Effective quality factor tuning mechanisms in micromechanical resonators

Miller, James M. Lehto; Ansari, Azadeh; Heinz, David B.; Chen, Yunhan; Flader, Ian B.; Shin, Dongsuk D.; Villanueva, L. Guillermo; Kenny, Thomas W.

doi:10.1063/1.5027850

Quality factor (Q) is an important property of micro- and nano-electromechanical (MEM/NEM) resonators that underlie timing references, frequency sources, atomic force microscopes, gyroscopes, and mass sensors. Various methods have been utilized to tune the effective quality factor of MEM/NEM resonators, including external proportional feedback control, optical pumping, mechanical pumping, thermal-piezoresistive pumping, and parametric pumping. This work reviews these mechanisms and compares the effective Q tuning using a position-proportional and a velocity-proportional force expression. We further clarify the relationship between the mechanical Q, the effective Q, and the thermomechanical noise of a resonator. We finally show that parametric pumping and thermal-piezoresistive pumping enhance the effective Q of a micromechanical resonator by experimentally studying the thermomechanical noise spectrum of a device subjected to both techniques.

I. INTRODUCTION

Micro- and nano-electromechanical (MEM/NEM) resonators are critical for timing references,¹ frequency sources,² atomic force microscopes (AFMs),³ gyroscopes,^4,5 and mass sensors.^6,7 The mechanical quality factor (⁠ $Q$ ⁠) of a MEM/NEM resonator is perhaps its most important property and is a measure of the energy decay rate in each cycle of vibrations. The higher the resonator $Q$ ⁠, the longer that coherent energy will remain in the mode prior to leaking into the environment. $Q$ is related to the thermomechanical displacement noise of a resonator, which is important for designing force sensors with a high signal-to-noise ratio (SNR), oscillators with low phase noise, and filters with large noise rejection. In force sensing applications, an increase in $Q$ of the resonator will improve the thermomechanical SNR.^8,9 In oscillators, the phase noise in the thermomechanical-noise-limited regime is inversely proportional¹⁰ to $Q$ ⁠. In cellular communication, resonators with high $Q$ factors are widely used as acoustic filters with sharp skirts. This has justified several decades of research into increasing the quality factor of resonators by reducing the underlying dissipation.

Instead of engineering the underlying dissipation in a MEM/NEM resonator, it is also possible to feed energy into or out of the mode to increase or decrease the decay rate. This artificially modifies the resonator dynamics in a similar manner to a change in $Q$ ⁠, without modifying the fluctuations that accompany the actual dissipation. We use the term “effective quality factor,” $Q_{e f f}$ ⁠, to distinguish this from changing the actual dissipation of the resonator. Several mechanisms have been demonstrated for enhancing or suppressing $Q_{e f f}$ of a MEM/NEM resonator. External feedback control,¹¹ optical pumping,¹² mechanical pumping,¹³ thermal-piezoresistive pumping,¹⁴ and parametric pumping¹⁵ are well-known methods that modify $Q_{e f f}$ by supplying an external time-varying (ac) or static (dc) energy source to the resonator. These techniques are used in resonators ranging in size from ton-scale gravitational wave detectors¹⁶ down to individual carbon nanotubes.¹⁷

This review discusses effective quality factor tuning mechanisms in MEM/NEM resonators. It first compares $Q$ and $Q_{e f f}$ tuning in terms of the thermal noise spectrum and mean-squared displacement noise of a resonator and differentiates between phase-dependent and phase-independent $Q_{e f f}$ tuning. It discusses the common techniques for measuring $Q$ of a resonator and illustrates how $Q$ versus $Q_{e f f}$ affects the phase noise in oscillators, the bandwidth and SNR in resonant sensors, and the phonon occupancy of cavity-cooled modes. Each $Q_{e f f}$ mechanism is delineated, with a summary of the corresponding experimental MEM/NEM literature. We conclude with experiments on a micromechanical resonator that compare degenerate parametric pumping, a phase-dependent $Q_{e f f}$ tuning mechanism, with thermal-piezoresistive pumping, a phase-independent $Q_{e f f}$ tuning mechanism, in terms of the resonator's transfer function, phase slope, and thermomechanical displacement noise.

II. THE QUALITY FACTOR AND THE THERMOMECHANICAL NOISE SPECTRUM

$Q$ of a resonator is defined as the ratio of the stored energy over the dissipated energy per vibration cycle¹⁸

Q = 2 π \frac{E_{stored}}{E_{dissipate d}} .

(1)

Equation (1) suggests that the $Q$ factor is inversely proportional to the losses in a resonant system, if no external pump adds or removes energy from the motion. Several dissipation-induced losses affect $Q$ in a MEM/NEM resonator

\frac{1}{Q} = \frac{1}{Q_{g a s}} + \frac{1}{Q_{T E D}} + \frac{1}{Q_{p h - p h}} + \frac{1}{Q_{surface}} + \dots,

(2)

where $Q_{g a s}$ is due to the collision of gas molecules with the resonator,¹⁹^, $Q_{T E D}$ is due to thermoelastic dissipation caused by strain-gradient-induced heat transfer,²⁰^, $Q_{p h - p h}$ comprises the Landau-Rumer damping and Akhieser damping and is due to the coupling between the resonant mode and the phonon bath,²¹ and $Q_{surface}$ is the damping due to surface loss mechanisms such as surface roughness or adsorbates and is often the dominant dissipation mechanism in nanomechanical resonators.²²

The quality factor can be described in terms of the linear damping, $b$ ⁠, in the mass-spring-damper equation or the resistance, $R$ ⁠, in the resistor-inductor-capacitor (RLC) equation. For a small vibration amplitude, the parameters of the resonant mode do not depend on the amplitude of vibration and the equation of motion remains in the linear regime. If we additionally assume velocity-proportional damping, we can model the displacement amplitude, $x$ ⁠, of a single mechanical mode of a resonator with the lumped mass-spring-damper model

m \ddot{x} + b \dot{x} + k x = f_{drive} + f_{t h},

(3)

where $m$ is the lumped mass, $b$ is the linear damping coefficient, $k$ is the lumped stiffness, $f_{drive}$ is the external driving force, and $f_{t h}$ is the thermomechanical noise force. The resonator natural frequency is given in terms of the lumped stiffness and mass as $ω_{0} = \sqrt{k / m}$ ⁠. The damping coefficient is the direct measure of linear dissipation in the mechanical mode and is inversely related to the mechanical quality factor, $Q$ ⁠, as

b = \frac{m ω_{0}}{Q} = \frac{\sqrt{k m}}{Q} .

(4)

The damping constant in Eqs. (3) and (4) assumes linear dissipation: $Q$ is strictly a linear parameter and does not depend on the amplitude. This is reasonable for MEM/NEM resonators at small amplitudes but breaks down for very small devices²³ or very large amplitudes²⁴ due to nonlinear damping.²⁵

$Q$ of a resonator can be increased by reducing the linear dissipation (reducing $b$ ⁠) or by increasing $\sqrt{k m}$ ⁠. For constant $\sqrt{k m}$ ⁠, $Q$ can only be modified by changing the dissipation. Engineering the anchor damping,²⁶ reducing the gas pressure²⁷ to increase $Q_{g a s}$ and reducing the operating temperature²⁸ to increase $Q_{T E D}$ are some of the many ways to increase $Q$ by reducing the dissipation. Increasing $\sqrt{k m}$ will also increase $Q$ ⁠. Liang et al. and Hoof et al. showed experimentally that by increasing $m$ while holding $k$ and $b$ constant, $Q$ increased.^29,30 Krause et al. showed that increasing $\sqrt{k m}$ and $b$ proportionally caused $Q$ to remain constant.³¹ Many works explore “dissipation dilution” of mechanical modes, whereby increasing $k$ while holding $m$ and $b$ constant causes $Q$ to increase.^32–35

For a mode in thermal equilibrium at a sufficiently high temperature (⁠ $k_{B} T ≫ ℏ ω_{0}$ ⁠), the position and momentum can be calculated from the partition function³⁶ to each have an equipartition energy, $\bar{E} = \frac{1}{2} k_{B} T$ ⁠, where $T$ is the temperature, $k_{B}$ is Boltzmann's constant, and $ℏ$ is Planck's constant. The overbar is used to denote the expectation value. Since the resonator is not driven when it is in thermal equilibrium, the mean noise displacement, ${\bar{x}}_{n}$ ⁠, and mean noise velocity, $\bar{\dot{x_{n}^{2}}}$ ⁠, are zero. By separately equating half of the total resonator equipartition energy to the average energy stored in the stiffness and mass of the mode

\bar{E} = \frac{1}{2} k_{B} T = \frac{1}{2} k \bar{x_{n}^{2}} = \frac{1}{2} m \bar{\dot{x_{n}^{2}}},

(5)

we compute the mean-squared displacement noise

\bar{x_{n}^{2}} = \frac{k_{B} T}{k},

(6)

and the mean-squared velocity noise

\bar{\dot{x_{n}^{2}}} = \frac{k_{B} T}{m} .

(7)

In thermal equilibrium, it is therefore the lumped stiffness and lumped mass of the resonator that determine the standard deviation in the thermal noise displacement and velocity, respectively. Even though the thermal noise force decreases with increasing $Q$ ⁠, the mean-squared noise displacement, $\bar{x_{n}^{2}}$ ⁠, and the mean-squared velocity, $\bar{\dot{x_{n}^{2}}}$ ⁠, of the resonator are not influenced by the quality factor. This is because the integrated area under the thermomechanical displacement and velocity power-spectral-densities (PSD) is constant at a given temperature.

The fluctuation-dissipation theorem (FDT) links the damping, $b$ ⁠, to the thermomechanical noise force, $f_{t h}$ ⁠, when the resonator is sufficiently close to thermal equilibrium.³⁷ $f_{t h}$ is treated as a stationary random process with an autocorrelation function, $K_{f} (τ)$ ⁠, given by³⁸

K_{f} (τ) = \bar{f (t) f (t + τ)} = 4 k_{B} T b δ (t - τ),

(8)

where $δ (t - τ)$ is the Dirac delta. The Wiener-Khintchine theorem asserts that the autocorrelation function of a stationary random variable and its PSD form a Fourier transform pair and can be used to obtain the thermomechanical noise force PSD as³⁹

F_{t h}^{2} (ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} K_{f} (τ) e^{i ω τ} d τ = 4 k_{B} T b,

(9)

where we use the property that the Fourier transform of the Dirac delta is unity. The thermal noise force therefore has a white amplitude-spectral-density (ASD) related to $Q$ as

F_{t h} = \sqrt{4 k_{B} T \frac{{m ω}_{0}}{Q}} .

(10)

The dissipation, and thus $Q$ ⁠, determines the thermal noise force, $F_{t h}$ ⁠, in the resonator. From the FDT, a larger dissipation (a lower $Q$ ⁠) corresponds to a larger fluctuation (a larger thermal noise force) when the resonator is in thermal equilibrium with the ambient environment. The thermal noise in a MEM/NEM resonator at a finite temperature is due to the interaction between the single degree of freedom, $x$ ⁠, associated with the mode of interest and the many other degrees of freedom in the system. In a resonator driven only by the thermal noise force, $x$ is a stationary random process with an autocorrelation function $K_{x} (τ) = \bar{x (t) x (t + τ)}$ ⁠. The displacement PSD is

X^{2} (ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} K_{x} (τ) e^{i ω τ} d τ .

(11)

The mechanical transfer function of the resonator, $H (ω)$ ⁠, shapes the spectrum by filtering the thermal noise force⁴⁰

X (ω) = |H (ω)| \times |F_{t h} (ω)|,

(12)

where $X (ω)$ is the displacement ASD. We substitute $X (ω) e^{i ω t}$ into Eq. (3) and set $f_{drive} = 0$ to obtain the displacement ASD due to the thermal noise force as

X (ω) = \frac{\sqrt{4 k_{B} T (\frac{ω_{0}}{m Q})}}{\sqrt{{({ω_{0}}^{2} - ω^{2})}^{2} + {(\frac{ω ω_{0}}{Q})}^{2}}} .

(13)

Figure 1 plots Eq. (13) near a resonance. As the $Q$ of the mode increases, the thermomechanical displacement ASD decreases for frequencies away from resonance and increases at resonance.

FIG. 1.

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Normalized thermomechanical displacement amplitude-spectral-density (ASD) in a linearly damped resonator versus normalized frequency, $ω / ω_{0}$ ⁠, at different $Q$ values, as in Eq. (13). Increasing $Q$ causes the ASD to increase for frequencies near resonance and to decrease for frequencies away from resonance.

At frequencies near and above resonance, the resonator thermal displacement noise ASD depends upon frequency. Sufficiently below resonance, the displacement ASD is approximately white and is given by the “noise floor”

X (ω ≪ ω_{0}) = \sqrt{\frac{4 k_{B} T}{{k ω}_{0} Q}} .

(14)

Force sensors are often used to detect signals at frequencies well below resonance. In these cases, Eq. (14) is often used as the thermomechanical noise floor in the SNR models.

In a linearly damped resonator, $b$ is the direct measure of dissipation, and $Q$ is proportional to the inverse dissipation with a proportionality factor, $\sqrt{k m}$ ⁠. Damping in a resonator can alternately be represented by a complex spring constant, $k_{anelastic} = k (1 + i ϕ (ω))$ ⁠, where $ϕ (ω)$ is the phase lag of the displacement behind the forcing.⁴¹ This model yields a thermal noise displacement spectrum that is steeper in one power of $ω$ than that predicted by the velocity-proportional damping model.⁴⁰ As we will show in Sec. VII, the linear velocity-proportional damping model is sufficient for comparing $Q$ and $Q_{e f f}$ ⁠, so we will use it moving forward.

Figure 2 summarizes Eqs. (6), (13), and (14) for the thermomechanical displacement noise of a resonator in thermal equilibrium, ignoring all other noise sources.

FIG. 2.

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Resonator thermomechanical motion and the relationship to its histogram and amplitude-spectral-density (ASD), neglecting all other noise sources. The histogram is obtained by counting for the trajectory, $x (t)$ ⁠, the number of occurrences of displacements within each displacement interval and converges to a Gaussian distribution in the limit of many samples. The ASD, $X (ω)$ ⁠, is obtained by taking the Fourier transform of the autocorrelation function of the displacement trajectory, as in Eq. (11). The mechanical quality factor can be obtained from the displacement noise floor below resonance using Eq. (14). From Sec. VII, we use $m$ =4.29 μg, $k$ =27.8 μN/nm, and $Q$ =13 k.

If any $Q_{e f f}$ enhancement or suppression technique is applied to the resonator, Eq. (1) no longer describes the measured $Q$ ⁠, because of the additional channel to an external source or sink of energy. In this case, Eq. (1) can be reformulated to define the effective quality factor, with the same $E_{stored}$ and $E_{dissipated}$ ⁠, but with an additional term to account for the external energy exchanged with the mode

Q_{e f f} = 2 π \frac{E_{stored}}{E_{d i s s ipated} + E_{e x t}},

(15)

where $E_{e x t}$ is the energy leaving the mode to an idealized noise-free sink. During $Q_{e f f}$ suppression, $E_{e x t} > 0$ ⁠, and so, $Q_{e f f} < Q$ ⁠. During $Q_{e f f}$ enhancement, $E_{e x t} < 0$ ⁠, and so, $Q_{e f f} > Q$ ⁠. When $E_{e x t} = - E_{dissipated}$ ⁠, $Q_{e f f} \to \infty$ and the mechanical mode self-oscillates.

In the linear driving regime, a resonator subjected to $Q_{e f f}$ tuning below the self-oscillation threshold will have the same response as a resonator with $Q = Q_{e f f}$ that has no $Q_{e f f}$ tuning. The linewidth of the resonance, as well as the ring-down response, will be the same in both cases. This is because $Q$ and $Q_{e f f}$ modify the resonator transfer function in the same manner, and it is this transfer function that determines the resonator behavior in the presence of an external driving force. The difference between $Q$ and $Q_{e f f}$ becomes critical when considering the thermal noise of the resonator or for understanding self-oscillations.

All phase-independent $Q_{e f f}$ tuning mechanisms can be represented by a feedback force, $f_{f b} = - (k^{'} x + b' \dot{x})$ ⁠, in their linear regime. We add $f_{f b}$ to Eq. (3) to modify the effective damping and stiffness of the lumped mass-spring-damper model²⁹

m \ddot{x} + (b + b^{'}) \dot{x} + (k + k^{'}) x = f_{drive} + f_{t h},

(16)

where $b^{'} > 0$ increases the effective damping and $b^{'} < 0$ reduces the effective damping. The feedback stiffness, $k^{'}$ ⁠, exists if a component of the feedback is in phase with the position instead of the velocity. The origin of the feedback force varies considerably from technique to technique. Equation (16) can be used to define the effective quality factor in terms of the effective linear damping constant as

Q_{e f f} = \frac{m ω_{0}}{b + b'},

(17)

and the total effective stiffness as

k_{e f f} = k + k^{'},

(18)

which enables Eq. (16) to be expressed as

m \ddot{x} + \frac{m ω_{0}}{Q_{e f f}} \dot{x} + k_{e f f} x = f_{drive} + f_{t h},

(19)

when $b^{'}$ is equal and opposite to $b$ in this linear model, the effective quality factor goes to infinity and the resonator initiates self-sustained oscillations, where nonlinearities instead of linear damping limit the vibration amplitude.

