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SEMICONDUCTOR PHYSICS

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

Wenhan Cao

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 Corresponding author: Wenhan Cao, E-mail: whcao11@fudan.edu.cn

DOI: 10.1088/1674-4926/36/4/042002

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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



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Fig. 1.  Region partitions for the series expansions for carrier density.

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Fig. 2.  Calculated electron densities $n$ by analytical formulas (10) when $\eta \in \left( -10, 30\right) $ and $y\in (0, 60)$.

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Fig. 3.  Relative errors between the analytical formulas (10) and the accurate numerical solutions of Equation ($3$).

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Fig. 4.  Calculated Fermi energy using the analytical formulas (12) for $\eta \in \left( -10, 30\right) $ and $y\in (0, 60)$.

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Fig. 5.  Relative errors between the analytical formulas (12) and the accurate numerical solutions of Equation (3).

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Fig. 6.  The left axis shows the electron densities calculated using the analytical formulas (10) (points) and using the accurate numerical solutions of Equation ($3$) (lines) for a GaAs/AlGaAs miniband superlattice system at an electron temperature of $T_{\rm e}=77$ K. The reduced Fermi energies are $\eta=-3$, 0, 3, and 6, respectively. The right axis shows the relative errors between the analytical formulas (10) and the accurate numerical solutions of Equation (3).

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Fig. 7.  TThe left axis shows the Fermi energy calculated using the analytical formulas (12) (points) and by the accurate numerical solutions of Equation ($3$) (lines) for a GaAs/AlGaAs miniband superlattice system at an electron temperature of $T_{\rm e}=77$ K. The miniband widths are $\Delta=20$, 50, and 120 meV, respectively. The right axis shows the relative errors between the analytical formulas (12) and the accurate numerical solutions of Equation (3).

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Table 1.   Coefficients used in Equation (10).

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    Received: 24 October 2014 Revised: Online: Published: 01 April 2015

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      Article Navigation >  Journal of Semiconductors  > 2015  >  36(4): 042002
      Wenhan Cao. Analytical formulas for carrier density and Fermi energy in semiconductors with a tight-binding band[J]. Journal of Semiconductors, 2015, 36(4): 042002. doi: 10.1088/1674-4926/36/4/042002 ****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.
      Citation:
      Wenhan Cao. Analytical formulas for carrier density and Fermi energy in semiconductors with a tight-binding band[J]. Journal of Semiconductors, 2015, 36(4): 042002. doi: 10.1088/1674-4926/36/4/042002 ****
      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.
      shu

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

      DOI: 10.1088/1674-4926/36/4/042002

      • Department of Communication Science and Engineering, Fudan University, Shanghai 200433, China

      More Information
      • Corresponding author: E-mail: whcao11@fudan.edu.cn
      • Received Date: 2014-10-24
      • Accepted Date: 2014-11-03
      • Published Date: 2015-01-25

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