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Article

First-Principles Investigation of Structural, Thermoelectric, and Optical Properties of Half-Heusler Compound ScRhTe under Varied Pressure

by
Junhong Wei
1,*,
Yongliang Guo
1 and
Guangtao Wang
2
1
School of Science, Henan Institute of Technology, Xinxiang 453003, China
2
School of Physics, Henan Normal University, Xinxiang 453007, China
*
Author to whom correspondence should be addressed.
Crystals 2022, 12(10), 1472; https://doi.org/10.3390/cryst12101472
Submission received: 28 September 2022 / Revised: 11 October 2022 / Accepted: 14 October 2022 / Published: 17 October 2022
(This article belongs to the Special Issue Advanced Materials and Composites for Thermoelectric Applications)
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Abstract

:
We thoroughly investigated the electronic structure and various properties of the half-Heusler compound ScRhTe using density functional theory calculations. The electronic structure shows that ScRhTe is a narrow-band-gap semiconductor. Owing to its characteristic conduction-band structure, ScRhTe has a higher Seebeck coefficient and a higher power factor for n-type doping than for p-type doping, with the maximum value of −493 µV K−1 appearing at 900 K. The optimal carrier concentration is approximately 5 × 1019 cm−3–1 × 1020 cm−3. In addition, ZTe is estimated as 0.95 at a doping level of approximately 1019 cm−3. Under pressure, the band structure changes from a direct to an indirect band gap, and the band gap increases as the pressure changes from tensile to compressive. The thermoelectric properties of ScRhTe improve under compressive pressure, whereas the optical properties improve greatly under tensile pressure. By varying the pressure, the electronic structure and various properties of ScRhTe can be effectively adjusted, which signifies that ScRhTe has the potential to become an important optoelectronic or thermoelectric material.

1. Introduction

Thermoelectric (TE) materials have garnered considerable interest in recent years owing to their potential applications in power generation from waste heat [1,2,3,4,5] because they enable the direct conversion between heat and electricity. They are expected to play an important role in promoting global energy sustainability and energy harvesting. However, the low efficiency of TE materials has been the main obstacle to the replacement of traditional power generation methods with TE methods [6,7]. Therefore, it is important to improve the efficiency of TE materials, which is described using the figure of merit ZT. ZT is given by ZT = S2σT/(κe + κL), where S is the Seebeck coefficient, T is the absolute temperature, σ is the electrical conductivity, κe is the electronic thermal conductivity, and κL is the lattice thermal conductivity [4,8]. In addition, the power factor is given by PF = S2σ. Good TE materials should have a high ZT value, i.e., a high Seebeck coefficient, high electrical conductivity, and low thermal conductivity [9,10]. However, these parameters are related to each other, and changing one parameter affects the others [4,11].
Narrow-band-gap semiconductors are good candidates for use as TE materials [12,13,14,15]; they must be inexpensive and capable of sustaining high temperatures, exhibit mechanical strength, and consist of nontoxic materials [16,17,18,19,20,21]. Ternary half-Heusler compounds, which have 18 or 8 valence electrons in the per formula unit, satisfy these criteria. Many half-Heusler alloys reportedly exhibit notable TE properties [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. Gautier et al. [37] recently reported ‘missing compounds’ in the 18-valence-electron ABX family; some of these materials have been found to exhibit interesting behavior, such as topological phase transitions, TE and phenomena, piezoelectric phenomena, and magnetic properties [38,39,40,41]. Ni-doped ZrPd1-xNixPb [35] decreases the lattice thermal conductivity, which suggests that ZrPd0.25Ni0.75Pb is a good TE material. Via strain engineering, ZrRhSb [36] obtains good TE properties, where ZTe value reaches 0.81 at room temperature. ZrNiPb [37] has been found to exhibit TE behavior, with Seebeck coefficient and power factor as large as −153.9 µV K−1 and 5.2 µW cm−2 K−1, respectively, at room temperature. Under pressure, topological transitions in HfIrX (X = As, Sb, Bi) [38] have been confirmed, and compressive stress in the ab plane causes HfIrBi to become a Weyl semimetal. The performance of ABPb (A = Hf, Zr; B = Ni, Pd) [39] has been estimated by theoretical calculations, yielding results in good agreement with experimental results [37]. TaCoSn [40] is considered to be an important photoelectric and TE material. FeRhCrZ (Z = Si and Ge) [41] is a newly synthesized alloy, as well as it has been predicted by theoretical calculation to be a good TE material with large power factors. Moreover, many half-Heusler compounds have been documented in the Inorganic Crystal Structure Database (ICSD) [42]. However, there is still an opportunity to search for new potential half-Heusler compounds. There has been little research on ScRhTe, one of the missing compounds identified by Gautier et al. [37]. To determine its suitability as a TE material, we calculated multiple properties of ScRhTe by theoretical simulations, investigated the effects of hydrostatic pressure on these properties and analyzed the results.

