Thin film Sn2Se3 is a promising new phase change memory material with a very high resistance contrast between insulating amorphous and conducting crystalline phases. We identify the structure of the Sn2Se3 crystalline phase using ab initio evolutionary structure search and report its properties. We find a structure based on Sn-Se ribbons with clear disproportionation of Sn into Sn(II) and Sn(IV) sites, similar to Sn2S3. The energy is only 9 meV/atom above the tie line between SnSe and SnSe2. Sn charge disproportionation is only marginally favored in this selenide, in contrast to sulfides. This leads to a semimetallic rather than semiconducting behavior. These findings, marginal stability of the crystalline phase and conducting character and close competition of disproportionated and non-disproportionated structures, are important for understanding the behavior of Sn2Se3 as a phase change memory material, specifically the fast low energy, the low temperature switching, and the high resistance contrast.

Phase change memory devices require materials with metastable amorphous and crystalline states having high resistance contrast, low switching energy, fast switching times, and long term stability.1–5 Ge-Sb-Te (GST) alloys have been widely employed because of their fast crystallization speed and demonstrated high cycle life.6–14 Sn selenide phases have also been investigated in this context, including SnSe and SnSe2, which correspond to known bulk compounds, as well as Sn2Se3.15 Recently, studies of Sn2Se3 have shown a very rapid low energy reversible switching between a high resistivity amorphous state and a very low resistivity crystalline state, with indications of performance that may exceed GST.15–18 The high conductivity of the crystalline phase is at odds with the known behavior of the sulfide Sn2S3, which is a semiconductor with a gap of ∼1 eV. This suggests a different behavior for the selenide system. However, there are no reported properties of bulk Sn2Se3, and the phase is not reported in phase diagrams. There is one report by Palatnik and Levitin who reported the phase but not a structure determination.19 The corresponding sulfide, Sn2S3, is a semiconductor,20 inconsistent with the low resistivities observed for Sn2Se3 crystalline films.

Here, we use first principles, genetic algorithm based crystal structure prediction to identify the Sn2Se3 phase. We then calculate the properties. We find two low energy structures, both very slightly above the SnSe-SnSe2 tie line and therefore marginally metastable consistent with the known phase diagram, as well as the fact that the crystalline phase can be formed in thin films,21 where phase separation may be kinetically inhibited.

The lowest energy structure has two formula units per cell with the Sn disproportionated into Sn (II) and Sn (IV) similar to Sn2S3.20,22–24 However, in contrast to the semiconducting sulfide, Sn2Se3 is semimetallic. We also find a non-disproportionated structure, which is metallic and very slightly higher in energy compared with the disproportionated structure. This competition of structures is consistent with the ready formation of an amorphous phase. As mentioned, semimetallic and metallic nature of the crystalline phases of Sn2Se3 is in contrast to semiconducting Sn2S3. This explains the low resistance of the crystalline material, which is essential for high resistance contrast.

We used the unbiased swarm structure search method as implemented in the CALYPSO code25,26 for the crystal structure determination. Our search for the Sn2Se3 compounds included cells up to 20 atoms (four formula units) and Np (Np = 30) random initial structures followed by full relaxation. The lowest energy structure found has two formula units per cell. We did the evolutionary search for 50 generations. At each step, we retained 60% of the lowest energy structures. The remaining 40% were replaced by new random structures with different symmetries. As seen in Fig. 1, although new metastable structures are introduced, no new ground states were generated in the final generations of the search. In other words, the lowest energy structure was already found in the final generations, and no better structures were produced by continuation of the search. We also tested the approach by applying it to the known compounds such as SnSe, SnS, and SnSe2. The correct ground state structures in accord with the experiment were predicted.

FIG. 1.

Energies of structures as a function of generation during the structure search for Sn2Se3. The lowest-energy structures are connected with a solid line. Note that by the end of the structure search, no new lowest energy potential ground state structures are being generated.

FIG. 1.

Energies of structures as a function of generation during the structure search for Sn2Se3. The lowest-energy structures are connected with a solid line. Note that by the end of the structure search, no new lowest energy potential ground state structures are being generated.

Close modal

The structure relaxations were performed with density functional theory (DFT) using the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA).27 We used the projector-augmented wave (PAW) method28 as implemented in the VASP code.29 Good convergence was obtained with an energy cut off of 400 eV for the plane-wave expansion and a Brillouin zone integration grid spacing of 2π × 0.032 Å−1. A higher 600 eV cut-off was used for the final total energy calculations to reliably compare the energies of the phases.

