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

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 ${\Gamma }_8\to {\Gamma }_6$, ${\Gamma }_8\to {\Gamma }_7$, and ${\Gamma }_8\to {\Gamma }_8$ of an acceptor in a group-IV semiconductor. Results are obtained for three different orientations of the magnetic field $\overrightarrow{\mathrm{B}}$ with respect to the crystal axes: $\overrightarrow{\mathrm{B}}\parallel \mathrm{}\left[001\right]\mathrm{}$, $\overrightarrow{\mathrm{B}}\parallel \mathrm{}\left[111\right]\mathrm{}$, and $\overrightarrow{\mathrm{B}}\parallel \mathrm{}\left[110\right]\mathrm{}$. For a ${\Gamma }_8\to {\Gamma }_8$ transition the relative intensities for $\overrightarrow{\mathrm{B}}\parallel \mathrm{}\left[001\right]\mathrm{}$ 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 $\overrightarrow{\mathrm{B}}\parallel \mathrm{}\left[111\right]\mathrm{}$ and $\overrightarrow{\mathrm{B}}\parallel \mathrm{}\left[110\right]\mathrm{}$ depend on energy splittings as well. When terms quadratic in $B$ are important, the relative intensities for $\overrightarrow{\mathrm{B}}\parallel \mathrm{}\left[110\right]\mathrm{}$ 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.