Journal of Electronic Materials

, Volume 48, Issue 2, pp 1031–1043 | Cite as

Using First-Principles Calculations in CALPHAD Models to Determine Carrier Concentration of the Binary PbSe Semiconductor

  • Matthew C. PetersEmail author
  • Jeff W. Doak
  • J. E. Saal
  • G. B. Olson
  • P. W. Voorhees


PbSe is a promising thermoelectric that can be further improved by nanostructuring, band engineering, and carrier concentration tuning; therefore, a firm understanding of the defects in PbSe is necessary. The formation energies of point defects in PbSe are computed via first-principles calculations under the dilute-limit approximation. We find that under Pb-rich conditions, PbSe is an n-type semiconductor dominated by doubly-charged Se vacancies. Conversely, under Se-rich conditions, PbSe is a p-type semiconductor dominated by doubly-charged Pb vacancies. Both of these results agree with previously performed experiments. Temperature- and chemical potential-dependent Fermi levels and carrier concentrations are found by enforcing the condition of charge neutrality across all charged atomic and electronic states in the system. The first-principles-predicted charge-carrier concentration is in qualitative agreement with experiment, but slightly varies in the magnitude of carriers. To better describe the experimental data, a CALPHAD assessment of PbSe is performed. Parameters determined via first-principles calculations are used as inputs to a five-sublattice CALPHAD model that was developed explicitly for binary semiconductors. This five sublattice model is in contrast to previous work which treated PbSe as a stoichiometric compound. The current treatment allows for experimental carrier concentrations to be accurately described within the CALPHAD formalism. In addition to the five-sublattice model, a two-sublattice model is also developed for use in multicomponent databases. Both models show excellent agreement with the experimental data and close agreement with first-principles calculations. These CALPHAD models can be used to determine processing parameters that will result in an optimized carrier concentration and peak zT value.


Thermoelectrics CALPHAD first-principles DFT defect chemistry 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors gratefully acknowledge thermoelectrics research at Northwestern University through the Center for Hierarchical Materials Design (CHiMaD) and financial support from the DARPA SIMPLEX program through SPAWAR (Contract #N66001-15-C-4036). M. Peters was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.


Funding was provided by Defense Advanced Research Projects Agency and National Institute of Standards and Technology (US).


