Assessing the thermoelectric properties of ScRhTe half-heusler compound

Accepted Manuscript Assessing the thermoelectric properties of ScRhTe half-heusler compound Sukhwinder Singh PII:

S2352-2143(17)30169-7

DOI:

10.1016/j.cocom.2017.10.002

Reference:

COCOM 103

To appear in:

Computational Condensed Matter

Received Date: 5 August 2017 Revised Date:

8 October 2017

Accepted Date: 10 October 2017

Please cite this article as: S. Singh, Assessing the thermoelectric properties of ScRhTe half-heusler compound, Computational Condensed Matter (2017), doi: 10.1016/j.cocom.2017.10.002. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Assessing the thermoelectric properties of ScRhTe half-heusler compound Sukhwinder Singh Department of Physics, Panjab University, Chandigarh-160014 Email id: [email protected]

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Abstract: We investigate structural, elastic, electronic and thermoelectric properties of ScRhTe with frame work of density functional theory and Boltzmann equations. The physical properties such as elastic constants, bulk modulus, shear modulus, Young's modulus and Poisson's ratio of ScRhTe compound are calculated. The elastic properties of ScRhTe reveal that this compound is mechanically stable. Further, the phonon calculation of ScRhTe indicates that it is also dynamically stable. The band structure as well as DOS spectra indicates that ScRhTe compound has a semiconductor nature. The calculated energy band gap of ScRhTe compound is 0.8eV.The thermoelectric properties such as Seebeck coefficient, electrical conductivity scaled by relaxation time, power factor scaled by relaxation time and electronic thermal conductivity has been investigated with variation of carrier concentration and temperature. The highest power factor obtained at 1200K is 11.9x1011V2SK-2ms-1 at optimum carrier concentration n~1.3x1021cm-3 for n-type ScRhTe. The maximum dimensionless figure of merit for n-type ScRhTe is 0.63. Using the slack equation, the lattice thermal conductivity of ScRhTe is also estimated. At room temperature, ScRhTe attains low thermal conductivity as compared to well know ZrNiPb half heusler compound. The lower lattice thermal conductivity of ScRhTe suggest that this compound may be act as good thermoelectric material. Our results suggested that power factor can’t improve further when electron carrier concentration reaches to n~1.3x1021cm-3 and hole carrier concentration reaches to n~7.5x1020cm-3. This is the first quantitative theoretical estimation of these properties. Keywords: Half heusler, Thermoelectric properties, DFT, Seebeck coefficient. 1. Introduction:

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Researches have shown the interest to improve alternative eco-friendly energy technologies for last three decades to overcome the expending energy demand. For production of energy, Scientists are trying to improve the efficiency of eco-friendly technologies such as wind turbine, biomass, solar photovoltaic and thermoelectric [1]. Thermoelectric technology is unique technology as compared to other technologies due to capability of conversion of waste heat into electricity. The efficiency of thermoelectric materials is gauged by dimensionless figure of merit (ZT) which is defined as [2]: ZT=


  

(1)

where S,σ,T and k represents Seebeck coefficient, electrical conductivity ,absolute temperature and total thermal conductivity(due to lattice(kl) and electron thermal conductivity(ke)).High efficient thermoelectric material needs high power factor (S σ) and low thermal conductivity (k). The thermoelectric properties of many half heusler compounds have been investigated by researches. According to literature, n-type XNiSn (X=Ti,Zr,Hf) and p-type XCoSb

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(X=Ti,Zr,Hf) half heusler compounds are good thermoelectric materials due to narrow band gap, low thermal conductivity and high power factor [3 ]. Recently, H.B.Ozisik [4] et al. studied the electronic and optical properties of NiXSn (X=Zr,Hf) under pressure. The electronic structure of RPtBi(R=La,Lu) was studied experimentally as well theoretically [5]. Thermoelectric properties of LaBiPt have been calculated by Jung et al. [6]. S.Singh [7] studied the thermoelectric properties of XAuPb(X=Lu,Y) compound under strain. FeXSb (X=V,Nb) [8-12 ] has also attracted the attention of researches due to high Seebeck coefficient and high power factor at 300K. C.G.Fu et al.[11] achieved the ZT~1.1 in p-type Ti doped FeNbSb at 1100K.The electronic and thermoelectric properties of p-type NbFeSb was studied by T.Fang et al. [13 ] and found that when hole concentration reaches to 2.6x1021cm-3 then power factor can’t improve further. Joshi et al. [14] achieved ZT~1 at 973K for p-type Nb0.6Ti0.4FeSb0.95Sn0.05 composition. D.Wang et al. [15] reported the thermoelectric properties of Ni doped ZrPdPb and found that Ni substitution decreases the lattice thermal conductivity from 26.4Wm-1K-1 to 6.37 26.4Wm-1K-1. D.wang et al. [16], studied the thermoelectric and elastic properties of pure and Hf doped ZrNiPb compound and reported the lattice thermal conductivity of these compounds using Slack equation. P.Hermet et al. [17], studied the mechanical and thermodynamic properties of TiNiSn compound by using the first principles calculations. R.Gautier et al. [18] reported the thermoelectric properties of newly developed ZrNiPb and they found that it has high power factor as compared to Zr0.5Hf0.5NiSn. Using elastic constants, B.Kong et al. [19] also studied the various physical quantities of FeVSb compound such as Debye temperature, Poisson’s ratio, Bulk modulus etc. Y.Wu et al. [20] derived the Debye temperatures of NiMnN (Si,Sb and As) compounds with the help of average sound velocities and they found that all these compounds are ductile in nature.

