First principles study of the structural, electronic, optical, elastic and thermodynamic properties of CdXAs2 (X=Si, Ge and Sn)

Materials Science in Semiconductor Processing 27 (2014) 79–96

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Materials Science in Semiconductor Processing journal homepage: www.elsevier.com/locate/mssp

First principles study of the structural, electronic, optical, elastic and thermodynamic properties of CdXAs2 (X ¼ Si, Ge and Sn) Sheetal Sharma a,b, A.S. Verma a,n, R. Bhandari c, Sarita Kumari d, V.K. Jindal b a

Department of Physics, Banasthali Vidyapith, Rajasthan 304022, India Department of Physics, Panjab University, Chandigarh 160014, India Department of Physics, Post Graduate Govt. College, Chandigarh 160011, India d Department of Physics, University of Rajasthan, Rajasthan 302004, India b c

a r t i c l e i n f o

PACS code: 31.15.A46.70.-P 62.20.D62.20.de 65.40.De 65.40.-b 65.60. þa 71.20.-b Keywords: First principles calculations Electronic properties Elastic constants Thermodynamic properties

abstract The first principles calculations were performed by the linearized augmented plane wave (LAPW) method as implemented in the WIEN2K code within the density functional theory to obtain the structural, electronic and optical properties for CdXAs2 (X¼ Si, Ge and Sn) in the body centered tetragonal (BCT) phase. Optical features, such as dielectric functions, refractive indices, extinction coefficient, optical reflectivity, absorption coefficients, and optical conductivities, were calculated for photon energies up to 40 eV. The six independent elastic parameters (C11, C12, C13, C33, C44 and C66) were evaluated and thermodynamic calculations within the quasi-harmonic approximation were used to give an accurate description of the pressure–temperature dependence of the thermal-expansion coefficient, bulk modulus, specific heat, Debye temperature, entropy and Grüneisen parameters. Based on the semi-empirical relation, we have determined the hardness of the materials for the first time at different temperature and pressure. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction Ternary cadmium chalcopyrite semiconductors have recently generated much interest due to their applications in the areas of optical parametric oscillators, solar cells, light emitting diodes up-converters and infrared detectors [1–4]. CdGeAs2 is known to have highest non-linear coefficient and is used as phase-matchable compound semiconductor [5–7]. These compounds can be formally derived from the well-known III–V compounds by cationic substitution, and crystallize in the chalcopyrite structure with I 42d (D12 2d ) space group. Koroleva et al. [8] identified

n

Corresponding author. Tel.: þ91 9412884655. E-mail address: [email protected] (A.S. Verma).

http://dx.doi.org/10.1016/j.mssp.2014.06.015 1369-8001/& 2014 Elsevier Ltd. All rights reserved.

Mn-doped chalcopyrites CdGeAs2, ZnGeAs2 and ZnSiAs2 as new materials for spintronics. Very recently [9] an attempt to improve the optical properties of the single crystals of CdGeAs2 on thermal annealing was performed using the modified vertical Bridgman method. The progress of Cd(Si, Ge, Sn)As2-based technologies has mostly been attained using scientific intuition rather than knowledge-based design: technology first and scientific explanation later. Nowadays, development of computational methods has led a new class of first principle's approaches. The complexities of experiments are removed as only the atomic numbers of the constituent atoms are the required inputs. Despite an increasing use of first principles approach in recent years, very little is known about cadmium chalcopyrites in comparison with Si or binary compounds. Also there are rare experimental studies

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focusing on the elastic and thermal properties of cadmium chalcopyrites. In order to mitigate this lack of data and to confront numerical results with the available experimental and theoretical data, the density functional theory (DFT) and the quasiharmonic Debye model have been employed to study the physical properties of CdXAs2 (X¼Si, Ge and Sn).

The outline of the paper is as follows. In Section 2, we have given a brief review of the computational scheme used. The calculations of the structural, electronic and optical properties along with the computed elastic and thermal properties are described in Section 3, while the summary and conclusions are drawn in Section 4.

Table 1 Structural equilibrium parameters a, c, u, B and B0 calculated in WC-GGA. Crystals

a (Å)

c (Å)

u

B (GPa)

B0

CdSiAs2 CdGeAs2 CdSnAs2

5.87, 5.88a 5.91, 5.94b 6.13, 6.08d

11.27, 10.88a 11.55, 11.2b 11.98, 11.91d

0.270 0.261, 0.279b 0.261, 0.262e

66, 72* 61, 70c 56, 55f

4.92, 5.49* 6.39, 6.18c 2.41

a

Experimental values from Reference [46]. Experimental values from Reference [47]. Experimental values from Reference [48]. d Experimental values from Reference [45]. e Experimental values from Reference [58]. f Theoretical values from Reference [49]. n LDA values from Reference [46]. b c

-664781.5 -566342.0

-664781.7

Energy (eV)

-566342.4

-566342.6

-664781.8 -664781.9 -664782.0 -664782.1

-566342.8 8000

8500

9000

9500

-664782.2 8000

10000

8500

Volume (Ang3)

9000

9500

Volume (Ang3)

-886815.0

CdSnAs 2 -886815.1

-886815.2

Energy (eV)

Energy (eV)

-566342.2

7500

CdGeAs 2

-664781.6

CdSiAs2

-886815.3

-886815.4

-886815.5

-886815.6

-886815.7 9000

9500

10000

10500

11000

11500

Volume (Ang3) Fig. 1. Calculated total energies as a function of volume of (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

10000

10500

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

Fig. 2. Band structures of (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2 using WC-GGA þ mBJ.

Fig. 3. The calculated partial and total density of states (DOS) for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2 using WC-GGA þmBJ.