2. Theoretical Methods

Using first-principles calculations, we studied the properties of ScRhTe via the full-potential linearized augmented plane-wave (FPLAPW) method in the WIEN2K code [43]. The properties include the electronic structure, optical properties, and TE properties under various pressures. The Perdew-Burke-Ernzerhof generalized gradient approximation [44,45] and the project-augmented wave method were employed in our study. To obtain accurate band gaps, we employed a modified Becke–Johnson (mBJ) potential [46,47]. We set the energy cutoff between the core and valence states to −8.0 Ry, and kept the plane-wave cutoff at RMT × KMAX = 10. A k-mesh of 20 × 20 × 20 was used for the self-consistency calculated in the Brillouin zone. The TE properties, such as S, electrical conductivity (σ/τ), κe, were evaluated from the semi-classical Boltzmann transport theory with the constant scattering time approximation (CSTA) [48] using BoltzTraP code [49,50]. Boltzmann transport calculations have been used for TEs for a long time, especially in the study of wide- and narrow-gap semiconductors [51,52,53,54]. Using the BoltzTraP code, we were able to obtain the TE properties with 20,000 k-points of the denser k-mesh. The relaxation time τ was assumed to be isotropic and constant with respect to the wave vector k and energy. This assumption is widely accepted for degenerately doped semiconductors. The spin–orbit interaction (SOC) effect, which can affect the electronic structure, and thus the properties of the material, was included in our calculations.

3. Results and Discussion

3.1. Effect of SOC on Structure and TE Properties

The half-Heusler compound ScRhTe has a cubic LiAlSi-type structure with space group Fm-43 (No.216), as shown in Figure 1a. Sc, Rh, and Te atoms are located in the Wyckoff positions of 4c (0.25, 0.25, 0.25), 4b (0.5, 0.5, 0.5) and 4a (0.0, 0.0, 0.0), respectively. We first optimized the crystal structure; the relaxed equilibrium lattice constant is a = 6.347 Å, which agrees well with the value reported by Gautier et al. [38]. Using the optimized lattice constant, we calculated the band structure and projected density of states (PDOS) of ScRhTe using both mBJ and mBJ + SOC, as shown in Figure 1b,c. ScRhTe is clearly a direct band gap [37], and the conduction-band minimum (CBM) and valence band maximum (VBM) are located at the Γ point. This structure is similar to those of half-Heusler compounds that are direct band gap semiconductors [55,56]. The band gap is 0.69 eV without SOC, and 0.73 eV with the SOC effect, which is in good agreement with the value reported by Gautier et al. [37]. Figure 1b clearly shows that the SOC affects the VBM at the high-symmetry Γ point, resulting in spin–orbit splitting, but has a negligible effect on the conduction bands. From the PDOS, we see that the CBM bands consist mainly of the Sc d states and the s states of Sc, Rh, Te, whereas the VBM near the Fermi level is dominated by the d states of Rh and Sc hybridized with the p-states of Rh and Te. It can be concluded that the d states of the constituent elements strongly affect the TE properties. Moreover, we find that the band at the CBM is rather flat, and that near the VBM is relatively sharp, indicating very low TE properties of the VBM. but very high TE properties of the CBM. Thus, we expect that ScRhTe has a higher S and larger PF for n-type doping.
We calculated the TE properties of ScRhTe using the Boltzmann theory to determine its TE performance. Previous reports have shown that SOC affects the structure of materials, particularly the S and PF [38,57,58,59,60,61,62]. To verify whether SOC affects the TE properties of ScRhTe, we first considered the effect of SOC on the transport coefficients of ScRhTe, including S and power factor with respect to the scattering time (S2σ/τ), for various carrier concentrations at 300 K using mBJ and mBJ + SOC. The results are shown in Figure 2. ScRhTe clearly has a larger S for n-type doping than for p-type doping, and S for p-type doping decreased when the SOC effect was included, whereas SOC had a negligible effect on S for n-type doping. However, S2σ/τ exhibited a great influence. The reason is that SOC splitting removed the degeneracy of the VBM and CBM, and modified the band structure, causing notable changes in the p-type and n-type doping. The properties of this material depend greatly on the band structure. Therefore, we considered the SOC effect in the subsequent calculations.

