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Journal of Shanghai University >

2026 , Vol. 32 >Issue 1: 33 - 43

DOI: https://doi.org/10.12066/j.issn.1007-2861.2700

Materials Science

Temperature-dependent band gap of thermoelectric PbTe from first-principles calculations

  • GAN Lu ,
  • YANG Jiong ,
  • XI Jinyang
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  • Materials Genome Institute, Shanghai University, Shanghai 200444, China

Received date: 2025-07-03

  Online published: 2026-03-16

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Abstract

The band gap is one of the most fundamental properties of semiconductors. Manystudies have demonstrated that temperature-induced vibrations have a significant impact on the band gap. In this paper, the band structure of the thermoelectric compound PbTe at finite temperature is investigated using first-principles calculations, focusing on the effects of electron-phonon renormalization (EPR) and spin-orbit coupling (SOC) on the temperature-dependentband gap. The results reveal that EPR leads to temperature- dependent variations in the band gap of PbTe, while the inclusion of SOC reverses the trend of band gap change (without SOC, the band gap decreases by 88 meV as the temperature rises from 0 K to 750 K, while the band gap increases by 144 meV with SOC). Furthermore, the zero-point renormalization contributes minimally to the band gap change. Therefore, both EPR and SOC effects have a significant influence on the band gap of PbTe and must be taken into account. This paper provides theoretical insight into the temperature dependence of PbTe’s band gap and ofiers guidance for optimizing its thermoelectric performance.

Key words： first-principles; thermoelectric PbTe; electron-phonon renormalization (EPR); spin-orbit coupling (SOC); temperature-dependent band gap

Cite this article

GAN Lu , YANG Jiong , XI Jinyang . Temperature-dependent band gap of thermoelectric PbTe from first-principles calculations[J]. Journal of Shanghai University, 2026 , 32(1) : 33 -43 . DOI: 10.12066/j.issn.1007-2861.2700

