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Ab-initio calculation of the structural, electronic, optical and transport properties of SbNSr3 ternary nitride compound
Computational Condensed Matter 21 (2019) e00395
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Computational Condensed Matter journal homepage: http://ees.elsevier.com/cocom/default.asp
Ab-initio calculation of the structural, electronic, optical and transport properties of SbNSr3 ternary nitride compound H. Salehi*, N. Mousavinezhad, P. Amiri Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 February 2019 Received in revised form 26 April 2019 Accepted 28 April 2019
In this research, the structural, electronic, optical and transport properties of SbNSr3 compound are predicted using the pseudopotential plane wave (PP-PW) method within density functional theory (DFT). Hybrid functionals and traditional approximations in DFT for exchange-correlation potential were used and compared together. To reparametrize Heyd-Scuseria-Ernzerh of (HSE) functional, we predicted the band gap value of SbNSr3 compound 1.17 eV that is in the better agreement with experiment than other theoretical results. The modified HSE functional calculated the lattice parameter with an acceptable accuracy in compared to experimental lattice constant. The optical properties such as the real and imaginary part of dielectric function and the electron energy loss were calculated by the modified HSE functional. Consideration of the transport properties like the electronic thermal and electrical conductivity, Seebeck coefficient as well as thermoelectric figure of merit show that this material can be considered as a good thermoelectric material in low temperatures. © 2019 Elsevier B.V. All rights reserved.
Keywords: Density functional theory Hybrid functional SbNSr3 Transport properties Thermoelectric materials
1. Introduction The ternary nitrides and carbides belong to antiperovskite family having the general formula ABX3 (A is group IIIeV element, B is carbon or nitrogen and X is group I-II element or transition metal) exhibit interesting and different physical properties. These compounds crystallize in a cubic structure with space group Pm3m (221). Their constituent atoms, A, B and X are located at (0, 0, 0), (0.5, 0.5, 0.5), and (0, 0.5, 0.5) coordinates of the unit cell, respectively. These compound showed very interesting properties due to the wide range of band gap that makes them applied materials in different fields of technology [1]. SbNSr3 and BiNSr3 compounds displayed special thermoelectric properties at room temperature theoretically [2] which can be led to applications in thermoelectric fields. For the first time, SbNSr3 antiperovskite was synthesised by €bler et al. [3] and introduced as a diamagnetic semiconductor Ga material. Optical band gap of SbNSr3 was measured 1.15 eV using diffuse reflectivity method. Theoretical researchers considered different properties of SbNSr3 compound within density functional theory (DFT) [2,4e6]. The nearest reported band gap to its experimental value was calculated 0.92 eV by Bilal et al. using improved
* Corresponding author. E-mail address: [email protected] (H. Salehi). https://doi.org/10.1016/j.cocom.2019.e00395 2352-2143/© 2019 Elsevier B.V. All rights reserved.
modified BeckeeJohnson (I-mBJ) potential [2]. Properties of materials such as optical, thermoelectric, etc. are strongly sensitive to the band gap. Thus, the accurate calculated band gaps are required in the theoretical considerations of materials for practical applications. In DFT, the explicit form of exchange-correlation functional is unknown, due to that it must be approximated. The common functionals e.g. the local density approximate (LDA) [7] and the semi-local generalized gradient (GGA) [8] often underestimate the band gaps of materials [9e11]. The band gap results are improved by the exact exchange density functionals (hybrid functional) which are based on combination of Hartree-Fock (HF) and DFT [12]. Indeed, hybrid functional mixes a part of local or semi-local exchange term with a part of the nonlocal exact HF. Hence, there is possibility of modifying the parameters of these functionals to reach a higher accuracy for calculating the band structure [13,14]. In the present work, the band gap of SbNSr3 compound was underestimated by LDA and GGA approximations. PBE0 and HSE hybrid functional overestimated the electronic properties. To reparametrize HSE functional, we tried to calculate accurate electronic and structural properties, simultaneously. The band gap value and the lattice parameter were predicted about 1.17 eV and 5.14 Å using the modified HSE functional, respectively. Then, with regarded to dependency of optical and thermoelectric properties on the band structure these properties are calculated with the modified functional.
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2. Method and computational details Calculations were performed within density functional theory [15]. Quantum espresso computational package was used where is based upon the pseudopotential plane wave (PP-PW) method [16]. The norm conserving pseudopotential has been employed for all the ioneelectron interactions [17]. Pseudo atomic calculations are performed for Sb 5s2 5p3, N 2s2 2p3 and Sr 5s2. A plane-wave basis set with an energy cutoff of 70 (Ry) is applied. For the Brillouin zone sampling, we used a 666 mesh of k-point according to the scheme proposed by Monkhorst and Pack [18]. Different functionals for exchange-correlation potential were used. First, we applied the local Density Approximation and Generalized Gradient Approximation introduced by Perdew-Burke-Ernzerho (GGA-PBE). Since the present local and semi local functional showed significant errors in the description of band gap with respect to the experimental result, we resorted to hybrid density functionals that have improved upon the GGA results. The hybrid density functional combines a portion of exact exchange from Hartree fock theory with the rest of the exchange-correlation energy from tradition functional of DFT [12]. We used from PBE0 and HSE functionals. In the PBE0 functional, the GGAePBE and HF exchange energy are mixed whit a ratio of 3 to 1 for exchange energy and the full PBE correlation energy are applied for its correlation energy contribution. It is given by the following relation [19]:
EPBE0 ¼ XC
1 HF 3 PBE E þ E þ EPBE C : 4 X 4 X
(1)
PBE where EHF is X is the HartreeeFock exact exchange functional, EX PBE the PBE exchange functional, and EC is the PBE correlation functional. The results showed that this functional overestimate band gap of SbNSr3 compound. Heyd, Scuseria and Ernzerhof developed a hybrid density functional based on a screened Coulomb potential (HSE) that splits the coulomb operator into short range (SR) and long range (LR) components [20,21].