3.2. Thermoelectric Properties

We calculated the TE properties of ScRhTe as a function of carrier concentration and temperature using the Boltzmann theory. Figure 3 shows S, S2σ/τ, σ/τ, and figure of merit ZTe at different temperatures. At T = 300 and 600 K, S is large at low carrier concentrations for both p- and n-type doping. At T = 900 and 1200 K, the S curves exhibit broad peaks resulting from the bipolar effect. The value for n-type doping is clearly higher than that for p-type doping at each temperature. For p-type doping, the highest S at 900 and 1200 K are 390 and 301 µV K−1 at concentrations of 8.6 × 1018 and 4.4 × 1019 cm−3, respectively. For n-type doping, the highest values at 900 and 1200 K are −493 and −387 µV K−1 at concentrations of 2 × 1019 cm−3 and 9.5 × 1019 cm−3, respectively.
Figure 3a shows a clear bipolar effect at high temperatures for both n- and p-type doping. The bipolar effect often appears in materials such as wide-gap semiconductors at high temperatures and narrow-gap semiconductors or semimetals at room temperature, because holes are major carriers and electrons are minor carriers for p-type materials. S is then defined as [63] S = (Se σe + Sh σh)/(σe + σh), where Se (Sh) and σe (σh) are the electron (hole) Seebeck coefficient and conductivity, respectively. For p-type materials, electrons make up a negligible proportion of the total carriers at low temperatures because there is little thermal excitation, but at high temperatures, the minor carrier concentration cannot be ignored. Thus, S reflects the bipolar effect at high temperatures. S2σ/τ is shown as a function of carrier concentration at 300, 600, 900 and 1200 K in Figure 3b. S2σ/τ increases with T for a given carrier concentration and exhibits a peak at approximately 1.6 × 1021 cm−3. The n-type S and S2σ/τ are clearly larger than the p-type values. This result indicates that n-type ScRhTe can be expected to have good TE properties.
Figure 3c shows the variation in electrical conductivity per second with respect to the scattering time (σ/τ) at different temperatures. At all temperatures, σ/τ exhibits similar behavior with increasing carrier concentration. As the temperature increases, the lattice vibration increases, which hinders the movement of carriers. Therefore, the conductivity decreases with increasing temperature. For a given temperature, the conductivity increases as the carrier concentration increases. The value of σ/τ is higher for p-type doping than for n-type doping. We can predict that the p-type doping has higher electrical conductivity than the n-type doping.
Figure 3d shows the figure of merit ZTe of ScRhTe as functions of T and carrier concentration for p- and n-type doping. The energy conversion efficiency η of TE devices in applications is essentially decided by the dimensionless figure of merit ZT of the material, which is defined as [64]: ZT = ZTe × κe/(κe + κL), where κe and κL are the electronic and lattice thermal conductivity, respectively. The ratio ZTe = S2σT/κe is independent of the relaxation time τ and is an upper limit on the TE figure of merit, which ignores only the lattice contribution to the thermal conductivity. At temperatures above room temperature, the large number of excited electrons causes an increase in the electrical thermal conductivity, whereas the lattice contribution decreases because the phonon scattering ratio is increased owing to severe lattice vibration. Thus, at higher temperatures, the effect of κL on ZTe is assumed to be insignificant [36]. Therefore, ZTe is a good proxy for ZT as the temperature increases. The figure of merit is calculated by ZTe = S2σT/κe. When the carrier concentration is fixed, ZTe increases with T, and when the temperature is fixed, ZTe decreases with the carrier concentration. It is higher for n-type doping than for p-type doping. ZTe is estimated to have a large value of 0.95 at a doping level of approximately 1019 cm−3 if the bipolar effect is not considered. The obtained TE properties clearly show that ScRhTe exhibits a nearly ideal TE performance and is a promising TE material.
Most materials typically exhibit optimal performance at carrier concentrations above those at which the maximum S occurs. To obtain a high-performance TE, we investigated the temperature dependence of S at four fixed electron (hole) concentrations, as shown in Figure 4. At concentrations of 5 × 1019 cm−3 and 1 × 1020 cm−3, the highest S are approximately −440 and −400 µV K−1 at 1000 and 1110 K, respectively. The optimal carrier concentration is clearly in the range 5 × 1019 cm−3 to 1 × 1020 cm−3. An optimal carrier concentration of approximately 1021 cm−3 has been reported for half-Heusler alloys [2,65]. Our results differ from most of those in the literature but are in good agreement with those reported by Singh [55].