The electronic structure calculations were carried out using the general potential linearized augmented planewave (LAPW) method,30 as implemented in the WIEN2K code,31 with well converged LAPW basis sets, including local orbitals for semicore states. The lattice parameters obtained in the VASP calculations as above were used. The electronic structures were then calculated using the modified Becke-Johnson potential of Tran and Blaha (mBJ)32 with spin-orbit coupling. The transport coefficients were obtained from the electronic structure using the BoltzTraP code.33 The phonon calculation was done using the PHONOPY code with the supercell finite difference method.34 For this purpose, 4 × 2 × 2 supercells were used. The elastic constants were calculated using finite strains with the VASP code.

The LAPW method implemented in WIEN2k is highly precise, and combined with the stable all electron implementation of the mBJ method, including spin-orbit provides more reliable predictions of the electronic properties. On the other hand, for the structure determination and relaxation, it is essential to have an efficient implementation of the stress tensor, which is available with the PAW method implemented in VASP. For this reason, we used two different electronic structure codes in the present study. We cross checked the energy differences between these two codes and found no significant differences, in particular, regarding the ground state. We also performed hybrid functional HSE calculations35 including spin-orbit with the PAW method to verify the band overlap predicted in the mBJ calculations (see below).

Figure 2 shows the lowest-energy structure P21/m and second lowest-energy structure P3¯m1 as obtained from the structure prediction. The structural parameters are given in Table I. The lowest energy structure has Sn disproportionated into Sn(II) and Sn(IV) sites. It consists of Sn2Se3 ribbons running along the c axis in a P21/m monoclinic phase. The structure contains two formula units per cell. These Sn2Se3 ribbons are similar to those in the mixed valence compound Sn2S3, which has an orthorhombic (Pnma) structure with four formula units per cell.20 The relaxation of Sn2Se3 in the Pnma structure of Sn2S3 yields an energy of 1.7 meV higher per atom than the P21/m structure that we find. The other crystal structure, a P3¯m1 phase as shown in Fig. 2(b) with one formula unit per cell, is formed by the Sn-Se polyhedra. Its energy is only 0.54 meV/atom higher than the P21/m phase. There is only one type of Sn atom, which is coordinated by Se and has valence, Sn(III).

FIG. 2.

The low-energy Sn2Se3 structures: the lowest energy structure, spacegroup, P21/m (a) and the non-disproportionated P3¯m1 (b) as obtained from the structure search. The P21/m (a) phase has Sn-Se ribbons with Sn (II) (blue) and Sn (IV) (grey) sites. Se is shown in green.

FIG. 2.

The low-energy Sn2Se3 structures: the lowest energy structure, spacegroup, P21/m (a) and the non-disproportionated P3¯m1 (b) as obtained from the structure search. The P21/m (a) phase has Sn-Se ribbons with Sn (II) (blue) and Sn (IV) (grey) sites. Se is shown in green.

Close modal
TABLE I.

Calculated lattice parameters and internal structural parameters in lattice units.

Space groupLattice parameters (Å)Wyckoff positionsAtomsxyz
P21/m a = 7.354  Sn(II) 0.7533 0.2500 0.9924 
 b = 4.050  Sn(IV) 0.4167 0.2500 0.3434 
 c = 9.839 2e Se1 0.4304 0.2500 0.8191 
 β = 92.05∘  Se2 0.2698 0.7500 0.4964 
   Se2 0.8965 0.7500 0.8298 
P3¯m1 a = 4.057 2d Sn 0.6667 0.3333 0.1886 
 c = 9.650 2d Se1 0.3333 0.6667 0.6569 
  1a Se2 0.0000 0.0000 0.0000 
Space groupLattice parameters (Å)Wyckoff positionsAtomsxyz
P21/m a = 7.354  Sn(II) 0.7533 0.2500 0.9924 
 b = 4.050  Sn(IV) 0.4167 0.2500 0.3434 
 c = 9.839 2e Se1 0.4304 0.2500 0.8191 
 β = 92.05∘  Se2 0.2698 0.7500 0.4964 
   Se2 0.8965 0.7500 0.8298 
P3¯m1 a = 4.057 2d Sn 0.6667 0.3333 0.1886 
 c = 9.650 2d Se1 0.3333 0.6667 0.6569 
  1a Se2 0.0000 0.0000 0.0000 