  1. 1.
    F.J. DiSalvo, Science 285, 703 (1999).Google Scholar
  2. 2.
    G.J. Tan, L.D. Zhao, and M.G. Kanatzidis, Chem. Rev. 116, 12123 (2016).Google Scholar
  3. 3.
    K. Biswas, J.Q. He, I.D. Blum, C.I. Wu, T.P. Hogan, D.N. Seidman, V.P. Dravid, and M.G. Kanatzidis, Nature 489, 414 (2012).Google Scholar
  4. 4.
    G.J. Tan, F.Y. Shi, S.Q. Hao, L.D. Zhao, H. Chi, X.M. Zhang, C. Uher, C. Wolverton, and V.P. Dravid, Nat. Commun. 7, 12167 (2016).Google Scholar
  5. 5.
    J.P. Heremans, C.M. Thrush, and D.T. Morelli, Phys. Rev. B 70, 5 (2004).Google Scholar
  6. 6.
    J. Androulakis, C.H. Lin, H.J. Kong, C. Uher, C.I. Wu, T. Hogan, B.A. Cook, T. Caillat, K.M. Paraskevopoulos, and M.G. Kanatzidis, J. Am. Chem. Soc. 129, 9780 (2007).Google Scholar
  7. 7.
    Y.Z. Pei, Z.M. Gibbs, A. Gloskovskii, B. Balke, W.G. Zeier, and G.J. Snyder, Adv. Energy Mater. 4, 12 (2014).Google Scholar
  8. 8.
    Y.Z. Pei, A.D. LaLonde, N.A. Heinz, and G.J. Snyder, Adv. Energy Mater. 2, 670 (2012).Google Scholar
  9. 9.
    L.D. Zhao, H.J. Wu, S.Q. Hao, C.I. Wu, X.Y. Zhou, K. Biswas, J.Q. He, T.P. Hogan, C. Uher, C. Wolverton, V.P. Dravid, and M.G. Kanatzidis, Energy Environ. Sci. 6, 3346 (2013).Google Scholar
  10. 10.
    Y.Z. Pei, A. LaLonde, S. Iwanaga, and G.J. Snyder, Energy Environ. Sci. 4, 2085 (2011).Google Scholar
  11. 11.
    H. Wang, Y.Z. Pei, A.D. LaLonde, and G.J. Snyder, Adv. Mater. 23, 1366 (2011).Google Scholar
  12. 12.
    H. Wang, Z.M. Gibbs, Y. Takagiwa, and G.J. Snyder, Energy Environ. Sci. 7, 804 (2014).Google Scholar
  13. 13.
    C.M. Jaworski, M.D. Nielsen, H. Wang, S.N. Girard, W. Cai, W.D. Porter, M.G. Kanatzidis, and J.P. Heremans, Phys. Rev. B 87, (4) (2013).Google Scholar
  14. 14.
    Q. Zhang, F. Cao, W.S. Liu, K. Lukas, B. Yu, S. Chen, C. Opeil, D. Broido, G. Chen, and Z.F. Ren, J. Am. Chem. Soc. 134, 10031 (2012).Google Scholar
  15. 15.
    J. Steininger, Metall. Trans. 1, 2939-+ (1970).Google Scholar
  16. 16.
    J.W. Doak, K.J. Michel, and C. Wolverton, J. Mater. Chem. C 3, 10630 (2015).Google Scholar
  17. 17.
    H.L. Lukas, Computational Thermodynamics The Calphad Method (Cambridge: Cambridge University Press, 2007)Google Scholar
  18. 18.
    Z.K. Liu, J. Phase Equilib. Diffus. 30, 517 (2009).Google Scholar
  19. 19.
    J.C. Lin, R.C. Sharma, and Y.A. Chang, J. Phase Equilib. 17, 253 (1996).Google Scholar
  20. 20.
    Y. Liu, Z. Kang, G. Sheng, L. Zhang, J. Wang, and Z. Long, J. Electron. Mater. 41, 1915 (2012).Google Scholar
  21. 21.
    J.C. Lin, T.L. Ngai, and Y.A. Chang, Metall. Trans. A 17, 1241 (1986).Google Scholar
  22. 22.
    J.C. Lin, K.C. Hsleh, R.C. Sharma, and Y.A. Chang, Bull. Alloy Phase Diagr. 10, 340 (1989).Google Scholar
  23. 23.
    J.C. Lin, R.C. Sharma, and Y.A. Chang, Bull. Alloy Phase Diagr. 7, 374 (1986).Google Scholar
  24. 24.
    W.F. Li, C.M. Fang, M. Dijkstra, and M.A. van Huis, J. Phys.: Condens. Matter 27, 14 (2015).Google Scholar
  25. 25.
    E.O. Wrasse, P. Venezuela, and R.J. Baierle, J. Appl. Phys. 116, 183703 (2014).Google Scholar
  26. 26.
    S. Bajaj, H. Wang, J.W. Doak, C. Wolverton, and G.J. Snyder, J. Mater. Chem. C 4, 1769 (2016).Google Scholar
  27. 27.
    A.N. Grundy, E. Povoden, T. Ivas, and L.J. Gauckler, CALPHAD 30, 33 (2006).Google Scholar
  28. 28.
    A. Saengdeejing, J.E. Saal, V.R. Manga, and Z.K. Liu, Acta Mater. 60, 7207 (2012).Google Scholar
  29. 29.
    J. Rogal, S.V. Divinski, M.W. Finnis, A. Glensk, J. Neugebauer, J.H. Perepezko, S. Schuwalow, M.H.F. Sluiter, and B. Sundman, Phys. Status Solidi B 251, 97 (2014).Google Scholar
  30. 30.
    P.W. Guan and Z.K. Liu, Scr. Mater. 133, 5 (2017).Google Scholar
  31. 31.
    P.W. Guan, S.L. Shang, G. Lindwall, T. Anderson, and Z.K. Liu, CALPHAD 59, 171 (2017).Google Scholar
  32. 32.
    K. Ozturk, Y. Zhong, L.Q. Chen, C. Wolverton, J.O. Sofo, and Z.K. Liu, Metall. Mater. Trans. A 36A, 5 (2005).Google Scholar
  33. 33.
    L.J. Zhang, J. Wang, Y. Du, R.X. Hu, P. Nash, X.G. Lu, and C. Jiang, Acta Mater. 57, 5324 (2009).Google Scholar
  34. 34.
    Y. Zhong, C. Wolverton, Y.A. Chang, and Z.K. Liu, Acta Mater. 52, 2739 (2004).Google Scholar
  35. 35.
    S. Bajaj, G.S. Pomrehn, J.W. Doak, W. Gierlotka, H.J. Wu, S.W. Chen, C. Wolverton, W.A. Goddard, and G.J. Snyder, Acta Mater. 92, 72 (2015).Google Scholar
  36. 36.
