# Stability of B2 compounds: Role of the $M$ point phonons

Shota Ono<EMAIL_ADDRESS>Daigo Kobayashi  Department of Electrical,
Electronic and Computer Engineering, Gifu University, Gifu 501-1193, Japan

###### Abstract

Although many binary compounds have the B2 (CsCl-type) structure in the
thermodynamic phase diagram, an origin of the structural stability is not
understood well. Here, we focus on 416 compounds in the B2 structure extracted
from the Materials Project, and study the dynamical stability of those
compounds from first principles. We demonstrate that the B2 phase stability
lies in whether the lowest frequency phonon at the $M$ point in the Brillouin
zone is endowed with a positive frequency. We show that the interatomic
interactions up to the fourth nearest neighbor atoms are necessary for
stabilizing such phonon modes, which should determine the minimum cutoff
radius for constructing the interatomic potentials of binary compounds with
guaranteed accuracy.

Introduction. Since the characterization of the crystal structure of CsCl in
1921 davey , many of the B2 (called CsCl-type) compounds have been synthesized
experimentally, resulting in that the B2 structure is the most common phase in
the thermodynamic phase diagram of binary compounds. Analyses of materials
database have shown how frequent they appear sluiter ; hart2007 ; kolli . More
recently, Kolli et al. identified 267 parent crystal structures that can
generate their derivative ordered phases, and showed that the body-centered
cubic (bcc) structure is the most common parent crystal structure kolli  by
using the Materials Project (MP) database MP . Among them, the B2 structure is
the most common ordering on bcc structure. Apart from the ambient condition,
the B2 structure may also appear: the B2 structure can be transformed from the
B1 (NaCl-type) structure under high pressure (such as alkali halides and
alkali-earth oxides sims ; florez ) and from the L10 (CuAu-type) structure in
warm dense matter regime ono_koba .

Although the elastic stability of the B2 structure has been studied in a wide
variety of compounds khenata ; wangB2 ; alsalmi , such a stability does not
always yield the dynamical stability against the zone boundary phonon
excitations grimvall . For the parent bcc structure, it is well known that the
transverse acoustic phonon at the $N$ point in the Brillouin zone (BZ),
propagating along the diagonal direction of any two axes in the cubic cell,
has relatively low frequencies frank ; persson . It has been shown that such a
phonon is stabilized by long-range interatomic interactions, allowing alkali
metals to form the bcc structure at the ambient condition ono2019 . By
considering the fact that the B2 structure is equivalent to the bcc structure
when two species are assumed to be the same element, we expect that a similar
scenario holds: the lowest frequency phonon at the $M$ point that is
stabilized by the long-range interactions determines the stability of the B2
compounds. We have recently confirmed that the long-range interatomic
interactions up to the fifth nearest neighbor (5NN) atoms are needed to
understand the dynamical stability of the CuAu in the L10 structure ono_koba .

The range of the interatomic interactions is of prime importance in the field
of atomistic modeling of condensed matters. For the bcc elemental metals, the
cutoff radii should be larger than the 3NN or 4NN distances adam ; seko ; zuo
. However, it has not been understood why more than 3NN distances are required
to describe the potential energy surface accurately.

In this paper, we study the dynamical stability of 416 compounds in the B2
structure from first principles, by assuming zero temperature and pressure. We
show that 266 out of 416 compounds are dynamically stable, and demonstrate
that the stability of the B2 compounds is mainly determined by whether the
lowest frequency phonon at the $M$ point in the BZ is endowed with a positive
frequency. In addition, we develop a force constant model taking into account
the interatomic interactions up to the 6NN atoms, and demonstrate that the
interatomic interactions up to the 4NN atoms are enough to stabilize the
lowest energy phonons at the $M$ point. This should determine a minimum cutoff
radius of the interatomic potentials for binary compounds. The present work
unveils the microscopic mechanism of the B2 compounds stability, which
stimulates other studies involving different crystal structures.

Figure 1: The phonon dispersions of the Cu-based B2 compounds. The dispersion
curves are colored red and blue for the dynamically stable and unstable
compounds, respectively.