The averaged Sn-Se bond lengths and calculated bond-valence sums of Sn atoms are listed in Table II. The P21/m phase has the bond-valence sums of 2.38 and 3.48 for the Sn(II) and Sn(IV) atoms, respectively. This is consistent with Sn2+ and Sn4+. Se is less electronegative than S. Therefore, it is expected that Sn2Se3 has more covalency than the sulfide. This would then work against the Sn disproportionation. The sulfide, Sn2S3, has bond valence sums of 2.0 and 4.4 for the different Sn atoms.20 The different Sn-Se bond lengths in P21/m phase are 2.83 Å for Sn(II) and 2.80 Å for Sn(IV) sites, as compared with the averaged bond length of 2.81 Å in SnSe (Pnma)36 for Sn(II)-Se bonds and 2.75 Å in SnSe2 (P3¯m1)37 compounds for Sn(IV)-Se bonds. It is worth noting that for the Sn(II) sites there is a cross-gap metal-ligand hybridization between the Se p derived valence bands and the nominally unoccupied Sn(II) p states in the conduction band. This cross-gap metal ligand hybridization is reminiscent of the cross-gap hybridization that occurs in GST.38 The P3¯m1 phase with one type of Sn with a bond-valence sum of 2.91, corresponding to Sn3+ as may be expected. The average bond length is 2.87 Å with 6 as the coordination number.

TABLE II.

Calculated averaged Sn-Se bond lengths and Sn bond valence sums.

Space groupBond typeBond length (Å)CoordinationBond valence sum
P21/m Sn(II) 2.830 2.38 
 Sn(IV) 2.795 3.48 
P3¯m1 Sn 2.872 2.91 
Space groupBond typeBond length (Å)CoordinationBond valence sum
P21/m Sn(II) 2.830 2.38 
 Sn(IV) 2.795 3.48 
P3¯m1 Sn 2.872 2.91 

The thermodynamic stabilities of the Sn-Se phases were evaluated by calculating the formation energies and generating the convex hull. The relative enthalpies of the different Sn-Se phases are shown in Fig. 3. Pnma SnSe and SnSe2 are the reference ground state structures. The other three reported phases of SnSe, Fm3¯m, Cmcm, and P4/nmm, though all synthesized experimentally, are non-ground state structures. The P21/m phase Sn2Se3 lies slightly above the SnSe-SnSe2 tie line. This means that it is not thermodynamically stable at 0 K. However, the relative energy is only 9 meV/atom higher than the phase separated ground state consisting of SnSe (Pnma) and SnSe2 (P3¯m1). Therefore, while Sn2Se3 is not a ground state, the driving force for phase separation is exceedingly weak. In addition, the energy by which it lies above the tie line is much lower than the energy of the experimentally observable non-ground state structures of SnSe. This low driving energy for phase separation means that while Sn2Se3 may not appear in the equilibrium bulk phase diagram, it is likely to exist in thin films where diffusion is kinetically inhibited as discussed in the context of other metastable materials.39,40 In the context of phase change memory, a phase separation would amount to a changing device geometry and parameters, which would presumably limit device cycle life.

FIG. 3.

Relative enthalpies of formation per atom with respect to Sn and Se for different Sn-Se phases. The solid line shows the convex hull.

FIG. 3.

Relative enthalpies of formation per atom with respect to Sn and Se for different Sn-Se phases. The solid line shows the convex hull.

Close modal

We calculated the phonon dispersion by the finite difference method to check the dynamic stability of P21/m Sn2Se3. These calculations were done using a 4 × 2 × 2 supercell. There are no imaginary phonon modes except perhaps in the acoustic modes near the zone center. These zone center acoustic modes are not fully converged around Γ due to the limited size of the supercell and the imposition of the acoustic sum rule in the PHONOPY code. In order to establish the stability of these modes, we calculated the elastic constants using finite strains. We find that the material is indeed elastically stable. The elastic constants are c11 = 12 GPa, c22 = 68 GPa, c33= 45 GPa, c44 = 21 GPa, c55 = 10 GPa, c66 = 6 GPa, c12 = 13 GPa, c13 = 13 GPa, c23 = 23 GPa, c15 = 9 GPa, c25 = 3 GPa, c35 = 12 GPa, and c46 = 1 GPa, with the setting of Table I. Thus, the P21/m phase Sn2Se3 is dynamically stable. It is interesting to note that compared with the Γ to Y direction, the acoustic modes in the Γ to X and Γ to Z directions of the Brillouin zone show a soft character, as also seen in the elastic constants. This is a consequence of weak inter-ribbon bonding (Fig. 4).

