Giant Enhancement in the Thermal Responsivity of Microelectromechanical Resonators by Internal Mode Coupling

We report on a giant enhancement in the thermal responsivity of doubly clamped $\mathrm{Ga}\mathrm{As}$ microelectromechanical-system- (MEMS) beam resonators by using the internal-mode-coupling effect. This is achieved by coupling the fundamental bending mode with the fundamental torsional mode of the MEMS-beam resonators through the cubic Duffing nonlinearity. In the mode-coupling regime, we find that when the input heat to the MEMS resonators is modulated at a particular frequency, the resonance-frequency shift caused by heating can be enhanced by almost 2 orders of magnitude. The observed effect is promising for realization of high-sensitivity thermal sensing by use of MEMS resonators, such as ultrasensitive terahertz detection at room temperature.

Figure 1

(a) The measurement setup. An ac voltage is applied to one of the piezoelectric capacitors to drive the beam and the induced resonant beam motion is monitored by a laser Doppler vibrometer. The motion signal is fed to a PLL circuit to provide feedback control to maintain self-oscillation. The PLL is also used as a lock-in amplifier to measure the resonance spectrum. (b) Measured resonance spectra of the first bending mode at various driving voltages (V_D = 10–50 mV) applied to the piezoelectric capacitor. (c) Measured spectrum for the first three modes (i.e., the first bending mode, the second bending mode, and the first torsional mode). The inset shows the mode shapes of the first bending mode and the first torsional mode calculated by the finite-element method. (d) Measured resonance spectra when V_D = 200, 300, 400 mV. (e) Enlargement of the spectrum in the mode-coupling region marked by a dotted rectangle in (d). Two clear drops in the amplitude appear in the spectrum at approximately 320.6 kHz and approximately 320.9 kHz, which are denoted as f_L and f_H, respectively. δf is the difference between f_L and f_H. SL, sacrificial layer.

Figure 2

Resonance frequency f_b for sample A measured as a function of the input heating power P at various values of V_D (283–424 mV). The red and black curves show the data when P is swept from 0 to 0.25 mW (forward) and from 0.25 to 0 mW (backward), respectively. The red and black arrows indicate the sweep directions in the measurements. The dashed red arrow indicates the position of the kink in f_b.

Figure 3

(a) Thermally induced frequency shift, Δf, for sample A as a function of the modulation frequency of the applied heat (approximately 25 nW) measured at various driving voltages, V_D. The black, blue, and red curves plot the results when V_D= 283, 354, and 424 mV, respectively. In the mode-coupling region (V_D= 354 and 424 mV), the frequency shift is vanishingly small when the heat modulation frequency is low (less than 100 Hz), but shows a large peak when the heat is modulated at a particular frequency (approximately 300 Hz). The black curve (V_D= 283 mV) shows the frequency shift measured outside the mode-coupling region. (b) Thermally induced Δf for sample B as a function of the modulation frequency of the applied heat (approximately 20 nW) measured at various values of V_D. The black and red curves plot the results when V_D= 650 mV (outside the mode-coupling region) and 820 mV (inside the mode-coupling region), respectively. Δf reaches its maximum when the heat is modulated at approximately 530 Hz.

Figure 4

(a) Waveform of the input heat pulses fed to the MEMS beam. The applied heat power is approximately 20 nW (ac amplitude) and is modulated at approximately 530 Hz. (b) Time trace of Δf measured when a small amount of heat is applied to sample B. The MEMS resonator is operated in the mode-coupling regime (V_D = 820 mV). (c) The beating effect between the f_L and f_H modes. In the mode-coupling condition, the bending mode and the torsional mode are renormalized into new coupled modes at frequencies f_L and f_H. The f_L and f_H modes consist of the bending component (A_b, $A_b^{\mathrm{\prime }}$) and the torsional component (A_t, $A_t^{\mathrm{\prime }}$). When the MEMS resonator is driven at f_L by our causing bending motion using the piezoelectric capacitor and, in addition, heat is applied periodically to the beam at frequency δf = f_H − f_L, the f_L mode coherently transfers its vibrational energy to the f_H mode through the parametric driving effect. The parametrically excited f_H mode generates a beating signal with the f_L mode, which periodically modulates the amplitude of the bending component.

Figure 5

(a) Thermally induced frequency shift in sample B measured in the mode-coupling region when the heating power P = 2, 4, 6, 8, 10, 16, and 20 nW and the heat modulation frequency, f_m, is swept from 450 to 600 Hz. (b) Peak frequency shift Δf in the mode-coupling region as a function of P. The dotted line indicates a linear fit to Δf for P ≤ 10 nW to estimate the thermal responsivity. (c) Spectral density of frequency noise, n_f, measured when V_D= 650 mV (outside the mode-coupling region; black) and when V_D= 820 mV (in the mode-coupling region; red). (d) NEP as a function of f_m when V_D = 650 mV (outside the mode-coupling region; black) and when V_D= 820 mV (in the mode coupling region; red). The dotted red line indicates the thermal fluctuation limit for NEP.

Figure 6

Fig. 6 Measured resonance spectrum of sample B for the first three modes (i.e., the first bending mode, the second bending mode, and the first torsional mode). (b) Enlargement of the region marked by a dotted rectangle in the spectrum shown in (a). Two clear drops in amplitude appear in the spectrum at approximately 316.5 kHz and approximately 317 kHz, which are denoted as f_L and f_H, respectively. δf is the difference between f_L and f_H. (c) Phase spectra measured at various driving voltages (V_D = 600, 700, and 800 mV). The phase sweep is performed with a PLL. (d) Resonance spectra measured at various driving voltages (V_D = 600, 700, and 800 mV). (e) Enlargement of the mode-coupling region marked by a dotted box in (d). Drops in the resonance amplitude are observed.