International Journal of Applied Nanotechnology eISSN: 2455-8524 Vol. 3: Issue 2
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Structural and Elastic Properties Calculation of CdX (X= S, Se, Te) Semiconductors from First-Principles Bramha P. Pandey Department of Electronics and Communication Engineering, GLA University, Mathura, India
ABSTRACT First-principle calculations have been performed to calculate the structural and elastic properties of wurzite (wz) CdX(X= S, Se, Te) chalcogenides semiconductor using the plane wave pseudo-potential method within the projected augmented wave approximation (PAW),which works directly with full all electron (AE) wavefunctions and AE potentials. The values of lattice constant (a) and (c), internal parameter (u), Bulk modulus (B), and reduce pressure derivative bulk modulus ( ) using equation of state(eos) have been calculated. The elastic stiffness constants Cij, i.e., C11, C12, C13, C33, C44, C66; Bulk modulus (B) using Cij, Young’s modulus (Y), Shear modulus (G), Poisson’s ratio ( ), Zener anisotropic factor (A), G/B ratio and Debye temperature (θD) have been calculated using stress–strain approach method. The calculated values of these parameters are compared with the available experimental and theoretical values, specially reported by P. Gopal et al. 2015 using ACBN0 functional. Reasonably good agreement has been obtained between them at low computational cost using PAW method. Keywords: Debye temperature, elastic constants, first-principle calculations, Wurzite II–VI semiconductors Corresponding Author E-mail:
[email protected] INTRODUCTION In the recent past, several attempts have been made to study the electronic, elastic and thermal properties of II–VI semiconductors which are of great technological importance having wide and direct band gaps. These chalcogenides have a potential application in the emerging field of light emitting diodes, solar cell, and high efficiency thin film based transistors, electro-optical and electro-acoustic devices [1–7]. The structural, elastic, and thermos dynamical properties are essential to study the stability and chemical reactivity at high temperature range [8, 9]. Structural, elastic, electronic, optical, and thermal properties of CdX (X = S, Se, Te) semiconductors have been investigated with the help of common traditional computational approach based on density IJAN (2017) 8–15 © JournalsPub 2017. All Rights Reserved
functional theory (DFT) and empirical pseudopotential method [8–14]. The local density approximation (LDA) and the generalized gradient approximation (GGA) pseudopotential method have been employed for the calculation of structural and elastic properties of ZB CdX(X=S, Se, Te) semiconductors [13, 15]. In this work we have calculated structural and ground state properties of wzCdX (X=S, Se, Te) semiconductors using projected augmented wave (PAW) method, which works directly with full all electron (AE) wavefunctions and AE potentials method at a fraction of computational cost of ACBN0 functional. Further, elastic constant and Debye temperature have been calculated using PAW method. The values of Bulk modulus (B) using Cij, Young’s modulus(Y), Shear modulus (G), Poisson’s ratio ( ), Zener anisotropic factor (A), Page 8
Structural and Elastic Properties Calculation of CdX (X= S, Se, Te) Semiconductors
G/B ratio and Debye temperature (θD) of these semiconductors have also been calculated. The calculated values of all parameters are in good agreement with the available experimental values in few cases where the experiments are performed, and the values reported by different workers. THEORY AND COMPUTATIONAL DETAILS The first-principle calculations have been performed to study the structural and elastic properties of wzCdX (X= S, Se, Te) semiconductors using plane wave pseudopotential using total energy method as implemented in the Quantum ESPRESSO simulation software [16]. The calculations are based on projected augmented wave approximation (PAW) [17] including exchange correlation Perdew-BurkeErnzerh of (PBE) functional potential [17, 18], with Ultra soft pseudo-potential [19]. The plane wave expansion cut-off is set at 612eV.The integrals over the Brillouin zone are replaced by a sum on a Monk horst-Pack grid of 8×8×4 special k-points [20].The convergence test shows that the Brillouin Zone sampling and the kinetic energy cut-off are sufficient for crystal optimization. The optimized crystal structures have been determined using the Broyden-Fletcher-Goldfarb-Shenno (BFGS) minimization technique [21] with total energy threshold of 1×10-10 eV/atom and Hellmann-Feynman ionic force threshold of 0.001 10–3eV/Ǻ. After the optimization of geometry, the relax structures have been used for calculation of the elastic stiffness constants. The elastic constants have been determined from DFT calculations by applying a set of given homogeneous deformations with a finite value of structural parameters. RESULTS AND DISCUSSION Geometry and Structural Details The wzCdX(X=S, Se, Te) semiconductors have a body-centered tetragonal (bct) structure having space group P6 mc:#186.
