Low-Temperature Softening Due to Vacancy Orbital with Γ8 Quartet Ground State in Boron-Doped Floating Zone Silicon

We have carried out low-temperature ultrasonic measurements using shear-mode ultrasound to clarify the quantum state of a vacancy orbital in boron-doped silicon grown by the floating zone (FZ) method. The elastic constants \((C_{11}-C_{12})/2\) and \(C_{44}\) of the transverse mode exhibit considerable softening below 2 and 5 K down to the base temperature of 30 mK, respectively. The elastic constant \(C_{44}\) measured by the three ultrasonic modes \((k_{x},u_{y})\), \((k_{z},u_{x})\), and \((k_{x},u_{z})\) shows the different magnetic field dependences among the configurations under applied magnetic fields along the \(z\)-axis. The elastic softening and the magnetic field dependence of the elastic constants are accounted for by the quadrupole susceptibility based on the energy level scheme of the vacancy orbital with a \(\Gamma_{8}\) quartet ground state and \(\Gamma_{7}\) doublet excited state located at an energy of 1 K. The difference in \(C_{44}\) between the two ultrasonic modes \((k_{z},u_{x})\) a...