Friedel Density Oscillations about a Coulombic Impurity in a High Magnetic Field

Abstract

The density perturbation $\delta \rho \left(\mathrm{r}\right)$ about a Coulombic impurity in strong magnetic fields is analyzed in the degenerate case. The results show that (a) anisotropic Friedel oscillatory behavior is discernible along the direction parallel to the magnetic field, and (b) exponential decay along the direction perpendicular to the magnetic field deprives the result of long-range character. These results are in accord with those of Rensink, save for the fact that the $\delta$-function impurity used by Rensink results in a Gaussian decay in the perpendicular direction, in contrast to the exponential decay in the perpendicular direction seen to result from a Coulombic impurity here. This indicates that the qualitative nature of the spatial decay in the perpendicular direction is dependent upon the character of the impurity potential. The results obtained here are valid at large distances from the impurity for a wide range of field strengths. A qualitative discussion of low-field behavior in the approach to the zero-field limit is presented, and the role of self-consistent aspects of response is briefly indicated.