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Carrier-Transport Equations

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Abstract

The current in semiconductors is carried by electrons and holes. Their lattice polarization modifies the effective mass, expressed as a change to polarons. While for large polarons the effect is small, semiconductors with narrow bands and large lattice polarization show a significant effect described by small polarons. The total current is composed of a drift and a diffusion current of electrons and holes. The drift current is determined by the electric field, and the energy obtained by carrier acceleration is given to the lattice by inelastic scattering, which opposes the energy gain, causing a constant carrier-drift velocity and Joule’s heating. The diffusion current is proportional to the gradient of carrier density up to a limit given by the thermal velocity. Proportionality factor of both drift and diffusion currents is the carrier mobility, which is proportional to a relaxation time and inverse to the mobility effective mass. Currents are proportional to negative potential gradients, with the conductivity as the proportionality factor. In spatially inhomogeneous semiconductors, both an external field, impressed by an applied bias, and a built-in field, due to space-charge regions, exist. Only the external field causes carrier heating by shifting and deforming the carrier distribution from a Boltzmann distribution to a distorted distribution with more carriers at higher energies.

The Boltzmann equation permits a detailed analysis of the carrier transport and the carrier distribution, providing well-defined values for transport parameters such as relaxation times. The Boltzmann equation can be integrated in closed form only for a few special cases, but approximations for small applied fields provide the basis for investigating scattering processes; these can be divided into essentially elastic processes with mainly momentum exchange and, for carriers with sufficient accumulated energy, into inelastic scattering with energy relaxation.

Karl W. Böer: deceased.

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Notes

  1. 1.

    A crystal with nonideal lattice periodicity is a solid which contains lattice defects (e.g., impurities) and oscillatory motions (i.e., phonons).

  2. 2.

    We are adopting here the picture of a localized electron. Such a localization can be justified in each scattering event. In this model, we use a gas-kinetic analogy with scattering cross-sections, e.g., for electron-phonon interaction. This is equivalent to a description of the interaction of delocalized electrons and phonons when calculating scattering rates.

  3. 3.

    This interaction is also commonly described as an electron-phonon interaction, however, of a different kind than that responsible for scattering. As a lattice deformation relates to phonons, the interaction of electrons with the lattice causing a specific deformation can formally be described by continuously absorbing and emitting phonons (see the following sections).

  4. 4.

    A decrease of lattice-bonding forces corresponds also to a decreased Debye temperature; see also Sect. 1.1.2 of chapter “Phonon-Induced Thermal Properties”.

  5. 5.

    This result is obtained from Fröhlich et al. (1950) and with a variational method from Lee et al. (1953). It can be used up to \( {\alpha}_{\mathrm{c}}\gtrsim 1 \). Earlier results from Pekar, using an adiabatic approximation, yielded \( {E}_{\mathrm{pol}}=-\left({\alpha}_{\mathrm{c}}^2/3\pi \right)\quad \hslash {\omega}_{\mathrm{LO}} \), which gives a lower self-energy than Eq. 11 in the range of validity \( {\alpha}_{\mathrm{c}}<1 \).

  6. 6.

    More precisely, these three energies are: Erelax, the energy given to phonons during lattice relaxation; J, the bandwidth of a band created by free, uncoupled carriers of the given density; and U, the energy necessary to put two carriers with opposite spin on the same lattice site.

  7. 7.

    Strictly speaking, steady-state carrier transport is due to external forces only. The diffusion current originates from a deformed carrier-density profile due to external forces and is a portion of the conventionally considered diffusion component. The major part of the diffusion is used to compensate the built-in field and has no part in the actual carrier transport: both drift and diffusion cancel each other and are caused by an artificial model consideration – see Sect. 3.4.

  8. 8.

    The rms (root mean square) velocity vrms, which is commonly used, should be distinguished from the slightly different average velocity \( {v}_{\mathrm{av}}=\left\langle \left|v\right|\right\rangle \) and from the most probable velocity vmp. Their ratios are \( {v}_{\mathrm{rms}}:{v}_{\mathrm{av}}:{v}_{\mathrm{mp}}=\sqrt{3/2}:\sqrt{4/\pi }:1=1.2247:1.1284:1 \), as long as the carriers follow Boltzmann statistics. For a distinction between these different velocities, see Fig. 3.

    Fig. 3
    figure 3

    Classical velocity distribution, with root mean square, average, and most probable velocities identified

    Full size image
  9. 9.