References

[1] Biswas K, He J, Blum I D, et al. High-performance bulk thermoelectrics with all-scale hierarchical architectures [J]. Nature, 2012, 489(7416): 414-418.
[2] Jia B H, Huang Y, Wang Y, et al. Realizing high thermoelectric performance in nonnanostructured n-type PbTe [J]. Energy & Environmental Science, 2022, 15(5): 1920-1929.
[3] Varshni Y P. Temperature dependence of the energy gap in semiconductors [J]. Physica, 1967, 34(1): 149-154.
[4] Gobel A, Ruf T, Cardona M, et al. Efiects of the isotopic composition on the fundamental gap of CuCl [J]. Physical Review B, 1998, 57(24): 15183-15190.
[5] D’innocenzo V, Grancini G, Alcocer M J P, et al. Excitons versus free charges in organolead tri-halide perovskites [J]. Nature Communications, 2014, 5(1): 3586.
[6] Giustino F, Louie S G, Cohen M L. Electron-phonon renormalization of the direct band gap of diamond [J]. Physical Review Letters, 2010, 105(26): 265501.
[7] Ponce S, Antonius G, Gillet Y, et al. Temperature dependence of electronic eigenenergies in the adiabatic harmonic approximation [J]. Physical Review B, 2014, 90(21): 214304.
[8] Ponce S, Gillet Y, Laflamme J J, et al. Temperature dependence of the electronic structure of semiconductors and insulators [J]. The Journal of Chemical Physics, 2015, 143(10): 102813.
[9] Giustino F. Electron-phonon interactions from first principles [J]. Reviews of Modern Physics, 2017, 89(1): 015003.
[10] Saidi W A, Ponce S, Monserrat B. Temperature dependence of the energy levels of methylammonium lead iodide perovskite from first-principles [J]. The Journal of Physical Chemistry Letters, 2016, 7(24): 5247-5252.
[11] Antonius G, Ponce S, Lantagne-Hurtubise E, et al. Dynamical and anharmonic effects on the electron-phonon coupling and the zero-point renormalization of the electronic structure [J]. Physical Review B, 2015, 92(8): 085137.
[12] Quarti C, Mosconi E, Ball J M, et al. Structural and optical properties of methylammonium lead iodide across the tetragonal to cubic phase transition: implications for perovskite solar cells [J]. Energy & Environmental Science, 2016, 9(1): 155-163.
[13] Zhang Y, Ke X, Kent P R C, et al. Anomalous lattice dynamics near the ferroelectric instability in PbTe [J]. Physical Review Letters, 2011, 107(17): 175503.
[14] Zacharias M, Giustino F. One-shot calculation of temperature-dependent optical spectra and phonon-induced band-gap renormalization [J]. Physical Review B, 2016, 94(7): 075125.
[15] Zhang Y M, Wang Z Y, Xi J Y, et al. Temperature-dependent band gaps in several semiconductors: from the role of electron{phonon renormalization [J]. Journal of Physics: Condensed Matter, 2020, 32(47): 475503.
[16] Pei Y Z, Shi X Y, Lalonde A, et al. Convergence of electronic bands for high performance bulk thermoelectrics [J]. Nature, 2011, 473: 66-69.
[17] An J M, Franceschetti A, Zunger A. Electron and hole addition energies in PbSe quantum dots [J]. Physical Review B, 2007, 76(4): 045401.
[18] Mahan G D. Figure of merit for thermoelectrics [J]. Journal of Applied Physics, 1989, 65(4): 1578-1583.
[19] Hook J R, Hall H E. Solid state physics [M]. New York: John Wiley & Sons, 2013.
[20] Wang Z Y, Xi J, Ning J, et al. Temperature-dependent band renormalization in CoSb3 skutterudites due to Sb-ring-related vibrations [J]. Chemistry of Materials, 2021, 33(3): 1046-1052.
[21] Ning J, Zheng L, Lei W, et al. Temperature-dependence of the band gap in the all-inorganic perovskite CsPbI3 from room to high temperatures [J]. Physical Chemistry Chemical Physics, 2022, 24(26): 16003-16010.
[22] Zhao Y H, Li Y X, Xi J Y, et al. Signiflcant temperature tunability of the band gap in two-dimensional materials [J]. Computational Materials Today, 2025, 5: 100019.
[23] Hohenberg P, Kohn W. Inhomogeneous electron gas [J]. Physical Review B, 1964, 136(3): 864-871.
[24] Kresse G, Furthmuller J. E-cient iterative schemes for ab initio total-energy calculations using a plane-wave basis set [J]. Physical Review B, 1996, 54(16): 11169.
[25] Kresse G, Joubert D. From ultrasoft pseudopotentials to the projector augmented-wave method [J]. Physical Review B, 1999, 59(3): 1758.
[26] Kresse G, Hafner J. Ab initio molecular dynamics for liquid metals [J]. Physical Review B, 1993, 47(1): 558-561.
[27] Blochl P E. Projector augmented-wave method [J]. Physical Review B, 1994, 50(24): 17953.
[28] Perdew J P, Burke K, Ernzerhof M. Generalized gradient approximation made simple [J]. Physical Review Letters, 1996, 77(18): 3865.
[29] Togo A, Tanaka I. First principles phonon calculations in materials science [J]. Scripta Materialia, 2015, 108: 1-5.
[30] Monkhorst H J, Pack J D. Special points for brillouin-zone integrations [J]. Physical Review B, 1976, 13(12): 5188-5192.
[31] Medeiros P V C, Stafstrom S, Bj ork J. Efiects of extrinsic and intrinsic perturbations on the electronic structure of graphene: retaining an effective primitive cell band structure by band unfolding [J]. Physical Review B, 2014, 89(4): 041407.
[32] Medeiros P V C, Tsirkin S S, Stafstrom S, et al. Unfolding spinor wave functions and expectation values of general operators: introducing the unfolding-density operator [J]. Physical Review B, 2015, 91(4): 041116.
[33] Kumar J, Tanwar P, Paliwal U, et al. Ab initio study of elastic, electronic, and vibrational properties of SnTe and PbTe [J]. Journal of Molecular Modeling, 2023, 29(11): 335.
[34] Albanesi E A, Okoye C M I, Rodriguez C O, et al. Electronic structure, structural properties, and dielectric functions of Ⅳ-Ⅵ semiconductors: PbSe and PbTe [J]. Physical Review B, 2000, 61(24): 16589-16595.
[35] Gibbs Z M, Kim H, Wang H, et al. Temperature dependent band gap in PbX (X=S, Se, Te) [J]. Applied Physics Letters, 2013, 103: 262109.
[36] Lu L, Sun Y Y. Temperature dependence of spin-orbit coupling effect in solids [J]. Physical Review B, 2025, 111(8): 085129.
[37] Querales-Flores J D, Aguado-Puente P, Dangic D, et al. Towards temperature-induced topological phase transition in SnTe: a first- principles study [J]. Physical Review B, 2020, 101(23): 235206.
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