Electronic structure and thermoelectric properties of Ta-based half-Heusler compounds with 18 valence electrons

Computational Materials Science 159 (2019) 470–477

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Electronic structure and thermoelectric properties of Ta-based half-Heusler compounds with 18 valence electrons

T

D.M. Hoat Benemérita Universidad Autónoma de Puebla, Instituto de Física ”Luis Rivera Terrazas”, Apdo. Postal J48, Col. San Manuel, Puebla, Pue. C.P. 72570, Mexico

A R T I C LE I N FO

A B S T R A C T

Keywords: Ta-based Half-Heusler compounds Structural Electronic structure Thermoelectric properties

Recently, the half-Heusler compounds with 18 valence electrons have been considered as promising candidate for thermoelectric applications due to their interesting properties. In this work, the structural, electronic and thermoelectric properties of Ta-based half-Heusler TaXY (X = Ru, Rh; X = Sb, Bi, Sn and Pb) have been investigated using the full-potential linearized augmented plane-wave (FP-LAPW) method within the density functional theory (DFT) and the classical Boltzmann transport theory. Generalized gradient approximation as parameterized by Wu-Cohen (GGA-WC) and Tran-Blaha modified Becke-Johnson exchange potential (mBJ) are employed for taking on account the exchange-correlation electron potential. All four studied materials are semiconductor with indirect band gap 0.815 eV for TaRuSb, 0.906 eV for TaRuBi, 1.109 eV for TaRhSn and 1.138 eV for TaRhPb. Additionally, the spin-orbit coupling (SOC) decreases slightly the band gap of these materials. Thermoelectric properties, such as, Seebeck coefficient, electrical conductivity, power factor, total thermal conductivity and figure of merit are calculated in function of chemical potential. Obtained results show that TaXY compounds have good thermoelectric performance with high figure of merit value 0.99.

1. Introduction Thermoelectric (TE) materials have received great attention from the researchers due to their ability of converting lost heat energy into electricity (Seebeck effect) and cooling with electricity in refrigerator devices (Peltier effect) [1,2]. In practice, the TE performance of materials is evaluated by the very important dimensionless parameter named figure of merit ZT which can be calculated by expression:

ZT =

S 2σT κ e + κl

(1)

where S is Seebeck coefficient, σ is electrical conductivity, T is temperature, κ e and κl are electrical and lattice contribution to the total thermal conductivity, respectively. The higher ZT is, the better TE performance is. Ideal TE materials must have large power factor PF = S 2σ and low thermal conductivity κ . Theoretically, the ZT value is no limited, hence, the most of scientific community’s efforts is focused on the optimization (with main approach of reducing thermal conductivity [3–6]) of existing materials and the search of new ones with high ZT value. In general, the semiconductors with moderate power factor and relatively low thermal conductivity are the most used in the TE generators [7]. Recently, half-Heusler (HH) compounds with 18 valence electrons

have been considered as promising candidates for TE applications due to their narrow band gap, thermal stability and tunable properties because of the similar crystal structure of semiconductors [8–16]. Experimentally, IVB group-based p-type MCoSb [17–20] and n-type MNiSn [20–24] (M = Ti, Zr and Hf) are TE HH compounds most extensively investigated, and their figure of merit has reached values close to unity at moderate temperatures. Otherwise, the presence of Ta atoms in HH has been found to be favorable for the TE properties of HH compounds. For examples, Zhow et al. [25] found that increasing the Ta dopant concentration in TiCoSb leads to the improvement of its TE performance by increasing electrical conductivity and reducing thermal conductivity simultaneously and ZT value of 0.3 was obtained for Ti0.92Ta0.08CoSb compound at 900 K. Zhao et al. [26] synthesized and characterized the TE properties of ZrNiSn HH alloy doped with Ta and concluded that ZT value was improved due to the decreased thermal conductivity and increased electrical conductivity, and the maximum ZT value of 0.60 was achieved for Zr0.97Ta0.03NiSn sample at 823 K. In 2012, Zakutayev et al. [27] reported the successful synthesis of TaCoSn HH compound and it was predicted theoretically to be semiconductor with band gap of 1.107 eV (using PBE functional) and 1.153 (using mBJ potential) and its thermoelectric properties as calculated by Haque et al. [28] suggested that it could be a potential candidate for thermoelectric device applications with merit of figure (ZT) is 0.731 at

E-mail address: [email protected]. https://doi.org/10.1016/j.commatsci.2018.12.039 Received 23 October 2018; Received in revised form 16 December 2018; Accepted 18 December 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.