$Q_{e f f}$ tuning methods modify the transfer function, $H (ω)$ ⁠, of the resonator but do not modify the thermal noise force. A resonator subjected to such $Q_{e f f}$ enhancement or suppression has a displacement noise ASD given by

X (ω) = \frac{\sqrt{4 k_{B} T (\frac{ω_{0}}{m Q})}}{\sqrt{{({ω_{0}}^{2} - ω^{2})}^{2} + {(\frac{ω ω_{0}}{Q_{e f f}})}^{2}}} .

(20)

Figure 3 plots Eq. (20) for a resonator with $Q$ =1 k that is subjected to $Q_{e f f}$ enhancement or $Q_{e f f}$ suppression. Changing $Q_{e f f}$ does not change the thermal noise floor for frequencies below or above resonance. It only changes the thermal displacement noise at frequencies close to resonance.

FIG. 3.

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Normalized thermomechanical displacement amplitude-spectral-density (ASD) in a linearly damped resonator subjected to $Q_{e f f}$ tuning, as in Eq. (20). Increasing $Q_{e f f}$ only changes the peak shape and does not reduce the thermal displacement noise ASD at frequencies away from resonance.

The mean-squared thermomechanical noise can be increased by increasing $Q_{e f f}$ or decreased by decreasing $Q_{e f f}$ ⁠. This is because most of the noise power is concentrated at resonance, and tuning $Q_{e f f}$ modifies the resonator displacement noise spectrum near resonance. The change in the peak height in Fig. 3 with changing $Q_{e f f}$ corresponds to a change in thermomechanical mean-squared displacement and velocity. Several groups report a change in the effective quality factor by defining an effective temperature for the resonant mode. This is given by

\bar{x_{n}^{2}} = \frac{Q_{e f f}}{Q} * \frac{k_{B} T}{k} = \frac{k_{B} T_{e f f}}{k},

(21)

\bar{\dot{x_{n}^{2}}} = \frac{Q_{e f f}}{Q} * \frac{k_{B} T}{m} = \frac{k_{B} T_{e f f}}{m},

(22)

where $T_{e f f} = Q_{e f f} T / Q$ is the effective temperature for the mode. $T_{e f f}$ is not a property of the system but describes a single mode that is pumped out of thermal equilibrium with the thermal reservoir. Temperature is strictly only defined for systems in thermal equilibrium,³⁶ and a mode subjected to $Q_{e f f}$ tuning is not in thermal equilibrium. The bulk temperature of the resonator is negligibly influenced by the “heating” or “cooling” of the resonant mode with $Q_{e f f}$ tuning because most of the degrees of freedom in the system are not modified by the tuning mechanism. Hammig and Wehe showed that effective cooling of the fundamental mode of a microcantilever down from room temperature to 11 K using $Q_{e f f}$ suppression has a negligible effect on the bulk temperature.⁴²

The preceding discussion, which considers the relationship between the feedback parameters in Eq. (19) and the thermomechanical noise ASD and mean-squared noise, is applicable for all phase-independent $Q_{e f f}$ tuning mechanisms in their linear regime. External proportional feedback control, optical pumping, and thermal pumping are example phase-independent $Q_{e f f}$ tuning mechanisms. There are also phase-dependent $Q_{e f f}$ tuning mechanisms, such as degenerate parametric pumping and back-action evasion. The effect of these mechanisms cannot strictly be represented by Eq. (19) because they simultaneously amplify the motion in one quadrature and suppress the motion in the other quadrature. To understand this, consider the narrow bandwidth decomposition of the resonator thermomechanical motion into two quadratures

x (t) = x_{1} (t) cos (ωt) + x_{2} (t) sin (ω t),

(23)

where $x_{1}$ is the amplitude of the first motion quadrature at a frequency $ω$ and $x_{2}$ is the amplitude of the second motion quadrature at a frequency $ω$ ⁠, which lags the first quadrature of motion by 90°.

An external harmonic force will excite only a single quadrature of motion. The thermomechanical noise force, on the other hand, excites the first and second quadrature of motion equally. During phase-dependent $Q_{e f f}$ tuning, the thermomechanical displacement noise ASD for $x_{1}$ resembles Fig. 3 with $Q_{e f f}$ enhancement (suppression), while the ASD for $x_{2}$ resembles Fig. 3 with $Q_{e f f}$ suppression (enhancement). Like phase-independent $Q_{e f f}$ tuning mechanisms, phase-dependent $Q_{e f f}$ tuning mechanisms will initiate self-oscillations of the mode when the pump exceeds a threshold, but unlike phase-independent $Q_{e f f}$ tuning mechanisms, applying sufficient $Q_{e f f}$ suppression will also induce self-oscillations, because of the $Q_{e f f}$ enhancement in the other quadrature.

III. MEASURING THE QUALITY FACTOR

In this section, we briefly review the various techniques for measuring $Q$ of a MEM/NEM resonator and highlight some scenarios where discrepancies may arise in the measured $Q .$ Care must be taken when comparing the measured $Q$ values reported in the literature, because sometimes the authors may not distinguish between the mechanical quality factor and the effective quality factor, and different techniques may measure different $Q$ values.

The most common techniques for measuring the $Q$ of a MEM/NEM resonator are the bandwidth (or 3-dB) method, the phase slope method, the frequency response fitting method, and the ring-down method. The first three $Q$ measurement techniques for microwave and acoustic-wave resonators are reviewed by Petersan and Anlage⁴³ and by Campanella.⁴⁴ $Q$ extraction methods are usually derived by assuming a one- or two-port passive network and cannot be applied to extract the mechanical $Q$ of a pumped system where energy flows into the device via a third terminal. The only method that can distinguish between $Q_{e f f}$ and $Q$ is by fitting Eq. (20) to the thermomechanical displacement ASD.

A. The bandwidth method

The bandwidth method estimates the $Q$ of a mode using the width of the peak in the amplitude-frequency response, such as is measured using a vector network analyzer. $Q$ is estimated using the resonant frequency, $ω_{0}$ ⁠, and the width of the peak, $Δ ω$ ⁠, at $1 / \sqrt{2}$ of the peak amplitude

Q = \frac{ω_{0}}{Δ ω} .

(24)

The bandwidth method is the most common method used to measure the $Q$ of MEM/NEM resonators and RLC circuits.⁴⁵ It is best suited to measure $Q < 10^{6}$ ⁠, in cases where the resonator anharmonicity and frequency fluctuations only negligibly broaden the measured displacement ASD.⁴⁶ When obtaining the amplitude-frequency curve, the time for the amplitude to stabilize after an increase in frequency is $\sim Q / ω_{0}$ ⁠. For a low frequency mode with a very high $Q$ ⁠, the amplitude does not have enough time to stabilize as the frequency is swept across resonance. Either the sweep should be slowed down to give time for transients to die down or another technique, such as the ring-down method, should be used.

B. The phase slope method

The bandwidth method can be challenging for high frequency and/or low- $Q$ MEM/NEM resonators since the peak can get buried in the capacitive feedthrough, which makes extraction of the $Q$ factor inaccurate, if not impossible. A more efficient technique is based on the slope of the phase-frequency response of the resonator.⁴⁴ When a sinusoidal driving force is swept across the resonant frequency, there is a phase lag, $φ (ω)$ ⁠, between the forcing and the displacement. From the phase slope at the resonant frequency, $Q$ can be estimated as

Q = \frac{ω_{0}}{2} \times {|\frac{\partial φ}{\partial ω}|}_{ω_{0}} .

(25)

For cases when the amplitude-frequency response is noisy or suffers from large feedthrough, the phase slope method may provide a better estimate of $Q$ than the bandwidth method. The phase slope method is less susceptible to electrical static capacitances and can measure an individual $Q$ factor (e.g., series and parallel quality factors) across all frequencies.

When an electrical readout is used with the bandwidth or phase slope methods, parasitic reactance can be an issue. For frequencies $ω_{0} > 10^{9}$ rad/s, the resonance peak may be buried under the reactive feedthrough, rendering the bandwidth method ineffectual. This can be resolved by reducing the parasitic capacitance, e.g., using microwave probe tips, applying differential techniques⁴⁷ or fitting the modified Butterworth-van Dyke model to the peak.⁴⁸

C. The frequency response fitting method

Another way to estimate $Q$ from the amplitude versus frequency response is to fit $|H (ω)|$ ⁠, scaled by the amplitude of the external forcing, to the data. $Q$ can be estimated from the fit that minimizes the least-squared error. In the absence of measurement uncertainties, the frequency response fitting method and the bandwidth method yield an identical estimate of $Q$ ⁠.

The resonator amplitude-frequency curve due to a frequency-independent external driving force is

X (ω) = \frac{F_{drive} / m}{\sqrt{{({ω_{0}}^{2} - ω^{2})}^{2} + {(\frac{ω ω_{0}}{Q})}^{2}}} .

(26)

Fitting Eq. (26) to the resonator frequency response can be used to extract $ω_{0}$ and $Q$ ⁠. In MEM/NEM resonators, the displacement is transduced into the electrical domain and amplified during its measurement. For linear transduction, the voltage at the output of the amplifier, $V (ω)$ ⁠, is related to the lumped displacement via the amplifier responsivity, $R$ ⁠. For frequencies near resonance and $Q ≫ 1$ ⁠, it is common to simplify Eq. (26) to

X (ω) = \frac{F_{drive} / m}{\sqrt{{({ω_{0}}^{2} - ω^{2})}^{2} + {(\frac{{ω_{0}}^{2}}{Q})}^{2}}} .

(27)

Equation (27) is symmetric about $ω_{0}$ and has a simpler dependence upon frequency than the true functional form for $|H (ω)|$ but may introduce error in the estimated $Q$ ⁠, especially for low $Q$ resonators.

It is possible to discern between $Q$ and $Q_{e f f}$ by studying the ASD of the thermomechanical displacement of the resonator directly. For a thermal noise-limited readout, we fit Eq. (20) to the thermal displacement noise ASD to extract $Q$ and $Q_{e f f}$ ⁠, presuming that we know the lumped mass of the mode and the bulk resonator temperature. This is fundamentally different from fitting Eq. (26) or Eq. (27) to the driven response because the external forcing, $F_{drive}$ ⁠, is not linked to the resonator mechanical quality factor as it is for the thermomechanical noise force in Eq. (10).

Very close to resonance, both $Q$ and $Q_{e f f}$ influence the thermal noise spectrum. Krause et al. were able to directly relate the thermal noise floor of their accelerometer to the $Q$ using an ultra-sensitive photonic cavity displacement readout.³¹ Both $Q$ and $Q_{e f f}$ can be estimated for a resonator by fitting Eq. (20) to the thermomechanical noise spectrum for various pump strengths (see Sec. VII).

D. The ring-down method

In the ring-down technique, the resonator is first externally forced at a resonant frequency, $ω_{0}$ ⁠, to induce vibrations. The drive voltage is then switched off (or reduced to a lower amplitude), and the vibrations are recorded as they decay. $Q_{e f f}$ can be extracted by fitting $x (t) = x_{0} exp (- ω_{0} t / 2 Q_{e f f})$ to the envelope.⁴⁹ In the absence of an external energy source or sink, the energy loss per oscillation cycle is due to the dissipation losses. The link between dissipation and $Q$ is made most clear using the ring-down method because dissipation in the resonator causes a decay in the vibration amplitude over time after cutting off the drive. Resonators with extremely high $Q$ ⁠, such as that recently reported in Ghadimi et al.,³⁵ use the ring-down method instead of the bandwidth method to characterize the dissipation.

IV. WHY DOES THE QUALITY FACTOR MATTER?

As previously mentioned, a large mechanical $Q$ factor is an indication of low dissipation in MEM/NEM resonators. For many applications, the resonator is incorporated into an oscillator, and the frequency stability of the oscillator is affected by the $Q$ of the resonator. In the oscillators used for timing applications, the figure of merit is oscillator phase noise spectral density. In the oscillators used for sensing, Allan deviation is traditionally used to describe the frequency stability. Force sensors based upon MEM/NEM oscillators use the SNR figure of merit. Phase noise, Allan deviation, and thermomechanical SNR all improve with higher $Q$ ⁠, justifying the work towards maximizing $Q$ in the underlying MEM/NEM resonators.

A. Phase noise in clocks

Phase noise is the critical noise parameter in oscillator circuits. Whether the frequency-selective component in the oscillator is an RLC circuit, a quartz crystal, or a MEM/NEM resonator, the amplitude noise is rejected when the signal passes through a limiting amplifier and is defined by the nonlinearities in the circuit. On the other hand, the phase of the oscillator is free-running and is subject to deviation with no restoring mechanism, which results in phase error. Many phase noise models have been developed to estimate this phase error and the corresponding clock frequency stability.

An oscillator is in its most general form a combination of a lossy resonator and an energy restoration element. For feedback oscillators, the energy restoration element is some kind of $Q_{e f f}$ enhancement mechanism, such as external feedback control⁵⁰ or parametric pumping,⁵¹ which feeds in energy $E_{e x t} = - E_{dissipated}$ to compensate for the energy lost during each cycle. To sustain oscillations, $Q_{e f f} = \infty$ ⁠. Different terminology is used to describe the energy restoration element, from an electronic amplifier in the case of oscillator circuits, to a pump with a frequency equal to the sum of two resonance modes in the case of optomechanical or non-degenerate mechanical amplifiers, to a thermal-piezoresistive pump in the case of thermal self-oscillators. The added noise of the energy restoration element or pump depends on the underlying $Q_{e f f}$ enhancement mechanism and is different in each case.

We consider a resonator mean-squared displacement amplitude $x_{s i g}^{2}$ ⁠, and use Eq. (6) for the thermal mean-squared displacement noise to define the oscillator SNR

S N R = \frac{x_{s i g}^{2}}{\bar{x_{n}^{2}}} = \frac{E_{Stored} / k}{k_{B} T / k} = \frac{E_{Stored}}{k_{B} T} .

(28)

The amount of power dissipated during each cycle of oscillation, $P_{loss}$ ⁠, includes the dissipation mechanisms in Eq. (2) as well as the power consumed to drive the external oscillator circuit.^52,53 This additional loss channel is irreversible and contributes the corresponding fluctuating thermomechanical noise force. Extending the definition of the quality factor in Eq. (1), we obtain the loaded quality factor

Q = \frac{ω_{0} E_{Stored}}{P_{loss}},

(29)

where this loaded $Q$ factor is lower than the mechanical $Q$ factor. The oscillator SNR simplifies to

S N R = \frac{Q P_{loss}}{ω_{0} k_{B} T} .

(30)

From this simple relation, we observe that the oscillator SNR can be improved by increasing the mechanical $Q$ in the underlying resonator.

We next derive the phase noise spectral density in an oscillator that arises from the thermomechanical noise in the underlying MEM/NEM resonator. In a practical system, the energy restoration contributes to the noise in the oscillator. By ignoring this noise, we can obtain the phase noise due to the MEM/NEM resonator, irrespective of the $Q_{e f f}$ enhancement mechanism used to sustain oscillations. The resulting equation is known as the Leeson model.^10,52,54 We present a derivation of the Leeson model using the mass-spring-damper framework that we introduced in Sec. II. See Hajimiri and Lee for an alternate derivation using a lumped circuit model.⁵⁵

The linear damping in the mechanical mode has an associated mean-squared thermomechanical noise force of

\bar{f_{t h}^{2}} = \frac{{4 k_{B} T m ω}_{0} Δ B}{Q},

(31)

where $Δ B$ is the noise bandwidth. Equation (31) is obtained by integrating Eq. (10) over frequency. During oscillations, the transfer function of the mechanical mode resembles that of a mass-spring model because $E_{e x t} = - E_{dissipated}$ ⁠. For a small frequency offset $δ ω$ from resonance, we simplify $H (ω)$ in Eq. (20) by letting $Q_{e f f} \to \infty$ ⁠, substituting in $ω = ω_{0} + δ ω$ ⁠, and ignoring terms of order $O ({δ ω}^{2})$ ⁠. The resulting magnitude-squared transfer function is

{| H (δ ω) |}^{2} = \frac{ω_{0}^{2}}{4 k^{2} {δ ω}^{2}},

(32)

where $k$ is the lumped spring constant of the mechanical mode. The mean-squared displacement noise spectral density is

\frac{\bar{x_{n}^{2}}}{Δ B} = {| H (δ ω) |}^{2} \times \bar{f_{t h}^{2}} = \frac{{k_{B} T ω}_{0}}{k {δ ω}^{2}} .

(33)

We next define the mean-squared signal power

P_{s i g} = f_{drive} {\dot{x}}_{s i g},

(34)

where $f_{drive}$ is the root-mean-squared force driving the mechanical mode and ${\dot{x}}_{s i g}$ is the root-mean-squared velocity of that mode. At resonance, the force is related to root-mean-squared displacement, $x_{s i g}$ ⁠, via

f_{drive} = \frac{k x_{s i g}}{Q},

(35)

where $Q$ is the loaded quality factor. ${\dot{x}}_{s i g}$ is given by

{\dot{x}}_{s i g} = ω_{0} x_{s i g} .

(36)

Combining these expressions, the signal power is given by

x_{s i g}^{2} = \frac{Q P_{s i g}}{k ω_{0}} .

(37)

A figure of merit for oscillators is the phase noise spectral density at a frequency offset of $δ ω$ from the resonance frequency, which is defined by

S_{φ} (δ ω) = 10 \log (\frac{\bar{x_{n}^{2}} / Δ B}{{2 x}_{s i g}^{2}}) .

