# Machine learning search for stable binary Sn alloys with Na, Ca, Cu, Pd, and
Ag

Aidan Thorn, Daviti Gochitashvili, Saba Kharabadze, and Aleksey N. Kolmogorov
<EMAIL_ADDRESS>Department of Physics, Applied Physics and
Astronomy,
Binghamton University, State University of New York,
PO Box 6000, Binghamton, New York 13902-6000, USA.

###### Abstract

We present our findings of a large-scale screening for new synthesizable
materials in five M-Sn binaries, M = Na, Ca, Cu, Pd, and Ag. The focus on
these systems was motivated by the known richness of M-Sn properties with
potential applications in energy storage, electronics packaging, and
superconductivity. For the systematic exploration of the large configuration
space, we relied on our recently developed MAISE-NET framework that constructs
accurate neural network interatomic potentials and utilizes them to accelerate
ab initio global structure searches. The scan of over two million candidate
phases at a fraction of the typical ab initio calculation cost has uncovered
29 possible intermetallics thermodynamically stable at different temperatures
and pressures (1 bar and 20 GPa). Notable predictions of ambient-pressure
materials include a simple hP6-NaSn2 phase, fcc-based Pd-rich alloys,
tI36-PdSn2 with a new prototype, and several high-temperature Sn-rich ground
states in the Na-Sn, Cu-Sn, and Ag-Sn systems. Our modeling work also involved
ab initio (re)examination of previously observed M-Sn compounds that helped
explain the entropy-driven stabilization of known Cu-Sn phases. The study
demonstrates the benefits of guiding structure searches with machine learning
potentials and significantly expands the number of predicted thermodynamically
stable crystalline intermetallics achieved with this strategy so far.

##  I Introduction

Ab initio screening of vast chemical spaces has become an integral part of
materials discovery. The AFLOW [1], Materials Project [2], OQMD [3], and other
open repositories contain ab initio results for hundreds of thousands of
compounds in observed structure types and demonstrate that density functional
theory (DFT) approximations offer a reliable determination of materials’
stability. Mining these databases for new synthesizable compounds or materials
with targeted properties has led to numerous interesting predictions [4, 5, 6,
7]. For example, correlations established with machine learning and data
mining methods helped identify new oxides [8], perovskites [9], and
intermetallics [10]. Global structure optimization methods have expanded the
exploration beyond known prototypes and resulted in prediction and
confirmation of unfamiliar motifs in various materials classes [11, 12, 13,
14, 15]. Unfortunately, the high cost of ab initio calculations limits the
scope of unconstrained searches. Evaluation of structure stability with less
expensive and fairly accurate machine learning potentials (MLPs) has shown
great promise for accelerating ab initio searches [16] but successful
predictions of stable compounds remain scarce [17, 18, 19, 20, 21, 22, 23]. In
particular, our recent re-examination of the Li-Sn binary with a MLP has
uncovered several stable alloys with large unit cells not detected in ab
initio searches [19]. In the present study extended to five metal-tin
binaries, we aim to demonstrate the applicability and benefit of the developed
predictive strategy on a larger scale in a materials class abundant with
potential applications.

Tin is a post-transition metal observed in various elemental and
multicomponent crystal structure phases [24, 25, 26, 27, 28, 29]. At ambient
temperature and pressure, ’white tin’ crystallizes in the $\beta$-Sn structure
and is known as a soft, malleable, and ductile metal. Below 13°C, ’grey tin’
adopts the $\alpha$-Sn diamond structure and exhibits a semimetallic behavior.
The allotropic $\beta\rightarrow\alpha$ transformation, a so-called ’tin pest’
process turning silvery tin objects into grey powder, has slow kinetics due to
the high activation energy associated with the change in the atomic
coordination from 6 to 4 and the volume expansion by 27% [29]. Under high
pressures and room temperature, tin undergoes a series of transformations to
more close-packed structures:
$\beta$-Sn$\rightarrow$bct$\rightarrow$bco$\rightarrow$bco+bcc$\rightarrow$bcc
at about 11, 32, 40, and 70 GPa, respectively [26, 30].

The high sensitivity of tin’s ground state to the external temperature and
pressure conditions can be traced back to the element’s particular placement
in the periodic table. Within group XIV, tin’s position defines the boundary
between the covalent bonding for the lighter elements and the metallic bonding
for the heavier lead. The propensity of the elements to form covalent bonds
can be quantified with a ratio between the $sp^{3}$ bond formation energy and
the $s{\rightarrow}p$ promotion energy cost. The steady 2.8:1.4:1.15:1.02:0.8
decrease of the ratio in the C:Si:Ge:Sn:Pb set [31] has been linked to the
decreasing bond integral strength between the $s$ and $p$ states [31, 32, 33].

The competitiveness of different bonding mechanisms in pure tin contributes to
the element’s readiness to form alloys. In this respect, tin shares a lot of
traits with boron which has been the subject of our past work [34]. This
metalloid with three valence electrons also occupies a borderline spot between
an insulator (carbon) and a metal (beryllium), assumes several elemental
configurations ($\alpha$-B, $\beta$-B, and $\gamma$-B [35, 36, 37, 38, 39])
due to the frustrated electronic structure, mixes with the majority of metals,
and forms extended 2D or 3D covalent networks. The distinction between the two
classes is tin’s versatility to be either a hosting or an alloying element in
compounds which defines the materials’ remarkable suite of demonstrated and
possible functions.

Tin’s large size and tendency to form extended frameworks give rise to
applications as a battery anode material [40, 41, 42]. Tin alloy anode
materials in general are more conductive and safer than graphite-based anodes,
and certain tin binary phases have been found to have larger theoretical
specific capacities than their commercially-available counterparts (Li22Sn5
[43] and Na15Sn4 [44] have 992 mA h g-1 and 847 mA h g-1 theoretical specific
capacities, respectively, while LiC6 has 372 mA h g-1). However, the large
volume change of over 250% (420%) upon Li (Na) insertion/extraction leads to
anode pulverization over just a few cycles [45, 46, 47].

Tin alloys have been extensively investigated as non-toxic alternatives to Pb-
free solders and durable joint materials in electronics interconnects [48, 49,
50, 29, 28, 51, 52, 53]. The efforts have focused on finding tin
intermetallics that can balance high mechanical stability, high thermal
conductivity, resistance to Sn-whisker formation, cost effectiveness, and
other factors important for next-generation integrated circuits [48, 50].

A number of tin-based materials have been studied for their non-trivial
topological behavior. They range from pure $\alpha$-Sn [54] and Sn-Te/Pb [55,
56, 57, 58] 3D materials to atom-thick stanene with a buckled honeycomb
morphology [25, 59, 60, 61]. Nevertheless, a known BaSn2 compound synthesized
first almost a decade ago [62] received little attention until its potential
as a strong TI with a wide 200-meV band gap has been demonstrated in our
previous studies [63, 64]. A recent experimental study provided insights into
synthesis and stability of the BaSn2 compound [65].

Given a large body of research dedicated to tin alloys over the past few
decades [66, 43, 67, 25, 68, 28, 44], it is surprising that new ambient-
pressure tin-based materials are still being synthesized in common metal-tin
binary systems. Recent discoveries include BaSn2 in 2008 [62], FeSn5 in 2011
[27], and NaSn2 in 2017 [69] predicted earlier by our group [63]. It is
evident that experimental observation of new phases in this materials class is
possible but requires a considerable effort to narrow down the starting
compositions and fine-tune the synthesis conditions.

Our present work involves a systematic exploration of M-Sn (M = Na, Ca, Cu,
Pd, and Ag) binaries at different pressures and temperatures. The evolutionary
ground state searches based on our neural network (NN) potentials were
performed at 0 and 20 GPa. All promising candidates uncovered in the
evolutionary searches at $T=0$ K were examined at the DFT level. The
thermodynamic stability of select candidates was further investigated in a
wide range of temperatures at both NN and DFT levels. As a result, we have
identified 11 (18) new intermetallics expected to be synthesizable at ambient
(high) pressure. The rest of the manuscript is organized as follows. Section
II provides details about the DFT calculations, the construction of the NN
potentials, and our compound prediction strategy. Section III overviews prior
work on the five M-Sn binaries, highlights their key applications, and
presents our findings on new stable alloys. Section IV summarizes our results
and discusses remaining challenges of materials prediction.

##  II Methods

###  II.1 Density functional theory calculations

All DFT calculations were performed with the Vienna ab-initio simulation
package (VASP) [70, 71, 72, 73]. We chose projector augmented wave potentials
[74] with semi-core electrons: Na_pv, Ca_sv, Cu_pv, Pd_pv, Ag, and Sn_d.
Unless specified otherwise, we used the Perdue-Burke-Ernzerhof (PBE) exchange-
correlation functional [75] within the generalized gradient approximation
(GGA) [76] and the energy cutoff of 500 eV. Select phases were examined within
the local density approximation (LDA) [77, 78] or with the strongly-
constrained and appropriately-normed (SCAN) functional in the meta-GGA [79].
All crystalline structures were evaluated with dense ($\Delta k\sim 2\pi\times
0.02$ $$\mathrm{\SIUnitSymbolAngstrom}$^{-1}$) Monkhorst-Pack $k$-point meshes
[80].