3.3. Influence of Pressure on Electronic Structure

The electronic structures of semiconductors are highly sensitive to pressure, which suggests the need for a strategy for tuning the band structure. Pressure-induced changes in the optical properties of β-Na0.33V2O5 have been reported [66]. The TE properties of some TE materials improve dramatically under compression [57,58,67,68,69]. As the deformation behavior of materials under compression could provide considerable information about the nature of the solids, such as phase transitions and changes in physical and chemical properties, investigations under pressure are very important. To explore the properties of ScRhTe, we attempted to predict the TE and optical properties of the compound under pressure. We varied the lattice constant between 96% and 104% of the calculated value to simulate hydrostatic pressure on the compound. The applied hydrostatic pressure is defined as ε = (a − a0)/a0, where a and a0 are the lattice constant under pressure and the equilibrium value, respectively. Positive and negative values of ε indicate tensile and compressive pressure, respectively.
The phonon dispersion of a crystal is a fundamental subject for identifying the phase dynamic stability of crystalline material. To verify the dynamic stability of ScRhTe under the applied hydrostatic pressure, we calculated the phonon dispersion spectra as shown in Figure 5a–i. If the phonon dispersion does not exhibit soft phonon modes or imaginary frequencies, the structure is dynamically stable [70].
ScRhTe is a direct band gap semiconductor at the equilibrium lattice constant, as shown in Figure 6b. Under pressure, the band gap increases as the pressure changes from tensile to compressive, as shown in Figure 6d. In addition, the band structure changes from a direct band gap to an indirect band gap under compressive pressure, where the CBM moves from Γ to a point between Γ and X. To understand this interesting change in the band structure under pressure, we focus on the projected band structure of ScRhTe in Figure 6b. The CBM (at the Γ point) consists mainly of s-orbitals of Sc, Rh, and Te, whereas the VBM consists mainly of the hybridized p–d states of the Rh and Sc atoms. The s-like orbital exhibits greater expansion than the localized d-like orbital. When compressive pressure is applied to the compound, the band energies of both the s-like and d-like orbitals increase. However, the energy of the s-like band increases much more than that of the d-like band because of the greater expansion of the s-orbital. Therefore, as shown in Figure 6a,c, the s-like band at the Γ-point moves upward, whereas the d-like band at the X point moves very little. When ε = −1%, the band structure of ScRhTe changes from a direct band gap (Figure 6c) to an indirect band gap (Figure 6a).