(38)

We assume per equipartition theory that half of $\bar{x_{n}^{2}}$ contributes to the phase noise of the oscillator, and the other half contributes to the amplitude noise. Substituting in $\bar{x_{n}^{2}}$ and $x_{s i g}^{2}$ ⁠, we derive the famous Leeson model for the phase noise due to white thermomechanical noise as

S_{φ} (δ ω) = 10 \log (\frac{k_{B} T}{{2 P}_{s i g}} {(\frac{ω_{0}}{Q δ ω})}^{2}),

(39)

where the $Q$ used in this equation is the loaded $Q$ factor. The ${δ ω}^{- 2}$ frequency dependence that we observe here is because the mass-spring transfer function rolls off as ${δ ω}^{- 1}$ on either side of resonance. Equation (39) clearly shows the improvement of $S_{φ} (δ ω)$ with higher $Q$ factors and higher signal powers.^10,55

Leeson added an empirical $F$ factor to the phase noise model to account for increased noise in the ${δ ω}^{- 2}$ region, a $(1 + \frac{{δ ω}_{c}}{|δ ω|})$ term to account for the ${δ ω}^{- 3}$ dependence very close to the resonant frequency, and a factor of unity to account for the ${δ ω}^{0}$ dependence far from resonance. These additional terms are not due to the thermomechanical noise force and lead to three regions of differing $S_{φ} (δ ω)$ slopes with respect to $δ ω$

S_{φ} (δ ω) = 10 \log (\frac{2 F k_{B} T}{P_{s i g}} [{1 + (\frac{ω_{0}}{Q δ ω})}^{2}] (1 + \frac{{δ ω}_{c}}{|δ ω|})),

(40)

where ${δ ω}_{c}$ is approximately the $ω^{- 1}$ corner frequency of the device noise. The noise contributed by the particular $Q_{e f f}$ enhancement mechanism during oscillations is captured by $F$ and ${δ ω}_{c}$ in Eq. (40) and varies among the mechanisms. For example, parametric feedback oscillators can have better phase noise than direct feedback oscillators because applying the pump at ${2 ω}_{0}$ minimizes the phase noise contributed by the feedback circuit.⁵¹

B. Noise in resonant sensors

MEM/NEM sensors have been fabricated to measure a wide range of phenomena, such as inertial forces,⁵⁶ heat transport,⁵⁷ and biomolecules.^7,58 Sensors transduce a phenomenon of interest into an electrical signal for subsequent processing. Resonant MEM/NEM sensors measure a signal as either a force, which induces a measurable displacement against the restoring force of the mechanical mode, or a shift in the resonant frequency due to a change in the lumped mass or stiffness.

MEM/NEM force sensors transduce the measured phenomenon into a mechanical displacement against a restoring force and include atomic force microscopes, accelerometers, pressure sensors, and Coriolis force gyroscopes. The thermomechanical SNR of the sensor scales with the square root of the mechanical quality factor.⁴⁰ The general expression for the SNR (in dimensionless units) of these sensors is

S N R = α S \sqrt{\frac{Q}{4 k_{B} T}},

(41)

where $S$ is the minimum signal, $α$ is a sensor-specific scaling factor, and $Q$ is the mechanical quality factor. This relationship between SNR and $Q$ in many sensors motivates continual efforts to characterize and reduce the dissipation of MEM/NEM resonators.

Table I delineates the $α$ scaling factor of Eq. (41) for some common sensors. For an accelerometer, $a_{s}$ is the acceleration, $m$ is the lumped mass, and $ω_{0}$ is the resonant frequency of the mode. For an optomechanical cavity gyroscope, $Ω_{s}$ is the angular velocity, $B$ is the bandwidth, and $r_{0}$ is the distance of the optomechanical cavity from the axis of rotation. For a pressure sensor, $P_{s}$ is the pressure, $A$ is the cross-sectional area of the membrane, and $m$ is the lumped mass of the fundamental flexural membrane mode. For an atomic force microscope operated using either the slope detection method or the frequency-modulation method, $δ F$ is the force gradient and $X_{d}$ is the mean-squared drive amplitude.

TABLE I.

Parameters of the thermomechanical signal-to-noise ratio (SNR) in Eq. (41) for a variety of resonant sensors.

Sensor	$S$	$α$
Accelerometer⁵⁹	$a_{s}$ (m/s² Hz^1/2)	$\sqrt{m / ω_{0}}$
Gyroscope⁶⁰	$Ω_{s}^{2}$ (Hz²)	$r_{0} \sqrt{2 m / ω_{0} B}$
Pressure sensor⁸	$P_{s}$ (Pa/Hz^1/2)	$\frac{A}{\sqrt{{m ω}_{0}}}$
Force microscope⁶¹	$δ F$ (N/m)	$\frac{X_{d}}{\sqrt{m ω_{0} B}}$

Sensor	$S$	$α$
Accelerometer⁵⁹	$a_{s}$ (m/s² Hz^1/2)	$\sqrt{m / ω_{0}}$
Gyroscope⁶⁰	$Ω_{s}^{2}$ (Hz²)	$r_{0} \sqrt{2 m / ω_{0} B}$
Pressure sensor⁸	$P_{s}$ (Pa/Hz^1/2)	$\frac{A}{\sqrt{{m ω}_{0}}}$
Force microscope⁶¹	$δ F$ (N/m)	$\frac{X_{d}}{\sqrt{m ω_{0} B}}$

In addition to the SNR, the bandwidth of a sensor is important. The bandwidth is the highest frequency signal that a sensor can transduce. A phenomenon that quickly varies in time requires a high bandwidth sensor to accurately measure. The amplitude of a high $Q$ resonator takes a long time to respond to changes in the external signal. The bandwidth of an amplitude modulated (AM) sensor is inversely related to the thermal SNR⁶¹. To maintain the large SNR without compromising the sensor bandwidth, either frequency modulation (FM)⁶¹ or $Q_{e f f}$ suppression¹¹ can be used. For an FM sensor, the resonator is maintained in a condition of oscillations, e.g., via $Q_{e f f}$ enhancement into the self-oscillation regime or embedding the resonator into a phase-locked loop. The signal of interest is transduced into a change in the frequency or phase of the oscillator. The sensor bandwidth is set by the oscillator control loop and is therefore independent of the resonator $Q$ ⁠. For an AM sensor, $Q_{e f f}$ suppression is often used to improve the sensor bandwidth. As per Fig. 3, the bandwidth can be improved by reducing $Q_{e f f}$ without increasing the noise floor.

The phase noise spectral density of an oscillator, discussed in Sec. IV A, best captures the close-to-carrier phase noise and gives a good visualization for harmonic frequency components. Allan deviation is often used to characterize the frequency stability for longer integration times and is defined in the time domain.⁶² The frequency stability is commonly predicted based on the dynamic range⁶³ (DR), expressed in dB, by

\frac{δ ω}{ω_{0}} = \frac{1}{2 Q} 10^{\frac{- D R}{20}} .

(42)

Equation (42) shows that frequency stability increases with a higher $Q$ or larger dynamic range. The maximum dynamic range typically corresponds to the onset of Duffing nonlinearity in a mode and increases as the $Q$ decreases. Roy et al. suggested that this increase in the dynamic range can enable the frequency stability to improve⁶⁴ with decreasing $Q$ ⁠.

The expression defining frequency stability using Allan deviation, $σ$ ⁠, as a function of integration time is given by

σ (τ) = \sqrt{\frac{1}{2 (M - 1)} \sum_{i}^{M - 1} {(\frac{{\bar{ω}}_{i + 1} - {\bar{ω}}_{i}}{ω_{0}})}^{2},}

(43)

where M is the number of samples of the resonant frequencies, ${\bar{ω}}_{1}$ ⁠,…, ${\bar{ω}}_{M}$ ⁠, each averaged over integration time, $τ$ ⁠, with zero dead time. Considering the regime where additive white noise dominates the frequency stability, the Allan deviation can be estimated as

σ_{A} \approx \frac{1}{2 Q_{e f f}} \sqrt{\frac{\bar{x_{n}^{2}}}{x_{s i g}^{2}}} \sqrt{\frac{1}{2 π τ}},

(44)

where $1 / 2 π τ$ is the measurement bandwidth with a first-order low-pass filter and $\bar{x_{n}^{2}}$ is the integrated displacement noise spectrum within the measurement bandwidth of resonance. $Q_{e f f}$ within Eq. (44) is determined using the phase slope at resonance.⁶⁵ Allan deviation, $σ (τ)$ ⁠, and phase noise spectral density, $S_{Ø} (ω)$ ⁠, are related by

σ (τ) = \frac{2}{ω τ} \sqrt{2 \int_{0}^{\infty} S_{φ} (ω) {sin}^{4} (\frac{ω τ}{2}) d ω} .

(45)

In MEM/NEM-based oscillators, dissipation-induced thermal fluctuations are not the ultimate limit to the phase noise because fluctuations due to temperature-dependent noise processes and other mechanisms also contribute.^63,66

V. EFFECTIVE QUALITY FACTOR TUNING METHODS

There are many methods for tuning the effective quality factor of a micromechanical resonator, as summarized in Fig. 4. For phase-independent feedback, the $Q_{e f f}$ tuning arises from a velocity-proportional force and can be represented by an effective damping constant, $b'$ ⁠, within Eq. (16). During external feedback control, the resonator position is externally monitored, phase-shifted, and applied back as a force. Optical pumping involves coupling the mechanical mode to an optical or microwave mode and exciting the optical mode in a manner such that energy is fed into or pulled out of the mechanical mode. Mechanical pumping is analogous to optical pumping but replaces the optical cavity with another mechanical mode in the resonator. Thermal-piezoresistive pumping utilizes the piezoresistive effect in semiconductor resonators to exchange energy between the mechanical motion and a direct electrical current. For acoustoelectric pumping, traveling waves in a piezoelectric resonator are amplified using a direct current. For degenerate parametric pumping, the spring constant or mass of a mode is modulated at $2 ω_{0} / N$ ⁠, where $N$ is a positive integer. Quantum back-action describes the intrinsic force exerted on the resonator during the measurement of its position.⁶⁷

FIG. 4.

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Techniques for tuning the effective quality factor of a mechanical mode.

Effective quality factor suppression is a routine procedure for increasing the bandwidth and reducing the mean-squared thermal displacement fluctuations of a resonant sensor. $Q_{e f f}$ suppression is often used to improve the imaging speed of amplitude modulated atomic force microscopes (AFMs).⁶⁸ Atomic force microscopy utilizes a vibrating cantilever to map out the nanoscale topography of a surface. The atomic-scale imaging resolution requires a very low thermal noise floor, and so, as per Fig. 1, high $Q$ cantilevers are used. The resulting excessive cantilever response times correspond to a low bandwidth, $B W = ω_{0} / 2 Q_{e f f}$ ⁠, when using the slope detection method, and thus excessive time to raster scan a surface.¹¹ By applying $Q_{e f f}$ suppression to the cantilever during imaging, the bandwidth, and therefore the imaging speed, can be improved while maintaining the large thermal SNR.

For experiments at the molecular scale, such as manipulating individual atoms,⁶⁹ or measuring the Seebeck coefficient of single molecules,⁷⁰ or measuring the magnetic field produced by a single electron spin,⁷¹ the mean-squared displacement of the microcantilever must be suppressed to well below its thermal equilibrium value using cryogenics or $Q_{e f f}$ suppression. A microcantilever⁷² with a lumped stiffness of 10 μN/m has a root-mean-squared cantilever tip displacement of 20 nm at room temperature, which is much larger than the lattice constant, and will damage the sample if the cantilever is placed in close contact with it. Reducing $Q_{e f f}$ of one or more modes via external feedback control, as in Eq. (20), can be used to minimize the cantilever thermomechanical noise motion.

$Q_{e f f}$ suppression also plays an important role in the experimental exploration of quantum mechanics. Micromechanical resonators are a promising platform for realizing quantum mechanical behavior such as superposition or entanglement within macroscopic objects.^73,74 A prerequisite for observing non-classical behavior in a resonator is to cool one of the modes into its quantum ground state. Quantum mechanics demands that the energy stored in the atomic vibrations of a solid be quantized as phonons of energy $ℏ ω$ ⁠, where $ω$ is the elastic wave frequency and $ℏ$ is Planck's constant. The average phonon occupancy,⁷⁴ $\bar{n}$ ⁠, of a mode of frequency $ω_{0}$ is given by $\bar{n} = \frac{k_{B} T_{e f f}}{ℏ ω_{0}} - \frac{1}{2}$ ⁠. In thermal equilibrium, the temperature of the mode is equal to the temperature of the bath. But when $Q_{e f f}$ suppression is applied, the mode is brought out of thermal equilibrium and its effective temperature decreases as per Eq. (21). For sufficiently low $T_{e f f}$ ⁠, the average phonon occupancy of the mode drops below unity (⁠ $\bar{n} < 1$ ⁠) and the probability of the resonator occupying its ground state becomes non-negligible. Conventional helium dilution refrigeration can be used to cool GHz-frequency resonators into their ground state.⁷⁵ To cool lower frequency modes into their ground state, conventional cooling approaches must be supplemented with $Q_{e f f}$ suppression techniques such as external feedback⁷⁶ or optical pumping.^77,78

Figure 5 summarizes several reports of $Q_{e f f}$ tuning in resonators ranging in mass over seven orders of magnitude and ranging in $Q_{e f f}$ over nine orders of magnitude. We select from experiments in Secs. V A–V F involving six different $Q_{e f f}$ tuning mechanisms.

FIG. 5.

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Several reports of effective quality factor tuning via external feedback control, parametric pumping, optical pumping, thermal-piezoresistive pumping, mechanical pumping, and acoustoelectric pumping, color-coded in terms of mechanism and labeled from low to high frequencies. Circles denote the device resonant frequency and mechanical quality factor, $Q$ ⁠. Open circles indicate that self-oscillations were reported. Upward (downward) arrows indicate the minimum (maximum) achieved effective quality factor, $Q_{e f f}$ ⁠. External feedback: Liu and Kenny,⁷⁹ Kleckner and Bouwmeester,⁷⁶ Tamayo,⁸⁰ Mertz et al.,¹¹ and Wilson et al.;⁸¹ Optical pumping: Metzger and Karrai,¹² Thompson et al.,⁸² Gigan et al.,⁸³ Zalalutdinov et al.,⁸⁴ Kippenberg et al.,⁸⁵ Teufel et al.,⁷⁷ Bahl et al.,⁸⁶ Chan et al.,⁸⁷ and Safavi-Naeini et al.;⁷⁸ Mechanical pumping: Dougherty et al.,⁸⁸ Venstra et al.,¹³ Mahboob et al.,⁸⁹ Okamoto et al.,⁹⁰ and Mathew et al.;⁹¹ Thermal-piezoresistive pumping: Miller et al.,⁹² Steeneken et al.,¹⁴ Rahafrooz and Pourkamali,⁹³ and Ramezany and Pourkamali;⁹⁴ Acoustoelectric pumping: Gokhale and Rais-Zadeh;⁹⁵ Degenerate parametric pumping: Chan et al.,⁹⁶ Rugar and Grütter,¹⁵ Turner et al.,⁹⁷ Mahboob and Yamaguchi,⁹⁸ Eichler et al.,¹⁷ and Karabalin et al.⁹⁹

In Secs. V A–V G, we will discuss external feedback, optical pumping, mechanical pumping, thermal-piezoresistive pumping, parametric pumping, and other reported $Q_{e f f}$ tuning mechanisms. These techniques all modify $Q_{e f f}$ ⁠, which induces a change in the thermal noise peak shown in Fig. 3. We discuss the physical principles underlying each technique and the extent of applications of these techniques to sensors, oscillators, and ground state cooling. We also review the experimental literature associated with each technique and summarize the type (enhancement, suppression, or both) and magnitude of $Q_{e f f}$ tuning for each reference.

A. External proportional feedback control

During external feedback, the resonator motion is first transduced into an electrical signal, then filtered, then phase shifted, and then applied back to force the device. Figure 6 illustrates an example implementation of external feedback control of the first flexural mode of a doubly clamped beam.

FIG. 6.

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Schematic for implementing external feedback control of a doubly clamped beam resonator. The beam thermomechanical motion is sensed capacitively using the right electrode and is applied back to force the beam with the left electrode.

The thermomechanical fluctuations of the first flexural mode create a tiny current out of the sense electrode, which is amplified into a voltage and then filtered to suppress noise at frequencies away from resonance. A tunable gain and phase shifter is used to amplify the filtered signal, and this voltage is applied back as an electrostatic force on the resonator using the drive electrode. For sufficient gain and a feedback force that is in phase with the thermomechanical velocity fluctuations, the resonator $Q_{e f f}$ will increase. By appropriately choosing the feedback gain and phase shift, $Q_{e f f}$ can be arbitrarily modified. $Q_{e f f}$ suppression is limited by the sensitivity of the measurement readout, the noise introduced by the feedback channel, and by the residual device heating.