uPBEh EXC ¼
þ ð1 aÞ
EXPBE; SR
ðuÞ þ ð1 aÞ
EXPBE; LR
ðuÞ (2)
where a is the mixing parameter (exchange fraction) and u is a tunable parameter that controls the short ranginess of the interPBE0 action (screening parameter). So that EPBEh XC degenerates to EXC for PBE u ¼ 0 and EXC for u / ∞. Standard values a ¼ 0.25 and u ¼ 0.2 (1/ Å) have been shown a good agreement with experimental results for most systems. The exchange fraction (a) and the screening parameter can be adjusted to find out the best possible match between results predicted by modified HSE functional and presented by experiment. In this work, screening parameter is taken to be u ¼ 0.2 (1/Å) while exchange fraction will be changed as a ¼ [0.120, 0.160, 0.170 0.250]. With regard to the experimental results, the structural and electronic properties of SbNSr3 antiperovskite achieve to an acceptable accurate for a ¼ 0.160, Simultaneously. Boltzmann transport equation is one of the methods that deal with charge and heat transport. The transport considerations were performed using BoltzTraP code [22] that is based on Boltzmann transport equation and the rigid band approach. In fact, it depends on smoothed Fourier interpolation to get analytical expression of bands. The transport coefficients, electrical conductivity, Seebeck coefficient and electronic thermal conductivity are given in the Boltzmann transport theory [23], respectively
vfm ðT; εÞ dε: Sab ðεÞ vε
1 V
Sab ðT; mÞ ¼
1 eTV sab
keab ðT; mÞ ¼
1 e2 TV
ð
ð Sab ðεÞ ðε mÞ
Sab ðεÞ ðε mÞ2
(3) vfm ðT; εÞ dε: vε
vfm ðT; εÞ dε: vε
(4)
(5)
where V is volume of the unit cell, a and b display Cartesian indices, m and fm are the chemical potential and Fermi Dirac distribution function, respectively. The energy projected conductivity tensors can be calculated by using the following equation
Sab ðεÞ ¼
d ε εi; k e2 X ti; k va ði; kÞ vb ði; kÞ : N dε
(6)
i; k
where N is the number of k-points for BZ integration. i, k are the indexes of the band and k-point, respectively. va(i, k) represent a component of group velocity. The relaxation time has been displayed by t that the constant relaxation time approximation is used in BoltzTraP code since it cannot be assigned from band structure calculations [22]. Thus, it calculates transport coefficients electrical conductivity and electronic thermal conductivity, with the relaxation time. For calculating the real values of transport coefficients, we need to evaluate the carrier relaxation time. To assume that scattering procedure occurs predominantly due to acoustic phonons, others type of scattering are neglected in this work. The electronephonon interactions are described by deformation potential (DP) theory developed by Bardeen and Shockley [24] that delocalized electrons scattered by longitudinal acoustic phonons as modeled by uniform lattice dilation. The carrier relaxation time could be evaluated by applying deformation potential theory and within relaxation time approximation [24]. The carrier relaxation time within DP theory is defined as [24,25]:
SR a EHF; ðuÞ X PBE þ EC :
ð
sab ðT; mÞ ¼
t¼
2 m* kB T
3=2
p1=2 Z4 cii 3E21
:
(7)
where m* is the carrier effective mass, E1 denote the deformation potential constant and cii is the direction-dependent elastic constant.
3. Results and discussion 3.1. Structural properties The structural parameters of SbNSr3 antiperovskite were calculated employing LDA and GGA-PBE approximations as well as the hybrid functionals such as PBE0 and HSE for exchangecorrelation functional. The exchange fraction in the HSE functional was altered from 0.125 to 0.250 while screening parameter was considered as a constant value. The ground state of cubic (Pm3m) phase with atomic coordinates as Sb (0, 0, 0), N (0.5, 0.5, 0.5) and Sr (0, 0.5, 0.5) was relaxed. For the different unit cell volumes, the total energies have been obtained by optimization process to calculate the ground state structural parameters. The lattice parameter was calculated by fitting energyevolume data to Murnaghan equation of state (EOS) [26]. The optimization plot of SbNSr3 using HSE-0.16 functional as a prototype for all functionals was shown in Fig. 1. The lattice parameter, equilibrium volume, bulk modulus, pressure derivative of bulk modulus and compressibility
H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
Fig. 1. Optimization plot of SbNSr3 using HSE-0.16 as a prototype for all functionals.
calculated by all functionals were presented in Table 1. As it is observed in Table 1, lattice parameter resulted from PBE0 functional has a better agreement with the experimental result. No experimental data for the bulk modulus (B) and its pressure derivative (B0 ) are not available for the SbNSr3 compound to be compared with the present calculations. B was predicted 53.700 using PBE0 functional, while other theoretical researches have been calculated it as 48.08 [4], 56.21 [4], 41.63 [5] 52.7407 [6] by LDA, GGA, GGA-EV, WC-GGA, respectively. 3.2. Electronic properties The electronic properties of SbNSr3 compound, band structure and density of states, were calculated using different functionals for exchange-correlation functional. Calculated band gaps of SbNSr3 compound in this research and other researches have been reported in Table 2. G€ abler et al. measured experimentally optical band gap 1.15 eV [3]. So far, the closest theoretical result to experimental band gap has been predicted by Bilal et al. [2] 0.92 eV using improved modified BeckeeJohnson (I-mBJ) potential. In our calculations, the LDA and GGA underestimated and the PBE0 and HSE
3
functionals overestimated the band gap. Fig. 2 has depicted process of shifting energy bands whit changing exchange-correlation functinlas. Direct band gap is observed in G point. From Fig. 2a, it is clear that the hybrid functional due to exact exchange contribution opens gap rather than traditional functionals. Difference of the PBE0 and HSE functionals is in the treatment of the long-range part of the HF exchange interaction. The HSE functional calculated the band gap lower than PBE0 functional while it is still larger than experimental result. With regarded to tunability of the HF and DFT exchange fraction in the hybrid functionals, we modified contribution of HF exchange in the HSE functional (Fig. 2b). As it can be seen, the band gap decreases with reducing the exact exchange contribution. We note that reduction of exact exchange contribution decreases the lattice parameter. Therefore, the variation of HF exchange contribution in functional was depended on an acceptable agreement with experiment for both the band gap and the lattice parameter. It is observed that both band gap and lattice parameter reduce with diminishing the HF exchange contribution. Hence, that reduction of the band gap is faster than that of the lattice parameter. Therefore among the applied methods the modified HSE functional, HSE-0.16 which combines 0.16 of HF and 0.84 of DFT exchange energy, predicted band gap and lattice parameter within 1.17 eV and 5.148 Å. Relative error percentages for the band gap and lattice parameter were 1.73% and 0.46%, respectively. It should be pointed out that modifying exact fraction shifted only the band lines while the overall band structure remained same for all functionals. The details of the peak structures and the relative heights of the peaks in the DOS spectra of all methods are similar. Hence, we have showed band structure and density of states calculated by HSE-0.16 functional as a prototype in Fig. 2. The plotted energy range is from 14 to 5 eV. From Fig. 2a and b it was found that the lowest states around 12 eV below Fermi level originate predominately from N-2s states with contribution from Sr-5s states. The structure at about 9 eV arises from Sb-s and Sr-s states, which are well separated from the upper valence bands. Sb-p and N-p states constitute the upper valence bands (UVB) mainly while Sr-s states make some small contributions in this structure. The bottom of the conduction band is dominantly Sr-s character and a little admixture from Sb-sp, N-sp states. 3.3. Optical properties The optical response of the material is described by the
Table 1 The calculated structural parameters of SbNSr3 in this work and compared with results of others.