Computational Materials Science 159 (2019) 470–477

D.M. Hoat

vector, α and β are the tensor indices (x, y, z), e is the electron charge, τi, k is wave vector dependent relaxation time and the component α of the group velocity ν (i, k ) is computed from band structure as:

600 K. Mentioned experimental and theoretical works have motivated us to investigate more about Ta-based HH alloys in order to explore their TE applicability. In the present work, the structural, electronic and thermoelectric properties of four Ta-based 18 valence electrons HH compounds TaRuSb, TaRuBi, TaRhSn and TaRhPb are predicted for the first time using theoretical calculations based on the density functional theory (DFT) and rigid band approach (RBA) within the constant relaxation time approximation (CRTA). The calculation of thermoelectric properties of materials are very sensitive to their electronic structure, hence, the accurate prediction of electronic band gap is required. Being aware of this issue, we employ the recently developed modified BeckeJohnson exchange potential for the electronic and thermoelectric calculations due to its ability of giving very accurate band gap of a lot of solid types. The paper is organized as follows: The used methodology and computational parameters are briefly described in Computational details section. The obtained results are presented and discussed in details in Results and discussion section. Finally, important results of the work are summarized in Conclusions section.

1 ∂Ei, k να ⎛⎜i, k ⎞⎟ = ℏ ∂k α ⎝ ⎠

Due to the difficulty of determining τi, k in Eq. (2), it is approximated to be constant. The Seebeck coefficient and electrical conductivity tensors in function of temperature T and chemical potential μ can be calculated by integrating the transport distribution as follow [38]:

1 σαβ ⎛⎜T , μ⎞⎟ = Ω ⎝ ⎠

∑ i, k

⎛ τi, k να ⎜i, ⎝

⎞ ⎛ k ⎟ νβ ⎜i, ⎠ ⎝

⎞ ∂ (∊ − ∊i, k ) k⎟ ∂∊ ⎠

⎦

⎣

(4)

∫ σαβ (∊) ⎛∊ − μ⎞ ⎡− ∂0 (T∂, ∊∊, μ) ⎤ d ∊ ⎜

⎟

⎝

⎠⎣

⎦

(5)

where Ω is unit cell volume, f0 is the Fermi-Dirac distribution function. It is important to mention that for calculations of thermoelectric properties, a much denser k-mesh of 42 × 42 × 42 which generates 1903 k-points is used. The thermoelectric properties calculated with mBJ and mBJ+SOC potentials are very similar, hence, in this work, only the thermoelectric results of the former are presented.

All-electron full-potential linearized augmented plane-wave (FPLAPW) method as implemented in WIEN2k package [29] is used for carrying out the structural optimization and calculation of electronic properties of considered materials. Exchange-correlation electron effect is treated with the generalized gradient approximation with Wu-Cohen scheme (GGA-WC) [30]. Theoretically, it is well known that the standard functionals can describe well the band structure profile of solids but they underestimate the band gap [31]. To overcome this issue, in this work, the Tran-Blaha modified Becke-Johnson exchange potential [32–34] with optimized parameterization for semiconductors with band gap up to 7 eV (mBJ) also is used. The energy cut-off −6 Ryd is used for separating core states from valence ones. The wave functions inside atomic spheres are expanded with spherical harmonic up to lmax = 10. The cut-off energy for plane wave expansion of wave functions in the interstitial region is taken to be RMT Kmax = 8, where RMT is minimum muffin-tin radius and that for charge density and potential in the interstitial region is set to be Gmax = 14 a.u−1. A 12 × 12 × 12 Monkhorst-Pack k-mesh [35,36] is employed for the integration in the first Brillouin zone. The self-consistent is considered to be converged if the system energy is stable within 10−4 Ry. Additionally, the spin-orbit coupling (SOC) effect on the electronic properties also is investigated setting the magnetization in [1 0 0] direction for TaRuSb, TaRuBi and TaRhPb and in [1 0 1] direction for TaRhSn due to that the lowest energy in found in these directions (Table 1). Based on the rigid band approach, the transport distribution tensors are calculated from electronic band structure as follows [37]:

