Skip to main content
SpringerLink
Log in

A Quasi-Classical Model of the Hubbard Gap in Lightly Compensated Semiconductors

  • Electronic Properties of Semiconductors
  • Published:
Semiconductors Aims and scope Submit manuscript

Abstract

A quasi-classical method for calculating the narrowing of the Hubbard gap between the A 0 and A + acceptor bands in a hole semiconductor or the D 0 and D – donor bands in an electron semiconductor is suggested. This narrowing gives rise to the phenomenon of a semiconductor transition from the insulator to metal state with an increase in doping level. The major (doping) impurity can be in one of three charge states (–1, 0, or +1), while the compensating impurity can be in states (+1) or (–1). The impurity distribution over the crystal is assumed to be random and the width of Hubbard bands (levels), to be much smaller than the gap between them. It is shown that narrowing of the Hubbard gap is due to the formation of electrically neutral acceptor (donor) states of the quasicontinuous band of allowed energies for holes (electrons) from excited states. This quasicontinuous band merges with the top of the valence band (v band) for acceptors or with the bottom of the conduction band (c band) for donors. In other words, the top of the v band for a p-type semiconductor or the bottom of the c band for an n-type semiconductor is shifted into the band gap. The value of this shift is determined by the maximum radius of the Bohr orbit of the excited state of an electrically neutral major impurity atom, which is no larger than half the average distance between nearest impurity atoms. As a result of the increasing dopant concentration, the both Hubbard energy levels become shallower and the gap between them narrows. Analytical formulas are derived to describe the thermally activated hopping transition of holes (electrons) between Hubbard bands. The calculated gap narrowing with increasing doping level, which manifests itself in a reduction in the activation energy ε2 is consistent with available experimental data for lightly compensated p-Si crystals doped with boron and n-Ge crystals doped with antimony.

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

Similar content being viewed by others

References

  1. N. F. Mott, Metal–Insulator Transitions (Taylor and Francis, London, 1990).

    Google Scholar 

  2. E. M. Gershenzon, A. P. Mel’nikov, R. I. Rabinovich, and N. A. Serebryakova, Sov. Phys. Usp. 23, 684 (1980).

    Article  ADS  Google Scholar 

  3. E. A. Schiff, Philos. Mag. B 45, 69 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  4. B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer, Berlin, 1984).

    Book  Google Scholar 

  5. N. F. Mott and E. A. Davis, Electronic Processes in Non- Crystalline Materials (Oxford Univ. Press, Oxford, 2012).

    Google Scholar 

  6. A. G. Zabrodskii, Philos. Mag. B 81, 1131 (2001).

    Article  ADS  Google Scholar 

  7. G. F. Neumark, Phys. Rev. B 20, 1519 (1979).

    Article  ADS  Google Scholar 

  8. H. Fritzsche, Phys. Rev. 99, 406 (1955).

    Article  ADS  Google Scholar 

  9. K. S. Shifrin, Zh. Tekh. Fiz. 14, 43 (1944).

    Google Scholar 

  10. P. D. Altukhov, K. N. El’tsov, and A. A. Rogachev, Sov. Phys. Solid State 23, 367 (1981).

    Google Scholar 

  11. G. E. Stillman and C. M. Wolfe, Thin Solid Films 31, 69 (1976).

    Article  ADS  Google Scholar 

  12. L. V. Berman and Sh. M. Kogan, Sov. Phys. Semicond. 21, 933 (1987).

    Google Scholar 

  13. H. Fritzsche, Philos. Mag. B 42, 835 (1980).

    Article  ADS  Google Scholar 

  14. L. P. Ginzburg, Semiconductors 27, 15 (1993).

    ADS  Google Scholar 

  15. N. A. Poklonski and A. I. Syaglo, Semiconductors 33, 391 (1999).

    Article  ADS  Google Scholar 

  16. N. A. Poklonski, S. A. Vyrko, and A. G. Zabrodskii, Semiconductors 40, 394 (2006).

    Article  ADS  Google Scholar 

  17. N. A. Poklonski, S. A. Vyrko, O. N. Poklonskaya, and A. G. Zabrodskii, J. Appl. Phys. 110, 123702 (2011).

    Article  ADS  Google Scholar 

  18. J. S. Blakemore, Semiconductor Statistics (Dover, New York, 2002).

    MATH  Google Scholar 

  19. V. A. Kul’bachinskii, V. G. Kytin, V. V. Abramov, A. B. Timofeev, A. G. Ul’yashin, and N. V. Shlopak, Sov. Phys. Semicond. 26, 1009 (1992).