GGA-PBE LDA PBE0 HSE-0.250 HSE-0.120 HSE-0.160 HSE-0.170 Theoretical results
Exp a b c d e
Ref. Ref. Ref. Ref. Ref.
[4]. [5]. [6]. [3]. [27].
GGA-PBE LDA GGA-EV WC-GGA
a0 (Å)
V0 (Å3)
B (GPa)
B0
K (GPa)1
5.131 4.979 5.161 5.157 5.144 5.148 5.149 5.172a 5.067a 5.123b 5.1408c 5.1725d 5.1722e
135.130 123.470 137.520 137.170 136.170 136.473 136.590 138.348 130.092a 134.453b 135.8487c 138.3822d 138.3648e
51.900 62.500 53.700 54.100 53.000 53.200 53.400 48.08a 56.21a 41.63b 52.7407c e
4.350 4.470 4.440 4.530 4.500 4.420 4.500 4.049a 4.00a 5.21b e e
0.0192 0.0160 0.0186 0.0184 0.0188 0.0187 0.0187 e e e e e
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H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
Table 2 The calculated band gaps of SbNSr3 compound in this research and other theoretical and experimental result. LDA
Band gap (eV) a b c d e
Ref. Ref. Ref. Ref. Ref.
0.163
GGA
0.320
PBE0
2.575
HSE-0.250
1.775
HSE-0.170
1.273
HSE-0.160
1.170
HSE-0.120
0.932
Theoretical results
Exp
GGA
GGA-EV
I-mbj
0.31a
0.55b 0.51c
0.92e 0.78c
1.15d
[4]. [5]. [6]. [3]. [2].
Fig. 2. (a) The band structure of SbNSr3 calculated by different hybrid and traditional functionals, (b) The predicted band structure of SbNSr3 for different exact exchange contribution (a ¼ 0.120, 0.160, 0.250) in HSE functional.
frequency dependent dielectric function [28]. We considered the optical properties of SbNSr3 applying the HSE-0.16 functional potential. Since the modified HSE-0.16 functional yields better band gap value than other functionals, it expects to make a better agreement with the real result of the optical spectra which are not available at the moment. The calculated real and imaginary parts of
the dielectric function of SbNSr3 compound have been displayed in Fig. 3. Because of the cubic structure of SbNSr3 compound only one component of ε(u) has been rendered in this essay (Fig. 4). The real part of dielectric function, ε1(u) has been shown in Fig. 4a. The zero energy of the real part of dielectric function called the static dielectric constant is 7.819. The highest peak of ε1(u) occurs at
Fig. 3. The calculated electronic properties of SbNSr3 using HSE-0.16 (a) Electronic band structure along some high symmetry directions of the Brillouin zone. Vertical lines represent high-symmetry points of the Brillouin zone. The horizontal dashed line designates the Fermi level (EF). (b) Total and partial densities of states.
H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
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Fig. 4. (a) Real and (b) Imaginary part of dielectric function of SbNSr3 antiperovskite within 0e14 eV.
2.01 eV and is about 17.405. The real part of dielectric function is negative from 2.490 eV to 8.175 eV. The static refractive index, n(0) was calculated 2.796 that is in agreement with the reported results of the others theoretical researches [6]. The absorptive behavior of the material under electromagnetic waves excitation is described by the imaginary part of the dielectric function, ε2(u). Fig. 4b has been plotted ε2(u) for the energy range within 0e14 eV. As it is shown, ε2(u) up to energy of 0.945 eV is very insignificant then increases rapidly and follows the several peaks. This range of energy is called optical gap that is consistent with the calculated band gap. The highest peak is 24.679 and locates at 2.445 eV. The available peaks in spectra were attributed to transitions from N-p and Sb-p states to Sr-s state. It cannot be realistic to give a single transition assignment to peaks because there are many transitions in the band structure that have energy corresponding to the peak. The electron energy loss function is given by following relation [29].
LðuÞ ¼
ε2 ðuÞ : ε21 ðuÞ þ ε22 ðuÞ
(8)
L(u) that describes the energy loss of a fast electron traversing in a material is an important factor for considering the macroscopic and microscopic properties of solids. Fig. 5 has depicted the calculated energy loss spectra. As shown there are several peaks in spectrum
Fig. 5. Electron energy loss spectrum of SbNSr3 compound within 0e14 eV.
that the sharpest peak locates at 8.490 eV and determines the plasma frequency, up. It was expectable because the contribution of real and imaginary of dielectric was very insignificant in this region.
