Strong Quadrupole-Strain Interaction of Vacancy Orbital in Boron-Doped Czochralski Silicon

We have carried out ultrasonic measurements of a boron-doped silicon ingot grown by the Czochralski method in order to determine the quadrupole-strain interaction constant of a vacancy orbital. The low-temperature softening of the elastic constant C₄₄ shows a remarkable variation depending on positions of the ingot, which reflects the distribution of vacancy concentration N in the ingot. An infrared laser scattering tomograph was employed to measure the density and size of voids in the silicon wafers by determining the vacancy concentration N_cons consumed in void formation. Using a combination of laser scattering tomography and low-temperature softening, we have found a sum rule in which the initially created vacancy concentration N_total corresponds to the sum of the residual vacancy concentration N and the consumed vacancy concentration N_cons as N_total = N + N_cons. Taking account of the sum rule, we deduce the interaction constant g_Γ5 = (2.8±0.2)×10⁵ K for the quadrupole-strain interaction H_QS = -g_Γ5O_zxε_zx of the vacancy orbital. The huge deformation energy of 1.6×10⁵ K per vacancy with the Γ₈ ground state for unit strain ε_zx = 1 verified the strong electron–lattice interaction of the vacancy orbital. Employing the one-to-one correspondence between the softening of ΔC₄₄/C₄₄ = 1.0×10^-4 down to 30 mK and the vacancy concentration of N = 1.5 ×10¹³ cm^-3, we can determine the vacancy concentration by low-temperature ultrasonic measurements. The present work surely puts forward a novel semiconductor technology based on low-temperature ultrasonic measurements for evaluating vacancy concentration in silicon wafers.