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 (C₁₁ - C₁₂)/2 and C₄₄ of the transverse mode exhibit considerable softening below 2 and 5 K down to the base temperature of 30 mK, respectively. The elastic constant C₄₄ 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 Γ₈ quartet ground state and Γ₇ doublet excited state located at an energy of 1 K. The difference in C₄₄ between the two ultrasonic modes (k_z, u_x) and (k_x, u_z) at fields along the z-axis indicates that the Γ₈ quartet ground state is slightly split by local strain in the silicon sample. The quantum state of the vacancy orbital is expected to be sensitive to strain because of the extremely large quadrupole-strain coupling energy of g_Γ ≈ 10⁵ K due to the extensively spreading orbital radius of r ≈ 1 nm. The differences in variation of the low-temperature softening and magnetic field dependence among eight samples cut out from different locations of the present boron-doped FZ silicon ingot evidence the inhomogeneous distribution of the vacancy concentration.