A matrix method for the calculation of nonlocal exchange potential and spin-orbit coupling in the first-principles calculation method

Adequate treatment of exchange interaction and inclusion of spin-orbit coupling are important in obtaining reliable electronic structures of semiconductors and magnetic materials. Here, a matrix method has been derived to calculate nonlocal exchange potential and spin-orbit coupling within a muffin-tin-orbital based first-principles calculation method. Good agreement between calculated and experimental band gaps for various zinc-blende and wurtzite semiconductors demonstrated that direct calculation of nonlocal exchange potential is feasible. This method may be an improvement over the currently used LSDA + U or LSDA + GGA + U method, which uses an adjustable U parameter, for wide-band-gap semiconductors and magnetic materials. In this method, the use of an extended Hamiltonian matrix, which contains off-diagonal matrix elements between up- and down-spin states due to spin-orbit coupling, enables mutual adjustment of the occupation of up- and down-spin states through atomic orbitals. In contrast, in the conventional spin-polarized calculation method up- and down-spin sub-matrices are diagonalized separately. The calculated spin-orbit splitting for the triplet states at the Γ point for GaAs and ZnS are in good agreement with experimental data.