An important aspect of controller design is resonator stability. For this reason, the Laplace domain with complex frequency, $s = i ω$ + $σ$ ⁠, should be used instead of the Fourier domain for modeling the dynamics of the resonator and the controller. The feedback controller in Fig. 6 can be represented by a transfer function, $G (s)$ ⁠, in the Laplace domain. The choice of controller transfer function for stable $Q_{e f f}$ suppression is a well-developed topic for the optimal control of force microscope cantilevers.^72,100–102 Given a maximum allowable variance in the resonator position, $\bar{x_{n, m a x}^{2}}$ ⁠, a maximum allowable variance in the control force, $\bar{f_{f b, m a x}^{2}}$ ⁠, a white measurement displacement noise PSD, $N_{a}^{2}$ ⁠, and a process noise dominated by the white thermal noise force, the optimal filter and optimal deterministic controller⁷² results in an effective spring constant for Eq. (16)

k^{'} = m ω_{0}^{2} [\frac{{(α + β)}^{2}}{2} + \frac{α + β}{Q}],

(46)

and an effective damping constant of

b^{'} = \frac{m ω_{0} (α + β + \frac{1}{Q})}{\sqrt{[\frac{{(α + β)}^{2}}{2}] + [\frac{(α + β)}{Q}] + 1}} - \frac{m ω_{0}}{Q},

(47)

where the parameters $α$ and $β$ are given by

α = {(\frac{1}{Q^{2}} + 2 ξ^{2})}^{1 / 2} - \frac{1}{Q},

(48)

β = {(\frac{1}{Q^{2}} + 2 ψ^{2})}^{1 / 2} - \frac{1}{Q},

(49)

and the parameters $ξ$ and $ψ$ are given by

ξ^{2} = {(1 + \frac{\bar{f_{f b, m a x}^{2}}}{m^{2} ω_{0}^{4} \bar{x_{n, m a x}^{2}}})}^{1 / 2} - 1,

(50)

ψ^{2} = {(1 + \frac{4 k_{B} T}{m ω_{0}^{3} Q N_{a}^{2}})}^{1 / 2} - 1.

(51)

External velocity-proportional feedback control is the $Q_{e f f}$ enhancement mechanism used in commercial MEM oscillators,¹ quartz crystal oscillators,¹⁰³ piezoelectric oscillators,¹⁰⁴ and a variety of NEM oscillators.^50,105 It is also widely used in MEM/NEM oscillators for studies of synchronization¹⁰⁶ and other nonlinear effects.¹⁰⁷ We refer the readers to van Beek and Puers¹⁰⁴ and Chen et al.¹⁰⁸ for in-depth reviews of MEM oscillators.

Table II summarizes the reports of $Q_{e f f}$ tuning via external feedback in MEM/NEM resonators. Mertz et al. was one of the first to publish reports of external proportional feedback control of a micromechanical resonator.¹¹ They showed that $Q_{e f f}$ suppression can be used to improve the bandwidth of an amplitude modulated cantilever without reducing the SNR. The microcantilever motion was measured using an interferometric readout, and the $Q_{e f f}$ suppression was implemented using a photo-thermal force.

The AFM community published many reports of $Q_{e f f}$ suppression using external feedback. Bruland et al. used external feedback to improve the imaging speed of a magnetic resonance force microscope (MRFM) via a magnetic torsional feedback force.^100,101 Stowe et al. demonstrated $Q_{e f f}$ suppression via capacitive forcing of torsional microcantilevers for MRFM.¹⁰⁹ Liang et al. suppressed the variance of their AFM cantilever position [per Eq. (21)] using $Q_{e f f}$ suppression.²⁹ Sulchek et al. showed that active damping improved the scanning speed of a tapping mode AFM cantilever.^110,111 Hammig et al. implemented $Q_{e f f}$ suppression in microcantilevers for improving charged particle impact detection.^42,112 Smullin et al. used $Q_{e f f}$ suppression to improve the detection bandwidth of a microcantilever for observing deviations from Newtonian gravitation at short distances.¹¹³ Tamayo et al.,¹¹⁴ Degen et al.,¹¹⁵ and Jacky et al.¹⁰² implemented digital feedback controllers for suppressing $Q_{e f f}$ of a microcantilever. Kageshima et al. implemented a piezoelectric actuator for suppressing $Q_{e f f}$ of a scanning probe microscope (SPM).¹¹⁶ Weld and Kapitulnik implemented feedback cooling of a microcantilever with radiation pressure.¹¹⁷ Jourdan et al. compared an increase in effective damping (without additional fluctuations) with an increase in actual damping (with added fluctuations) in a microcantilever.¹¹⁸ Ruppert and Moheimani suppressed $Q_{e f f}$ of multiple modes in an AFM using external feedback.¹¹⁹ Kawamura and Kanegae suppressed $Q_{e f f}$ of an aluminum-coated silicon cantilever by several orders of magnitude using feedback¹²⁰ and independently tuned $Q_{e f f}$ of two modes in a silicon nitride cantilever.¹²¹

There are also many reports of AFM $Q_{e f f}$ enhancement. Anczykowski et al. showed that $Q_{e f f}$ enhancement can beneficially modify the interaction forces between a scanning force microscope and the adjacent surface during scanning.¹²² Humphris et al. used $Q_{e f f}$ enhancement to improve the sensitivity of their AFM cantilever to the elongation of a single dextran molecule.¹²³ Tamayo et al. improved the elastic modulus measurement of an agarose gel sample in liquid with their $Q_{e f f}$ controlled AFM cantilever.¹²⁴ Tamayo et al. next showed that $Q_{e f f}$ enhancement improved the frequency resolution of a force cantilever embedded in a phase-locked loop, due to the increase in the phase slope at resonance.¹²⁵ Tamayo et al. next used this technique to scan living cells with better image quality.¹²⁶ Lei et al. also showed an improvement in image quality with increased $Q_{e f f}$ for shear force topographical imaging of human aortic tissue.¹²⁷ Ebeling et al. improved the image quality of DNA adsorbed on mica using $Q_{e f f}$ enhancement of an AM-AFM.¹²⁸ Chen et al. showed that $Q_{e f f}$ enhancement extends the attractive sensing regime and $Q_{e f f}$ suppression extends the repulsive regime in an AM-AFM.¹²⁹ Gunev et al. implemented real-time $Q_{e f f}$ tuning during surface scanning to simultaneously optimize scan speed and probe sensitivity.¹³⁰ Orun et al. demonstrated simultaneous frequency and $Q_{e f f}$ tuning of a tapping mode AFM cantilever.¹³¹ Moore et al. suppressed $Q_{e f f}$ of a magnet-tipped cantilever for electron spin resonance detection.¹³² Manzaneque et al. enhanced $Q_{e f f}$ in a piezoelectric cantilever and microbridge.¹³³ Gavartin et al. used external feedback to suppress $Q_{e f f}$ of a doubly clamped beam that was optically coupled to a microtoroid resonator.¹³⁴ Huefner et al. implemented $Q_{e f f}$ tuning of a conductive cantilever by applying the feedback as a voltage to the environment.¹³⁵ Fairbairn and Moheimani developed a resonant controller to adjust $Q_{e f f}$ of a microcantilever with guaranteed closed-loop stability.¹³⁶ Harris et al. tuned $Q_{e f f}$ of a microtoroid using electrical gradient force feedback.¹³⁷ Vitorino et al. reported simultaneous large tuning of resonant frequency and modest tuning of $Q_{e f f}$ using external feedback on a microcantilever.¹³⁸

While most external feedback schemes have been implemented in microcantilevers for AFM applications, there are several demonstrations of $Q_{e f f}$ tuning using other resonator geometries. To improve the bandwidth of an AM electron-tunneling accelerometer, Liu and Kenny used feedback $Q_{e f f}$ suppression to critically damp a high $Q$ accelerometer.⁷⁹ Arcizet et al. used feedback cooling to lower the effective temperature of a micro-mirror.¹³⁹ Lee et al. applied feedback cooling to an optically transduced microtoroid resonator.¹⁴⁰ Anthony et al. demonstrated $Q_{e f f}$ enhancement of a folded spring comb-drive structure.¹⁴¹ Poot et al. applied feedback cooling to a 2 MHz resonator embedded in a superconducting quantum interference device.¹⁴² Hosseini suppressed $Q_{e f f}$ in a nanowire using laser-induced thermal expansion feedback.¹⁴³ Lee et al. simultaneously tuned $Q_{e f f}$ and the resonant frequency of a quartz tuning fork resonator via feedback control.¹⁴⁴ Ohta et al. used a single feedback loop to simultaneously tune $Q_{e f f}$ of six interconnected beam resonators.¹⁴⁵ Buters et al. damped the motion of an outer “mechanical low-pass filter” resonator to isolate an inner trampoline mode resonator from the environment.¹⁴⁶

Feedback cooling has also been explored for preparing a mechanical mode in its quantum ground state. Kleckner and Bouwmeester used feedback cooling to reduce the effective temperature of a microcantilever from room temperature to 135 mK and suggested that combining conventional cooling and external feedback could be used to reach sub-unity phonon occupancy.⁷⁶ Poggio et al. showed that feedback cooling can “squash” the noise intensity at resonance below the detector shot noise limit.¹⁴⁷ Wilson et al. used external feedback to cool a mode of a silicon nanobeam down to a phonon occupation of $\bar{n} \sim 5$ ⁠, which is perhaps the closest that external feedback has brought a nanomechanical mode to its ground state.⁸¹

The intrinsic limits to feedback cooling imposed by the detector noise have so far prevented external feedback from cooling a resonator mode into sub-unity phonon occupancy. Optical pumping, to be discussed, has been more successful than external feedback control for cooling MEM/NEM resonators into their quantum ground state.

TABLE II.

Reports of $Q_{e f f}$ tuning via external feedback in chronological order, neglecting external feedback oscillators.

Reference	$Q_{e f f}$ type	$Q$	$Q_{e f f}$
Mertz et al.¹¹	Suppress	1.8 k	2
Bruland et al.¹⁰⁰	Suppress	2 k	5
Stowe et al.¹⁰⁹	Suppress	5 k	50
Bruland et al.¹⁰¹	Suppress	15 k	220
Anczykowski et al.¹²²	Enhance	450	2.5 k
Sulchek et al.¹¹⁰	Both	68	40, 120
Humphris et al. ¹²³	Enhance	1	300
Liang et al.²⁹	Suppress	100	32
Tamayo et al.¹²⁴	Enhance	1	1 k
Liu and Kenny⁷⁹	Suppress	100	700 m
Tamayo et al.¹²⁵	Enhance	2	625
Tamayo et al.¹²⁶	Enhance	2	100
Sulchek et al.¹¹¹	Suppress	90	15
Tamayo et al. ¹¹⁴	Enhance	2	72
Lei et al.¹²⁷	Enhance	100	1 k
Tamayo⁸⁰	Enhance	45	7.8 k
Smullin et al.¹¹³	Suppress	75 k	10 k
Hammig et al.¹¹²	Suppress	34 k	24
Arcizet et al.¹³⁹	Suppress	15 k	250
Degen et al.¹¹⁵	Suppress	10 k	5
Ebeling et al.¹²⁸	Enhance	34	200
Kleckner and Bouwmeester⁷⁶	Suppress	137 k	290μ
Weld and Kapitulnik¹¹⁷	Suppress	12 k	700
Kageshima et al.¹¹⁶	Suppress	29	750 m
Jourdan et al.¹¹⁸	Suppress	180	68
Poggio et al.¹⁴⁷	Suppress	44 k	8
Jacky et al.¹⁰²	Suppress	10 k	75
Chen et al.¹²⁹	Both	780	160, 2 k
Hammig and Wehe⁴²	Suppress	50	24
Gunev et al.¹³⁰	Enhance	310	2.5 k
Orun et al.¹³¹	Both	360	10, 620
Moore et al.¹³²	Suppress	38 k	3 k
Lee et al.¹⁴⁰	Suppress	550	25
Anthony et al.¹⁴¹	Enhance	500	3.5 k
Manzaneque et al.¹³³	Enhance	1 k	200 k
Poot et al.¹⁴²	Both	24 k	16 k, 43 k
Gavartin et al. ¹³⁴	Suppress	480 k	8.4 k
Huefner et al.¹³⁵	Both	16 k	3.2 k, 140 k
Fairbairn and Moheimani¹³⁶	Both	180	38, 990
Harris et al.¹³⁷	Both	1.8 k	180, 260 k
Hosseini et al.¹⁴³	Suppress	380	8
Vitorino et al.¹³⁸	Both	11 k	10 k, 21 k
Wilson et al.⁸¹	Suppress	760 k	190
Lee et al.¹⁴⁴	Both	950	410, 20 k
Ruppert and Moheimani¹¹⁹	Suppress	290	60
Kawamura and Kanegae¹²⁰	Suppress	3 k	810 m
Ohta et al.¹⁴⁵	Enhance	2.3 k	48 k
Buters et al.¹⁴⁶	Suppress	90 k	20
Kawamura and Kanegae¹²¹	Suppress	290	58

Reference	$Q_{e f f}$ type	$Q$	$Q_{e f f}$
Mertz et al.¹¹	Suppress	1.8 k	2
Bruland et al.¹⁰⁰	Suppress	2 k	5
Stowe et al.¹⁰⁹	Suppress	5 k	50
Bruland et al.¹⁰¹	Suppress	15 k	220
Anczykowski et al.¹²²	Enhance	450	2.5 k
Sulchek et al.¹¹⁰	Both	68	40, 120
Humphris et al. ¹²³	Enhance	1	300
Liang et al.²⁹	Suppress	100	32
Tamayo et al.¹²⁴	Enhance	1	1 k
Liu and Kenny⁷⁹	Suppress	100	700 m
Tamayo et al.¹²⁵	Enhance	2	625
Tamayo et al.¹²⁶	Enhance	2	100
Sulchek et al.¹¹¹	Suppress	90	15
Tamayo et al. ¹¹⁴	Enhance	2	72
Lei et al.¹²⁷	Enhance	100	1 k
Tamayo⁸⁰	Enhance	45	7.8 k
Smullin et al.¹¹³	Suppress	75 k	10 k
Hammig et al.¹¹²	Suppress	34 k	24
Arcizet et al.¹³⁹	Suppress	15 k	250
Degen et al.¹¹⁵	Suppress	10 k	5
Ebeling et al.¹²⁸	Enhance	34	200
Kleckner and Bouwmeester⁷⁶	Suppress	137 k	290μ
Weld and Kapitulnik¹¹⁷	Suppress	12 k	700
Kageshima et al.¹¹⁶	Suppress	29	750 m
Jourdan et al.¹¹⁸	Suppress	180	68
Poggio et al.¹⁴⁷	Suppress	44 k	8
Jacky et al.¹⁰²	Suppress	10 k	75
Chen et al.¹²⁹	Both	780	160, 2 k
Hammig and Wehe⁴²	Suppress	50	24
Gunev et al.¹³⁰	Enhance	310	2.5 k
Orun et al.¹³¹	Both	360	10, 620
Moore et al.¹³²	Suppress	38 k	3 k
Lee et al.¹⁴⁰	Suppress	550	25
Anthony et al.¹⁴¹	Enhance	500	3.5 k
Manzaneque et al.¹³³	Enhance	1 k	200 k
Poot et al.¹⁴²	Both	24 k	16 k, 43 k
Gavartin et al. ¹³⁴	Suppress	480 k	8.4 k
Huefner et al.¹³⁵	Both	16 k	3.2 k, 140 k
Fairbairn and Moheimani¹³⁶	Both	180	38, 990
Harris et al.¹³⁷	Both	1.8 k	180, 260 k
Hosseini et al.¹⁴³	Suppress	380	8
Vitorino et al.¹³⁸	Both	11 k	10 k, 21 k
Wilson et al.⁸¹	Suppress	760 k	190
Lee et al.¹⁴⁴	Both	950	410, 20 k
Ruppert and Moheimani¹¹⁹	Suppress	290	60
Kawamura and Kanegae¹²⁰	Suppress	3 k	810 m
Ohta et al.¹⁴⁵	Enhance	2.3 k	48 k
Buters et al.¹⁴⁶	Suppress	90 k	20
Kawamura and Kanegae¹²¹	Suppress	290	58

B. Optical pumping

Optical pumping (also called optomechanical back-action or cavity cooling) is a technique for tuning $Q_{e f f}$ by coupling the mechanical resonator to an optical or microwave cavity.¹⁴⁸ The coupling between the optical and mechanical degrees of freedom can arise from radiation pressure,¹² bolometric forcing,⁸⁴ or electron-hole generation.¹⁴⁹ Often, the coupling is achieved by engineering the resonator and the cavity so that the displacement of the mechanical mode changes the cavity length, thus modulating the resonant frequency of the optical mode, $ω_{o p}$ ⁠. The resonant frequency of the mechanical mode, $ω_{0}$ ⁠, is in the kHz to GHz range. $ω_{o p}$ is typically in the GHz range for cavities operating at microwave wavelengths and in the THz range for cavities at infrared wavelengths. The coupling between the mechanical mode and the optical mode creates sidebands in the optical spectrum at the sum and difference frequencies, $ω_{s u m} = ω_{o p} + ω_{0}$ and $ω_{diff} = ω_{o p} - ω_{0}$ ⁠. The peak at the sum frequency is called the blue sideband, and the peak at the difference frequency is called the red sideband. By pumping the optical cavity at the red sideband (i.e., by using a pump with a wavelength near $λ_{r e d} = 2 π c / ω_{diff}$ ⁠, where $c$ is the speed of light), phonons in the mechanical mode will up-convert to photons in the optical mode, lowering $Q_{e f f}$ of the mechanical mode. By pumping the optical cavity at the blue sideband (i.e., by using a laser or microwave pump with a wavelength near $λ_{blue} = 2 π c / ω_{s u m}$ ⁠), photons from the laser or microwave pump will down-convert into phonons in the mechanical mode, raising $Q_{e f f}$ ⁠.

