What is the Valence of Mn in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{N}$?

We investigate the current debate on the Mn valence in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{N}$, a diluted magnetic semiconductor (DMS) with a potentially high Curie temperature. From a first-principles Wannier-function analysis, we unambiguously find the Mn valence to be close to $2+$ ($d^5$), but in a mixed spin configuration with average magnetic moments of $4{\mu }_B$. By integrating out high-energy degrees of freedom differently, we further derive for the first time from first-principles two low-energy pictures that reflect the intrinsic dual nature of the doped holes in the DMS: (1) an effective $d^4$ picture ideal for local physics, and (2) an effective $d^5$ picture suitable for extended properties. In the latter, our results further reveal a few novel physical effects, and pave the way for future realistic studies of magnetism. Our study not only resolves one of the outstanding key controversies of the field, but also exemplifies the general need for multiple effective descriptions to account for the rich low-energy physics in many-body systems in general.

Figure 1

(a) Total and partial densities of states (DOSs) of ${\mathrm{Ga}}_{31}{\mathrm{MnN}}_{32}$ with the Fermi energy ($E_F$) at 0 eV. The partial DOSs have been scaled up in units of “per atom of this kind.” (b) The DOS of the impurity bands around ${\mathrm{E}}_{\mathrm{F}}$ in units of “per GaN primitive unitcell.” (c) Illustration of the hybridization of Mn-N orbitals. Up and down arrows represent spin majority and minority, respectively. UHB (LHB) denotes upper (lower) Hubbard bands, whereas BD (ABD) denotes bonding (antibonding). Note that two electrons residing in the spin-majority $e_g$ orbitals are not shown here.

Figure 2

Illustration of the WOs used in (a) low-energy effective $d^4$ and (b),(c) the effective $d^5$ picture. The upper panels show the local crystal structure, while the lower panels plot the isosurface of (a) $\widetilde{\mathrm{Mn}}\text{−}{t}_{2g}$, and (b),(c) $\tilde{N}\text{−}s{p}^3$ WOs at $0.07{\mathrm{bohr}}^{-3/2}$.

Figure 3

The spin-majority band structure of ${\mathrm{Ga}}_{31}{\mathrm{MnN}}_{32}$ (a) with complete parameters, (b) with the leading orbital energy shift $T_{\mathbf{j}\mathbf{j}\mathbf{j}}^{mm}=2488\mathrm{meV}$ removed, and (c) with the two leading exchange-assisted parameters $J_{\mathbf{j}\mathbf{j}\mathbf{j}}^{m{m}^{\prime }\ne m}=-633$ and $J_{\mathbf{j}\mathbf{j}{\mathbf{i}}^{\prime }}^{m{m}^{\prime }\ne m}=800\mathrm{meV}$ with ${\mathbf{i}}^{\prime }=\mathrm{NN}(\mathbf{j})$ removed.