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 (x cosx-sinx)x⁴ dependence of the oscillations in a three-dimensional metal is replaced by a sinxx² dependence in a two-dimensional system, where x=2kFr, kF 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 1r³ (times an oscillating function) dependence in three dimensions to a 1r² dependence in two dimensions.