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Semiconductor Physics pp 1–32Cite as

Equilibrium Statistics of Carriers

Abstract

Electrons and holes are the carriers of currents in semiconductors. The density of these carriers in equilibrium is obtained from the Fermi–Dirac statistics. The Fermi energy E F as a key parameter can be obtained from quasineutrality; it lies near the middle of the bandgap for intrinsic and near the donor or acceptor level for doped semiconductors. The difference between the respective band edge and the Fermi level represents the activation energy of a Boltzmann factor, whose product with the joint density of energy states yields the carrier density. The density of minority carriers may be frozen-in in semiconductors with a large bandgap and represented by a quasi-Fermi energy.

At high quasi-particle densities and low temperature, phase transitions take place with substantial changes in the optical and electronic behavior. An insulator-metal transition occurs above a critical Mott density of dopants. A similar process is initiated by sufficient optical generation of electrons and holes, leading to an electron–hole plasma and – at suitable conditions – to a condensation into an electron–hole liquid.

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Notes

  1. 1.

    Or cyclic boundary conditions; here energy and particle number are conserved by demanding, that with the passage of a particle out of a surface, an identical one enters from the opposite surface (Born-von Karman boundary condition). The two conditions are mathematically equivalent.

  2. 2.

    The values of ν i are positive integers. Negative values do not yield linear independent waves and \( {\nu}_i=0 \) yields waves which cannot be normalized.

  3. 3.

    In the hole picture, an increasing energy corresponds to a decreasing electron energy. The energy axis in Fig. 2 refers to an electron energy.

  4. 4.

    This is justified since m n does not change much near the bottom of the conduction band (see Fig. 15 of chapter “Quantum Mechanics of Electrons in Crystals”); moreover, m n increases with increasing E, which usually renders higher energy electrons less important for a number of low-field transport properties.

  5. 5.

    When a hot wire touches the semiconductor, the wire becomes charged oppositely to the carrier type. For example, when contacted to an n-type semiconductor, the wire becomes positively charged with respect to the semiconductor since electrons are emitted into the semiconductor.

  6. 6.

    The graphical presentation is similar to the Brouwer diagram introduced in Sect. 2.6 of chapter “Crystal Defects.” In the Brouwer diagram, ln(n) is plotted versus 1/kT, while here we plot ln(n) versus E. In both presentations, small contributions are neglected due to the logarithmic density scale.

  7. 7.

    Justification in steady state, e.g., with optical excitation, is reasonable. The use, however, of quasi-Fermi levels close to a frozen-in situation becomes questionable when different types of defects are involved.

  8. 8.

    Through mutual generation and recombination of electrons and holes, a change in minority carrier density also causes a change in majority carrier density; but, because of the much larger density of majority carriers, the change is truly negligible.

  9. 9.

    With strong photon coupling, however, one has polaritons rather than excitons. They cannot accumulate near K = 0 because of their photon nature. Therefore, Bose–Einstein condensation of these quasi-particles in bulk material is impossible. However, for exciton–polaritons coupled to the mode of a microcavity mode, such a condensation appears meanwhile established as discussed by Deng et al. (2010).

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Authors and Affiliations

  1. Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstr. 36, Berlin, 10623, Germany

    Udo W. Pohl

  2. 138 Moorings Park Rd. O 301, Naples, FL, 34105, USA

    Karl W. Böer

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  1. Karl W. Böer
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Correspondence to Udo W. Pohl .

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Böer, K.W., Pohl, U.W. (2016). Equilibrium Statistics of Carriers. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06540-3_21-1

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  • DOI: https://doi.org/10.1007/978-3-319-06540-3_21-1

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  • Print ISBN: 978-3-319-06540-3

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  1. Latest

    Equilibrium Statistics of Carriers
    Published:
    01 June 2022

    DOI: https://doi.org/10.1007/978-3-319-06540-3_21-4

  2. Equilibrium Statistics of Carriers
    Published:
    29 February 2020

    DOI: https://doi.org/10.1007/978-3-319-06540-3_21-3

  3. Equilibrium Statistics of Carriers
    Published:
    28 September 2017

    DOI: https://doi.org/10.1007/978-3-319-06540-3_21-2

  4. Original

    Equilibrium Statistics of Carriers
    Published:
    19 January 2017

    DOI: https://doi.org/10.1007/978-3-319-06540-3_21-1

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