Stability and superconductivity properties of metal substituted aluminum diborides (M0.5Al0.5B2)

Computational Materials Science 154 (2018) 234–242

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Stability and superconductivity properties of metal substituted aluminum diborides (M0.5Al0.5B2)

T

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Sezgin Aydin , Mehmet Şimşek Department of Physics, Faculty of Sciences, Gazi University, 06500 Teknikokullar, Ankara, Turkey

A R T I C LE I N FO

A B S T R A C T

Keywords: Density functional theory Magnesium diboride Stability analysis Phonon modes Superconductivity

The stability table and superconducting properties such as screened Coulomb potential, electron-phonon coupling and critical temperature of planar boron sheet included ternary crystalline compounds of MgB2-type M0.5Al0.5B2 (M = Li, Be, Na, Mg, K, Ca, Sc, Ti, V, Y, Zr, Nb, Mo, Tc, Ru, Hf, Ta, W, Re, and Os) have been investigated by first-principles density functional theory calculations with spin-orbit coupling. It is shown that the M = Li, Mg, Ca, Sc, Ti, V, Y, Zr, Nb, Mo, Tc, Hf, Ta, and W-substituted compounds are thermodynamically stable at the ambient conditions. Among them, Nb0.5Al0.5B2 has the lowest cohesive energy while Ti0.5Al0.5B2 has the lowest formation enthalpy. Except for Be0.5Al0.5B2, all compounds are mechanically stable, and fourteen of them are dynamically stable also. So, a dozen of M0.5Al0.5B2 (M = Li, Mg, Ca, Sc, Ti, V, Zr, Nb, Mo, Tc, Hf, and Ta) compounds are satisfying all three stability conditions. The calculated electronic properties show that all structures have metallic character and interestingly the radii of the substituted atoms correlate with surface area of regular B6 hexagons and volumes of metal-boron pyramids. All of them are hard materials, and V0.5Al0.5B2 has the highest semi-empirical microhardness (27 GPa). According to the precise phonon dispersion and electronphonon coupling calculations, Tc’s of the stable compounds are lower than that of MgB2 (39 K), and Tc0.5Al0.5B2 has higher Tc (∼7 K) than the others.

1. Introduction Due to the exclusive property of superconductivity character [1–8], the structures with stacking layers are recently becoming interest [9–17]. One of well-known extraordinary layered structure is MgB2 [1], and its unexpectedly high superconductivity Tc (∼39 K) stimulated the extensive searches on understanding how these kinds of materials behave structurally and electronically [18–29]. Recent observations and calculations showed that boron-based ternary alkali, alkaline-earth and transition metal borides in the form of honeycomb network systems confident the researchers for increasing critical temperature (Tc) [1,9,10,28–31]. However, to develop the superconductivity of binary, ternary, quaternary borides and borocarbides, the impressive results have been recently reported by experiment and theory [28,29,32–35]. For instance, it was shown that hole doping to LiBC, which is a layered semiconductor structure, leads to superconductivity character [16,29,33,35–37]. Moreover, numerous elemental boron allotropes [26,38,39] may gain superconductivity character by compressing [34,40]. And also, a number of honeycomb network systems of metal borides are improving the superconducting characters with doping of 1A group elements such as lithium, sodium, potassium, etc.

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[8,27,41–46]. However, using the knowledge of being the boron p-electrons were dominant at near the Fermi energy in the electronic structure of MgB2, Slusky et al. [47] shown that the addition of electrons to the MgB2 with partial substation of Al for Mg loss the superconductivity. Also, recent experiments and calculated Eliashberg functions showed that it depends very sensitively on concentrations of Al and C [12,48–50]. It has been primarily observed that the superconductivity in Mg1−xAlxB2 decrease with Al doping [51–58], and Tc(x) fall with increasing Al concentration x and diminish for x > 0.5 [59–61]. On the other hand, it was shown that electron-phonon coupling in MgB2 and the strong covalent nature of the σ bands lead the interaction of the bond-stretching modes [23,62–66], and phonon anomalies can be used to predict superconducting Tc for AlB2-type structures [67–69]. In this concept, the structural, electronic band structure, and full phonon dispersion properties were analyzed for the phases of Mg1−xAlxB2 (in the range of 0 < x < 1) within the framework of density-functional theory using the self-consistent virtual-crystal approximation [48,49,70,71]. Moreover, with the Raman [72] and EPR [73] study, it was shown that the optimal composition of the superstructure phase is M0.5Al0.5B2, a superconductor with Tc ∼ 12 K. Bianconi et al. [74] studied the

Corresponding author. E-mail address: [email protected] (S. Aydin).

https://doi.org/10.1016/j.commatsci.2018.08.005 Received 19 June 2018; Received in revised form 1 August 2018; Accepted 2 August 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.

Computational Materials Science 154 (2018) 234–242

S. Aydin, M. Şimşek

Fig. 1(a)). The unit cell contains one formula unit of M0.5Al0.5B2, and the structure includes parallel boron layers which form of interconnected six-member regular honey-comb boron hexagons intercalated with aluminum and other metal atoms (see Fig. 1(a, b)). However, the location of the boron layer are not exactly the middle of metal-aluminum distance. Aluminum and other metal atoms occupy the positions directly above and below respectively on the center of boron hexagons. Our calculated lattice constants of M0.5Al0.5B2 compounds are listed in Table 1. From Table 1, the USPP and NCPP results are in good agreement with maximum difference of 0.04 Å for a-lattice. However, some considerable differences between the USPP and NCPP results in clattice and formation enthalpy are observed for M = Li, Be, K, V, Y, Nb, Mo, Ru, Hf, and Ta. M = Y, Hf, and Ta compounds have higher differences than the others. Besides, the lattice constants of Mg0.5Al0.5B2 are a = 3.043 Å, c = 6.791 Å with NCPP, and a = 3.038 Å, c = 6.773 Å with USPP respectively. The calculated Wyckoff positions of the atoms are following: Al at 1a (0.0, 0.0, 0.0), Mg at 1b (0.0, 0.0, 1/2), and B at 4 h (1/3, 2/3, 0.233) for both USPP and NCPP. These results agree well with the experimental results at 16 K [54] of the lattice parameters, a = 3.044 Å, c = 6.712 Å, and Wyckoff positions of boron atoms B (1/ 3, 2/3, 0.241). It is seen from Table 1 that M = Y-substituted compound has the highest a-lattice, while Be-substituted compound has the smallest one. And, M = K-substituted compound has the highest c-lattice, while Be-substituted compound has the smallest one. Significantly, while the lattice parameters of the hexagonal AlB2-type unit cell of M0.5Al0.5B2 compounds depend on the substituted metal atom, the unit cell volume changes very little through a group and a period of Periodic Table. As seen from Table 1, it decreases from left to right across a period, and increases from top to bottom in a group, these results are important for understanding and altering the mechanical response of the novel M0.5Al0.5B2 material. Furthermore, the transition metal compounds can have considerable spin-orbit coupling (SOC) effects in some cases [101,102]. Therefore, new high precise geometry optimizations with SOC are performed by using Quantum Espresso. It is not observed any remarkable deviation in the lattice parameters with SOC, except for small changing in fractional coordinate z of boron atoms (zB, see Table 2).