    Google Scholar 

  20. N. A. Bogoslovskii and K. D. Tsendin, Semiconductors 46, 559 (2012).

    Article  ADS  Google Scholar 

  21. N. A. Poklonski, S. A. Vyrko, and A. G. Zabrodskii, Semiconductors 42, 1388 (2008).

    Article  ADS  Google Scholar 

  22. Yu. A. Astrov, S. A. Lynch, V. B. Shuman, L. M. Portsel’, A. A. Makhova, and A. N. Lodygin, Semiconductors 47, 247 (2013).

    Article  ADS  Google Scholar 

  23. V. N. Aleksandrov, E. M. Gershenzon, A. P. Mel’nikov, and N. A. Serebryakova, Sov. Phys. Semicond. 11, 306 (1977).

    Google Scholar 

  24. N. A. Poklonski, V. F. Stelmakh, V. D. Tkachev, and S. V. Voitikov, Phys. Status Solidi B 88, K165 (1978).

    Article  ADS  Google Scholar 

  25. K. Ya. Shtivel’man, Sov. Phys. Semicond. 8, 528 (1974).

    Google Scholar 

  26. D. C. Look, Phys. Rev. B 24, 5852 (1981).

    Article  ADS  Google Scholar 

  27. N. A. Poklonski, S. A. Vyrko, A. G. Zabrodskii, and S. V. Egorov, Phys. Solid State 45, 2053 (2003).

    Article  ADS  Google Scholar 

  28. V. E. Ogluzdin, Phys. Usp. 49, 401 (2006).

    Article  ADS  Google Scholar 

  29. B. L. Gel’mont and A. V. Rodina, Sov. Phys. Semicond. 25, 1319 (1991).

    Google Scholar 

  30. B. V. Gnedenko, Theory of Probability (Editorial URSS, Moscow, 2005; CRC, Boca Raton, FL, 1998).

    MATH  Google Scholar 

  31. N. A. Poklonski, S. A. Vyrko, and A. G. Zabrodskii, Phys. Solid State 46, 1101 (2004).

    Article  ADS  Google Scholar 

  32. L. P. Ginzburg, Sov. Phys. Semicond. 12, 326 (1978).

    Google Scholar 

  33. V. M. Mikheev, Phys. Solid State 39, 1765 (1997).

    Article  ADS  Google Scholar 

  34. G. Bethe and E. Salpeter, Quantum Mechanics of Oneand Two-Electron Atoms (Springer, Berlin, 1977; Nauka, Moscow, 1960).