FIG. 4.

Calculated phonon dispersion curves of the P21/m Sn2Se3 structure.

FIG. 4.

Calculated phonon dispersion curves of the P21/m Sn2Se3 structure.

Close modal

The band structures of the two phases are shown in Fig. 5. In addition, band structures, as obtained with the mBJ potential, are shown for Sn2S3 (calculations as in Ref. 20) and SnSe2 in Fig. 6. As seen, both compounds are semiconductors in accord with the previous reports.20,22,41 The density of states (DOS) and the projections of s and p character on the Sn(II) and Sn(IV) atoms of Sn2Se3 is shown in Fig. 7. The band structure of the P3¯m1 phase shows that it is metallic. The P21/m phase is semimetallic with band extrema that are not on the symmetry lines. It is semimetallic due to a band overlap. This seen in the existence of a Fermi surface, with carrier pockets located in the zone as shown in Fig. 8. We checked the band overlap using different approaches, in all cases, including spin-orbit. We find an overlap of 0.147 eV with the PBE functional, which is known to give too small gaps. With the more reliable mBJ functional, we obtain 0.049 eV, while with the HSE hybrid functional, we obtain a very similar semimetallic overlap of 0.053 eV.

FIG. 5.

Band structures including spin orbit of the P21/m and P3¯m1 Sn2Se3. The dashed lines denote the Fermi level at 0 eV. Calculations are shown for the mBJ potential and the HSE hybrid functional for the P21/m structure and for the PBE functional and the mBJ potential for the P3¯m1 structure.

FIG. 5.

Band structures including spin orbit of the P21/m and P3¯m1 Sn2Se3. The dashed lines denote the Fermi level at 0 eV. Calculations are shown for the mBJ potential and the HSE hybrid functional for the P21/m structure and for the PBE functional and the mBJ potential for the P3¯m1 structure.

Close modal
FIG. 6.

Band structures including spin orbit with the mBJ potential for Sn2S3 and SnSe2.

FIG. 6.

Band structures including spin orbit with the mBJ potential for Sn2S3 and SnSe2.

Close modal
FIG. 7.

Total DOS (top) and Sn projections (bottom) as obtained with the mBJ potential for P21/m phase Sn2Se3 on a per unit formula basis.

FIG. 7.

Total DOS (top) and Sn projections (bottom) as obtained with the mBJ potential for P21/m phase Sn2Se3 on a per unit formula basis.

Close modal
FIG. 8.

Fermi surface of P21/m (left) and P3¯m1 (right) Sn2Se3. The plot shows the electron pockets (blue) and hole pockets (red) for the monoclinic phase. The results shown are for the mBJ potential for the semimetallic P21/m structure and the PBE functional for the metallic P3¯m1 structure.

FIG. 8.

Fermi surface of P21/m (left) and P3¯m1 (right) Sn2Se3. The plot shows the electron pockets (blue) and hole pockets (red) for the monoclinic phase. The results shown are for the mBJ potential for the semimetallic P21/m structure and the PBE functional for the metallic P3¯m1 structure.

Close modal

For the P21/m phase, the valence and conduction bands show different orbital character. The valence bands within ∼0.5 eV of the valence band maximum (VBM) are derived from the s states of the divalent Sn(II) site, while the conduction bands within ∼0.5 eV of the conduction band minimum (CBM) are derived from the s states of the tetravalent Sn(IV) site. The behavior of two different Sn sites is similar with the Sn in Sn2S3. However, the energy range of the valence bands and conduction bands at the band edges in Sn2S3 (∼1 eV)20 is almost two times larger than in P21/m phase of Sn2Se3. As mentioned, the calculated bond valence sums of the different Sn sites in Sn2S3 compound are 2.0 and 4.4, while for the selenide, the corresponding values are 2.38 and 3.48, respectively.