Methods. By using the MP MP  and the pymatgen pymatgen , we first extracted
the list of the B2 compounds. By setting the space group to $Pm\bar{3}m$ with
the number of atoms in a unit cell being two and excluding the atoms in the
actinide series, 416 B2 compounds having the inorganic crystal structure
database (ICSD) IDs were found.

We next optimized the lattice parameter $a$ for the 416 compounds. All the
density-functional theory (DFT) calculations were performed with the Quantum
ESPRESSO (QE) qe  using the Perdew-Burke-Ernzerhof (PBE) pbe  functionals of
the generalized gradient approximation for the exchange-correlation energy and
using the ultrasoft pseudopotentials provided in the pslibrary.1.0.0 dalcorso
. Spin-polarized approximation was used for all calculations. The cutoff
energies for the wavefunction and the charge density are 60 Ry and 600 Ry,
respectively, and 20$\times$20$\times$20 $k$ grid was used in the self-
consistent field (SCF) calculations MK . The SCF convergence threshold was set
to be $10^{-8}$ Ry and the smearing parameter of $\sigma=0.02$ Ry smearing
was used for all calculations. The total energy and forces were converged
within $10^{-5}$ Ry and $10^{-4}$ a.u., respectively.

The accuracy of the present calculations was checked by comparing the
optimized $a$’s with those in the MP MP . We have confirmed that the optimized
$a$’s agree with the reference values within an error of 1 % except for the 11
Ce-based compounds, ClO, and NCl. The optimized $a$’s in the Ce-based
compounds are larger than the reference values by a few percent. This may be
due to the absence of the $f$-electrons in the present calculations, resulting
in no magnetic moments, whereas the Ce-based compounds show ferromagnetic
phase in the MP MP . The error of the $a$’s for ClO and NCl were 4.6 and 11.1
%, respectively.

Although the formation energies $E_{\rm form}$ of the 416 compounds can be
obtained from the MP MP , the dynamical stability properties are not always
obtained. We thus performed phonon dispersion calculations based on density-
functional perturbation theory (DFPT) dfpt  implemented in QE. The threshold
parameter for the self-consistency (tr2_ph) was set to be $10^{-14}$, and
4$\times$4$\times$4 $q$ grid (10 $q$ points) was used. We calculated the
phonon dispersions along the symmetry lines $\Gamma$-$X$-$M$-$\Gamma$-$R$-$X$.
When the phonon frequency $\omega$ is imaginary, the phonon energy is
represented as a negative value, $-\hbar|\omega|$, with the Planck constant
$\hbar$. In the present work that adapts a finite size $q$ grid, we identify
the B2 compound as dynamically stable if the lowest phonon energy is larger
than $\varepsilon_{\rm min}=-1$ meV.

Figure 2: The minimum phonon energy plotted as the $M$ point phonon energy.
The solid line indicates the relation of $\omega_{\rm M}=\omega_{\rm min}$.
The color of the data points indicates the value of the formation energy
extracted from the MP database MP .

Phonon dispersions. We have found that 266 out of 416 compounds are identified
to be dynamically stable. For example, we show the phonon dispersions for the
15 Cu-based compounds in Fig 1. The 12 compounds (see Fig. 1) out of 15 are
dynamically stable, whereas the others (CuN, ZnCu, and ZrCu) are unstable.
Although the former 12 compounds are stable, the phonon energy of the
transverse acoustic branch at the $M$ point tends to be small, compared to
that at the other symmetry points (except for $\Gamma$) in the BZ. In a
similar manner, the instabilities of the ZnCu and ZrCu are due to the phonon
softening around the $M$ point, leading to negative phonon energies. The CuN
shows strong phonon softening over the entire BZ. This may be ascribed to the
positive formation energy (1.22 eV/atom MP ), implying that the B2 CuN is
decomposed into the Cu crystal in the face-centered cubic structure and the N2
molecules.

