Skip to main content

Advertisement

SpringerLink
Log in

Semiclassical electron and phonon transport from first principles: application to layered thermoelectrics

  • Published:
Journal of Computational Electronics Aims and scope Submit manuscript

Abstract

Thermoelectrics are a promising class of materials for renewable energy owing to their capability to generate electricity from waste heat, with their performance being governed by a competition between charge and thermal transport. A detailed understanding of energy transport at the nanoscale is thus of paramount importance for developing efficient thermoelectrics. Here, we provide a comprehensive overview of the methodologies adopted for the computational design and optimization of thermoelectric materials from first-principles calculations. First, we introduce density-functional theory, the fundamental tool to describe the electronic and vibrational properties of solids. Next, we review charge and thermal transport in the semiclassical framework of the Boltzmann transport equation, with a particular emphasis on the various scattering mechanisms between phonons, electrons, and impurities. Finally, we illustrate how these approaches can be deployed in determining the figure of merit of tin and germanium selenides, an emerging family of layered thermoelectrics that exhibits a promising figure of merit. Overall, this review article offers practical guidelines to achieve an accurate assessment of the thermoelectric properties of materials by means of computer simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Notes

  1. Besides plane waves, localized basis sets consisting of atomic-like orbitals (e.g., Gaussian- or Slater-type functions) have found a widespread use, in particular in the computational chemistry community. Contrary to plane waves, fewer basis functions are often needed to achieve a reasonable accuracy, hence significantly decreasing the computational effort. However, localized basis sets are controlled by many parameters in addition to the energy cutoff, in a way that no systematic convergence can be attained.

References

  1. Yang, L., Chen, Z.-G., Dargusch, M.S., Zou, J.: High performance thermoelectric materials: progress and their applications. Adv. Energy Mater. 8, 1701797 (2018). https://doi.org/10.1002/aenm.201701797

    Article  Google Scholar 

  2. Hasan, M.N., Wahid, H., Nayan, N., Mohamed Ali, M.S.: Inorganic thermoelectric materials: a review. Int. J. Energy Res. 44, 6170 (2020). https://doi.org/10.1002/er.5313

    Article  Google Scholar 

  3. Zoui, M.A., Bentouba, S., Stocholm, J.G., Bourouis, M.: A review on thermoelectric generators: progress and applications. Energies 13, 3606 (2020). https://doi.org/10.3390/en13143606

    Article  Google Scholar 

  4. Gutiérrez Moreno, J.J., Cao, J., Fronzi, M., Assadi, M.H.N.: A review of recent progress in thermoelectric materials through computational methods. Mater. Renew. Sustain. Energy (2020). https://doi.org/10.1007/s40243-020-00175-5

    Article  Google Scholar 

  5. Giustino, F.: Materials Modelling Using Density Functional Theory: Properties and Predictions. Oxford University Press (2014)

    Google Scholar 

  6. Szabo, A., Ostlund, N.S.: Modern Quantum Chemistry. Introduction to Advanced Electronic Structure Theory. Dover (1996)

    Google Scholar 

  7. Hohenberg, P., Kohn, W.: Inhomogeneous electron gas. Phys. Rev. 136, B864 (1964). https://doi.org/10.1103/PhysRev.136.B864

    Article  MathSciNet  Google Scholar 

  8. Kohn, W., Sham, L.J.: Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133 (1965). https://doi.org/10.1103/PhysRev.140.A1133

    Article  MathSciNet  Google Scholar 

  9. Sholl, D., Steckel, J.A.: Density Functional Theory: A Practical Introduction. Wiley (2009)

    Book  MATH  Google Scholar 

  10. Parr, R.G., Yang, W.: Density-Functional Theory of Atoms and Molecules. Oxford University Press (1994)

    Google Scholar 

  11. Ceperley, D.M., Alder, B.J.: Ground state of the electron gas by a stochastic method. Phys. Rev. Lett. 45, 566 (1980). https://doi.org/10.1103/PhysRevLett.45.566

    Article  Google Scholar 

  12. Perdew, J.P., Burke, K., Ernzerhof, M.: Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996). https://doi.org/10.1103/PhysRevLett.77.3865

    Article  Google Scholar 

  13. Perdew, J.P., Wang, Y.: Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 45, 13244 (1992). https://doi.org/10.1103/PhysRevB.45.13244

    Article  Google Scholar 

  14. Perdew, J.P., Ruzsinszky, A., Csonka, G.I., Vydrov, O.A., Scuseria, G.E., Constantin, L.A., Zhou, X., Burke, K.: Restoring the density-gradient expansion for exchange in solids and surfaces. Phys. Rev. Lett. 100, 136406 (2008). https://doi.org/10.1103/PhysRevLett.100.136406

    Article  Google Scholar 

  15. Zhang, Y., Yang, W.: Comment on Generalized gradient approximation made simple. Phys. Rev. Lett. 80, 890 (1998). https://doi.org/10.1103/PhysRevLett.80.890

    Article  Google Scholar 

  16. Sun, J., Marsman, M., Csonka, G.I., Ruzsinszky, A., Hao, P., Kim, Y.-S., Kresse, G., Perdew, J.P.: Self-consistent meta-generalized gradient approximation within the projector-augmented-wave method. Phys. Rev. B 84, 035117 (2011). https://doi.org/10.1103/PhysRevB.84.035117

    Article  Google Scholar 

  17. Sun, J., Ruzsinszky, A., Perdew, J.P.: Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 115, 036402 (2015). https://doi.org/10.1103/PhysRevLett.115.036402

    Article  Google Scholar 

  18. Perdew, J.P., Ernzerhof, M., Burke, K.: Rationale for mixing exact exchange with density functional approximations. J. Chem. Phys. 105, 9982 (1996). https://doi.org/10.1063/1.472933

    Article  Google Scholar 

  19. Heyd, J., Scuseria, G.E., Ernzerhof, M.: Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys. 118, 8207 (2003). https://doi.org/10.1063/1.1564060

    Article  Google Scholar 

  20. Krukau, A.V., Vydrov, O.A., Izmaylov, A.F., Scuseria, G.E.: Influence of the exchange screening parameter on the performance of screened hybrid functionals. J. Chem. Phys. 125, 224106 (2006). https://doi.org/10.1063/1.2404663

    Article  Google Scholar 

  21. Görling, A.: Density-functional theory for excited states. Phys. Rev. A 54, 3912 (1996)

    Article  Google Scholar 

  22. Cohen, A.J., Mori-Sánchez, P., Yang, W.: Insights into current limitations of density functional theory. Science 321, 792 (2008)

    Article  Google Scholar 

  23. Gonze, X., Amadon, B., Anglade, P.-M., Beuken, J.-M., Bottin, F., Boulanger, P., Bruneval, F., Caliste, D., Caracas, R., Côté, M., Deutsch, T., Genovese, L., Ghosez, P., Giantomassi, M., Goedecker, S., Hamann, D., Hermet, P., Jollet, F., Jomard, G., Leroux, S., Mancini, M., Mazevet, S., Oliveira, M., Onida, G., Pouillon, Y., Rangel, T., Rignanese, G.-M., Sangalli, D., Shaltaf, R., Torrent, M., Verstraete, M., Zerah, G., Zwanziger, J.: Abinit: first-principles approach to material and nanosystem properties. Comput. Phys. Commun. 180, 2582 (2009). https://doi.org/10.1016/j.cpc.2009.07.007

    Article  Google Scholar 

  24. Clark, S.J., Segall, M.D., Pickard, C.J., Hasnip, P.J., Probert, M.I.J., Refson, K., Payne, M.C.: First principles methods using castep. Z. Krist. Cryst. Mater. 220, 567 (2005). https://doi.org/10.1524/zkri.220.5.567.65075

    Article  Google Scholar 

  25. Hutter, J., Iannuzzi, M., Schiffmann, F., VandeVondele, J.: cp2k: atomistic simulations of condensed matter systems. WIREs Comput. Mol. Sci. 4, 15 (2014). https://doi.org/10.1002/wcms.1159

    Article  Google Scholar 

  26. Enkovaara, J., Rostgaard, C., Mortensen, J.J., Chen, J., Dułak, M., Ferrighi, L., Gavnholt, J., Glinsvad, C., Haikola, V., Hansen, H.A., Kristoffersen, H.H., Kuisma, M., Larsen, A.H., Lehtovaara, L., Ljungberg, M., Lopez-Acevedo, O., Moses, P.G., Ojanen, J., Olsen, T., Petzold, V., Romero, N.A., Stausholm-Møller, J., Strange, M., Tritsaris, G.A., Vanin, M., Walter, M., Hammer, B., Häkkinen, H., Madsen, G.K.H., Nieminen, R.M., Nørskov, J.K., Puska, M., Rantala, T.T., Schiøtz, J., Thygesen, K.S., Jacobsen, K.W.: Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method. J. Phys. Condens. Matter 22, 253202 (2010). https://doi.org/10.1088/0953-8984/22/25/253202