3.4. Transport properties Thermoelectric generators convert directly thermal energy into electrical energy. The transport properties such as thermal conductivity (k), electrical conductivity (s) and Seebeck coefficient (S) are important in the consideration of the thermoelectric efficiency of material. We employed BoltzTraP code based on semi classical Boltzmann equation to calculate transport parameters. It applies the constant relaxation time and calculates electronic thermal and electrical conductivity with the relaxation time. Hence, we evaluated the carrier relaxation time using equ. 7. within the deformation potential formalism to extract the real value of electronic thermal and electrical conductivity. Due to that, it was needed to calculate parameters such as dependent direction elastic constant, cii, the deformation potential constant, and the effective mass of charge carrier. In order to obtain the elastic constant, one can stretch and compress the crystal's lattice vectors along the external field's direction. Then, through fitting the total energy with respect to volume change, (E-E0)/V0 to dilation Da/a0 with (E-E0)/V0 ¼ Cii (Da/a0)2/2, the elastic constant, Cii, can be determined [25]. Where V0 and a0 are the equilibrium cell volume and lattice constant, respectively. We calculated Cii in the [100] direction. The deformation potential constant was calculated according to relation El ¼ dE/d(a/a0) [24]. Where, dE is the change of the conduction band minimum (CBM) and the valence band maximum (VBM) energy due to the applied stretch and compress. Since HSE-0.16 functional gives the best results for the band gap, the electronic properties predicted by HSE-0.16 functional were used. The changes of the total energy and the valence band maximum (VBM) energy due to the applied stretch and compress have been shown in Fig. 6a and b. According to equation m* ¼ Z2/(d2ε/dk2) where ε is the band edge energy as a function of wave vector k, the effective mass, m* can be obtained by fitting parabolic functions around the CBM and VBM for electrons and holes, respectively. Fig. 6c has been depicted temperature dependent relaxation time of hole and showed that it first decreases rapidly and then decreases slowly with temperature. The short hole relaxation time at the high temperatures indicates large holes scattering in SbNSr3. One of the main factors for a good thermoelectric material is called figure of merit, ZT ¼ sS2T/k, where s, S, T and k displayed electrical conductivity, Seebeck coefficient, temperature and
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H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
Figure 6. (a) the total energy of a unit cell versus lattice deformation along the [100] direction which gives the elastic constant, (b) The band edge shifts of valence and conduction band with respect to the lattice dilation along the [100] direction which give the deformation potential constant (El) according to the relation of El ¼ dE/d(a/a0), (c)Temperature dependent hole relaxation time of SbNSr3.
thermal conductivity consist of the electronic (kel) and lattice (kl) contribution. A good thermoelectric material has ZT equal to unite or higher than unit. The Seebeck coefficient or thermopower of a material is a measure of the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material. The use of materials with a high Seebeck coefficient (regarded to relation of ZT) is one of many important factors for the efficient behavior of thermoelectric generators. Seebeck coefficient dependent to chemical potential and temperature has been shown in Fig. 7. From the left panel, it is clear that S has nonzero value in the range of 0.3 to 0.3 of chemical potential for p-type and n-type SbNSr3, respectively. At zero point, middle of the band gap, is zero that is expectable because there is no carrier in region. Seebeck coefficient for both p-type and n-type region has maximum value. The changes of Seebeck coefficient with temperature for p-type SbNSr3 in this point is shown in the right panel. It is observable that low temperature Seebeck coefficient has high value and it decreases exponentially with temperature. The electronic thermal and the electrical conductivity per unit relaxation time as function of temperature have been calculated using Boltztrap code. Then, by multiplying the relaxation time with them, electronic thermal and the electrical conductivity were obtained and shown in Fig. 8. Calculations were performed within a
temperature range of 50e1200 K. Thermal conductivity is the flow of heat in materials due to electrons and phonons. In metals, it arises from free electrons and originates mainly from lattice vibrations in semiconductors. From Fig. 8a, it is clear that the electronic thermal conductivity increase with temperature. Up to 400 K, the increase of thermal conductivity is slower than those in the temperature range of 400e1200 K. Plot of the electrical conductivity (Fig. 8b) shows an increase versus temperature. Our calculations showed that ZT of SbNSr3 compound at 300 K have peaks around 0.93 and 0.87 at 0.088 eV and 0.074 eV chemical potential for p-type and n-type region, respectively (showed in Fig. 9a). The temperature dependency of figure of merit for p-type SbNSr3 in the point of 0.088 eV chemical potential has been presented in Fig. 9b. In the low temperature figure of merit is unit that is due to high S and low kel in this range. On the other hand, the low electrical conductivity reduces the figure of merit. Then ZT decreases with temperature and so reduction of S and increase of s and kel. Finally, it reaches to 0.3 at 1200 K. It was observed that SbNSr3 can be effective in thermoelectric application particularly at low temperatures. Also, one can be hoped that thermoelectric efficiency of SbNSr3 is improved through alloying with the suitable element or Nano structuring. There is no experimental result to compare. Our results are in a
Fig. 7. Seebeck coefficient of SbNSr3 compound as function of (a) chemical potential at T ¼ 300 K, (b) temperature.
H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
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Fig. 8. (a) Electronic thermal conductivity and (b) Electrical conductivity of SbNSr3 compound as function of temperature.
Fig. 9. Figure of merit of SbNSr3 compound as function of (a) chemical potential at 300 K, (b) temperature.
good agreement with the other theoretical consideration [2]. 4. Conclusion The structural, electronic, optical and transport properties of SbNSr3 antiperovskite are predicted employing the pseudopotential plane wave approach based on density functional theory by Hybrid functionals and traditional approximations for exchangecorrelation potential. Local density and generalized gradient approximation underestimate, and PBE0 and HSE functionals overestimated the electronic properties. To reparametrize the HSE exact functional, our calculation showed that the reduction of the contribution of HF exchange in HSE functional diminishes the band gap value. Such that, we taken the exchange fraction equal a ¼ 0.160 and calculated the closer band gap values to the experimental than the previous theoretical results. The optical properties were considered by the modified HSE-0.16 functional. Also, we studied the temperature dependency of the thermoelectric properties which showed interesting properties in low temperature. The figure of merit close to unit at 300 K showed that can be considered as a good thermoelectric material in low temperatures. Also, it may be possible to improve thermoelectric efficiency by alloying with the suitable element or Nano structuring. Unfortunately, the experimental results for the optical and transport properties of SbNSr3 compound were not available to compare our results. Our results were in accordance with other theoretical studies.
