Linear-response function of a semi-infinite degenerate plasma in the presence of an external magnetic field

The static linear-response function ${\chi }_0(\overrightarrow{\mathrm{r}},{\overrightarrow{\mathrm{r}}}^{\prime })$ of a semi-infinite, degenerate plasma of noninteracting electrons is calculated in the presence of an external magnetic field perpendicular to the surface. The surface is simulated by an infinite-barrier model. The work is based on the density-matrix formalism for an impurity embedded in a plasma and its image in the surface. The induced electron number density is expressed as a sum over Landau levels by a Laplace transformation of the density matrix. The Laplace-transform representation allows an analytic evaluation of the linear-response function for various magnetic field regimes. Results are presented for ${\chi }_0(\overrightarrow{\mathrm{r}},{\overrightarrow{\mathrm{r}}}^{\prime })$ in the quantum strong-field limit, such that electrons occupy only the lowest Landau eigenstate, as well as the lowfield limit where the response function is expressed as a series expansion in powers of the applied magnetic field strength. The intermediate-field regime for de Haas—van Alphen oscillations is also briefly discussed. These results provide a useful representation of the nonlocal static linear-response properties of a quantum plasma since ${\chi }_0(\overrightarrow{\mathrm{r}},{\overrightarrow{\mathrm{r}}}^{\prime })$ is expressed in terms of known elementary and special functions.