    Article  Google Scholar 

  27. Prentice, J.C., Aarons, J., Womack, J.C., Allen, A.E., Andrinopoulos, L., Anton, L., Bell, R.A., Bhandari, A., Bramley, G.A., Charlton, R.J., et al.: The onetep linear-scaling density functional theory program. J. Chem. Phys. 152, 174111 (2020). https://doi.org/10.1063/5.0004445

    Article  Google Scholar 

  28. Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni, C., Ceresoli, D., Chiarotti, G.L., Cococcioni, M., Dabo, I., Corso, A.D., de Gironcoli, S., Fabris, S., Fratesi, G., Gebauer, R., Gerstmann, U., Gougoussis, C., Kokalj, A., Lazzeri, M., Martin-Samos, L., Marzari, N., Mauri, F., Mazzarello, R., Paolini, S., Pasquarello, A., Paulatto, L., Sbraccia, C., Scandolo, S., Sclauzero, G., Seitsonen, A.P., Smogunov, A., Umari, P., Wentzcovitch, R.M.: QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009). https://doi.org/10.1088/0953-8984/21/39/395502

    Article  Google Scholar 

  29. Kresse, G., Furthmüller, J.: Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169 (1996). https://doi.org/10.1103/PhysRevB.54.11169

    Article  Google Scholar 

  30. Kresse, G., Furthmüller, J.: Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15 (1996). https://doi.org/10.1016/0927-0256(96)00008-0

    Article  Google Scholar 

  31. Marzari, N., Vanderbilt, D.: Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev. B 56, 12847 (1997). https://doi.org/10.1103/PhysRevB.56.12847

    Article  Google Scholar 

  32. Marzari, N., Mostofi, A.A., Yates, J.R., Souza, I., Vanderbilt, D.: Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419 (2012). https://doi.org/10.1103/RevModPhys.84.1419

    Article  Google Scholar 

  33. Mostofi, A.A., Yates, J.R., Lee, Y.-S., Souza, I., Vanderbilt, D., Marzari, N.: wannier90: a tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun. 178, 685 (2008). https://doi.org/10.1016/j.cpc.2007.11.016

    Article  MATH  Google Scholar 

  34. Mostofi, A.A., Yates, J.R., Pizzi, G., Lee, Y.-S., Souza, I., Vanderbilt, D., Marzari, N.: An updated version of wannier90: a tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun. 185, 2309 (2014). https://doi.org/10.1016/j.cpc.2014.05.003

    Article  MATH  Google Scholar 

  35. Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Clarendon Press (1966)

    MATH  Google Scholar 

  36. Hellmann, H.: Einfuhrung in Die Quantenchemie. F. Deuticke, Leipzig (1937)

    Google Scholar 

  37. Feynman, R.P.: Forces in molecules. Phys. Rev. 56, 340 (1939). https://doi.org/10.1103/PhysRev.56.340

    Article  MATH  Google Scholar 

  38. DeCicco, P., Johnson, F.: The quantum theory of lattice dynamics. IV. Proc. Lond. A R. Soc. Math. Phys. Sci. 310, 111 (1969). https://doi.org/10.1098/rspa.1969.0066

    Article  Google Scholar 

  39. Pick, R.M., Cohen, M.H., Martin, R.M.: Microscopic theory of force constants in the adiabatic approximation. Phys. Rev. B 1, 910 (1970). https://doi.org/10.1103/PhysRevB.1.910

    Article  Google Scholar 

  40. Maradudin, A.A., Vosko, S.H.: Symmetry properties of the normal vibrations of a crystal. Rev. Mod. Phys. 40, 1 (1968). https://doi.org/10.1103/RevModPhys.40.1

    Article  Google Scholar 

  41. Baroni, S., Giannozzi, P., Testa, A.: Elastic constants of crystals from linear-response theory. Phys. Rev. Lett. 59, 2662 (1987). https://doi.org/10.1103/PhysRevLett.59.2662

    Article  Google Scholar 

  42. Levine, Z.H., Allan, D.C.: Linear optical response in silicon and germanium including self-energy effects. Phys. Rev. Lett. 63, 1719 (1989). https://doi.org/10.1103/PhysRevLett.63.1719

    Article  Google Scholar 

  43. Giannozzi, P., De Gironcoli, S., Pavone, P., Baroni, S.: Ab initio calculation of phonon dispersions in semiconductors. Phys. Rev. B 43, 7231 (1991). https://doi.org/10.1103/PhysRevB.43.7231

    Article  Google Scholar 

  44. de Gironcoli, S., Baroni, S., Resta, R.: Piezoelectric properties of III–V semiconductors from first-principles linear-response theory. Phys. Rev. Lett. 62, 2853 (1989). https://doi.org/10.1103/PhysRevLett.62.2853

    Article  Google Scholar 

  45. de Gironcoli, S., Giannozzi, P., Baroni, S.: Structure and thermodynamics of Si\(_x\) Ge\(_{1-x}\) alloys from ab initio Monte Carlo simulations. Phys. Rev. Lett. 66, 2116 (1991). https://doi.org/10.1103/PhysRevLett.66.2116

    Article  Google Scholar 

  46. Dal Corso, A., Baroni, S., Resta, R.: Density-functional theory of the dielectric constant: gradient-corrected calculation for silicon. Phys. Rev. B 49, 5323 (1994). https://doi.org/10.1103/PhysRevB.49.5323

    Article  Google Scholar 

  47. Quong, A.A., Eguiluz, A.G.: First-principles evaluation of dynamical response and plasmon dispersion in metals. Phys. Rev. Lett. 70, 3955 (1993). https://doi.org/10.1103/PhysRevLett.70.3955

    Article  Google Scholar 

  48. Stengel, M.: Flexoelectricity from density-functional perturbation theory. Phys. Rev. B 88, 174106 (2013). https://doi.org/10.1103/PhysRevB.88.174106

    Article  Google Scholar 

  49. Dreyer, C.E., Stengel, M., Vanderbilt, D.: Current-density implementation for calculating flexoelectric coefficients. Phys. Rev. B 98, 075153 (2018). https://doi.org/10.1103/PhysRevB.98.075153

    Article  Google Scholar 

  50. Royo, M., Stengel, M.: First-principles theory of spatial dispersion: dynamical quadrupoles and flexoelectricity. Phys. Rev. X 9, 021050 (2019). https://doi.org/10.1103/PhysRevX.9.021050

    Article  Google Scholar 

  51. Stott, M., Zaremba, E.: Linear-response theory within the density-functional formalism: application to atomic polarizabilities. Phys. Rev. A 21, 12 (1980). https://doi.org/10.1103/PhysRevA.21.12

    Article  Google Scholar 

  52. Zangwill, A., Soven, P.: Resonant photoemission in barium and cerium. Phys. Rev. Lett. 45, 204 (1980). https://doi.org/10.1103/PhysRevLett.45.204

    Article  Google Scholar 

  53. Mahan, G.: Modified Sternheimer equation for polarizability. Phys. Rev. A 22, 1780 (1980). https://doi.org/10.1103/PhysRevA.22.1780

    Article  MathSciNet  Google Scholar 

  54. Ghosh, S.K., Deb, B.M.: Dynamic polarizability of many-electron systems within a time-dependent density-functional theory. Chem. Phys. 71, 295 (1982). https://doi.org/10.1016/0301-0104(82)87030-4

    Article  Google Scholar 

  55. Zein, N.: On density functional calculations of crystal elastic modula and phonon spectra. Fiz. Tverd. Tela 26, 3028 (1984)

    Google Scholar 

  56. Baroni, S., Giannozzi, P., Testa, A.: Green’s-function approach to linear response in solids. Phys. Rev. Lett. 58, 1861 (1987). https://doi.org/10.1103/PhysRevLett.58.1861

    Article  Google Scholar 

  57. Gonze, X., Allan, D.C., Teter, M.P.: Dielectric tensor, effective charges, and phonons in \(\alpha\)-quartz by variational density-functional perturbation theory. Phys. Rev. Lett. 68, 3603 (1992). https://doi.org/10.1103/PhysRevLett.68.3603

    Article  Google Scholar 

  58. Gonze, X., Vigneron, J.-P.: Density-functional approach to nonlinear-response coefficients of solids. Phys. Rev. B 39, 13120 (1989). https://doi.org/10.1103/PhysRevB.39.13120

    Article  Google Scholar 

  59. Hirschfelder, J.O., Brown, W.B., Epstein, S.T.: Recent developments in perturbation theory. In: Advances in Quantum Chemistry. Academic Press Inc., pp. 255–374 (1964)

  60. Baroni, S., De Gironcoli, S., Dal Corso, A., Giannozzi, P.: Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys. 73, 515 (2001). https://doi.org/10.1103/RevModPhys.73.515