For the B2 compounds including the platinum-group metals (Os, Ru, Ir, Rh, Pt,
and Pd), Hart et al. have identified that 16 compounds have the B2 structure
as the ground state by using the DFT calculations within the PBE hart . We
have also identified that the 16 compounds are dynamically stable. In
addition, we find that the B2 FeRh is dynamically stable, which is consistent
with the experimental synthesis FeRh , whereas the FeRh in the B2 structure
has been predicted to be unstable hart .

Overall, the dynamical stability of the B2 compounds is determined by the $M$
point phonon energy. The phonon dispersions for the 416 compounds are shown in
the Supplemental Material SM .

The $M$ point phonons. In order to understand to what extent the $M$ point
phonon determines the dynamical stability of the B2 compounds, we plot the
minimum phonon energy within the entire BZ ($\hbar\omega_{\rm min}$) as a
function of the lowest phonon energy at the $M$ point ($\hbar\omega_{\rm M}$)
in Fig. 2. When $\omega_{\rm M}\leq 0$, the relationship of $\omega_{\rm
M}=\omega_{\rm min}$ holds, whereas when $\omega_{\rm M}>0$, the equality of
$\omega_{\rm min}=0$ holds. These show that the stability of the $M$ point
phonon is a good descriptor for identifying the dynamical stability of the B2
compounds.

In Fig. 2, we can find some exceptions that do not follow the B2 stability
criteria above. There are 16 compounds (AlPt, CsN, KS, LuMg, MgHg, MgTl, MnAu,
MnHg, NaS, PrMg, SiRh, TcB, TiRh, VRu, YMg, and ZrRh) that satisfy both
$\omega_{\rm M}>0$ and $\hbar\omega_{\rm min}<\varepsilon_{\rm min}$. Although
the $M$ point phonon does not have the minimum phonon energy in these
compounds, negative phonon energies appear along the $\Gamma$-$M$ line in the
BZ (see the Supplemental Material SM ). The instability of VRu is anomalous
because negative phonon energies are observed only along the $\Gamma$-$R$ line
in the BZ, while the experimental synthesis of the VRu in the B2 and
tetragonally distorted B2 structures VRu1 ; VRu2  has been reported.

The value of $E_{\rm form}$ may be another descriptor for understanding the
stability of the B2 compounds: the dynamically stable compounds have negative
$E_{\rm form}$ and, in turn, the unstable compounds have positive $E_{\rm
form}$. However, many exceptions have been found: (i) the six compounds of
CaNi (0.024), CeMg (0.087), CrCo (0.154), LiBe (0.366), MnZn (0.062), and YbRu
(0.209) have positive $E_{\rm form}$ (eV/atom), where the figure in a
parenthesis indicates the magnitude of $E_{\rm form}$, although they are
dynamically stable. In contrast, (ii) 103 out of 150 unstable compounds have
negative $E_{\rm form}$: for example, CsF ($-2.734$), LiF ($-2.878$), RbF
($-2.775$), and SrO ($-2.662$) for strongly bonded systems and AlRe
($-0.010$), CrN ($-0.013$), and MnAu ($-0.010$) for weakly bonded systems. A
similar issue has been found in a wide variety of materials, such as ordered
alloys sun ; ono_meta  and many two-dimensional materials wang ; ono_simple ;
ono_satomi . The metastability of materials has to be studied in detail.

It is noteworthy that the B2 compounds including a semiconducting element
(group 14-17) tend to be unstable (see Table S1 and Figure S1 SM ). Such
compounds include the strongly bonded systems mentioned above. The instability
of these may be due to the presence of different ground state structure such
as the B1 structure. The understanding for the group dependence of the
dynamical stability is left for future work.

Figure 3: The displacement vectors of the $M$ point phonon modes having the
frequencies of $\omega_{\pm}$ (doubly degenerated). The cases that the atoms
$A$ and $B$ move along the $x$ and $y$ directions, respectively, are
illustrated.  Figure 4: The variation of the relative error of the $M$ point
phonon energies between the $p$NN models and the DFPT for the 15 Cu-based
compounds.

