Effect of doping on intermode Kerr nonlinearity in semiconductor-based nanomechanical resonators

Low-lying mechanical eigenmodes in nano- and micromechanical resonators are usually nondegenerate. A major effect of the mode-mode coupling is the change of the frequency of one mode due to vibrations of other modes. This change results from the quartic vibration nonlinearity, more specifically, from the nonlinear coupling, which is bi-quadratic in the coordinates of the vibrational modes. Such coupling is often called cross-Kerr or cross-Duffing nonlinearity. It allows one to modify the frequency of one mode by exciting another mode. It also leads to frequency fluctuations of the modes due to thermal vibrations of other modes. We study the cross-Kerr nonlinearity in the resonators based on p-doped semiconductors with diamond structure and A₃B₅ semiconductors. We show that this nonlinearity can be strong, which is a consequence of the strong hole-strain coupling. It results from the back-action of the strain-induced redistribution of the holes between different energy bands. The nonlinearity nonmonotonically depends on temperature, which is a consequence of the freezing out of the free holes at low temperature and broadening of the hole distribution over the energy bands with the increasing temperature. The results are applied to Si-based micromechanical resonators, which are broadly used in various applications, and the limitations on the frequency stability of high-Q modes in such resonators due to the cross-Kerr coupling are considered.