Influence of surface stress in the determination of the (2 2 0) lattice spacing of silicon

Abstract

We have examined the influence of surface stress in the measurement of the (2 2 0) lattice plane spacing of a Si crystal by combined x-ray and optical interferometry. By means of a finite element simulation we have obtained first a quantitative description of various non-linear displacement fields of the diffracting planes; the simulation indicates that surface stress does not cause a significant difference between bulk and surface lattice spacing. Subsequently, to quantify the effect of the above fields on the period of the interference fringes, we have solved numerically the perturbed Takagi–Taupin equations; these results indicate that the fringe period is determined by the atom spacing on the crystal entrance surface, whereas, intuitively, we might have expected this period to be a kind of average of the (2 2 0) lattice plane spacing. To corroborate our numerical calculations, we have found analytical expressions which allow us to isolate the contribution of the non-linear superficial terms on the measurement of the plane spacing.