Group-Theoretical Study of the Zeeman Effect of Acceptors in Silicon and Germanium

Abstract

Group-theoretical techniques are used to deduce the selection rules, energy splittings, and relative intensities of the Zeeman components of the electric dipole absorption lines ${\ensuremath{\Gamma}}_{8}\ensuremath{\rightarrow}{\ensuremath{\Gamma}}_{6}$, ${\ensuremath{\Gamma}}_{8}\ensuremath{\rightarrow}{\ensuremath{\Gamma}}_{7}$, and ${\ensuremath{\Gamma}}_{8}\ensuremath{\rightarrow}{\ensuremath{\Gamma}}_{8}$ of an acceptor in a group-IV semiconductor. Results are obtained for three different orientations of the magnetic field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}$ with respect to the crystal axes: $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}[001]$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}[111]$, and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}[110]$. For a ${\ensuremath{\Gamma}}_{8}\ensuremath{\rightarrow}{\ensuremath{\Gamma}}_{8}$ transition the relative intensities for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}[001]$ are expressed in terms of two real parameters, which are essentially ratios of matrix elements of the electric-dipole-moment operator. The relative intensities for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}[111]$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}[110]$ depend on energy splittings as well. When terms quadratic in $B$ are important, the relative intensities for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}[110]$ become dependent on $B$. The results obtained are quite general, being based on symmetry considerations alone. They are applicable to an impurity located at a site of tetrahedral symmetry, provided that the Zeeman splitting of a given level is small in comparison with its distance from the nearest zero-field level. Our treatment proves particularly useful for studying acceptor states in group-IV semiconductors. As an example, we discuss the case of boron impurity in germanium.