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

Abstract

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 \(C_{44}\) 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 \(N_{\text{cons}}\) 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 \(N_{\text{total}}\) corresponds to the sum of the residual vacancy concentration \(N\) and the consumed vacancy concentration \(N_{\text{cons}}\) as \(N_{\text{total}} = N + N_{\text{cons}}\). Taking account of the sum rule, we deduce the interaction constant gΓ5 = (2.8±0.2)×105 K for the quadrupole-strain interaction HQS = -gΓ5Ozxεzx of the vacancy orbital. The huge deformation energy of \(1.6\times 10^{5}\) K per vacancy with the Γ8 ground state for unit strain \(\varepsilon_{zx} = 1\) 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 \(N = 1.5 \times 10^{13}\) 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.