superconducting properties of Mg1−xAlxB2 and it was pointed out that by changing the Al/Mg, the boron σ bands cross Fermi surface and they have also indicated the origin of Tc amplification in Al1−xMgxB2 from 5 K in AlMgB4 to 39 K in MgB2. Superconducting and structural properties of a Al1−xMgxB2 series were studied systematically and superconducting gaps variation as a function of Al substance investigated experimentally [69,75–81], and superconducting phenomena described within a multiband version into intraband and interband contributions [82–85]. In this study, the structural, mechanical and superconducting properties of MgB2-type ternary M0.5Al0.5B2 (M = Li, Be, Na, Mg, K, Ca, Sc, Ti, V, Y, Zr, Nb, Mo, Tc, Ru, Hf, Ta, W, Re, and Os) compounds are systematically studied by using first-principles of density functional theory with and without spin-orbit coupling. Detailed stability analysis which includes thermodynamic, mechanic and dynamic properties is presented. Possible correlations are inquired between the calculated results and the available literature such as atomic size, data of MgB2 and Mg0.5Al0.5B2. 2. Computational details The first-principles plane-wave pseudopotential calculations are performed by using CASTEP (CAmbridge Serial Total Energy Package) [86]. In the calculations, atomic coordinates and unit cell parameters are fully relaxed with BFGS (Broyden, Fletcher, Goldfarb, Shanno) scheme [87]. The Vanderbilt ultrasoft (USPP) [88] and norm-conserving (NCPP) pseudopotentials [89] are used to model the ion-electron interactions, and exchange-correlation effects are treated within the generalized gradient approximation (GGA) [90] by the Perdew-BurkeErnzerhof functional (PBE) [91]. The plane wave cut-off energy of 400 is employed for USPP. 13 × 13 × 6 k-points set generated by Monkhorst-Pack (MP) scheme is used (separation of 0.03 Å−1). During the calculations, the following convergence criteria are applied: (i) the maximum ionic Hellman-Feynman force on an atom is below 0.01 eV/ Å, (ii) maximum displacement is below 5.0 × 10−4 Å, (iii) maximum energy change was below 5.0 × 10−6 eV/atom, and (iv) maximum stress is below 0.02 GPa. SCF tolerance is 5 × 10−7 eV/atom, and the maximum strain amplitude in the elastic constants calculations is 0.003. And, Mulliken analysis is executed on the 2 × 2 × 1 supercell. Finally, after the successfully completed and well-converged geometry optimizations with norm-conserving pseudopotentials, the linear response theory [92] is used to calculate phonons/dispersion curves with the cut-off energy of 750 eV. q-vector grid spacing for interpolation is set to 0.05 Å−1, the convergence tolerance for the force constants is 1 × 10–5 eV. Å−2, and the separation between neighboring q-vectors in the MP grid to calculate dispersion curves is 0.01 Å−1. Here, it is remembered that the MP grid separation for the q-vectors is crucial to get accurate dispersion curves [67,68,93]. Additionally, the superconducting properties such as electron-phonon coupling constants (λ), critical temperature (Tc), etc. are calculated by Quantum ESPRESSO package [94,95] with Goedecker-Hartwigsen-Hutter-Teter type [96,97] NCPP for the same functional. In order to overcome spin-orbit coupling (SOC) effects for the transition metal elements, fully relativistic optimized norm-conserving pseudopotentials are generated by using ONCVPSP code [98,99] (http://www.mat-simresearch.com) and the atomistic input files are taken from ABINIT [100] (https://www.abinit. org).

3.2. Thermodynamic stability Furthermore, to obtain the phase stability of AlB2-type hexagonal structures of M0.5Al0.5B2 compounds, it is carried out the calculations of the formation enthalpy (ΔH) and also cohesive energy (Ecoh) respectively defined,

ΔH (eV/atom) = [EFinal−(EAl + EM + 4 × EB )]/6, iso iso Ecoh (eV/atom) = [EFinal−(EAl + EM + 4 × EBiso)]/6.

where EFinal is total energy of the unit cell and EAl, EM , and EB are energies of an aluminum, metal and boron atoms, respectively, which are taken from ground state phases of them (fcc for Al, and α-B12 for boron iso iso , EM , and EBiso are the isolated single atom energies of alu[103]). EAl minum, metal and boron atoms, respectively. As known, the negative value of the formation enthalpy indicates the thermodynamic stability of the structure, which depending on the chemical equilibrium with its environment. Thermodynamically stable structures can be synthesized under appropriate conditions easily. The reflector of the energetic stability of the structure is cohesive energy, which indicates the gained energy the structure by arranging the atoms in given crystalline phases, or the required energy of the structure to break all of the bonds of its constituent atoms. The calculated values for formation enthalpy and cohesive energy are also listed in Table 1 with USPP and NCPP. It is shown that USPP results are consistent with NCPP results, and all compounds are energetically stable due to the negative cohesive energies. From schematic stability analysis shown in Fig. 2, M = Be, Na,

3. Results and discussion 3.1. Crystal structure In this study, the X-ray powder diffraction data [54] for Mg0.5Al0.5B2 is used to begin the simulations and to design the crystal structure of M0.5Al0.5B2 compounds. The crystal structure resembles AlB2-type (or MgB2) hexagonal structure of P6/mmm space group (see 235

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Fig. 1. (a) The unitcell, (b) 2 × 2 × 1 supercell of Mg0.5Al0.5B2, (c) AlB6 (pink colored) and MB6 (green colored) hexagonal pyramids in the structure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Calculated structural parameters, formation enthalpies (ΔH) and cohesive energies (Ecoh) with USPP and NCPP of M0.5Al0.5B2 compounds. M

Li Be Na Mg K Ca Sc Ti V Y Zr Nb Mo Tc Ru Hf Ta W Re Os

a (Å)

ΔH (eV/atom)

V (Å3)

c (Å)

Ecoh (eV/atom)

USPP

NCPP

USPP

NCPP

USPP

NCPP

USPP

NCPP

USPP

NCPP

2.997 2.964 3.031 3.038 3.049 3.098 3.062 2.994 2.953 3.136 3.057 3.005 2.978 2.972 2.992 3.058 3.017 2.972 2.967 2.983

2.993 2.950 3.030 3.043 3.059 3.101 3.062 2.990 2.955 3.102 3.058 3.008 2.980 2.977 3.003 3.021 2.975 2.964 2.962 2.978

6.715 6.178 7.429 6.773 8.090 7.267 6.797 6.624 6.525 7.331 6.977 6.844 6.758 6.686 6.591 6.976 6.915 6.823 6.756 6.691

6.637 6.107 7.455 6.791 8.118 7.267 6.797 6.631 6.583 7.036 7.019 6.888 6.846 6.717 6.648 6.817 6.781 6.843 6.752 6.646

52.240 46.989 59.117 54.147 65.137 60.413 55.191 51.410 49.268 62.451 56.486 53.509 51.904 51.140 51.110 56.485 54.522 52.208 51.498 51.560