e2 N

∫ σαβ (∊) ⎡− ∂0 (T∂, ∊∊, μ) ⎤ d ∊

1 Sαβ ⎛⎜T , μ⎞⎟ = eT Ω σ (T , μ) αβ ⎝ ⎠

2. Computational details

σαβ (∊) =

(3)

3. Results and discussion 3.1. Structural properties TaXY (X = Ru, Rh; Y = Sb, Bi, Sn Pb) compounds adopt the MgAgAs-type structure with space group F 43m (No. 216) in which Ta, X and Y atoms are located at 4a(0; 0; 0), 4b(0.25; 0.25;0.25) and 4c(0.5; 0.5; 0.5) Wyckoff positions [28], respectively. This structure can be considered as a combination of rock-salt structure (Ta4Y4) and zincblende structure (Ta4X4). The crystal structure of TaXY HH compounds is illustrated in Fig. 1. The structural optimization of studied materials is performed by calculating the total system energy at different volumes, then all obtained energy-volume data are fitted to the BirchMunarghan equation of state [39]: 2

E (V ) = E0 +

3

9V0 B ⎧ ⎡ V0 3 ⎛ ⎞ − 1⎤ ⎥ B′ + 16 ⎨ ⎢ ⎝ V ⎠ ⎦ ⎣ ⎩

2

2

2

V 3 ⎤⎫ ⎤ ⎡ ⎡ ⎛ V0 ⎞ 3 − 1⎥ ⎢6 − 4 ⎛ 0 ⎞ ⎥ ⎢ V V⎠ ⎬ ⎝ ⎝ ⎠ ⎦⎭ ⎦ ⎣ ⎣ (6)

the equilibrium geometry is found out by minimizing the energy in function of volume. Obtained lattice constant and bulk modulus are given in Table 2. From the table, it is observed that the lattice

(2)

where N is the number of k-points, i is the band index, k is the wave Table 1 Energy difference in different magnetization directions (meV). TaRuSb

TaRuBi

TaRhSn

TaRhPb

E[001] − E[001]

0

0

0

0

E[010] − E[001]

12.42

57.07

16.18

37.33

E[100] − E[001] E[110] − E[001]

−19.25

−46.86

8.06

−34.81

−0.81

13.44

10.83

10.90

E[101] − E[001] E[011] − E[001]

−9.32

−16.03

−4.12

−15.65

10.87

60.19

10.92

49.80

E[111] − E[001]

6.52

31.77

6.79

15.92

Fig. 1. Crystal structure of TaXY half-Heusler (yellow ball: Ta; red ball: X and gray ball: Y). 471

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exothermic nature of synthesis process of materials, that is, they can be experimentally formed without difficulty, whereas cohesive energy represents the necessary energy to break all bonds associated to each constituent atoms and negative value of Ec implies the structural stability of materials. From the table, it is observed that both ΔHf and Ec of all four studied materials are negative. This results suggest that they can be prepared and stabilized experimentally, that is, they will not be descomposed into free atoms after the formation. Between these HH compounds, the synthesis of TaRhSn should be most favorable thermodynamically due to its most negative ΔHf value and TaRuBi should be most structurally stable due to its highest binding energy.

Table 2 Calculated lattice constant a, bulk modulus B, formation enthalpy ΔHf and cohesive energy Ec of TaXY compounds.