The 5s states of the different Sn sites are substantially hybridized with the Se 4p states in both valence bands and conduction bands as shown in Fig. 7. This strong hybridization originates from the region of –1.5 eV in valence band to 1 eV in conduction band. The valence bands within ∼1.5 eV of the band edge are mainly hybridized Sn(II) (divalent, see Table I for the atom labels) antibonding 5s states with the Se2 4p states and Se3 4p states. The conduction bands within ∼1 eV of the band edge are formed by the hybridization of Sn(IV) antibonding 5s states with Se1 4p states. We note that the occurrence of antibonding s states of Sn(II) site in the valence bands is qualitatively similar to SnO compounds.42–45 Note also the asymmetric structure with all three coordinating seleniums on the same side of the Sn(II) which may be viewed as resulting from the lone pair occupying the position on the opposite side. This antibonding feature would be attributed to a lone pair with the Sn 5s2 electrons occupying the sp hybridized orbital, which would be interpreted as being stereochemically active in directing the layered structures. This is also reflected in the cross-gap metal-ligand hybridization mentioned above.

The electrical properties are controlled by the band structures in the vicinity of the Fermi energy. Figure 8 shows the Fermi surfaces of the phases. The Fermi surface of the metallic P3¯m1 phase shows significant anisotropy for the carrier pockets at the Γ point. The hole (blue) and electron (red) carrier pockets of the semimetallic P21/m phase are along the ky direction. Both the electron and hole pockets have ellipsoidal shapes with considerable anisotropy. Based on the carrier pocket shapes, the conductivity for the P21/m phase may be the highest along the b-axis. Figure 9 shows the calculated thermopower as a function of temperature. There is a large anisotropy of the thermopower, which is different from the normal semiconductors46 and reflects the semimetallic nature of the material. Specifically, there is compensation between positive and negative contributions from hole and electron pockets, and these are weighted by the anisotropic conductivities for holes and electrons.

FIG. 9.

Calculated thermopower as a function of temperature as obtained in the constant scattering time approximation, for P21/m Sn2Se3 without doping.

FIG. 9.

Calculated thermopower as a function of temperature as obtained in the constant scattering time approximation, for P21/m Sn2Se3 without doping.

Close modal

In any case, the electronic structure, specifically the absence of a band gap, will lead to metallic conductivity for crystalline Sn2Se3 independent of doping. This is in contrast to SnSe2, which has a semiconducting gap, as seen in Fig. 6, amounting to 1.1 eV in our calculations. In the experiments of Chung and co-workers, high conductivity is found for the crystalline state of both Sn2Se3 and SnSe2 alloys.15 While, as mentioned, our results explain metallic conductivity in crystalline Sn2Se3, the conductivity of SnSe2 cannot be explained in the same way. SnSe2 as a Sn(IV) compound grows naturally n-type presumably due to Se vacancies.15 The degenerate doping by such vacancies is a likely origin for the conductivity of the reported SnSe2 films. This explanation is not likely for Sn2Se3 where the lower average Sn valence would work against Se vacancies.

We report the structure of Sn2Se3 as obtained using first principles structure prediction. We find that the lowest energy structure is a monoclinic P21/m semimetallic phase. This lowest energy structure has disproportionated Sn similar to Sn2S3. The Sn-Se bonds are more covalent than the corresponding bonds in the sulfide. This reflects the lower electronegativity of Se, relative to S. The relative enthalpy of P21/m Sn2Se3 is only 9 meV/atom higher than the ground state consisting of phase separated SnSe (Pnma) and SnSe2 (P3¯m1). We also find a competing non-disproportionated Sn2Se3 phase, slightly higher in energy. This phase is metallic. We find strong anisotropy in the electronic structure with complex shaped Fermi surfaces for electron and hole pockets in the semimetallic phase. The conducting semimetallic and metallic nature of the crystalline phases no doubt underlies the low resistivity of the crystalline material, which makes possible the high resistivity contrast in Sn2Se3 based phase change devices. We note as an aside that conducting phases on the borderline of charge disproportion are potentially of interest as superconductors in analogy with the (K,Ba)BiO3 system,47,48 suggesting that low temperature measurements of crystalline films should be made. The results also provide insights into other aspects of Sn2Se3 as a phase change material. Specifically, the close competition of disproportionated and non-disproportionated Sn may underlie the ready formation of an amorphous phase with low energy switching, while the very weak driving force for phase separation allows for long term stable devices. Experimental investigation of the structural and spectroscopic properties of thin film crystalline Sn2Se3 will be of considerable interest.