###  II.2 Neural network interatomic potentials

The MLPs employed in this study rely on the Behler-Parrinello (BP) descriptor
for converting local atomic environments into NN input signals [81]. The
parameter sets defining the BP pair and triplet symmetry functions within an
extended 7.5 $\mathrm{\SIUnitSymbolAngstrom}$ cutoff radius can be found in
our previous studies [82, 11]. The NN models were constructed using an
automated iterative scheme implemented in our MAISE-NET framework [11] that
features an evolutionary sampling algorithm to generate representative
reference structures and a stratified training protocol to build
multicomponent models on top of elemental ones [83]. The resulting datasets
generated in four MAISE-NET cycles consist of (i) partially optimized crystal
structures with up to 10 atoms for elements and 32 atoms for binaries in the
$0.125\leq x\leq 0.875$ composition range obtained in short evolutionary runs
at 0, 10, 30, and 50 GPa and (ii) a small fraction (10-20%) of compressed and
expanded configurations of basic prototypes and found minima. About 10% of
structures randomly selected from the former group were set aside for testing.

In the stratified parameterization of models for the Sn binary systems, we
relied on our previously developed Na, Ca, Cu, Pd, Ag, and Sn NNs [11] with
the 51-10-10-1 architecture and 641 adjustable parameters (the total of
$(8+43+1)\times 10+(10+1)\times(10)+(10+1)\times 1$ weights corresponding to 8
BP pair functions, 43 BP triplet functions, and 1 bias in the input layer, 10
neurons and 1 bias in each of the two hidden layers, and 1 output). The binary
M-Sn NNs with a 145-10-10-1 architecture and 1,880 adjustable parameters (the
total of $(8_{\text{MSn}}+43_{\text{MMSn}}+43_{\text{MSnSn}})\times
10_{\text{M}}+(8_{\text{SnM}}+43_{\text{SnSnM}}+43_{\text{SnMM}})\times
10_{\text{Sn}}$ interspecies weights connecting the input symmetry functions
with 10 neurons in the first hidden layer of each elemental NN) were fitted to
datasets of only binary structures.

Information about the dataset sizes, NN root-mean-square errors, and error
distributions is given in Table 1 and Figs. S1-S5. As discussed in our
previous studies [11, 19], the relatively modest accuracy of about 10 meV/atom
results from the inclusion of unfamiliar configurations identified during NN-
based structure search test runs at the end of each MAISE-NET cycle. The
addition of diverse high-energy configurations generally increases the total
error by about a factor of two but greatly reduces the number of artificial
minima. This strategy has been found to ensure a desired balance between
accuracy and robustness for the intended NN application in global structure
searches. The Na-Sn system proved to be the most challenging binary to model
because the experimentally observed ground states have large unit cell sizes
outside our standard sampled range and the NN could not learn how to
accurately describe the exotic Sn frameworks [63]. Incorporation of the
equation-of-state data for the nine structures into the training set resulted
in only a marginal NN improvement, as the original 19-96 meV/atom errors
decreased to 4-68 meV/atom for this subset.

Considering that this study focuses on Sn alloys, we examined the description
of pure Sn in more detail. Fig. S6 displays the relative stability of relevant
Sn phases in the 0-20 GPa range evaluated with the reference PBE flavor, our
NN potential, and the most accurate modified embedded atom model (MEAM)
developed for this element [84]. First, it is worth reviewing how the DFT
predictions agree with experimental observations. The likely overestimation of
the $\Delta E_{\beta-\alpha}$ energy difference in common DFT approximations
along with the limitations of the standard harmonic approximation for
evaluating free energies leads to notably higher estimates of the
$T_{\alpha\rightarrow\beta}$ transition temperature [33, 84, 24, 85], 540 K in
our case versus measured 286 K [86]. This discrepancy makes it difficult to
make definitive predictions about possible high-$T$ ground states at the Sn-
rich end of the phase diagram. The PBE results at $T=0$ K also indicate
stability of hcp-Sn in the 5.5-15 GPa window not detected experimentally and
favorability of bco over bct up to about 20 GPa in disagreement with reported
data [30]. Our NN model reproduces $\Delta E_{\beta-\alpha}$ within 3.5
meV/atom and predicts $T_{\alpha\rightarrow\beta}=530$ K but does not
differentiate bco, bct, and bcc above 14 GPa. Overall, the NN offers a
satisfactory description of relative stabilities in the considered pressure
range given that all structures were fully relaxed with the corresponding
methods and that the absolute error doubles in the calculation of enthalpy
differences between two phases. The MEAM agrees well with DFT describing
phases at ambient pressure. Since it was not parameterized to model compressed
Sn configurations, it is not unexpected to see its less accurate performance
resolving competing phases under high pressures. At the same time, the MEAM
significantly disfavors the viable but elusive $\gamma$-Sn with the simple
hexagonal structure [24] and our MEAM-based evolutionary searches uncovered an
artificial ground state (oI4-$Immm$, $a=3.2485$ Å, $b=4.5366$ Å, $c=8.3219$ Å,
and a single $4j$ (1/2,0, 0.3276) Wyckoff position) 11 meV/atom below
$\alpha$-Sn at 0 K and 0 GPa. Finally, an interesting hybrid model, a
combination of EAM and rapid artificial NN potential, has been recently
developed and demonstrated to have a much-improved description of Sn phases in
a wide range of temperatures and pressures [85].

|  | Training set | Testing set | Testing errors
---|---|---|---|---
|  | $E$ | $F$ | $E$ | $F$ | $E$ | $F$
|  | size | size | size | size | meV/at. | meV/$\mathrm{\SIUnitSymbolAngstrom}$
Na |  | 4584 | 34059 | 509 | 3882 | 0.7 | 3
Ca |  | 4483 | 33327 | 498 | 4011 | 6.2 | 30
Cu |  | 4316 | 32199 | 479 | 4053 | 1.8 | 10
Pd |  | 4398 | 33225 | 488 | 4101 | 4.6 | 40
Ag |  | 4421 | 35889 | 491 | 4416 | 1.9 | 11
Sn |  | 4346 | 29694 | 482 | 3687 | 8.3 | 35
Na-Sn |  | 6953 | 33501 | 772 | 2610 | 12.5 | 43
Ca-Sn |  | 7211 | 29376 | 801 | 3327 | 14.5 | 65
Cu-Sn |  | 7685 | 69579 | 853 | 7578 | 9.5 | 45
Pd-Sn |  | 5507 | 46272 | 611 | 4812 | 9.6 | 51
Ag-Sn |  | 5410 | 37548 | 601 | 4236 | 8.4 | 33
Table 1: Dataset sizes and root-mean-square errors of constituent elemental
and combined binary NN models.

###  II.3 Evolutionary structure searches

Global structure optimizations were performed for each binary system with our
MAISE package [11]. Evolutionary searches were carried out for selected fixed
M1-xSnx compositions with $0.125\leq x\leq 0.875$. We considered up to 8
formula units and limited the structure sizes to 24 atoms per unit cell.
Randomly generated populations of 32 members were evolved for up to 168
generations with standard evolutionary operations. The number of generations
was set to $g=N+N^{2}/4$ as a function of the unit cell size that has been
found to be usually sufficient to reach search convergence in our previous
study on Li-Sn [19]. Evolutionary operations consisted of mutations of 4
single parents (random atom displacements, atom swaps, and unit cell
distortions), injections of 4 new random structures, and crossovers of 24
pairs of parents (combination of two roughly equal parts obtained with planar
cuts) [87, 11]. Child structures were locally relaxed with the NN potentials
for up to 300 Broyden–Fletcher–Goldfarb–Shanno minimization steps and assigned
a fitness based on the final enthalpy. Our fingerprint method based on the
radial distribution function (RDF) [87, 88, 11] was used to identify and
eliminate similar structures.

For select compositions, we carried out additional searches with our hybrid
NN+DFT approach that ensured more reliable convergence to putative ground
states in the study of Au nanoparticles [82]. The method involves structural
relaxations with a NN model followed by static DFT calculations to better
resolve candidate’s favorability during evolutionary runs. None of the 21
hybrid searches completed for the Na-Sn and Pd-Sn binaries improved on the
best candidates obtained with the standard NN-based runs.

###  II.4 Vibrational property analysis

Most of the phonon calculations were performed in the harmonic approximation
with the finite displacement method as implemented in Phonopy [89]. We used
symmetry-preserving expansions of primitive or conventional unit cells to
generate supercells with 72-256 atoms and applied 0.1
$\mathrm{\SIUnitSymbolAngstrom}$ displacements. The Gibbs free energy
corrections due to the vibrational entropy were included via summation over
$20\times 20\times 20$ grids in the Brillouin zone approximating the integral
$\Delta F_{\text{vib.}}=k_{b}T\int_{0}^{\infty}d\omega
g(\omega)ln[2sinh(\hbar\omega/2k_{b}T)]$.

In some cases, we performed quasi-harmonic approximation (QHA) calculations to
determine the significance of anharmonic effects at elevated temperatures. For
a given structure, the procedure involves (i) creation of a uniform grid of
expanded and compressed volumes about the equilibrium; (ii) relaxation of each
structure with VASP under the constant-volume constraint; (iii) phonon
calculation in the harmonic approximation at each volume using Phonopy and
VASP; (iv) fitting the free energy points at each temperature with a third-
order polynomial; and (v) finding the minimum of the free energy as a function
of volume at each temperature.

