Quantum Theory of Galvanomagnetic Effects

A study is made of the effect of the quantization of the electron orbits in a magnetic field on the galvanomagnetic properties of an isotropic semiconductor or semimetal in the phonon-scattering range. The conductivity tensor is calculated using the quantum-mechanical density operator. A generalized Boltzmann equation is derived, in which the scattering for the density matrix elements of interest in the calculation of the current can be described by a relaxation time $\tau \left(\epsilon \right)$, that is for most energies inversely proportional to the density of electronic states in the magnetic field.