Nonlinear damping and dephasing in nanomechanical systems

We present a microscopic theory of nonlinear damping and dephasing of low-frequency eigenmodes in nanomechanical and micromechanical systems. The mechanism of the both effects is scattering of thermally excited vibrational modes off the considered eigenmode. The scattering is accompanied by energy transfer of $2\hslash {\omega }_0$ for nonlinear damping and is quasielastic for dephasing. We develop a formalism that allows studying both spatially uniform systems and systems with a strong nonuniformity, which is smooth on the typical wavelength of thermal modes but is pronounced on their mean free path. The formalism accounts for the decay of thermal modes, which plays a major role in the nonlinear damping and dephasing. We identify the nonlinear analogs of the Landau-Rumer, thermoelastic, and Akhiezer mechanisms and find the dependence of the relaxation parameters on the temperature and the geometry of a system.

Figure 1

Elementary phonon-phonon interaction processes leading to nonlinear damping. (a) Three-phonon scattering: two quanta of the mode of interest, which has frequency ${\omega }_0$, scatter into a phonon of a continuous spectrum with frequency ${\omega }_{\kappa }\approx 2{\omega }_0$. (b) Four-phonon scattering: a phonon of a continuous spectrum is scattered off the considered mode into another phonon, with the frequency change ${\omega }_{{\kappa }^{\prime }}-{\omega }_{\kappa }\approx 2{\omega }_0$. The modes $\kappa ,{\kappa }^{\prime }$ are not assumed to be plane waves.

Figure 2

The reciprocal effective quality factor for nonlinear decay $Q_{\mathrm{fl}}^{\left(\mathrm{nl}\right)}$ [Eq. (63)] for the fundamental flexural mode of a doubly clamped beam. The material parameters refer to silicon at room temperature, $D=1{\mathrm{cm}}^2{\mathrm{s}}^{-1}$. The ratio ${\omega }_0/{\mathrm{\Lambda }}_D$ can be controlled by varying the beam thickness $H$. The dashed lines show partial contributions from thermal diffusion modes, which are described by the individual terms in the sum in Eq. (62). The solid line includes all terms in the sum.

Figure 3

A part of the diagrams that describes resonant mode scattering for the correlation function $\langle b_{\kappa }^{†}{b}_{{\kappa }^{\prime }}{|}_{\omega }$ for small frequency $\omega$. The upper and lower lines refer to the modes $\kappa$ and ${\kappa }^{\prime }$, the arrow direction indicates the creation (right arrow) and annihilation (left arrow) operators in the correlation function. (a), (b) show the self-energy contribution that comes from resonant scattering of the $\kappa$ mode only, the diagrams for the mode ${\kappa }^{\prime }$ are similar. (c)–(e) show three types of the vertex terms that correspond to scattering processes in which both modes are involved.