###  II.5 Post-search stability analysis

The protocol of selecting viable candidate phases found in NN-based
evolutionary searches for further examination with DFT approximations was
detailed in our previous studies [82, 19]. For creating pools of possible
$T=0$ K ground states, we adopted similar enthalpy windows of up to 20
meV/atom, approximately double of the NN model errors, to include distinct
minima above the lowest-enthalpy configuration obtained in each evolutionary
run. We used the same 0.95 RDF dot product cutoff to exclude similar
structures and a 0.1 Åtolerance to symmetrize each unit cell before their re-
optimization at the DFT-PBE level [19]. For making and evaluating pools of
possible high-$T$ ground states, we adhered to the following steps designed to
reduce the error and computational time [19]: (i) calculate $\Delta
F^{\textrm{NN}}_{\textrm{vib.}}$ for all candidate structures in the $T=0$ K
pools within the 20 meV/atom DFT enthalpy window; (ii) construct a convex hull
at 600 K using $\Delta G^{\textrm{DFT+NN}}=\Delta H^{\textrm{DFT}}+\Delta
F^{\textrm{NN}}_{\textrm{vib.}}$ and select all structures with Gibbs free
formation energies no further than 10 meV/atom from the tie-line; and (iii)
evaluate thermal corrections for these structures with DFT and determine
high-$T$ ground states via convex hull construction at every 10 K up to 2000
K.

|  |  Pool size |  $\Delta H^{\textrm{DFT}}_{\textrm{relax}}$ |  Pool size |  $\Delta F^{\textrm{NN-DFT}}_{\textrm{vib.}}$
---|---|---|---|---|---
|  |  at 0 K |  meV/at. |  at high $T$ |  meV/at.
Na-Sn |  |  3370 |  6.9 |  43 |  1.4
Ca-Sn |  |  1780 |  4.2 |  25 |  1.4
Cu-Sn |  |  2590 |  1.7 |  27 |  1.3
Pd-Sn |  |  1539 |  3.4 |  39 |  0.8
Ag-Sn |  |  4602 |  1.5 |  39 |  0.5
Table 2: Correspondence between NN and DFT results evaluated for pools of NN
minima selected at zero and high temperatures. The $\Delta
H^{\textrm{DFT}}_{\textrm{relax}}$ values represent average drops in enthalpy
upon re-optimization of NN minima with DFT at $T=0$ K. The $\Delta
F^{\textrm{NN-DFT}}_{\textrm{vib.}}$ values are average deviations between NN
and DFT relative free energy corrections from vibrational entropy at $T=600$ K
(see text for more detail).

Table 2 presents information on the pool sizes and the correspondence between
NN and DFT results for each M-Sn binary. The relatively low $\Delta
H^{\textrm{DFT}}_{\textrm{relax}}$ values of the average enthalpy drops during
DFT re-optimizations, ranging from 1.5 meV/atom in Ag-Sn to 6.9 meV/atom in
Na-Sn, demonstrate the proximity of the starting configurations in the NN
pools to the corresponding DFT minima. This observation lends further support
to the aforementioned NN+DFT hybrid method as an effective approach of
predicting the DFT ranking for NN candidate phases without DFT relaxations at
$T=0$ K. To quantify the agreement between the NN and DFT methods for
assessing vibrational entropy corrections at elevated temperatures, we
determined average deviations $\Delta F^{\textrm{NN-
DFT}}_{\textrm{vib.}}=\Delta F^{\textrm{NN}}_{\textrm{vib.}}-\Delta
F^{\textrm{DFT}}_{\textrm{vib.}}$. The free energy differences at the two
levels of theory were calculated separately for each pool with respect to the
pool’s mean free energy and then averaged, as root-mean-square, over all pools
for each binary. The low deviation values of less than 1.4 meV/atom justify
the use of the NN-based prescreening as a means of finding viable high-$T$
ground states.

##  III Results

###  III.1 The Na-Sn binary

Investigation of the compound-rich Na-Sn binary system extends over a century.
In 1920, the existence of four alloys at the 4:1, 2:1, 1:1, and 1:2
compositions was demonstrated with electrochemical experiments [90]. Further
synthesis and characterization work uncovered complex phases at or near the
15:4, 5:2, 7:3, 9:4, 7:12, 5:13, 1:3, 1:4, and 1:5 stoichiometries [91]. The
recent research on these alloys has been motivated primarily by their
potential use in energy storage applications based on abundant and inexpensive
sodium [40, 42]. As the graphite anode materials utilized in commercial
lithium-ion batteries are not suitable for storing larger sodium ions, Na-Sn
alloys have been considered as promising anode alternatives with a high
theoretical capacity of about 847 mA h g-1 when converting Sn into Na15Sn4
[69]. A new layered NaSn2 material was also predicted to be a topologically
nontrivial metal [63]. Stratford et al. [69] detected this phase and proposed
solutions to previously unidentified intermediates in their comprehensive
experimental and DFT study of the full Na-Sn composition range.

Figure 1: Calculated stability of Na-Sn intermetallics at ambient pressure.
The three panels show (a) the convex hull constructed at $T=0$ K, (b) energy
distances to the tie-line at $T=0$ K, and (c) temperature ranges of
thermodynamic stability. The solid, hollow, and crossed diamonds correspond to
phases found in our calculations to be stable at $0$ K, stabilizing at
elevated temperatures, and metastable at all considered temperatures,
respectively. The green, blue, and red colors denote experimentally observed,
previously considered, and our predicted phases.

The presence of phases with large unit cells, partial occupancies, and unique
morphologies made the identification of the full set with unconstrained
structure searches impractical. Following the strategy used in our Li-Sn
study, we performed the thermodynamic stability analysis of the Na-Sn binary
by combining the previously observed phases and low-enthalpy ones found in our
NN-based evolutionary searches (see Figs. 1 and 2). At ambient pressure and
zero temperature, the resulting convex hull is comprised of the cI76-Na15Sn4
[92], hP60-Na7Sn3 [93], and tI64-NaSn [94] ordered ground states and an
aP76-Na7Sn12 representation [63] of the disordered monoclinic phase [95]. As
discussed in our previous study [63] and below, the related ordered mS48-NaSn2
and mP78-Na5Sn8 variants [69] are actually metastable by 2.6 and 1.8 meV/atom,
respectively. The known hR21-Na5Sn2 [96, 97], oS52-Na9Sn4 [92], and
oS288-Na5Sn13 [98] phases hover about 5, 7, and 23 meV/atom above the
corresponding tie-lines.

Figure 2: Calculated stability of Na-Sn intermetallics at 20 GPa. The three
panels show (a) the convex hull constructed at $T=0$ K, (b) energy distances
to the tie-line at $T=0$ K, and (c) temperature ranges of thermodynamic
stability. The symbol and line styles are the same as in Fig. 1.

Our screening for high-$T$ ground states identified three new viable
candidates, mP20-Na4Sn, oI12-NaSn5, and hP6-NaSn2, that are within 0.4
meV/atom, 32.6 meV/atom, and 2.8 meV/atom of stability at 0 K in our PBE
calculations (see Table S1). With vibrational entropy corrections included,
mP20-Na4Sn stabilizes at 250 K and lies 1.7 meV/atom below the hcp-
Na$\leftrightarrow$cI76-Na15Sn4 tie-line at 650 K, which is the approximate
melting temperature at this Na-Sn composition [91]. The experimentally
observed tP12-NaSn5 phase [99] is stabilized at 240 K (note that the choice of
the metastable $\beta$-Sn in the calculation of formation energies makes this
alloy stable at $T=0$ K [69]). Our proposed oI12-NaSn5 polymorph becomes the
ground state at this stoichiometry above 450 K. We did not attempt to
calculate phonons for the metastable oS288-Na5Sn13 or stable aP76-Na7Sn12 due
to their exceptionally large sizes and/or low symmetry. We did approximate the
free energy of the latter by adding a linear interpolation of the vibrational
entropy terms evaluated for the related mS48-NaSn2 and mP78-Na5Sn8 at the
adjacent compositions. This allowed us to estimate that the new hP6-NaSn2
phase should become the ground state at 70 K and, in turn, destabilize
aP76-Na7Sn12 above 240 K. The configuration entropy contribution in Na7Sn12 is
expected to be insignificant because only $M=4$ out of 30 Na atom sites in the
$N=78$-atom unit cell have fractional occupancies. As shown in Fig. 4(b), the
$\frac{M/N}{1-x_{d}M/N}kT[x_{d}\ln(x_{d})+(1-x_{d})\ln(1-x_{d})]=-kT/19\ln(2)$
correction evaluated for half-filled $4g$ sites ($x_{d}=0.5$) in the
assumption that all configurations are equiprobable shifts the hP6
stabilization temperature up by just $\sim 15$ K.

Figure 3: Structures of select Na-Sn phases. The polyhedra in (a,d) show the
local environments of the minority species in the predicted high-temperature
oS20-Na4Sn and oI12-NaSn5 ground states, respectively. Figure 4: Stability
analysis of NaSn2 phases. (a) Relative energy of the hexagonal NaSn2
structures along the hP3 to hP6 transformation path following an out-of-plane
Sn phonon mode. At each point, the magnitude of the Sn layer distortion was
kept fixed while the unit cell shape was optimized. (b) Free energies of NaSn2
polymorphs relative to hP3 as a function of temperature. The solid points
correspond to free energies with vibrational entropy contributions calculated
in the harmonic approximation. The crossed red hexagons show the significance
of the quasi-harmonic (QH) corrections on the relative stability between hP6
and hP3. The crossed black circles illustrate the effect of the
configurational entropy in the disordered aP76-Na7Sn12 on the distance to the
aP76-Na7Sn${}_{12}\leftrightarrow\alpha$-Sn tie-line.