    Book  Google Scholar 

  35. B. M. Smirnov, Physics of Atoms and Ions (Energoatomizdat, Moscow, 1986), Chap. 3 [in Russian].

    Google Scholar 

  36. B. A. Volkov and V. V. Matveev, Sov. Phys. Solid State 8, 577 (1966).

    Google Scholar 

  37. O. I. Loiko, Sov. Phys. Semicond. 21, 797 (1987).

    Google Scholar 

  38. V. L. Bonch-Bruevich, Izv. Vyssh. Uchebn. Zaved., Fiz. 28 (9), 98 (1985).

    Google Scholar 

  39. N. A. Poklonski and S. Yu. Lopatin, Phys. Solid State 43, 2219 (2001).

    Article  ADS  Google Scholar 

  40. N. A. Poklonski, S. A. Vyrko, and A. G. Zabrodskii, Solid State Commun. 149, 1248 (2009).

    Article  ADS  Google Scholar 

  41. N. A. Poklonski, S. A. Vyrko, and A. G. Zabrodskii, Semicond. Sci. Technol. 25, 085006 (2010).

    Article  ADS  Google Scholar 

  42. N. A. Poklonski, S. A. Vyrko, and A. G. Zabrodskii, Semiconductors 41, 30 (2007).

    Article  ADS  Google Scholar 

  43. N. A. Poklonski, S. A. Vyrko, and A. G. Zabrodskii, Semiconductors 41, 1300 (2007).

    Article  ADS  Google Scholar 

  44. A. G. Andreev, V. V. Voronkov, G. I. Voronkova, A. G. Zabrodskii, and E. A. Petrova, Semiconductors 29, 1157 (1995).

    ADS  Google Scholar 

  45. Yu. M. Gal’perin, E. P. German, and V. G. Karpov, Sov. Phys. JETP 72, 193 (1991).

    Google Scholar 

  46. A. F. Barabanov, Yu. M. Kagan, L. A. Maksimov, A. V. Mikheenkov, and T. V. Khabarova, Phys. Usp. 58, 446 (2015).

    Article  ADS  Google Scholar 

  47. T. M. Lifshits, Prib. Tekh. Eksp., No. 1, 10 (1993).

    Google Scholar 

  48. Semiconductors: Data Handbook, Ed. by O. Madelung (Springer, Berlin, 2004).

  49. A. Rogalski, Progr. Quant. Electron. 36, 342 (2012).

    Article  ADS  Google Scholar 

  50. E. M. Gershenzon, Yu. A. Gurvich, A. P. Mel’nikov, and L. N. Shestakov, Sov. Phys. Semicond. 25, 95 (1991).

    Google Scholar 

  51. J. A. Chroboczek, F. H. Pollak, and H. F. Staunton, Philos. Mag. B 50, 113 (1984).

    Article  ADS  Google Scholar 

  52. F. M. Ismagilova, L. B. Litvak-Gorskaya, G. Ya. Lugovaya, and I. E. Trofimov, Sov. Phys. Semicond. 25, 154 (1991).

    Google Scholar 

  53. E. M. Gershenzon, F. M. Ismagilova, L. B. Litvak-Gorskaya, and A. P. Mel’nikov, Sov. Phys. JETP 73, 568 (1991).

    Google Scholar 

  54. E. M. Gershenzon, L. B. Litvak-Gorskaya, and G. Ya. Lugovaya, Sov. Phys. Semicond. 15, 742 (1981).

    Google Scholar 

  55. E. M. Gershenzon, L. B. Litvak-Gorskaya, G. Ya. Lugovaya, and E. Z. Shapiro, Sov. Phys. Semicond. 20, 58 (1986).

    Google Scholar 

  56. H. Fritzsche, J. Phys. Chem. Solids 6, 69 (1962).

    Article  ADS  Google Scholar 

  57. H. Fritzsche, Phys. Rev. 125, 1552 (1962).

    Article  ADS  Google Scholar 

  58. E. A. Davis and W. D. Compton, Phys. Rev. A 140, 2183 (1965).

    Article  ADS  Google Scholar 

  59. N. V. Agrinskaya, V. I. Kozub, T. A. Polyanskaya, and A. S. Saidov, Semiconductors 33, 135 (1999).

    Article  ADS  Google Scholar 

  60. M. Kobayashi, Y. Sakaida, M. Taniguchi, and Sh. Narita, J. Phys. Soc. Jpn. 47, 138 (1979).

    Article  ADS  Google Scholar 

  61. J. Bethin, T. G. Castner, and N. K. Lee, Solid State Commun. 14, 1321 (1974).

    Article  ADS  Google Scholar 

  62. T. G. Castner, N. K. Lee, H. S. Tan, L. Moberly, and O. Symko, J. Low Temp. Phys. 38, 447 (1980).

    Article  ADS  Google Scholar 

  63. R. K. Ray and H. Y. Fan, Phys. Rev. 121, 768 (1961).

    Article  ADS  Google Scholar 

  64. H. Fritzsche, J. Phys. Chem. Solids 6, 69 (1958).

    Article  ADS  Google Scholar 

  65. K. Somogyi, Phys. Status Solidi A 35, 659 (1976).

    Article  ADS  Google Scholar 

  66. N. A. Poklonski, S. A. Vyrko, O. N. Poklonskaya, and A. G. Zabrodskii, Phys. Status Solidi B 246, 158 (2009).

    Article  ADS  Google Scholar 

  67. T. T. Mnatsakanov, M. E. Levinshtein, L. I. Pomortseva, and S. N. Yurkov, Semiconductors 38, 56 (2004).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Belarusian State University, Minsk, 220030, Belarus

    N. A. Poklonski, S. A. Vyrko & A. I. Kovalev

  2. Ioffe Physical-Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021, Russia

    A. G. Zabrodskii

Authors
  1. N. A. Poklonski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. A. Vyrko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. I. Kovalev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. G. Zabrodskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Zabrodskii.

Additional information

Original Russian Text © N.A. Poklonski, S.A. Vyrko, A.I. Kovalev, A.G. Zabrodskii, 2016, published in Fizika i Tekhnika Poluprovodnikov, 2016, Vol. 50, No. 3, pp. 302–312.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Poklonski, N.A., Vyrko, S.A., Kovalev, A.I. et al. A Quasi-Classical Model of the Hubbard Gap in Lightly Compensated Semiconductors. Semiconductors 50, 299–308 (2016). https://doi.org/10.1134/S1063782616030192

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063782616030192

Keywords

Search

Navigation