Strongly coupled modes in a weakly driven micromechanical resonator

Abstract

We demonstrate a strong coupling between the flexural vibration modes of a clamped-clamped micromechanical resonator vibrating at low amplitudes. This coupling enables the direct measurement of the frequency response via amplitude-and phase modulation schemes using the fundamental mode as a mechanical detector. In the linear regime, a frequency shift of 0.8 Hz is observed for a mode with a line width of 5.8 Hz in vacuum. The measured response is well-described by the analytical model based on the Euler-Bernoulli beam including tension. Calculations predict an upper limit for the roomtemperature Q-factor of 4:5 Â 10 5 for our top-down fabricated micromechanical beam resonators. Nonlinear interactions between the vibration modes in micro-and nanomechanical resonators have attracted significant interest recently. In extensional structures, such as clamped-clamped bridges, the modes are coupled by the displacement-induced tension, 1 which yields a quadratic relation between the resonance frequency of the mode considered and the amplitudes of the other modes. In singly clamped cantilevers the displacement-induced tension is absent; here, the inextensionality condition couples the horizontal and vertical displacements of all modes, resulting in qualitatively similar dynamics. 2-5 Several applications and consequences have been put forward based on the modal interactions, such as enhancement of the dynamic range, 1 modification of the resonator damping by employing phonon-phonon cavities, 6,7 frequency stabilization, 8 the study of relaxation mechanisms, 9 and linear frequency conversion. 10 In thermal equilibrium, via the modal interactions the displacement fluctuations in one mode give rise to frequency fluctuations in the other modes, thus broadening their spectral line. In recent theoretic work these frequency fluctuations were quantified for a carbon nanotube, 11 yielding a boundary for the experimental Q-factor of such resonators at finite temperature. Experiments on suspended carbon nanotube resonators in the Coulomb blockade regime demonstrate that single-electron-tunneling processes provide a strong electrostatic coupling between the modes, in addition to the mechanical mode coupling. 12,13 While the recent experimental work has focused on the mode coupling in strongly driven resonators exhibiting nonlinear vibrations, in this letter we investigate these interactions in the low-amplitude regime. We demonstrate that the modal interactions play a significant role in the dynamic behavior of a linear resonator, as the vibrations of a weakly driven mode modulate the motion of a second vibration mode. We employ this coupling to perform swept-frequency type measurements of the linear frequency response of a high frequency (target) mode, by measuring the induced amplitude (AM) and phase modulation (PM) in a low frequency (probe) mode which is weakly driven at a fixed frequency. This provides a practical way to measure the frequency response, and it should be contrasted to the scheme presented earlier, 1 where the freq...

Spectral effects of dispersive mode coupling in driven mesoscopic systems

“…1, 7,8,16,17 The other is the decay of the "invisible" mode that is dispersively coupled to the studied mode. The double-peak structure of the driving-induced power spectrum sensitively depends both on the strength of the dispersive coupling and the parameters of the invisible mode.…”

“…The boundaries between the regions τ and τ ′ and the values ofk m are specified in Eq. (16). We enumerate the regions in the order of decreasing time, that is, the region τ < t ′′ < t corresponds to m = 1, etc.…”

“…Refs. 3,7,16, and 17 and can be used for quantum nondemolition measurements of the oscillator Fock states. 18,19 In many mesoscopic systems only one or a few modes can be directly accessed and controlled.…”

“…The approach does not require access to other modes, in contrast to Refs. 3,7,16, and 17, for example. It relies on the fact that, quite generally, frequency fluctuations lead to the features in the power spectrum of a periodically driven mode, which do not occur without such fluctuations.…”

Considering Nonlinearities to Design Micro System's Vibrational Sensitivity

“…Recently, more attention has drawn to increase the performance metrics of MEMS considering nonlinear behavior in the design process. Among others, the utilization of internal resonances [2], frequency hardening and softening [3] as well as parametric resonances [4] have been applied to increase the bandwidth, sensitivity and to reduce the signal to noise ratio driving the system in the nonlinear regime [5]. However, concepts of applying such an approach to SPM have been limited and a direct correlation to a sensitivity increase is missing.…”