Limits to Thermal-Piezoresistive Cooling in Silicon Micromechanical Resonators

We study thermal-piezoresistive cooling in silicon micromechanical resonators at large currents and high temperatures. Crossing a thermal transition region corresponds to a steep reduction in resonance frequency, an abrupt plateauing in the effective quality factor, and a large increase in thermomechanical fluctuations. Comparing measurements with simulations suggests that the second-order temperature coefficients of elasticity of doped silicon are not sufficient to capture the drop in resonance frequency at large currents. Overall, our results show that there are clear thermal limits to cooling a resonant mode using current-controlled thermal-piezoresistive feedback in silicon.

SECTION I. Introduction

Micro- and nano- electromechanical (MEM/NEM) resonators have wide utility as resonant sensors [1], [2], oscillators [3], and filters [4]. The quality factor ($Q$ ), or inverse damping of a resonator, is perhaps the most important property of these systems since it determines the dynamic response and thermomechanical signal-to-noise ratio (SNR). A wide range of phenomena are under investigation for tuning the effective quality factor ($Q_{\textit {eff}}$ ) and inducing oscillations in micro- and and nano- scale resonators [5]–​[12]. Feedback-based schemes can suppress $Q_{\textit {eff}}$ , improving the resonator bandwidth [13]–​[16], or enhance $Q_{\textit {eff}}$ to the onset of self-sustained oscillations. One such mechanism utilizes the thermal-piezoresistive effect, whereby flowing a direct current through thermal actuator beams connected to the vibrating geometry exerts feedback on the mode via the actuator piezoresistance and thermal expansion coefficient [17]–​[19]. Compared to the other feedback mechanisms in MEM/NEM resonators, the thermal-piezoresistive effect is distinguished by its ease of implementation; flowing a simple direct current through an appropriately designed resonator can enhance or suppress $Q_{\textit {eff}}$ , relying on a thermo-electro-mechanical feedback mechanism intrinsic to the device. If a direct current flows through actuators with a negative longitudinal piezoresistive coefficient, the thermal-piezoresistive effect amplifies the vibrations and increases $Q_{\textit {eff}}$ . Thermal-piezoresistive $Q_{\textit {eff}}$ enhancement is finding applications in oscillators for e.g. mass-sensing [20]–​[23], tuning resonator nonlinearities [24], pre-amplifying signals in resonant sensors [25], [26], and nanoscale radio-frequency amplifiers [27], [28]. Work is ongoing to increase the operating frequency [29], [30], reduce the power consumption [31], and increase the frequency stability [32] of thermal-piezoresistive oscillators and sensors.

If the sign of the thermal-piezoresistive effect is reversed, the feedback acts to suppress the effective quality factor. The sign change can be achieved by utilizing current control in a p-type doped resonator [19], or tuning the lumped electrical parameters in the biasing circuit [17], [33]. The reversed feedback suppresses the thermal fluctuations of the mode, effectively cooling it. Thermal-piezoresistive cooling offers a scheme for simple real-time adjustments in resonant sensor bandwidth [34]. This technique has potential for temporarily and reversibly switching resonant sensors and oscillators into a “safe” mode in preparation for high acceleration shocks, whereby an integrated sensor flows a direct current through the thermal-piezoresistive actuators prior to impact, suppressing $Q_{\textit {eff}}$ to a fraction of the intrinsic $Q$ , thus mitigating the risk for pull-in associated with high $Q$ structures [35], [36]. An important consideration with thermal-piezoresistive cooling is the limiting behavior at large currents and high temperatures, since these limits must be taken into account during device design to achieve some desired $Q_{\textit {eff}}$ reduction in practice.