It has been discussed that the Na-Sn and related Li-Si/Ge/Sn binaries exhibit
similar morphological trends, as different intercalated covalent frameworks
appearing in Sn-rich alloys give way to Sn-Sn dimers and eventually to
isolated Sn atoms with the increase of the alkali-metal concentration [100,
101, 69, 63]. The tP12-NaSn5 phase features a complex 3D framework with square
Sn nets crosslinked by a percolating web of Sn bonds, from 4 to 7 per atom,
that has well-defined channels conducive for Na migration (see Fig. 3). In
fact, Stratford et al. [69] identified metastable NaSn3 and NaSn4 phases with
similar 3D morphologies, which could explain the reported but not identified
alloy at the latter composition [91]. Our predicted oI12-NaSn5 is comprised of
fused elongated hexagonal bipyramids that effectively trap Na ions. In
tI64-NaSn, the Sn framework consisting of tetrahedra about 3.70 Å apart was
found with a combination of ab initio molecular dynamics and a reverse Monte
Carlo refinement to be described better with an amorphous structure [69]. The
predicted Na-rich mP20-Na4Sn phase has a fairly uniform distribution of Sn
atoms with Na9 and Na11 local coordinations.

The phases at and around the 1:2 composition deserve a closer look. It has
been discussed previously that the mS48-NaSn2, aP76-Na7Sn12, and mP78-Na5Sn8
phases are closely related and can be obtained by different population of the
available Na sites between disjointed polyanion Sn layers stacked along the
$c$-axis [63]. The most stable aP76 structure was obtained from mP78 by
removing two $4g$ Na atoms that results in the largest separation between the
vacancies. Our original motivation for studying NaSn2 was to examine whether
the compound could crystalize in the AlB2 structure with weakly interacting
honeycomb layers and be subsequently exfoliated into stanene. We found that
the phase does stabilize over the known phases at elevated temperatures and
pressures but owes its stability to strong interlayer covalent bonds that
prevent the material from being a good precursor for stanene synthesis. The
following ab initio analysis by Stratford et al. [69] supported our
conclusions regarding the hP3-NaSn2 stability and showed that a derived
metastable NaSn3 phase with partial substitution of Na for Sn is consistent
with experimental observations.

The new hP6-NaSn2 ground state identified in our unconstrained searches has an
unexpectedly simple CaIn2 prototype. It can be constructed from hP3 by
doubling the unit cell and distorting the Sn honeycomb framework in and out of
the basal plane. Our examination of the transformation path along the
corresponding phonon mode in Fig. 4 shows that both hexagonal structures
represent local minima at the PBE and SCAN levels, which is further supported
with a phonon dispersion analysis that indicates their dynamical stability
(see Fig. S7). The presence and depth of the hP6 minimum do depend on the
level of theory. Namely, previous PBE calculations revealed a slow convergence
of the out-of-plane phonon mode in hP3 [69], our NN model predicts a 7
meV/atom drop along the barrier-free transition path, while the LDA produces a
plateau 7 meV/atom above the starting configuration. Our further PBE
calculations demonstrated that thermodynamic corrections evaluated in the QHA
have little effect on the relative free energy (see Fig. 4(b) and Fig. S8).
Hydrostatic compression, on the other hand, disfavors the hP6 minimum at 0.4
GPa and the distorted structure relaxes back to hP3 above 2 GPa (see Fig. S9).
If the proposed phase does form it should be easily distinguishable from hP3
because the distortion induces noticeable changes in structural and
vibrational properties. The contraction of the $a$ lattice constant by 2.4%
and the expansion of the $c$-axis by 9.1% results in considerable shifts of
powder XRD peak positions (see Fig. S10). The change in local Sn atom
environments from three 3.102-Å in-plane and two 3.255-Å out-of-plane bonds in
hP3 to one out-of-plane 3.056-Å and three 3.071-Å bonds in hP6 softens the
highest optical modes by 17%.

At 20 GPa pressure, we find that the convex hull is defined by nine phases
(Fig. 2). Some of the ambient-pressure ground states, e.g., tI64-NaSn and
mS48-NaSn2, destabilize by more than 0.1 eV/atom and only one, hR21-Na5Sn2,
remains stable under this pressure. The eight new ground states found in our
searches exhibit compact morphologies more favorable under compression. On the
Na-rich side, the hR15-Na4Sn, hR42-Na11Sn3, hP12-Na3Sn, and hR21-Na5Sn2 phases
with large $c/a$ ratios resemble the Li-rich bcc alloys with different
stacking sequences along the [111] direction investigated in our previous
study [19]. According to our notation that specifies the separation between Sn
layers along the $c$ axis [19], these structures can be described simply as
$|5|$, $|455|$, $|345|$, and $|34|$, respectively. In the middle of the
composition range, the mP6-NaSn phase isostructural to mP6-LiSn [102] is a
different bcc-based alloy that exhibits zig-zag Sn nets. At the Sn-rich end,
oS16-Na3Sn5 displays square Sn nets bridged by rows of Sn atoms and a
cP4-NaSn3 phase has the common L12 prototype. The observed conformity between
several compressed Na-Sn and ambient-conditions Li-Sn ground state structures
indicates the similarity of effective size ratios between the alkali metal and
tin ions under the corresponding 20 GPa and 1 bar pressures.

###  III.2 The Ca-Sn binary

Early work on the Ca-Sn binary materials indicated the existence of
oP12-Ca2Sn, oS8-CaSn, cP4-CaSn3, and tI204-Ca31Sn20 phases [103, 104, 105]. In
2000, Palenzona and Fornasini [106] systematically re-examined the binary
system in the full composition range with a combination of differential
thermal analysis, metallographic analysis, and single-crystal and powder X-ray
diffraction. The study led to the discovery of new oP52-Ca7Sn6,
tP118-Ca36Sn23, and tI32-Ca5Sn3 intermetallics and the compilation of the most
complete phase diagram to date.

Figure 5: Select structures of Ca-Sn phases observed under ambient pressure
(a,b) and predicted to be stable at 20 GPa (c,d). The Ca and Sn atoms are
shown as large and small spheres, respectively. The tI32 structure has pure Sn
layers shown with coordination polyhedra and mixed Sn-Ca layers shown with
shaded planes. The oS8 structure features stretched honeycomb layers linked
via zig-zag Sn-Sn bonds.

Structural, bonding, electronic, topological, and other properties of Ca-Sn
compounds have been analyzed in a number of detailed DFT studies. Ohno et al.
[107] used a combination of DFT and CALPHAD methods to obtain the
thermodynamic model of the binary system. Yang et al. [108] compared ab initio
and previously measured heats of formation, constructed the convex hull, and
analyzed the intermetallics’ electronic and elastic properties. Engelkemier et
al. [109] used their DFT-chemical pressure approach to rationalize how the
atomic sizes determine the favorability of Ca-Sn superstructures derived from
the W5Cr3 prototype. A recent demonstration of the Ca7Sn6 function as an anode
material with a high working voltage of 4.45 V, excellent cyclability with 95%
retention after 350 cycles, and good capacity of 85 mA h g-1 for Ca-ion
batteries [41] spurred computational studies of the binary compounds’
electrochemical, stability, and elastic properties [110, 111] and further
experimental investigations of the Ca-Sn materials’ energy storage potential
[112]. CaSn3 with the Cu3Au-type (L12) structure has been shown to have
nontrivial topological and superconducting properties [113, 114, 115, 116,
117], while CaSn has been found to be a nodal-line semimetal with potential
for topological superconductivity [118].

Figs. 6 and 7 summarize the outcome of our Ca-Sn screening under different
$(T,P)$ conditions. In line with our modeling of the other binaries, we relied
on prior experimental data to include phases with more than 24 atoms per
primitive cell but reproduced all other known binary intermetallics with our
evolutionary structure searches. In agreement with previous DFT findings [107,
108, 110], the ambient-pressure convex hull in our calculations is defined by
four oP12-Ca2Sn, tI204-Ca31Sn20, oS8-CaSn, and cP4-CaSn3 phases, while the
remaining three, oP52-Ca7Sn6, tP118-Ca36Sn23, and tI32-Ca5Sn3, are metastable
by less than 10 meV/atom.

Figure 6: Calculated stability of Ca-Sn intermetallics at ambient pressure.
The three panels show (a) the convex hull constructed at $T=0$ K, (b) energy
distances to the tie-line at $T=0$ K, and (c) temperature ranges of
thermodynamic stability. The symbol and line styles are the same as in Fig. 1.
Figure 7: Calculated stability of Ca-Sn intermetallics at 20 GPa. The three
panels show (a) the convex hull constructed at $T=0$ K, (b) energy distances
to the tie-line at $T=0$ K, and (c) temperature ranges of thermodynamic
stability. The symbol and line styles are the same as in Fig. 1.

The compositional and formation energy proximity of the tI32-Ca5Sn3,
tI204-Ca31Sn20, and tP118-Ca36Sn23 phases arises from their morphological
connection discussed in previous studies [106, 109]. The last two are members
of the R5n+6(T,M)3n+5 family with $n=5$ and $n=6$ representing intergrown
segments of different lengths from the parent W5Cr3 prototype (see Fig. 5(a)).
In addition to being metastable, the 36:23 phase was determined previously to
be mechanically unstable with $C_{44}=-122.7$ GPa [108]. The existence of the
stoichiometric tI32-Ca5Sn3 phase itself seems to not have been established
conclusively, and its appearance has been attributed to hydrogen-impurity
and/or entropy-driven stabilization [106, 107, 109].

