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Generally, the full band structure of semiconductors incorporates an
anisotropic dispersion relation
, which is only accessible by
numerical methods. Fig. 3.4 illustrates iso-energy surfaces of the
first conduction band of silicon within one octant of the Brillouin zone,
where the lowest energy minima are located close to the
-points. Around
these energy minima valleys are formed, whose shape deviates strongly from the
conventionally used elliptic approximation with increasing energies. Beside
the valleys at the
-points, further valleys are located at the
-points,
whose energy minima are higher than that of the
-valleys. However, in
order to describe the transport in the semiconductor in a closed analytical
way, simplified expressions for the rather complicated full-band structure are
commonly introduced. In the following derivations, isotropic bands are
assumed, which imply the dispersion relation to depend only on the magnitude of
the wave vector
. For the sake of convenience, the dispersion relation
is separated into its parabolic and non-parabolic contributions
[76]
M. Wagner: Simulation of Thermoelectric Devices