Carrier-Transport Equations

  • Karl W. Böer
  • Udo W. PohlEmail author
Living reference work entry

Later version available View entry history



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 carrier gradient 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.


Boltzmann transport equation Built-in electric field Carrier heating Collision integral Conductivity Diffusion current Drift current Drift velocity Effective mass Einstein relation Energy relaxation Fröhlich coupling Inelastic and elastic scattering Joule’s heating Mean free path Mobility effective mass Momentum relaxation Polaron Polaron mass Polaron self-energy Quasi-Fermi level Relaxation time Relaxation-time approximation 


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.NaplesUSA
  2. 2.Institut für Festkörperphysik, EW5-1Technische Universität BerlinBerlinGermany

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