This work explores the limiting thermal behavior of thermal-piezoresistive cooling at high temperatures, highlighting the importance of boundary conditions in practice. At large currents, the Joule heating in the resonator results in substantial device heating, with the heating and corresponding temperature rise concentrated in one or more narrow actuator beams where the electrical and thermal resistance is highest. At the large currents typical of the thermal-piezoresistive effect, the current density through the thermal actuator can approach [37], [38] or exceed [39], [40] the ≈1 GA/m² breakdown current density in silicon, and the properties in the thermal actuator govern the device behavior. The elastic moduli of silicon decrease with temperature, which results in a reducing actuator stiffness with current [41]–​[43]. In both n-type-doped and p-type-doped silicon, the electrical conductivity and thermal conductivity both decrease substantially with temperature [44], [45]. These three effects feed on each other at large currents to substantially reduce the extensional stiffness of the thermal actuator, as depicted in Fig. 1(b), thus resulting in a highly nonlinear temperature dependence of resonance frequency. Thermal-piezoresistive resonators exhibit several interesting effects at these large currents.

Fig. 1.

(a) The measurement setup for the thermal-piezoresistive resonators. Both anchors of the device are voltage-biased at $V_{b}=60$ V for capacitive transduction, and a direct current $I_{dc}$ additionally flows through the structure, raising the voltage of the actuator beam anchor with respect to the support beam anchor. The resonator is driven near resonance with $V_{ac}\approx1$ mV and sensed using a transimpedance amplifier (TIA) connected to the opposing anchor. A $500~\Omega$ resistor is connected in-series with the output of the current source. (b) A depiction of the large current behavior in the actuator of a thermal-piezoresistive resonator. (c) A scanning electron microscope (SEM) image of the device layer of a fabricated thermal-piezoresistive resonator prior to encapsulation. (d) A chip “floated” above the substrate with wirebonds.

SECTION II. Method and Results

We investigate the limits to thermal-piezoresistive cooling in the cantilevered thermal-piezoresistive resonators depicted in Fig. 1(a, c). The devices are fabricated in a wafer-scale encapsulation process, which produces stable silicon microresonators in an ultra-clean vacuum environment [46], [47]. A fabricated device is shown before and after encapsulation in Figs. 1(c) and 1(d), respectively. The high temperature annealing step inherent to this fabrication process makes our devices well-suited for studying large current effects. We study p-type, moderately boron doped ($N_{a} \approx 3.4\times 10^{18}$ cm⁻³) resonators, with the actuator beam aligned with the $\left < { 100}\right >$ -direction. The relatively large electrical resistivity corresponds to a large Joule heating, while the longitudinal piezoresistive coefficient in the $\left < { 100}\right >$ direction is low so the thermal-piezoresistive feedback is relatively weak compared to devices fabricated in the $\left < { 110}\right >$ direction [48]. We previously experimentally and theoretically characterized the low-current thermal-piezoresistive behavior for thermally-grounded devices with a variety of other dopings and orientations [19]. The devices in this study are thermally “floated” above the bond pads with $50~\mu \text{m}$ diameter wire-bond wires [49], as depicted in Fig. 1(d), which increases the thermal resistance to the ambient temperature, thus enabling a larger temperature rise in the thermal actuators for a given current. Using thermally-floated devices with relatively weak thermal-piezoresistive feedback enables us to access the limiting thermal behavior. The device layer thickness is $40~\mu \text{m}$ , and the proof mass has a length and width of $360~\mu \text{m}$ and $100~\mu \text{m}$ , respectively. The proof mass is supported by a $12~\mu \text{m}$ wide support beam and a 3 $\mu \text{m}$ wide actuator beam. We measure the fundamental in-plane cantilever mode of the “long-beam” devices with an actuator/support beam length of $50~\mu \text{m}$ and “short-beam” devices with a $20~\mu \text{m}$ actuator/support beam length. We use the ring-down technique to extract the $Q_{\textit {eff}}$ and angular resonance frequency, $\omega _{0}$ , as a function of direct current, since this approach can quickly and accurately obtain the resonator dynamical properties [50]. At each direct current value, a harmonic drive is applied near the resonance frequency, and then shut off while monitoring the vibrations. An exponential function is fit to the decaying amplitude response for each direct current to extract $Q_{\textit {eff}}$ , and a Fast Fourier Transform (FFT) is utilized to extract $\omega _{0}$ from the decaying vibrations.

