  _et al._ [2013] W. Zhang, A. R. Oganov, A. F. Goncharov, Q. Zhu, S. E. Boulfelfel, A. O. Lyakhov, E. Stavrou, M. Somayazulu, V. B. Prakapenka, and Z. Konopkova, Science 342, 1502 (2013).
  * Onoda  _et al._ [1990] M. Onoda, K. Kato, Y. Gotoh, and Y. Oosawa, Acta Crystallographica, Section B: Structural Science 46, 487 (1990).
  * Wiegers  _et al._ [1990] G. Wiegers, A. Meetsma, R. Haange, S. van Smaalen, J. de Boer, A. Meerschaut, P. Rabu, and J. Rouxel, Acta Crystallographica, Section B: Structural Science 46, 324 (1990).
  * Meerschaut  _et al._ [1992] I. Meerschaut, C. Auriel, and J. Rouxel, Journal of Alloys and Compounds 183, 129 (1992).
  * Sakthi Sudar Saravanan  _et al._ [2015] R. Sakthi Sudar Saravanan, M. Meena, P. D., and C. Mahadevan, Journal of Alloys and Compounds 627, 69 (2015).
  * Zagorac  _et al._ [2011] D. Zagorac, K. Doll, J. Schoen, and M. Jansen, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 84, 045206 (2011).
  * van Smaalen  _et al._ [1991] S. van Smaalen, A. Meetsma, G. Wiegers, and J. de Boer, Acta Crystallographica, Section B: Structural Science 47, 314 (1991).
  * Ren  _et al._ [1997] Y. Ren, A. Meetsma, A. Spijkerman, and G. Wiegers, Zeitschrift fuer Kristallographie (1979-2010) 212, 586 (1997).
  * Meerschaut  _et al._ [1991] A. Meerschaut, C. Auriel, A. Lafond, C. Deudon, P. Gressier, and J. Rouxel, European Journal of Solid State Inorganic Chemistry 28, 581 (1991).
  * Wastin  _et al._ [1995] F. Wastin, J. Spirlet, and J. Rebizant, Journal of Alloys and Compounds 219, 232 (1995).
  * Min  _et al._ [2012] N. K. Min, K. Yong-Il, J. Younghun, L. S. Mi, K. B. G., C. Ran, C. Sang-Il, S. Hyunjoon, and P. J. T., Journal of the American Chemical Society 134, 8392 (2012).
  * Trzebiatowski  _et al._ [1964] W. Trzebiatowski, A. Sepichowska, and A. Zygmunt, Bulletin de l’Academie Polonaise des Sciences, Serie des Sciences Chimiques 12, 687 (1964).
  * Ettmayer  _et al._ [1979] P. Ettmayer, J. Waldhart, and A. Vendl, Monatshefte fuer Chemie 110, 1109 (1979).
  * Cynn  _et al._ [2010] H. Cynn, M. Lipp, W. Evans, and Y. Ohishi, Journal of Physics: Conference Series 215, 012010 (2010).
  * Isobe  _et al._ [2007] M. Isobe, M. Onoda, M. Shizuya, M. Tanaka, and E. Takayama Muromachi, Journal of the Physical Society of Japan 76, 014602 (2007).
  * Oftedal [1927] I. Oftedal, Zeitschrift fuer Physikalische Chemie (Leipzig) 128, 154 (1927).
  * Vegard [1921] L. Vegard, Zeitschrift fuer Physik 5, 17 (1921).
  * Piermarini and Weir [1962] G. Piermarini and C. Weir, Journal of Research of the National Bureau of Standards, Section A. Physics and Chemistry 66, 325 (1962).
  * Singh  _et al._ [2011] R. Singh, R. Singh, and M. Rajagopalan, Physica B, Condensed Matter 406, 1717 (2011).
  * Ungelenk [2004] J. Ungelenk, Zeitschrift fuer Kristallographie (1979-2010) 219, 348 (2004).
  * Taylor  _et al._ [1974] J. Taylor, L. Calvert, J. Despault, E. Gabe, and J. Murray, Journal of the Less-Common Metals 37, 217 (1974).
  * West [1934] C. West, Zeitschrift fuer Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie (-144,1977) 88, 94 (1934).
  * Swanson and Fuyat [1953] H. Swanson and R. Fuyat, National Bureau of Standards (U.S.), Circular 539, 44 (1953).
  * Goldschmidt [1926] V. Goldschmidt, Skrifter utgitt av det Norske Videnskaps-Akademi i Oslo 1: Matematisk-Naturvidenskapelig Klasse 1926, 1 (1926).
  * Kruger and Moser [1967] O. Kruger and J. Moser, Journal of Physics and Chemistry of Solids 28, 2321 (1967).
  * Beeken and Schweitzer [1981] R. Beeken and J. Schweitzer, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 23, 3620 (1981).