51.492 46.030 59.277 54.454 65.766 60.518 55.191 51.337 49.792 58.626 56.830 53.957 52.669 51.543 51.924 53.873 51.987 52.055 51.318 51.057

−0.117 0.031 0.015 −0.132 0.103 −0.157 −0.444 −0.492 −0.359 −0.212 −0.458 −0.394 −0.179 −0.054 0.044 −0.398 −0.320 −0.092 0.067 0.218

−0.104 0.071 0.044 −0.108 0.158 −0.133 −0.404 −0.447 −0.306 −0.405 −0.421 −0.327 −0.139 −0.055 0.022 −0.492 −0.380 −0.061 0.095 0.233

−5.549 −5.699 −5.381 −5.528 −5.190 −5.600 −6.337 −6.754 −6.916 −6.086 −6.801 −7.225 −7.146 −6.932 −6.659 −6.733 −6.989 −7.142 −7.045 −6.771

−5.540 −5.777 −5.283 −5.458 −5.114 −5.564 −6.286 −6.672 −6.801 −6.449 −6.754 −7.094 −6.988 −6.823 −6.494 −6.905 −7.243 −7.113 −6.964 −6.764

K, Ru, Re, and Os-substituted compounds are thermodynamically unstable due to the positive formation enthalpy, while the others are stable due to the negative formation enthalpy. Among the stable

compounds, Ti-substituted one has the lower formation enthalpy, and Tc- and W-substituted compounds have higher than the others, which are thermodynamically less stable material.

Table 2 Calculated density of states at Fermi level (N(EF), electrons/eV), electron-phonon coupling constants (λ), logarithmic average frequencies (〈wlog 〉, in K), and critical temperatures (Tc, K), screened Coulomb potentials ( μ∗) and fractional coordinate z for the boron atoms (zB) of M0.5Al0.5B2 compounds by using Quantum Espresso with spin-orbit (SOC) and without spin-orbit (NSOC). λ

N(EF)

Li Mg K Ca Sc Ti V Zr Nb Mo Tc Hf Ta

μ∗

Tc

〈wlog 〉

zB

NSOC

SOC

NSOC

SOC

NSOC

SOC

NSOC

SOC

NSOC

SOC

NSOC

SOC

6.90 8.59 9.22 11.50 8.67 8.61 8.17 10.56 10.37 10.88 10.91 9.54 9.27

– – – – 10.08 9.97 9.52 11.50 11.55 11.41 11.59 10.21 10.14

0.64 0.62 0.55 0.66 0.38 0.34 0.33 0.52 0.50 0.66 0.80 0.48 0.47

– – – – 0.52 0.45 0.45 0.58 0.54 0.65 0.80 0.53 0.51

319.48 451.56 416.35 377.05 652.16 802.61 794.44 407.11 407.91 323.54 283.23 475.64 443.51

– – – – 363.09 439.71 293.78 423.39 473.72 400.02 311.48 473.87 484.95

2.28 2.16 1.15 3.23 0.03 0.00 0.00 0.96 0.64 3.93 6.85 0.80 0.59

– – – – 0.87 0.24 0.17 2.19 1.48 4.46 7.49 1.58 1.26

0.20 0.21 0.20 0.20 0.19 0.19 0.18 0.19 0.19 0.18 0.18 0.18 0.18

– – – – 0.19 0.19 0.19 0.19 0.19 0.18 0.18 0.18 0.18

0.2509 0.2336 0.1918 0.2158 0.2336 0.2428 0.2463 0.2317 0.2381 0.2397 0.2385 0.2336 0.2375

– – – – 0.2349 0.2422 0.2452 0.2314 0.2377 0.2392 0.2389 0.2322 0.2349

236

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S. Aydin, M. Şimşek

Fig. 2. Schematic representation of the thermodynamic, mechanic and dynamic stabilities of M0.5Al0.5B2 compounds.

(AlB6)), and surface area of B6 hexagon (S(B6)) (see Fig. 1(c)). The surface area of B6 is calculated as S (B6) = 3 × (dB − B )2 /1.154 . The volumes of MB6 and AlB6 hexagonal pyramids are given by

3.3. Bonding nature and the effect of atomic size The characteristic bond lengths as some particular distances can be used for alternative structural analysis, and their Mulliken bond overlap populations (MP) can give useful information about the actual binding character. Calculated bond lengths, their Mulliken populations, and atomic charges are listed in Table S1 (see Supplemental Material [104]). The bond lengths of BeMg, BeAl and BeB distances are 2.520 Å, 2.359 Å, and 1.754 Å with USPP for Mg0.5Al0.5B2, and of 2.529 Å, 2.360 Å, 1.757 Å with NCPP, respectively, agree well with the experimental results [54] of 2.471 Å, 2.390 Å, and 1.757 Å, respectively. However, these lengths are comparable with MgeB (2.505 Å), and BeB (1.781 Å) lengths in MgB2 [105]. Also, it is seen from Table S1 that the BeB bonds forming the planar boron sheet are shorter than the other BeM and BeAl bonds for all compounds. Except for V-substituted compound, BeM bonds are longer than the BeAl bonds. Among the thermodynamic stable compounds, the Y-substituted compound has highest lengths of BeB, BeAl, and BeM bonds, while V-substituted compound has the smallest ones. However, the lengths of BeB bonds and BeAl bonds decrease from left to right across a period, which is caused by decreasing in radius of the substituted metal atom and metallic character. As seen from Table S1, according to the Mulliken analysis, BeB bonds have strong covalent nature in all structures due to the higher positive population values than the others, and BeAl and BeM bonds have ionic character due to the population values close to zero. Also, the BeM bonds are more ionic than the BeAl bonds. Besides, the ionic binding nature of the compounds can also be detected from the calculated atomic charges individually. Then, it can be concluded that there is a charge transfer supporting ionic nature between boron atoms and Al, and other substituted metal atoms. Generally, the substituted metal atoms and aluminum atoms behave as a cation, which are the ions with positive charges, and boron atoms behave as an anions with a negative charges for all compounds. It is useful to remember here that metallic character of an atom relates to electron-losing ability (cation character), while nonmetallic character relates to electrongaining ability (anion character). However, atomic charges of the transition metals decrease from left to right across a period (see Table S1). And, the atomic charges of the metals in the thermodynamically unstable compounds are lower than those of stable ones. Alternative structural and bonding analysis can be performed by using the volumes of MB6 and AlB6 hexagonal pyramids (V(MB6) and V