Figure 2 presents the measured effective quality factor and resonance frequency versus direct current, $I_{dc}$ , in devices with an actuator/support beam length of $50~\mu \text{m}$ and $20~\mu \text{m}$ . In both devices, the resonance frequency and effective quality factor decrease with increasing current. In the long-beam device at 30 mA, the resonance frequency drops off precipitously while the $Q_{\textit {eff}}$ plateaus. Increasing the direct current beyond 30 mA results in the slope of the resonance frequency becoming linear with current about a new low frequency state. The behavior in the short beam device is similar, but the current at which the frequency drops off increases to 45 mA.

Fig. 2.

(a) The measured resonance frequency ($\omega_{0}/2\pi$ ) of the $50~\mu\text{m}$ actuator device for increasing (downward triangles) and decreasing (upward triangles) direct current $I_{dc}$ , with the predicted resonance frequency (solid line). Inset: the corresponding measured and simulated effective quality factor ($Q_{\textit{eff}}$ ), with the simulated contributions from thermoelastic dissipation ($Q_{TED}$ ) and thermal-piezoresistive pumping ($Q_{TPP}$ ). (b) The same measurements and simulations as in (a), except with the $20~\mu\text{m}$ actuator device.

The abrupt drop in resonance frequency and plateauing in $Q_{\textit {eff}}$ at large direct current corresponds to a thermal transition in the resonators. We confirm that the transition is a thermal phenomenon with hysteresis measurements and direct thermal infrared (IR) imaging of the chip. For the hysteresis measurements, we measure the resonance frequency while sweeping from low to high direct current, as well as high to low direct current, without giving the device sufficient time to settle to its steady-state temperature at each current. In the thermal transition region, e.g., at $I_{dc} \approx 30$ mA in Fig. 2(a), the resonance frequency is lower for the downward current sweep than the upward current sweep. This indicates that the very large shift in resonance frequency in the transition region likely arises from thermal effects. Giving the system sufficient time to equilibrate reduces the frequency hysteresis between the upward and downward current sweeps. The thermal transition region occurs at a larger current for the short-beam device than the long-beam device perhaps because of the lower total electrical and thermal resistance of the shorter actuator beam,

which would correspond to a lower actuator temperature for a given direct current.

We employ a FLIR A615 thermal IR camera with a 25 $\mu \text{m}$ resolution lens to image the chip surface temperature as a function of direct current through the long-beam device. These measurements are presented in Fig. 3, and reveal that in the thermal transition region, the chip surface temperature departs from a simple quadratic dependence on direct current. Plotting the shift in resonance frequency for the long-beam device versus measured chip temperature in Fig. 3(d) illustrate two regimes with differing slopes where the change in resonance frequency is proportional to the change in chip surface temperature, demarcated by a thermal transition region. We develop finite-element (FE) simulations based on COMSOL Multiphysics to model $\omega _{0}$ and $Q_{\textit {eff}}$ at various $I_{dc}$ levels in Fig. 2. The FE simulation is divided into two steps. In the first step, a stationary analysis is performed to obtain the elevated temperature profile ($T_{dc}$ ) due to Joule heating, as well as the direct current density ($J_{dc}$ ). Given that $T_{dc}$ can significantly exceed the anchor temperature at high $I_{dc }$ (over 100 K), the temperature-dependent material properties (i.e., resistivity [45] and thermal conductivity [51]) are used in our model to predict $T_{dc}$ , hence $\omega _{0}$ and $Q_{\textit {eff}}$ versus $I_{dc}$ . We additionally compute the deformation of the device induced by thermal expansion at $T_{dc}$ , and treat this as the pre-stressed condition for the subsequent analysis. In the second step, the eigenfrequency analysis is performed to simulate $\omega _{0}$ as well as the thermoelastic damping (TED) and the thermal-piezoresistive pumping (TPP). $Q_{TED}$ and $Q_{TPP}$ are the two major loss mechanisms for these devices. Based on the simulated $T_{dc}$ and the temperature coefficients of elasticity (TCEs), the $\omega _{0}$ shift at various $I_{dc}$ values can be obtained. The TCEs up to the second order are adopted from [52]. For the TED simulation, the heat source, thermal conductivity and elastic moduli are set to be $T_{dc}$ -dependent so that the $Q_{TED}$ varies with $I_{dc}$ . For the TPP simulation, we adapt the approach that we reported in [19].