81

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2. Computational methods The calculations were done using full potential linearized augmented plane wave (FP-LAPW) computational scheme [10,11] as implemented in the WIEN2K code [12]. In this method, the space is divided into an interstitial region (IR) and non-overlapping muffin-tin (MT) spheres centered at the atomic sites. In the IR, the basis set consists of plane waves. Inside the MT spheres, the basis set is described by radial solutions of the one-particle Schrödinger equation (at fixed energy) and their energy derivatives multiplied by spherical harmonics without spin–orbit interactions. The FP-LAPW method expands the Kohn– Sham orbitals in atomic-like orbitals inside the MT atomic spheres and plane waves in the interstitial region. The Kohn–Sham equations were solved using the recently developed Wu–Cohen generalized gradient approximation (WC-GGA) [13,14] for the exchange-correlation (XC) potential. It has been shown that this new functional is more accurate for solids than any existing GGA and metaGGA forms. For a variety of materials, it improves the equilibrium lattice constants and bulk moduli significantly over local-density approximation [15] and Perdew–Burke– Ernzerhof (PBE) [16] and therefore should also perform better for the CdXAs2 (X¼Si, Ge and Sn). For this reason and for testing purposes, we adopted the new WC-GGA approximation for the XC potential for studying the present systems. Furthermore, modified Becke–Johnson potential (mBJ) [17] as coupled with WC-GGA is used for electronic structure and optical calculations. The valence wave functions inside the atomic spheres were expanded up to l ¼10 partial waves. In the interstitial region, a plane wave expansion with RMTKmax equal to seven was used for all the investigated systems, where RMT is the minimum radius of the muffin-tin spheres and Kmax gives the magnitude of the largest K vector in the plane wave expansion. The potential and the charge density were Fourier expanded up to Gmax ¼12. Convergence tests were carried out for the charge-density Fourier expansion using higher Gmax values. The RMT (muffin-tin radii) are taken to be 2.32, 2.1, 1.9, 2.11 and 2.22 a.u. (atomic unit) for Cd, As, Si, Ge and Sn respectively. The modified tetrahedron method [18] was applied to integrate inside the Brillouin zone (BZ) with a dense mesh of 5000 uniformly distributed k-points (equivalent to 405 in irreducible BZ) where the total energy converges to less than 10  6 Ry.

3. Results and discussion 3.1. Geometry and electronic structure The ternary chalcopyrite semiconductor crystallizes in the chalcopyrite structure with space group I 42d (D12 2d ). The Cd atoms are located at (0,0,0);(0,1/2,1/4), the X atom (X ¼Si, Ge and Sn) at (1/2,1/2,0); (1/2,0,1/4) and As at (u,1/4,1/8); (  u,3/4,1/8); (3/4,u,7/8); (1/4,  u,7/8). Two unequal bond lengths dCd–As and dX–As result in two structural deformations. The first is characterized by u 2 2 parameter defined as u¼0.25 þ(dCd–As  dX–As)/a2 where a is the lattice parameter in x and y directions, and the

second parameter η ¼c/2a, where c is lattice parameter in z direction which is generally different from 2a. To determine the best minimum total energy for the considered systems as a function of volume, the total energy of the system was minimized with respect to the other geometrical parameters. Typically the process involves a non-linear regression approach and the quantity to be minimized is the lattice enthalpy. The minimization is done in two steps, first parameter u is minimized by the calculation of the internal forces acting on the atoms within the unit cell until the forces become negligible; for this, MINI task is used which is included in the WIEN2K code. Second, the total energy of the crystal is calculated for a grid of volume of the unit cell (V) and c/a values, where each point in the grid involves the minimization with respect to u. Five values of c/a are used for each volume and a polynomial is then fitted to the calculated energies to calculate the best c/a ratio. The result is an optimal curve (c/a, u) as a function of volume. Further, a final optimal curve of total energy is obtained by minimizing energy versus [V,c/a(V),u(V)] by FPLAPW calculations and Murnaghan equation of state [19]. Table 1 presents the lattice constants along with the bulk modulus and its pressure derivative. The calculated total energy vs. volume of unit cell of the CdSiAs2, CdGeAs2, and CdSnAs2 crystals is plotted in Fig. 1. The imperative features are: 1. Compared with the experimental measurements, our calculated values for lattice constants are well within acceptable error from the experiment measurements. Specifically for CdSiAs2, the percent deviation in c/a value as calculated by WC-GGA and experimental method is 3.8%, while it is  0.1% for the LDA calculation performed by Boukabrine et al. [46]. 2. Although for CdGeAs2, the FP-LAPW method predicts better structural parameters with WC-GGA (c/a ¼1.95 (3.6%) in comparison to LDA and GGA-PBE (c/a ¼1.96 (4.1%) with LDA and c/a¼ 1.98 (5.3%) with GGA, not presented in the paper). But the results are not improved over LDA and GGA calculations performed by Zapol et al. [48] and Yu et al. [43]. 3. The WC-GGA calculated values for internal parameter u are lower than the experimentally measured values, whereas c/a values are always overestimated for each compound. Table 2 The calculated minimum band gaps Eg (eV), refractive index (n) and dielectric constant (ε) for CdSiAs2, CdGeAs2 and CdSnAs2 compared with other experimental and theoretical data. Crystals

Eg (eV)

n

ε1

CdSiAs2 CdGeAs2 CdSnAs2

1.27, 1.55a, 0.54b 0.45, 0.6e, 0.57f 0.44, 0.26f

3.22, 3.5c 3.57, 3.5c 3.37, 3.8c

10.35, 10.01d 12.78, 10.3d 11.38, 11.3d

a

Experimental values from Reference [50]. Values from Reference [46]. c Experimental values from Reference [51]. d Theoretical values from Reference [52]. e Experimental values from Reference [48]. f Theoretical values from Reference [53]. b