TaRuSb TaRuBi TaRhSn TaRhPb

a (Å)

B (GPa)

ΔHf (eV/atom)

Ec (eV/atom)

6.143 6.256 6.142 6.229

197.058 177.614 184.024 170.990

−0.578 −0.294 −3.103 −0.401

−8.807 −8.451 −5.999 −7.639

parameter of TaRuBi(TaRhPb) is 1.84%(1.42%) larger than that of TaRuSb(TaRhSn), this is due to the increase of atomic radii of atoms in VA(IVA) group as increasing atomic number. Bulk modulus in an important parameter which is used to characterize the hardness of materials. Bulk modulus of TaXY HH alloys increases slightly in order: TaRhPb < TaRuBi < TaRhSn < TaRuSb, which indicate that the resistance to contraction also increases in this order. Additionally, the formation enthalpy ΔHf and cohesive energy Ec of materials of interest also are estimated in order to examine their stability. Theoretically, they are calculated as follow:

ΔHf =

Ec =

ET − zE b (Ta) − xE b (X ) − yE b (Y ) x+y+z

ET − zE (Ta) − xE (X ) − yE (Y ) x+y+z

3.2. Electronic properties At their respective optimized lattice constant, the electronic band structure of TaXY HH compounds is calculated along high symmetry direction W − L − Γ − X − W − K in energy range from −6 eV to 6 eV using mBJ potential as displayed in Fig. 2. The profile of band structures reveals that the maximum of valence band (MVB) and minimum of conduction band (MCB) occurs at L point and X(Γ ) point in case of TaRuSb(TaRuBi), whereas for both TaRhSn-TaRhPb, the MVB and MCB are found at Γ and X point, respectively, indicating that all four investigated materials have indirect band gap. The calculated band gap values are 0.815 eV, 0.906 eV, 1.109 eV and 1.138 eV for TaRuSb, TaRuBi, TaRhSn and TaRhPb, respectively, indicating the semiconductor nature of these materials. When the SOC is included in the calculations, the band gap of these materials decreases slightly being 0.783 eV, 0.866 eV, 1.013 eV and 1.017 eV, respectively. This parameter increases as X goes from Ru to Rh and Y in order of increasing atomic number in VA and IVA group. In Fig. 2, total and partial density of state (TDOS and PDOS) of

(7)

(8)

where ET is total energy of unit cell, E b (Ta), E b (X ) and E b (Y ) are energy per atom of Ta, X and Y in bulk, respectively, E(Ta), E(X) and E(Y) are energy of isolated Ta, X and Y atoms, respectively. x, y and z are number of X, Y and Ta atoms in unit cell, respectively. Obtained ΔHf and Ec are listed in Table 2. Negative value of ΔHf indicates the

Fig. 2. Band structure (mBJ: black line and mBJ+SOC: red line); Total and partial density of state (States/eV) of (a) TaRuSb; (b) TaRuBi; (c) TaRhSn and (d) TaRhPb calculated with mBJ potential. 472

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Fig. 3. Seebeck coefficient in function of temperature at different fixed doping levels of (a) TaRuSb; (b) TaRuBi; (c) TaRhSn and (d) TaRhPb.

Fig. 4. Seebeck coefficient in function of chemical potential at different temperatures of (a) TaRuSb; (b) TaRuBi; (c) TaRhSn and (d) TaRhPb.

large thermopower. Hence, one can expect TaRhSn(Pb) to have higher Seebeck coefficient than TaRuSb(Bi), which will be discussed in the following subsection. Otherwise, no significant difference on the band structure around Fermi level is noted when the SOC is taken on account, this is the reason why the thermoelectric results obtained with mBJ and mBJ+SOC potentials are similar.

studied compounds also are illustrated in order to gain better understanding on their electronic properties. Very similar features are observed in the DOS plots of TaXY alloys. The conduction band is formed mainly from the Ta-5d orbital which hybridizes with Ru(Rh)-4d orbital. The increase of electronic band gap presented above is due to that the MCB is located at higher energy as results of increasing the orbital overlaps of these two mentioned orbitals. The upper partition of valence band is originated from the hybridization between 3 orbitals: Ta5d; X-4d and Y-5p, while the lower part is dominated mainly from the Y5p orbital. The dense electronic states around Fermi level suggests the

3.3. Thermoelectric properties Potential difference induced by the difference of temperature across 473

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Fig. 5. Electrical conductivity in function of chemical potential at different temperatures of (a) TaRuSb; (b) TaRuBi; (c) TaRhSn and (d) TaRhPb.

