Electronic Effects in the Elastic Constants of $n$-Type Silicon

The temperature dependence of the second-order elastic moduli, and the 25°C third-order elastic constants, are reported for pure silicon and for silicon doped with ∼2×${10}^{19}$ ($P$ atoms)/${\mathrm{cm}}^3$. Certain combinations of the elastic constants are unaffected by the doping; others show large changes which are interpreted in the light of Keyes's theory of the electronic contribution to the elastic constants. The agreement between theory and experiment is excellent, and permits a thermodynamic determination of the carrier concentration of the doped sample and of the shear deformation potential constant ${\Xi }_u$ for the silicon conduction band of 8.6 eV at 25°C.