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

83

20 16

E ⊥ c E || c CdSiA s2

15

E ⊥c E || c

14

CdSiAs 2

12 10

ε2

ε1

10

5

8 6 4

0

2 -5 0

2

4

6

0

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

0

2

4

6

Energy (eV )

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV)

20

E⊥c E || c CdGeAs2

15

16

E ⊥ c E || c CdG eAs 2

14 12

10

ε2

ε1

10

5

8 6 4

0

2 -5

0

2

4 6

0

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

0

2

4

6

16

16

E⊥c E || c CdSnAs 2

14 12

E⊥c E || c CdSnAs 2

14 12

10

10

8 6

ε2

ε1

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV)

Energy (eV)

8

4 6 2 4

0

2

-2 -4 0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV)

0

0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV)

Fig. 4. The calculated real ε1(ω) and imaginary ε2(ω) parts of complex dielectric constant for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

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3.0 4.5

E⊥ c E || c CdSiAs2

4.0

E⊥ c E || c CdSiAs2

2.5

Extinction coefficient

Refractive index

3.5 3.0 2.5

2.0 1.5

2.0

1.5

1.0

1.0

0.5 0.5 0.0

0.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV)

Energy (eV)

3.0

4.5

E⊥ c E || c CdGeAs 2

4.0

E⊥ c E || c CdGeAs 2

2.5

Extinction coefficient

Refractive index

3.5 3.0 2.5 2.0 1.5 1.0

2.0

1.5

1.0

0.5

0.5 0.0

0.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV)

Energy (eV) 3.0

4.5

E⊥c E || c CdSnAs 2

4.0

E⊥c E || c CdSnAs 2

2.5

Extinction coefficient

Refractive index

3.5 3.0 2.5 2.0 1.5 1.0

2.0

1.5

1.0

0.5

0.5 0.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV)

0.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV)

Fig. 5. The calculated refractive index and extinction coefficient for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

3.2. Electronic and optical properties The band structure of CdSiAs2, CdGeAs2 and CdSnAs2 (using WC-GGA þmBJ) without spin–orbit interaction is shown in Fig. 2. As is clear from the energy band diagram,

0.6

CdSiAs2, CdGeAs2 and CdSnAs2 have a direct band gap at Γ (1.27 eV, 0.45 eV and 0.44 eV of CdSiAs2, CdGeAs2 and CdSnAs2, respectively). For a more in-depth analysis of bonding, the partial and total density for states (PDOS and DOS) for CdSiAs2, CdGeAs2 and CdSnAs2 are calculated using the mBJ potential together

E⊥c E || c CdSiAs2

0.5

180

E⊥c E || c CdSiAs2

160

120

α(ω) (10 /cm)

0.3

4

Reflectivity

140 0.4

0.2

100 80 60

0.1

0.0

85

40 20 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

0

Energy (eV)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV)

0.6

E⊥c E || c CdGeAs2

E⊥c E || c CdGeAs2

140 120

0.4

α(ω) (10 /cm)

0.3

100

4

Reflectivity

0.5

160

0.2

80 60 40

0.1

20 0.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

0

Energy (eV)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV) 0.6

160

E⊥c E || c CdSnAs2

0.5

120 100

4

α(ω) (10 /cm)

0.4

Reflectivity

E⊥c E || c CdSnAs2

140

0.3

0.2

80 60 40

0.1

20 0.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV) Fig. 6. The calculated reflectivity (R(ω)) for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV) Fig. 7. The calculated absorption coefficient (α(ω)) for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

with WC-GGA for the correlation (Fig. 3). The position of the Fermi level is at 0 eV. It can be seen in Fig. 3 that Cd-d states are localized deeper in valence band along with Si/Ge/Sn-s states, and hence has no effect on deciding the magnitude of the semi-conducting band gap. The valence band maxima (VBM) is mainly derived from hybridized As-p and Si/Ge/Sn-p states. The conduction band consists essentially of Si/Ge/Sn-s and As-p with a minor presence of Si/Ge/Sn-p and As-s states. The Cd-s and As-s states have negligible contribution near the Fermi region. The comparison of the theoretical band gaps with available experimental data in Table 2 shows that the mBJ correlation potential allows the prediction of band gap values much closer to the experimental values. The mBJ potential gives results in good agreement with experimental values that are similar to those produced by more sophisticated methods but at much lower computational costs [17,20]. The linear response to an external electromagnetic field with a small wave vector is measured through the complex dielectric function εðωÞ ¼ ε1 ðωÞ þ iε2 ðωÞ

dielectric function ε2(ω) up to 40 eV. Our analysis shows that the critical points of the ε2(ω) occur at 1.21 eV, 0.41 and 0.44 eV for CdSiAs2, CdGeAs2 and CdSnAs2, respectively, corresponding to the main energy gap. Our onset values of energies have a small deviation from the experimental band gap (Table 2).