    Article  Google Scholar 

  61. Gonze, X.: Adiabatic density-functional perturbation theory. Phys. Rev. A 52, 1096 (1995). https://doi.org/10.1103/PhysRevA.52.1096

    Article  Google Scholar 

  62. Gonze, X.: Perturbation expansion of variational principles at arbitrary order. Phys. Rev. A 52, 1086 (1995). https://doi.org/10.1103/PhysRevA.52.1086

    Article  Google Scholar 

  63. Lam, P.K., Cohen, M.L.: Ab initio calculation of phonon frequencies of Al. Phys. Rev. B 25, 6139 (1982). https://doi.org/10.1103/PhysRevB.25.6139

    Article  Google Scholar 

  64. Togo, A.: First-principles phonon calculations with phonopy and phono3py. J. Phys. Soc. Jpn. 92, 012001 (2023). https://doi.org/10.7566/JPSJ.92.012001

    Article  Google Scholar 

  65. McGaughey, A.J., Kaviany, M.: Phonon transport in molecular dynamics simulations: formulation and thermal conductivity prediction. Adv. Heat Transf. 39, 169 (2006). https://doi.org/10.1016/S0065-2717(06)39002-8

    Article  Google Scholar 

  66. Kong, L.T.: Phonon dispersion measured directly from molecular dynamics simulations. Comput. Phys. Commun. 182, 2201 (2011). https://doi.org/10.1016/j.cpc.2011.04.019

    Article  Google Scholar 

  67. Hellman, O., Abrikosov, I., Simak, S.: Lattice dynamics of anharmonic solids from first principles. Phys. Rev. B 84, 180301 (2011)

    Article  Google Scholar 

  68. Unke, O.T., Chmiela, S., Sauceda, H.E., Gastegger, M., Poltavsky, I., Schütt, K.T., Tkatchenko, A., Müller, K.-R.: Machine learning force fields. Chem. Rev. 121, 10142 (2021)

    Article  Google Scholar 

  69. Haug, H., Jauho, A.-P., Cardona, M.: Quantum Kinetics in Transport and Optics of Semiconductors, vol. 2. Springer (2008)

    Google Scholar 

  70. Stefanucci, G., Van Leeuwen, R.: Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction. Cambridge University Press (2013)

    Book  MATH  Google Scholar 

  71. Mahan, G.D.: Condensed matter in a nutshell. In: Condensed Matter in a Nutshell. Princeton University Press (2010)

  72. Kubo, R.: Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn. 12, 570 (1957). https://doi.org/10.1143/JPSJ.12.570

    Article  MathSciNet  Google Scholar 

  73. Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255 (1966). https://doi.org/10.1088/0034-4885/29/1/306

    Article  MATH  Google Scholar 

  74. Thouless, D.: Relation between the Kubo–Greenwood formula and the Boltzmann equation for electrical conductivity. Phil. Mag. 32, 877 (1975). https://doi.org/10.1080/14786437508221628

    Article  Google Scholar 

  75. Poncé, S., Li, W., Reichardt, S., Giustino, F.: First-principles calculations of charge carrier mobility and conductivity in bulk semiconductors and two-dimensional materials. Rep. Prog. Phys. 83, 036501 (2020). https://doi.org/10.1088/1361-6633/ab6a43

    Article  MathSciNet  Google Scholar 

  76. Sangalli, D., Marini, A.: Ultra-fast carriers relaxation in bulk silicon following photo-excitation with a short and polarized laser pulse. Europhys. Lett. 110, 47004 (2015). https://doi.org/10.1209/0295-5075/110/47004

    Article  Google Scholar 

  77. Landau, L.: On the theory of the fermi liquid. Sov. Phys. JETP 8, 70 (1959)

    MathSciNet  Google Scholar 

  78. Pines, D.: Theory of Quantum Liquids: Normal Fermi Liquids. CRC Press (2018)

    Book  Google Scholar 

  79. Pottier, N.: Nonequilibrium Statistical Physics: Linear Irreversible Processes. Oxford University Press (2009)

    MATH  Google Scholar 

  80. Peierls, R.: Some simple remarks on the basis of transport theory. In: Transport Phenomena. Springer, pp. 1–33 (1974)

  81. Hussey, N.E., Takenaka, K., Takagi, H.: Universality of the Mott–Ioffe–Regel limit in metals. Phil. Mag. 84, 2847 (2004). https://doi.org/10.1080/14786430410001716944

    Article  Google Scholar 

  82. Emery, V.J., Kivelson, S.A.: Superconductivity in bad metals. Phys. Rev. Lett. 74, 3253 (1995). https://doi.org/10.1103/PhysRevLett.74.3253

    Article  Google Scholar 

  83. Hartnoll, S.A.: Theory of universal incoherent metallic transport. Nat. Phys. 11, 54 (2015). https://doi.org/10.1038/nphys3174

    Article  Google Scholar 

  84. Chang, B.K., Zhou, J.-J., Lee, N.-E., Bernardi, M.: Intermediate polaronic charge transport in organic crystals from a many-body first-principles approach. npj Comput. Mater. 8, 63 (2022). https://doi.org/10.1038/s41524-022-00742-6

    Article  Google Scholar 

  85. Kohn, W., Luttinger, J.M.: Quantum theory of electrical transport phenomena. Phys. Rev. 108, 590 (1957). https://doi.org/10.1103/PhysRev.108.590

    Article  MathSciNet  MATH  Google Scholar 

  86. Luttinger, J.M., Kohn, W.: Quantum theory of electrical transport phenomena. II. Phys. Rev. 109, 1892 (1958). https://doi.org/10.1103/PhysRev.109.1892

    Article  MathSciNet  MATH  Google Scholar 

  87. Protik, N.H., Li, C., Pruneda, M., Broido, D., Ordejón, P.: The elphbolt ab initio solver for the coupled electron–phonon Boltzmann transport equations. npj Comput. Mater. 8, 28 (2022). https://doi.org/10.1038/s41524-022-00710-0

    Article  Google Scholar 

  88. Xiao, D., Chang, M.-C., Niu, Q.: Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  89. von Neumann, J.: Proof of the ergodic theorem and the H-theorem in quantum mechanics. Eur. Phys. J. H 35, 201 (2010). https://doi.org/10.1140/epjh/e2010-00008-5

    Article  Google Scholar 

  90. Kadanoff, L.P.: Entropy is in flux V3.4. J. Stat. Phys. 167, 1039 (2017). https://doi.org/10.1007/s10955-017-1766-2

    Article  MathSciNet  MATH  Google Scholar 

  91. Allen, P.: Boltzmann theory and resistivity of metals. Kluwer International Series In Engineering And Computer Science, p. 219 (1996)

  92. Poncé, S., Margine, E.R., Giustino, F.: Towards predictive many-body calculations of phonon-limited carrier mobilities in semiconductors. Phys. Rev. B 97, 121201 (2018). https://doi.org/10.1103/PhysRevB.97.121201

    Article  Google Scholar 

  93. Liu, Y., Yuan, Z., Wesselink, R., Starikov, A.A., Van Schilfgaarde, M., Kelly, P.J.: Direct method for calculating temperature-dependent transport properties. Phys. Rev. B 91, 220405 (2015). https://doi.org/10.1103/PhysRevB.91.220405

    Article  Google Scholar 

  94. Ziman, J.M.: Electrons and Phonons: The Theory of Transport Phenomena in Solids. Oxford University Press (2001)

    Book  MATH  Google Scholar 

  95. Grimvall, G.: The electron–phonon interaction in metals. North-Holland, Amsterdam (1981)

  96. Askerov, B.M., Figarova, S.: Thermodynamics, Gibbs Method and Statistical Physics of Electron Gases. Springer Series on Atomic, Optical and Plasma Physics, vol. 57. Springer (2009)

    MATH  Google Scholar 

  97. Chaves, A.S., González-Romero, R.L., Meléndez, J.J., Antonelli, A.: Investigating charge carrier scattering processes in anisotropic semiconductors through first-principles calculations: the case of p-type SnSe. Phys. Chem. Chem. Phys. 23, 900 (2021). https://doi.org/10.1039/D0CP05022A

    Article  Google Scholar 

  98. Ahmad, S., Mahanti, S.: Energy and temperature dependence of relaxation time and Wiedemann–Franz law on PbTe. Phys. Rev. B 81, 165203 (2010). https://doi.org/10.1103/PhysRevB.81.165203

    Article  Google Scholar 

  99. Ravich, Y.I., Efimova, B., Tamarchenko, V.: Scattering of current carriers and transport phenomena in lead chalcogenides. Phys. Status Solidi (B) 43, 11 (1971). https://doi.org/10.1002/pssb.2220430102

    Article  Google Scholar 

  100. Li, W.: Electrical transport limited by electron–phonon coupling from Boltzmann transport equation: an ab initio study of Si, Al, and MoS\(_2\). Phys. Rev. B 92, 075405 (2015). https://doi.org/10.1103/PhysRevB.92.075405