    C.G. Van de Walle and J. Neugebauer, J. Appl. Phys. 95, 3851 (2004).Google Scholar
  37. 37.
    C. Freysoldt, B. Grabowski, T. Hickel, J. Neugebauer, G. Kresse, A. Janotti, and C.G. Van de Walle, Rev. Mod. Phys. 86, 253 (2014).Google Scholar
  38. 38.
    S. Lany and A. Zunger, Phys. Rev. B 78, 235104 (2008).Google Scholar
  39. 39.
    A. Goyal, P. Gorai, E.S. Toberer, and V. Stevanovic, Npj Comput. Mater. 3, Article number 42 (2017).Google Scholar
  40. 40.
    C. Freysoldt, J. Neugebauer, and C.G. Van de Walle, Phys. Rev. Lett. 102, (1) (2009).Google Scholar
  41. 41.
    Y. Kumagai and F. Oba, Phys. Rev. B 89, 195205 (2014).Google Scholar
  42. 42.
    T.R. Durrant, S.T. Murphy, M.B. Watkins, and A.L. Shluger, J. Chem. Phys. 149, (2) (2018).Google Scholar
  43. 43.
    G. Makov and M.C. Payne, Phys. Rev. B 51, 4014 (1995).Google Scholar
  44. 44.
    W. Kohn and L.J. Sham, Phys. Rev. 140, 1133 (1965).Google Scholar
  45. 45.
    P. Hohenberg and W. Kohn, Phys. Rev. B 136, B864 (1964).Google Scholar
  46. 46.
    G. Kresse and J. Hafner, Phys. Rev. B 49, 14251 (1994).Google Scholar
  47. 47.
    G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996).Google Scholar
  48. 48.
    G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).Google Scholar
  49. 49.
    J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).Google Scholar
  50. 50.
    A. Belsky, M. Hellenbrandt, V.L. Karen, and P. Luksch, Acta Crystallogr. Sect. B Struct. Sci. 58, 364 (2002).Google Scholar
  51. 51.
    H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13, 5188 (1976).Google Scholar
  52. 52.
    M. Gajdos, K. Hummer, G. Kresse, J. Furthmuller, and F. Bechstedt, Phys. Rev. B 73, 045112 (2006).Google Scholar
  53. 53.
    S. Baroni and R. Resta, Phys. Rev. B 33, 7017 (1986).Google Scholar
  54. 54.
    X. Wu, D. Vanderbilt, and D.R. Hamann, Phys. Rev. B 72, 035105 (2005).Google Scholar
  55. 55.
    Q. Chen, M. Hillert, B. Sundman, W.A. Oates, S.G. Fries, and R. Schmid-Fetzer, J. Electron. Mater. 27, 961 (1998).Google Scholar
  56. 56.
    W.A. Oates, G. Eriksson, and H. Wenzl, J. Alloys Compd. 220, 48 (1995).Google Scholar
  57. 57.
    Q. Chen and M. Hillert, J. Alloys Compd. 245, 125 (1996).Google Scholar
  58. 58.
    J.B. Li and J.C. Tedenac, J. Electron. Mater. 31, 321 (2002).Google Scholar
  59. 59.
    J.B. Li and L.L. Kerr, Opt. Mater. 35, 1213 (2013).Google Scholar
  60. 60.
    B. Jansson, Ph.D. thesis, Royal Institute of Technology, Stockholm, Sweden (1984).Google Scholar
  61. 61.
    B. Sundman, B. Jansson, and J.O. Andersson, CALPHAD 9, 153 (1985).Google Scholar
  62. 62.
    A.T. Dinsdale, CALPHAD 15, 317 (1991).Google Scholar
  63. 63.
    A.T. Dinsdale, A.V. Khvan, and A. Watson, Mater. Sci. Technol. 30, 1715 (2014).Google Scholar
  64. 64.
    P. Franke, J. Phase Equilib. Diffus. 35, 780 (2014).Google Scholar
  65. 65.
    N. Ohashi and K. Igaki, Trans. Jpn. Inst. Met. 5, 94 (1964).Google Scholar
  66. 66.
    N. Chou, K. Komarek, and E. Miller, Trans. Metall. Soc. AIME 245, 1553 (1969).Google Scholar
  67. 67.
    A.F. Kohan, G. Ceder, D. Morgan, and C.G. Van de Walle, Phys. Rev. B 61, 15019 (2000).Google Scholar
  68. 68.
    O. Madelung, Semiconductors: Data Handbook, 3rd ed. (Berlin: Springer, 2004).Google Scholar
  69. 69.
    M.C. Peters, J.W. Doak, W.-W. Zhang, J.E. Saal, G.B. Olson, and P.W. Voorhees, CALPHAD 58, 17 (2017).Google Scholar
  70. 70.
    A.R. Calawa, T.C. Harman, M. Finn, and P. Youtz, Trans. Metall. Soc. AIME 242, 374 (1968).Google Scholar
  71. 71.
    B.J. Sealy and A.J. Crocker, J. Mater. Sci. 8, 1737 (1973).Google Scholar
  72. 72.
    R.F. Brebrick and E. Gubner, J. Chem. Phys. 36, 170 (1962).Google Scholar
  73. 73.
    N. Wang, D. West, J.W. Liu, J. Li, Q.M. Yan, B.L. Gu, S.B. Zhang, and W.H. Duan, Phys. Rev. B 89, 045142 (2014).Google Scholar
  74. 74.
    J.W. Doak, C. Wolverton, and V. Ozolins, Phys. Rev. B 92, 174306 (2015).Google Scholar
  75. 75.
    H. Wang, Y.Z. Pei, A.D. LaLonde, and G.J. Snyder, Proc. Natl. Acad. Sci. U.S.A. 109, 9705 (2012).Google Scholar

Copyright information

© The Minerals, Metals & Materials Society 2018

Authors and Affiliations

  1. 1.Department of Materials Science and EngineeringNorthwestern UniversityEvanstonUSA
  2. 2.QuesTek Innovations LLCEvanstonUSA

Personalised recommendations