Fig. 6. Power factor in function of chemical potential at different temperatures of (a) TaRuSb; (b) TaRuBi; (c) TaRhSn and (d) TaRhPb.

behavior just is seen at low temperatures for carrier concentration of 1017 − 1018 which drops fastly at high temperatures. Besides, the dependence of S on chemical potential μ also is investigated. In Fig. 4, the plots show the variation of S on μ function at temperatures 300 K, 600 K and 900 K. It is seen that the maximum values of S are obtained with temperature 300 K. These values are 1404(−1285) (μ V/K), 1576(−1459) (μ V/K), 1875(−1783) (μ V/K) and 1880(−1900) (μ V/K)

material is represented by Seebeck coefficient S. For TE materials, high S values is desirable. The sign of S depends on the carrier type, it is positive(negative) if holes(electrons) are majority carriers in material corresponding to p- and n-type, respectively. In Fig. 3, the variation of S in function of temperature at different fixed charge-carrier concentrations. From the figure, it is observed that for high carrier concentration (1019 to 1022 ), S varies almost linearly with temperature, while this 474

Computational Materials Science 159 (2019) 470–477

D.M. Hoat

Fig. 7. Thermal conductivity in function of chemical potential at different temperatures of (a) TaRuSb; (b) TaRuBi; (c) TaRhSn and (d) TaRhPb.

[−0.1 eV to 1.1 eV], and beyond it increases rapidly. In considered chemical potential range, there is no important difference on response of electrical conductivity for TaXY in p-region ( μ negative) and n-region ( μ positive). Interestingly, σ of TaRuSb(Bi) is clearly higher than that of TaRhSn(Pb), which may be due to their smaller effective mass of charge carriers. As mentioned, large values of S and σ are desirable in TE materials. But as observed in Figs. 4 and 5, S takes its significant values in chemical range in which σ becomes zero, and vice versa. Therefore, the power factor defined as: PF = S 2σ , is used to reflect the TE performance of materials. The PF of TaXY compounds is plotted in function of chemical potential at different temperatures 300 K, 600 K and 900 K in Fig. 6. From the figure, it is perceived that PF decreases with temperature, and it reaches its maximum at 900 K, whose value is 17.46 (1011 W/ms K2) and 9 (1011 W/ms K2) for TaRuSb, 18.4 (1011 W/ms K2) and 12.2 (1011 W/ms K2) for TaRuBi, 24.2 (1011 W/ms K2) and 9.24 (1011 W/ms K2) for TaRhSn, 19.12 (1011 W/ms K2) and 11.12 (1011 W/ ms K2) for TaRhPb in p-region and n-region, respectively. It is obvious p-type TaXY compounds have higher PF value than n-type ones. Total thermal conductivity κ is defined as the sum of electronic κ e and lattice κl thermal conductivity. In practice, the TE materials should have low κ . In Fig. 7, the variation of κ in function of chemical potential at temperatures 300 K, 600 K and 900 K is illustrated for TaXY HH compounds. It is observed that κ increases significantly with increasing temperature, this result is due to the increase of both free electrons energy and vibrational energy in these materials as temperature is raised. In the considered chemical potential range, there is not remarkable difference of κ in p- and n-region for TaRhSn and TaRhPb, while in case of TaRuSb and TaRuBi, κ in n-region is found to be higher than that in p-region. Figs. 5 and 7 also show that κ and σ exist in the same chemical potential range, which do obey the Wiedemann-Franz law which states the κ e is proportional to σ [40,41]. While in this work, the lattice thermal conductivity κl is estimated using the model proposed by Slack and Berman [42]:

Fig. 8. Lattice thermal conductivity in function of temperature of TaRuSb; TaRuBi; TaRhSn and TaRhPb.