3.4. Influence of Pressure on Thermoelectric Properties

The TE properties depend strongly on the electronic structure of the material. As the band structures changed under the applied pressure, the TE properties were also affected by the pressure. Thus, we examined S and S2σ/τ under different pressures, which were obtained using Boltzmann theory with the CSTA. As the variation of the properties under pressure does not depend on the temperature, we present only the properties at room temperature, as shown in Figure 7. It is interesting that the behavior of S, σ/τ, and S2σ/τ is the same as that of the band gaps for n- and p-type doping as the pressure increases from –4% to 4%. As shown in Figure 7a, S increases gradually as the pressure increases from 4% to –4% for n-and p-type doping. As S makes the largest contribution to the power factor, the effects of pressure on S2σ/τ and S are the same, as shown in Figure 7b. The effects of the pressure for n-type doping are greater than for p-type doping. The CBM band becomes increasingly flat as the pressure changes from 4% to −4% (from Figure 6c to Figure 6a). The flat band indicates very high TE properties; thus, S for n-type doping increases as the pressure changes from tensile to compressive. The results also show that the pressure has a negligible effect on the S and S2σ/τ values for p-type doping because the pressure only slightly affects the electronic structure of the VBM. The strong pressure dependence of S and S2σ/τ for n-type doping shows that the TE properties of ScRhTe increase with increasing compressive pressure; specifically, ScRhTe may become a more efficient TE material under pressure.
Figure 7d shows the calculated ZTe as a function of carrier concentration for p- and n-type doping at room temperature. The effect of pressure on ZTe is clearly negligible for p-type doping at strains of –4% to 4%. For n-type doping, ZTe decreases gradually with increasing tensile pressure from 0% to 4%. Overall, ZTe maintains a high value of approximately 0.97. When the carrier concentration is less than 1019 cm−3, the pressure has no effect on ZTe and the maximum value is approximately 0.97. When the carrier concentration exceeds 1019 cm−3, ZTe increases with increasing compressive pressure from 0% to −4%. This result shows that the application of compressive pressure is useful for improving the ZTe value of ScRhTe, whereas the application of tensile pressure is unfavorable.

3.5. Influence of Pressure on Optical Properties

Semiconductor materials are known for their important technological applications, especially in the manufacture of electronic and electro-optical devices [71]. They may have direct or indirect band gaps. In direct band gap materials, the top and bottom of the valence band appear at the same wave vector value, while for indirect band gap materials, they appear at different wave vector values. Indirect band gap materials have low absorption and therefore are optically inactive [72]. By contrast, direct band gap materials have high absorption and are optically active. Direct band gap materials should be used for optoelectronic devices such as solar cells and detectors to obtain a fast response and high efficiency. ScRhTe is a direct band gap semiconductor and an optically active material; its band structure is highly sensitive to pressure. Pressure-induced changes in optical properties have been reported [58,66]. The optical properties play a crucial role in the optoelectronic behavior of a material. We assumed that the optical properties of ScRhTe would be affected by pressure. Hence, we investigated the optical properties of ScRhTe under different pressures; the frequency-dependent dielectric function and absorption coefficient are shown in Figure 8.
The optical properties can be described in terms of the dielectric function ε(ω). WIEN2K code is used to calculate the imaginary and real parts of the frequency-dependent dielectric function in the ground-state electronic configuration of ScRhTe. The real and imaginary parts of ε(ω) are often denoted as ε1(ω) and ε2(ω), respectively [73], as shown in Figure 8a,b.
The frequency-dependent dielectric function ε1(ω) is shown in Figure 8a. The zero-frequency limit ε1(0), which is the electric part of ε1(ω) without pressure is 15.4. Under pressure, ε1(0) decreases monotonically from 13.82 to 17.92 as the pressure changes from −4% to 4% because it is inversely proportional to the band gap [74]. Beyond the zero-frequency limit, it increases smoothly around 18.9 eV and reaches its maximum value; it then decreases, with some notable variation, decreasing below zero in certain energy ranges. Finally, it increases from negative to positive, and the curve becomes smooth.
Figure 8b shows a plot of ε2(ω). Without pressure, a threshold appears at 2.59 eV, which is the optical gap of the compound. As the compressive pressure increases, the threshold shifts towards higher energies, whereas under increasing tensile pressure, the threshold shifts toward lower energies. The shift in the threshold is clearly reflected in the variation of the band gap with pressure.
As pressure modified the band structure, the optical properties of ScRhTe changed. Figure 8c shows the optical absorption under various pressures. Without pressure, ScRhTe exhibits an optical absorption peak at approximately 307 nm in the visible region, which corresponds to a band gap energy of 4.04 eV. Under compressive pressure, the band structure changes from a direct band gap to an indirect band gap. As indirect band gap materials are optically inactive, the absorption coefficient of ScRhTe decreases as the compressive pressure ε is varied from 0% to −4%. Under tensile pressure, the absorption coefficient of ScRhTe increases as the pressure is increased from 0% to 4%, and the absorption edge is clearly red-shifted, which is opposite to the behavior of the calculated band gaps in Figure 6d. Therefore, the results indicate that tensile pressure greatly improves the optical absorption properties of ScRhTe but that compressive pressure degrades them.