V (MB6) =

1 × S (B6) h1 3

and V (AlB6 ) =

1 × S (B6) h2 3

where h1 and h2 are heights of MB6 and AlB6 pyramids, respectively, and they are defined by h1 = c × (0.5−zB ) and h2 = c × zB (c represents lattice parameter c in Å, zB is fractional coordinate z of boron atoms for 4 h position). The calculated V(MB6) and V(AlB6) of M0.5Al0.5B2 compounds are shown in Fig. 3(a) (it is note that V(MB6) and V(AlB6) are equivalent in MgB2). M = Na, K, Ca, Y, and Hf-substituted compounds have higher V(MB6) than that of MgB2, while the others have lower. And, the changing in V(MB6) affects V(AlB6) due to the incorporation of metal atoms into the boron layer. Moreover, the calculated S(B6) are demonstrated in Fig. 3(b) with the atomic radii of the metal elements. It is clearly seen that surface area and volume of MB6 pyramid correlates with the atomic radii/size. Also, this interesting trend for surface area and volumes can be interpreted as a measure of interactions between metal atom and boron layer, because the volumes will become lower (higher) in case of strong (weak) bonding with the stacking effect originated from atomic radii. It can be concluded that there are stronger interactions between metal layer and boron layer than those of MgB2 for M = Li, Be, Ti, V, Nb, Mo, Tc, Ru, Ta, W, Re, and Os-substituted compounds. 3.4. Mechanical stability Furthermore, the energetic parameters such as formation enthalpy and cohesive energy related to energetic/thermodynamic stability are not enough to announce the stability of a given compound. In addition to energetic analysis above, the mechanical and dynamic stabilities of the compounds must be examined. There are five independent elastic constants Cij (C11, C33 , C44 , C12 , and C13 ) for the hexagonal crystalline phase which are determined by using the stress-strain method. For the mechanical stability, the criteria [106] are given by, C44 > 0 , 2 C11 > |C12 |, (C11 + C12) C33 > 2C13 . The calculated elastic constants and associated mechanical properties are listed in Table S2. It is concluded from Table S2 that except for M = Be-substituted compound, the other compounds satisfy the mechanical stability criteria, namely they are mechanically stable. Among the elastic constants, C11 and C33 values are 237

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Fig. 3. (a) Volumes of MB6 and AlB6 pyramids V(MB6) and V(AlB6), (b) surface area of B6 hexagon S(B6) for M0.5Al0.5B2 compounds and atomic radii for the elements. Blue dashed lines represent data of MgB2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

higher than the others. This implies that the actual binding/interactions along x- and z-axis are stronger than the other directions. It is noted here that if the substituted metal is in the nd3 group (V, Nb, and Ta) the C11 values are higher than the others. Moreover, the mechanical properties such as bulk modulus (B), shear modulus (G) and Young’s modulus (E) are calculated as the functions of these elastic constants within Reuss-Voigt-Hill approximation [106]. Reuss (R, lower limit) and Voigt (V, upper limit) bulk modulus and shear modulus of a hexagonal system are given as,

BV =

1 [2(C11 + C12) + 4C13 + C33], 9

GV =

1 [M + 12C44 + 12C66], 30

and the results are listed in Table S3. The Poisson ratio of covalent compounds are generally small and around 0.1. For ionic compounds, it reaches 0.25, and 0.33 for metallic compounds [107]. The M = Li, Na, Mg, K, Ca (alkali or earth alkali metals), Ti, V, Nb, and Ta-substituted compounds have the Poisson ratio close to 0.2, and it can be said that these compounds have covalent and ionic bonding nature together. And, the M = Sc, Zr, Mo, Hf, and W-substituted ones have Poisson ratio close to 0.25, then the ionic component in these compounds are more dominant than the covalent character. The M = Y, Tc, and Re-substituted compounds have dominant ionic and metallic nature. Also, the M = Ru- and Os-substituted compounds have higher Poisson ratio implying higher metallic component. Moreover, AU = 0 for a locally isotropic single crystal, and the larger value means the larger anisotropy [108,109]. It is seen from Table S3 that M = K, Y, Tc, and Os-substituted compounds have a higher anisotropic nature while M = Ca, V, Nb, Mo, and Ta-substituted compounds have a lower anisotropic nature. Furthermore, zero value for AB and AG means isotropic crystal, and value of 100% means the maximum anisotropy [110]. Generally, it is clear from Table S3 that AG values are larger than AB values for all M0.5Al0.5B2 compounds, indicating that the shear anisotropy or directional shear effects of the compounds are more dominant than the compression (or bulk) anisotropy. The M = Li-substituted compound has the highest AB while M = Y-substituted compound has the lowest one. And, the M = Ossubstituted compound has the highest AG while M = Ta-substituted compound has the lowest one. The other crucial mechanical classification is Pugh’s indicator (or G/B ratio) [76], which is used to define brittle/ductile behavior of the material, and it is related to the compressibility and fracture ability of the material. If G/B < 0.5, the material behaves in a ductile manner, if G/B > 0.5, the material demonstrates brittleness. As seen from Table S3, among the mechanically stable compounds, the M = Y, Tc, Ru, and Os-substituted compounds have ductile nature, while the others are brittle. And, Os.5Al0.5B2 is very ductile due to the lowest G/B ratio of 0.23, and Li0.5Al0.5B2 is very brittle with the highest G/B = 0.83. To present a better understanding of the mechanical properties of alkali, earth alkali and transition metal substituted aluminum diborides (M0.5Al0.5B2), especially heavy 4d and 5d metal compounds, which could be potential superhard and ultra-incompressible systems [111], hardness properties of the compounds are investigated. The hardnesses of the M0.5Al0.5B2 compounds are calculated by three different

BR = C 2/ M ,

GR =

5 2 (C C44 C66)/[3BV C44 C66 + C 2 (C44 + C66)], 2

B = (BV + BR)/2,

G = (GV + GR)/2,

E = 9BG /(3B + G )

2 C 2 = (C11 + C12) C33−2C13 where the abbreviations are , M = C11 + C12 + 2C33−4C13 and c66 = (C11−C12)/2 . Among the thermodynamically and mechanically stable compounds, M = Tc and W-substituted compounds are larger bulk modulus than the others, while M = Ta, Nb, and V-substituted compounds are larger shear and Young's moduli than the others. Namely, M = Tc and W-substituted compounds have larger resistance to volume deviation, while M = Ta, Nb, and Vsubstituted compounds have larger resistance to shear and directional strain deformations, and shape deformation. Additionally, employing the moduli above, Poisson's ratio (v) [107], universal anisotropy index (AU) [108,109], percentage bulk (AB) and shear (AG) anisotropy factors [110] are calculated by following expressions,

v = (3B−2G )/[2(3B + G )],

AU = 5

GV B + V −6 ⩾ 0, GR BR

AB =

BV −BR × 100, BV + BR

AG =

GV −GR × 100, GV + GR 238

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S. Aydin, M. Şimşek

3.6. Superconducting properties

theoretical models, because different models can generate reasonably different results due to the different physical background, concept, and parameters, especially in the absence of any experimental result. So, the various calculations can be useful to remove some contradictions. In this study, to determine the hardness of the compounds, Xue method [112] based on electronegativity, Simunek method [113,114] based on bond strength and Chen method [115] based on the relation between bulk and shear moduli are used, and their average which is including different physical properties spreading in a wide range can be used as a good approximation. The calculated hardness values for the thermodynamically and mechanically stable compounds are also given in Table S3. It is seen from Table S3 that M = V, Nb, and Ta-substituted compounds have higher hardness than the others, among the compounds, while K-substituted one has the lowest hardness. And, the alkali-metal-substituted compounds have less hardness while the transition-metal-substituted ones have higher hardness. Also, among the transition metal-substituted compounds, the M = Y-substituted compound has the lowest hardness. Because of nature of the hardness methods of Xue [112] and Simunek methods [113,114], the hardness increases from left to right in a period, and generally, as seen from Table S3 average hardness decreases from top to bottom in a group.