Fig. 3.

(a) A thermal infrared (IR) image of the thermally “floated” chip with no direct current in a $25^{\circ}\text{C}$ ambient environment, with the corresponding temperature scale. The emissivity is calibrated to the top silicon nitride passivation layer (dark gray color) of the chip, and thus the measured temperature for the aluminum bond pads and gold substrate (light gray color) erroneously differ from the ambient temperature. (b) The IR image when 40 mA of direct current flows through the $L=50~\mu\text{m}$ device (white arrow), as in Fig. 2(a). (c) The measured chip surface temperature ($T_{s}$ ) directly above the $L=50~\mu\text{m}$ device as a function of direct current. A quadratic model, $T_{s} = T_{s,0} + c I_{dc}^{2}$ , for some constant $c$ , is fit to the low current behavior to illustrate the thermal transition. The actual device temperature is much higher than the chip surface temperature. (d) The measured resonance frequency offset $\Delta\omega_{0}/2\pi=\left[{\omega_{0}(I_{dc})-\omega_{0}(0 \textrm{mA} )}\right]/2\pi$ versus the change in surface temperature, $\Delta T_{s}=T_{s}(I_{dc})-T_{s}(0 \textrm{mA} )$ , for the measurement in (c). Two linear models are fit to the resonance frequency below and above the thermal transition region.

The predicted resonance frequency and effective quality factor from our model is overlaid on the measurements in Fig. 2. For both devices, the models reproduce the large shift in resonance frequency up to moderate currents, and agree with $Q_{\textit {eff}}$ up to the thermal transition region. The model does not capture the steep drop in resonance frequency, or the plateauing in $Q_{\textit {eff}}$ beyond the thermal transition current. We hypothesize that at this point, the actuator material properties become extremely nonlinear, and the TCEs up to second-order and extracted based on a relatively small temperature range (-40 to 85°C) [52] are not sufficient to reproduce the behavior.

The plateauing in the quality factor at the thermal transition region results from the resonator temperature, as opposed to the current density through the actuator. To confirm this, we perform additional measurements of the resonance frequency and quality factor of the long-beam device for chamber temperatures varying from −20°C to 70°C, as presented in Fig. 4. The chamber temperature is increased in steps from −20°C to 70°C, and the device is given thirty minutes to thermally equilibrate at each chamber temperature before performing the measurements over $I_{dc}$ . The device $Q$ at low currents decreases monotonically with increasing device temperature, providing a clear indication that this device is TED-limited. Increasing the chamber temperature shifts the temperature hysteresis regime to lower direct currents, as both the resonance frequency and quality factor measurements reveal. For all chamber temperatures, the quality factor plateaus to the same constant level. The plateauing in the quality factor appears to be a thermal phenomenon, but its exact origins remain unclear. The simulations in Fig. 2 suggest that thermal-piezoresistive pumping only influences the quality factor at large currents, beyond the thermal transition region. Neither the temperature-dependent $Q_{TED}$ or $Q_{TPP}$ simulations predict the observed plateauing in $Q_{\textit {eff}}$ . Thermoelastic dissipation likely saturates at the high device temperatures, but it is difficult to prove this with the present measurements.

Fig. 4.