    Article  Google Scholar 

  101. Poncé, S., Margine, E.R., Verdi, C., Giustino, F.: Epw: electron–phonon coupling, transport and superconducting properties using maximally localized Wannier functions. Comput. Phys. Commun. 209, 116 (2016). https://doi.org/10.1016/j.cpc.2016.07.028

    Article  MathSciNet  Google Scholar 

  102. Li, W., Carrete, J., Katcho, N.A., Mingo, N.: ShengBTE: a solver of the Boltzmann transport equation for phonons. Comput. Phys. Commun. 185, 1747–1758 (2014). https://doi.org/10.1016/j.cpc.2014.02.015

    Article  MATH  Google Scholar 

  103. Zhou, J.-J., Park, J., Lu, I.-T., Maliyov, I., Tong, X., Bernardi, M.: Perturbo: a software package for ab initio electron–phonon interactions, charge transport and ultrafast dynamics. Comput. Phys. Commun. 264, 107970 (2021). https://doi.org/10.1016/j.cpc.2021.107970

    Article  MathSciNet  MATH  Google Scholar 

  104. Cepellotti, A., Coulter, J., Johansson, A., Fedorova, N.S., Kozinsky, B.: Phoebe: a high-performance framework for solving phonon and electron Boltzmann transport equations. J. Phys. Mater. 5, 035003 (2022). https://doi.org/10.1088/2515-7639/ac86f6

    Article  Google Scholar 

  105. Onsager, L.: Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405 (1931). https://doi.org/10.1103/PhysRev.37.405

    Article  MATH  Google Scholar 

  106. Onsager, L.: Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265 (1931). https://doi.org/10.1103/PhysRev.38.2265

    Article  MATH  Google Scholar 

  107. Callen, H.B.: The application of Onsager’s reciprocal relations to thermoelectric, thermomagnetic, and galvanomagnetic effects. Phys. Rev. 73, 1349 (1948). https://doi.org/10.1103/PhysRev.73.1349

    Article  MATH  Google Scholar 

  108. Groot, S.R.: Thermodynamics of Irreversible Processes, vol. 3. North-Holland Publishing Company (1963)

    MATH  Google Scholar 

  109. Callen, H.B.: Thermodynamics and an Introduction to Thermostatistics. Wiley (1995)

    MATH  Google Scholar 

  110. Goupil, C., Seifert, W., Zabrocki, K., Müller, E., Snyder, G.J.: Thermodynamics of thermoelectric phenomena and applications. Entropy 13, 1481 (2011). https://doi.org/10.3390/e13081481

    Article  MATH  Google Scholar 

  111. Feldhoff, A.: Thermoelectric material tensor derived from the Onsager–de Groot–Callen model. Energy Harvest. Syst. 2, 5 (2015). https://doi.org/10.1515/ehs-2014-0040

    Article  Google Scholar 

  112. Chaikin, P.: An introduction to thermopower for those who might want to use it to study organic conductors and superconductors. In: Organic Superconductivity. Springer, pp. 101–115 (1990)

  113. Robinson, J.E.: Thermoelectric power in the nearly-free-electron model. Phys. Rev. 161, 533 (1967). https://doi.org/10.1103/PhysRev.161.533

    Article  Google Scholar 

  114. Feldhoff, A., Geppert, B.: A high-temperature thermoelectric generator based on oxides. Energy Harvest. Syst. 1, 69 (2014). https://doi.org/10.1515/ehs-2014-0016

    Article  Google Scholar 

  115. Antončík, E.: On the theory of temperature shift of the absorption curve in non-polar crystals. Cechoslov. Fiz. Z. 5, 449 (1955). https://doi.org/10.1007/BF01687209

    Article  Google Scholar 

  116. Lautenschlager, P., Allen, P., Cardona, M.: Phonon-induced lifetime broadenings of electronic states and critical points in Si and Ge. Phys. Rev. B 33, 5501 (1986). https://doi.org/10.1103/PhysRevB.33.5501

    Article  Google Scholar 

  117. Giustino, F.: Electron–phonon interactions from first principles. Rev. Mod. Phys. 89, 015003 (2017). https://doi.org/10.1103/RevModPhys.89.015003

    Article  MathSciNet  Google Scholar 

  118. Keating, P.: Dielectric screening and the phonon spectra of metallic and nonmetallic crystals. Phys. Rev. 175, 1171 (1968). https://doi.org/10.1103/PhysRev.175.1171

    Article  Google Scholar 

  119. Marini, A., Poncé, S., Gonze, X.: Many-body perturbation theory approach to the electron-phonon interaction with density-functional theory as a starting point. Phys. Rev. B 91, 224310 (2015). https://doi.org/10.1103/PhysRevB.91.224310

    Article  Google Scholar 

  120. Baym, G.: Field-theoretic approach to the properties of the solid state. Ann. Phys. 14, 1 (1961). https://doi.org/10.1016/0003-4916(61)90050-1

    Article  MathSciNet  MATH  Google Scholar 

  121. Hedin, L., Lundqvist, S.: Effects of electron–electron and electron–phonon interactions on the one-electron states of solids. In: Solid State Physics, vol. 23. Elsevier, pp. 1–181 (1970)

  122. Migdal, A.: Interaction between electrons and lattice vibrations in a normal metal. Sov. Phys. JETP 7, 996 (1958)

    Google Scholar 

  123. Allen, P.B., Mitrović, B.: Theory of superconducting \(T_c\). Solid State Phys. 37, 1 (1983). https://doi.org/10.1016/S0081-1947(08)60665-7

    Article  Google Scholar 

  124. Mustafa, J.I., Bernardi, M., Neaton, J.B., Louie, S.G.: Ab initio electronic relaxation times and transport in noble metals. Phys. Rev. B 94, 155105 (2016). https://doi.org/10.1103/PhysRevB.94.155105

    Article  Google Scholar 

  125. Gonze, X., Lee, C.: Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory. Phys. Rev. B 55, 10355 (1997). https://doi.org/10.1103/PhysRevB.55.10355

    Article  Google Scholar 

  126. Verdi, C., Giustino, F.: Fröhlich electron–phonon vertex from first principles. Phys. Rev. Lett. 115, 176401 (2015). https://doi.org/10.1103/PhysRevLett.115.176401

    Article  Google Scholar 

  127. Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Oxford University Press, London (1954)

    MATH  Google Scholar 

  128. Frölich, H.: Electrical breakdown in solid crystals. Proc. R. Soc. 160, 230–238 (1937)

    Google Scholar 

  129. Callen, H.B.: Electric breakdown in ionic crystals. Phys. Rev. 76, 1394 (1949). https://doi.org/10.1103/PhysRev.76.1394

    Article  MATH  Google Scholar 

  130. Howarth, D., Sondheimer, E.: The theory of electronic conduction in polar semi-conductors. Proc. R. Soc. Lond. A 219, 53 (1953). https://doi.org/10.1098/rspa.1953.0130

    Article  MATH  Google Scholar 

  131. Vogl, P.: Microscopic theory of electron–phonon interaction in insulators or semiconductors. Phys. Rev. B 13, 694 (1976). https://doi.org/10.1103/PhysRevB.13.694

    Article  Google Scholar 

  132. Lawaetz, P.: Long-wavelength phonon scattering in nonpolar semiconductors. Phys. Rev. 183, 730 (1969). https://doi.org/10.1103/PhysRev.183.730

    Article  Google Scholar 

  133. Rohlfing, M., Louie, S.G.: Electron-hole excitations and optical spectra from first principles. Phys. Rev. B 62, 4927 (2000). https://doi.org/10.1103/PhysRevB.62.4927

    Article  Google Scholar 

  134. Sjakste, J., Vast, N., Calandra, M., Mauri, F.: Wannier interpolation of the electron–phonon matrix elements in polar semiconductors: Polar-optical coupling in GAAS. Phys. Rev. B 92, 054307 (2015). https://doi.org/10.1103/PhysRevB.92.054307

    Article  Google Scholar 

  135. Brunin, G., Miranda, H.P.C., Giantomassi, M., Royo, M., Stengel, M., Verstraete, M.J., Gonze, X., Rignanese, G.-M., Hautier, G.: Electron–phonon beyond Fröhlich: dynamical quadrupoles in polar and covalent solids. Phys. Rev. Lett. 125, 136601 (2020). https://doi.org/10.1103/PhysRevLett.125.136601

    Article  Google Scholar 

  136. Brunin, G., Miranda, H.P.C., Giantomassi, M., Royo, M., Stengel, M., Verstraete, M.J., Gonze, X., Rignanese, G.-M., Hautier, G.: Phonon-limited electron mobility in Si, GAAS, and gap with exact treatment of dynamical quadrupoles. Phys. Rev. B 102, 094308 (2020). https://doi.org/10.1103/PhysRevB.102.094308

