Localized Orbital Moments of Defects in Insulators—TheFCenter

The factors determining the orbit-Zeeman splitting of a localized ${\Gamma }_{4u}$ electronic state in an insulating solid are discussed in an idealized model and the results applied to $F$ centers in the alkali halides. The model used involves explicit orthogonalization of an approximate one-electron function for the localized defect state to the core states of the host-lattice ions. The expectation value of $L_z$ (or orbital $g$ factor, $g_{\mathrm{orb}}$) and the electron currents surrounding the defect are calculated. It is shown that the requirement of orthogonality to core-electron states may cause significant deviations of the orbital moment from unity in ${\Gamma }_{4u}$ states derived from $P$-like defect states. For neighboring ions with only $S$-like core states, overlap corrections decrease the moment; this may explain the low value of $g_{\mathrm{orb}}$ observed for the LiF $F$ center. For neighbors with $S$ and $P$-like core states, corrections to the orbital moment may be either positive or negative. Localization of defect electron charge on neighboring ions tends to decrease orbital angular momentum, while "hopping" of charge from one neighbor to the next tends to augment the moment. For the NaCl $F$ center, explicit calculations yield values of $g_{\mathrm{orb}}$ between 1.2 and 1.4 suggesting a predominance of the "hopping" motion. Various experimental determinations of $g_{\mathrm{orb}}$ are discussed and a physical picture for its deviation from unity is developed in terms of electron currents.