  * Mitchell and Lam [1971] A. Mitchell and D. Lam, Journal of Nuclear Materials 39, 219 (1971).
  * Dewaele  _et al._ [2012] A. Dewaele, A. Belonoshko, G. Garbarino, F. Occelli, P. Bouvier, M. Hanfland, and M. Mezouar, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 85, 214105 (2012).
  * Finch and Fordham [1936] G. Finch and S. Fordham, Proceedings of the Physical Society, London 48, 85 (1936).
  * Ott [1926] H. Ott, Zeitschrift fuer Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie (-144,1977) 63, 222 (1926).
  * Olsen  _et al._ [1988] J. Olsen, L. Gerward, U. Benedict, H. Luo, and O. Vogt, High Temperatures-High Pressures 20, 553 (1988).
  * d’Eye  _et al._ [1952] R. d’Eye, P. Sellman, and J. Murray, Journal of the Chemical Society 1952, 2555 (1952).
  * Beintema [1937] J. Beintema, Zeitschrift fuer Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie (-144,1977) 97, 300 (1937).
  * Feng  _et al._ [2012] W. Feng, H. Hu, X. Xiao, S. Cui, Z. Lv, G. Zhang, and C. Wu, Computational Materials Science, Elsevier 53, 75 (2012).
  * Abdusalyamova  _et al._ [1988] M. Abdusalyamova, O. Rakhmatov, and K. Shokirov, Russian Metallurgy 1988, 183 (1988).
  * Hull [1919] A. Hull, Transactions of the American Institute for Electrical Engineers 38, 1445 (1919).
  * Cario  _et al._ [1998] L. Cario, A. Meerschaut, C. Deudon, and J. Rouxel, C.R. Acad. Sci. Paris, T. 1, Serie II 1, 269 (1998).
  * Eliseev  _et al._ [1974] A. Eliseev, O. Sadovskaya, and N. V. Tam, Izvestiya Akademii Nauk SSSR, Neorganicheskie Materialy 10, 2134 (1974).
  * Troitskaya and Pinsker [1961] N. Troitskaya and Z. Pinsker, Kristallografiya 6, 43–48 (1961).
  * Dong and Range [2000] Z. X. Dong and K. Range, Journal of Alloys and Compounds 296, 72 (2000).
  * Yang  _et al._ [2014] Z.-X. Yang, X.-Y. Kuang, M.-M. Zhong, and X.-F. Huang, Solid State Sciences 28, 20 (2014).
  * Schoenberg [1954] N. Schoenberg, Acta Chemica Scandinavica (1-27,1973-42,1988) 8, 208 (1954).
  * Wiegers and Zhou [1991] G. Wiegers and W. Zhou, Materials Research Bulletin 26, 879–885 (1991).
  * Badachhape and Goswami [1962] S. Badachhape and A. Goswami, Journal of the Physical Society of Japan Supplement B2 17, 251–253 (1962).
  * Hoistad  _et al._ [1995] L. Hoistad, A. Meerschaut, P. Bonneau, and J. Rouxel, Journal of Solid State Chemistry 114, 435 (1995).
  * Meetsma  _et al._ [1989] A. Meetsma, G. Wiegers, R. Haange, and J. de Boer, Acta Crystallographica. Section A: Foundations of Crystallography 45, 285 (1989).
  * Wiegers and Meerschaut [1992] G. Wiegers and I. Meerschaut, Journal of Alloys and Compounds 178, 351 (1992).
  * Nelmes  _et al._ [1995] R. Nelmes, M. McMahon, N. Wright, and D. Allan, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 51, 15724 (1995).
  * Karbanov  _et al._ [1968] S. Karbanov, V. Zlomanov, and A. Novoselova, Doklady Chemistry 182, 862 (1968).
  * Onodera  _et al._ [1997] A. Onodera, I. Sakamoto, Y. Fujii, N. Mori, and S. Sugai, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 56, 7935 (1997).
  * Ott [1923] H. Ott, Physikalische Zeitschrift 24, 209 (1923).
  * Chihi  _et al._ [2011] T. Chihi, M. Fatmi, and B. Ghebouli, Solid State Communications 151, 1672 (2011).
  * Brauer  _et al._ [1972] G. Brauer, E. Mohr, A. Neuhaus, and A. Skogan, Monatshefte fuer Chemie 103, 794 (1972).
  * Adams and Davis [1962] L. Adams and B. Davis, Proceedings of the National Academy of Sciences, U.S.A. 48, 982 (1962).
  * Aimin  _et al._ [2009] H. Aimin, Y. X. Cui, Y. Ruomeng, G. C. Xiao, H. Yonghao, L. R. Ping, and T. Yongjun, Journal of Physics and Chemistry of Solids 70, 433 (2009).