Effect of long-range interatomic interactions. We next study an origin of the
positive value for the phonon energies at the $M$ point and identify the role
of the long-range interatomic interactions. Based on standard lattice dynamics
maradudin , we derived analytical expressions for the phonon frequencies for
the wavevector $\bm{q}=(\pi/a,\pi/a,0)$, at the $M$ point in the BZ, by
assuming the B2 compounds that consist of atoms $\kappa=A$ and $B$ with the
masses $M_{\kappa}$. By using the translational symmetry of the crystal, the
dynamical matrix $\tilde{D}$ is given by

$\displaystyle\tilde{D}_{\alpha\beta}^{\kappa\kappa^{\prime}}(\bm{q})=\frac{1}{\sqrt{M_{\kappa}M_{\kappa^{\prime}}}}\sum_{j}D_{\alpha\beta}^{\kappa\kappa^{\prime}}(\bm{R}_{j},\bm{0})e^{-i\bm{q}\cdot\bm{R}_{j}},$
(1)

where $D_{\alpha\beta}^{\kappa\kappa^{\prime}}(\bm{R}_{j},\bm{0})$ is the
force constant matrix, i.e., the force along the direction of $\alpha$ acting
on the atom $\kappa$ in a unit cell characterized by the lattice vector
$\bm{R}_{j}$ when the atom $\kappa^{\prime}$ in the cell of $\bm{R}=\bm{0}$ is
displaced along the direction of $\beta$. With the $q$ grid used in the
present work, the $\bm{R}_{j}$ can take the vectors of
$\sum_{i=1}^{3}m_{i}\bm{a}_{i}$ with $m_{i}=-1,0,1,2$, where $\bm{a}_{i}$’s
are the primitive lattice vectors in the cubic cell. We thus considered the
force constants up to the 6NN sites of B2 $AB$. By assuming that the atom $A$
is located at the origin, the position of the first, second, third, fourth,
fifth, and sixth NNs are $B(a/2,a/2,a/2)$, $A(a,0,0)$, $A(a,a,0)$,
$B(3a/2,a/2,a/2)$, $A(a,a,a)$, and $A(2a,0,0)$, respectively, and these
equivalent sites, where the numbers of the equivalent sites are 8, 6, 12, 24,
8, and 6, respectively. Due to the equivalence between the $x$ and $y$
directions, four different phonon modes are present: the $z$-polarized modes
having $\omega_{z\kappa}^{2}=\tilde{D}_{zz}^{\kappa\kappa}(\bm{q})$ ($\kappa=$
A or B) and the $x$\- and $y$-polarized modes having

$\displaystyle\omega_{\pm}^{2}=\frac{1}{2}\left[\tilde{D}_{xx}^{AA}(\bm{q})+\tilde{D}_{yy}^{BB}(\bm{q})\pm\tilde{C}_{xy}(\bm{q})\right],$
(2)

with

$\displaystyle\tilde{C}_{\alpha\beta}(\bm{q})$ $\displaystyle=$
$\displaystyle\sqrt{\left[\tilde{D}_{\alpha\alpha}^{AA}(\bm{q})-\tilde{D}_{\beta\beta}^{BB}(\bm{q})\right]^{2}+\left[2\tilde{D}_{\alpha\beta}^{AB}(\bm{q})\right]^{2}}.$
(3)

The value of $\omega_{\pm}^{2}$s in Eq. (2) does not change when the indexes
$x$ and $y$ are replaced with $y$ and $x$, respectively, leading to the doubly
degenerated modes. The displacement patterns of the normal modes with
$\omega_{\pm}$ are described by a combination of the $x\ (y)$-polarized
vibration of the atom $A$ and the $y\ (x)$-polarized vibration of the atom
$B$, which are shown in Fig. 3. From the expressions of Eqs. (2) and (3), one
can expect that the coupling term of $\tilde{D}_{\alpha\beta}^{AB}(\bm{q})$
plays an important role to yield the positive value of $\omega_{-}^{2}$. The
derivations of these expressions are provided in the Supplemental Material SM
.

