Magnetic and nonmagnetic impurities in two-dimensional metals

We examine the spatial dependence of the spin polarization arising from a magnetic impurity as well as the impurity-impurity interaction for impurities dissolved in an idealized two-dimensional metal. We find that the usual $\frac{(x cosx-sinx)}{x^4}$ dependence of the oscillations in a three-dimensional metal is replaced by a $\frac{sinx}{x^2}$ dependence in a two-dimensional system, where $x=2\mathrm{}{k}_{F}r, k_F$ is the Fermi wave vector, and $r$ is the distance from the impurity. Using this result we show that the temperature and concentration dependence of the thermodynamic properties of a magnetic-impurity system dissolved in a two-dimensional metal will, in the molecular-field approximation, be identical with that dissolved in a three-dimensional system. Similarly, the distance dependence of the screening charge from a nonmagnetic impurity in a metal changes from a $\frac{1}{r^3}$ (times an oscillating function) dependence in three dimensions to a $\frac{1}{r^2}$ dependence in two dimensions.