Microscopic mobility

An explicit formal expression is obtained for the energy-dependent microscopic mobility of a disordered system of independent electrons. It is this mobility which gives the average drift velocity of electrons of a given initial energy in response to an electric field. Both this microscopic mobility and the more familiar microscopic conductivity are local in energy. The microscopic mobility is proportional to the energy derivative of the microscopic conductivity, not to the microscopic conductivity itself, which leads to the scaling relation $s=\tau +1\mathrm{}$, where $s$ is the conductivity exponent and $\tau$ the mobility exponent. The absurdity of a mobility divergent at the mobility edge leads to the restriction $s\ge 1$. In the presence of interactions, values of $s<\mathrm{}1\mathrm{}$ can occur. If a quasiparticle representation is valid, $0\le \tau <s$ and the density of states diverges as the mobility edge is approached from the extended-state side with an exponent $u=-[\tau +(1-s\left)\right]$. The microscopic mobility provides a dynamical basis for the introduction of the hole concept into the theory of strongly disordered materials. Comparison with the thermopower then leads to a deeper understanding of the nature of holes in disordered materials. A carrier representation can then be derived for transport in disordered semiconductors.