    Article  Google Scholar 

  137. Jhalani, V.A., Zhou, J.-J., Park, J., Dreyer, C.E., Bernardi, M.: Piezoelectric electron–phonon interaction from ab initio dynamical quadrupoles: impact on charge transport in wurtzite GaN. Phys. Rev. Lett. 125, 136602 (2020). https://doi.org/10.1103/PhysRevLett.125.136602

    Article  Google Scholar 

  138. Park, J., Zhou, J.-J., Jhalani, V.A., Dreyer, C.E., Bernardi, M.: Long-range quadrupole electron–phonon interaction from first principles. Phys. Rev. B 102, 125203 (2020). https://doi.org/10.1103/PhysRevB.102.125203

    Article  Google Scholar 

  139. Martin, R.M.: Piezoelectricity. Phys. Rev. B 5, 1607 (1972). https://doi.org/10.1103/PhysRevB.5.1607

    Article  Google Scholar 

  140. Poncé, S., Macheda, F., Margine, E.R., Marzari, N., Bonini, N., Giustino, F.: First-principles predictions of hall and drift mobilities in semiconductors. Phys. Rev. Res. 3, 043022 (2021). https://doi.org/10.1103/PhysRevResearch.3.043022

    Article  Google Scholar 

  141. Ren, Q., Fu, C., Qiu, Q., Dai, S., Liu, Z., Masuda, T., Asai, S., Hagihala, M., Lee, S., Torri, S., Kamiyama, T., He, L., Tong, X., Felser, C., Singh, D.J., Zhu, T., Yang, J., Ma, J.: Establishing the carrier scattering phase diagram for ZrNiSn-based half-Heusler thermoelectric materials. Nat. Commun. 11, 1 (2020). https://doi.org/10.1038/s41467-020-16913-2

    Article  Google Scholar 

  142. Macheda, F., Barone, P., Mauri, F.: Electron–phonon interaction and longitudinal-transverse phonon splitting in doped semiconductors. Phys. Rev. Lett. 129, 185902 (2022). https://doi.org/10.1103/PhysRevLett.129.185902

    Article  Google Scholar 

  143. Ehrenreich, H.: Screening effects in polar semiconductors. J. Phys. Chem. Solids 8, 130 (1959). https://doi.org/10.1016/0022-3697(59)90297-5

    Article  Google Scholar 

  144. Chaves, A.S., Larson, D.T., Kaxiras, E., Antonelli, A.: Microscopic origin of the high thermoelectric figure of merit of n-doped SnSe. Phys. Rev. B 104, 115204 (2021). https://doi.org/10.1103/PhysRevB.104.115204

    Article  Google Scholar 

  145. Chaves, A.S., Larson, D.T., Kaxiras, E., Antonelli, A.: Out-of-plane thermoelectric performance for p-doped GeSe. Phys. Rev. B 105, 205201 (2022). https://doi.org/10.1103/PhysRevB.105.205201

    Article  Google Scholar 

  146. Radisavljevic, B., Kis, A.: Mobility engineering and a metal-insulator transition in monolayer MoS\(_2\). Nat. Mater. 12, 815 (2013). https://doi.org/10.1038/nmat3687

    Article  Google Scholar 

  147. Li, S.-L., Tsukagoshi, K., Orgiu, E., Samorì, P.: Charge transport and mobility engineering in two-dimensional transition metal chalcogenide semiconductors. Chem. Soc. Rev. 45, 118 (2016). https://doi.org/10.1039/C5CS00517E

    Article  Google Scholar 

  148. Bergmann, G.: Weak localization in thin films: a time-of-flight experiment with conduction electrons. Phys. Rep. 107, 1 (1984). https://doi.org/10.1016/0370-1573(84)90103-0

    Article  Google Scholar 

  149. Lee, P.A., Stone, A.D.: Universal conductance fluctuations in metals. Phys. Rev. Lett. 55, 1622 (1985). https://doi.org/10.1103/PhysRevLett.55.1622

    Article  Google Scholar 

  150. Datta, S.: Electronic Transport in Mesoscopic Systems. Cambridge University Press (1997)

    Google Scholar 

  151. Dewandre, A., Hellman, O., Bhattacharya, S., Romero, A.H., Madsen, G.K., Verstraete, M.J.: Two-step phase transition in SnSe and the origins of its high power factor from first principles. Phys. Rev. Lett. 117, 276601 (2016). https://doi.org/10.1103/PhysRevLett.117.276601

    Article  Google Scholar 

  152. Gunlycke, D., White, C.T.: Graphene valley filter using a line defect. Phys. Rev. Lett. 106, 136806 (2011). https://doi.org/10.1103/PhysRevLett.106.136806

    Article  Google Scholar 

  153. Koenraad, P.M., Flatté, M.E.: Single dopants in semiconductors. Nat. Mater. 10, 91 (2011). https://doi.org/10.1038/nmat2940

    Article  Google Scholar 

  154. Zheng, Y., Slade, T.J., Hu, L., Tan, X.Y., Luo, Y., Luo, Z.-Z., Xu, J., Yan, Q., Kanatzidis, M.G.: Defect engineering in thermoelectric materials: what have we learned? Chem. Soc. Rev. (2021). https://doi.org/10.1039/D1CS00347J

    Article  Google Scholar 

  155. Brooks, H.: Theory of the electrical properties of germanium and silicon. In: Advances in Electronics and Electron Physics, vol. 7. Elsevier, pp. 85–182 (1955)

  156. Chattopadhyay, D., Queisser, H.J.: Electron scattering by ionized impurities in semiconductors. Rev. Mod. Phys. 53, 745 (1981). https://doi.org/10.1103/RevModPhys.53.745

    Article  Google Scholar 

  157. Moore, E.J.: Quantum-transport theories and multiple scattering in doped semiconductors. I. Formal theory. Phys. Rev. 160, 607 (1967). https://doi.org/10.1103/PhysRev.160.607

    Article  Google Scholar 

  158. Papanikolaou, N., Zeller, R., Dederichs, P., Stefanou, N.: Lattice distortion in Cu-based dilute alloys: a first-principles study by the KKR Green-function method. Phys. Rev. B 55, 4157 (1997). https://doi.org/10.1103/PhysRevB.55.4157

    Article  Google Scholar 

  159. Settels, A., Korhonen, T., Papanikolaou, N., Zeller, R., Dederichs, P.: Ab initio study of acceptor-donor complexes in silicon and germanium. Phys. Rev. Lett. 83, 4369 (1999). https://doi.org/10.1103/PhysRevLett.83.4369

    Article  Google Scholar 

  160. Höhler, H., Atodiresei, N., Schroeder, K., Zeller, R., Dederichs, P.: Cd-vacancy and Cd-interstitial complexes in Si and Ge. Phys. Rev. B 70, 155313 (2004). https://doi.org/10.1103/PhysRevB.70.155313

    Article  Google Scholar 

  161. Ebert, H., Koedderitzsch, D., Minar, J.: Calculating condensed matter properties using the KKR-Green’s function method-recent developments and applications. Rep. Prog. Phys. 74, 096501 (2011). https://doi.org/10.1088/0034-4885/74/9/096501

    Article  Google Scholar 

  162. Restrepo, O., Varga, K., Pantelides, S.: First-principles calculations of electron mobilities in silicon: phonon and coulomb scattering. Appl. Phys. Lett. 94, 212103 (2009). https://doi.org/10.1063/1.3147189

    Article  Google Scholar 

  163. Lordi, V., Erhart, P., Åberg, D.: Charge carrier scattering by defects in semiconductors. Phys. Rev. B 81, 235204 (2010). https://doi.org/10.1103/PhysRevB.81.235204

    Article  Google Scholar 

  164. Lu, I.-T., Zhou, J.-J., Bernardi, M.: Efficient ab initio calculations of electron-defect scattering and defect-limited carrier mobility. Phys. Rev. Mater. 3, 033804 (2019). https://doi.org/10.1103/PhysRevMaterials.3.033804

    Article  Google Scholar 

  165. Lu, I.-T., Park, J., Zhou, J.-J., Bernardi, M.: Ab initio electron-defect interactions using Wannier functions. npj Comput. Mater. 6, 1 (2020). https://doi.org/10.1038/s41524-020-0284-y

    Article  Google Scholar 

  166. Fugallo, G., Lazzeri, M., Paulatto, L., Mauri, F.: Ab initio variational approach for evaluating lattice thermal conductivity. Phys. Rev. B 88, 045430 (2013). https://doi.org/10.1103/PhysRevB.88.045430

    Article  Google Scholar 

  167. Feng, T., Ruan, X.: Quantum mechanical prediction of four-phonon scattering rates and reduced thermal conductivity of solids. Phys. Rev. B 93, 045202 (2016). https://doi.org/10.1103/PhysRevB.93.045202

    Article  Google Scholar 

  168. Han, Z., Yang, X., Li, W., Feng, T., Ruan, X.: Fourphonon: an extension module to Shengbte for computing four-phonon scattering rates and thermal conductivity. Comput. Phys. Commun. 270, 108179 (2022). https://doi.org/10.1016/j.cpc.2021.108179