Figure 8: Calculated $\omega^{2}$ dependence on pressure for a soft phonon
mode defining the hP6 to oP12 transformation. The red arrows in the
orthorhombic representation of the hP6 structure illustrate the atomic
displacements along the eigenvector.

Our phonon calculations indicate that the three related phases are dynamically
stable and that the vibrational entropy corrections stabilize the 36:23 alloy
above 1010 K but change the free energy distance of the 5:3 alloy from 7.8
meV/atom at 0 K to 5.4 meV/atom above the convex hull at 1400 K. All new Ca-Sn
phases based on $\alpha$-Sn, $\beta$-Sn, and fcc-Ca unit cells proposed in
Ref. [111] are at least 22 meV/atom above the convex hull boundary (e.g., the
blue points at $x=0.25$, 0.375, and 0.5 in Fig. 6).

Only one of the ambient-pressure phases, cP4-CaSn3, remains thermodynamically
stable at 20 GPa, which offers an opportunity to better understand and tune
its intriguing topological and superconducting properties via compression. The
new set of ground states identified with our global searches consists of
hP8-Ca3Sn, hP6-Ca2Sn, and tP2-CaSn phases with simple Ni3Sn, Ni2In, and
$\delta$-CuTi prototypes (see Fig. 5 (c,d)). At the 2:1 composition, the known
hP6 structure comprised of closely spaced heteronuclear Ca-Sn honeycomb layers
intercalated with Ca is isoelectronic and morphologically related to the C32
structure adopted by MgB2 but not expected to exhibit the iconic quasi-2D
superconductivity defined by hole-doped B states and hard B phonon modes. The
high-pressure hP6 polymorph does help appreciate the morphology of its
ambient-pressure oP12 counterpart known to be a derivative of the honeycomb
lattice [119, 120]. Fig. 8 illustrates that the frequency of the phonon mode
transforming hP6 to oP12 obtained in our linear response calculations becomes
imaginary below 1.9 GPa, while the linear slope of the squared frequency
dependence on pressure points to a common soft-mode phase transition described
by the Landau theory [121, 88]. The considerable distortions reaching 0.7 Å
for out-of-plane Ca displacements at 0 GPa dramatically change the local
atomic environments, e.g., changing the number of Ca neighbors around Sn atoms
from 3 in hP6 to 7 in oP12 within the 3.5-Å radius cutoff and from 11 in hP6
to 9 in oP12 within the 3.7-Å radius cutoff. The eigenmode displacements in
hP6 and the remnant honeycomb connections in oP12 displayed in Fig. 8 help
visualize the pressure-induced structural changes in Ca2Sn.

###  III.3 The Cu-Sn binary

In 1990, Saunders et al. published a detailed review of the Cu-Sn phase
diagram [122] with all (meta)stable phases discovered since the beginning of
the 20th century: $\beta$ (cI2-Cu17Sn3), $\delta$ (cF416-Cu41Sn11), $\gamma$
(cF16-Cu3Sn), $\zeta$ (hP26-Cu10Sn3), $\epsilon$ (oS80-Cu3Sn), $\eta$
(hP4-Cu6Sn5), and $\eta^{\prime}$ (mS44-Cu6Sn5). In 1995, Larsson et al. [123]
indexed a phase appearing above 350∘C as a monoclinic $\eta^{6}$
(mS54-Cu5Sn4). In 2009, a hP8-Cu3Sn (D019) phase, labeled here as
$\epsilon^{\prime}$, was observed by Sang et al. [124], which was much simpler
than the previously synthesized Cu3Ti type eight- [125] and ten-fold [126]
hcp-based superstructures. In 2012, Wu et al. [127] characterized a new
$\eta^{4+1}$ phase consisting of four $\eta^{8}$ (mP36-Cu5Sn4) [123] and one
$\eta^{\prime}$ unit cells. Other notable investigations of the binary system
with an updated phase diagram were reported a year later [128, 129]. In 2014,
Müller and Lidin [130] performed a comprehensive characterization of Cu3Sn and
proposed an off-stoichiometry modulated structure to explain missing
reflections in the XRD data. In our recent work [131], we analyzed the
thermodynamic stability of the previously reported phases at and near the 3:1
composition and found hP8 to have the lowest energy at $T=0$ K. In 2020,
Leineweber et al. [132] reported an experimental study of the ordered
$\eta^{\prime\prime}$-Cu1.235Sn phase along with a detailed analysis of all
previously reported NiAs-type derivatives. The most recent comprehensive
review of the Cu-Sn system was published in 2023 [133] with updated phase
diagrams.

The binary system has found several technological applications due to the
excellent mechanical and electronic properties of Cu-Sn intermetallics. The
high melting temperature, electrical conductivity, resistance to
electromigration, and other beneficial characteristics have made the Cu3Sn and
Cu6Sn5 compounds prime candidates for Pb-free interconnects in high-
performance electronic devices [134, 135, 136, 137, 138, 52, 139, 131]. The
Cu-Sn alloys have also been investigated as negative electrode candidate
materials in lithium-ion batteries [140, 141, 67, 142, 143].

Our constructed convex hull in Fig. 9 agrees with the current results in the
Materials Project and AFLOW databases that the ordered hP4-CuSn is the only
ground state with $E_{\textrm{form}}=-24$ meV/atom under ambient pressure and
zero temperature in this binary system. All other observed alloys are located
at least 5 meV/atom above the fcc-Cu$\leftrightarrow$hP4-CuSn tie-line, and
only a handful of phases have negative formation energies. Given that only two
Cu-Sn compounds, around the 6:5 and 3:1 compositions, have been reported to
have stability regions extending down to the lowest displayed temperature of
373 K [122, 128], it is evident that entropy plays an important role in
stabilizing the known Cu-rich alloys. The presence of the disordered $\beta$
and $\eta$ phases and the high sensitivity of transition temperature estimates
to the DFT approximations [63, 64] make it difficult to accurately map out the
phase diagram. The following analysis focuses on assessing the importance of
different contributions for the relative phase stability in this complex
binary.

Figure 9: Calculated stability of Cu-Sn intermetallics at ambient pressure.
The three panels show (a) the convex hull constructed at $T=0$ K, (b) energy
distances to the tie-line at $T=0$ K, and (c) temperature ranges of
thermodynamic stability. The symbol and line styles are the same as in Fig. 1.
Figure 10: Calculated stability of Cu-Sn intermetallics at 20 GPa. The three
panels show (a) the convex hull constructed at $T=0$ K, (b) energy distances
to the tie-line at $T=0$ K, and (c) temperature ranges of thermodynamic
stability. The symbol and line styles are the same as in Fig. 1.

We find that the vibrational contributions indeed lower the formation free
energies of all examined Cu-Sn binary phases. The results in Fig. 9 indicate
that cF416-Cu41Sn11 and hP26-Cu10Sn3, metastable by 25 meV/atom and 22
meV/atom at $T=0$ K without zero point energy, move more than half-way toward
the convex hull at 550 K but do not become stable at the respective
experimentally established $\sim 623$ K and $\sim 860$ K transition
temperatures [128]. The cF16-Cu3Sn phase undergoes only a minor stabilization,
from 75 meV/atom above the convex hull at 0 K down to 67 meV/atom at 600 K,
and is certainly metastable under typical synthesis conditions. The best phase
at this composition, hP8 shown in Fig. 11(b), has a positive formation energy
of 3.6 meV/atom at 0 K but becomes a true ground state at 550 K. As discussed
in our recent study [131], the oS64 or oS80 superstructures with long-period
anti-phase boundaries (APBs) remain a few meV/atom above hP8 at all considered
temperatures and their formation is likely defined by kinetic factors. Our
searches did not produce any particularly favorable bcc-based phases around
$x=0.15$ to explain the stabilization of the $\beta$ alloy with entropic
terms. At the Sn-rich side, the high-$T$ screening yielded a possible
tP10-CuSn4 ground state stabilizing near the reported
$\eta\leftrightarrow$liquid boundary at 500 K [128]. It belongs to a family of
layered structures featuring metal-intercalated Sn building blocks and has the
A${}^{+}\square^{-}$ stacking in our notation described in the Pd-Sn
subsection.

Figure 11:  Structures of known and simulated Cu-Sn phases. (a,b) Observed
hcp-based Cu3Sn polymorphs, with anti-phase boundaries between blocks shaded
in blue and grey. hP8 is shown in an orthorhombic representation to illustrate
its relation to oS80. (c,d) Monoclinic derivatives of hP4-CuSn with ordered
populations of interstitial sites. (e,f) Representation of ordered (hP4) and
disordered (hP4*) phases. Figure 12:  Stability analysis of Cu-Sn phases. (a)
Formation free energies at 0 K (in black) and 800 K (in olive). The solid
lines mark the boundaries of the convex hulls; the dashed lines are linear
fits to data sets representing hP4* disordered phases with interstitial Cu
defects; the dotted line connects particular interstitial configurations
leading to the known mS44 monoclinic phase; the dash-dotted lines show the
configurational entropy contribution with different fractions $f$ of available
interstitial sites in hP4* at 800 K; and the short dash line is the tangent to
the free energy curve for $f=1/6$. (b) Free energy distances to the convex
hull for select Cu-Sn phases with the arrows illustrating the estimated
temperature ranges of stability. Due the difficulty of modeling the disordered
hP4* phase, the free energy and estimated transition temperatures were
evaluated for a fixed-composition hP4*-Cu6Sn5 phase with $f=1/7$ occupation
factor explained in the main text.

