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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-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(Gamma 5) = (2.8 +/- 0.2) x 10(5) K for the quadrupole-strain interaction H-QS = -g(Gamma 5)O(zx)epsilon(zx) of the vacancy orbital. The huge deformation energy of 1.6 x 10(5) K per vacancy with the Gamma(8) ground state for unit strain epsilon(zx) = 1 verified the strong electron-lattice interaction of the vacancy orbital. Employing the one-to-one correspondence between the softening of Delta C-44/C-44 = 1.0 x 10(-4) down to 30 mK and the vacancy concentration of N = 1.5 x 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.