To study how the $M$ point phonons are stabilized by the interatomic
interactions, we introduce the $p$NN model taking into account the force
constants up to the $p$th NN atoms and compare the $M$ point phonon energies
with the DFPT results. The errors of the phonon energies with restricted
interactions, with respect to the DFPT results, are plotted as a function of
$p$ in Fig. 4. For clarity, the cases of the 15 Cu-based compounds are shown.
The phonon frequencies are denoted as $\omega_{k}$ with $k=1,\cdots,6$ in an
ascending order. The lowest energy phonons ($\omega_{1}$ and $\omega_{2}$)
correspond to the normal modes having the frequency $\omega_{-}$ except for
CuN. For the low energy phonon modes, the deviation from the DFPT is large
when $p\leq 3$. The $p=4$ is a critical value for the convergence of the
phonon energy, and such a $p$ corresponds to the interatomic interactions
between different atoms $A$ and $B$, which is consistent with our expectation
above. On the other hand, $p=2$ is found to suffice for the high energy
phonons. These results indicate that the low and high energy phonons at the
$M$ point are stabilized by the long-range and short-range interatomic
interactions, respectively. The instability of the $M$ point phonon modes in
the CuN was due to the negative value of
$\omega_{z\kappa}^{2}=\tilde{D}_{zz}^{\kappa\kappa}(\bm{q})$ with $\kappa=$
Cu.

The interatomic distance with the $p=4$ atoms is $\sqrt{11}a/2\simeq 1.66a$,
and the total number of the NN atoms up to $p=4$ is 50. This might be a
minimum criteria for determining the cutoff radius $(R_{c})$ of the
interatomic potentials in the $AB$-type compounds with guaranteed accuracy.
Interestingly, Seko et al. have constructed an interatomic potential for the
bcc K with $a=5.284$ Å, and shown that the energy, force, and stress
calculated by using the interatomic potential agree well with those obtained
by the DFT calculations when the value of $R_{c}$ is set to be more than 9 Å
seko . Such a $R_{c}$ is quite similar to $\sqrt{11}a/2=8.762$ Å.

Some compounds have a large discrepancy of the lowest phonon energy between
the 6NN model and the DFPT. For example, the CdAg, MnHg, ZnCu, ZnAg, and ZnAu
systems have $(\omega_{1},\omega_{\rm DFPT})=(-3.8,-0.9)$, $(-2.0,1.5)$,
$(-5.7,-1.7)$, $(-3.5,3.0)$, and $(-3.2,2.4)$, respectively, in units of meV.
This implies that more long-range interactions are required to achieve the
convergence to the DFPT results. It should be noted that the stability of the
$d$ electron compounds might not be described accurately within the PBE
approximation. The value of $E_{\rm form}$ tends to be overestimated within
the PBE when one studies the B2 compound that consists of the atoms having the
completely filled $d$ orbitals such as Cu, Ag, Au, Zn, and Cd ruzsinszky2020 .
More analysis using other functionals is beyond the scope of this work. The
comparisons between the $p$NN and the DFPT for the 416 compounds are provided
in the Supplemental Material SM .

Conclusions. In conclusion, we studied an origin of the dynamical stability
for 416 B2 compounds extracted from the MP MP , and identified that the 266
out of 416 compounds are dynamically stable. The lowest phonon energy at the
$M$ point determines whether the B2 compound is dynamically stable, and the
interatomic interactions up tp the 4NN atoms are necessary for an accurate
description of the B2 phase stability, which should determine the minimum
cutoff radius for constructing the interatomic potentials for the $AB$-type
compounds. The instability of the other compounds may be attributed to (i) the
positive formation energy of the B2 structure (such as CuN) and/or (ii) the
presence of different structures (such as CsF).

The present work assumes the B2 ordering on bcc structure. A competition
between the B2 ordering and the finite temperature effects play an important
role to understand the stability of binary turchi  and high-entropy alloys
neugebauer . Exploring a phonon stabilization under such situations is left
for future work. In addition, the role of the spin-orbit coupling on the
metastability schonecker  needs to be studied when heavy elements are
included. We hope that this work stimulates other studies involving different
crystal structures and gives a useful insight for developing the interatomic
potentials of binary systems.

