All-optical measurement of in-plane and out-of-plane Young's modulus and Poisson's ratio in silicon wafers by means of vibration modes

An all-optical technique was developed to determine the in-plane and out-of-plane Young's modulus, E, and Poisson's ratio, sigma, in silicon wafers through excitation and detection of vibration modes. The technique is remote, non-destructive and works on-line, making it an attractive inspection tool for use in semiconductor foundries. Vibration modes were generated by a pulse from a frequency-doubled Q-switched Nd:YAG laser and detected by a commercial heterodyne Michelson interferometer. Because of their importance as component substrates in the electronic and MEMS industries, both the (100) and (111) silicon wafers were investigated. It was found that the resonance frequencies of the first pure torsional and of the first hybrid (torsional and flexural) modes varied continuously and symmetrically with respect to a 90degrees rotation of the wafer around the centre, while the resonance frequencies of the first and second pure flexural modes remained constant. On the basis of this experimental observation, we could correlate the resonance frequencies of the two former and two latter modes with the in-plane and out-of-plane mechanical parameters E and sigma, respectively. A novel inverse method for circular anisotropic plates was then derived to extract the in-plane and out-of-plane parameters from the resonance frequency measurements. The results obtained by this approach were in good agreement with the standard values calculated by the theory of elasticity.