VARIATIONAL METHOD FOR TENSILE-STRESS EVALUATION AND APPLICATION TO HEAVILY BORON-DOPED SQUARE-SHAPED SILICON DIAPHRAGMS

Motivated by the need for tensile stress evaluation in ultrathin diaphragms, and especially in heavily boron-doped silicon membranes, a new approach for the determination of this physical property is proposed. The technique employs equipment common to most IC laboratories. It combines deflection measurements and an appropriate theoretical relation between deflection and mechanical tension. Condenser microphones incorporating 0.8 mu m thick p(+) silicon membranes are used as test devices. Experimental data are obtained from the optical measurement of membrane deflection profiles under the application of electrostatic pressures. This idea is supported by appropriate theory describing the load-deflection behaviour of built-in, square-shaped, stretched membranes under non-uniform loading (of electrostatic nature). The exact solution of the equations governing this physical system being unattainable, an approximate solution is derived using a variational method. The mechanical tension is then easily deduced from deflection measurements. Applying this technique to three samples, the tensile stress in p(+) silicon membranes is found to be about 65 MPa.