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In the semiclassical model, an electron in band n with crystal momentum ħk moves with the group velocity v = (1/ħ)∇ₖε(k), and external forces change its crystal momentum according to ħ dk/dt = −e(E + v × B). The lattice's effect is contained entirely in the band energy ε(k). Near a band minimum or maximum the electron responds as if it had an effective mass m*, set by the curvature of the band, which is negative near the top of a band. A nearly full band is best described by holes, which behave as particles with positive charge and positive mass.
Work with the one-dimensional tight-binding band ε(k) = −2t cos(ka). Plot ε, the velocity v(k), and the effective mass along the Brillouin zone, and trace an electron's motion in a constant electric field. Show that a completely filled band carries no current, then remove one electron and identify the hole. Compare with measured effective masses and the sign of the Hall coefficient in real materials.
A free electron obeys F = dp/dt and v = p/m. In a crystal the periodic potential changes both relations, but Bloch's theorem keeps a useful structure. Each electron state is labelled by a band index n and a wavevector k, with energy εₙ(k). The semiclassical model treats a wave packet built from states near one k as a particle and gives two rules.
First, the velocity is the group velocity of the band: v = (1/ħ)∇ₖε(k). Second, an external force changes the wavevector: ħ dk/dt = F, where for an electron F = −e(E + v × B). The force acts on ħk, called the crystal momentum. The force from the lattice itself never appears; it is already built into ε(k). The model assumes that fields vary slowly on the scale of the lattice and are too weak to push electrons into other bands.
Combining the rules gives the acceleration. Near a band extremum, ε(k) is close to a parabola, and the electron responds as if it had mass m*, where 1/m* = (1/ħ²) d²ε/dk² in one dimension. In three dimensions this becomes an effective mass tensor. Flat bands give heavy electrons and sharply curved bands give light ones. Electrons in the conduction band of gallium arsenide, for example, have m* close to 0.067 times the free-electron mass. Near the top of a band the curvature is negative, so m* is negative: a force pushes the electron the opposite way.
A filled band carries no current, because velocities at k and −k cancel. A band with one electron missing therefore carries the current of that one missing electron, with the sign reversed. It is simpler to describe this as a single hole with charge +e. Holes sit near the top of a band, where the electron effective mass is negative, so the hole's effective mass is positive. Holes explain positive Hall coefficients in some metals and are the main carriers in p-type semiconductors.
The model makes a surprising prediction. In a constant electric field, k increases steadily and crosses the Brillouin zone. When it reaches the zone boundary it reappears at the opposite edge, and the velocity, which follows sin(ka) in a tight-binding band, oscillates. The electron moves back and forth instead of accelerating. These Bloch oscillations have period h/(eEa). In metals, scattering destroys them long before one period passes, but they have been seen in semiconductor superlattices and in cold atoms in optical lattices. Ordinary conduction arises because scattering keeps resetting the electrons, which the Boltzmann transport equation describes.
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