10000

which is related to the interaction of photons with electrons [21]. The imaginary part ε2(ω) of the dielectric function could be obtained from the momentum matrix elements between the occupied and unoccupied wave functions and is given by [22]. Z    2 4π 2 e2 3 ε2 ðωÞ ¼ 2 2 ∑ iM j f i ð1  f i Þ  δ½Ef Ei  ωd k: ð2Þ m ω

4000

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV)

10000

E⊥c E || c CdGeAs2

σ (ω) (1/Ohm.cm)

8000

6000

4000

2000

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV) 10000

E⊥c E || c CdSnAs2

8000

σ (ω) (1/Ohm.cm)

P represents the principal integral. All of the other optical properties, including the absorption coefficient α (ω), the refractive index n(ω), the extinction coefficient k (ω), and the energy-loss spectrum L(ω), can be directly calculated from ε1(ω) and ε2(ω) [22,24]. Fig. 4 displays the real and imaginary parts of the electronic dielectric function ε(ω) spectrum for the photon energy ranging up to 40 eV. We remark that the CdSiAs2, CdGeAs2 and CdSnAs2 spectra have some features in common. The main peaks of the real part of the electronic dielectric function ε1(ω), which is mainly generated by electronic transition from the top of the valence band to the bottom of conduction band, occur at 2.305 eV, 1.65 eV and 1.87 eV for CdSiAs2, CdGeAs2 and CdSnAs2, respectively. Then, ε1(ω) spectra further decrease up to 6.79 eV, 7.01 eV and 6.90 eV for CdSiAs2, CdGeAs2 and CdSnAs2, respectively. Optical spectra exhibit anisotropy in two directions (along basal-plane and z-axis) with a very small difference (0.0802, 0.5258 and 0.3625 for CdSiAs2, CdGeAs2 and CdSnAs2, respectively) in the static limit. The imaginary part of the dielectric constant ε2(ω) is the fundamental factor of the optical properties of a material. Fig. 4 displays the imaginary (absorptive) part of the

6000

2000

ð1Þ

M is the dipole matrix, i and j are the initial and final states, respectively, fi is the Fermi distribution function for the ith state, and Ei is the energy of the electron in the ith state. The real part ε1(ω) can be evaluated from ε2(ω) using the Kramers–Kronig relations and is given by [23] Z 1 2 ω'ε2 ðω'Þ ε1 ðωÞ ¼ 1 þ P dω': ð3Þ π 0 ω'2  ω2

E⊥c E || c CdSiAs2

8000

σ (ω) (1/Ohm.cm)

86

6000

4000

2000

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (eV) Fig. 8. The calculated photoconductivity (b) CdGeAs2 and (c) CdSnAs2.

(σ(ω))

for

(a)

CdSiAs2,

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

Fig. 5 presents the refractive index n(ω) along with the extinction coefficient k(ω). The refractive index spectrum shows an anisotropic behavior (Δn(0 eV)¼0.02493, 0.07358, and 0.05374 for CdSiAs2, CdGeAs2 and CdSnAs2, respectively) [25]; hence only the averages are listed in Table 2. The refractive index increases with energy reaching a maximum value in the visible region for all compounds. The peak values for CdSiAs2, CdGeAs2 and CdSnAs2 are 4.21 at 2.33 eV, 4.27 at 1.68 eV and 3.96 at 2.08 eV, respectively. Fig. 5 also shows extinction coefficient k(ω) of CdSiAs2, CdGeAs2 and CdSnAs2. It is related to the decay or damping of the oscillation amplitude of the incident electric field, the extinction coefficient k(ω) decreases with increasing incident photon energy. The calculated optical reflectivity R(ω) is displayed in Fig. 6. The maximum reflectivity occurs in region 4.6–11.4 eV for CdSiAs2, 4.2–11.2 eV for CdGeAs2 and 4.5–10.9 eV for CdSnAs2. The absorption coefficient α(ω) is a parameter that indicates the fraction of light lost by the electromagnetic wave when it passes through a unit thickness of the material. These have been plotted for CdSiAs2, CdGeAs2 and CdSnAs2 in Fig. 7. It is clear that polarization has a minor influence on the spectrum. From the absorption spectrum, we can easily find the absorption edges of 1.54 eV for CdSiAs2, 0.66 eV for CdGeAs2 and 0.55 eV for CdSnAs2. When the photon energy is more than the absorption edge value, the adsorption coefficient increases. The absorption coefficients further decrease rapidly in the high energy region, which is the typical characteristic of semiconductors. Optical conductivity parameters are closely related to the photo-electric conversion efficiency and are mainly used to measure the change caused by the illumination. Fig. 8 shows the optical conductivities of CdSiAs2, CdGeAs2 and CdSnAs2, respectively. As is clear from Fig. 8, these materials have finite conductivities in the visible light region (1.65 eV–3.1 eV) and therefore can be used in photovoltaic technology. The major peaks of the conductivity spectra occur at 4.27 eV, 4.16 eV and 4.05 eV for CdSiAs2, CdGeAs2 and CdSnAs2, respectively (i.e., in the UV region of the electromagnetic spectrum).

The determination of the elastic constants requires knowledge of the curvature of the energy curve as a function of strain for selected deformations of the unit cell. The deformations [26,27] are shown in Table 3 and chosen such that the strained systems have the maximum possible symmetry. The system has been optimized for each deformed cell geometry. The WIEN2K package [12] facilitates this task by providing a force-driven optimization of the internal cell geometry. The elastic stiffness tensor of chalcopyrite compounds has six independent components because of the symmetry properties of the D12 2d space group, namely C11, C12, C13, C33, C44 and C66 in the Young notation. The calculated elastic constants for the tetragonal phase of Cd-chalcopyrite's are listed in Tables 4 and 5. In general, our results are in good agreement with the experimental data and theoretical data. In particular if we consider shear constants (C44 and C66) appear to be no worse than the rest of the elastic constants, even though the inner strain component is particularly difficult in those constants. The comparison with other theoretical calculations also shows an important dispersion of values. The calculated elastic constants fulfill the mechanical stability criteria for the tetragonal systems [28]: 2 C11 4|C12|, (C11 þC12) C33 4 2C13, C44 40, and C66 40. In order to check the internal consistency of calculated elastic constants, we can compare the bulk modulus

Table 4 Elastic constants Cij (in GPa) of the Cd-chalcopyrites compared with available data. Crystals

C11

C12

C13

C33

C44

C66

CdSiAs2 CdGeAs2 CdSnAs2

96, 84b 88, 95a 80, 79b

51, 52b 51, 60a 50, 49b

53, 49b 51, 60a 50, 47b

92, 79b 91, 83a 76, 74b

41, 29b 38, 42a 32, 27b

40, 26b 38, 4a 31, 24b

a b

Experimental values derived from Reference [47]. Theoretical values derived from Reference [55].