    Article  MathSciNet  Google Scholar 

  169. Garg, J., Bonini, N., Kozinsky, B., Marzari, N.: Role of disorder and anharmonicity in the thermal conductivity of silicon–germanium alloys: a first-principles study. Phys. Rev. Lett. 106, 045901 (2011). https://doi.org/10.1103/PhysRevLett.106.045901

    Article  Google Scholar 

  170. Liao, B., Qiu, B., Zhou, J., Huberman, S., Esfarjani, K., Chen, G.: Significant reduction of lattice thermal conductivity by the electron–phonon interaction in silicon with high carrier concentrations: a first-principles study. Phys. Rev. Lett. 114, 115901 (2015). https://doi.org/10.1103/PhysRevLett.114.115901

    Article  Google Scholar 

  171. Madsen, G.K., Singh, D.J.: Boltztrap. a code for calculating band-structure dependent quantities. Comput. Phys. Commun. 175, 67 (2006). https://doi.org/10.1016/j.cpc.2006.03.007

    Article  MATH  Google Scholar 

  172. Bardeen, J., Shockley, W.: Deformation potentials and mobilities in non-polar crystals. Phys. Rev. 80, 72 (1950). https://doi.org/10.1103/PhysRev.80.72

    Article  MATH  Google Scholar 

  173. Xi, J., Long, M., Tang, L., Wang, D., Shuai, Z.: First-principles prediction of charge mobility in carbon and organic nanomaterials. Nanoscale 4, 4348 (2012). https://doi.org/10.1039/C2NR30585B

    Article  Google Scholar 

  174. Xi, L., Pan, S., Li, X., Xu, Y., Ni, J., Sun, X., Yang, J., Luo, J., Xi, J., Zhu, W., et al.: Discovery of high-performance thermoelectric chalcogenides through reliable high-throughput material screening. J. Am. Chem. Soc. 140, 10785 (2018). https://doi.org/10.1021/jacs.8b04704

    Article  Google Scholar 

  175. Ganose, A.M., Park, J., Faghaninia, A., Woods-Robinson, R., Persson, K.A., Jain, A.: Efficient calculation of carrier scattering rates from first principles. Nat. Commun. 12, 1 (2021). https://doi.org/10.1038/s41467-021-22440-5

    Article  Google Scholar 

  176. Ma, J., Nissimagoudar, A.S., Li, W.: First-principles study of electron and hole mobilities of Si and GaAs. Phys. Rev. B 97, 045201 (2018). https://doi.org/10.1103/PhysRevB.97.045201

    Article  Google Scholar 

  177. Giustino, F., Cohen, M.L., Louie, S.G.: Electron–phonon interaction using Wannier functions. Phys. Rev. B 76, 165108 (2007). https://doi.org/10.1103/PhysRevB.76.165108

    Article  Google Scholar 

  178. Agapito, L.A., Bernardi, M.: Ab initio electron–phonon interactions using atomic orbital wave functions. Phys. Rev. B (2018). https://doi.org/10.1103/PhysRevB.97.235146

    Article  Google Scholar 

  179. Samsonidze, G., Kozinsky, B.: Accelerated screening of thermoelectric materials by first-principles computations of electron–phonon scattering. Adv. Energy Mater. 8, 1800246 (2018). https://doi.org/10.1002/aenm.201800246

    Article  Google Scholar 

  180. Deng, T., Wu, G., Sullivan, M.B., Wong, Z.M., Hippalgaonkar, K., Wang, J.-S., Yang, S.-W.: EPIC STAR: a reliable and efficient approach for phonon- and impurity-limited charge transport calculations. npj Comput. Mater. 6, 1 (2020). https://doi.org/10.1038/s41524-020-0316-7

    Article  Google Scholar 

  181. Yao, M., Wang, Y., Li, X., Sheng, Y., Huo, H., Xi, L., Yang, J., Zhang, W.: Materials informatics platform with three dimensional structures, workflow and thermoelectric applications. Sci. Data 8, 1 (2021). https://doi.org/10.1038/s41597-021-01022-6

    Article  Google Scholar 

  182. Engel, M., Marsman, M., Franchini, C., Kresse, G.: Electron-phonon interactions using the projector augmented-wave method and Wannier functions. Phys. Rev. B 101, 184302 (2020). https://doi.org/10.1103/PhysRevB.101.184302

    Article  Google Scholar 

  183. Brouder, C., Panati, G., Calandra, M., Mourougane, C., Marzari, N.: Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98, 046402 (2007). https://doi.org/10.1103/PhysRevLett.98.046402

    Article  Google Scholar 

  184. Souza, I., Marzari, N., Vanderbilt, D.: Maximally localized Wannier functions for entangled energy bands. Phys. Rev. B 65, 035109 (2001). https://doi.org/10.1103/PhysRevB.65.035109

    Article  Google Scholar 

  185. Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. Courier Corporation, Berlin (2012)

    Google Scholar 

  186. Chaves, A.S., Antonelli, A., Larson, D.T., Kaxiras, E.: Boosting the efficiency of ab initio electron–phonon coupling calculations through dual interpolation. Phys. Rev. B 102, 125116 (2020). https://doi.org/10.1103/PhysRevB.102.125116

    Article  Google Scholar 

  187. Chadi, D.J., Cohen, M.L.: Special points in the Brillouin zone. Phys. Rev. B 8, 5747 (1973). https://doi.org/10.1103/PhysRevB.8.5747

    Article  MathSciNet  Google Scholar 

  188. Shankland, D.G.: Interpolation in k-space with functions of arbitrary smoothness. In: Computational Methods in Band Theory. Springer, pp. 362–367 (1971). https://doi.org/10.1007/978-1-4684-1890-328

  189. Koelling, D., Wood, J.: On the interpolation of eigenvalues and a resultant integration scheme. J. Comput. Phys. 67, 253 (1986). https://doi.org/10.1016/0021-9991(86)90261-5

    Article  Google Scholar 

  190. Pickett, W.E., Krakauer, H., Allen, P.B.: Smooth Fourier interpolation of periodic functions. Phys. Rev. B 38, 2721 (1988). https://doi.org/10.1103/PhysRevB.38.2721

    Article  Google Scholar 

  191. Togo, A., Chaput, L., Tanaka, I.: Distributions of phonon lifetimes in Brillouin zones. Phys. Rev. B 91, 094306 (2015). https://doi.org/10.1103/PhysRevB.91.094306

    Article  Google Scholar 

  192. Carrete, J., Mingo, N., Curtarolo, S.: Low thermal conductivity and triaxial phononic anisotropy of SnSe. Appl. Phys. Lett. 105, 101907 (2014). https://doi.org/10.1063/1.4895770

    Article  Google Scholar 

  193. Pei, Y., Shi, X., LaLonde, A., Wang, H., Chen, L., Snyder, G.J.: Convergence of electronic bands for high performance bulk thermoelectrics. Nature 473, 66 (2011). https://doi.org/10.1038/nature09996

    Article  Google Scholar 

  194. Pei, Y., Wang, H., Snyder, G.J.: Band engineering of thermoelectric materials. Adv. Mater. 24, 6125 (2012). https://doi.org/10.1002/adma.201202919

    Article  Google Scholar 

  195. Liu, W., Tan, X., Yin, K., Liu, H., Tang, X., Shi, J., Zhang, Q., Uher, C.: Convergence of conduction bands as a means of enhancing thermoelectric performance of n-type Mg\(_2\)Si\(_{1-x}\)Sn\(_x\) solid solutions. Phys. Rev. Lett. 108, 166601 (2012). https://doi.org/10.1103/PhysRevLett.108.166601

    Article  Google Scholar 

  196. Dehkordi, A.M., Zebarjadi, M., He, J., Tritt, T.M.: Thermoelectric power factor: enhancement mechanisms and strategies for higher performance thermoelectric materials. Mater. Sci. Eng. R. Rep. 97, 1 (2015). https://doi.org/10.1016/j.mser.2015.08.001

    Article  Google Scholar 

  197. Parker, D.S., May, A.F., Singh, D.J.: Benefits of carrier-pocket anisotropy to thermoelectric performance: the case of p-type AgBiSe\(_2\). Phys. Rev. Appl. 3, 064003 (2015). https://doi.org/10.1103/PhysRevApplied.3.064003

    Article  Google Scholar 

  198. Morelli, D., Jovovic, V., Heremans, J.: Intrinsically minimal thermal conductivity in cubic I–V–VI\(_2\) semiconductors. Phys. Rev. Lett. 101, 035901 (2008). https://doi.org/10.1103/PhysRevLett.101.035901