Depending on whether one or two pumps are used, optical pumping is either a phase-independent or a phase-dependent $Q_{e f f}$ tuning mechanism, respectively. Pumping the optical mode at both $ω_{s u m}$ and $ω_{diff}$ ⁠, with identical pump strengths in each sideband, is a form of back-action-evasion.^67,150 Like degenerate parametric pumping, back-action-evasion reduces the displacement noise in only one quadrature, but unlike degenerate parametric pumping, the squeezing can exceed half of the thermal equilibrium mean-squared displacement, $\bar{x_{n}^{2}}$ ⁠, prior to self-oscillations in the other quadrature. In principle, this technique can squeeze the uncertainty in one displacement quadrature below the mechanical zero-point motion.¹⁵¹ In practice, spontaneous degenerate parametric oscillations of the mechanical mode make this limit difficult to achieve.^152,153 In the remainder of Sec. V B, we will concentrate on phase-independent optical pumping, whereby the optical mode is pumped either at $ω_{s u m}$ or $ω_{diff}$ ⁠, but not at both.

Optical pumping is thus far the only $Q_{e f f}$ suppression technique that has successfully cooled a mode of a mechanical resonator into its quantum ground state,^77,87 thus enabling the first observation of zero-point fluctuations in a mechanical object.⁷⁸ The superior cooling capability is enabled by its passive nature and the low intrinsic heating from the radiation field. Work to further increase the coupling, the $Q$ ⁠, and the mechanical resonance frequency may enable room temperature optical pumping of mechanical objects into their quantum ground state.^34,154

We can define effective feedback coefficients for optical pumping. Starting from the coupled optomechanical model in Schliesser et al.,¹⁵⁵ we define $k^{'}$ and $b^{'}$ for Eq. (16) as

k^{'} = 2 τ C (\frac{{Δ ω}_{s}}{4 τ^{2} {Δ ω}_{s}^{2} + 1} + \frac{{Δ ω}_{a s}}{4 τ^{2} {Δ ω}_{a s}^{2} + 1}) P_{i n},

(52)

b^{'} = - \frac{C}{ω_{0}} (\frac{1}{4 τ^{2} {Δ ω}_{s}^{2} + 1} - \frac{1}{4 τ^{2} {Δ ω}_{a s}^{2} + 1}) P_{i n},

(53)

where the constant of proportionality, $C$ ⁠, is given as

C = \frac{8 n^{2} F^{2} ω_{o p} C}{c^{2}},

(54)

where the coupling parameter, $C$ ⁠, is defined as

C = \frac{τ}{τ_{e x} (4 τ^{2} {Δ ω}^{2} + 1)},

(55)

where $ω_{o p}$ is the optical cavity resonance frequency, $ω_{0}$ is the mechanical resonance frequency, $F$ is the optical cavity finesse, $n$ is the refractive index, $c$ is the speed of light, $m$ is the lumped mass of the mechanical mode, $τ = Q_{o p} / ω_{o p}$ is the optical cavity photon decay time, $Q_{o p}$ is the quality factor of the optical cavity, ${1 / τ}_{e x}$ is the rate of coupling from the optical fiber into the cavity, $P_{i n}$ is the laser power launched into the fiber, $Δ ω$ is the laser detuning from the optical resonance, ${Δ ω}_{s} = Δ ω - ω_{m}$ is the detuning of the Stokes photons, and ${Δ ω}_{a s}$ = $Δ ω + ω_{m}$ is the detuning of the anti-Stokes photons.

Table III summarizes the reports of $Q_{e f f}$ tuning via optical pumping in MEM/NEM resonators. Optical pumping has often been implemented to tune $Q_{e f f}$ of MEM/NEM cantilevers. Metzger and Karrai were one of the first to report optomechanical $Q_{e f f}$ suppression of a microcantilever.¹² Harris et al. demonstrated optomechanical cooling with a similar setup.¹⁵⁶ Favero et al. demonstrated $Q_{e f f}$ enhancement and suppression of a microcantilever supporting a 1 μm in diameter mirror.¹⁵⁷ Gröblacher et al. cooled a microcantilever from 35 K to an effective temperature of 290 mK using radiation pressure.¹⁵⁸ Jourdan et al. tuned $Q_{e f f}$ of two different modes of an AFM cantilever in opposite directions using optical pumping.¹⁵⁹ Metzger et al. compared effective cooling using radiation pressure versus photothermal pressure in a microcantilever¹⁶⁰ and studied the nonlinear dynamics of optical pumping-induced self-oscillations in a similar device.¹⁶¹ Okamoto et al. studied $Q_{e f f}$ tuning in gallium arsenide (GaAs) microcantilevers induced by piezoelectric stress-mediated optical pumping.¹⁶² Hölscher et al. studied how the Fabry-Pérot setup influences the optomechanical $Q_{e f f}$ tuning of an AFM cantilever.¹⁶³ Fu et al. studied optomechanical bolometric $Q_{e f f}$ tuning in a microcantilever,¹⁶⁴ demonstrated self-oscillations of higher order modes,¹⁶⁵ and investigated the role of the laser spot position in the cantilever on the effective cooling.¹⁶⁶ Okamoto et al. demonstrated $Q_{e f f}$ suppression and self-oscillations in an n-type/intrinsic GaAs bilayer cantilever.¹⁶⁷ Laurent et al. studied $Q_{e f f}$ tuning in a microcantilever due to coherent coupling to the noise in the interferometer.¹⁶⁸ Watanabe et al. observed self-oscillations in an aluminum gallium arsenide/gallium arsenide (AlGaAs/GaAs) microcantilever.¹⁶⁹ Vanner et al. cooled a mode in a cantilever using pulses of light with a duration much shorter than the mechanical vibration period.¹⁷⁰ Li-Ping et al. showed that the optomechanical effective cooling factors in a microcantilever strongly depend on the ambient temperature.¹⁷¹ Okamoto et al. studied the electron-hole generation back-action mechanism in an AlGaAs/GaAs microcantilever.¹⁷²

The doubly clamped beam has also been a common platform for investigating optical pumping. Arcizet et al. observed radiation-pressure-induced self-oscillations in a silicon doubly clamped beam.¹⁷³ Gigan et al. cooled the fundamental mode of a clamped-clamped beam from room temperature down to an effective temperature of 10 K using radiation pressure.⁸³ Heidmann et al. self-oscillated a resonator consisting of a mirror mounted to a doubly clamped beam.¹⁷⁴ Teufel et al. cooled a mode in a clamped-clamped nanobeam down to a phonon occupancy of 140 by coupling it to a microwave cavity.^33,175 Anetsberger et al. self-oscillated a nanomechanical beam by coupling it to an optical mode in a toroidal structure.¹⁷⁶ Gröblacher et al. effectively cooled a microbeam mode down to an occupancy of 30 phonons.¹⁷⁷ Hertzberg et al. demonstrated single tone optical pumping and two-tone back-action-evading measurement of a nanomechanical beam, achieving a sensitivity near the beam quantum zero-point motion.¹⁵² Massel et al. used optomechanical pumping of a NEM beam to amplify microwave signals.¹⁷⁸ Khurgin et al. modeled and demonstrated phonon lasing in a set of beams using optical pumping.^179,180 Faust et al. demonstrated self-oscillations in a silicon nitride beam by coupling it to a microwave cavity.¹⁸¹ Massel et al. studied optomechanical coupling between two nanobeams and a microwave circuit.¹⁸² Bagheri et al. demonstrated synchronization between two optomechanically self-oscillating NEM resonators.¹⁸³ Yuvaraj et al. observed self-oscillations in their buckled beam resonator with red detuned pumping, which they attributed to an additional phase shift in the bolometric forcing.¹⁸⁴ Blocher et al. studied frequency entrainment of an optomechanically self-oscillating microbeam.¹⁸⁵ Thijssen et al. enhanced $Q_{e f f}$ in an array of silicon nitride beams using optical pumping.¹⁸⁶ Shlomi et al. studied synchronization of fiber-mounted gold mirror self-oscillations with the laser frequency.¹⁸⁷ Khanaliloo et al. demonstrated optomechanical $Q_{e f f}$ suppression and self-oscillations in a diamond nanomechanical beam to induce electron spin transitions in nitrogen vacancy centers.¹⁸⁸ de Alba et al. studied the nonlinear dynamics of an optomechanically self-oscillating silicon nitride and niobium nanowire.¹⁸⁹

The centrally supported microtoroid (disk with a stem) is an excellent geometry for studying optical pumping because the structure has simultaneous high $Q$ optical and mechanical vibrational modes. Zalalutdinov et al. were one of the first to observe $Q_{e f f}$ enhancement due to interactions between the mechanical motion of a centrally supported disk and the interferometric laser standing wave pattern⁸⁴ and studied synchronization between the optomechanical self-oscillations and a direct or parametric drive.¹⁹⁰ Aubin et al. experimentally and theoretically studied self-oscillations in a similar structure.¹⁹¹ Kippenberg et al. demonstrated self-oscillations in a centrally supported microtoroid with a lip around the circumference for supporting optical modes.⁸⁵ Rokhsari et al. studied the optomechanical coupling mechanism in a similar microtoroid geometry.¹⁹² Carmon et al. characterized the time domain behavior of self-oscillations in a microtoroid optomechanical resonator.¹⁹³ Hossein-Zadeh et al. measured the sub-threshold mechanical linewidth and the phase noise of an optomechanical self-oscillator and discussed the prospects for optomechanical self-oscillators as photonic frequency references.¹⁹⁴ Pandey et al. experimentally validated their modeling of entrainment of a self-oscillating disk resonator.¹⁹⁵ Schliesser et al. cooled a mode in a microtoroid down from room temperature to 11 K using optical pumping¹⁵⁵ and combined cryogenics and optomechanical back-action to cool the same device down to an occupancy of 63 phonons.^196,197 Lin et al. dramatically improved the $Q_{e f f}$ tuning efficiency using a pair of concentric microtoroids.¹⁹⁸ Grudinin et al. studied optomechanical self-oscillations in a pair of adjacent microtoroids.¹⁹⁹ Verhagen et al. combined cryogenics and optomechanical back-action to cool a microtoroid to an occupancy of below 2 phonons.²⁰⁰ Rivière et al. cooled a similar device to a phonon occupancy of 9 using optical pumping and cryogenics.²⁰¹ Harris et al. studied the influence of self-oscillations on the resonator read-out sensitivity.²⁰² Taylor et al. studied the phase noise of optomechanical self-oscillations in a microtoroid.²⁰³ Zhang et al. self-oscillated and synchronized two microtoroids using optical pumping²⁰⁴ and demonstrated phase noise reduction in arrays of up to seven optically synchronized microtoroids.²⁰⁵ Gil-Santos studied synchronization in an array of microtoroids that were coupled via a shared optical waveguide.²⁰⁶ Suzuki et al. studied the interactions between four-wave mixing and optomechanical self-oscillations in a microtoroid.²⁰⁷

Optical pumping has been demonstrated in a variety of membrane resonators. Thompson et al. optomechanically cooled a silicon nitride membrane from 300 mK to an effective temperature of 7 mK.⁸² Teufel et al. coupled an LC microwave circuit to a membrane resonator for optomechanical $Q_{e f f}$ suppression of more than 300-fold.²⁰⁸ In the following year, Teufel et al. combined conventional cooling and optical pumping to cool a similar device down to a phonon expectation value of 0.34, reaching the quantum ground state.⁷⁷ Barton et al. self-oscillated a graphene membrane using optical pumping.²⁰⁹ Purdy et al. optomechanically cooled a silicon nitride membrane from a bath temperature of 5 K to an occupancy of below 10 phonons.²¹⁰ Flowers-Jacobs et al. suppressed $Q_{e f f}$ of a silicon nitride membrane.²¹¹ Usami et al. observed back-action in a GaAs membrane due to optomechanically induced charge carrier generation and recombination.¹⁴⁹ Suh et al. suppressed $Q_{e f f}$ of a silicon nitride membrane by coupling it to a superconducting microwave circuit.²¹² Adiga et al. enhanced $Q_{e f f}$ of a graphene-coated silicon nitride membrane via optical pumping.²¹³ Fainstein et al. observed optomechanical self-oscillations in an AlGaAs/GaAs stack due to photon-phonon confinement.²¹⁴ Dhayalan et al. characterized the phase noise of an optomechanically self-oscillating gold membrane.²¹⁵ Pirkkalainen et al. demonstrated quadrature-dependent squeezing of the mechanical motion of a drum resonator by simultaneously pumping at the sum and difference frequencies.²¹⁶ Peterson et al. optically cooled a membrane mode down to a phonon occupancy of 0.2 and studied the coupling of the mechanical mode to the optical mode while both were in their quantum regime.²¹⁷ Clark et al. optomechanically cooled a drum resonator down to a phonon occupancy of 0.19 using squeezed light.²¹⁸ Houri et al. studied synchronization of an optomechanically self-oscillating graphene membrane to a signal near resonance and near twice the resonant frequency.²¹⁹ Inoue et al. studied the nonlinear dynamics of an optomechanically self-oscillating graphene resonator.²²⁰

A membrane supported by four beams is another common structure for implementing optical pumping. Zaitsev et al. showed optomechanical $Q_{e f f}$ suppression and self-oscillations of a gold palladium mirror suspended by four aluminum beams.²²¹ Suchoi et al. observed self-oscillations in an aluminum resonator²²² and studied transitions from a cooled to a self-oscillating state in the same structure.²²³ Yang et al. optomechanically self-oscillated an indium-phosphide membrane supported by four narrow beams.²²⁴

There are several demonstrations of optical pumping using micro-spherical resonators. Tomes and Carmon optomechanically self-oscillated a silica microsphere.²²⁵ Park and Wang optomechanically cooled a silica microsphere from 1.4 K down to an occupancy of 37 phonons.²²⁶ Bahl et al. tuned $Q_{e f f}$ of a silica microsphere using Brillouin light-scattering-mediated optical pumping.⁸⁶ Kim and Bahl demonstrated optical pumping of a silica microsphere using two simultaneous optical modes.²²⁷

Embedding a photonic crystal into a mechanical resonator enables the high resolution measurement of GHz frequency mechanical modes and strong optical pumping. The optomechanical coupling at high frequencies afforded by this design dramatically eases the cryostat temperature required prior to optically pumping a mechanical mode into its ground state. Chan et al. cooled a photonic crystal nanobeam down from 20 K into the quantum ground state using optical pumping,⁸⁷ the highest cryostat temperature reported to date. Safavi-Naeini et al. studied the zero-point fluctuations of a similar resonator that was optomechanically cooled to near its quantum ground state.⁷⁸ Krause et al. demonstrated $Q_{e f f}$ suppression in an accelerometer with an ultra-sensitive photonic cavity displacement readout.³¹ Woolf et al. tuned $Q_{e f f}$ in a membrane with an embedded photonic cavity.²²⁸ Zhu et al. suppressed $Q_{e f f}$ and induced self-oscillations in a gold/silicon nitride membrane with a patterned photonic crystal.²²⁹ Patel et al. observed $Q_{e f f}$ suppression in the mechanical modes of a phononic waveguide when they optically pumped the adjacent photonic crystal nanobeam mode at its red sideband and suggested that this was due to resonant coupling of the phononic waveguide modes to the localized mechanical mode in the nanobeam.²³⁰

TABLE III.

Reports of $Q_{e f f}$ tuning via optical pumping in chronological order. Papers that do not mention their device $Q$ are denoted by “N.M.”. Reports of self-oscillations are denoted by “self-osc.”.