Table 5 Elastic moduli of the Cd-chalcopyrites. Crystals

3.3. Elastic properties The elastic properties of a solid are among the most fundamental properties that can be predicted from the first-principles ground-state total-energy calculations.

87

B (GPa)

CdSiAs2 66 CdGeAs2 63 CdSnAs2 59

G (GPa)

Y (GPa)

ν

κa (GPa  1)

31 28 23

81 74 61

0.30 0.0049 0.31 0.0054 0.33 0.0053

κc (GPa  1)

B/G

0.0053 0.0049 0.0062

2.11 2.24 2.56

Table 3 The lattice parameters of the deformed tetragonal unit cell, the expression relating the δ and ε variables, the finite Lagrangian strain tensor (Voigt notation) and the value of the second derivative, (1/2V)(d2E/dε2), in terms of the elastic constants (ε being deformation coordinate and E the energy) [54]. Strained cell   a þ δ; a þ δ; c þacδ; 90; 90; 90 ðaþ δ; a þ δ; c; 90; 90; 90Þ ða; a; c þacδ; 90; 90; 90Þ ða; a þ δ; c; 90; 90; 90Þ ða; a; c; 90; 90 þ δ; 90Þ ða; a; c; 90; 90; 90þ δÞ

ε ða þ δÞ2 1 a2 ða þ δÞ2 1 a2 ðc þ δÞ2  1 c2 ða þ δÞ2  1 a2

sin δ sin δ

dE2/d ε2

Strain (η) ε



ε cε 2; 2; 2a; 0; 0; 0 ε ε ð2; 2; 0; 0; 0; 0Þ

1 1 1 4 ðC 11 þ C 12 Þ þ 8 C 33 þ 2 C 13

ð0; 0; 2ε ; 0; 0; 0Þ

1 8 C 33

ð0; 2ε ; 0; 0; 0; 0Þ

1 8 C 11

ð0; 0; 0; 0; ε; 0Þ ð0; 0; 0; 0; 0; εÞ

C 44 C 66

1 4 ðC 11 þ C 12 Þ

88

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

where C 2 ¼ ðC 11 þC 12 ÞC 33  2C 213 and M ¼ C 11 þ C 12 þ 2C 33 4C 13 . In the Voigt–Reuss–Hill approximation [31], B and G of the polycrystalline material are approximated as the arithmetic mean of the Voigt and Reuss limits:

reported in Table 1 with an equivalent combination of Cijs. Bulk modulus should be bound from above by the Voigt approximation (uniform strain assumption) [29]: BV ¼ 19 ð2C 11 þ C 33 þ 2C 12 þ 4C 13 Þ

ð4Þ

Reuss found lower bounds for all lattices [30]: ðC 11 þ C 12 ÞC 33 2C 213 BR ¼ C 11 þ C 12 þ 2C 33  4C 13

B¼

BV þBR 2

ð8Þ

G¼

GV þ GR 2

ð9Þ

ð5Þ

The approximations of Voigt and Reuss provide, in fact, an estimation of the elastic behavior of an isotopic material, for instance a polycrystalline sample. Such a medium would have a single shear constant, G, upper bounded by

Finally, the Poisson ratio and the Young modulus are obtained as

1 ðM þ3C 11  3C 12 þ 12C 44 þ6C 66 Þ GV ¼ 30

ð6Þ

υ¼

3B  2G 2ð3B þ GÞ

ð10Þ

ð7Þ

Y¼

9BG 3B þ G

ð11Þ

and lower bounded by   18BV 6 6 3 1 þ þ þ GR ¼ 15 ðC 11  C 12 Þ C 44 C 66 C2

9200

0 GPa 4 GPa 8 GPa

0 GPa 4 GPa 8 GPa

9600 9400

Volume (Ang3)

8800

8600

8400

2 GPa 6 GPa

9200 9000 8800 8600

8200 8400 8000

0

200

400

600

800

0

200

Temperature (K)

10400

400

Temperature (K)

0 GPa 4 GPa 7 GPa

10600

2 GPa 6 GPa

10200

Volume (Ang3)

Volume (Ang3)

9000

2 GPa 6 GPa

10000

9800

9600

9400

9200

0

200

400

600

800

Temperature (K) Fig. 9. Volume vs. temperature at various pressures for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

600

800

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

was recently exploited in the study of brittle vs. ductile transition in intermetallic compounds from first-principles calculations [33,34]. A high B/G ratio is associated with ductility, whereas a low value corresponds to the brittle nature. The critical value which separates ductile and brittle material is around 1.75, i.e., if B/G 41.75, the material behaves in a ductile manner, otherwise the material behaves in a brittle manner. We have found that B/G ratios are 2.11, 2.24 and 2.56 for CdSiAs2, CdGeAs2 and CdSnAs2, respectively, classifying these materials as ductile. Consequently, the Ba/Gb reflects the competition between the shear and cohesive strengths at the crack tip of fracture.

