Quadrupole Effects of Vacancy Orbital in Boron-Doped Silicon

We have investigated low-temperature properties at a longitudinal elastic constant $C_{\text{L[111]}}$ of a boron-doped silicon crystal grown by a floating zone (FZ) method. The softening of $C_{\text{L[111]}}$ depending on magnetic fields indicates that a magnetic charge state $V^{+}$ of a vacancy orbital accommodating three electrons is stable. Appreciable anisotropy in the softening of $C_{\text{L[111]}}$ at magnetic fields along the [111] and $[1\bar{1}0]$ axes up to 10 T is described in terms of a quadrupole susceptibility for the vacancy orbital consisting of a $\Gamma_{8}$ quartet ground state and $\Gamma_{6}$ doublet excited state due to the spin--orbit interaction.