  * Clarke and Freedman [2015] S. M. Clarke and D. E. Freedman, Inorganic Chemistry 54, 2765 (2015).
  * Aurivillius  _et al._ [1960] B. Aurivillius, O. von Heidenstamm, and I. Jonsson, Acta Chemica Scandinavica (1-27,1973-42,1988) 14, 944 (1960).
  * Mimasaka  _et al._ [1987] M. Mimasaka, I. Sakamoto, K. Murata, J. Fujii, and A. Onodera, Journal of Physics C 20, 4689 (1987).
  * Yu  _et al._ [1978] S.-C. Yu, I. Spain, and E. Skelton, Journal of Applied Physics 49, 4741 (1978).
  * Vanderborgh  _et al._ [1989] C. Vanderborgh, Y. Vohra, and A. Ruoff, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 40, 12450 (1989).
  * Fei  _et al._ [2011] Y. Fei, S. J. Xun, and C. T. Hong, Physica B, Condensed Matter 406, 1789 (2011).
  * Boudemagh  _et al._ [2011] D. Boudemagh, D. Fruchart, R. Haettel, E. Hill, A. Lacoste, L. Ortega, N. Skryabina, J. Tobola, and P. Wolfers, Diffusion and Defect Data - Solid State Data B: Solid State Phenomena 170, 253 (2011).
  * Fischmeister [1962] H. Fischmeister, Monatshefte fuer Chemie 93, 420 (1962).
  * Stoewe [2004] K. Stoewe, Zeitschrift fuer Kristallographie (1979-2010) 219, 359 (2004).
  * Eldridge  _et al._ [1965] J. M. Eldridge, E. Miller, and K. Komarek, Transactions of the Metallurgical Society of Aime 233, 1303 (1965).
  * Sommer and Hoppe [1977] H. Sommer and R. Hoppe, Zeitschrift fuer Anorganische und Allgemeine Chemie (1950) (DE) 429, 118 (1977).
  * Liu  _et al._ [2011] G. T. Liu, H. Wang, Y. M. Ma, and Y. M. Ma, Solid State Communications 151, 1899 (2011).
  * Kjekshus and Skaug [1972] A. Kjekshus and K. Skaug, Acta Chemica Scandinavica (1-27,1973-42,1988) 26, 2554 (1972).
  * LaBotz  _et al._ [1963] R. LaBotz, D. Mason, and D. OḰane, Journal of the Electrochemical Society 110, 127 (1963).
  * Grzechnik  _et al._ [2000] A. Grzechnik, A. Vegas, K. Syassen, I. Loa, M. Hanfland, and M. Jansen, Journal of Solid State Chemistry 154, 603 (2000).
  * Idiri  _et al._ [2004] M. Idiri, T. Le Bihan, S. Heathman, and J. Rebizant, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 70, 014113 (2004).
  * Gerward  _et al._ [1992] L. Gerward, J. Olsen, S. Streenstrup, M. Malinovskii, S. Asbrink, and A. Waskowska, Journal of Applied Crystallography 25, 578 (1992).
  * Morris  _et al._ [2001] E. Morris, T. Groy, and K. Leinenweber, Journal of Physics and Chemistry of Solids 62, 1117 (2001).
  * Wang  _et al._ [2012] J. Wang, C. Ma, D. Zhou, Y. Xu, M. Zhang, W. Gao, H. Zhu, and Q. Cui, Journal of Solid State Chemistry 186, 231 (2012).
  * Santamaria-Perez  _et al._ [2011] D. Santamaria-Perez, A. Vegas, C. Muehle, and M. Jansen, Acta Crystallographica, Section B: Structural Science 67, 109 (2011).
  * Vegas  _et al._ [2001] A. Vegas, A. Grzechnik, K. Syassen, I. Loa, M. Hanfland, and M. Jansen, Acta Crystallographica, Section B: Structural Science 57, 151 (2001).
  * Leger  _et al._ [1995] J. Leger, J. Haines, A. Atouf, O. Schulto, and S. Hull, Physical Review, Serie 3. B - Condensed Matter (18,1978-) 52, 13247 (1995).
  * Hull  _et al._ [2011] S. Hull, S. Norberg, I. Ahmed, S. Eriksson, and C. Mohn, Journal of Solid State Chemistry 184, 2925 (2011).
  * Haase and Brauer [1978] A. Haase and G. Brauer, Zeitschrift fuer Anorganische und Allgemeine Chemie (1950) (DE) 441, 181 (1978).
  * Ehm  _et al._ [2003] L. Ehm, K. Knorr, F. Maedler, H. Voigtlaender, E. Busetto, A. Cassetta, A. Lausi, and B. Winkler, Journal of Physics and Chemistry of Solids 64, 919 (2003).