Reference	$Q_{e f f}$ type	$Q$	$Q_{e f f}$
Zalalutdinov et al.⁸⁴	Enhance	10 k	Self-osc.
Zalalutdinov et al.¹⁹⁰	Enhance	10 k	Self-osc.
Aubin et al.¹⁹¹	Enhance	7.5 k	Self-osc.
Metzger and Karrai¹²	Both	2 k	120, self-osc.
Carmon et al.¹⁹³	Enhance	1.2 k	Self-osc.
Rokhsari et al.¹⁹²	Enhance	630	Self-osc.
Kippenberg et al.⁸⁵	Enhance	3.5 k	Self-osc.
Pandey et al.¹⁹⁵	Enhance	10 k	Self-osc.
Schliesser et al.¹⁵⁵	Both	2.9 k	110, self-osc
Arcizet et al.¹⁷³	Both	10 k	330, self-osc.
Gigan et al.⁸³	Suppress	10 k	330
Rokhsari et al.¹⁹²	Enhance	630	Self-osc.
Hossein-Zadeh et al. ¹⁹⁴	Enhance	2 k	Self-osc.
Heidmann et al.¹⁷⁴	Both	2 k	67, self-osc.
Harris et al.¹⁵⁶	Suppress	1.9 k	370
Favero et al.¹⁵⁷	Both	1.1 k	640, 3.2 k
Jourdan et al.¹⁵⁹	Both	2 k	950, 8.3 k
Teufel et al.³³	Both	3.8 k	2.6 k, 5.6 k
Metzger et al.¹⁶⁰	Suppress	260	28
Metzger et al.¹⁶¹	Enhance	290	Self-osc.
Thompson et al.⁸²	Suppress	1.1 M	25 k
Teufel et al.¹⁷⁵	Suppress	500 k	14 k
Schliesser et al.¹⁹⁶	Suppress	30 k	46
Gröblacher et al.¹⁵⁸	Suppress	2.1 k	200 m
Anetsberger et al.¹⁷⁶	Enhance	70 k	Self-osc.
Lin et al.¹⁹⁸	Both	4	170 m, self-osc.
Gröblacher et al.¹⁷⁷	Suppress	30 k	8
Okamoto et al.¹⁶²	Both	20 k	5.7 k, 90 k
Tomes and Carmon²²⁵	Enhance	770	Self-osc.
Schliesser et al.¹⁹⁷	Suppress	2 M	160 k
Park and Wang²²⁶	Suppress	1.5 k	430
Hölscher et al.¹⁶³	Both	180 k	110 k, 410 k
Hertzberg et al.¹⁵²	Both	280 k	56 k, 2.8 M
Fu et al.¹⁶⁴	Both	4.1 k	1.2 k, 14 k
Grudinin et al.¹⁹⁹	Enhance	1 k	Self-osc.
Teufel et al.²⁰⁸	Suppress	360 k	11 k
Bahl et al.⁸⁶	Both	12 k	810, 480 k
Rivière et al.²⁰¹	Suppress	10 k	450
Verhagen et al.²⁰⁰	Suppress	5.2 k	45
Teufel et al.⁷⁷	Suppress	330 k	35
Okamoto et al.¹⁶⁷	Both	6.5 k, 8.5 k	2.1 k, self-osc.
Chan et al.⁸⁷	Suppress	100 k	250
Zaitsev et al.²²¹	Both	240 k	80 k, self-osc.
Laurent et al.¹⁶⁸	Both	15 k	7.5 k, self-osc.
Fu et al.¹⁶⁵	Enhance	1.6 k	Self-osc.
Massel et al.¹⁷⁸	Enhance	27 k	Self-osc.
Krause et al.³¹	Suppress	1.4 M	14 k
Fu et al.¹⁶⁶	Both	1.5 k	750, 1.7 k
Faust et al.¹⁸¹	Both	290 k	150 k, self-osc.
Barton et al.²⁰⁹	Both	500	240, self-osc.
Massel et al.¹⁸²	Suppress	35 k	1.6 k
Harris et al.²⁰²	Enhance	320	Self-osc.
Watanabe et al.¹⁶⁹	Enhance	1.3 k	Self-osc.
Safavi-Naeini et al.⁷⁸	Suppress	93 k	2.6 k
Purdy et al.²¹⁰	Suppress	14 M	290
Khurgin et al.¹⁷⁹	Enhance	19 k	Self-osc.
Taylor et al.²⁰³	Enhance	600	Self-osc.
Usami et al.¹⁴⁹	Both	2.3 M	31 k, self-osc.
Flowers-Jacobs et al.²¹¹	Suppress	65 k	3.3 k
Zhang et al.²⁰⁴	Enhance	3.4 k.	Self-osc.
Suh et al.²¹²	Enhance	74 k	460 k
Fainstein et al.²¹⁴	Enhance	100 k	Self-osc.
Vanner et al.¹⁷⁰	Suppress	31 k	450
Blocher et al.¹⁸⁵	Enhance	N.M.	Self-osc.
Yuvaraj et al.¹⁸⁴	Enhance	N.M.	Self-osc.
Bagheri et al.¹⁸³	Enhance	6 k	Self-osc.
Woolf et al.²²⁸	Both	3.6 k	640, self-osc.
Adiga et al.²¹³	Enhance	17 k	80 k
Li-Ping et al.¹⁷¹	Suppress	25 k	2.1 k
Suchoi et al.²²²	Enhance	7.4 k	Self-osc.
Thijssen et al.¹⁸⁶	Enhance	1 k	5 k
Dhayalan et al.²¹⁵	Enhance	N.M.	Self-osc.
Suchoi et al.²²³	Enhance	270 k	Self-osc.
Shlomi et al.¹⁸⁷	Enhance	N.M.	Self-osc.
Yang et al.²²⁴	Enhance	3.6 k	Self-osc.
Okamoto et al.¹⁷²	Both	5.6 k	930, 19 k
Zhang et al.²⁰⁵	Enhance	3.4 k	Self-osc.
Pirkkalainen et al.²¹⁶	Suppress	39 k	370
Khanaliloo et al.¹⁸⁸	Both	720 k	7.2 k, self-osc.
Zhu et al.²²⁹	Enhance	61 k	18 k, Self-osc.
Kim and Bahl²²⁷	Suppress	8.3 k	3.8 k
Peterson et al.²¹⁷	Suppress	8.2 M	1.6 k
Clark et al.²¹⁸	Suppress	670 k	1.7 k
Inoue et al.²²⁰	Enhance	300	Self-osc.
Suzuki et al.²⁰⁷	Enhance	100	Self-osc.
Houri et al.²¹⁹	Enhance	430	Self-osc.
de Alba et al.¹⁸⁹	Enhance	5 k	Self-osc.
Gil-Santos et al.²⁰⁶	Enhance	1 k	Self-osc.
Patel et al.²³⁰	Suppress	200 k	33 k

Reference	$Q_{e f f}$ type	$Q$	$Q_{e f f}$
Zalalutdinov et al.⁸⁴	Enhance	10 k	Self-osc.
Zalalutdinov et al.¹⁹⁰	Enhance	10 k	Self-osc.
Aubin et al.¹⁹¹	Enhance	7.5 k	Self-osc.
Metzger and Karrai¹²	Both	2 k	120, self-osc.
Carmon et al.¹⁹³	Enhance	1.2 k	Self-osc.
Rokhsari et al.¹⁹²	Enhance	630	Self-osc.
Kippenberg et al.⁸⁵	Enhance	3.5 k	Self-osc.
Pandey et al.¹⁹⁵	Enhance	10 k	Self-osc.
Schliesser et al.¹⁵⁵	Both	2.9 k	110, self-osc
Arcizet et al.¹⁷³	Both	10 k	330, self-osc.
Gigan et al.⁸³	Suppress	10 k	330
Rokhsari et al.¹⁹²	Enhance	630	Self-osc.
Hossein-Zadeh et al. ¹⁹⁴	Enhance	2 k	Self-osc.
Heidmann et al.¹⁷⁴	Both	2 k	67, self-osc.
Harris et al.¹⁵⁶	Suppress	1.9 k	370
Favero et al.¹⁵⁷	Both	1.1 k	640, 3.2 k
Jourdan et al.¹⁵⁹	Both	2 k	950, 8.3 k
Teufel et al.³³	Both	3.8 k	2.6 k, 5.6 k
Metzger et al.¹⁶⁰	Suppress	260	28
Metzger et al.¹⁶¹	Enhance	290	Self-osc.
Thompson et al.⁸²	Suppress	1.1 M	25 k
Teufel et al.¹⁷⁵	Suppress	500 k	14 k
Schliesser et al.¹⁹⁶	Suppress	30 k	46
Gröblacher et al.¹⁵⁸	Suppress	2.1 k	200 m
Anetsberger et al.¹⁷⁶	Enhance	70 k	Self-osc.
Lin et al.¹⁹⁸	Both	4	170 m, self-osc.
Gröblacher et al.¹⁷⁷	Suppress	30 k	8
Okamoto et al.¹⁶²	Both	20 k	5.7 k, 90 k
Tomes and Carmon²²⁵	Enhance	770	Self-osc.
Schliesser et al.¹⁹⁷	Suppress	2 M	160 k
Park and Wang²²⁶	Suppress	1.5 k	430
Hölscher et al.¹⁶³	Both	180 k	110 k, 410 k
Hertzberg et al.¹⁵²	Both	280 k	56 k, 2.8 M
Fu et al.¹⁶⁴	Both	4.1 k	1.2 k, 14 k
Grudinin et al.¹⁹⁹	Enhance	1 k	Self-osc.
Teufel et al.²⁰⁸	Suppress	360 k	11 k
Bahl et al.⁸⁶	Both	12 k	810, 480 k
Rivière et al.²⁰¹	Suppress	10 k	450
Verhagen et al.²⁰⁰	Suppress	5.2 k	45
Teufel et al.⁷⁷	Suppress	330 k	35
Okamoto et al.¹⁶⁷	Both	6.5 k, 8.5 k	2.1 k, self-osc.
Chan et al.⁸⁷	Suppress	100 k	250
Zaitsev et al.²²¹	Both	240 k	80 k, self-osc.
Laurent et al.¹⁶⁸	Both	15 k	7.5 k, self-osc.
Fu et al.¹⁶⁵	Enhance	1.6 k	Self-osc.
Massel et al.¹⁷⁸	Enhance	27 k	Self-osc.
Krause et al.³¹	Suppress	1.4 M	14 k
Fu et al.¹⁶⁶	Both	1.5 k	750, 1.7 k
Faust et al.¹⁸¹	Both	290 k	150 k, self-osc.
Barton et al.²⁰⁹	Both	500	240, self-osc.
Massel et al.¹⁸²	Suppress	35 k	1.6 k
Harris et al.²⁰²	Enhance	320	Self-osc.
Watanabe et al.¹⁶⁹	Enhance	1.3 k	Self-osc.
Safavi-Naeini et al.⁷⁸	Suppress	93 k	2.6 k
Purdy et al.²¹⁰	Suppress	14 M	290
Khurgin et al.¹⁷⁹	Enhance	19 k	Self-osc.
Taylor et al.²⁰³	Enhance	600	Self-osc.
Usami et al.¹⁴⁹	Both	2.3 M	31 k, self-osc.
Flowers-Jacobs et al.²¹¹	Suppress	65 k	3.3 k
Zhang et al.²⁰⁴	Enhance	3.4 k.	Self-osc.
Suh et al.²¹²	Enhance	74 k	460 k
Fainstein et al.²¹⁴	Enhance	100 k	Self-osc.
Vanner et al.¹⁷⁰	Suppress	31 k	450
Blocher et al.¹⁸⁵	Enhance	N.M.	Self-osc.
Yuvaraj et al.¹⁸⁴	Enhance	N.M.	Self-osc.
Bagheri et al.¹⁸³	Enhance	6 k	Self-osc.
Woolf et al.²²⁸	Both	3.6 k	640, self-osc.
Adiga et al.²¹³	Enhance	17 k	80 k
Li-Ping et al.¹⁷¹	Suppress	25 k	2.1 k
Suchoi et al.²²²	Enhance	7.4 k	Self-osc.
Thijssen et al.¹⁸⁶	Enhance	1 k	5 k
Dhayalan et al.²¹⁵	Enhance	N.M.	Self-osc.
Suchoi et al.²²³	Enhance	270 k	Self-osc.
Shlomi et al.¹⁸⁷	Enhance	N.M.	Self-osc.
Yang et al.²²⁴	Enhance	3.6 k	Self-osc.
Okamoto et al.¹⁷²	Both	5.6 k	930, 19 k
Zhang et al.²⁰⁵	Enhance	3.4 k	Self-osc.
Pirkkalainen et al.²¹⁶	Suppress	39 k	370
Khanaliloo et al.¹⁸⁸	Both	720 k	7.2 k, self-osc.
Zhu et al.²²⁹	Enhance	61 k	18 k, Self-osc.
Kim and Bahl²²⁷	Suppress	8.3 k	3.8 k
Peterson et al.²¹⁷	Suppress	8.2 M	1.6 k
Clark et al.²¹⁸	Suppress	670 k	1.7 k
Inoue et al.²²⁰	Enhance	300	Self-osc.
Suzuki et al.²⁰⁷	Enhance	100	Self-osc.
Houri et al.²¹⁹	Enhance	430	Self-osc.
de Alba et al.¹⁸⁹	Enhance	5 k	Self-osc.
Gil-Santos et al.²⁰⁶	Enhance	1 k	Self-osc.
Patel et al.²³⁰	Suppress	200 k	33 k

C. Mechanical pumping

The study of energy transfer between different modes of a mechanical resonator has recently become a very active area of research. Coupling of a mechanical mode to other modes in the resonator can tune the effective quality factor or even the mechanical quality factor of the mode. To tune $Q_{e f f}$ ⁠, a technique analogous to optical pumping is used, where the electromagnetic mode in a microwave or optical cavity is replaced by a mechanical mode in the resonator. The second mechanical mode usually has a higher frequency than the mode that experiences the $Q_{e f f}$ tuning. The resonator is pumped at either the sum (blue sideband) or the difference (red sideband) in frequency of the modes.

Effective feedback parameters for Eq. (16) can be derived for mechanical pumping. From de Alba et al.,²³¹ the conservative nonlinear dynamics of two mechanical modes can be represented by the following Hamiltonian:

H = \sum_{j = 0}^{1} H_{j} + T_{01} x_{0} x_{1}^{2} + T_{10} x_{1} x_{0}^{2} + α_{01} x_{0}^{2} x_{1}^{2},

(56)

where

H_{j} = \frac{m_{j} {\dot{x}}_{j}^{2}}{2} + \frac{m_{j} ω_{j}^{2} x_{j}^{2}}{2} + L_{j} x_{j} + S_{j} x_{j}^{2} + T_{j} x_{j}^{3} + α_{j} x_{j}^{4},

(57)

where the index $j = 0$ corresponds to the lower frequency mode and $j = 1$ corresponds to the higher frequency mode. $L_{j}$ exerts a constant force on mode $j$ ⁠. $S_{j}$ modifies the linear stiffness, and $T_{j}$ and $α_{j}$ modify the nonlinear stiffness of mode $j$ ⁠. $α_{01}$ shifts the resonant frequency of the lower frequency mode by an amount proportional to the squared amplitude of the higher frequency mode. $T_{01}$ accounts for the $Q_{e f f}$ tuning of the lower frequency mode, and $T_{10}$ accounts for the $Q_{e f f}$ tuning of the higher frequency mode

k^{'} = m_{0} {(Ω_{0} + \frac{2 G^{2} x_{p}^{2} {Δ ω}_{p} [\frac{ω_{1}^{2}}{4 Q_{1}^{2}} - ω_{0}^{2} + {Δ ω}_{p}^{2}]}{[\frac{ω_{1}^{2}}{4 Q_{1}^{2}} + {(ω_{0} - {Δ ω}_{p})}^{2}] [\frac{ω_{1}^{2}}{4 Q_{1}^{2}} + {(ω_{0} + {Δ ω}_{p})}^{2}]})}^{2},

(58)

b^{'} = - \frac{4 G^{2} x_{p}^{2} m_{0} ω_{1} {Δ ω}_{p} Ω_{0}}{Q_{1} [\frac{ω_{1}^{2}}{4 Q_{1}^{2}} + {(ω_{0} - {Δ ω}_{p})}^{2}] [\frac{ω_{1}^{2}}{4 Q_{1}^{2}} + {(ω_{0} + {Δ ω}_{p})}^{2}]},

(59)

where

Ω_{0} - ω_{0} = \frac{1}{m_{0} ω_{0}} [2 S_{0} - \frac{12 T_{0} (T_{01} x_{p}^{2} + \frac{L_{0}}{2})}{(m_{0} ω_{0}^{2} + 4 S_{0})} + 4 α_{01} x_{p}^{2}],

(60)

and where $Δ ω_{p} = ω_{p} - ω_{1}$ is the frequency detuning of the pump off the higher frequency mode, $ω_{0}$ is the frequency of the lower frequency mode, $G = \frac{d ω_{1}}{d x}$ is the linearized coupling rate between the two modes, $x_{p}$ is the pump amplitude, $x_{1}$ is the amplitude of the higher frequency mode, $m_{0}$ is the lumped mass of the lower frequency mode, and $Q_{1}$ is the mechanical quality factor of the higher frequency mode.