###### Acknowledgements.

This work was supported by JSPS KAKENHI (Grant No. JP21K04628). A part of
numerical calculations has been done using the facilities of the Supercomputer
Center, the Institute for Solid State Physics, the University of Tokyo.

## References

  * (1) W. P. Davey and F. G. Wick, The crystal structure of two rare halogen salts, Phys. Rev. 17, 403 (1921).
  * (2) M. H. F. Sluiter, Some observed bcc, fcc, and hcp superstructures, Phase Transitions 80, 299 (2007).
  * (3) G. L. W. Hart, Where are nature’s missing structures?, Nat. Mater. 6, 941 (2007).
  * (4) S. K. Kolli, A. R. Natarajan, J. C. Thomas, T. M. Pollock, and A. Van der Ven, Discovering hierarchies among intermetallic crystal structures, Phys. Rev. Mater. 4, 113604 (2020).
  * (5) 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, The Materials Project: A materials genome approach to accelerating materials innovation, APL Mater. 1, 011002 (2013).
  * (6) C. E. Sims, G. D. Barrera, N. L. Allan, and W. C. Mackrodt, Thermodynamics and mechanism of the $B1$-$B2$ phase transition in group-I halides and group-II oxides, Phys. Rev. B 57, 11164 (1998).
  * (7) M. Flórez, J. M. Recio, E. Francisco, M. A. Blanco, and A. Martín Pendás, First-principles study of the rocksalt–cesium chloride relative phase stability in alkali halides, Phys. Rev. B 66, 144112 (2002).
  * (8) S. Ono and D. Kobayashi, Lattice stability of ordered Au-Cu alloys in the warm dense matter regime, Phys. Rev. B 103, 094114 (2021).
  * (9) R. Khenata and M. Sahnoun and H. Baltache and M. Rérat and D. Rached and M. Driz and B. Bouhafs, Structural, electronic, elastic and high-pressure properties of some alkaline-earth chalcogenides: An ab initio study, Physica B: Condens. Matter 371, 12 (2006).
  * (10) X. F. Wang, Travis E. Jones, W. Li, and Y. C. Zhou, Extreme Poisson’s ratios and their electronic origin in B2 CsCl-type AB intermetallic compounds, Phys. Rev. B 85, 134108 (2012).
  * (11) O. Alsalmi, M. Sanati, R. C. Albers, T. Lookman, and A. Saxena, First-principles study of phase stability of bcc $X\mathrm{Zn}$ ($X=\mathrm{Cu}$, Ag, and Au) alloys, Phys. Rev. Materials 2, 113601 (2018).
  * (12) G. Grimvall, B. Magyari-Köpe, V. Ozoliņš, and K. A. Persson, Lattice instabilities in metallic elements, Rev. Mod. Phys. 84, 945 (2012).
  * (13) W. Frank, C. Elsässer, and M. Fähnle, Ab initio Force-Constant Method for Phonon Dispersions in Alkali Metals, Phys. Rev. Lett. 74, 1791 (1995).
  * (14) K. Persson, M. Ekman, and V. Ozoliņš, Phonon instabilities in bcc Sc, Ti, La, and Hf, Phys. Rev. B 61, 11221 (2000).
  * (15) S. Ono, Lattice dynamics for isochorically heated metals: A model study, J. Appl. Phys. 126, 075113 (2019).
  * (16) J. B. Adams and S. M. Foiles, Development of an embedded-atom potential for a bcc metal: Vanadium, Phys. Rev. B 41, 3316 (1990).
  * (17) A. Seko, A. Takahashi, and I. Tanaka, First-principles interatomic potentials for ten elemental metals via compressed sensing, Phys. Rev. B 92, 054113 (2015).
  * (18) Y. Zuo, C. Chen, X. Li, Z. Deng, Y. Chen, J. Behler, G. Csányi, A. V. Shapeev, A. P. Thompson, M. A. Wood, and S. P. Ong, Performance and cost assessment of machine learning interatomic potentials, J. Phys. Chem. 124, 731 (2020).
  * (19) S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia, D. Gunter, V. L. Chevrier, K. A. Persson, and G. Ceder, Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis, Comput. Mater. Sci. 68, 314 (2013).