Furthermore, the one of main property of aluminum metal borides (like M0.5Al0.5B2 compounds in this study) which have a metallic character with covalent B-B bonds is superconductivity [6,47,74,116]. Thus, the logarithmic average frequency 〈wlog 〉, the electron-phonon coupling constants (λ) and critical temperature (Tc) of M0.5Al0.5B2 compounds are calculated. To determine the superconducting temperature, McMillan equation [117] is used,

Tc =

〈wlog 〉 1.2

−1.04(1 + λ ) ⎤ exp ⎡ ∗ ∗ ⎢ λ ⎣ (1−0.62μ )−μ ⎥ ⎦

where μ∗ is Coulomb pseudopotential (or screened Coulomb term), and can be approximated by [118]

μ∗ ≈ 1/ ln (EF / kB θD ) where EF is Fermi energy, kB is Boltzmann constant, and θD is Debye temperature [119]. The calculated results for the dynamically stable compounds are listed in Table 2 with and without spin-orbit coupling. It is note here that the screened Coulomb potential is one of the key parameters to calculate Tc. Tc is highly sensitive to deviations of μ∗. It is shown from Table 2 that μ∗ changes on the range of 0.18–0.19 for the transition metal compounds, while it changes on the range of 0.20–0.21. Among the dynamically stable compounds, V0.5Al0.5B2 has the smallest λ and Tc, while Tc0.5Al0.5B2 has the highest ones, and its λ is very close to 0.73 of MgB2 [64]. It is a remarkable observation that the compounds which have lower value of N(EF) and higher value 〈wlog 〉 have lower critical temperatures, while Tc0.5Al0.5B2 has higher value of N(EF) and lower value of 〈wlog 〉. Namely, increasing metallic character and the electron-phonon coupling, and decreasing vibration frequencies can produce high critical temperature in M0.5Al0.5B2 compounds. But, it is not forgotten that due to the effects of substituted aluminum [120] and substituted metal atoms on the metallicity and vibrational nature of the compound, all of them have lower Tc than that of MgB2 (∼39 K). However, it was already shown that aluminum doping on the magnesium site cause to loss of superconductivity in Al1−xMgxB2 structures [54,69,72]. Then, in the scope of this study, substituted aluminum and substituted metal atoms can cause to decrease or loss superconductivity with the new existing phonon modes (E1g, A1g, B1u, and E2u), which are different from those of MgB2. These modes appear the existence of the second metal layer, and then, with the changing of the actual interactions between the intrinsic boron layer and metal layers. At the same time, it is another significant observation that electron-phonon couplings of E1g, E1u, and A2u modes for all compounds are higher than the others (see Fig. 4). Differently, E2g mode has higher λ values in MgB2, and it is responsible for the superconductivity. Here, it can be said that these modes cause to break down the vibrational harmony between the metal layers, and also the mutual parallel vibrations between the metal layers and boron layers. On the other hand, the existence of superconductivity in AlB2-type structures is related to E2g mode anomalies in the vicinity of Γ point, and critical temperature can be approximated by the size of anomaly [67,68]. The calculated phonon dispersion curves promote this idea, and this-type anomaly is observed in the M = Tc, Sc, Li, Mg, and Ca-substituted compounds which have relatively higher Tc (see Fig. S1). Additionally, the effects of spin-orbit coupling on the dynamic properties for transition metal substituted compounds can be analyzed from Table 2. It is arise as a general trend that N(EF), λ and Tc are increasing in presence of spin-orbit coupling. And, 〈wlog 〉 is increasing for the relative heavy transition metals Zr, Nb, Mo, Tc, Hf, Ta, conversely, 〈wlog 〉 is decreasing for the light transition metals Sc, Ti, and V.

3.5. Dynamic stability and phonon modes As the final step of the stability analysis, the dynamic stabilities of the compounds are examined, and the phonon dispersion curves are calculated. The results are presented in Fig. S1. It is precisely seen that the M = Be, Na, Y, Ru, W, Re and Os-substituted compounds are dynamic unstable due to the imaginary phonon frequency, while the others are stable. However, it is interesting that if the structure involves 4d-second transition elements such as yttrium (4d1, Y) and ruthenium (4d7, Ru), and also 5d-second transition elements such as tungsten (5d4, W), rhenium (5d5, Re), and osmium (5d6, Os), it is becoming dynamically unstable, but molybdenum (4d5, Mo), and technetium (4d5, Tc) become dynamically stable. The more detailed effects of 4d-5d electrons in AlB2-type metal diborides can be found in Ref. [111]. And, to present a complete visualization for the stability of the M0.5Al0.5B2 compounds, all stability cases are summarized in Fig. 2. To give a detailed vibrational analysis utilizing the calculated phonon spectra, the phonon modes (3A2u + 3E1u + E1g + A1g + B1u + B2g + E2g + E2u) of MgB2-type M0.5Al0.5B2 compounds appeared at Γ point are shown in Fig. 4 with those of MgB2, and the corresponding atomic vibrations are depicted as vectors. The frequencies of the phonon modes at Γ are listed in Table S4. It is concluded from the accurate vibrational analysis that the origin of the dynamic instability in M = Be, Ru, Re, and Os-substituted compounds is nondegenerate E1u phonon mode with negative frequency. For the remaining dynamically unstable compounds, the origin of the instability is studied by considering the stable ones. It is observed from Table S4 that the vibration frequencies of the unstable M = Na-substituted compound has the same trend with stable M = K-substituted one, but E2g mode breaks this trend. Otherwise, the vibration frequencies of the unstable M = W-substituted compound has the same trend with stable M = Tc-substituted one, but E1g mode breaks this trend. Also, the vibration frequencies of the unstable M = Y-substituted compound has the same trend with stable M = Ca-substituted one, but A2u(III) mode breaks this trend. Therefore, E1u, A2u, E1g and E2g modes play an essential role in the stability of the compounds. Besides, it can be seen from Table S4 that the calculated frequencies of E1u(III), E1g, A1g, and E2g modes at Γ are higher than those of Mg0.5Al0.5B2, and the others are smaller. Moreover, for the stable compounds, E1g, A1g, and E2g modes are Raman active, while A2u and E1u modes are IR active. Hence, IR active modes display a dipole moment that complies with an inner macroscopic field.