The known phases just below the 1:1 composition have a common morphology of
the NiAs structure with Cu linear chains arranged on a triangular lattice
shown in Fig. 11(c-f). The mP36-Cu5Sn4, mS44-Cu6Sn5, and hP4*-Cu6Sn5
derivatives of the never-observed stoichiometric CuSn have different
distributions of additional Cu atoms in trigonal bipyramidal interstices. The
inclusion of the vibrational entropy does stabilize the ordered monoclinic
$\eta^{\prime}$ above 410 K and $\eta^{8}$ above 760 K as illustrated in Fig.
12. The high-$T$ boundaries for these phases depend on the free energy of the
disordered $\eta$ that benefits from an additional configurational entropy
term. Its evaluation proved to be a challenge because of the sizable
interaction between defects. Indeed, every CuSn unit cell with four atoms can
accommodate up to two interstitial Cu atoms, which means that the
compositional shifts down to $x\sim 0.45$ correspond to the trigonal
bipyramidal site occupancies of $\sim 30$%. The results in Fig. 12(a) for
$T=0$ K demonstrate that at this occupancy level the average defect formation
free energies, $\Delta F_{\textrm{d}}$, spread over a significant
$0.096-0.213$ eV/defect range. Seeing that the most stable monoclinic
decorations have fairly uniform defect distributions (Fig. 12), it is apparent
that only a fraction of possible configurations, $f$, effectively contribute
to the partition function. Moreover, the values change dramatically if the
vibrational entropy contributions are included: for three representative
configurations, $\Delta F_{\textrm{d}}$ was determined to decrease from 0.158,
0.154, and 0.111 eV/defect at 0 K down to 0.042, 0.024, and 0.003 eV/defect at
800 K, respectively. In light of these findings, we chose to approximate the
hP4* free energy correction as $F_{\textrm{conf}}=\Delta
F_{\textrm{d}}(1-2x)+k_{\textrm{B}}\frac{f}{(1+fx_{d})}\\{x_{d}\ln{(x_{d})}+(1-x_{d})\ln{(1-x_{d})}\\}$,
where $x_{d}=\frac{(1-2x)}{2fx}$, $\Delta F_{\textrm{d}}\approx 0.024$
eV/defect, and $f$ is treated as an adjustable parameter to check
correspondence with experiment. The sample curve and corresponding tangent
constructed for $f=1/6$ at 800 K illustrate that hP4* can indeed easily
destabilize both monoclinic phases at elevated temperatures. While this
analysis does not allow us to make accurate estimates of transition
temperatures, it helps rationalize the kinetics of the ordered $\eta^{\prime}$
and $\eta^{8}$ phase formation from the disordered $\eta$ precursor. Namely,
it appears likely that only selected trigonal bipyramidal sites are
effectively available for interstitial Cu occupation and the well-spaced
defects do not need to migrate far to precipitate in the favorable ordered
monoclinic configurations. The excess Cu atoms apparently do not diffuse out
of the lattice at low temperatures, which may explain the absence of the
stoichiometric hP4. It is worth noting that Leineweber pointed out a large
variance of $E(\eta)-E(\eta’)$ values reported in several DFT studies (from
-94 meV/atom to 360 meV/atom) and their discrepancy with the measured values
(up to 4.5 meV/atom) [133]. Our proposed approach of estimating the
$\eta-\eta’$ transition temperature highlights the importance of including the
entropic terms in the evaluation of the free energy difference.

The application of pressure reduces the number of viable ground states down to
two at the 3:1 and 1:2 compositions as shown in Fig. 10. It is interesting to
see that the oS80-Cu3Sn phase commonly obtained in ambient-pressure
experiments but only metastable in DFT calculations [131] does stabilize over
the hP8 polymorph above 10 GPa. Compression is the only factor identified in
our studies so far that promotes the formation of the specific long-period
superstructure. At the Sn-rich end, our NN-guided searches identified a
thermodynamically stable tI12-CuSn2 phase. This layered structure type is
discussed further in the Pd-Sn and Ag-Sn subsections.

###  III.4 The Pd-Sn binary

Figure 13: Calculated stability of Pd-Sn intermetallics at ambient pressure.
The three panels show (a) the convex hull constructed at $T=0$ K, (b) energy
distances to the tie-line at $T=0$ K, and (c) temperature ranges of
thermodynamic stability. The symbol and line styles are the same as in Fig. 1.

The Pd-Sn binary has been attractive primarily due to its applications in
nanocatalysis [144, 145, 146, 147]. In the bulk crystalline form, Pd-Sn alloys
have been observed at the 3:1, 2:1, 1:1, 5:7, 1:2, 1:3, and 1:4 compositions
[148, 149, 150, 151, 152, 153]. The oS20-PdSn4 phase with space group $Ccca$
has received most attention due its unusual topological properties. This Sn-
rich phase has the same crystal structure and electron count as PtSn4 [154,
153], known for its extremely large magnetoresistance [155] and Dirac node
arcs [156, 157], but features gapped out Dirac node arcs [157].

Figure 14: Calculated stability of Pd-Sn intermetallics at 20 GPa. The three
panels show (a) the convex hull constructed at $T=0$ K, (b) energy distances
to the tie-line at $T=0$ K, and (c) temperature ranges of thermodynamic
stability. The symbol and line styles are the same as in Fig. 1.

Our screening and modeling of Pd-Sn alloys indicate that the binary may have
several additional phases synthesizable at both ambient and high pressures, as
illustrated in Figs. 13 and 14. The known phases defining parts of the
ambient-pressure convex hull are oP8-PdSn discovered in 1946 [148], cP4-Pd3Sn
and oP12-Pd2Sn reported in 1957 [149], and oS20-PdSn4 observed in 2004 [153].
The mP24-Pd5Sn7 phase was shown in 2010 to have partial occupancies [150]. We
simulated six different decorations of the 4h sites in unit cells with Sn
atoms and observed that the fully relaxed structures lie 24-40 meV/atom above
the tie-line. The tI48-PdSn2 [151] and oS32-PdSn3 [152] phases observed in the
1950s were found to be metastable by about 1 and 5 meV/atom, respectively.

The majority of new competitive compounds found in our searches are located at
the Pd-rich end of the composition range. The tI18-Pd8Sn phase matches the
bf0bcb581de49d90 entry listed in AFLOW but not discussed previously. It is
based on the Pd bcc lattice with Sn substitutions that lead to a $c/a=1.055$
tetragonal distortion and a $-21.3$ meV/atom stabilization with respect to the
fcc-Pd$\leftrightarrow$cP4-Pd3Sn tie-line. Although 8:1 was outside the range
of standard compositions we chose to scan for the Sn binaries, the presence of
the tI18 putative ground state prompted us to perform a search at this
stoichiometry as well. Our resulting best candidate, aP9-Pd8Sn, is an
improvement on tI18 by 1.9 meV/atom. The identified aP8-Pd7Sn, hR21-Pd6Sn, and
mS12-Pd5Sn phases happen to be different decorations of the fcc lattice that
break the fcc-Pd$\leftrightarrow$cP4-Pd3Sn tie-line by similar $-24.3$,
$-27.0$, and $-22.2$ meV/atom, and lie $0.6$, $-3.8$, and $-1.2$ meV/atom from
the aP9-Pd8Sn$\leftrightarrow$cP4-Pd3Sn tie-line, respectively. The mS12-Pd5Sn
phase found in our searches is 8.1 meV/atom more stable than the proposed
mS12-Pd5Sn reported by Wang et al. [158]. The relative energies are consistent
in other DFT approximations (see Table S4), which suggests that at least some
of these four ground states might be obtained via standard synthesis routes.
We also find that an mP32-Pd5Sn3 phase with isolated Sn atoms breaks the
oP12-Pd2Sn$\leftrightarrow$oP8-PdSn tie-line by $-1.7$ meV/atom and a
tI36-PdSn2 phase is below tI48-PdSn2 experimental phase by $-1.3$ meV/atom.
Fig. 13(c) shows that the vibrational contributions do not cause any changes
in the convex hull up to 600 K.

Figure 15: Layer stacking of Pd-rich phases at ambient pressure. In this
notation, A or B determines if the Pd atoms are in the (0,0) and (1/2,1/2)
positions or in the (1/2,0) and (0,1/2) positions, while an empty square
symbol denotes the absence of a Pd layer. For the Sn layers, the ”o”
superscript denotes a layer of squares (44 Kepler nets), while ”+” and ”$-$”
superscripts specify the square rotation direction within the layer of squares
and rhombi (32434 Kepler nets) as described in the text.

In order to better understand stable structural motifs defining the Sn-rich
ground states, we compared the proposed and known phases with tetragonal
(tI36-PdSn2 and tI48-PdSn2) or near-tetragonal (oS32-PdSn3 and oS20-PdSn4)
symmetry featuring eight-fold coordinated Pd atoms (see Fig. 15). According to
the detailed analysis of MSnn ($n=2-4$) alloys by Nylen et al. [153], observed
Sn-rich phases in several binary systems can be described as sequences of
closely related building blocks. The Sn layers appear as either unrotated
squares (44 Kepler nets in Schläfli notation) or a combination of rotated
squares by 15-20∘ and rhombi (32434 Kepler nets in Schläfli notation). Blocks
formed by adjacent Sn layers along the stacking axis can also be shifted by
(1/2,0), (0,1/2), or (1/2,1/2) within the basal plane.