(a) The measured resonance frequency ($\omega_{0}/2\pi$ ) of the $L = 50~\mu\text{m}$ device for increasing (downward triangles) and decreasing (upward triangles) direct current $I_{dc}$ , for an ambient temperature of −20°C, 0°C, 25°C, 50°C, and 70°C (light to dark shade). Inset: the corresponding measurements of the effective quality factor ($Q_{\textit{eff}}$ ). (b) The resonance frequency offset $\Delta\omega_{0}/2\pi=\left[{\omega_{0}(I_{dc})-\omega_{0}(0 \textrm{mA} )}\right]/2\pi$ versus direct current, in the hysteresis regime. (c) The resonance frequency difference $\delta\omega_{0}/2\pi=\left[{\omega_{0\uparrow}(I_{dc})-\omega_{0\downarrow}(I_{dc})}\right]/2\pi$ between the upward and downward sweep as a function of direct current.

The steep drop in resonance frequency at the thermal transition region results from a change in the mechanical stiffness of the resonator, as opposed to a change in the electrostatic softening. To confirm this, we measure the resonance frequency and quality factor as a function of device bias voltage for the $L = 50~\mu \text{m}$ and $L = 20~\mu \text{m}$ devices. Figure 5(a) plots $Q_{\textit {eff}}$ and the change in resonance frequency for bias voltages varying from $V_{b} = 20$ V to $V_{b} = 60$ V. Increasing the bias voltage increases the slope of the drop in resonance frequency above the thermal transition region, but has very little impact on the current at which the resonance frequency starts to abruptly decrease. Separate FE simulations suggest that at large currents, the proof mass shifts towards one of the side electrodes, increasing the corresponding electrostatic softening. Fig. 5(a) suggests that electrostatic softening plays a role in the long-beam device resonance frequency, but only for currents beyond the thermal transition. The measurements of the short-beam device in Fig. 5(b) seem to suggest that electrostatics play a minimal role in the device behavior. The negligible impact of electrostatics in Fig. 5(b) can be explained by the increased mechanical stiffness in the $L = 20~\mu \text{m}$ device compared to the $L = 50~\mu \text{m}$ device. For both devices, the bias voltage voltage has a negligible impact on the thermal transition current and the quality factor. This provides additional evidence that the $Q_{\textit {eff}}$ results from a thermal effect, as opposed to another source such as electrical damping [53], [54]. Below $V_{b} = 40$ V, the short-beam resonator has too poor of a capacitive readout signal to establish a PLL and perform ringdown measurements.

Fig. 5.

(a) The measured resonance frequency offset $\Delta\omega_{0}/2\pi=\left[{\omega_{0}(I_{dc})-\omega_{0}(0)}\right]/2\pi$ for the $50~\mu\text{m}$ actuator device as a function of direct current $I_{dc}$ , and different bias voltages, $V_{b}$ . Inset: the corresponding measured effective quality factor $Q_{\textit{eff}}$ versus direct current. (b) The same measurements and simulations as in (a), except with the $20~\mu\text{m}$ actuator device.

To provide additional support for the conclusion that the resonance frequency and quality factor behavior result mainly from thermal effects instead of current effects, we performed ringdowns on the long-beam device except after shutting the direct current off for pre-defined intervals, as shown in Fig. 6. At each direct current value, the PLL establishes lock on the resonator, and the direct current is shut off while keeping the harmonic drive on and maintaining lock. After the pause, the harmonic drive is shut off and the ringdown envelope is measured to extract $\omega _{0}/2\pi $ and $Q_{\textit {eff}}$ . We repeat the entire current sweep up to 40 mA for pause times of 0 seconds, then 1 second, and finally 2 seconds. At each direct current value, the resonance frequency in Fig. 6(a) increases very abruptly to near its thermal equilibrium value after shutting off the direct current (${\ll }1$ ms timescale) followed by a more gradual increase (${\approx }1$ s timescale). The quality factor in Fig. 6(b) increases at a slower timescale than the initial drop in resonance frequency. These experiments suggest that the device temperature decreases non-uniformly after shutting off the direct current: the fast actuator/support beam timescale and the slow timescale for the anchors and chip. The resonance frequency is determined by the stiffness in the actuator/support beams and thus the beam temperatures, therefore following their ${\approx }10~\mu \text{s}$ thermal timescale [38]. The resonator quality factor is determined more broadly by the phonon scattering between the mode of interest and the thermal bath, and thus follows the slower timescale of the anchors and the entire floated chip.