4. Conclusion In conclusion, the stability characteristics and superconductivity 239

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Fig. 4. Vibrational modes for Mg0.5Al0.5B2 compound with those of MgB2 in parenthesis.

appearing new phonon modes (E1g, A1g, B1u, and E2u), due to the changing of actual interactions between boron and metal layers in the structure.

properties of M0.5Al0.5B2 (M = Li, Be, Na, Mg, K, Ca, Sc, Ti, V, Y, Zr, Nb, Mo, Tc, Ru, Hf, Ta, W, Re, and Os) compounds have been studied by first-principles density functional calculations. With the thermodynamic, mechanic and dynamic stability analysis, it is concluded that M = Li, Mg, Ca, Sc, Ti, V, Zr, Nb, Mo, Tc, Hf, and Ta-substituted compounds are stable. In all structures, the actual binding properties consist of the covalent, ionic and metallic components, and including dominant metallic components are unstable due to the higher Poisson’s ratio. Spin-orbit coupling is an important major if substituted atom is a transition metal to compare the alkali or earth-alkali atoms. This difference confirms the effect of 4d- and 5d-electrons explicitly. Furthermore, the radius of substituted metal atoms correlates with surface area of boron layer. Comparing the results with those of MgB2, it is seen that the existence of the second metal layer plays an essential role on the structural and mechanical properties, and stability characteristics. Finally, this metal layer causes to change the actual binding nature and vibrational harmony between the intrinsic boron layer and metal layers in the structure. Therefore, it has a considerable effect on the electronic and superconducting properties such as existing new phonon modes. The origin of the decreasing or losing of superconductivity in this-type materials family is break down of E2g-mechanism in MgB2, worsening electron-phonon couplings, and the

Author contribution All authors contribute equally in all processes of the work.

Acknowledgment All first-principles calculations were performed by high-performance computing center (HPCC) at Gazi University. Also, the some numerical calculations including superconducting properties with spinorbit coupling (SOC) reported in this paper were partially performed at TUBITAK ULAKBIM, High Performance and Grid Computing Center (TRUBA resources).

Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.commatsci.2018.08.005. 240

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References

K. Inumaru, R.J. Cava, Phys. Rev. Lett. 87 (2001) 037001. [53] J.Y. Xiang, D.N. Zheng, J.Q. Li, L. Li, P.L. Lang, H. Chen, C. Dong, G.C. Che, Z.A. Ren, H.H. Qi, H.Y. Tian, Y.M. Ni, Z.X. Zhao, Phys. Rev. B 65 (2002) 214536. [54] S. Margadonna, K. Prassides, I. Arvanitidis, M. Pissas, G. Papavassiliou, A.N. Fitch, Phys. Rev. B 66 (2002) 014518. [55] A. Bianconi, D. Di Castro, S. Agrestini, G. Campi, N.L. Saini, A. Saccone, S. De Negri, M. Giovannini, J. Phys.: Condens. Matter 13 (2001) 7383–7390. [56] B. Renker, K.B. Bohnen, R. Heid, D. Ernst, H. Schober, M. Koza, P. Adelmann, P. Schweiss, T. Wolf, Phys. Rev. Lett. 88 (2002) 067001. [57] L.D. Cooley, A.J. Zambano, A.R. Moodenbaugh, R.F. Klie, J.C. Zheng, Y.M. Zhu, Phys. Rev. Lett. 95 (2005) 267002. [58] R.F. Klie, J.C. Zheng, Y. Zhu, A.J. Zambano, L.D. Cooley, Phys. Rev. B 73 (2006) 014513. [59] P. Postorino, A. Congeduti, P. Dore, A. Nucara, A. Bianconi, D. Di Castro, S. De Negri, A. Saccone, Phys. Rev. B 65 (2002) 020507. [60] H. Luo, C.M. Li, H.M. Luo, S.Y. Ding, J. Appl. Phys. 91 (2002) 7122–7124. [61] S.V. Barabash, D. Stroud, Phys. Rev. B 66 (2002) 012509. [62] A.Y. Liu, I.I. Mazin, J. Kortus, Phys. Rev. Lett. 87 (2001) 087005. [63] Y. Kong, O.V. Dolgov, O. Jepsen, O.K. Andersen, Phys. Rev. B 64 (2001) 020501. [64] K.P. Bohnen, R. Heid, B. Renker, Phys. Rev. Lett. 86 (2001) 5771–5774. [65] A.Q.R. Baron, H. Uchiyama, S. Tsutsui, Y. Tanaka, D. Ishikawa, J.P. Sutter, S. Lee, S. Tajima, R. Heid, K.P. Bohnen, Physica C 456 (2007) 83–91. [66] A. Floris, A. Sanna, M. Luders, G. Profeta, N.N. Lathiotakis, M.A.L. Marques, C. Franchini, E.K.U. Gross, A. Continenza, S. Massidda, Physica C 456 (2007) 45–53. [67] J.A. Alarco, P.C. Talbot, I.D.R. Mackinnon, PCCP 17 (2015) 25090–25099. [68] I.D.R. Mackinnon, P.C. Talbot, J.A. Alarco, Comput. Mater. Sci. 130 (2017) 191–203. [69] O. de la Pena, A. Aguayo, R. de Coss, Phys. Rev. B 66 (2002) 012511. [70] S. Brutti, G. Gigli, J. Chem. Theory Comput. 5 (2009) 1858–1864. [71] O. de la Peña, A. Aguayo, R. de Coss, Phys. Rev. B 66 (2002) 012511. [72] J.Q. Li, L. Li, F.M. Liu, C. Dong, J.Y. Xiang, Z.X. Zhao, Phys. Rev. B 65 (2002) 132505. [73] A. Bateni, E. Erdem, S. Repp, S. Weber, M. Somer, Appl. Phys. Lett. 108 (2016) 202601. [74] A. Bianconi, S. Agrestini, D. Di Castro, G. Campi, G. Zangari, N.L. Saini, A. Saccone, S. De Negri, M. Giovannini, G. Profeta, A. Continenza, G. Satta, S. Massidda, A. Cassetta, A. Pifferi, M. Colapietro, Phys. Rev. B 65 (2002) 174515. [75] R. Schneider, A.G. Zaitsev, O. De la Pena-Seaman, R. de Coss, R. Heid, K.P. Bohnen, J. Geerk, Phys. Rev. B 81 (2010) 054519. [76] P. Szabo, P. Samuely, J. Kacmarcik, T. Klein, J. Marcus, D. Fruchart, S. Miraglia, C. Marcenat, A.G.M. Jansen, Phys. Rev. Lett. 87 (2001) 137005. [77] M. Iavarone, G. Karapetrov, A.E. Koshelev, W.K. Kwok, G.W. Crabtree, D.G. Hinks, W.N. Kang, E.M. Choi, H.J. Kim, H.J. Kim, S.I. Lee, Phys. Rev. Lett. 89 (2002) 187002. [78] H. Schmidt, J.F. Zasadzinski, K.E. Gray, D.G. Hinks, Phys. Rev. Lett. 88 (2002) 127002. [79] R.S. Gonnelli, D. Daghero, G.A. Ummarino, V.A. Stepanov, J. Jun, S.M. Kazakov, J. Karpinski, Phys. Rev. Lett. 89 (2002) 247004. [80] J. Karpinski, N.D. Zhigadlo, G. Schuck, S.M. Kazakov, B. Batlogg, K. Rogacki, R. Puzniak, J. Jun, E. Muller, P. Wagli, R. Gonnelli, D. Daghero, G.A. Ummarino, V.A. Stepanov, Phys. Rev. B 71 (2005) 174506. [81] S.A. Kuzmichev, T.E. Shanygina, S.N. Tchesnokov, S.I. Krasnosvobodtsev, Solid State Commun. 152 (2012) 119–122. [82] F. Bernardini, S. Massidda, Phys. Rev. B 74 (2006) 014513. [83] A.A. Golubov, J. Kortus, O.V. Dolgov, O. Jepsen, Y. Kong, O.K. Andersen, B.J. Gibson, K. Ahn, R.K. Kremer, J. Phys.: Condens. Matter 14 (2002) 1353–1360. [84] G. Profeta, A. Continenza, S. Massidda, Phys. Rev. B 68 (2003) 144508. [85] J. Geerk, R. Schneider, G. Linker, A.G. Zaitsev, R. Heid, K.P. Bohnen, H. von Lohneysen, Phys. Rev. Lett. 94 (2005) 227005. [86] S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.J. Probert, K. Refson, M.C. Payne, Z. Kristallogr. 220 (2005) 567–570. [87] B.G. Pfrommer, M. Cote, S.G. Louie, M.L. Cohen, J. Comput. Phys. 131 (1997) 233–240. [88] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892–7895. [89] D.R. Hamann, M. Schlüter, C. Chiang, Phys. Rev. Lett. 43 (1979) 1494–1497. [90] J.S. Lin, A. Qteish, M.C. Payne, V. Heine, Phys. Rev. B 47 (1993) 4174–4180. [91] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [92] K. Refson, P.R. Tulip, S.J. Clark, Phys. Rev. B 73 (2006) 155114. [93] J.A. Alarco, A. Chou, P.C. Talbot, I.D.R. Mackinnon, PCCP 16 (2014) 24443–24456. [94] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G.L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A.P. Seitsonen, A. Smogunov, P. Umari, R.M. Wentzcovitch, J. Phys.: Condens. Matter 21 (2009). [95] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M.B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A.D. Corso, S.D. Gironcoli, P. Delugas, J.R.A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N.L. Nguyen, H.V. Nguyen, A. Otero-de-la-Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A.P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, S. Baroni, J. Phys.: Condens. Matter 29 (2017) 465901.