Table IV summarizes the reports of $Q_{e f f}$ tuning via mechanical pumping in MEM/NEM resonators. While many papers report optical pumping (see Sec. V B), there remain very few studies of mechanical pumping. Dougherty et al. studied mechanical pumping of a magnetic force microscope cantilever.⁸⁸ Olkhovets et al. observed self-oscillations in a pair of coupled torsional resonators when they pumped the resonators at their sum frequency.²³² Napoli et al. observed self-oscillations of two coupled cantilevers with frequencies $ω_{1}$ and $ω_{2}$ when they pumped the system either at $2 ω_{1}$ or $2 ω_{2}$ (parametric pumping) or when they pumped the system at $ω_{1} + ω_{2}$ (mechanical pumping).²³³ Baskaran and Turner demonstrated mechanically pumped self-oscillations in a torsional comb-drive resonator.²³⁴ Inspired by recent developments in optical pumping, Venstra et al. demonstrated $Q_{e f f}$ suppression and enhancement by coupling two flexural modes of the same microcantilever.¹³ Mahboob et al. observed $Q_{e f f}$ suppression and parametric mode-splitting,⁸⁹ systematically studied the influence of coupling seven different modes on $Q_{e f f},$ ²³⁵ and studied self-oscillations,²³⁶ all in a gallium arsenide heterostructure beam. Okamoto et al. demonstrated self-oscillations by pumping two coupled gallium arsenide beams at their sum frequency.⁹⁰ Patil et al. self-oscillated a silicon nitride membrane using mechanical pumping.²³⁷ Mahboob et al. demonstrated phase-dependent $Q_{e f f}$ tuning and thermal noise squeezing in two coupled doubly clamped beam resonators.²³⁸ Mahboob et al. next considered the interactions between a doubly clamped MEM beam and an embedded NEM beam and observed self-oscillations²³⁹ and multistability²⁴⁰ in the MEM beam when the NEM beam was driven at one of its resonances. de Alba et al. observed $Q_{e f f}$ suppression and self-oscillations by coupling various modes of a graphene membrane.²³¹ In the same month, Mathew et al. published a report of self-oscillations and mode splitting in a graphene drum resonator.⁹¹ Sun et al. discovered perfect phase noise anti-correlation in two self-oscillating modes of a torsional resonator and proposed a feedback scheme for improving the phase noise of an oscillator using the anti-correlation.²⁴¹ Renault et al. demonstrated parity time symmetry breaking and Rabi self-oscillations using three different modes of a piezoelectric doubly clamped beam resonator.²⁴² Mahboob et al. studied vibration correlations of two mechanical modes that were pumped into self-oscillations.²⁴³

Mechanical modal interactions comprise many more phenomena than mechanical pumping alone. Lin et al. demonstrated passive enhancement of the mechanical quality factor of a wineglass disk resonator by coupling it to an array of high $Q$ resonators.²⁴⁴ Karabalin et al. studied the chaotic dynamics of two coupled beams driven to large amplitudes.²⁴⁵ van der Avoort et al. observed amplitude saturation in a double cantilever due to coupling between the in-plane and out-of-plane modes.²⁴⁶ Westra et al. showed that driving the fundamental mode of a clamped-clamped beam shifted the resonant frequencies of the other modes.²⁴⁷ Faust et al. studied coupling and avoided crossing between the two transverse modes of a doubly clamped nanobeam.²⁴⁸ Eichler et al. demonstrated nonlinear coupling between different modes in a carbon nanotube resonator.²⁴⁹ Huang et al. developed a transduction scheme for measuring the motion of a target resonator by coupling it to a detector resonator.²⁵⁰ Zhu et al. observed a tunable tenfold reduction in $Q$ of a square-extensional resonator, possibly due to coupling to other modes.²⁵¹ Okamoto et al. rapidly switched the $Q$ of a doubly clamped beam by controlling the coupling to another beam.²⁵² Flader et al. tuned the quality factor of a disk resonating gyroscope mode by controlling the coupling to other modes in the resonator.²⁵³ Verbiest et al. studied frequency tuning due to the coupling between a silicon beam and a comb-drive actuator.²⁵⁴ Ilyas et al. demonstrated amplitude enhancement due to the coupling between two polyimide beams.²⁵⁵ Chen et al. observed constant amplitude vibrations in a mode after cutting off the drive due to energy transfer from a second mode.²⁵⁶ Taheri-Tehrani et al. demonstrated 3:1 frequency synchronization in a disk resonating gyroscope.²⁵⁷ Güttinger et al. tuned the mechanical $Q$ of a graphene resonator by controlling the fluctuation contributions from other modes.²⁵⁸ Gajo et al. studied modal coupling between the transverse modes of two silicon nitride resonators.²⁵⁹

TABLE IV.

Reports of $Q_{e f f}$ tuning via mechanical pumping in chronological order. Reports of self-oscillations are denoted by “self-osc.”.

Reference	$Q_{e f f}$ type	$Q$	$Q_{e f f}$
Dougherty et al.⁸⁸	Enhance	2 k	Self-osc.
Olkhovets et al.²³²	Enhance	270	Self-osc.
Napoli et al.²³³	Enhance	3 k	Self-osc.
Baskaran and Turner²³⁴	Enhance	570	Self-osc.
Venstra et al.¹³	Both	4.6 k	230, 5.9 k
Mahboob et al.⁸⁹	Suppress	156 k	78 k
Mahboob et al.²³⁵	Suppress	82 k	2.9 k
Mahboob et al.²³⁶	Enhance	159 k	Self-osc.
Okamoto et al.⁹⁰	Enhance	14 k	Self-osc.
Patil et al.²³⁷	Enhance	50 M	Self-osc.
Mahboob et al.²³⁸	Enhance	1.3 k	Self-osc.
Mahboob et al.²³⁹	Both	290 k	30 k, Self-osc.
de Alba et al.²³¹	Both	57	40, Self-osc.
Mahboob et al.²⁴⁰	Enhance	175 k	Self-osc.
Mathew et al.⁹¹	Enhance	1.1 k	Self-osc.
Sun et al.²⁴¹	Both	110 k	45 k, self-osc.
Renault et al.²⁴²	Enhance	150 k	Self-osc.
Mahboob et al.²⁴³	Enhance	141 k, 230 k	Self-osc.

Reference	$Q_{e f f}$ type	$Q$	$Q_{e f f}$
Dougherty et al.⁸⁸	Enhance	2 k	Self-osc.
Olkhovets et al.²³²	Enhance	270	Self-osc.
Napoli et al.²³³	Enhance	3 k	Self-osc.
Baskaran and Turner²³⁴	Enhance	570	Self-osc.
Venstra et al.¹³	Both	4.6 k	230, 5.9 k
Mahboob et al.⁸⁹	Suppress	156 k	78 k
Mahboob et al.²³⁵	Suppress	82 k	2.9 k
Mahboob et al.²³⁶	Enhance	159 k	Self-osc.
Okamoto et al.⁹⁰	Enhance	14 k	Self-osc.
Patil et al.²³⁷	Enhance	50 M	Self-osc.
Mahboob et al.²³⁸	Enhance	1.3 k	Self-osc.
Mahboob et al.²³⁹	Both	290 k	30 k, Self-osc.
de Alba et al.²³¹	Both	57	40, Self-osc.
Mahboob et al.²⁴⁰	Enhance	175 k	Self-osc.
Mathew et al.⁹¹	Enhance	1.1 k	Self-osc.
Sun et al.²⁴¹	Both	110 k	45 k, self-osc.
Renault et al.²⁴²	Enhance	150 k	Self-osc.
Mahboob et al.²⁴³	Enhance	141 k, 230 k	Self-osc.

D. Thermal-piezoresistive pumping

Thermal-piezoresistive pumping uses an internal feedback mechanism in a resonator fabricated from a semiconductor with appreciable material piezoresistivity.¹⁴ A thermal-piezoresistive resonator is designed with one or more thermal actuators that feed energy into the motion via thermal expansion when a direct current flows through them. During motion at resonance, the actuator beam undergoes periodic elongation and contraction. This modulates the actuator electrical resistance, which modulates the Joule heating due to the constant current, and thus the thermal expansion. The thermal expansion in the actuator beam contributes a position-proportional force and velocity-proportional force to the motion, shifting the resonant frequency and modifying the effective quality factor. The necessary phase lag required for converting the position-dependent feedback to a velocity-proportional force is provided by the finite heat capacity of the actuator beams.¹⁴ Thermal pumping feeds energy into all modes that have anti-nodes at the contact points of the actuator beams with the vibrating mass. With increasing current, thermal pumping will simultaneously tune $Q_{e f f}$ of multiple modes in the resonator.

Thermal-piezoresistive pumping can be represented by effective feedback parameters whose magnitude can be tuned with a direct current. Starting from Steeneken et al.,¹⁴ we define $k^{'}$ and $b^{'}$ for Eq. (16) as

k^{'} = - \frac{4 π k_{t h} γ_{k} γ_{z} ρ_{d c} π_{l} α_{t e} Y k I_{d c}^{2}}{L λ_{h} A^{2} [{(\frac{4 π k_{t h}}{L λ_{h}})}^{2} + {(\frac{4 π k_{t h}}{L λ_{h}} + ω_{0} ρ_{d} c_{p})}^{2}]},

(61)

b^{'} = \frac{γ_{k} γ_{z} ρ_{d c} π_{l} α_{t e} Y m ω_{0} (\frac{4 π k_{t h}}{L λ_{h}} + ω_{0} ρ_{d} c_{p}) I_{d c}^{2}}{A^{2} [{(\frac{4 π k_{t h}}{L λ_{h}})}^{2} + {(\frac{4 π k_{t h}}{L λ_{h}} + ω_{0} ρ_{d} c_{p})}^{2}]},

(62)

where $γ_{k}$ accounts for the fraction of the strain energy that is concentrated in the engine beam and is given by

γ_{k} = \frac{\int_{V_{engine}} ε_{a c}^{2} (r) d r}{\int_{V} ε_{a c}^{2} (r) d r} .

(63)

$L$ is the thermal actuator length, $A$ is the thermal actuator cross-sectional area, $α_{t e}$ is the thermal expansion coefficient, $k_{t h}$ is the thermal conductivity, $ρ_{d c}$ is the unstressed electrical resistivity, $π_{l}$ is the longitudinal piezoresistive coefficient parallel to the actuator, $Y$ is the Young's modulus parallel to the actuator, $c_{p}$ is the specific heat, $ρ_{d}$ is the mass density, $I_{d c}$ is the direct current flowing through the actuator, $ω_{0}$ is the angular resonant frequency, $γ_{z}$ is a degradation factor to account for a finite parallel electrical impedance to ground at $ω_{0}$ ⁠, and $ε_{a c} (r)$ is the ac strain at a position $r$ in the geometry. $m$ is the lumped mass, and $k$ is the lumped stiffness of the mode.

The advantages of thermal pumping include the simplicity of using a direct current for $Q_{e f f}$ tuning, the ease of integration into commercial MEM/NEM sensor and oscillator designs, and its self-oscillation capabilities. The main disadvantages of thermal pumping include the large power consumption associated with flowing a current through the device and the pronounced temperature dependence of the effect. The temperature rise which accompanies thermal pumping also changes the mechanical properties and dissipation in the resonator in such a way as to make the mechanism less effective.²⁶⁰ This leads to a pump saturation at large currents. The significant power consumption is not unique to thermal pumping; external feedback, optical pumping, and any $Q_{e f f}$ tuning technique that utilizes an optical readout will experience some residual heating due to photon absorption.

Thermal-piezoresistive self-oscillators have promise as ultra-high frequency, ultra-low power timing references. Commercial MEM oscillators use an external feedback loop to sustain the resonator oscillations. This approach has yielded excellent power consumption for kHz-range resonators; the lowest power 32 kHz oscillator used in consumer electronics operates on less than 1 μW [SiTime SiT1532]. However, the power consumption substantially increases at higher frequencies: the power consumption of a 1 MHz oscillator is more than 50 μW [SiTime SiT1576], and a 725 MHz oscillator consumes more than 200 mW of power [SiTime SiT9367]. Conversely, thermal pumping becomes more efficient at higher frequencies,²⁶¹ which opens up the possibility of MHz or even GHz MEM/NEM oscillators within wireless consumer electronics. Different groups are working to reduce the power consumption and increase the operating frequencies of thermal-piezoresistive oscillators. Lehée et al. suppressed $Q_{e f f}$ nearly 70-fold in an accelerometer using as little as 50 μW of power.²⁶² Li et al. demonstrated 840 kHz self-oscillators with a power consumption of 70 μW, comparable to commercial oscillators at similar frequencies.²⁶³ Hall et al. demonstrated 161 MHz self-oscillators, the highest frequency to date, with a power consumption below 20 mW.²⁶⁴

Thermal-piezoresistive oscillators may be helpful as in-cryostat signal generators for quantum computers. Each quantum bit (qubit) in a present-day quantum computer requires an external microwave signal generator to program its state, which introduces cryostat feedthrough cabling with parasitic reactance.²⁶⁵ If GHz frequency, sub-μW power consumption, low phase noise thermal-piezoresistive oscillators are developed, they could potentially be integrated directly with the qubits in the cryostat, eliminating the need for microwave feedthroughs and potentially aiding in scaling up the number of qubits.

Table V summarizes the reports of $Q_{e f f}$ tuning via thermal-piezoresistive pumping in MEM/NEM resonators. Thermal pumping was first demonstrated in a silicon proof mass supported by two beams of different widths.¹⁴ Phan et al. observed self-oscillations in this geometry and proposed a finite element model that predicted the threshold current for self-oscillations.²⁶⁶ Steeneken et al. developed a model for the feedback parameters and showed how the parallel impedance to the thermal actuator beam can be modified to control how the $Q_{e f f}$ changes with current.¹⁴ Using a similar geometry, Miller et al. theoretically and experimentally studied the ambient temperature dependence of the threshold current for self-oscillations.²⁶⁰ Miller, Zhu et al. extended the finite-element model of Phan et al.²⁶⁶ to account for the temperature- and doping-dependence of the piezoresistivity and used this model to predict $Q_{e f f}$ tuning in fabricated devices for varying dopant types, concentrations, geometries, and crystallographic directions.⁹²

The most common implementation of thermal-piezoresistive pumping involves a dual-plate geometry, which consists of two proof masses connected by one or more thermal actuator beams. At resonance, the two plates move towards and away from each other, exerting a periodic stress on the actuators and inducing the internal feedback. Rahafrooz and Pourkamali demonstrated self-oscillations in a dual-plate structure with a variety of different dimensions.²⁶⁷ Using a dual-plate geometry, Hajjam et al. were the first to demonstrate the use of thermal-piezoresistive self-oscillators for mass sensing.²⁶⁸ Their devices operated for nearly 2 h of continuous oscillations using only a direct current, with a resolution good enough to detect the adsorption of individual micro-particles. In a similar device, Rahafrooz and Pourkamali experimentally studied $Q_{e f f}$ enhancement below the self-oscillation threshold and analytically showed that thermal-piezoresistive oscillator power reduces quadratically with the reducing linear scaling parameter.⁹³ Rahafrooz and Pourkamali also characterized the frequency jitter for this self-oscillator geometry.²⁶⁹ Iqbal et al. observed more than a threefold increase in $Q_{e f f}$ with direct current in a dual-plate geometry with long connecting beams.²⁷⁰ Hall et al. observed thermal-piezoresistive self-oscillations of the out-of-plane flapping mode of a dual-plate resonator.²⁶⁴ Zhu et al. studied the influence of direct current on $Q_{e f f}$ ⁠, transconductance, and resonant frequency in a long beam dual-plate geometry.²⁷¹ Guo et al. investigated the sensitivity of their self-oscillator to the molecule concentration for a variety of different gases²⁷² and showed that thermal-piezoresistive $Q_{e f f}$ enhancement increased the sensitivity of a MEM gyroscope.²⁷³ Zhu et al. used a direct current to cancel out the frequency nonlinearity in a thermal-piezoresistive resonator.²⁷⁴ Mehdizadeh et al. increased the sensitivity of a Lorentz force magnetometer using the thermal-piezoresistive effect.²⁷⁵ Hall et al. tuned the threshold current for self-oscillations via laser illumination.²⁷⁶ Kumar et al. set a record in MEM Lorentz force magnetometry using thermal pumping to improve the sensitivity into the single picotesla regime.²⁷⁷ Ramezany et al. implemented a dual-plate design for narrow bandwidth amplification of electrical signals²⁷⁸ and scaled down the geometry to push the resonant frequency up to 730 MHz.⁹⁴ Chang et al. demonstrated a dual-plate mass sensor with a mass resolution of 100 fg.²⁷⁹ Chu et al. demonstrated a three plate thermal-piezoresistive mass sensor with a supplementary external amplifier to reduce the threshold current and demonstrated a mass resolution of 3 fg.²⁸⁰ Chu et al. next delineated piezoresistive feedthrough reduction and benchmarked their self-oscillator's performance against other MEM oscillators and mass-sensors.²⁸¹

Thermal pumping has been demonstrated using several other interesting geometries. Li et al. fabricated a flapping-mode self-oscillator with ultra-narrow thermal actuators to reduce the power consumption to 70 μW.²⁶³ Ansari and Rais-Zadeh observed plausible thermal-piezoresistive $Q_{e f f}$ enhancement in piezoelectric bulk acoustic resonators.²⁸² Lehée et al. demonstrated tunable bandwidth control of an accelerometer by flowing a current through a pair of p-type doped silicon nanowires tethered to the proof mass²⁶² and demonstrated self-oscillations by placing an appropriate capacitance in parallel to the actuators.²⁸³ Liu et al. demonstrated a 30 fg mass resolution in a combined complementary metal-oxide-MEM mass sensor.²⁸⁴

TABLE V.

Reports of $Q_{e f f}$ tuning via thermal-piezoresistive pumping in chronological order. Papers that do not mention their device $Q$ are denoted by “N.M.”. Reports of self-oscillations are denoted by “self-osc.”.