This work was supported by the U.S. Department of Energy, Basic Energy Sciences through the Computational Materials Science Program, MAGICS center, Award No. DE-SC0014607. G.X. gratefully acknowledges support from the China Scholarship Council.

1.
S. R.
Ovshinsky
,
Phys. Rev. Lett.
21
,
1450
(
1968
).
2.
S.
Raoux
,
F.
Xiong
,
M.
Wuttig
, and
E.
Pop
,
MRS Bull.
39
,
703
(
2014
).
3.
H.-S. P.
Wong
,
S.
Raoux
,
S.
Kim
,
J.
Liang
,
J. P.
Reifenberg
,
B.
Rajendran
,
M.
Asheghi
, and
K. E.
Goodson
,
Proc. IEEE
98
,
2201
(
2010
).
4.
G. W.
Burr
,
M. J.
Breitwisch
,
M.
Franceschini
,
D.
Garetto
,
K.
Gopalakrishnan
,
B.
Jackson
,
B.
Kurdi
,
C.
Lam
,
L. A.
Lastras
,
A.
Padilla
,
B.
Rajendran
,
S.
Raoux
, and
R. S.
Shenoy
,
J. Vac. Sci. Technol. B
28
,
223
(
2010
).
5.
M.
Wuttig
,
Nature Mater.
4
,
265
(
2005
).
6.
K.
Kim
and
G.
Jeong
,
Microsyst. Technol.
13
,
145
(
2007
).
7.
A. V.
Kolobov
,
P.
Fons
,
A. I.
Frenkel
,
A. L.
Ankudinov
,
J.
Tominaga
, and
T.
Uraga
,
Nature Mater.
3
,
703
(
2004
).
8.
J.
Hegedus
and
S. R.
Elliott
,
Nature Mater.
7
,
399
(
2008
).
9.
N.
Yamada
,
E.
Ohno
,
K.
Nishiuchi
,
N.
Akahira
, and
M.
Takao
,
J. Appl. Phys.
69
,
2849
(
1991
).
10.
I.
Friedrich
,
V.
Weidenhof
,
W.
Njoroge
,
P.
Franz
, and
M.
Wutting
,
J. Appl. Phys.
87
,
4130
(
2000
).
11.
D.-H.
Kang
,
N. Y.
Kim
,
H.
Jeong
, and
B.-K.
Cheong
,
Appl. Phys. Lett.
100
,
063508
(
2012
).
12.
D.
Loke
,
T. H.
Lee
,
W. J.
Wang
,
L. P.
Shi
,
R.
Zhao
,
Y. C.
Yeo
,
T. C.
Chong
, and
S. R.
Elliott
,
Science
336
,
1566
(
2012
).
13.
M.
Hase
,
P.
Fons
,
K.
Mitrofanov
,
A. V.
Kolobov
, and
J.
Tominaga
,
Nat. Commun.
6
,
8367
(
2015
).
14.
X.
Yu
and
J.
Robertson
,
Sci. Rep.
5
,
12612
(
2015
).
15.
K.-M.
Chung
,
D.
Wamwangi
,
M.
Woda
,
M.
Wuttig
, and
W.
Bensch
,
J. Appl. Phys.
103
,
083523
(
2008
).
16.
Y.
Hu
,
X.
Zhu
,
H.
Zou
,
Y.
Lu
,
J.
Xue
,
Y.
Sui
,
W.
Wu
,
L.
Yuan
,
S.
Song
, and
Z.
Song
,
J. Mater. Sci.: Mater. Electron.
26
,
7757
(
2015
).
17.
N. K.
Abrikosov
,
Semiconducting II–VI, IV–VI, and V–VI Compounds
(
Springer
,
2013
).
18.
Y.
Hu
,
S.
Li
,
T.
Lai
,
S.
Song
,
Z.
Song
, and
J.
Zhai
,
J. Alloys Compd.
581
,
515
(
2013
).
19.
L. S.
Palatnik
and
V. V.
Levitin
,
Dokl. Akad. Nauk SSSR
96
,
975
(
1954
).
20.
D. J.
Singh
,
Appl. Phys. Lett.
109
,
032102
(
2016
).
21.
M.
Aguiar
,
R.
Caram
,
M.
Oliveira
, and
C.
Kiminami
,
J. Mater. Sci.
34
,
4607
(
1999
).
22.
L. A.