Using the single crystal Cij data, one can evaluate the linear compressibilities along the principle axis of the lattice. For the tetragonal structure, the linear compressibilities κa and κc along the a- and c-axis, respectively, are given in terms of elastic constants by the following relations: κa ¼ 

1 ∂a C 33  C 13 ¼ a ∂p C 33 ðC 11 þ C 12 Þ  2C 213

ð12Þ

κc ¼ 

1 ∂c C 11 þ C 12 2C 13 ¼ c ∂p C 33 ðC 11 þC 12 Þ 2C 213

ð13Þ

89

Pugh [32] proposed that the resistance to plastic deformation is related to the product Gb, where ‘b’ is the Burgers vector, and that the fracture strength is proportional to the product Ba, where ‘a’ corresponds to the lattice parameter. As b and a are constants for specific materials, Ba/Gb can be simplified into B/G. This formula

3.4. Thermal properties To investigate the thermodynamic properties of Cd-chalcopyrite, we have used Gibbs program [35].

95

0 GPa 4 GPa 8 GPa

100 95

2 GPa 6 GPa

90 85

90

80

B (GPa)

80 75

75 70 65

70

60

0 GPa 4 GPa 8 GPa

65 55 60 0

100

200

300

400

500

600

700

50

800

0

100

2 GPa 6 GPa 200

300

Temperature (K)

400

500

Temperature (K)

88

0 GPa 4 GPa 7 GPa

84 80

2 GPa 6 GPa

76 72

B (GPa)

B (GPa)

85

68 64 60 56 52 48 0

100

200

300

400

500

600

700

800

Temperature (K) Fig. 10. Bulk modulus vs. temperature at various pressures (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

600

700

800

90

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

is expressed as [36]

The obtained set of total energy versus primitive cell volume determined in the previous section has been used to derive the macroscopic properties as a function of temperature and pressure from the standard thermodynamic relations. Gibbs program is based on the quasiharmonic Debye model [35], in which the non-equilibrium Gibbs function Gn(V; P, T) can be written in the form of Gn ðV; P; TÞ ¼ EðVÞ þPV þ Avib ½θD ; T

θD ¼

ð14Þ

2

BS ffiBðVÞ ¼ V

200 K 600 K

360

300 K

330

350

320

θD (K)

360

340

300

320

290

310

280 2

4

6

8

100 K 400 K 700 K

200 K 500 K

310

330

0

0K 300 K 600 K 800 K

350 340

300

ð17Þ

dV 2

By solving Eq. (18), one can obtain the thermal equation of state (EOS) V(P, T). The heat capacity CV, entropy S and

370

270

0

2

4

6

Pressure (GPa)

Temperature (K)

320

0K 400 K 700 K

310

100 K 500 K 800 K

200 K 600 K

300 K

300

θD (K)

θD (K)

380

100 K 500 K 800 K

d EðVÞ

f(σ) is given by Refs. [35,38]; σ is the Poisson ratio [39]. Therefore, the non-equilibrium Gibbs function Gn(V; P, T) as a function of (V; P, T) can be minimized with respect to volume V n

∂G ðV; P; TÞ ¼0 ð18Þ ∂V P;T

where n is the number of atoms per formula unit, D(θ/T) represents the Debye integral, and for an isotropic solid, θ

0K 400 K 700 K

ð16Þ

M being the molecular mass per unit cell and BS the adiabatic bulk modulus, approximated by the static compressibility [35]:

where E(V) is the total energy per unit cell, PV corresponds to the constant hydrostatic pressure condition, θD is the Debye temperature, and Avib is the vibrational term, which can be written using the Debye model of the phonon density of states as [36,37]  9θ θ Avib ½θD ; T  ¼ nkT þ 3 lnð1 e  θ=T Þ  D ð15Þ 8T T

390

rffiffiffiffiffi  1 3 1 ℏ BS 6π 2 V 2 n f ðσ Þ k M

290 280 270 260 250 240

0

1

2

3

4

5

6

7

Temperature (K) Fig. 11. Debye temperature vs. pressure at various temperatures for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

8

6

6

5

5

4

4

-5

α (10 /K)

-5

α (10 /K)

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

3

2

0

0

100

200

300

400

500

600

700

3

2

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa

1

91

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa

1

0

800

0

100

200

Temperature (K)

300

400

500

600

700

800

Temperature (K)

6

4

-5

α (10 /K)

5

3

0 GPa 2 GPa 4 GPa 6 GPa 7 GPa

2

1

0

0

100

200

300

400

500

600

700

800

Temperature (K) Fig. 12. Thermal expansion coefficients vs. temperature at various pressures for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

250 200

150

150

CV (J/mol.K)

CP (J/mol.K)

200

100

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa

50

0

0

100

200

300

400

500

Temperature (K)

600

700

100

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa

50

800

0

0

100

200

300

400

500

Temperature (K)

Fig. 13. Heat capacity vs. temperature at various pressures for CdSiAs2.

600

700

800

92

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

the thermal expansion coefficient α are given by [31] 

θ 3θ=T  θ=T C V ¼ 3nk 4D ð19Þ T e 1 

θ 3 lnð1  e  θ=T Þ S ¼ nk 4D T

Ivanovskii [41] studied the hardness by using semiempirical relations for binary crystals. We have provided a prediction for the hardness of chalcopyrites by using the semi-empirical equation developed by Verma et al. [42] H ¼ KBK þ 1

ð20Þ

γC V α¼ BT V

B and K are the bulk modulus and constant, respecV tively. The value of K is 0.59 for AIIBIVC2 (A ¼Cd, B¼Si, Ge and Sn, C ¼As). Through the quasi-harmonic Debye model, the thermodynamic properties are determined in the temperature range from 0 to 800 K (because the melting temperature (Tm) of CdSnAs2 is less than 900 K [45]) and pressure effects are studied in different ranges. Fig. 9 presents relationships between the equilibrium volume V(Å3) and pressure at T¼0 to 800 K. Meanwhile, the equilibrium volume V decreases dramatically as the pressure P increases at a given temperature. This account suggests that the CdXAs2 (X¼Si, Ge, Sn) under loads turns to be more compressible with increasing pressure than decreasing temperature.

