Effect of Doping on the Elastic Constants of Silicon

Abstract

Keyes' theory of the dependence of the shear elastic constants of degenerate $p$-type Ge on hole concentration has been extended to degenerate $p$-type Si by including in the theory the light-hole band and the split-off band. It is shown that the total effect in the elastic constants is the sum of effects arising from the light-hole, heavy-hole, and split-off bands and originates in a change of the Fermi level with strain. Using the ultrasonic pulse-echo technique, measurements were made at 78\ifmmode^\circ\else\textdegree\fi{}K of the elastic constant ${C}^{\ensuremath{'}}=\frac{1}{2}({C}_{11}\ensuremath{-}{C}_{12})$ of Si samples containing: (a) 7.0\ifmmode\times\else\texttimes\fi{}${10}^{15}$, (b) 3.2\ifmmode\times\else\texttimes\fi{}${10}^{18}$, (c) 1.4\ifmmode\times\else\texttimes\fi{}${10}^{19}$, and (d) 1.1\ifmmode\times\else\texttimes\fi{}${10}^{20}$ B atoms ${\mathrm{cm}}^{\ensuremath{-}3}$. Relative to the purest sample (a), the change in ${C}^{\ensuremath{'}}$ is -12, -42, and -193 (in units of ${10}^{8}$ dyne ${\mathrm{cm}}^{\ensuremath{-}2}$), respectively. Equating these values with the calculated changes in ${C}^{\ensuremath{'}}$ and using effective masses from the top of the various valence bands, the values of 5.8, 8.5, and 11.8 eV were obtained for the deformation potential constant ${{\ensuremath{\Xi}}_{s}}^{\ensuremath{'}}$. These values were corrected to 4.9, 7.1, and 9.94 eV by estimating the dependence on the wave vector of the effective masses as inferred from Kane's work on the band structure of Si. Possible sources of the dependence of ${{\ensuremath{\Xi}}_{s}}^{\ensuremath{'}}$ on doping are discussed.