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In this work, we present an investigation on the structural, elastic, phonon, electronic and thermoelectric properties of newly predicted ScRhTe half-heusler compound using the first principles calculations and Boltzmann equations. As per our knowledge, no theoretical and experimental work has been reported yet. 2. Computational methodology:

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First principles calculations were performed by using the Quantum Espresso package [21], which implements Density Functional theory (DFT). The Local Density approximation (LDA) of Perdew-Wang [22] was used for exchange –correlation functional .The cutoff for the kinetic energy was fixed to 75RY for the plane-wave expansion of the electronic wave functions. The charge-density cutoff was kept at 300RY and the marzari-vanderbilt cold smearing size was fixed at 0.003 RY. The Brillouin zone integration was performed using Monkhorst–Pack scheme [23] with 9× 9 × 9 K-point mesh for ScRhTe compound. In order to calculate the density of state (DOS), we used tetrahedron method with 12x12x12 denser kpoint mesh. The elastic and thermodynamics properties were calculated using thermo_pw package [24, 25]. The phonon band structure has been calculated using density functional perturbation theory (DFPT) [26]. The dynamical matrices have been calculated for 4x4x4 qpoints grid generated according to Monkhorst and pack [23]. Using Slack numerical method, [27] we estimated the lattice thermal conductivity of ScRhTe compound.In order to calculate the thermoelectric properties, we have used BoltzTraP code [28].This code is based on constant relaxation time and rigid band approximations [29, 30]. With the help of this code we are able to calculate the thermoelectric coefficients such as Seebeck coefficient(S), electrical conductivity scaled by relaxation time (σ/ꞇ ) and electronic thermal conductivity scaled by relaxation time (ke/ꞇ ), where ꞇ is relaxation time. The structural parameters i.e

ACCEPTED MANUSCRIPT lattice constant and bulk modulus have been calculated by computing the total energy for different volumes and fitted to Murnaghan’s equation of state [31]: ( ) =



  

 ⌊ 





− 1⌋

(2)

3. Results and discussion: (a) Structural properties and lattice thermodynamics:

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Where P is the pressure, V0 is the reference volume, V is the deformed volume, B0 is the bulk modulus and , is the derivative of the bulk modulus with respect to pressure.

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The half heusler are ternary compounds with LiAlSi-type structure. These compounds crystallized in face-centered cubic structure with space group F43m (space group no.216). The unit cell of ScRhTe contains four formula unities with Sc,Rh and Te atoms locating at the 4b (0.5,0.5,0.5),4c(0.25,0.25,0.25) and 4a(0,0,0) Wyckoff sites respectively.The variation of total energy as a function of volume of ScRhTe compound is depicted in Fig.1. This plot is fitted to the Murnaghan’s equation of state [31] (EOS) in order to calculate the various structural parameters such as equilibrium lattice constant a, Bulk modulus (B), derivative of bulk modulus (B’), minimum volume (V0) and ground state energy (E0).The calculated value of lattice constant is 6.07Å and this value is in the good agreement with value reported by R.Gautier et al. [18]. These calculated parameters are summarized in Table 1.

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Table1. Calculated values of lattice constant (a), bulk modulus (B) (GPa), pressure derivative of bulk modulus (B’), minimum volume (V0) and ground state energy (E0) Compound

a( )

B(GPa)

ScRhTe

6.07,6.35[18] 139.9

B’

V0(a.u3)

E0 (Ry)

4.97

378.01

-64.45

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In order to investigate the stability of ScRhTe, we evaluated the phonon band structure. The phonon band structure for ScRhTe has been calculated by plotting vibrational frequencies along high symmetry directions as shown in Fig.2. The ScRhTe compound contains three atoms per unit cell, thus there are nine normal modes of vibrations, which includes three acoustic and six optical modes. In Fig.2, there is no imaginary frequency, which indicates the dynamically stability of ScRhTe compound. At low frequencies, on X and W high symmetric points some information is missing. Furthermore, in fig.2 there is absence of LO-TO phonon splitting at gamma point.

