Magnetic interactions of $\mathrm{Cr}\text{−}\mathrm{Cr}$ and $\mathrm{Co}\text{−}\mathrm{Co}$ impurity pairs in $\mathrm{ZnO}$ within a band-gap corrected density functional approach

The well-known “band-gap” problem in approximate density functionals is manifested mainly in an overly low energy of the conduction band (CB). As a consequence, the localized gap states of $3d$ impurities states in wide-gap oxides such as $\mathrm{ZnO}$ occur often incorrectly as resonances inside the CB, leading to a spurious transfer of electrons from the impurity state into the CB of the host, and to a physically misleading description of the magnetic $3d\text{−}3d$ interactions. A correct description requires that the magnetic coupling of the impurity pairs be self-consistently determined in the presence of a correctly positioned CB (with respect to the $3d$ states), which we achieve here through the addition of empirical nonlocal external potentials to the standard density functional Hamiltonian. After this correction, both Co and Cr form occupied localized states in the gap and empty resonances low inside the CB. In otherwise undoped $\mathrm{ZnO}$, Co and Cr remain paramagnetic, but electron-doping instigates strong ferromagnetic coupling when the resonant states become partially occupied.

Figure 1

(Color online) Orbital and spin configuration of (a) ${\mathrm{Co}}_{\mathrm{Zn}}$ and (b) ${\mathrm{Cr}}_{\mathrm{Zn}}$ in $\mathrm{ZnO}$ and the resulting magnetic moment $m$ in GGA, in $\mathrm{GGA}+U$ (for $\mathrm{Zn}\text{−}d$, $\mathrm{Co}\text{−}d$, and $\mathrm{Cr}\text{−}d$), and in the fully gap-corrected NLEP method. Charge transfer from occupied TM-$d$ states into the host conduction band (e.g., $e_{-}^2\to e_{-}^{1.9}{c}^{0.1}$ for ${\mathrm{Co}}_{\mathrm{Zn}}$ in GGA) is indicated by open arrow symbols. Dashed lines in (b) indicate the average level energy before the Jahn-Teller splitting.

Figure 2

(Color online) The ferromagnetic stabilization energy $\mathrm{\Delta }{E}_{\mathrm{FM}}=E_{\mathrm{FM}}-E_{\mathrm{AFM}}$ in eV for (a) the $\mathrm{Co}\text{−}\mathrm{Co}$ and (b) the $\mathrm{Cr}\text{−}\mathrm{Cr}$ pairs in a 72-atom $\mathrm{ZnO}$ supercell, as a function of the pair distance $d$. Results are given for the uncorrected GGA and for the gap-corrected NLEP methods, and for different levels of additional electron doping up to $1e$ per TM pair $(\sim {10}^{21}{\mathrm{cm}}^{-3})$.