    Article  Google Scholar 

  199. He, J., Amsler, M., Xia, Y., Naghavi, S.S., Hegde, V.I., Hao, S., Goedecker, S., Ozoliņš, V., Wolverton, C.: Ultralow thermal conductivity in full Heusler semiconductors. Phys. Rev. Lett. 117, 046602 (2016). https://doi.org/10.1103/PhysRevLett.117.046602

    Article  Google Scholar 

  200. González-Romero, R.L., Antonelli, A., Chaves, A.S., Meléndez, J.J.: Ultralow and anisotropic thermal conductivity in semiconductor As\(_2\)Se\(_3\). Phys. Chem. Chem. Phys. 20, 1809 (2018). https://doi.org/10.1039/C7CP07242B

    Article  Google Scholar 

  201. Hochbaum, A.I., Chen, R., Delgado, R.D., Liang, W., Garnett, E.C., Najarian, M., Majumdar, A., Yang, P.: Enhanced thermoelectric performance of rough silicon nanowires. Nature 451, 163 (2008). https://doi.org/10.1038/nature06381

    Article  Google Scholar 

  202. Boukai, A.I., Bunimovich, Y., Tahir-Kheli, J., Yu, J.-K., Goddard, W.A., III., Heath, J.R.: Silicon nanowires as efficient thermoelectric materials. Nature 451, 168 (2008). https://doi.org/10.1038/nature06458

    Article  Google Scholar 

  203. Kanatzidis, M.G.: Nanostructured thermoelectrics: the new paradigm? Chem. Mater. 22, 648 (2009). https://doi.org/10.1021/cm902195j

    Article  Google Scholar 

  204. Zhao, L.-D., Hao, S., Lo, S.-H., Wu, C.-I., Zhou, X., Lee, Y., Li, H., Biswas, K., Hogan, T.P., Uher, C., Wolverton, C., Dravid, V.P., G, K.M.: High thermoelectric performance via hierarchical compositionally alloyed nanostructures. J. Am. Chem. Soc. 135, 7364 (2013). https://doi.org/10.1021/ja403134b

    Article  Google Scholar 

  205. McKinney, R.W., Gorai, P., Stevanović, V., Toberer, E.S.: Search for new thermoelectric materials with low Lorenz number. J. Mater. Chem. A 5, 17302 (2017). https://doi.org/10.1039/C7TA04332E

    Article  Google Scholar 

  206. Mahan, G., Sofo, J.: The best thermoelectric. Proc. Natl. Acad. Sci. 93, 7436 (1996). https://doi.org/10.1073/pnas.93.15.7436

    Article  Google Scholar 

  207. Baranowski, L.L., Jeffrey Snyder, G., Toberer, E.S.: Effective thermal conductivity in thermoelectric materials. J. Appl. Phys. 113, 204904 (2013). https://doi.org/10.1063/1.4807314

    Article  Google Scholar 

  208. Ortiz, B.R., Gorai, P., Krishna, L., Mow, R., Lopez, A., McKinney, R., Stevanović, V., Toberer, E.S.: Potential for high thermoelectric performance in n-type Zintl compounds: a case study of Ba doped KAlSb\(_4\). J. Mater. Chem. A 5, 4036 (2017). https://doi.org/10.1039/C6TA09532A

    Article  Google Scholar 

  209. Putatunda, A., Singh, D.J.: Lorenz number in relation to estimates based on the Seebeck coefficient. Mater. Today Phys. 8, 49 (2019). https://doi.org/10.1016/j.mtphys.2019.01.001

    Article  Google Scholar 

  210. He, J., Tritt, T.M.: Advances in thermoelectric materials research: looking back and moving forward. Science 357, eaak9997 (2017). https://doi.org/10.1126/science.aak9997

    Article  Google Scholar 

  211. Biswas, K., He, J., Blum, I.D., Wu, C.-I., Hogan, T.P., Seidman, D.N., Dravid, V.P., Kanatzidis, M.G.: High-performance bulk thermoelectrics with all-scale hierarchical architectures. Nature 489, 414 (2012). https://doi.org/10.1038/nature11439

    Article  Google Scholar 

  212. Liu, H., Shi, X., Xu, F., Zhang, L., Zhang, W., Chen, L., Li, Q., Uher, C., Day, T., Snyder, G.J.: Copper ion liquid-like thermoelectrics. Nat. Mater. 11, 422 (2012). https://doi.org/10.1038/nmat3273

    Article  Google Scholar 

  213. Fu, T., Yue, X., Wu, H., Fu, C., Zhu, T., Liu, X., Hu, L., Ying, P., He, J., Zhao, X.: Enhanced thermoelectric performance of PbTe bulk materials with figure of merit zT \(>\) 2 by multi-functional alloying. J. Mater. 2, 141 (2016). https://doi.org/10.1016/j.jmat.2016.05.005

    Article  Google Scholar 

  214. Olvera, A., Moroz, N., Sahoo, P., Ren, P., Bailey, T., Page, A., Uher, C., Poudeu, P.: Partial indium solubility induces chemical stability and colossal thermoelectric figure of merit in Cu\(_2\)Se. Energy Environ. Sci. 10, 1668 (2017). https://doi.org/10.1039/C7EE01193H

    Article  Google Scholar 

  215. Cheng, Y., Yang, J., Jiang, Q., He, D., He, J., Luo, Y., Zhang, D., Zhou, Z., Ren, Y., Xin, J.: New insight into InSb-based thermoelectric materials: from a divorced eutectic design to a remarkably high thermoelectric performance. J. Mater. Chem. A 5, 5163 (2017). https://doi.org/10.1039/C6TA10827J

    Article  Google Scholar 

  216. Ma, N., Li, Y.-Y., Chen, L., Wu, L.-M.: \(\alpha\)-CsCu\(_5\)Se\(_3\): discovery of a low-cost bulk selenide with high thermoelectric performance. J. Am. Chem. Soc. 142, 5293 (2020). https://doi.org/10.1021/jacs.0c00062

    Article  Google Scholar 

  217. Roychowdhury, S., Ghosh, T., Arora, R., Samanta, M., Xie, L., Singh, N.K., Soni, A., He, J., Waghmare, U.V., Biswas, K.: Enhanced atomic ordering leads to high thermoelectric performance in AgSbTe\(_2\). Science 371, 722 (2021). https://doi.org/10.1126/science.abb3517

    Article  Google Scholar 

  218. Terasaki, I., Sasago, Y., Uchinokura, K.: Large thermoelectric power in NaCo\(_2\)O\(_4\) single crystals. Phys. Rev. B 56, R12685 (1997). https://doi.org/10.1103/PhysRevB.56.R12685

    Article  Google Scholar 

  219. Rhyee, J.-S., Lee, K.H., Lee, S.M., Cho, E., Kim, S.I., Lee, E., Kwon, Y.S., Shim, J.H., Kotliar, G.: Peierls distortion as a route to high thermoelectric performance in In\(_4\)Se\(_{3-\delta }\) crystals. Nature 459, 965 (2009). https://doi.org/10.1038/nature08088

    Article  Google Scholar 

  220. Ohta, H., Kim, S.W., Kaneki, S., Yamamoto, A., Hashizume, T.: High thermoelectric power factor of high-mobility 2D electron gas. Adv. Sci. 5, 1700696 (2018). https://doi.org/10.1002/advs.201700696

    Article  Google Scholar 

  221. Cheng, L., Zhang, C., Liu, Y.: The optimal electronic structure for high-mobility 2D semiconductors: exceptionally high hole mobility in 2D antimony. J. Am. Chem. Soc. 141, 16296 (2019). https://doi.org/10.1021/jacs.9b05923

    Article  Google Scholar 

  222. Li, Z., Xiao, C., Xie, Y.: Layered thermoelectric materials: structure, bonding, and performance mechanisms. Appl. Phys. Rev. 9, 011303 (2022). https://doi.org/10.1063/5.0074489

    Article  Google Scholar 

  223. Zhao, L.-D., Lo, S.-H., Zhang, Y., Sun, H., Tan, G., Uher, C., Wolverton, C., Dravid, V.P., Kanatzidis, M.G.: Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals. Nature 508, 373 (2014). https://doi.org/10.1038/nature13184

    Article  Google Scholar 

  224. Zhao, L.-D., Tan, G., Hao, S., He, J., Pei, Y., Chi, H., Wang, H., Gong, S., Xu, H., Dravid, V.P., et al.: Ultrahigh power factor and thermoelectric performance in hole-doped single-crystal SnSe. Science 351, 141 (2016). https://doi.org/10.1126/science.aad3749

    Article  Google Scholar 

  225. Chang, C., Wu, M., He, D., Pei, Y., Wu, C.-F., Wu, X., Yu, H., Zhu, F., Wang, K., Chen, Y., et al.: 3D charge and 2D phonon transports leading to high out-of-plane ZT in n-type SnSe crystals. Science 360, 778 (2018). https://doi.org/10.1126/science.aaq1479

