Spin structure of impurity band of semiconductors in two and three-dimensional cases

The exchange interaction between electrons located at different randomly distributed impurities is studied for small density of impurities. The singlet-triplet splitting 2J(R) is calculated for two Coulomb centers at a distance R. Interpolated formulas are found which work for all distances R from zero to infinity. The data from atomic physics are used for the interpolation in three-dimensional case. For two-dimensional case the original calculations are performed to find asymptotic behavior of the splitting at large R, the splitting for the ``two-dimensional helium atom'' (R=0) and the splitting at $R=a_B$, where $a_B$ is the effective Bohr radius. The spin structure of impurity band is described by the Heisenberg Hamiltonian. The ground state of a system consists of localized singlets. The new results are obtained for the distribution of the singlet pairs in the ground state. These results are exact at low density. The problem is reduced to a non-trivial geometric problem which is solved in the mean field approximation and by computer modeling. The density of free electrons is found as a function of temperature and the distribution function of the singlet-triplet transitions energies is calculated. Both functions are given in an analytical form.