    This motion resembles a random walk (Chandrasekhar 1943).

  10. 10.

    In chapters dealing with carrier transport, F is chosen for the field, since E is used for the energy.

  11. 11.

    Often a minimum scattering angle of 90° is used to distinguish scattering events with loss of memory from forward scattering events – see Sect. 4.6.1.

  12. 12.

    As a reminder: here and in all following sections, |e| is used when not explicitly stated differently.

  13. 13.

    The atoms in the alloyed lattice (or here in the anion sublattice) are statistically arranged. Strictly speaking, this causes random fluctuation of the composition in a microscopic volume element of the crystal and results in a local fluctuation of the bandgap energy due to a locally varying parameter x; as a result, extended or localized states with energies Ec,v(x) close to the band edge \( {E}_{\mathrm{c},\mathrm{v}}\left(\overline{x}\right) \) of the mean composition \( \overline{x} \) are formed, leading to some tailing of the band edges (Sect. 3.1 of chapter “Optical Properties of Defects”). The composition parameter u used in the text actually refers to the mean composition, and the effect of band tailing is neglected here.

  14. 14.

    Major deviations from linearity of Eg with composition are observed when the conduction-band minimum lies at a different point in the Brillouin zone for the two end members. One example is the alloy of Ge and Si. Other deviations (bowing – see Sect. 2.1 of chapter “Bands and Bandgaps in Solids”) are observed when the alloying atoms are of substantially different size and electronegativity.

  15. 15.

    pn junctions are the best studied intentional space-charge regions. Inhomogeneous doping distributions – especially near surfaces, contacts, or other crystal inhomogeneities – are often unintentional and hard to eliminate.

  16. 16.

    This argument no longer holds with a bias, which will modify the space charge; partial heating occurs, proportional to the fraction of external field. This heating can be related to the tilting of the quasi-Fermi levels.

  17. 17.

    With bias, the Fermi level in a junction is split into two quasi-Fermi levels which are tilted, however, with space-dependent slope. Regions of high slope within the junction region will become preferentially heated. The formation of such regions depends on the change of the carrier distribution with bias and its contribution to the electrochemical potential (the quasi-Fermi level). Integration of transport, Poisson, and continuity equations yields a quantitative description of this behavior (Böer 1985b).

  18. 18.

    The most severe approximation is the linear relation in time, which eliminates memory effects in the Boltzmann equation (Nag 1980).

  19. 19.

    Here discussed for electrons, although with a change of the appropriate parameters, it is directly applicable to holes, polarons, etc; applications to excitons, polaritons, and e-h plasmas are reviewed by Snoke (2011). The influence of other fields, such as thermal or magnetic fields, is neglected here; for such influences, see chapter “Carriers in Magnetic Fields and Temperature Gradients”.

  20. 20.

    An alternative approach considers a hydrodynamic transport as, e.g., discussed for electrons by Bringuier (2019) and for phonons by Sendra et al. (2021).

  21. 21.

    The quantities τm and f0 in Eq. 108 are functions of E only and hence do not change over a constant-energy surface.

  22. 22.

    A recent study indicates that the widely applied relaxation-time approximation to the relativistic Boltzmann equation is not compatible with conservation laws if the relaxation time depends on energy (Rocha et al. 2021); see also Bringuier (2019).

  23. 23.

    In two-dimensional structures, significant momentum relaxation is found due to near-surface acoustic phonon scattering; see Pipa et al. (1999).

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  1. Naples, FL, USA

    Karl W. Böer

  2. Institut für Festkörperphysik, EW5-1, Technische Universität Berlin, Hardenbergstr. 36, 10623, Berlin, Germany

    Udo W. Pohl

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Böer, K.W., Pohl, U.W. (2022). Carrier-Transport Equations. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06540-3_22-4

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    Carrier-Transport Equations
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    18 June 2022

    DOI: https://doi.org/10.1007/978-3-319-06540-3_22-4

  2. Carrier-Transport Equations
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    28 March 2020

    DOI: https://doi.org/10.1007/978-3-319-06540-3_22-3

  3. Carrier-Transport Equations
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    DOI: https://doi.org/10.1007/978-3-319-06540-3_22-2

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    Carrier-Transport Equations
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    DOI: https://doi.org/10.1007/978-3-319-06540-3_22-1

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