Burton
and
A.
Walsh
,
J. Phys. Chem. C
116
,
24262
(
2012
).
23.
L. S.
Price
,
I. P.
Parkin
,
A. M.
Hardy
,
R. J.
Clark
,
T. G.
Hibbert
, and
K. C.
Molloy
,
Chem. Mater.
11
,
1792
(
1999
).
24.
G.
Amthauer
,
J.
Fenner
,
S.
Hafner
,
W.
Holzapfel
, and
R.
Keller
,
J. Chem. Phys.
70
,
4837
(
1979
).
25.
Y.
Wang
,
J.
Lv
,
L.
Zhu
, and
Y.
Ma
,
Phys. Rev. B
82
,
094116
(
2010
).
26.
Y.
Wang
,
J.
Lv
,
L.
Zhu
, and
Y.
Ma
,
Comput. Phys. Commun.
183
,
2063
(
2012
).
27.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
28.
G.
Kresse
and
D.
Joubert
,
Phys. Rev. B
59
,
1758
(
1999
).
29.
G.
Kresse
and
J.
Furthmüller
,
Phys. Rev. B
54
,
11169
(
1996
).
30.
D. J.
Singh
and
L.
Nordstrom
,
Planewaves, Pseudopotentials, and the LAPW Method
, 2nd ed. (
Springer
,
Berlin
,
2006
).
31.
P.
Blaha
,
K.
Schwarz
,
G.
Madsen
,
D.
Kvasnicka
, and
J.
Luitz
, WIEN2k, An augmented plane wave+local orbitals program for calculating crystal properties (
2001
).
32.
F.
Tran
and
P.
Blaha
,
Phys. Rev. Lett.
102
,
226401
(
2009
).
33.
G. K.
Madsen
and
D. J.
Singh
,
Comput. Phys. Commun.
175
,
67
(
2006
).
34.
A.
Togo
and
I.
Tanaka
,
Scr. Mater.
108
,
1
(
2015
).
35.
J.
Heyd
,
G. E.
Scuseria
, and
M.
Ernzerhof
,
J. Chem. Phys.
118
,
8207
(
2003
).
36.
K.
Adouby
,
C.
Pérez-Vicente
,
J.
Jumas
,
R.
Fourcade
, and
A. A.
Touré
,
Z. Kristallogr.
213
,
343
(
1998
).
37.
B.-Z.
Sun
,
Z.
Ma
,
C.
He
, and
K.
Wu
,
Phys. Chem. Chem. Phys.
17
,
29844
(
2015
).
38.
S.
Mukhopadhyay
,
J.
Sun
,
A.
Subedi
,
T.
Siegrist
, and
D. J.
Singh
,
Sci. Rep.
6
,
25981
(
2016
).
39.
H.
Sellinschegg
,
S. L.
Stuckmeyer
,
M. D.
Hornbostel
, and
D. C.
Johnson
,
Chem. Mater.
10
,
1096
(
1998
).
40.
L.
Fister
and
D. C.
Johnson
,
J. Am. Chem. Soc.
114
,
4639
(
1992
).
41.
42.
Y.
Li
,
D. J.
Singh
,
M.-H.
Du
,
Q.
Xu
,
L.
Zhang
,
W.
Zheng
, and
Y.
Ma
,
J. Mater. Chem. C
4
,
4592
(
2016
).
43.
A.
Walsh
and
G. W.
Watson
,
J. Phys. Chem. B
109
,
18868
(
2005
).
44.
G. W.
Watson
,
J. Chem. Phys.
114
,
758
(
2001
).
45.
A.
Walsh
and
G. W.
Watson
,
Phys. Rev. B
70
,
235114
(
2004
).
46.
J.
Sun
,
S.
Mukhopadhyay
,
A.
Subedi
,
T.
Siegrist
, and
D. J.
Singh
,
Appl. Phys. Lett.
106
,
123907
(
2015
).
47.
R. J.
Cava
,
B.
Batlogg
,
J. J.
Krajewski
,
R.
Farrow
,
L. W.
Rupp
, Jr.
,
A. E.
White
,
W. F.
Peck
, and
T.
Kometani
,
Nature
332
,
814
(
1988
).
48.
L. F.
Mattheiss
and
D. R.
Hamann
,
Phys. Rev. Lett.
60
,
2681
(
1988
).