ð21Þ

γ is the Grüneisen parameter, which is defined as γ¼

d ln θðVÞ d ln V

ð23Þ

ð22Þ

Through the quasi-harmonic Debye model, one could calculate the thermodynamic quantities of any temperatures and pressures of compounds from the calculated E–V data at T ¼0 and P¼0. Hardness is one of the most important properties of materials which often determine their technological and industrial applications. Recently, Tian et al. [40] and

250 200 200

CV (J/mol.K)

CP (J/mol.K)

150 150

100

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa

50

0

0

100

200

300

400

500

600

700

100

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa

50

0

800

0

100

200

Temperature (K)

300

400

500

600

700

800

Temperature (K)

Fig. 14. Heat capacity vs. temperature at various pressures for CdGeAs2.

250 200

200

CV (J/mol.K)

CP (J/mol.K)

150

150

100

0 GPa 2 GPa 4 GPa 6 GPa 7 GPa

50

0

0

100

200

300

400

500

Temperature (K)

600

700

100

0 GPa 2 GPa 4 GPa 6 GPa 7 GPa

50

800

0

0

100

200

300

400

500

Temperature (K)

Fig. 15. Heat capacity vs. temperature at various pressures for CdSnAs2.

600

700

800

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

93

at zero pressure below the temperature of 300 K. After the sharp increase, the volume thermal expansion coefficient of the CdXAs2 (X ¼Si, Ge, Sn) is nearly insensitive to the temperature above 300 K due to the electron contributions. As very important parameters, the heat capacities of a substance not only provide essential insight into the vibrational properties but are also mandatory for many applications. Our calculation of the heat capacities CP and CV of CdXAs2 (X ¼Si, Ge, Sn) versus temperature at pressure range 0–8 GPa is shown in Figs. 13–15. From these figures, we can see that the constant volume heat capacity CV and the constant pressure capacity CP are very similar in appearance and both of them are proportional to T3 at low temperatures. For higher temperatures, the anharmonic approximations of the Debye model are used in which the anharmonic effect on CV is suppressed and it is very close to the Dulong Petit limit (CV(T) 3 R for mono-atomic solids), which is common to all solids at

Furthermore, the relationship between the bulk modulus and temperature for CdXAs2 (X¼Si, Ge, Sn) is shown in Fig. 10. The bulk modulus, signifying the average strength of the coupling between the neighboring atoms, slightly decreases with increasing temperature at a given pressure and increases with increasing pressure at a given temperature; the gradual change with temperature is consistent with the trend of volume for these compounds. The variation of the Debye temperature θD(K) as a function of pressure and temperature is displayed in Fig. 11. With the applied pressure increasing, the Debye temperatures are almost linearly increasing. The temperature and pressure dependence of θD(K) reveals that the thermal vibration frequency of atoms in chalcopyrites changes with temperature and pressure. Fig. 12 shows the volume thermal expansion coefficient α (10  5/K) of CdXAs2 (X ¼Si, Ge, Sn) at various pressures, from which it can be seen that the volume thermal expansion coefficient α increases quickly at a given temperature particularly

500 450

450 400

400 350

S (J/mol.K)

300 250 200

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa

150 100 50 0

0

100

200

300

400

500

600

700

300 250 200

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa

150 100 50 800

0

0

100

200

300

400

500

Temperature (K)

Temperature (K)

500 450 400 350

S (J/mol.K)

S (J/mol.K)

350

300 250 200

0 GPa 2 GPa 4 GPa 6 GPa 7 GPa

150 100 50 0

0

100

200

300

400

500

600

700

800

Temperature (K) Fig. 16. Entropy vs. temperature at various pressures for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

600

700

800

94

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

In Fig. 18, we have presented the values of hardness (H in GPa) at different temperatures and pressures. It shows the hardness decreases as the temperature increases at a given pressure and increases as the pressure increases at a given temperature. The values of hardness are reported for the first time at different pressures and temperatures. Table 6 presents the thermal properties such as isothermal bulk modulus, hardness, Grüneisen parameter, Debye temperature and thermal expansion coefficient at 300 K.

high temperatures. Fig. 16 shows the entropy vs. temperature at various pressures for CdXAs2 (X ¼Si, Ge, Sn). The entropies are variable by power exponent with increasing temperature, but the entropies are higher at low pressure than at high pressure for the same temperature. At 30 K the obtained value of entropy is 3.15 Jmol  1 K  1 for CdGeAs2 that matches well with the 2.85 Jmol  1 K  1 and 2.23 Jmol  1 K  1 as calculated by Yu et al. [43] and Bohmhammel et al. [44]. The Grüneisen parameter γ is another important quantity for the materials that describes the effect of change of the volume of the crystal lattice on the dynamics of the crystal. In Fig. 17, we have shown the values of Grüneisen parameter γ at different temperatures and pressures. It shows the γ value remains constant at low temperature and increases linearly as the temperature increases at a given pressure. Also, γ value decreases as the pressure increases at a given temperature. It is well-known that hardness of the materials depends on methods of measurement, temperature, etc. [41].