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Pandey
The structural properties have been calculated using the PAW approximation pseudo-potential for wzCdX (X=S, Se, Te) semiconductors. The structures are optimized to find the ground state properties by fitting the equation of state(eos) [22] to calculate the equilibrium parameters such as lattice constant (a) and (c), internal parameter (u), bulk modulus(B),and reduce pressure derivative ( ). The calculated equilibrium lattice parameters ‘a’ are 4.17, 4.37 and 4.67 Å, and ‘c’ are 6.84, 7.14 and 7.63, internal parameters are 0.374, 0.375 and 0.375, bulk moduli are 53.8, 49.9 and 36 GPa, and reduce pressure derivative are 4.43, 3.19 and 4.23 for wzCdS, CdSe and CdTe semiconductors respectively. The calculated equilibrium structural properties are compared with available experimental and reported values given in Table 1. Elastic Properties The elastic properties of materials are important as they provide information about the interatomic potentials and correlate the various fundamental solid state phenomena such as interatomic bonding, equations of state, thermal expansion, Debye temperature and phonon spectra as well as specific heat capacity [23–25]. Elastic constants are defined by means of a Taylor expansion of the total energy namely the derivative of the energy as a function of a lattice strain [26–29]. The tetragonal crystal has six independent elastic coefficients Cij, i.e., C11, C12 C13, C33, C44, and C66. The calculated values of the elastic constants Cij of wzCdX(X=S, Se, Te) semiconductors at equilibrium are listed in Table 2 long with the available experimental and reported values. Table 2 shows that the elastic constants (C11 = C22) C33, C33 (C44 = C55) and C66, which shows that wzCdS and CdSe semiconductors are mechanically anisotropic and stable at equilibrium (P=0 GPa) whereas in case of wzCdTe, we find that (C11 = C22) C33. The elastic
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International Journal of Applied Nanotechnology eISSN: 2455-8524 Vol. 3: Issue 2
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constants also provide the information about the stability and stiffness of materials. For elastic stability of tetragonal structure, the necessary and sufficient four conditions should satisfy the following equation [30–32]:
And
where B is the bulk modulus and G is the shear modulus, which are related to the elastic constants by the relation [35]: (5)
(1) The knowledge of the Young’s modulus (Y), Poisson’s ratio ( ) and Zener anisotropic factor (A) are important for the industrial and technological applications. The Young’s modulus provides information about the measure of the stiffness of the solids, i.e. the larger the value of Y, the stiffer is the material, therefore, wzCdS is stiffer than wzCdSe and CdTe semiconductors. The Poisson’s ratio gives the information about the characteristics of the bonding forces. The values of are typically between 0.1 and 0.25 for covalent materials and interatomic forces are non-central forces [33]. For ionic crystals, the lower and upper limits of are 0.25 and 0.5, respectively, and interatomic forces are central forces [34]. The calculated values of are in between 0.25 to 0.5 for wzCdX(X=S, Se, Te). The Zener anisotropy factor is used for measuring the degree of elastic anisotropy in solids. The value of A =1 for a completely isotropic material and if A is smaller or greater than one, it shows that the material is anisotropic. Table 2 shows that calculated values of A are less than 1 represents wzCdX(X=S, Se, Te) are anisotropic semiconductors. The values of Young’s modulus, Poisson’s ratio, and Zener anisotropic factor of wzCdX(X=S, Se, Te) semiconductors have been calculated using the relations: (2) (3)
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(4)
where
(5a) (5b)
(6) The Voigt’s shear modulus( GV ) and the Reuss’s shear modulus (GR ) both are related to the elastic constant by the relations proposed by Chen et al. [32]. The bulk modulus B is a measure of resistance to volume change by applied pressure, whereas shear modulus G is a measure of resistance to reversible deformations upon the shear stress and they are related by Poisson’s ratio, [35]. Hence the ratio of G/B also gives the information about covalent and ionic behavior of materials on the basis of their brittle and ductile character in solids [36, 37]. We have also calculated the values of B and G using first principles calculations within DFT and calculated results are presented in Table 2, together with the available experimental and other theoretical results. The upper limits of G/B are 1.1 and 0.6, respectively, for brittle and ductile character, i.e., if G/B ≤ 0.6, the materials are ductile (ionic), otherwise brittle (covalent) in nature. Based on the calculated values of G/B, the bond nature of wzCdX(X=S, Se, Te) semiconductors are ductile (ionic). The calculated values of B, G,θD, Y, , A and G/B ratio of these semiconductors are listed in Table 2. The
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Structural and Elastic Properties Calculation of CdX (X= S, Se, Te) Semiconductors
calculated values of these parameters for all wzCdX(X=S, Se, Te) semiconductors are in good agreement with the experimental and theoretical results.