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Fig.1 Total energy as a function of volume for ScRhTe compound.

Fig.2 The phonon band structure of ScRhTe compound.

ACCEPTED MANUSCRIPT (b).Elastic constants and Debye temperature For ScRhTe cubic compound, there are three independent elastic constants (Cij) and their calculated values are displayed in Table 2. The mechanical stability of ScRhTe can be checked by using Born-Huang stability criteria [32]: C11-C12>0, C11>0, C44>0 and (C11+2C12)>0

(3)

$=

(3 − 2") (4) 2(3 + ")

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It is evident that elastic constant in Table 2 satisfies the Born-Huang stability criteria. Therefore, according to this criteria ScRhTe is mechanically stable. With the help of VoigtReuss-Hill approximations (VRH) [33,34], we estimate the bulk modulus (B) and shear modulus (G) of ScRhTe compound .The value of VRH bulk modulus (138.8GPa) is in good agreement with EOS Bulk modulus (139.9GPa) which indicates the good self-consistency of this work. The Poisson’s ratio (ν) and Young’s modulus (Y) are estimated with the help of following relations:

9" (5) 3 + "

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All above calculated parameters are given in Table 2. The obtained value of Poisson’s ratio is 0.29.The calculated density of ScRhTe compound is 8.16 g/cm3. The Debye temperature is associated with the specific heat, elastic constants and the melting point [35]. The Debye temperature of ScRhTe is estimated from average sound velocity. The shear (vs) and compressional (vp) sound velocities are calculated using following relations [36]: G v( = ) (6) ρ

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3B + 4G v- = ) (7) 3ρ

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Where B, G and ρ are bulk modulus, shear modulus and density of material respectively. The average sound velocity (vm) in terms of compressional (vp) and shear (vs) sound velocities is given as [37]: 3 3/4

01 = 245



3

3/4

78 : + 8 : < 9

;

(8)

The Debye temperature (ɵ> ) in terms of average sound velocity is written as: ħ 4B EFG

ɵ> = @ ACD 2

H

3/4

5I

01 (9)

Where ħ is Planck’s constant, k is Boltzmann’s constant, NA is Avogadro’s number, n is the number of atoms per formula unit and ρ is the density.

ACCEPTED MANUSCRIPT The calculated quantities from Eqs. (6-9) are given in Table 2. The calculated value of Debye temperature of ScRhTe compound is 352.7 K. Table 2. Elastic and thermodynamic properties of ScRhTe compound. Values

1

C11(GPa)

218.5

2

C12 (GPa)

99.1

3

C44(GPa)

4

C11-C12(GPa)

5

C11+2C12(GPa)

6

B(GPa)

7

G(GPa)

8

Y(GPa)

9

Poisson’s Ratio( ν)

0.29

10

ρ(g.cm-3)

8.16

0J (ms-1)

11

0K (ms-1)

13 14

ɵ> (L)

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(c).Electronic properties

1 (ms-1)

68.4

119.4

416.7

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138.8

64.7

168.1

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Properties

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2816.1

3143.5 352.7

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The calculated band structure of ScRhTe compound is depicted in fig.3 (a). It is seen that the top of valence band maxima lies at Г symmetry point and conduction band minima also lies at Г symmetry point. Hence, ScRhTe compound is a direct band gap semiconductor [18]. The calculated energy band gap of this compound is 0.8 eV and it gives good agreement with the value reported by R.Gautier et al. [18]. The total DOS and partial DOS of ScRhTe compound from -10eV to 10eV are plotted in fig.3. (b). Since the DOS and partial DOS shows an energy gap near the Fermi level (Ef), therefore ScRhTe compound should exhibit a semiconducting nature. The partial DOS confirms that the valence band of ScRhTe compound mainly consists of the Rh_d, Sc_d and Te_p orbitals while all remaining orbitals give minor contribution in valence band. The conduction band of this compound primarily consists of Sc_d, Te_p orbitals while all remaining orbital gives small contribution in conduction band.