The stacking sequences in such phases can be represented conveniently with an
alternative notation that explicitly specifies the location and orientation of
the Pd and Sn layers. The ”A”, ”B”, and ”$\square$” symbols denote A-centered,
B-centered, and missing Pd layers, respectively (see Fig. 15). The ”$+$”,
”$-$”, and ”o” superscripts indicate clockwise, counterclockwise, and null
rotations of Sn squares, respectively. The lateral placement of Sn squares is
defined by the centering of the adjacent Pd layer(s) because metal atoms are
never found directly above or below the Sn rhombi. The rotation sign is
defined for Sn squares centered at (1/2,0) or (1/2,1/2). With this convention,
the new tI36-PdSn2 member of the PdSnn family can be represented as
A∘B∘A-A∘B∘A+ (Fig. 15(a)), which illustrates the presence of near-cubic ∘B∘ Sn
coordinations around Pd atoms not seen in the known Pd-Sn prototypes.

Figure 16: Relative stability of MSn2 polymorphs referenced to the known tI48
structure. The red solid diamonds correspond to the proposed tI36 structure.
The green solid circles denote the most stable known phase for each existing
MSn2 compound. The red hollow circle marks the predicted tI12-AgSn2 discussed
in the main text.

Considering the ubiquity of observed MSn2 compounds, it seemed fitting to
examine the relative stability of the competing prototypes across the block of
noble metals. We chemically substituted and fully relaxed binary Sn alloys
with Ru, Rh, Pd, Ag, Os, Ir, Pt, and Au metals in the tI36 and tI48 unit
cells. Our results presented in Fig. 16 show that tI36 becomes favored over
tI48 at the electron-rich end of both $4d$ and $5d$ sets. Comparison of these
phases to the ground states appearing in the Materials Project database [2]
shows that only PdSn2 has the potential to form in the tI36 configuration and
that RhSn2 [151] has virtually identical oS24 and tI24 energies due to a close
relationship between the two prototypes with A∘B+B∘A+ and A∘B+B∘A+A∘B-B∘I-
representations.

The hydrostatic compression to 20 GPa induces few changes in the calculated
convex hull (Fig. 14). Our predicted ambient-pressure mS32-Pd5Sn3 ground state
is replaced with a new one, oS20-Pd3Sn2, at a nearby composition. At the 1:2
stoichiometry, the tI12-PdSn2 polymorph stabilizes over tI48 by 58.4 meV/atom.
This structure with the A+A- sequence (see Fig. 19) is also predicted to be
favored at elevated pressure for CuSn2 (Fig. 10) and at elevated temperature
for AgSn2 (Fig. 17). On the Sn-rich side, the known oS32-PdSn3 phase remains
metastable and the PdSn4 compound is no longer favored.

###  III.5 The Ag-Sn binary

The most comprehensive Ag-Sn phase diagram was compiled by Karakaya and
Thompson in 1987 [159]. It summarized an extensive body of research on Ag-Sn
alloys carried out since the 1920s. The accrued data indicated that the binary
system has a high solubility of Sn in fcc-Ag up to $x=0.115$, a disordered
$\zeta$ alloy between $x=0.118$ and $x=0.228$, an ordered Ag3Sn compound
$\epsilon$ with a relatively narrow stability range, and a eutectic point at
the Sn-rich end with $x=0.962$ and 221°C. The $\zeta$ alloy was identified as
an hcp-based solid solution with a near-ideal $c/a=1.626$. Ag3Sn has also been
determined to have an underlying hcp lattice but its exact crystal structure
has been the subject of a long debate. Structural solutions proposed since the
compound’s discovery in 1926 [160] include a disordered hcp phase [161], an
orthorhombic displacive homeotype $\beta$-TiCu ($Cmcm$) [162], and the
D0a-Cu3Ti prototype ($Pmmn$) [163]. The latest studies [51, 164, 165, 166, 53]
point to a general consensus that the equilibrium $\epsilon$ phase has the
ordered Cu3Ti-type structure (oP8).

Figure 17: Calculated stability of Ag-Sn intermetallics at ambient pressure.
The three panels show (a) the convex hull constructed at $T=0$ K, (b) energy
distances to the tie-line at $T=0$ K, and (c) temperature ranges of
thermodynamic stability. The symbol and line styles are the same as in Fig. 1.
Figure 18: Calculated stability of Ag-Sn intermetallics at 20 GPa. The three
panels show (a) the convex hull constructed at $T=0$ K, (b) energy distances
to the tie-line at $T=0$ K, and (c) temperature ranges of thermodynamic
stability. The symbol and line styles are the same as in Fig. 1.

Study of Ag-Sn compounds is of significant importance in the ongoing
development of Pb-free solders for high-performance electronic devices [167,
168, 169, 51, 53]. The binary eutectic intermetallic is an integral part of
the Sn-rich multicomponent alloys with Cu, Sb, Bi, and In optimized for robust
operation under mechanical and thermal stresses. In particular, the extensive
work discussed in Ref. [53] has demonstrated that joints’ resistance to creep
and thermal fatigue is strongly affected by the morphology of Ag3Sn forming
during solder solidification. A recent transport study has also indicated that
Ag3Sn is a topological material with a nontrivial Berry phase and a possible
candidate for valleytronics [166].

Our screening and modeling findings (see Fig. 17 and 18) are consistent with
the experimental observations. The unconstrained searches identified a number
of hcp-based Ag-Sn phases in the $0.12<x<0.23$ range that lie within 1-2
meV/atom to the fcc-Ag$\leftrightarrow$oP8-Ag3Sn tie line. At the 7:1
composition, for instance, a single Ag substitution for Sn in the hexagonal
$2\times 2\times 1$ hcp-Ag supercell ($c/a=1.638$) produced an hP8 structure
($c/a=1.620$) located essentially on the tie-line (0.2 meV/atom in PBE, -1.1
meV/atom in LDA and 0.7 meV/atom in SCAN). An alternative monoclinic
decoration resulted in a nearly degenerate mP16-Ag7Sn phase (0.4 meV/atom in
PBE, $-1.8$ meV/atom in LDA and 1.2 meV/atom in SCAN). At the 7:2 composition,
a monoclinic mP18 phase was found to be similarly close to stability (0.2
meV/atom in PBE, $-0.7$ meV/atom in LDA and 1.5 meV/atom in SCAN). It is
evident that the hcp derivatives have comparable energies, which explains
their appearance as a solid solution promoted by the configurational entropy
in this Ag-rich part of the phase diagram. In agreement with the previous ab
initio analysis of the random alloy, the most stable hcp configurations have
fairly uniform distributions of Sn atoms [165].

To better understand the favorability of oP8 at the 3:1 stoichiometry, we
compared the relative stability of competing hcp supercells for the binary
M3Sn compounds with Cu, Ag, and Au. Fig. S11 illustrates that only Cu3Sn
benefits from the appearance of APBs and has a slight preference, by $2.4$
meV/atom, for the hP8 configuration with the highest APB number per formula
unit over oP8. In the M3Sn compounds with the larger Ag and Au metals, the
trend is reversed and oP8 is favored over hP8 by 4.9 meV/atom and 19 meV/atom,
respectively. We also considered the simplest orthorhombic representation of
hcp at the Ag3Sn composition with the reported unit cell dimensions of
$a=2.98$ Å, $b=5.15$ Å, and $c=4.77$ Å [162]. In the ordered form, this four-
atom unit cell cannot have the $Cmcm$ symmetry. We found the fully relaxed oP4
structure with the $Pmm2$ symmetry to be less stable than oP8 by 43 meV/atom.

Figure 19:  Structures of relevant Ag-Sn phases. (a) oP8-Ag3Sn is the only
experimentally observed phase in the Ag-Sn binary system and a subject of
multiple studies to date. (b) mP18-Ag7Sn2 is one of the ordered
representations of the disordered $\zeta$ phase. (c) A+A- tI12-AgSn2 is a
predicted high-$T$ ground state with the stacking notation explained in the
Pd-Sn subsection.

The inclusion of the vibrational entropy has little effect on the relative
free energies for phases with $x\leq 0.25$ but stabilizes two new phases at
the Sn-rich end in our calculations. The tI12–AgSn2 phase, shown to be
metastable by Saleh et al. [165], is found 29 (6) meV/atom above the
oP8-Ag3Sn$\leftrightarrow\alpha$-Sn ($\beta$-Sn) tie-line in our $T$=0 K
calculations as well. This structure has the familiar A+A- stacking (see Fig.
19(c)) appearing as the high-$P$ ground states in the Cu-Sn and Pd-Sn
binaries. In the tI12-AgSn2 case, the entropic contribution makes it the
ground state above 360 K. The tI10-AgSn4 phase breaks the
tI12-AgSn${}_{2}\leftrightarrow\beta$-Sn tie-line at much higher 570 K
temperature that falls between the $\sim 500$ K and $\sim 680$ K boundaries
that define the stability range of the $\epsilon$-liquid mixture at this
composition. The tall tetragonal unit cell with $a=3.310$ Å and $c=23.51$ Å
has two Ag atoms regularly spaced along the $c$-axis comprised of five stacked
bct blocks. The application of 20 GPa compression (Fig. 18) destabilizes the
two Sn-rich phases and leads to minor reordering of relative enthalpies, by
1-2 meV/atom, for competing hcp derivatives with $0.12<x<0.25$. Some of these
ordered phases, such as mP16 and oI40, become marginally stable at elevated
pressures and temperatures.