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Ab-initio calculation of the structural, electronic, optical and transport properties of SbNSr3 ternary nitride compound H. Salehi*, N. Mousavinezhad, P. Amiri Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 February 2019 Received in revised form 26 April 2019 Accepted 28 April 2019
In this research, the structural, electronic, optical and transport properties of SbNSr3 compound are predicted using the pseudopotential plane wave (PP-PW) method within density functional theory (DFT). Hybrid functionals and traditional approximations in DFT for exchange-correlation potential were used and compared together. To reparametrize Heyd-Scuseria-Ernzerh of (HSE) functional, we predicted the band gap value of SbNSr3 compound 1.17 eV that is in the better agreement with experiment than other theoretical results. The modified HSE functional calculated the lattice parameter with an acceptable accuracy in compared to experimental lattice constant. The optical properties such as the real and imaginary part of dielectric function and the electron energy loss were calculated by the modified HSE functional. Consideration of the transport properties like the electronic thermal and electrical conductivity, Seebeck coefficient as well as thermoelectric figure of merit show that this material can be considered as a good thermoelectric material in low temperatures. © 2019 Elsevier B.V. All rights reserved.
Keywords: Density functional theory Hybrid functional SbNSr3 Transport properties Thermoelectric materials
1. Introduction The ternary nitrides and carbides belong to antiperovskite family having the general formula ABX3 (A is group IIIeV element, B is carbon or nitrogen and X is group I-II element or transition metal) exhibit interesting and different physical properties. These compounds crystallize in a cubic structure with space group Pm3m (221). Their constituent atoms, A, B and X are located at (0, 0, 0), (0.5, 0.5, 0.5), and (0, 0.5, 0.5) coordinates of the unit cell, respectively. These compound showed very interesting properties due to the wide range of band gap that makes them applied materials in different fields of technology [1]. SbNSr3 and BiNSr3 compounds displayed special thermoelectric properties at room temperature theoretically [2] which can be led to applications in thermoelectric fields. For the first time, SbNSr3 antiperovskite was synthesised by €bler et al. [3] and introduced as a diamagnetic semiconductor Ga material. Optical band gap of SbNSr3 was measured 1.15 eV using diffuse reflectivity method. Theoretical researchers considered different properties of SbNSr3 compound within density functional theory (DFT) [2,4e6]. The nearest reported band gap to its experimental value was calculated 0.92 eV by Bilal et al. using improved
* Corresponding author. E-mail address: [email protected] (H. Salehi). https://doi.org/10.1016/j.cocom.2019.e00395 2352-2143/© 2019 Elsevier B.V. All rights reserved.
modified BeckeeJohnson (I-mBJ) potential [2]. Properties of materials such as optical, thermoelectric, etc. are strongly sensitive to the band gap. Thus, the accurate calculated band gaps are required in the theoretical considerations of materials for practical applications. In DFT, the explicit form of exchange-correlation functional is unknown, due to that it must be approximated. The common functionals e.g. the local density approximate (LDA) [7] and the semi-local generalized gradient (GGA) [8] often underestimate the band gaps of materials [9e11]. The band gap results are improved by the exact exchange density functionals (hybrid functional) which are based on combination of Hartree-Fock (HF) and DFT [12]. Indeed, hybrid functional mixes a part of local or semi-local exchange term with a part of the nonlocal exact HF. Hence, there is possibility of modifying the parameters of these functionals to reach a higher accuracy for calculating the band structure [13,14]. In the present work, the band gap of SbNSr3 compound was underestimated by LDA and GGA approximations. PBE0 and HSE hybrid functional overestimated the electronic properties. To reparametrize HSE functional, we tried to calculate accurate electronic and structural properties, simultaneously. The band gap value and the lattice parameter were predicted about 1.17 eV and 5.14 Å using the modified HSE functional, respectively. Then, with regarded to dependency of optical and thermoelectric properties on the band structure these properties are calculated with the modified functional.
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H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
2. Method and computational details Calculations were performed within density functional theory [15]. Quantum espresso computational package was used where is based upon the pseudopotential plane wave (PP-PW) method [16]. The norm conserving pseudopotential has been employed for all the ioneelectron interactions [17]. Pseudo atomic calculations are performed for Sb 5s2 5p3, N 2s2 2p3 and Sr 5s2. A plane-wave basis set with an energy cutoff of 70 (Ry) is applied. For the Brillouin zone sampling, we used a 666 mesh of k-point according to the scheme proposed by Monkhorst and Pack [18]. Different functionals for exchange-correlation potential were used. First, we applied the local Density Approximation and Generalized Gradient Approximation introduced by Perdew-Burke-Ernzerho (GGA-PBE). Since the present local and semi local functional showed significant errors in the description of band gap with respect to the experimental result, we resorted to hybrid density functionals that have improved upon the GGA results. The hybrid density functional combines a portion of exact exchange from Hartree fock theory with the rest of the exchange-correlation energy from tradition functional of DFT [12]. We used from PBE0 and HSE functionals. In the PBE0 functional, the GGAePBE and HF exchange energy are mixed whit a ratio of 3 to 1 for exchange energy and the full PBE correlation energy are applied for its correlation energy contribution. It is given by the following relation [19]:
EPBE0 ¼ XC
1 HF 3 PBE E þ E þ EPBE C : 4 X 4 X
(1)
PBE where EHF is X is the HartreeeFock exact exchange functional, EX PBE the PBE exchange functional, and EC is the PBE correlation functional. The results showed that this functional overestimate band gap of SbNSr3 compound. Heyd, Scuseria and Ernzerhof developed a hybrid density functional based on a screened Coulomb potential (HSE) that splits the coulomb operator into short range (SR) and long range (LR) components [20,21].