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Fig.3 (a) The band structure (b) The total DOS and partial DOS of ScRhTe compound. (d). Thermoelectric properties: The calculated thermoelectric properties such as as Seebeck coefficient (S), electrical conductivity (σ/ꞇ ) scaled by relaxation time, power factor (S2 σ/ꞇ ) scaled by relaxation time and electronic thermal conductivity (ke) as the function of electron carrier concentration at different values of temperature for n-type doping are presented in Fig.4 and as the function

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of hole carrier concentration at different values of temperature for p-type doping are presented in Fig.5. As the concentration of hole and electron carrier increases the Seebeck coefficient for n-type (fig.4 (a)) as well as for p-type doping (fig.5 (a)) decreases. The electrical conductivity increases in both cases linearly with carrier concentration. The power factor over relaxation time as a function of carrier concentration indicate that it is higher for n-type and lower for p-type ScRhTe at 1100K. The electronic thermal conductivity increases with n-type and p-type doping concentration shown in Fig.4 (d) and Fig.5 (d). The power factor, Seebeck coefficient, electrical conductivity corresponding to optimal carrier concentrations at different values of temperature are summarized in Table.3. The optimal carrier concentration of n-type doping for power factor increases from n=1.1x1020cm-3at 300K to n=1.31x1021cm-3 at 1100K and for p-type doping its increases from n=1.5x1020cm-3 at 300K to n= 7.5x1020cm-3 at 1100K. Therefore, from this discussion we found that the ntype ScRhTe compound may be serve as good thermoelectric material due to high power factor as compared to p-type ScRhTe compound.

Fig.4 (a)Seebeck coefficient (S),(b)electrical conductivity (σ/ꞇ ),(c)power factor (S2σ/ꞇ ),(d)electronic thermal conductivity (ke) as a function of electron concentration for ntype ScRhTe in temperature range 300K-1100K.

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Fig.5 (a)Seebeck coefficient (S),(b)electrical conductivity (σ/ꞇ ),(c)power factor (S2σ/ꞇ ),(d)electronic thermal conductivity (ke) as a function of electron concentration for ptype ScRhTe in temperature range 300K-1100K.

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Table.3 Optimal carrier concentration (n), corresponding Seebeck coefficient (S), Electrical conductivity (σ/ꞇ) and power factor (S2σ/ꞇ) for n-type and p-type doping respectively.

300 500 700 900

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Optimal carrier Seebeck concentration(n) coefficient (cm-3) (µV/K)

Power factor (1011 V2SK-2ms1)

n

1.1x1020

p

20

n

Electrical conductivity (1019 Sms-1)

1.5x10

-152

1.09

2.36

138

0.89

1.69

20

-155

1.83

4.36

20

152

1.40

3.24

2.90x10

p

2.40x10

n

20

5.5x10

-157

2.68

6.57

p

4.0x1020

148

2.25

4.92

n

9.5x1020

-155

3.71

8.86

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p

5.5x1020

152

2.93

6.74

n

1.3x1021

-160

4.32

11.1

p

7.5x1020

152

3.75

8.70

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The thermoelectric properties as function of temperature for n-type ScRhTe are depicted in Fig.6 and for p-type ScRhTe are depicted in Fig.7 at different optimal carrier concentration which are summarized in Table.3. To confirm the value of optimum carrier concentration corresponding to highest power factor we consider further values of concentration for both cases (for n-type are 1.7x1021, 2.2x1021 and for p-type are 9.0x1020 ,1.1x1021 ). The Seebeck coefficient of n-type fig6. (a) and p-type fig7. (a), increases with increasing temperature and decreases with increasing carrier concentrations. Fig.6.(b) and Fig.7(b),shows the variation of electrical conductivity scaled by relaxation time for n-type and p-type ScRhTe compound respectively at different carrier concentration. The electrical conductivity in both cases show slightly decrease with increasing temperature due to decrease in mobility of carriers. From S2σ/ꞇ - T curve (fig.6 (c)), we found that if we increases the electron concentration above 1.3x1021cm-3, then at high temperatures the magnitude of power factor remains unchanged. Therefore, n~ 1.3x1021cm-3 is optimum electron concentration for achieving maximum power factor in n-type ScRhTe .At 1200K, the power factor corresponding to n~1.3x1021cm-3 is 11.9x1011V2SK-2ms-1. The variation of power factor scaled by relaxation time as the function of temperature at different hole concentration in fig. 7(c) indicates that enhancement in carrier concentration above n~7.5x1020cm-3 can’t improve the power factor further. This indicates that we will be able to achieve maximum power factor at n~7.5x1020cm-3, this concentration is optimum hole concentration for p-type ScRhTe. The power factor corresponding to n~7.5x1020cm-3 is 9.7x1011V2SK-2ms-1 at 1200K.Therefore, these results indicate that when electron carrier concentration reaches to n~1.3x1021cm-3 and hole carrier concentration reaches to n~7.5x1020cm-3 power can’t improve further. The electronic thermal conductivity in both cases increases with increasing temperature due to addition of thermally excited charge carriers.