4. Conclusions

The electronic structure and transport properties of the ternary half-Heusler compound ScRhTe were investigated using first-principles calculations and the mBJ potential. The results indicated that ScRhTe is a direct band gap semiconductor. The TE properties S and S2σ/τ for p- and n-type doping were calculated using the Boltzmann transport theory. A high S was obtained for ScRhTe, particularly for n-type doping. The optimal carrier concentration was approximately from 5 × 1019 cm−3 to 1 × 1020 cm−3. In addition, the highest value of S was −493 µV K−1 at a concentration of 2 × 1019 cm−3, at 900 K for n-type doping. ZTe, a proxy for the figure of merit, had a large value of 0.95 at a doping level of approximately 1019 cm−3. The band structure changed under pressure; specifically, the band gap decreased as the pressure was varied from −4% to 4%. As the TE and optical properties depend strongly on the electronic structure of the material, the TE and optical properties are also affected by pressure. The S and S2σ/τ values for n-type doping increased as the compressive pressure increased from 0% to −4%. The absorption coefficient of ScRhTe increased as the tensile pressure ε was increased from 0% to 4%, and the absorption edge was clearly red-shifted. The results showed that compressive pressure improves the TE properties of ScRhTe, whereas tensile pressure improved the optical properties. In summary, our research indicated that ScRhTe is a promising material for TE and thin-film photovoltaic absorption applications, and the application of pressure was expected to be an effective method of improving the TE and optical properties. The theoretical prediction of new materials is easier than experimental realization, and it is hoped that our simulation can provide theoretical guidance for the preparation and application of this type of new material.

Author Contributions

J.W. designed the research, wrote and revised the manuscript, and conducted data analysis and details of the work. Y.G. and G.W. designed and guided the direction of the work. All authors have read and agreed to the published version of the manuscript.