in p(n)-type for TaRuSb, TaRuBi, TaRhSn and TaRhPb, respectively. The peak value decreases as increasing temperature. Both figures suggest that the p-type compound title is more favorite than n-type one in TaRuSb, TaRuBi and TaRhSn as at constant temperature and carrier concentration (or chemical potential), their S values with p-type are higher than the absolute values of S with n-type, while very similar response for both types is found in case of TaRhPb. Otherwise, it is also seen that S value of TaRuSb(Bi) compounds is lower than that of TaRhSn(Pb) ones, this result agree well with the density of state form near Fermi level as analyzed above. In TE materials, the electrons move from high-temperature regions to low-temperature ones generating flow of electrons which is characterized by electrical conductivity σ . σ values should be large for good TE performance. In Fig. 5, the σ of TaXY HH compounds is plotted in function of chemical potential μ at temperatures 300 K, 600 K and 900 K. One can see that σ of all four considered compounds does not vary significantly with temperature. The σ is very small for μ range

κl = 475

γ2

2.43 × 10−8 M θ3δ − 0.514γ + 0.228 Tn23

(9)

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Fig. 9. Figure of merit in function of chemical potential at different temperatures of (a) TaRuSb; (b) TaRuBi; (c) TaRhSn and (d) TaRhPb.

to be semiconductor with indirect band and the degree of overlap between Ta-5d and X-4d orbitals is responsible for the variation of the band gap values. Thermopower of TaRhSn(Pb) is higher than that of TaRuSb(Bi) due to their denser electronic states near Fermi level. These materials have high power factor PF in p-region with maximum at chemical potential about −0.25 eV. Obtained results show that considered compounds are promising candidates for thermoelectric applications with high figure of merit value 0.99 which suggests their good thermoelectric performance.

where γ and θ are Grüneisen parameter and Debye temperature which are calculated using quasi-harmonic Debye model [43]; M is average atomic mass; δ 3 is volume per atom; T is temperature and n is number of atoms in primitive cell. Obtained results are displayed in Fig. 8. From the figure, one can see that κl of all four studied materials decreases considerably with temperature. In throughout considered temperature range, TaRuSb has lowest κl , while that of the rest of the compounds is very similar. At room temperature, κl value is 14.68 (W/mK), 17.39 (W/ mK), 17.18 (W/mK) and 16.79 (W/mK) for TaRuSb, TaRuBi, TaRhSn and TaRhPb, respectively. Once obtained all transport coefficients, the thermoelectric performance of considered half-Heusler compounds can be predicted by means of the figure of merit ZT calculated with Eq. (1). Fig. 9 show the variation of ZT in function of chemical potential at different temperatures 300 K, 600 K and 900 K. The profile of shown plots suggests the same behavior of ZT in all four investigated materials. At 300 K, the maximum value of ZT is about 0.99 which occurs at chemical potential 0.3 eV and 0.55 eV for TaRuSb, 0.35 eV and 0.6 eV for TaRuBi, 0.45 eV and 0.7 eV for TaRhSn and TaRhPb. This high ZT value may suggest the good thermoelectric performance of these compounds. Large ZT around mentioned points is due to the large Seebeck coefficient and low thermal conductivity. The value of ZT peaks decreases slightly with increasing temperature. With chemical potentials lower than −0.5 eV and higher than 2 eV, ZT practically approaches to zero due to the very small Seebeck coefficient.

Conflict of interest I declare that there is no conflict. CRediT authorship contribution statement D.M. Hoat: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. Acknowledgement D.M. Hoat gratefully acknowledges the doctoral fellowship CVU 640142 granted by Consejo nacional de ciencia y tecnología de México (CONACyT). The calculations have been performed in Laboratorio Nacional de Supercómputo del Sureste de México with access provided by Cuerpo Académico de Física Computational de la Materia CondensadaIFUAP.

4. Conclusions Structural, electronic and thermoelectric properties of four Ta-based half-Heusler with 18 valence electrons, namely, TaRuSb, TaRuBi, TaRhSn and TaRhPb, have been investigated by means of theoretical calculations based on the FP-LAPW method and Boltzmann transport theory. Lattice constant increases according to the increase of atomic number of atoms in IVA and VA group, and the compressibility is found to increase in order TaRuSb < TaRhSn < TaRuBi < TaRhPb. All studied materials can be experimentally synthesized and stabilized as their negative enthalpy formation and cohesive energy. They are found

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