uPBEh EXC ¼
þ ð1 aÞ
EXPBE; SR
ðuÞ þ ð1 aÞ
EXPBE; LR
ðuÞ (2)
where a is the mixing parameter (exchange fraction) and u is a tunable parameter that controls the short ranginess of the interPBE0 action (screening parameter). So that EPBEh XC degenerates to EXC for PBE u ¼ 0 and EXC for u / ∞. Standard values a ¼ 0.25 and u ¼ 0.2 (1/ Å) have been shown a good agreement with experimental results for most systems. The exchange fraction (a) and the screening parameter can be adjusted to find out the best possible match between results predicted by modified HSE functional and presented by experiment. In this work, screening parameter is taken to be u ¼ 0.2 (1/Å) while exchange fraction will be changed as a ¼ [0.120, 0.160, 0.170 0.250]. With regard to the experimental results, the structural and electronic properties of SbNSr3 antiperovskite achieve to an acceptable accurate for a ¼ 0.160, Simultaneously. Boltzmann transport equation is one of the methods that deal with charge and heat transport. The transport considerations were performed using BoltzTraP code [22] that is based on Boltzmann transport equation and the rigid band approach. In fact, it depends on smoothed Fourier interpolation to get analytical expression of bands. The transport coefficients, electrical conductivity, Seebeck coefficient and electronic thermal conductivity are given in the Boltzmann transport theory [23], respectively
vfm ðT; εÞ dε: Sab ðεÞ vε
1 V
Sab ðT; mÞ ¼
1 eTV sab
keab ðT; mÞ ¼
1 e2 TV
ð
ð Sab ðεÞ ðε mÞ
Sab ðεÞ ðε mÞ2
(3) vfm ðT; εÞ dε: vε
vfm ðT; εÞ dε: vε
(4)
(5)
where V is volume of the unit cell, a and b display Cartesian indices, m and fm are the chemical potential and Fermi Dirac distribution function, respectively. The energy projected conductivity tensors can be calculated by using the following equation
Sab ðεÞ ¼
d ε εi; k e2 X ti; k va ði; kÞ vb ði; kÞ : N dε
(6)
i; k
where N is the number of k-points for BZ integration. i, k are the indexes of the band and k-point, respectively. va(i, k) represent a component of group velocity. The relaxation time has been displayed by t that the constant relaxation time approximation is used in BoltzTraP code since it cannot be assigned from band structure calculations [22]. Thus, it calculates transport coefficients electrical conductivity and electronic thermal conductivity, with the relaxation time. For calculating the real values of transport coefficients, we need to evaluate the carrier relaxation time. To assume that scattering procedure occurs predominantly due to acoustic phonons, others type of scattering are neglected in this work. The electronephonon interactions are described by deformation potential (DP) theory developed by Bardeen and Shockley [24] that delocalized electrons scattered by longitudinal acoustic phonons as modeled by uniform lattice dilation. The carrier relaxation time could be evaluated by applying deformation potential theory and within relaxation time approximation [24]. The carrier relaxation time within DP theory is defined as [24,25]:
SR a EHF; ðuÞ X PBE þ EC :
ð
sab ðT; mÞ ¼
t¼
2 m* kB T
3=2
p1=2 Z4 cii 3E21
:
(7)
where m* is the carrier effective mass, E1 denote the deformation potential constant and cii is the direction-dependent elastic constant.
3. Results and discussion 3.1. Structural properties The structural parameters of SbNSr3 antiperovskite were calculated employing LDA and GGA-PBE approximations as well as the hybrid functionals such as PBE0 and HSE for exchangecorrelation functional. The exchange fraction in the HSE functional was altered from 0.125 to 0.250 while screening parameter was considered as a constant value. The ground state of cubic (Pm3m) phase with atomic coordinates as Sb (0, 0, 0), N (0.5, 0.5, 0.5) and Sr (0, 0.5, 0.5) was relaxed. For the different unit cell volumes, the total energies have been obtained by optimization process to calculate the ground state structural parameters. The lattice parameter was calculated by fitting energyevolume data to Murnaghan equation of state (EOS) [26]. The optimization plot of SbNSr3 using HSE-0.16 functional as a prototype for all functionals was shown in Fig. 1. The lattice parameter, equilibrium volume, bulk modulus, pressure derivative of bulk modulus and compressibility
H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
Fig. 1. Optimization plot of SbNSr3 using HSE-0.16 as a prototype for all functionals.
calculated by all functionals were presented in Table 1. As it is observed in Table 1, lattice parameter resulted from PBE0 functional has a better agreement with the experimental result. No experimental data for the bulk modulus (B) and its pressure derivative (B0 ) are not available for the SbNSr3 compound to be compared with the present calculations. B was predicted 53.700 using PBE0 functional, while other theoretical researches have been calculated it as 48.08 [4], 56.21 [4], 41.63 [5] 52.7407 [6] by LDA, GGA, GGA-EV, WC-GGA, respectively. 3.2. Electronic properties The electronic properties of SbNSr3 compound, band structure and density of states, were calculated using different functionals for exchange-correlation functional. Calculated band gaps of SbNSr3 compound in this research and other researches have been reported in Table 2. G€ abler et al. measured experimentally optical band gap 1.15 eV [3]. So far, the closest theoretical result to experimental band gap has been predicted by Bilal et al. [2] 0.92 eV using improved modified BeckeeJohnson (I-mBJ) potential. In our calculations, the LDA and GGA underestimated and the PBE0 and HSE
3
functionals overestimated the band gap. Fig. 2 has depicted process of shifting energy bands whit changing exchange-correlation functinlas. Direct band gap is observed in G point. From Fig. 2a, it is clear that the hybrid functional due to exact exchange contribution opens gap rather than traditional functionals. Difference of the PBE0 and HSE functionals is in the treatment of the long-range part of the HF exchange interaction. The HSE functional calculated the band gap lower than PBE0 functional while it is still larger than experimental result. With regarded to tunability of the HF and DFT exchange fraction in the hybrid functionals, we modified contribution of HF exchange in the HSE functional (Fig. 2b). As it can be seen, the band gap decreases with reducing the exact exchange contribution. We note that reduction of exact exchange contribution decreases the lattice parameter. Therefore, the variation of HF exchange contribution in functional was depended on an acceptable agreement with experiment for both the band gap and the lattice parameter. It is observed that both band gap and lattice parameter reduce with diminishing the HF exchange contribution. Hence, that reduction of the band gap is faster than that of the lattice parameter. Therefore among the applied methods the modified HSE functional, HSE-0.16 which combines 0.16 of HF and 0.84 of DFT exchange energy, predicted band gap and lattice parameter within 1.17 eV and 5.148 Å. Relative error percentages for the band gap and lattice parameter were 1.73% and 0.46%, respectively. It should be pointed out that modifying exact fraction shifted only the band lines while the overall band structure remained same for all functionals. The details of the peak structures and the relative heights of the peaks in the DOS spectra of all methods are similar. Hence, we have showed band structure and density of states calculated by HSE-0.16 functional as a prototype in Fig. 2. The plotted energy range is from 14 to 5 eV. From Fig. 2a and b it was found that the lowest states around 12 eV below Fermi level originate predominately from N-2s states with contribution from Sr-5s states. The structure at about 9 eV arises from Sb-s and Sr-s states, which are well separated from the upper valence bands. Sb-p and N-p states constitute the upper valence bands (UVB) mainly while Sr-s states make some small contributions in this structure. The bottom of the conduction band is dominantly Sr-s character and a little admixture from Sb-sp, N-sp states. 3.3. Optical properties The optical response of the material is described by the
Table 1 The calculated structural parameters of SbNSr3 in this work and compared with results of others.