Funding

This project was supported by the National Natural Science Foundation of China (Grant No. 11904081), the Basic Research Program of Education Bureau of Henan Province (Grant No. 20A140007) and the Research Initiation Fund of Henan Institute of Technology (Grant No. KQ1853). The calculations are supported by the High Performance Computing Center of Henan Normal University.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Crystals 12 01472 g001 550
Figure 1. (a) Crystal structure of ScRhTe; Sc, Rh, and Te atoms are shown by green, blue, and grayfilled spheres, respectively. Band structure and projected density of states (PDOS) of ScRhTe (b) without spin–orbit interaction (SOC) and (c) with SOC.
Figure 1. (a) Crystal structure of ScRhTe; Sc, Rh, and Te atoms are shown by green, blue, and grayfilled spheres, respectively. Band structure and projected density of states (PDOS) of ScRhTe (b) without spin–orbit interaction (SOC) and (c) with SOC.
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Figure 2. (a) S (µV K−1) and (b) S2σ/τ (1 × 1010 W m−1 K−2 s−1) presented with mBJ (black dotted line) and mBJ + SOC (red dotted line) at 300 K.
Figure 2. (a) S (µV K−1) and (b) S2σ/τ (1 × 1010 W m−1 K−2 s−1) presented with mBJ (black dotted line) and mBJ + SOC (red dotted line) at 300 K.
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Figure 3. S (µV K−1) (a), S2σ/τ (1 × 1010 W m−1 K−2 s−1) (b), σ/τ (c), and ZTe (d) of ScRhTe at different temperatures. The black, red, green, and blue dashed lines represent T = 300, 600, 900, and 1200 K, respectively.
Figure 3. S (µV K−1) (a), S2σ/τ (1 × 1010 W m−1 K−2 s−1) (b), σ/τ (c), and ZTe (d) of ScRhTe at different temperatures. The black, red, green, and blue dashed lines represent T = 300, 600, 900, and 1200 K, respectively.
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Figure 4. Calculated S for ScRhTe at various fixed concentrations as a function of temperature.
Figure 4. Calculated S for ScRhTe at various fixed concentrations as a function of temperature.
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Figure 5. Phonon dispersion curves of ScRhTe at (a) ε = −4%, (b) ε = −3%, (c) ε = −2%, (d) ε = −1%, (e) ε = 0%, (f) ε = 1%, (g) ε = 2%, (h) ε = 3%, (i) ε = 4% different pressure. The value of ε is positive for tensile pressure and negative for compressive pressure and zero for unstrained.
Figure 5. Phonon dispersion curves of ScRhTe at (a) ε = −4%, (b) ε = −3%, (c) ε = −2%, (d) ε = −1%, (e) ε = 0%, (f) ε = 1%, (g) ε = 2%, (h) ε = 3%, (i) ε = 4% different pressure. The value of ε is positive for tensile pressure and negative for compressive pressure and zero for unstrained.
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Figure 6. Band structure of ScRhTe under varying pressure. (a–c) correspond to ε = −2%, 0%, 2%, respectively. (b) Symbol sizes correspond to the projected weight of Bloch states onto the s-like (red circle) orbit and d-like (blue circle) orbit. (d) Trend of the band gap of ScRhTe under varying strain. ε represents applied hydrostatic pressure.
Figure 6. Band structure of ScRhTe under varying pressure. (a–c) correspond to ε = −2%, 0%, 2%, respectively. (b) Symbol sizes correspond to the projected weight of Bloch states onto the s-like (red circle) orbit and d-like (blue circle) orbit. (d) Trend of the band gap of ScRhTe under varying strain. ε represents applied hydrostatic pressure.
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Figure 7. Seebeck coefficient S (µV K−1) (a), power factor with respect to scattering time S2σ/τ (1 × 1010 W m−1 K−2 s−1) (b), conductivity with respect to scattering time σ/τ (c), and figure of merit ZTe (d) under varying pressure of ScRhTe at room temperature.
Figure 7. Seebeck coefficient S (µV K−1) (a), power factor with respect to scattering time S2σ/τ (1 × 1010 W m−1 K−2 s−1) (b), conductivity with respect to scattering time σ/τ (c), and figure of merit ZTe (d) under varying pressure of ScRhTe at room temperature.
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Figure 8. Real part of complex dielectric function (a), imaginary part of complex dielectric function (b), and absorption coefficient (c) for ScRhTe under varying pressure.
Figure 8. Real part of complex dielectric function (a), imaginary part of complex dielectric function (b), and absorption coefficient (c) for ScRhTe under varying pressure.
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Wei, J.; Guo, Y.; Wang, G. First-Principles Investigation of Structural, Thermoelectric, and Optical Properties of Half-Heusler Compound ScRhTe under Varied Pressure. Crystals 2022, 12, 1472. https://doi.org/10.3390/cryst12101472

AMA Style

Wei J, Guo Y, Wang G. First-Principles Investigation of Structural, Thermoelectric, and Optical Properties of Half-Heusler Compound ScRhTe under Varied Pressure. Crystals. 2022; 12(10):1472. https://doi.org/10.3390/cryst12101472

Chicago/Turabian Style

Wei, Junhong, Yongliang Guo, and Guangtao Wang. 2022. "First-Principles Investigation of Structural, Thermoelectric, and Optical Properties of Half-Heusler Compound ScRhTe under Varied Pressure" Crystals 12, no. 10: 1472. https://doi.org/10.3390/cryst12101472

APA Style

Wei, J., Guo, Y., & Wang, G. (2022). First-Principles Investigation of Structural, Thermoelectric, and Optical Properties of Half-Heusler Compound ScRhTe under Varied Pressure. Crystals, 12(10), 1472. https://doi.org/10.3390/cryst12101472

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