Thermoelastic damping and frequency shift in micro/nano-ring resonators considering the nonlocal single-phase-lag effect in the thermal field

Estimating thermoelastic damping (TED) and frequency shift with sufficient accuracy is of importance for the quality factor prediction of micro/nano-resonators operating in certain special cases containing the ultra-high frequency excitation and the ultra-instantaneous thermal heating etc. At the design stage before the engineering application of micro/nano-resonators, the mathematical modelling of the frequency-shift ratio and TED gets challenging when considering the non-Fourier effect and size-dependent effect in the thermal field. In this paper, one/two-dimensional (1D/2D) analytical formulations of frequency-shift ratio and TED for micro/nano-ring resonators are firstly developed by adopting the nonlocal single-phase-lag (SPL) model. Basing on the present proposed models, the nonlocal thermal-relaxation time τN is firstly introduced in terms of the nonlocal thermal-length lQ and the thermal diffusion coefficient χ. In simulation cases, the typical semiconductor material silicon is selected, the SPL time τQ is computed basing on the second sound, and lQ refers to the mean-free path of energy carriers. The convergence performance of the proposed models is analyzed. Meanwhile, the results of existed TED models for rings are presented for comparison. The impacts of the nonlocal-SPL effect on the TED spectrum and fluctuation temperature are examined emphatically. And the peak phenomenon of TED spectrum characterized by the peak damping, the equivalent relaxation time associated with the peak frequency, and the frequency-shift ratio are explored. Results demonstrate that the thermo-dynamical behaviors of micro/nano-ring resonators are conjunctively determined by the amount relationship of τQ, τN, and the thermal relaxation time. Especially, when the radial thickness of the ring operating at ultra-high frequencies is comparable to lQ, the nonlocal-SPL effect of heat conduction should be considered to estimate the frequency-shift ratio and TED.