Pandey
mass per formula unit, V0 is the volume of per formula unit and vm the average wave velocity which can be approximated by the relation [38]:
Debye Temperature The Debye temperature(θD) of wzCdX(X=S, Se, Te) semiconductors can be calculated using relations [38]:
where and
are
the
transverse and longitudinal elastic wave velocities, respectively [25]. Using Eq. (9) the calculated values of θD of wzCdX(X=S, Se, Te) semiconductors are listed in Table 2 along with the experimental and reported values.
where h is the Planck constant, k is the Boltzmann constant, NA is the Avogadro number, is the density, n is the number of atoms per formula unit, M is the molecular
Table 1. Ground state properties wz structures such as Lattice constant of (a) and (c) are in Å, internal parameter (u) in Å, bulk modulus (B) calculations using equation of state (eos) in GPa, and reduce pressure derivative (B’) are presented with available experimental and reported results by others. Present work
Expt
Others
wzCdS a
4.17
4.1367a, 4.1348e
4.183b
4.154c
c
6.84
6.7161a, 6.749e
6.682b
6.762c
u
0.374
B
53.8
61c, 65d
53b
74c
4.43 wzCdSe a
4.37
4.2999a,e
4.33b
4.292c
c
7.14
7.0109a,e
7.07b
7.021c
u
0.375
B
49.9
52b
62c
3.19 wzCdTe a
4.67
4.57f
4.60b
c
7.63
7.47f
7.498b
u
0.375
B
36.0
57.4b
4.23 aRef.
[39] at 300 K,
bref.
[40],
cref.
[41], dref. [42], eref. [43], fref. [44].
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International Journal of Applied Nanotechnology eISSN: 2455-8524 Vol. 3: Issue 2
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Table 2. Elastic stiffness constants C11, C12, C13, C33, C44 and C66 have been calculated and given in GPa. The calculated values of bulk modulus (B) in GPa, Young modulus (Y) in GPa, Shear modulus (G) in GPa, Poisson ratio ( ), Zener anisotropic factor (A), G/B ratio and Debye temperature (θD) in K, with available experimental and reported results by others of these semiconductors are presented. Semiconductors
C11
C12
C13
C33
C44
C66
26.1 37.1 88.4
22.5
35.4
Bulk Young Shear Poisson Zener Debye G/B modulus, modulus, modulus, Ratio, anisotropic temperature, ratio B Y G factor (A) K
wzCdS Present work a
Expt
97.1 86.5
54
47.3 94.4
15
16.3
53.7
70.7
62.7
27.6
0.2803
0.6344
0.514
0.330.37
48.1-62.5
273.6 265
e
Others
83.1i
50.4i 46.2i 94.8i
15.3i
16.3i
53 61c 65d
18.8
30.1
45.6
wzCdSe Present work a
Expt
82.4 74.1
Others
74.6f74.9i
22.1 31.6 45.2
39
75 84.3
13.4
14.5
59.8
53.1
23.3
0.2813
0.623
0.511
0.310.37
43.1-59.2
223.3 181.7f
46.1f 81.7f f 39.26i 13 13.15i 14.3f14.41i 52e 62c 46.09i 84.51i wzCdTe
Present work
64.7
15.9 20.3 60.9
15.4
24.4
33.7
49.1
19.5
0.2572
0.631
188 153a
Expt Others aRef.
0.578
62.2b
[39] at 300 K,
bref.
35.9b 29.1b 68.9b
[45],
cref.
[41],
11.6b dref.
[42],
13.1b eref.
[40],
57.4e fref.
200g, 158h
[43],
CONCLUSIONS The first-principle calculations have been successfully performed to study the structural and elastic properties of wzCdX(X=S, Se, Te) semiconductors using Quantum ESPRESSO simulation package. The structural properties such as lattice constant (a) and (c), internal parameter (u), bulk modulus (B) and reduce pressure derivative ( ) using equation of state (eos) have been calculated and listed in Table 1. Further, elastic stiffness constants C11, C12, C13, C33, C44 and C66, have been calculated and listed in Table 2 along with the available experimental data and other theoretical results. The calculated values of B, Y, G, ,A, G/B ratio and θD of these semiconductors are presented in Table 2. The calculated values of all 18 parameters are in reasonable agreement with the experimental values and values reported by different workers, specially reported by P. Gopal et al. 2015 using ACBN0
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gref.
[44],
href.
[46],
iref.
[47].
functional. Reasonably good agreement except bulk modulus of wzCdTe, have been obtained between them at low computational cost using PAW method. The computational approach presented here is of immense importance which guides further study of structural and elastic properties for new heterostructure materials of this family recently developed by replacing X (S, Se, Te) type atom with different composition of S, Se and Te atoms simultaneously. ACKNOWLEDGEMENTS The author would like to thank C-DAC, Pune for providing the high-performance computing facility (NPSF) available in CDAC center for this research article. REFERENCES [1]. J.L. Shay, L.M. Schiavone, E. Buehler, J.H. Wernick. Spontaneousand stimulated-emission spectra of CdSnP2, J Appl Phys. 1972; 43: 2805–10p. doi:10.1063/1.1661599. Page 12
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