    Article  Google Scholar 

  226. Ding, G., Gao, G., Yao, K.: High-efficient thermoelectric materials: the case of orthorhombic IV–VI compounds. Sci. Rep. 5, 1 (2015). https://doi.org/10.1038/srep09567

    Article  Google Scholar 

  227. Guo, R., Wang, X., Kuang, Y., Huang, B.: First-principles study of anisotropic thermoelectric transport properties of IV–VI semiconductor compounds SnSe and SnS. Phys. Rev. B 92, 115202 (2015). https://doi.org/10.1103/PhysRevB.92.115202

    Article  Google Scholar 

  228. Skelton, J.M., Burton, L.A., Parker, S.C., Walsh, A., Kim, C.-E., Soon, A., Buckeridge, J., Sokol, A.A., Catlow, C.R.A., Togo, A., et al.: Anharmonicity in the high-temperature Cmcm phase of SnSe: soft modes and three-phonon interactions. Phys. Rev. Lett. 117, 075502 (2016). https://doi.org/10.1103/PhysRevLett.117.075502

    Article  Google Scholar 

  229. Li, S., Tong, Z., Bao, H.: Resolving different scattering effects on the thermal and electrical transport in doped SnSe. J. Appl. Phys. 126, 025111 (2019). https://doi.org/10.1063/1.5098340

    Article  Google Scholar 

  230. Aseginolaza, U., Bianco, R., Monacelli, L., Paulatto, L., Calandra, M., Mauri, F., Bergara, A., Errea, I.: Phonon collapse and second-order phase transition in thermoelectric SnSe. Phys. Rev. Lett. 122, 075901 (2019). https://doi.org/10.1103/PhysRevLett.122.075901

    Article  Google Scholar 

  231. Zhang, X., Shen, J., Lin, S., Li, J., Chen, Z., Li, W., Pei, Y.: Thermoelectric properties of GeSe. J. Mater. 2, 331 (2016). https://doi.org/10.1016/j.jmat.2016.09.001

    Article  Google Scholar 

  232. Shaabani, L., Aminorroaya-Yamini, S., Byrnes, J., Akbar Nezhad, A., Blake, G.R.: Thermoelectric performance of Na-doped GeSe. ACS Omega 2, 9192 (2017). https://doi.org/10.1021/acsomega.7b01364

    Article  Google Scholar 

  233. Hao, S., Shi, F., Dravid, V.P., Kanatzidis, M.G., Wolverton, C.: Computational prediction of high thermoelectric performance in hole doped layered GeSe. Chem. Mater. 28, 3218 (2016)

    Article  Google Scholar 

  234. Roychowdhury, S., Samanta, M., Perumal, S., Biswas, K.: Germanium chalcogenide thermoelectrics: electronic structure modulation and low lattice thermal conductivity. Chem. Mater. 30, 5799 (2018). https://doi.org/10.1021/acs.chemmater.8b02676

    Article  Google Scholar 

  235. Yuan, K., Sun, Z., Zhang, X., Tang, D.: Tailoring phononic, electronic, and thermoelectric properties of orthorhombic GeSe through hydrostatic pressure. Sci. Rep. 9, 1 (2019). https://doi.org/10.1038/s41598-019-45949-8

    Article  Google Scholar 

  236. Asen-Palmer, M., Bartkowski, K., Gmelin, E., Cardona, M., Zhernov, A., Inyushkin, A., Taldenkov, A., Ozhogin, V., Itoh, K.M., Haller, E.: Thermal conductivity of germanium crystals with different isotopic compositions. Phys. Rev. B 56, 9431 (1997). https://doi.org/10.1103/PhysRevB.56.9431

    Article  Google Scholar 

  237. Omini, M., Sparavigna, A.: An iterative approach to the phonon Boltzmann equation in the theory of thermal conductivity. Phys. B 212, 101 (1995). https://doi.org/10.1016/0921-4526(95)00016-3

    Article  Google Scholar 

  238. Simoncelli, M., Marzari, N., Mauri, F.: Unified theory of thermal transport in crystals and glasses. Nat. Phys. 15, 809 (2019). https://doi.org/10.1038/s41567-019-0520-x

    Article  Google Scholar 

  239. Zhou, C., Lee, Y.K., Yu, Y., Byun, S., Luo, Z.-Z., Lee, H., Ge, B., Lee, Y.-L., Chen, X., Lee, J.Y., et al.: Polycrystalline SnSe with a thermoelectric figure of merit greater than the single crystal. Nat. Mater. 20, 1378 (2021). https://doi.org/10.1038/s41563-021-01064-6

    Article  Google Scholar 

  240. Ibrahim, D., Vaney, J.-B., Sassi, S., Candolfi, C., Ohorodniichuk, V., Levinsky, P., Semprimoschnig, C., Dauscher, A., Lenoir, B.: Reinvestigation of the thermal properties of single-crystalline SnSe. Appl. Phys. Lett. 110, 032103 (2017). https://doi.org/10.1063/1.4974348

    Article  Google Scholar 

  241. Sarkar, D., Ghosh, T., Roychowdhury, S., Arora, R., Sajan, S., Sheet, G., Waghmare, U.V., Biswas, K.: Ferroelectric instability induced ultralow thermal conductivity and high thermoelectric performance in rhombohedral p-type GeSe crystal. J. Am. Chem. Soc. 142, 12237 (2020). https://doi.org/10.1021/jacs.0c03696

    Article  Google Scholar 

  242. Xia, Y.: Revisiting lattice thermal transport in PbTe: the crucial role of quartic anharmonicity. Appl. Phys. Lett. 113, 073901 (2018). https://doi.org/10.1063/1.5040887

    Article  Google Scholar 

  243. Wei, P.-C., Bhattacharya, S., He, J., Neeleshwar, S., Podila, R., Chen, Y., Rao, A.: The intrinsic thermal conductivity of SnSe. Nature 539, E1 (2016). https://doi.org/10.1038/nature19832

    Article  Google Scholar 

  244. Zhao, L.-D., Lo, S.-H., Zhang, Y., Sun, H., Tan, G., Uher, C., Wolverton, C., Dravid, V.P., Kanatzidis, M.G.: The intrinsic thermal conductivity of SnSe: Reply. Nature 539, E2 (2016). https://doi.org/10.1038/nature19833

    Article  Google Scholar 

  245. Wu, D., Wu, L., He, D., Zhao, L.-D., Li, W., Wu, M., Jin, M., Xu, J., Jiang, J., Huang, L., et al.: Direct observation of vast off-stoichiometric defects in single crystalline SnSe. Nano Energy 35, 321 (2017). https://doi.org/10.1016/j.nanoen.2017.04.004

    Article  Google Scholar 

  246. Lee, Y.K., Luo, Z., Cho, S.P., Kanatzidis, M.G., Chung, I.: Surface oxide removal for polycrystalline SnSe reveals near-single-crystal thermoelectric performance. Joule 3, 719 (2019). https://doi.org/10.1016/j.joule.2019.01.001

    Article  Google Scholar 

Download references

Acknowledgements

The calculations performed for this work used resources of CCJDR-IFGW-UNICAMP in Brazil, the National Energy Research Scientific Computing Center (NERSC), a US Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231, as well as the FASRC Cannon cluster supported by the FAS Division of Science Research Computing Group at Harvard University.

Funding

M.P. is supported by the Swiss National Science Foundation (SNSF) through the Early Postdoc.Mobility program (Grant No. P2ELP2-191706). A.A. gratefully acknowledges support from the Brazilian agencies CNPq and FAPESP under Grants No. 2010/16970-0, No. 2013/08293-7, No. 2015/26434-2, No. 2016/23891-6, No. 2017/26105-4, and No. 2019/26088-8. We acknowledge funding from the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319; NSF Award No. DMR-1922172; the Army Research Office under Cooperative Agreement Number W911NF-21-2-0147; and the Simons Foundation, Award No. 896626.

Author information

Authors and Affiliations

  1. School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA

    Anderson S. Chaves, Michele Pizzochero & Efthimios Kaxiras

  2. Department of Physics, Harvard University, Cambridge, MA, 02138, USA

    Daniel T. Larson & Efthimios Kaxiras

  3. Gleb Wataghin Institute of Physics and Centre for Computational Engineering and Sciences, University of Campinas, UNICAMP, 13083-859, Campinas, São Paulo, Brazil

    Alex Antonelli

Authors
  1. Anderson S. Chaves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michele Pizzochero
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel T. Larson
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Alex Antonelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Efthimios Kaxiras
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. Calculations and analysis of the data presented in the text were performed by ASC and DTL. The first draft of the manuscript was written by MP, ASC, and DTL and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Daniel T. Larson.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chaves, A.S., Pizzochero, M., Larson, D.T. et al. Semiclassical electron and phonon transport from first principles: application to layered thermoelectrics. J Comput Electron 22, 1281–1309 (2023). https://doi.org/10.1007/s10825-023-02062-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10825-023-02062-4

Keywords

Search

Navigation