Analytical formulas for carrier density and Fermi energy in semiconductors with a tight-binding band

Abstract: Analytical formulas for evaluating the relation of carrier density and Fermi energy for semiconductors with a tight-binding band have been proposed. The series expansions for a carrier density with fast convergency have been obtained by means of a Bessel function. A simple and analytical formula for Fermi energy has been derived with the help of the Gauss integration method. The results of the proposed formulas are in good agreement with accurate numerical solutions. The formulas have been successfully used in the calculation of carrier density and Fermi energy in a miniband superlattice system. Their accuracy is in the order of 10^-5.

Key words: analytical formulas, carrier density, Fermi energy, tight-binding band

References:

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[2]	Diez M, Dahlhaus J P, Wimmer M. Emergence of massless Dirac Fermions in graphene's Hofstadter butterfly at switches of the quantum Hall phase connectivity[J]. Phys Rev Lett, 2014, 112: 196602.
[3]	Lei X L, Horing N J M, Cui H L. Theory of negative differential conductivity in a superlattice miniband[J]. Phys Rev Lett, 1991, 66: 3277.
[4]	Keay X B J, Allen S J, Galan J. Photon-assisted electric field domains and multiphoton-assisted tunneling in semiconductor superlattices[J]. Phys Rev Lett, 1995, 75: 4098.
[5]	Brandl S, Schomburg E, Scheuerer R. Millimeter wave generation by a self-sustained current oscillation in an InGaAs/InAlAs superlattice[J]. Appl Phys Lett, 1998, 73: 3117.
[6]	Ignatov A A, Klappenberger F, Schomburg E. Detection of THz radiation with semiconductor superlattices at polar-optic phonon frequencies[J]. J Appl Phys, 2002, 91: 1281.
[7]	Scheuerer R, Haeussler M, Renk K F. Frequency multiplication of microwave radiation by propagating space-charge domains in a semiconductor superlattice[J]. Appl Phys Lett, 2003, 82: 2826.
[8]	Joyce W B. Expression for the Fermi energy in narrow-bandgap semiconductors[J]. IEEE J Quantum Electron, 1983, 19: 1625.
[9]	Aymerich-Humet X, Serra-Mestres F, Millán J. A generalized approximation of the Fermi-Dirac integrals[J]. J Appl Phys, 1983, 54: 2850.
[10]	Press W H, Teukolsky S A, Vetterling W T. Numerical recipes: the art of scientific computing[J]. Cambridge University Press, 1986.

[1]	Nakayama M, Kawabata T. Electric field effects on reduced effective masses of minibands at the mini-Brillouin-zone center and edge in a GaAs/AlAs superlattice[J]. J Appl Phys, 2012, 111: 053523.
[2]	Diez M, Dahlhaus J P, Wimmer M. Emergence of massless Dirac Fermions in graphene's Hofstadter butterfly at switches of the quantum Hall phase connectivity[J]. Phys Rev Lett, 2014, 112: 196602.
[3]	Lei X L, Horing N J M, Cui H L. Theory of negative differential conductivity in a superlattice miniband[J]. Phys Rev Lett, 1991, 66: 3277.
[4]	Keay X B J, Allen S J, Galan J. Photon-assisted electric field domains and multiphoton-assisted tunneling in semiconductor superlattices[J]. Phys Rev Lett, 1995, 75: 4098.
[5]	Brandl S, Schomburg E, Scheuerer R. Millimeter wave generation by a self-sustained current oscillation in an InGaAs/InAlAs superlattice[J]. Appl Phys Lett, 1998, 73: 3117.
[6]	Ignatov A A, Klappenberger F, Schomburg E. Detection of THz radiation with semiconductor superlattices at polar-optic phonon frequencies[J]. J Appl Phys, 2002, 91: 1281.
[7]	Scheuerer R, Haeussler M, Renk K F. Frequency multiplication of microwave radiation by propagating space-charge domains in a semiconductor superlattice[J]. Appl Phys Lett, 2003, 82: 2826.
[8]	Joyce W B. Expression for the Fermi energy in narrow-bandgap semiconductors[J]. IEEE J Quantum Electron, 1983, 19: 1625.
[9]	Aymerich-Humet X, Serra-Mestres F, Millán J. A generalized approximation of the Fermi-Dirac integrals[J]. J Appl Phys, 1983, 54: 2850.
[10]	Press W H, Teukolsky S A, Vetterling W T. Numerical recipes: the art of scientific computing[J]. Cambridge University Press, 1986.

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W H Cao. Analytical formulas for carrier density and Fermi energy in semiconductors with a tight-binding band[J]. J. Semicond., 2015, 36(4): 042002. doi: 10.1088/1674-4926/36/4/042002.

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