[1] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. Akimitsu, Nature 410 (2001) 63–64. [2] J. Kortus, I.I. Mazin, K.D. Belashchenko, V.P. Antropov, L.L. Boyer, Phys. Rev. Lett. 86 (2001) 4656–4659. [3] S. Suzuki, S. Higai, K. Nakao, J. Phys. Soc. Jpn. 70 (2001) 1206–1209. [4] J.M. An, W.E. Pickett, Phys. Rev. Lett. 86 (2001) 4366–4369. [5] K.D. Belashchenko, M. van Schilfgaarde, V.P. Antropov, Phys. Rev. B 64 (2001) 092503. [6] G. Satta, G. Profeta, F. Bernardini, A. Continenza, S. Massidda, Phys. Rev. B 64 (2001) 104507. [7] A. Bussmann-Holder, H. Keller, J. Phys.: Condens. Matter 24 (2012) 233201. [8] K. Togano, P. Badica, Y. Nakamori, S. Orimo, H. Takeya, K. Hirata, Phys. Rev. Lett. 93 (2004) 247004. [9] B. Albert, H. Hillebrecht, Angew. Chem. Int. Ed. 48 (2009) 8640–8668. [10] S. Tsuchiya, T. Toshima, H. Nobukane, K. Inagaki, S. Tanda, Phys. Rev. B 80 (2009) 094502. [11] J. Kortus, O.V. Dolgov, R.K. Kremer, A.A. Golubov, Phys. Rev. Lett. 94 (2005) 027002. [12] M.S. Park, H.J. Kim, B. Kang, S.K. Lee, Supercond. Sci. Tech. 18 (2005) 183–186. [13] V.P.S. Awana, A. Vajpayee, M. Mudgel, H. Kishan, Supercond. Sci. Tech. 22 (2009) 034105. [14] Z.C. Pan, H. Sun, C.F. Chen, Phys. Rev. B 73 (2006) 193304. [15] T. Bazhirov, Y. Sakai, S. Saito, M.L. Cohen, Phys. Rev. B 89 (2014) 045136. [16] S.Y. Lu, H.Y. Liu, I.I. Naumov, S. Meng, Y.W. Li, J.S. Tse, B. Yang, R.J. Hemley, Phys. Rev. B 93 (2016) 104509. [17] S.K. Saha, G. Dutta, Phys. Rev. B 94 (2016) 125209. [18] T. Oguchi, J. Phys. Soc. Jpn. 71 (2002) 1495–1500. [19] I.I. Mazin, O.K. Andersen, O. Jepsen, O.V. Dolgov, J. Kortus, A.A. Golubov, A.B. Kuz'menko, D. van der Marel, Phys. Rev. Lett. 89 (2002) 107002. [20] A. Brinkman, A.A. Golubov, H. Rogalla, O.V. Dolgov, J. Kortus, Y. Kong, O. Jepsen, O.K. Andersen, Phys. Rev. B 65 (2002) 180517. [21] H.J. Choi, D. Roundy, H. Sun, M.L. Cohen, S.G. Louie, Phys. Rev. B 66 (2002) 020513. [22] U. Welp, A. Rydh, G. Karapetrov, W.K. Kwok, G.W. Crabtree, C. Marcenat, L.M. Paulius, L. Lyard, T. Klein, J. Marcus, S. Blanchard, P. Samuely, P. Szabo, A.G.M. Jansen, K.H.P. Kim, C.U. Jung, H.S. Lee, B. Kang, S.I. Lee, Physica C 385 (2003) 154–161. [23] J. Kortus, Physica C 456 (2007) 54–62. [24] P. Zhang, S. Saito, S.G. Louie, M.L. Cohen, Phys. Rev. B 77 (2008) 052501. [25] H.J. Choi, S.G. Louie, M.L. Cohen, Phys. Rev. B 79 (2009) 094518. [26] Y. Zhao, S. Zeng, J. Ni, Phys. Rev. B 93 (2016) 014502. [27] S.Y. Wu, P.H. Shih, J.Y. Ji, T.S. Chan, C.C. Yang, Supercond. Sci. Tech. 29 (2016) 045001. [28] M.M.D. Esfahani, Q. Zhu, H.F. Dong, A.R. Oganov, S.N. Wang, M.S. Rakitin, X.F. Zhou, PCCP 19 (2017) 14486–14494. [29] M. Gao, Z.Y. Lu, T. Xiang, Phys. Rev. B 91 (2015) 045132. [30] J.H. Liao, Y.C. Zhao, Y.J. Zhao, H. Xu, X.B. Yang, PCCP 19 (2017) 29237–29243. [31] S.Y. Xie, X.B. Li, W.Q. Tian, N.K. Chen, Y.L. Wang, S.B. Zhang, H.B. Sun, PCCP 17 (2015) 1093–1098. [32] A.N. Kolmogorov, M. Calandra, S. Curtarolo, Phys. Rev. B 78 (2008) 094520. [33] D.C. Johnston, H.F. Braun, Superconductivity in Ternary Compounds II, first ed., Springer-Verlag Berlin Heidelberg, New York, 1982. [34] C. Buzea, T. Yamashita, Supercond. Sci. Tech. 14 (2001) R115–R146. [35] H. Rosner, A. Kitaigorodsky, W.E. Pickett, Phys. Rev. Lett. 88 (2002) 127001. [36] A. Lazicki, R.J. Hemley, W.E. Pickett, C.S. Yoo, Phys. Rev. B 82 (2010) 180102. [37] P.P. Singh, Solid State Commun. 124 (2002) 25–28. [38] M.I. Eremets, V.V. Struzhkin, H.-K. Mao, R.J. Hemley, Science 293 (2001) 272–274. [39] Y.M. Ma, J.S. Tse, D.D. Klug, R. Ahuja, Phys. Rev. B 70 (2004) 214107. [40] S.K. Bose, T. Kato, O. Jepsen, Phys. Rev. B 72 (2005) 184509. [41] P. Badica, K. Togano, H. Takeya, K. Hirata, S. Awaji, K. Watanabe, Physica C 460 (2007) 91–94. [42] P. Badica, T. Kondo, K. Togano, J. Phys. Soc. Jpn. 74 (2005) 1014–1019. [43] H. Takeya, M. El Massalami, R.E. Rapp, K. Hirata, K. Yamaura, K. Togano, T. Nagata, Physica C 445–448 (2006) 484–487. [44] M.S. Bailey, E.B. Lobkovsky, D.G. Hinks, H. Claus, Y.S. Hor, J.A. Schlueter, J.F. Mitchell, J. Solid State Chem. 180 (2007) 1333–1339. [45] N. Emery, C. Herold, M. d'Astuto, V. Garcia, C. Bellin, J.F. Mareche, P. Lagrange, G. Loupias, Phys. Rev. Lett. 95 (2005) 087003. [46] F. Peng, M.S. Miao, H. Wang, Q. Li, Y.M. Ma, J. Am. Chem. Soc. 134 (2012) 18599–18605. [47] J.S. Slusky, N. Rogado, K.A. Regan, M.A. Hayward, P. Khalifah, T. He, K. Inumaru, S.M. Loureiro, M.K. Haas, H.W. Zandbergen, R.J. Cava, Nature 410 (2001) 343–345. [48] O. De la Pena-Seaman, R. de Coss, R. Heid, K.P. Bohnen, Phys. Rev. B 79 (2009) 134253. [49] O. De la Peña-Seaman, R. de Coss, R. Heid, K.P. Bohnen, Phys. Rev. B 82 (2010) 224508. [50] M. Angst, S.L. Bud'ko, R.H.T. Wilke, P.C. Canfield, Phys. Rev. B 71 (2005) 144512. [51] M. Putti, M. Affronte, P. Manfrinetti, A. Palenzona, Phys. Rev. B 68 (2003) 094514. [52] T. Yildirim, O. Gulseren, J.W. Lynn, C.M. Brown, T.J. Udovic, Q. Huang, N. Rogado, K.A. Regan, M.A. Hayward, J.S. Slusky, T. He, M.K. Haas, P. Khalifah,

