Linear-scaling DFT+U applied to hole localization and Friedel oscillations in very dilute (Ga,Mn)As

System-size and strong electronic correlation are two factors inhibiting the routine first-principles simulation of transition-metal doped nanostructures. Tackling these issues simultaneously, we have developed a linear-scaling implementation of the DFT+$U$ method within the ONETEP code,\footnote{Hine, Haynes, Mostofi, Skylaris \& Payne, {\it Comp. Phys. Commun.,} {\bf 180}, 1041 (2009).} demonstrating scaling upto $7,000$ atoms. Our implementation allows for nonorthogonal projectors,\footnote{O'Regan, Payne \& Mostofi, {\it PRB} {\bf 83}, 245124 (2011).} which may be self-consistently optimized.\footnote{O'Regan, Hine, Payne \& Mostofi, {\it PRB} {\bf 82}, 081102(R) (2010).} We apply our approach to the prototypical dilute magnetic semiconductor (Ga,Mn)As. The ferromagnetic interaction between distant localized magnetic moments in (Ga,Mn)As is mediated by defect-induced holes, whose long-range character is critical. Our large-scale calculations on $1,728$ atom super-cells enable us to study the localization and symmetry of the magnetization and hole in the very dilute ($0.1\%$) limit, and to analyze the long-range Friedel oscillations.