  * (20) P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, and M. Cococcioni et al., Advanced capabilities for materials modeling with Quantum ESPRESSO, J. Phys.: Condens. Matter 29, 465901 (2017).
  * (21) J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996).
  * (22) A. Dal Corso, Pseudopotentials periodic table: From H to Pu, Computational Material Science 95, 337 (2014).
  * (23) H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).
  * (24) N. Marzari, D. Vanderbilt, A. De Vita, and M. C. Payne, Thermal Contraction and Disordering of the Al(110) Surface, Phys. Rev. Lett. 82, 3296 (1999).
  * (25) S. Baroni, S. Gironcoli, A. Dal Corso, and P. Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73, 515 (2001).
  * (26) G. L. W. Hart, S. Curtarolo, T. B. Massalski, and O. Levy, Comprehensive Search for New Phases and Compounds in Binary Alloy Systems Based on Platinum-Group Metals, Using a Computational First-Principles Approach, Phys. Rev. X 3, 041035 (2013).
  * (27) J. M. Lommel and J. S. Kouvel, Effects of mechanical and thermal treatment on the structure and magnetic transitions in FeRh, J. Appl. Phys. 38, 1263 (1967).
  * (28) See Supplemental Material at (URL) for the derivation of Eqs. (2) and (3), the list of the 416 compounds, the phonon dispersions of the 416 compounds, and the $M$ point phonon energies within the $p$NN models and the DFPT calculations for the 416 compounds.
  * (29) R. M. Waterstrat and R. C. Manuszewski, The vanadium-ruthenium constitution diagram, J. Less-Common Met. 48, 151 (1976).
  * (30) M. Marezio, P. D. Dernier, and C. W. Chu, Low-temperature X-Ray diffraction studies of near-equiatomic VRu alloys, Phys. Rev. B 4, 2825 (1971).
  * (31) W. Sun, S. T. Dacek, S. P. Ong, G. Hautier, A. Jain, W. D. Richards, A. C. Gamst, K. A. Persson, and G. Ceder, The thermodynamic scale of inorganic crystalline metastability, Sci. Adv. 2, e1600225 (2016).
  * (32) S. Ono, Metastability relationship between two- and three-dimensional crystal structures: a case study of the Cu-based compounds, Sci. Rep. 11, 14588 (2021).
  * (33) B. Wang and G. Frapper, Prediction of two-dimensional Cu2C with polyacetylene-like motifs and Dirac nodal line, Phys. Rev. Mater. 5, 034003 (2021).
  * (34) S. Ono, Dynamical stability of two-dimensional metals in the periodic table, Phys. Rev. B 102, 165424 (2020).
  * (35) S. Ono and H. Satomi, High-throughput computational search for two-dimensional binary compounds: Energetic stability versus synthesizability of three-dimensional counterparts, Phys. Rev. B 103, L121403 (2021).
  * (36) A. A. Maradudin, E. W. Montroll, G. H. Weiss, and I. P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation (Academic Press, New York, 1971).
  * (37) N. K. Nepal, S. Adhikari, B. Neupane, and A. Ruzsinszky, Formation energy puzzle in intermetallic alloys: Random phase approximation fails to predict accurate formation energies, Phys. Rev. B 102, 205121 (2020).
  * (38) P. E. A. Turchi, M. Sluiter, F. J. Pinski, D. D. Johnson, D. M. Nicholson, G. M. Stocks, and J. B. Staunton, First-Principles Study of Phase Stability in Cu-Zn Substitutional Alloys, Phys. Rev. Lett. 67, 1779 (1991).
  * (39) F. Körmann, T. Kostiuchenko, A. Shapeev, and J. Neugebauer, B2 ordering in body-centered-cubic AlNbTiV refractory high-entropy alloys, Phys. Rev. Materials 5, 053803 (2021).
  * (40) S. Schönecker, X. Li, M. Richter, and L. Vitos, Lattice dynamics and metastability of fcc metals in the hcp structure and the crucial role of spin-orbit coupling in platinum, Phys. Rev. B 97, 224305 (2018).