241

Computational Materials Science 154 (2018) 234–242

S. Aydin, M. Şimşek

[96] [97] [98] [99] [100]

[101] [102] [103] [104]

[105]

[106] Z.J. Wu, E.J. Zhao, H.P. Xiang, X.F. Hao, X.J. Liu, J. Meng, Phys. Rev. B 76 (2007) 054115. [107] M. Khazaei, M. Arai, T. Sasaki, M. Estili, Y. Sakka, J. Phys.: Condens. Matter 26 (2014) 505503. [108] S.I. Ranganathan, M. Ostoja-Starzewski, Phys. Rev. Lett. 101 (2008) 055504. [109] J. Yang, M. Shahid, C.L. Wan, F. Jing, W. Pan, J. Eur. Ceram. Soc. 37 (2017) 689–695. [110] S.X. Cui, D.Q. Wei, H.Q. Hu, W.X. Feng, Z.Z. Gong, J. Solid State Chem. 191 (2012) 147–152. [111] A.L. Ivanovskii, Prog. Mater Sci. 57 (2012) 184–228. [112] K.Y. Li, X.T. Wang, F.F. Zhang, D.F. Xue, Phys. Rev. Lett. 100 (2008) 235504. [113] A. Simunek, J. Vackar, Phys. Rev. Lett. 96 (2006) 085501. [114] A. Simunek, Phys. Rev. B 75 (2007) 172108. [115] X.Q. Chen, H.Y. Niu, D.Z. Li, Y.Y. Li, Intermetallics 19 (2011) 1275–1281. [116] M. Nishiyama, Y. Inada, G.Q. Zheng, Phys. Rev. B 71 (2005) 202505. [117] W.L. McMillan, Phys. Rev. 167 (1968) 331–344. [118] T. Klein, P. Achatz, J. Kacmarcik, C. Marcenat, F. Gustafsson, J. Marcus, E. Bustarret, J. Pernot, F. Omnes, B.E. Sernelius, C. Persson, A. Ferreira da Silva, C. Cytermann, Phys. Rev. B 75 (2007) 165313. [119] O.L. Anderson, J. Phys. Chem. Solids 24 (1963) 909–917. [120] D.A. Tompsett, Z.P. Yin, G.G. Lonzarich, W.E. Pickett, Phys. Rev. B 82 (2010) 235101.

C. Hartwigsen, S. Goedecker, J. Hutter, Phys. Rev. B 58 (1998) 3641–3662. S. Goedecker, M. Teter, J. Hutter, Phys. Rev. B 54 (1996) 1703–1710. D.R. Hamann, Phys. Rev. B 88 (2013) 085117. D.R. Hamann, Phys. Rev. B 95 (2017) 239906. X. Gonze, F. Jollet, F. Abreu Araujo, D. Adams, B. Amadon, T. Applencourt, C. Audouze, J.M. Beuken, J. Bieder, A. Bokhanchuk, E. Bousquet, F. Bruneval, D. Caliste, M. Côté, F. Dahm, F. Da Pieve, M. Delaveau, M. Di Gennaro, B. Dorado, C. Espejo, G. Geneste, L. Genovese, A. Gerossier, M. Giantomassi, Y. Gillet, D.R. Hamann, L. He, G. Jomard, J. Laflamme Janssen, S. Le Roux, A. Levitt, A. Lherbier, F. Liu, I. Lukačević, A. Martin, C. Martins, M.J.T. Oliveira, S. Poncé, Y. Pouillon, T. Rangel, G.M. Rignanese, A.H. Romero, B. Rousseau, O. Rubel, A.A. Shukri, M. Stankovski, M. Torrent, M.J. Van Setten, B. Van Troeye, M.J. Verstraete, D. Waroquiers, J. Wiktor, B. Xu, A. Zhou, J.W. Zwanziger, Comput. Phys. Commun. 205 (2016) 106–131. Y. Liang, B. Zhang, Phys. Rev. B 76 (2007) 132101. J.B. Goodenough, Phys. Rev. 171 (1968) 466–479. B.F. Decker, J.S. Kasper, Acta Crystallogr. A 12 (1959) 503–506. Supplemental Material. Calculated phonon dispersion curves (Fig. S1), bond lengths and their Mulliken bond overlap populations (Table S1), elastic and mechanical properties (Table S2), anisotropy, Pugh’s ratio and hardness (Table S3), and phonon frequencies at Γ (Table S4). M.E. Jones, R.E. Marsh, J. Am. Chem. Soc. 76 (1954) 1434–1436.

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