Semiclassical approach to the density of states of the disordered electron gas in a quantum wire

A theory is given of the density of states (DOS) of the quasi-one-dimensional electron gas (1DEG) in a semiconductor quantum wire in the presence of some random field. For a smooth random field, the derivation is carried out within Gaussian statistics and a semiclassical model. The DOS is then obtained in a simple analytic form, where the input function for disorder interaction is the autocorrelation function of the random field. This allows one to take completely into account the geometry of the wire, the origin of disorder, and the many-body screening by 1D electrons. The DOS is demonstrated to be composed of the classical DOS and the quantum correction, which are connected with fluctuations in the random potential and in the random force, respectively. The disorder is found to smear out the square-root singularity of the DOS of the ideal 1DEG into a finite peak tailing below the subband edge. The disorder effects from impurity doping and surface roughness on the DOS of the 1DEG in a cylindrical GaAs wire of radius R are thoroughly examined. It is shown that for $R<~a^{*}/2\mathrm{}$ (with $a^{*}$ as the effective Bohr radius) surface roughness with a radius fluctuation equal to 10% of R overwhelms impurity doping with a density $n_i{=10\mathrm{}}^6{\mathrm{cm}}^{\mathrm{-}1},$ but for $R>~{2a}^{*}$ the latter is dominant.