New method for an accurate determination of residual strain in polycrystalline silicon films by analysing resonant frequencies of micromachined beams

The residual strain of polycrystalline silicon (poly-Si) film is evaluated using the strain-sensitive nature of the resonant frequency of a doubly-sup ported beam. By introducing the length dependence curve of the resonant fre quency, the effect of the residual strain is discussed. Both measurements a nd finite-element calculations are performed to accurately determine the re sidual strain in two types of phosphorus-doped poly-Si films. From the fini te-element calculations including nonlinear buckling, the post-buckling fre quency is more sensitive to compressive residual strain than the frequency before buckling. By also considering these post-buckling frequencies, a mor e accurate estimation of the residual strain can be obtained. A test chip consisting of 97 poly-Si beams of different lengths was fabrica ted for residual-strain evaluation. By fitting the calculations with measur ed resonant frequencies, the residual strains of the films are determined t o be -48 +/- 5 x 10(-6) and -400 +/- 100 x 10(-6) for two types of films. T he difference in the residual strain is explained by the heat treatment ste ps of phosphorus-doped poly-Si films. This method is extremely suitable for monitoring small fluctuations in the film residual strain between differen t process runs.