GGA-PBE LDA PBE0 HSE-0.250 HSE-0.120 HSE-0.160 HSE-0.170 Theoretical results
Exp a b c d e
Ref. Ref. Ref. Ref. Ref.
[4]. [5]. [6]. [3]. [27].
GGA-PBE LDA GGA-EV WC-GGA
a0 (Å)
V0 (Å3)
B (GPa)
B0
K (GPa)1
5.131 4.979 5.161 5.157 5.144 5.148 5.149 5.172a 5.067a 5.123b 5.1408c 5.1725d 5.1722e
135.130 123.470 137.520 137.170 136.170 136.473 136.590 138.348 130.092a 134.453b 135.8487c 138.3822d 138.3648e
51.900 62.500 53.700 54.100 53.000 53.200 53.400 48.08a 56.21a 41.63b 52.7407c e
4.350 4.470 4.440 4.530 4.500 4.420 4.500 4.049a 4.00a 5.21b e e
0.0192 0.0160 0.0186 0.0184 0.0188 0.0187 0.0187 e e e e e
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H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
Table 2 The calculated band gaps of SbNSr3 compound in this research and other theoretical and experimental result. LDA
Band gap (eV) a b c d e
Ref. Ref. Ref. Ref. Ref.
0.163
GGA
0.320
PBE0
2.575
HSE-0.250
1.775
HSE-0.170
1.273
HSE-0.160
1.170
HSE-0.120
0.932
Theoretical results
Exp
GGA
GGA-EV
I-mbj
0.31a
0.55b 0.51c
0.92e 0.78c
1.15d
[4]. [5]. [6]. [3]. [2].
Fig. 2. (a) The band structure of SbNSr3 calculated by different hybrid and traditional functionals, (b) The predicted band structure of SbNSr3 for different exact exchange contribution (a ¼ 0.120, 0.160, 0.250) in HSE functional.
frequency dependent dielectric function [28]. We considered the optical properties of SbNSr3 applying the HSE-0.16 functional potential. Since the modified HSE-0.16 functional yields better band gap value than other functionals, it expects to make a better agreement with the real result of the optical spectra which are not available at the moment. The calculated real and imaginary parts of
the dielectric function of SbNSr3 compound have been displayed in Fig. 3. Because of the cubic structure of SbNSr3 compound only one component of ε(u) has been rendered in this essay (Fig. 4). The real part of dielectric function, ε1(u) has been shown in Fig. 4a. The zero energy of the real part of dielectric function called the static dielectric constant is 7.819. The highest peak of ε1(u) occurs at
Fig. 3. The calculated electronic properties of SbNSr3 using HSE-0.16 (a) Electronic band structure along some high symmetry directions of the Brillouin zone. Vertical lines represent high-symmetry points of the Brillouin zone. The horizontal dashed line designates the Fermi level (EF). (b) Total and partial densities of states.
H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
5
Fig. 4. (a) Real and (b) Imaginary part of dielectric function of SbNSr3 antiperovskite within 0e14 eV.
2.01 eV and is about 17.405. The real part of dielectric function is negative from 2.490 eV to 8.175 eV. The static refractive index, n(0) was calculated 2.796 that is in agreement with the reported results of the others theoretical researches [6]. The absorptive behavior of the material under electromagnetic waves excitation is described by the imaginary part of the dielectric function, ε2(u). Fig. 4b has been plotted ε2(u) for the energy range within 0e14 eV. As it is shown, ε2(u) up to energy of 0.945 eV is very insignificant then increases rapidly and follows the several peaks. This range of energy is called optical gap that is consistent with the calculated band gap. The highest peak is 24.679 and locates at 2.445 eV. The available peaks in spectra were attributed to transitions from N-p and Sb-p states to Sr-s state. It cannot be realistic to give a single transition assignment to peaks because there are many transitions in the band structure that have energy corresponding to the peak. The electron energy loss function is given by following relation [29].
LðuÞ ¼
ε2 ðuÞ : ε21 ðuÞ þ ε22 ðuÞ
(8)
L(u) that describes the energy loss of a fast electron traversing in a material is an important factor for considering the macroscopic and microscopic properties of solids. Fig. 5 has depicted the calculated energy loss spectra. As shown there are several peaks in spectrum
Fig. 5. Electron energy loss spectrum of SbNSr3 compound within 0e14 eV.
that the sharpest peak locates at 8.490 eV and determines the plasma frequency, up. It was expectable because the contribution of real and imaginary of dielectric was very insignificant in this region.