##  IV Summary

The NN-guided materials prediction approach introduced in our previous study
and tested on a single Li-Sn binary system [19] has been used in this work to
screen a larger configuration space of five M-Sn binaries. Over 2.0 million
candidate phases were fully optimized with NN models in our evolutionary
searches, requiring over 260,000 CPU hours, and almost 14,000 of them were
further re-examined with DFT, costing about 0.98 million CPU hours. Compared
to our previous study of Li-Sn alloys [19], the NN simulations for similar-
sized structures were cheaper because of the lower number of nearest neighbors
within the cutoff radius for the Sn binaries with larger metals, while the DFT
calculations were more expensive because of the higher number of included
(semi-core) electrons. Therefore, an equivalent DFT-level scan of the M-Sn
phases with the considered structure sizes would have necessitated about 100
million CPU hours, which indicates that the utilization of NN potentials
accelerates global structure searches at $T=0$ K by two orders of magnitude.
An even more considerable benefit was seen in the performed screening for
high-$T$ ground states, as our phonon calculations at the NN level for over
6,000 phases (230 CPU hours) helped narrow the pool down to 173 viable
candidates for subsequent phonon calculations at the DFT level (1.2 million
CPU hours).

Tin | Pearson | Space |  $T$ range (K) |  AFLOW
---|---|---|---|---
alloy | symbol | group | 0 GPa | 20 GPa |  prototype
Na4Sn | mP20 | 11 | 250 | - |  |  |  |  |  -
NaSn2 | hP6 | 194 | 70 | - |  |  |  |  |  CaIn2
NaSn5 | oI12 | 71 | 450 | - |  |  |  |  |  -
Na4Sn | hR15 | 166 |  |  |  | 0 | - |  |  -
Na11Sn3 | hR42 | 166 |  |  |  | 0 | - |  |  -
Na7Sn2 | oS36 | 65 |  |  |  | 0 | - |  |  -
Na3Sn | hP12 | 156 |  |  |  | 0 | - | 630 |  -
Na2Sn | hP3 | 191 |  |  |  | 0 | - |  |  Os2Pt
NaSn | mP6 | 10 |  |  |  | 0 | - |  |  LiSn
Na3Sn5 | oS16 | 65 |  |  |  | 0 | - | 420 |  -
NaSn3 | cP4 | 221 |  |  |  | 0 | - |  |  Cu3Au (L12)
Ca3Sn | hP8 | 194 |  |  |  | 0 | - |  |  Ni3Sn (D019)
Ca2Sn | hP6 | 194 |  |  |  | 0 | - |  |  Ni2In (B82)
CaSn | tP2 | 123 |  |  |  | 0 | - |  |  $\delta$-CuTi (L2a)
CuSn4 | tP10 | 125 | 550 | - |  |  |  |  |  PtPb4 (D1d)
CuSn2 | tI12 | 140 |  |  |  | 0 | - |  |  Al2Cu (C16)
Pd8Sn | aP9 | 2 | 0 | - |  | 0 | - |  |  -
Pd6Sn | hR21 | 148 | 0 | - |  | 0 | - |  |  -
Pd5Sn | mS12 | 12 | 0 | - | 660 | 0 | - |  |  -
Pd5Sn3 | mS32 | 15 | 0 | - |  |  |  |  |  -
PdSn2 | tI36 | 140 | 0 | - |  |  |  |  |  -
Pd3Sn2 | oS20 | 36 |  |  |  | 0 | - | 710 |  -
PdSn2 | tI12 | 140 |  |  |  | 0 | - |  |  Al2Cu (C16)
AgSn2 | tI12 | 140 | 360 | - |  |  |  |  |  Al2Cu (C16)
AgSn4 | tI10 | 139 | 570 | - |  |  |  |  |  Al4Ba (D13)
Ag7Sn | mP16 | 13 |  |  |  | 0 | - | 880 |  -
Ag6Sn | mS28 | 15 |  |  |  | 0 | - |  |  -
Ag9Sn2 | mS44 | 15 |  |  |  | 100 | - |  |  -
Ag4Sn | oI40 | 44 |  |  |  | 0 | - |  |  -
Table 3: Compilation of all new thermodynamically stable M-Sn phases predicted
in this study. The columns from left to right denote the composition, Pearson
symbol, space group number, stable temperature ranges at ambient and elevated
pressures, and prototype if available in the AFLOW database.

The systematic scan has uncovered a surprisingly large number of possible
thermodynamically stable M-Sn intermetallics summarized in Table 3. A total of
11 phases have been shown to be below tie-lines formed by known alloys at
ambient pressure and an additional 18 phases have been determined to define
convex hulls at 20 GPa. In terms of structural complexity, the identified
ground states range from 2- and 3-atom known prototypes with high symmetry
(tP2-CaSn and hP3-Na2Sn) to 20- and 22-atom unknown low-symmetry unit cells
(mP20-Na4Sn and mS44-Ag9Sn2). According to our experience [88, 11, 170, 171,
19, 172], identification of configurations with over 16 atoms per primitive
unit cell is challenging and the conventional global searches performed
directly at the DFT level could have easily missed the largest ones located in
this study. A common approach for dealing with the problem is to collect
relevant known prototypes and carry out a high-throughput chemical
substitution screening. We have checked the AFLOW library [1, 173] and found
that only 12 out of the 29 predicted structures match the known prototypes in
this widely used database. Phases with particular underlying lattices, such as
the proposed fcc-based Pd-rich Sn alloys, could also be found with the cluster
expansion method but exhaustive screening of possible configurations becomes
computationally demanding for large unit cells [19].

The main findings for each M-Sn binary include the following. In the Na-Sn
system, we uncovered three possible high-$T$ ground states at ambient
pressure. The identification of the stable hP6-NaSn2 derivative of the
hP3-NaSn2 phase is particularly unexpected because the system had been
previously explored with global structure searches [69]. The Ca-Sn binary was
the only considered system that did not show any new viable ground states at 1
bar. Application of 1.9 GPa pressure induces a transformation of Ca2Sn from
oP12 to a more symmetric hP6 polymorph with Ca-Sn honeycomb layers. With the
known cP4-CaSn3 remaining stable up to at least 20 GPa, it would be
interesting to probe the response of its topological and superconducting
properties to compression. In the Cu-Sn binary, we identified a possible
high-$T$ ambient-pressure tP10-CuSn4 ground state and observed that the known
oS80-Cu3Sn, metastable at ambient conditions [131], becomes favored at 20 GPa.
We also performed a detailed ab initio analysis of the known alloys and were
able to demonstrate how vibrational and configurational entropy terms
stabilize off-stoichiometry derivatives of the hP4-CuSn phase at high
temperatures. In the Pd-Sn binary, we uncovered several viable ground states
across the full composition range, including three fcc-based phases around
$x=0.15$ and tI36-PdSn2. To rationalize the stability of the former, we
introduced a notation illustrating its morphology in terms of Pd and Sn
building blocks and examined the favorability trend of this A∘B∘A-A∘B∘A+ MSn2
configuration across a set of noble metals. In the Ag-Sn system, we detected
several ordered representations of the known disordered alloy in the
$0.12<x<0.23$ composition range and identified two possible Sn-rich phases
thermodynamically stable at high temperatures. While a number of our previous
DFT-based predictions have been validated experimentally, such as the revision
of misidentified CrB4 [174, 175], MnB4 [34], and Na2IrO3 [176] crystal
structures or the discovery of new LiB [177, 178], CaB6 [88], FeB4 [87, 179],
and NaSn2 [63, 69] materials, the feasibility of obtaining the identified
thermodynamically stable M-Sn phases may also depend on various kinetic
factors that cannot yet be modeled reliably. It will be interesting to see if
the appropriate choice of starting materials and/or synthesis conditions could
help expand the set of known M-Sn alloys.

We find it important to note that the reliability of these predictions depends
strongly on the accuracy of the PBE functional in the GGA chosen for this
study. Our additional LDA and SCAN tests presented in Tables S1-S5 illustrate
that most of our conclusions are consistent across the considered DFT
approximations. Nevertheless, the unexpectedly high number of phases found to
be stable in our calculations but never observed experimentally in these
extensively studied binaries may still be a consequence of the limited
accuracy of standard DFT flavors. Therefore, the dramatically expanded scope
of NN-based searches can be instrumental for detecting potential DFT
approximation artifacts. Further improvement of NN models may also be needed
to ensure a robust identification of all viable candidates. The 8-15 meV/atom
accuracy of NN potentials appears to have been appropriate for the considered
set of Sn alloys, as our NN+DFT hybrid tests for select Na-Sn and Pd-Sn
compositions did not reveal any missed ground states, but more studies should
be carried out for materials with diverse bonding types. The encouraging
performance of our NN-guided structure prediction approach in the exploration
of simple bulk alloys [83, 17], metal nanoparticles [180, 82], and more
complex Sn alloys studied in Ref. [19] and here indicates that it may also be
suitable for broader materials classes.

## Data availability statement

The M-Sn NN models, IDs 002BD87A, 0552A1B2, 01121374, 036C3A42, 016C1E4A, can
be downloaded at https://github.com/maise-guide/maise/. Relevant M-Sn
structures are given in the supplementary material. Other data supporting the
ﬁndings of this study is available from the corresponding author upon request.

## Code availability statement

The MAISE and MAISE-NET codes are freely available for download at
https://github.com/maise-guide/.

## Acknowledgements

We acknowledge the NSF support (Award No. DMR-1821815) and the Extreme Science
and Engineering Discovery Environment computational resources [181] (NSF Award
No. ACI-1548562, Project No. TG- PHY190024).

## Competing interests

The authors declare no competing interests.

## Additional information

Supplementary information The online version contains supplementary material.

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