Fig. 6.

(a) The measured resonance frequency ($\omega_{0}/2\pi$ ) of the $L = 50~\mu\text{m}$ device for increasing direct current $I_{dc}$ , with the direct current on ($I_{dc}$ ) and off (0 to 3 s) during the ringdown. Inset: the corresponding measurements of the effective quality factor ($Q_{\textit{eff}}$ ).

We finally consider the noise dynamics of the thermal-piezoresistive resonator in the vicinity of the thermal transition region. We turn off the harmonic drive, and monitor the thermomechanical vibrations directly using a low noise capacitive sensing scheme [55]. A spectrum analyzer is used to measure the scalar spectrum in the vicinity of the resonance frequency, and the noise is simultaneously demodulated at resonance with a lock-in amplifier to extract the two voltage quadratures at a given direct current. Fig. 7 summarizes these measurements for the long-beam device. At $I_{dc} = 0$ mA, the system is in thermal equilibrium, and the demodulated noise consists of the Gaussian noise contributions from the thermomechanical vibrations of the mode and the amplifier Johnson noise. At moderate current, the resonance linewidth increases as $Q_{\textit {eff}}$ decreases. When the direct current is increased beyond the thermal transition region, the thermomechanical noise spectrum near resonance increases while maintaining the same broad linewidth, and the variance in the thermomechanical noise cloud grows dramatically. The substantial increase in demodulated thermomechanical noise near resonance beyond the thermal transition region could result from a steep increase in the device temperature and corresponding thermomechanical noise force.

Fig. 7.

The (a, c, e, g) amplitude spectral density (ASD) of the thermomechanical noise spectrum for the $L = 50~\mu\text{m}$ device as function of frequency detuning $\Delta\omega=\omega-\omega_{0}$ away from resonance, and the (b, d, f, h) corresponding demodulated noise at resonance, for increasing direct current $I_{dc}$ . $v_{x}$ and $v_{y}$ correspond to the two voltage quadratures of the noise, as measured using a lock-in amplifier. The harmonic drive is shut off for these measurements.

SECTION III. Conclusion and Future Work

These experiments and simulations reveal the limiting thermal behavior that can impact thermal-piezoresistive cooling in the large current regime: a steep reduction in resonance frequency, an abrupt plateauing in the effective quality factor, and a large increase in thermomechanical fluctuations. Additional experiments and simulations are required to fully understand the highly nonlinear behavior in the thermal transition region. However, our work highlights the importance of chip-scale boundary conditions and their impact on device performance. Thermal-piezoresistive cooling is one application that would benefit from third-order and higher temperature coefficients of elasticity in doped silicon [52], [56]. The experiments reported here could be repeated in non-encapsulated devices, and the mechanical vibrations and microscale heat transport can be imaged at large currents using laser Doppler vibrometry and time-domain thermoreflectance, respectively [57], [58]. Large current experiments could be interesting in nanowire-based thermal-piezoresistive resonators [33], [34]. The behavior of silicon beams at large currents and high temperatures has additional practical implications in micro- tether design for improving inertial sensor fabrication yields [59].

ACKNOWLEDGMENT

The author James M. L. Miller is grateful to Tanya Liu, Farid Soroush, and Woosung Park for assisting with the thermal infrared camera and software. The authors are grateful for the helpful suggestions provided by the anonymous reviewers. The data that support the findings of this study are available from the corresponding authors upon reasonable request.