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• Posted on 25 July, 2008 by Ben Haley

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  k is usually called the wave vector. A crystalline material, like a metal wire or a semiconductor device, has a periodic structure. Each point in the periodic material can be
  written in terms of a set of vectors {R} which describes the crystal. The set of vectors {R} has
  a related set of reciprocal lattice vectors {K}, which is defined such that exp(i*K*R) = 1. The
  periodicity of the lattice also implies that an electron traveling through a crystal will encounter an electrostatic potential with the same periodicity. The periodic nature of the potential is due to the regular arrangement of the positively charged nuclei in the crystal atoms. The potential can be written in a Fourier series whose only non-zero terms exp(i*k*r) are those for which k is a vector on the reciprocal lattice {K} of the crystal. The motion of the electron in a periodic potential can also be represented in terms of a function
  of the form exp(i*k*r) (this is called Bloch's theorem), so the wave vector k is a very useful mathematical formalism for expressing both the motion of electrons and the nature of an electrostatic potential in a crystal.
  
  The E-k diagram (or bandstructure) converts the mathematical formalism of the wave vector into useful information about the allowed energy levels of an electron in a crystal. For a given k (which corresponds to motion in a certain direction in the crystal), only certain energy levels E are accessible to an electron. Correspondingly, electrons with a certain energy E can only move in certain directions. This information is very useful in understanding the fundamental motion of electrons in a crystal, especially in the presence of an external electrostatic potential.

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