3.4. Transport properties Thermoelectric generators convert directly thermal energy into electrical energy. The transport properties such as thermal conductivity (k), electrical conductivity (s) and Seebeck coefficient (S) are important in the consideration of the thermoelectric efficiency of material. We employed BoltzTraP code based on semi classical Boltzmann equation to calculate transport parameters. It applies the constant relaxation time and calculates electronic thermal and electrical conductivity with the relaxation time. Hence, we evaluated the carrier relaxation time using equ. 7. within the deformation potential formalism to extract the real value of electronic thermal and electrical conductivity. Due to that, it was needed to calculate parameters such as dependent direction elastic constant, cii, the deformation potential constant, and the effective mass of charge carrier. In order to obtain the elastic constant, one can stretch and compress the crystal's lattice vectors along the external field's direction. Then, through fitting the total energy with respect to volume change, (E-E0)/V0 to dilation Da/a0 with (E-E0)/V0 ¼ Cii (Da/a0)2/2, the elastic constant, Cii, can be determined [25]. Where V0 and a0 are the equilibrium cell volume and lattice constant, respectively. We calculated Cii in the [100] direction. The deformation potential constant was calculated according to relation El ¼ dE/d(a/a0) [24]. Where, dE is the change of the conduction band minimum (CBM) and the valence band maximum (VBM) energy due to the applied stretch and compress. Since HSE-0.16 functional gives the best results for the band gap, the electronic properties predicted by HSE-0.16 functional were used. The changes of the total energy and the valence band maximum (VBM) energy due to the applied stretch and compress have been shown in Fig. 6a and b. According to equation m* ¼ Z2/(d2ε/dk2) where ε is the band edge energy as a function of wave vector k, the effective mass, m* can be obtained by fitting parabolic functions around the CBM and VBM for electrons and holes, respectively. Fig. 6c has been depicted temperature dependent relaxation time of hole and showed that it first decreases rapidly and then decreases slowly with temperature. The short hole relaxation time at the high temperatures indicates large holes scattering in SbNSr3. One of the main factors for a good thermoelectric material is called figure of merit, ZT ¼ sS2T/k, where s, S, T and k displayed electrical conductivity, Seebeck coefficient, temperature and
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H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
Figure 6. (a) the total energy of a unit cell versus lattice deformation along the [100] direction which gives the elastic constant, (b) The band edge shifts of valence and conduction band with respect to the lattice dilation along the [100] direction which give the deformation potential constant (El) according to the relation of El ¼ dE/d(a/a0), (c)Temperature dependent hole relaxation time of SbNSr3.
thermal conductivity consist of the electronic (kel) and lattice (kl) contribution. A good thermoelectric material has ZT equal to unite or higher than unit. The Seebeck coefficient or thermopower of a material is a measure of the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material. The use of materials with a high Seebeck coefficient (regarded to relation of ZT) is one of many important factors for the efficient behavior of thermoelectric generators. Seebeck coefficient dependent to chemical potential and temperature has been shown in Fig. 7. From the left panel, it is clear that S has nonzero value in the range of 0.3 to 0.3 of chemical potential for p-type and n-type SbNSr3, respectively. At zero point, middle of the band gap, is zero that is expectable because there is no carrier in region. Seebeck coefficient for both p-type and n-type region has maximum value. The changes of Seebeck coefficient with temperature for p-type SbNSr3 in this point is shown in the right panel. It is observable that low temperature Seebeck coefficient has high value and it decreases exponentially with temperature. The electronic thermal and the electrical conductivity per unit relaxation time as function of temperature have been calculated using Boltztrap code. Then, by multiplying the relaxation time with them, electronic thermal and the electrical conductivity were obtained and shown in Fig. 8. Calculations were performed within a
temperature range of 50e1200 K. Thermal conductivity is the flow of heat in materials due to electrons and phonons. In metals, it arises from free electrons and originates mainly from lattice vibrations in semiconductors. From Fig. 8a, it is clear that the electronic thermal conductivity increase with temperature. Up to 400 K, the increase of thermal conductivity is slower than those in the temperature range of 400e1200 K. Plot of the electrical conductivity (Fig. 8b) shows an increase versus temperature. Our calculations showed that ZT of SbNSr3 compound at 300 K have peaks around 0.93 and 0.87 at 0.088 eV and 0.074 eV chemical potential for p-type and n-type region, respectively (showed in Fig. 9a). The temperature dependency of figure of merit for p-type SbNSr3 in the point of 0.088 eV chemical potential has been presented in Fig. 9b. In the low temperature figure of merit is unit that is due to high S and low kel in this range. On the other hand, the low electrical conductivity reduces the figure of merit. Then ZT decreases with temperature and so reduction of S and increase of s and kel. Finally, it reaches to 0.3 at 1200 K. It was observed that SbNSr3 can be effective in thermoelectric application particularly at low temperatures. Also, one can be hoped that thermoelectric efficiency of SbNSr3 is improved through alloying with the suitable element or Nano structuring. There is no experimental result to compare. Our results are in a
Fig. 7. Seebeck coefficient of SbNSr3 compound as function of (a) chemical potential at T ¼ 300 K, (b) temperature.
H. Salehi et al. / Computational Condensed Matter 21 (2019) e00395
7
Fig. 8. (a) Electronic thermal conductivity and (b) Electrical conductivity of SbNSr3 compound as function of temperature.
Fig. 9. Figure of merit of SbNSr3 compound as function of (a) chemical potential at 300 K, (b) temperature.
good agreement with the other theoretical consideration [2]. 4. Conclusion The structural, electronic, optical and transport properties of SbNSr3 antiperovskite are predicted employing the pseudopotential plane wave approach based on density functional theory by Hybrid functionals and traditional approximations for exchangecorrelation potential. Local density and generalized gradient approximation underestimate, and PBE0 and HSE functionals overestimated the electronic properties. To reparametrize the HSE exact functional, our calculation showed that the reduction of the contribution of HF exchange in HSE functional diminishes the band gap value. Such that, we taken the exchange fraction equal a ¼ 0.160 and calculated the closer band gap values to the experimental than the previous theoretical results. The optical properties were considered by the modified HSE-0.16 functional. Also, we studied the temperature dependency of the thermoelectric properties which showed interesting properties in low temperature. The figure of merit close to unit at 300 K showed that can be considered as a good thermoelectric material in low temperatures. Also, it may be possible to improve thermoelectric efficiency by alloying with the suitable element or Nano structuring. Unfortunately, the experimental results for the optical and transport properties of SbNSr3 compound were not available to compare our results. Our results were in accordance with other theoretical studies.
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