Excited Cr impurity states in Al${}_2$O${}_3$ from constraint density functional theory

The excited states, ${}^4{T}_{2g}$ and ${}^2{E}_g$, of a Cr impurity in Al${}_2$O${}_3$ were treated by constraint density functional theory by imposing a density matrix constraint (constraint field) to control the electron occupation numbers of the $d$ orbitals. The calculated excitation energies, directly calculated from the self-consistent total energies of the ${}^4{A}_{2g}$ ground states and the various excited states, correctly reproduce the experimental ordering. In addition, we find that there is no stationary solution for the excited ${}^4{T}_{2g}$ state corresponding to the crystal-field transition state in the usual Kohn-Sham equation, i.e., with no constraint field. By contrast, the excited ${}^2{E}_g$ state of the spin-flip transition state is a (meta-) stable stationary solution, and may be responsible for the long radiative decay lifetime observed in experiments on ruby.

Figure 1

(a) Schematic electronic structure of the ${}^4{A}_{2g}$ ground state of a Cr impurity with a $C_3$ site symmetry in Al${}_2$O${}_3$, exhibiting cubiclike $O_h$ splitting into lower $t_{2g}$ and upper $e_g$ levels, which under the trigonal distortion split into $a$ and $e$ irreducible representations. (b) Two-particle excitations from the ground ${}^4{A}_{2g}$ state to ${}^4{T}_{2g}$ (U band) and ${}^2{E}_g$ (R line) states, which correspond to the electronic configurations $t_{2g,\uparrow }^2{e}_{g,\uparrow }$ and $t_{2g,\uparrow }^2{t}_{2g,\downarrow }$, respectively.

Figure 2

Density of states (DOS) for Al${}_2$O${}_3$:Cr${}^{3+}$ in (a) LSDA and (b) $\mathrm{LSDA}+U$ for $U=4.0$ eV and $J=0.58$ eV; zero energy corresponds to the Fermi level. The narrow peaks in the plots correspond to the Cr $d$ impurity states.

Figure 3

Calculated total energy differences $\Delta E$ of the excited ${}^4{T}_{2g}$ and ${}^2{E}_g$ states with respect to ${}^4{A}_{2g}$ ground state, in (a) LDSA and (b) $\mathrm{LSDA}+U$, as a function of the constraint field ${\mu }_n$ for a Cr impurity in Al${}_2$O${}_3$. Calculations were started with a set of constraint fields and then all ${\mu }_n$s were gradually set back to zero. For the ${}^4{T}_{2g}$ state, there is no stationary solution at zero constraint field. Triangles represent those in experiments (Expt).

Figure 4

Calculated charge density difference, $\Delta \rho \left(\mathbf{r}\right)={\rho }_{\mathrm{E}}\left(\mathbf{r}\right)-{\rho }_{\mathrm{G}}\left(\mathbf{r}\right)$, between the ${}^4{T}_{2g}$ excited state and the ground state of a Cr impurity in Al${}_2$O${}_3$ in (a) LSDA and (b) $\mathrm{LSDA}+U$. Dark (blue) and light (yellow) contour planes represent positive (excited electron) and negative (hole) values of the charge difference, 1.4$\times$10${}^{-2}$ electrons/bohr${}^3$. Oxygen and aluminum ions are represented by red and gray circles, respectively, and chromium ions are located in the center.