0 GPa 4 GPa 8 GPa

1.90

4. Summary and conclusion To conclude, results have been presented for the solid state properties such as structural, electronic, optical, elastic and thermal properties of the ternary pnictides semiconductors CdXAs2 (X ¼Si, Ge, Sn) using the firstprinciples calculation. The structural properties in the chalcopyrite structure are obtained using the total energy

1.95

2 GPa 6 GPa

0 GPa 4 GPa 8 GPa

1.90

2 GPa 6 GPa

1.85

γ

1.80

1.80

1.75

1.75

1.70 1.70 1.65 0

100

200

300

400

500

600

700

800

0

100

200

300

Temperature (K)

400

500

Temperature (K)

1.95

0 GPa 4 GPa 7 GPa

1.90

2 GPa 6 GPa

1.85

γ

γ

1.85

1.80

1.75

1.70

1.65

0

100

200

300

400

500

600

700

800

Temperature (K) Fig. 17. Gruneisen parameter vs. temperature at various pressures for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

600

700

800

S. Sharma et al. / Materials Science in Semiconductor Processing 27 (2014) 79–96

95

9.0

0 GPa 4 GPa 8 GPa

8.5 8.0

2 GPa 6 GPa

7.5

Hardness (H in GPa)

0 GPa 4 GPa 8 GPa

8

2 GPa 6 GPa

7

Hardness (H in GPa)

7.0 6.5 6.0 5.5 5.0

6

5

4

4.5 4.0 3.5

0

100

200

300

400

500

600

700

3

800

0

100

200

300

Hardness ( H in GPa)

Temperature (K)

7.00 6.75 6.50 6.25 6.00 5.75 5.50 5.25 5.00 4.75 4.50 4.25 4.00 3.75 3.50 3.25 3.00 2.75 2.50

400

500

600

700

800

Temperature (K)

0 GPa 4 GPa 7 GPa

0

100

200

300

400

500

600

2 GPa 6 GPa

700

800

Temperature (K) Fig. 18. Hardness vs. temperature at various pressures for (a) CdSiAs2, (b) CdGeAs2 and (c) CdSnAs2.

Table 6 Selection of thermal properties at 300 K; isothermal bulk modulus (B in GPa), Hardness (H in GPa), Gruneisen parameter (γ), Debye temperature (θD in K) and thermal expansion coefficient (α in 10  5/K). Crystals

B (GPa)

H (GPa)

γ

θD (K)

α (10-5/K)

CdSiAs2 CdGeAs2 CdSnAs2

66 61 54

4.51, 6.1a 3.96, 4.6a 3.3, 3.3a

1.863 1.864 1.864

321, 316b 290, 252b 261, 255b

4.51 4.73 4.84

a b

Experimental values from Reference [56]. Theoretical values from Reference [57].

as a function of volume; the derived equilibrium parameters are compared with experimental data. We have calculated electronic and optical properties by the mBJ functional and compared with available experimental and theoretical data. We find that the mBJ functional provides an accurate description of the electronic and optical properties. The CdSiAs2, CdGeAs2 and CdSnAs2 have a

direct band gap. We have also derived the static refractive index. Thermal properties such as Grüneisen parameter, volume expansion coefficient, bulk modulus, specific heat, entropy, Debye temperature and hardness are calculated successfully at various temperatures and pressures, and trends are discussed. The ground state parameters of interest were obtained and showed good agreement with

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published experimental and theoretical data. To the best of our knowledge, most of the investigated parameters are reported for the first time and hoped to stimulate the succeeding studies. Acknowledgments We express our gratitude to the computing facilities at IUAC (New Delhi, India) and the departmental computing facilities at Department of Physics, PU, Chandigarh, India. SS wishes to acknowledge CSIR (New Delhi, India) for financial support. References [1] A.S. Verma, Solid State Commun. 149 (2009) 1236. [2] J.E. Jaffe, A. Zunger, Phys. Rev. B 28 (1983) 5822. [3] A.G. Jackson, M.C. Ohmer, S.R. LeClair, Infrared Phys. Technol. 38 (1997) 233. [4] L.A. Maltseva, Y.V. Rud, Sov. Tech. Phys. Lett. 2 (1976) 266. [5] R.L. Byer, H. Kildal, R.S. Feigelson, Appl. Phys. Lett. 19 (1971) 237. [6] G.D. Boyd, E. Buehler, F. Storz, J.H. Wernick, IEEE J. Quantum Electron. 8 (1972) 419. [7] V.G. Dmitriev, G.G. Gurzadyan, D.N. Nikogosyan, Handbook of Nonlinear Optical Crystals, Springer, New York, 1997. [8] L.I. Koroleva, D.M. Zashchirinskii, T.M. Khapaeva, A.I. Morozov, S.F. Marenkin, I.V. Fedorchenko, R. Szymczak, J. Magn. Magn. Mater. 323 (2011) 2923. [9] W. Huang, B. Zhao, S. Zhu, Z. He, B. Chen, J. Li, Y. Yu, J. Tang, W. Liu, J. Cryst. Growth 362 (2013) 291. [10] G.K.H. Madsen, P. Blaha, K. Schwarz, E. Sjöstedt, L. Nordström, Phys. Rev. B 64 (2001) 195134. [11] K. Schwarz, P. Blaha, G.K.H. Madsen, Comput. Phys. Commun. 147 (2002) 71. [12] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k: An Augmented Plane Waveþ Local Orbitals Program for Calculating Crystal Properties, Karlheinz Schwarz/Techn. Universität Wien, Austria, 2001. [13] Z. Wu, R.E. Cohen, Phys. Rev. B 73 (2006) 235116. [14] F. Tran, R. Laskowski, P. Blaha, K. Schwarz, Phys. Rev. B 75 (2007) 115131. [15] W. Kohn, L.J. Sham, Phys. Rev 140 (1965) A1133. [16] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [17] F. Tran, P. Blaha, Phys. Rev. Lett. 102 (2009) 226401. [18] P.E. Blöchl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223. [19] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244. [20] W. Hetaba, P. Blaha, F. Tran, P. Schattschneider, Phys. Rev. B 85 (2012) 205108. [21] J. Sun, H.T. Wang, N.B. Ming, J. He, Y. Tian, Appl. Phys. Lett. 84 (2004) 4544. [22] S.A. Korba, H. Meradji, S. Ghemid, B. Bouhafs, Comput. Mater. Sci. 44 (2009) 1265.

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