Modification of Friedel Oscillations by a Magnetic Field

The electron density near an impurity or "test particle" in an electron gas is investigated for the case in which a uniform magnetic field is applied to the system. The electron gas is at zero temperature and the Coulomb interaction between electrons is neglected. A $\delta$-function potential is used for the interaction of the electrons with the test particle. The induced electron density along a line passing through the test particle parallel to the magnetic field is of the form ${\mathcal{r}}^{-3\mathrm{}}cos2k_F\mathcal{r}$ for large $\mathcal{r}$, where $k_F$ is the Fermi wave number and $\mathcal{r}$ is the distance from the test particle. The induced electron density along a line passing through the test particle perpendicular to the magnetic field is qualitatively different. It exhibits only a finite number of oscillations in space and then falls off monotonically with increasing distance from the test particle. The number of complete oscillations corresponds to the number of occupied Landau levels in the electron gas. Similar results may be expected for the electron spin density near a magnetic impurity.