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Fig.6 (a)Seebeck coefficient(S), (b)electrical conductivity(σ/ꞇ ), (c)power factor(S2σ/ꞇ ) ,(d)electronic thermal conductivity (ke) as a function of temperature of n-type ScRhTe from 100K-1200K .

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Fig.7 (a)Seebeck coefficient(S), (b)electrical conductivity(σ/ꞇ ), (c)power factor(S2σ/ꞇ ) ,(d)electronic thermal conductivity (ke) as a function of temperature of p-type ScRhTe from 100K-1200K .

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At optimal carrier concentrations, we have calculated the dimensionless figure of merit as the function of temperature for both types of ScRhTe compound which is depicted in fig.8. As the temperature increases the figure of merit also increases. The maximum value of dimensionless figure of merit for n-type ScRhTe is 0.63 and for p-type ScRhTe is 0.58. The thermoelectric performance of n-type ScRhTe is better as compared to p-type due to high value of power factor. Below 600K, the thermoelectric performance of both cases approximately same. The calculated results demonstrate that n-type ScRhTe has more potential to act as a good thermoelectric material. The lattice thermal conductivity (LM ) of ScRhTe is determined using Slack method [38]. The lattice thermal conductivity is given as: NM =

W

O.Q R.SCQR3T Hɵ: ^  : [ .VWX .[[\ [32 5Z2 [ 5] _B: ` Y

Y

(10)

where n is the number of atoms in the primitive unit cell, V is volume of unit cell,ɵ> is Debye temperature M is average mass of atoms, T is an absolute temperature, γ is Grüneision parameter. Further γ is calculated using Poison ratio (v) by using following formula [39]. But this formula is valid only for acoustic modes. a =

3 (1 + 0) 2 (2 − 30)

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The calculated value of γ is 1.71. The variation of lattice thermal conductivity of ScRhTe is found to be function of temperature which is depicted in fig.9. As the temperature increases the thermal conductivity decrease. At 300K, the lattice thermal conductivity of ScRhTe is 24.77 Wm-1K-1. ScRhTe has low thermal conductivity as compared to well-known ZrNiPb (26.4 W-1K-1 ) [15] half heusler compound. This result suggests that ScRhTe compound has potential to act as good thermoelectric performer as compared to ZrNiPb.

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Fig.8 The Dimensionless figure of merit of ScRhTe as function of temperature at optimal carrier concentrations.

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Fig.9 Lattice thermal conductivity as the function of temperature for ScRhTe compound. 4. Conclusion:

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The structural, elastic, electronic and thermoelectric properties of ScRhTe compound have been studied with frame work of density functional theory and Boltzmann equations. The calculated equilibrium lattice constant of the ScRhTe compound agree with the previous reported value. ScRhTe compound is mechanically stable according to the Born-Huang stability criteria, while no experimental results of elastic moduli for comparison are available. The phonon calculation indicates that ScRhTe compound is dynamically stable. The calculated Debye temperature of this compound is 352.7 K. The band structure as well as DOS indicate that ScRhTe compound has a semiconductor behaviour.Our results suggested that power factor can’t improve further when electron carrier concentration reaches to n~1.3x1021cm-3 and hole carrier concentration reaches to n~7.5x1020cm-3. The room temperature lattice thermal conductivity of ScRhTe is 24.77 Wm-1K-1 that is lower thermal conductivity as compared to ZrNiPb. The maximum dimensionless figure of merit of n-type ScRhTe is 0.63 and p-type is 0.58. We hope that our theoretically calculated results may provide a help to experimentalists. Acknowledgment Sukhwinder Singh, thanks the SHE DST Inspire under no. 792/2012 India for providing fellowship. Reference 1. X.F.Zheng, C.X.Liu, Y.Y.Yan and Q.Wang Renew. Sustainable Energy Rev.32 (2014) 486.

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• The elastic properties of ScRhTe are reported for the first time. • The thermoelectric coefficients as the function of temperature are reported for the first time. • The maximum dimensionless figure of merit of n-type ScRhTe is 0.63 and p-type ScRhTe is 0.58. • We found that ScRhTe compound is mechanically as well as dynamically stable. • At room temperature, ScRhTe attains low lattice thermal conductivity as compared to ZrNiPb. • The first time calculated value of Debye temperature of ScRhTe is 352.7 K.