Hubbard $U$ Calculations for Dilute Magnetic Semiconductors

Based on constrained density functional theory, we present ab initio calculations for the Hubbard $U$ parameter of transition metal impurities in dilute magnetic semiconductors, by choosing Mn in GaN as an example. The calculations are performed by two methods: (i) the Korringa-Kohn-Rostoker (KKR) Green function method for a single Mn impurity in GaN and (ii) the Full-potentian Linear Augmented Plane-Wave (FLAPW) method for large supercell of GaN with a single Mn impurity in each cell. By changing the occupancy of the majority $t_{2}$ gap state of Mn, we determine the $U$ parameter either from the total energy differences $E(N+1)$ and $E(N-1)$ of the $N+1$ and $N-1$ excited states with respect to the ground state $E(n)$, or by using the single particle energies for $N+1/2$ and $N-1/2$ occupancies (Janak's transition state model). Both methods give nearly identical results. Moreover the values calculated by the Green function technique and the supercell method agree quite well.