Reference	$Q_{e f f}$ type	$Q$	$Q_{e f f}$
Phan et al.²⁶⁶	Enhance	13 k	Self-osc.
Steeneken et al.¹⁴	Both	9 k	2 k, self-osc.
Rahafrooz and Pourkamali²⁶⁷	Enhance	49 k	Self-osc.
Hajjam et al.²⁶⁸	Enhance	N.M.	Self-osc.
Rahafrooz and Pourkamali⁹³	Enhance	2 k	Self-osc.
Rahafrooz and Pourkamali²⁶⁹	Enhance	N. M.	Self-osc.
Iqbal et al.²⁷⁰	Enhance	100 k	360 k
Hall et al.²⁶⁴	Enhance	23 k	Self-osc.
Zhu et al.²⁷¹	Enhance	130 k	240 k
Guo et al.²⁷²	Enhance	N.M.	Self-osc.
Guo et al.²⁷³	Enhance	22 K	11M
Zhu et al.²⁷⁴	Enhance	340 k	440 k
Mehdizadeh et al.²⁷⁵	Enhance	1.1 k	17 k
Hall et al.²⁷⁶	Enhance	N. M.	Self-osc.
Li et al.²⁶³	Enhance	480	Self-osc.
Kumar et al.²⁷⁷	Enhance	680	1.1M
Ramezany et al.²⁷⁸	Enhance	9.8 k	260 k
Ansari and Rais-Zadeh²⁸²	Enhance	1.7 k	13.9 k
Chang et al.²⁷⁹	Enhance	N.M.	Self-osc.
Chu et al.²⁸⁰	Enhance	9 k	Self-osc.
Liu et al.²⁸⁴	Enhance	600	Self-osc.
Lehée et al.²⁶²	Suppress	30 k	450
Lehée et al.²⁸³	Enhance	28 k	Self-osc.
Miller et al.²⁶⁰	Enhance	15 k	Self-osc.
Chu et al.²⁸¹	Enhance	4.3 k	Self-osc.
Ramezany and Pourkamali⁹⁴	Enhance	1 k	89 k
Miller et al.⁹²	Both	7 k	3 k, 68 k

Reference	$Q_{e f f}$ type	$Q$	$Q_{e f f}$
Phan et al.²⁶⁶	Enhance	13 k	Self-osc.
Steeneken et al.¹⁴	Both	9 k	2 k, self-osc.
Rahafrooz and Pourkamali²⁶⁷	Enhance	49 k	Self-osc.
Hajjam et al.²⁶⁸	Enhance	N.M.	Self-osc.
Rahafrooz and Pourkamali⁹³	Enhance	2 k	Self-osc.
Rahafrooz and Pourkamali²⁶⁹	Enhance	N. M.	Self-osc.
Iqbal et al.²⁷⁰	Enhance	100 k	360 k
Hall et al.²⁶⁴	Enhance	23 k	Self-osc.
Zhu et al.²⁷¹	Enhance	130 k	240 k
Guo et al.²⁷²	Enhance	N.M.	Self-osc.
Guo et al.²⁷³	Enhance	22 K	11M
Zhu et al.²⁷⁴	Enhance	340 k	440 k
Mehdizadeh et al.²⁷⁵	Enhance	1.1 k	17 k
Hall et al.²⁷⁶	Enhance	N. M.	Self-osc.
Li et al.²⁶³	Enhance	480	Self-osc.
Kumar et al.²⁷⁷	Enhance	680	1.1M
Ramezany et al.²⁷⁸	Enhance	9.8 k	260 k
Ansari and Rais-Zadeh²⁸²	Enhance	1.7 k	13.9 k
Chang et al.²⁷⁹	Enhance	N.M.	Self-osc.
Chu et al.²⁸⁰	Enhance	9 k	Self-osc.
Liu et al.²⁸⁴	Enhance	600	Self-osc.
Lehée et al.²⁶²	Suppress	30 k	450
Lehée et al.²⁸³	Enhance	28 k	Self-osc.
Miller et al.²⁶⁰	Enhance	15 k	Self-osc.
Chu et al.²⁸¹	Enhance	4.3 k	Self-osc.
Ramezany and Pourkamali⁹⁴	Enhance	1 k	89 k
Miller et al.⁹²	Both	7 k	3 k, 68 k

E. Effective quality factor tuning in piezoelectric semiconductors

Piezoelectric semiconductor materials, such as GaAs, gallium nitride (GaN), zinc oxide (ZnO), and cadmium sulfide (CdS), offer a unique platform for studying the interplay between acoustic phonons and electrons. Under certain conditions, energy can be transferred from the free electrons to the phonons in a piezoelectric medium, as described by the acoustoelectric effect.²⁸⁵ When an elastic wave propagates in semiconducting media, it induces current and space charge via direct interaction with the free electrons, resulting in acoustic loss.^286,287 While extensive theoretical and experimental work in the 1960s focused on amplification and attenuation of traveling acoustic waves in piezoelectric semiconductors,^285,286,288 very few works reported on $Q$ tuning in piezoelectric semiconductor resonators by varying the electric field.

Gokhale and Rais-Zadeh demonstrated acoustoelectric pumping in a standing wave piezoelectric resonator and suggested that the acoustoelectric effect improves the mechanical $Q$ by reducing the underlying dissipation.⁹⁵ However, a systematic approach depicting $Q$ enhancement by reduction of the dissipation-induced losses (such as phonon-electron scattering) has not been shown to date. As discussed in Sec. II, an improvement in measured $Q$ using the bandwidth method is not sufficient to conclude that a reduction in the internal dissipation-induced loss mechanisms is the cause of $Q$ enhancement, particularly in the presence of a third terminal that can couple energy into the system. Fitting Eq. (20) to the thermomechanical ASD of the resonator is the only experimental method to check whether $Q$ is effectively or intrinsically enhanced in the presence of a pump, and the thermomechanical noise peak provides the best estimation of the mechanical $Q$ ⁠.

In piezoelectric semiconductor materials, the application of a dc electric field can affect the measured $Q$ in various ways. Besides the motional loss and structural losses, other loss sources contribute to the measured $Q$ ⁠, including²⁸⁹

Electromechanical coupling loss: A dc voltage can deplete the transducer of charges and thus reduce the electromechanical coupling loss, including resistive heating losses. This manifests itself as an increase in the resonator's measured $Q$ ⁠. $Q$ of depletion-mediated piezoelectric resonators is often tuned by such a mechanism. Masmanidis et al. demonstrated effective $Q$ enhancement in p-i-n GaAs beam resonators,²⁹⁰ and Ansari and Rais-Zadeh showed $Q$ enhancement as an AlGaN/GaN transducer was depleted of charges.²⁹¹ Tonisch et al. showed the dependency of piezoelectric coefficients on the dc electric field.²⁹² Non-idealities in electrical-to-mechanical energy conversion and vice versa are also included in this loss mechanism and are captured by the complex piezoelectric coefficients.²⁹³
Electrical losses: These include resistive losses associated with the finite resistance of the conductive material or the two-dimensional electron gas sheet used as electrodes and the dielectric losses associated with the generation of heat in the static capacitances of the device. A dc voltage impacts the static capacitances and the transducer diode characteristics. Applying an electric field changes the metal-semiconductor energy band structure, which in turn induces stress in the thin films. The induced stress increases the $Q$ by increasing $\sqrt{km}$ in Eq. (4). This is more pronounced in piezoelectric low dimensional quantum systems, such as AlGaN/GaN heterostructures, where the quantum well band structures shift significantly with the electric field, and the electric field distribution is very sensitive to the applied voltage. Care must be taken in decoupling the contributions of each loss mechanism, since in most cases, several mechanisms contribute simultaneously. A fundamental $Q$ tuning mechanism that targets phonon-mediated losses has yet to be shown.

F. Parametric pumping

Degenerate parametric pumping is a technique for feeding energy into a dynamical system by modulating some reactive parameter of the system at twice the resonant frequency. The phenomenon was first observed in the mid-19th century by Michael Faraday, who observed that the surface waves in a vertically excited cylinder moved at half the frequency of excitation.²⁹⁴ In the early 20th century, parametric amplification was utilized to amplify signals in radio telephony by modulating the capacitance or inductance of an LC filter at twice the resonant frequency.²⁹⁵ The resulting “magnetic amplifier” was useful as a high-power amplifier until it was superseded by vacuum tube amplifiers and then transistor amplifiers.

With Rugar and Grütter's paper demonstrating parametric pumping of a micromechanical cantilever in the early 1990s,¹⁵ the topic reemerged as a vigorous area of research. Micromechanical resonators have proven themselves to be an excellent platform for studying parametric amplification and parametric resonance.²⁹⁶ In the foreseeable future, parametric pumping could be used for improving the SNR of commercial MEM resonant sensors, such as magnetometers²⁹⁷ and gyroscopes.^4,298–300

Degenerate parametric amplification causes excitations at frequencies $ω = 2 ω_{0} / N$ ⁠, where $ω_{0}$ is the resonant frequency of the mode and $N$ is an integer greater than or equal to one.²⁹⁴ Most demonstrations only study the first instability region (i.e., $N = 1$ ⁠) because damping exponentially narrows the instability regions for higher values of $N$ ⁠. Turner et al. demonstrated five parametric instability regions in a torsional resonator.⁹⁷ Jia et al. observed over twenty instability regions³⁰¹ and later observed over one hundred instability regions,³⁰² both in engineered membrane structures.

Degenerate parametric pumping is qualitatively different from “true” feedback techniques because the amplification is phase-sensitive: the parametric pump amplifies the motion in one quadrature and squeezes the motion in the other quadrature. Sufficient parametric pumping can still lead to self-oscillations, like the other $Q_{e f f}$ enhancement techniques, but parametric suppression cannot reduce $Q_{e f f}$ in one quadrature below $Q / \sqrt{2}$ (the 6 dB limit) without self-oscillations in the other quadrature. Because parametric pumping is phase-dependent, it also modifies the slope of the resonator phase lag in the opposite direction from phase-independent $Q_{e f f}$ tuning techniques, as we demonstrate in Sec. VII.

Table VI summarizes the reports of $Q_{e f f}$ tuning via parametric pumping in MEM/NEM resonators. The fixed-fixed beam has been the most common geometry for studying parametric pumping. Kraus et al. demonstrated parametrically resonating silicon-metal nanobridges with wide frequency tunability.³⁰³ Mahboob and Yamaguchi demonstrated piezoelectric parametric pumping in a GaAs/AlGaAs clamped-clamped beam,⁹⁸ demonstrated mechanical bit storage using the binary phase of parametric resonance,³⁰⁴ and improved the SNR of a charge detector by over 20-fold using parametric amplification.³⁰⁵ Karabalin et al. demonstrated parametric resonance at 130 MHz of a nanobeam using a Lorentz force.⁹⁹ Suh et al. showed that capacitively coupling a NEM beam to a Cooper pair box qubit can be much more effective for parametric noise squeezing than capacitive coupling to a nearby electrode.³⁰⁶ Karabalin et al. explored parametric amplification in an array of GaAs nanobeams.³⁰⁷ Yie et al. showed that a parametrically driven methanol vapor sensor was insensitive to added noise.³⁰⁸ Westra et al. studied the interactions between parametric and harmonic resonance in a piezoelectrically actuated beam.³⁰⁹ Karabalin et al. devised an amplifier in which the detected signal modifies the stable branches of the bifurcation diagram for a pair of parametrically resonating beams.³¹⁰ Villanueva et al. demonstrated a novel parametric feedback loop for sustaining a nanobeam in parametric oscillations.⁵¹ Mahboob et al. implemented a multibit logic circuit in a parametrically resonating clamped-clamped beam.³¹¹ Cho et al. parametrically resonated a silicon nitride beam using dielectric gradient modulation.³¹² Thomas et al. tuned $Q_{e f f}$ using parametric amplification in micro-bridges of varying lengths.³¹³ Li et al. demonstrated real-time explosive gas sensing using a parametrically resonating beam.³¹⁴ Mahboob et al. encoded two bits of information using the phase of two parametrically resonating modes in a beam.³¹⁵ Ramini et al. studied parametric resonance of the first four flexural modes of an arched beam resonator.³¹⁶ Mouro et al. parametrically resonated a hydrogenated amorphous silicon beam.³¹⁷ Mahboob et al. squeezed the thermal noise of two modes simultaneously using parametric pumping.³¹⁸ Seitner et al. studied the dynamics of two hybridized modes of a doubly clamped beam subjected to dielectric parametric pumping.³¹⁹

Parametric pumping has often been implemented in cantilevers for applications in AFMs and SPMs. Rugar and Grütter proposed parametric amplification as a means for improving the force sensitivity of AFMs.¹⁵ Dana et al. increased the force sensitivity of an AFM cantilever using parametric amplification.³²⁰ Napoli et al. studied a combination of harmonic forcing and parametric pumping in a microcantilever.³²¹ Patil and Dharmadhikari used a parametrically resonating AFM cantilever to image a crystalline surface with atomic-scale resolution.³²² Ono et al. used parametric noise squeezing to improve the SNR of thermal infrared detectors.³²³ Ouisse et al. observed spontaneous parametric pumping in an electrostatic force microscope when the cantilever tip was brought close to the sample.³²⁴ Requa and Turner demonstrated parametric resonance in a Lorentz force actuated cantilever³²⁵ and showed that parametric resonance offers superior frequency resolution over harmonic resonance.³²⁶ Moreno-Moreno et al. imaged a silicon grating and a DNA strand with a parametrically resonating SPM.³²⁷ Krylov et al. demonstrated large amplitude parametric resonance in a pair of cantilevers connected at their ends.³²⁸ Collin et al. parametrically pumped a cantilever at cryogenic temperatures into its nonlinear regime.³²⁹ Prakash et al. parametrically resonated the higher order flexural modes of a microcantilever.³³⁰ Yie et al. demonstrated parametric resonance of an array of microcantilevers for mass sensing.³³¹ Szorkovszky et al. combined parametric pumping and feedback control to squeeze the thermal noise of a microcantilever by 6.2 dB, beyond the limit of degenerate parametric pumping.³³² Soon after, Vinante and Falferi pushed the noise squeezing of this method down to 11.5 dB.³³³ Linzon et al. demonstrated large amplitude parametric resonance of a microcantilever using fringing fields.³³⁴ Wang et al. improved the magnetic field sensitivity of a multiferroic cantilever sixfold using parametric amplification.³³⁵

There are many demonstrations of parametric pumping using resonators with comb-drive actuators or folded-beam structures. Turner et al. demonstrated parametric resonance in a torsional resonator via comb-drive fringing fields for scanning probe microscopes.⁹⁷ Zhang et al. independently tuned the linear and cubic stiffness terms of a parametric resonator.³³⁶ DeMartini et al. demonstrated a tunable single frequency band-pass filter using a parametric oscillator.³³⁷ Koskenvuori and Tittonen used parametric amplification to improve a MEM signal mixer/filter.³³⁸ Thompson and Horsley enhanced the sensitivity of a Lorentz force magnetometer more than 80-fold using parametric amplification.²⁹⁷ Guo and Fedder demonstrated large amplitude parametric resonance in a comb-drive folded-beam structure.³³⁹ Lee et al. parametrically pumped a dog-bone resonator.³⁴⁰ Poot et al. demonstrated 15 dB of noise squeezing in a photonic crystal resonator using a combined parametric pump and linear feedback.^341,342 Shmulevich et al. devised a comb-drive resonator with a stiffness that is independent of the amplitude.^343,344 Ganesan et al. studied parametric resonance of a beam that was pinned at its center.³⁴⁵ Pallay and Towfighian incorporated folded beam supports into a cantilever for large amplitude parametric resonance.³⁴⁶

There are several reports of parametric pumping of MEM gyroscopes. Harish et al. amplified the rotation rate signal of a ring gyroscope by fourfold using parametric amplification.²⁹⁹ Oropeza-Ramos et al. studied parametric resonance in a proof mass gyroscope.³⁴⁷ Hu et al. improved the sensitivity of their ring gyroscope by 80-fold using parametric amplification³⁴⁸ and implemented parametric pumping of their device using a digital signal processing scheme.²⁹⁸ Sharma et al. studied parametric enhancement and damping of a proof mass gyroscope.³⁰⁰ Ahn et al. demonstrated parametric $Q_{e f f}$ tuning of a disk resonating gyroscope and a threefold improvement in the noise equivalent rotation rate with parametric pumping.⁴ Zega et al. theoretically and experimentally studied the oscillation amplitude and stability of a parametrically pumped disk resonating gyroscope.³⁴⁹ Nitzan et al. discovered self-induced parametric amplification in a disk resonating gyroscope due to the nonlinear coupling between the drive and sense modes,³⁵⁰ which was subsequently modeled by Polunin and Shaw.³⁵¹

Parametric pumped has also been studied in a variety of torsional resonator geometries. Carr et al. demonstrated parametric $Q_{e f f}$ tuning of the out-of-plane torsional mode of a silicon square anchored by opposing beams.³⁵² Chan and Stambaugh studied noise induced switching between the two phases of a torsional parametric oscillator,³⁵³ and Chan et al. subsequently modeled this switching.^96,354 Arslan et al. parametrically excited a torsional micro-scanner.³⁵⁵ Droogendijk et al. parametrically actuated a microfluidic Coriolis mass flow sensor for measuring water flow rates.³⁵⁶ Kawai et al. parametrically amplified the vibration amplitude of a torsional micro-mirror.³⁵⁷

Parametric pumping has been applied to several disk and membrane geometries. Zalalutdinov et al. focused a laser on the periphery of a silicon disk supported by a central silica stem to induce parametric suppression and enhancement in the disk.³⁵⁸ Suh et al. observed spontaneous parametric pumping and instability of a membrane during back-action-evasion due to dissipation-induced shifts in the resonant frequency.¹⁵³ Jia et al. designed a piezoelectric membrane for energy harvesting from multiple parametric resonance subharmonics.³⁰¹ Pontin et al. demonstrated a degenerate parametric feedback scheme for surpassing the 6 dB noise squeezing limit in a micro-mirror resonator.³⁵⁹ Chowdhury studied parametric resonance in a membrane with an embedded photonic crystal for motion detection.³⁶⁰ Ozdogan et al. parametrically resonated a micro-mirror.³⁶¹ Prasad et al. parametrically pumped a graphene membrane and studied how the cubic nonlinearity limited the gain.³⁶² Dolleman et al. parametrically resonated 14 different mechanical modes in a graphene membrane by opto-thermally modulating the membrane tension.³⁶³

Parametric pumping has been demonstrated in several nanowire/nanotube structures fabricated using bottom-up techniques. Yu et al. excited a cantilevered boron nanowire into parametric resonance electrostatically using a nearby probe.³⁶⁴ Nichol et al. combined position-proportional control and parametric amplification to extend the dynamic range of a parametrically resonating silicon nanowire.³⁶⁵ Midtvedt et al. parametrically resonated a clamped-clamped carbon nanotube resonator,³⁶⁶ and Eichler et al. studied the amplitude saturation mechanism during parametric resonance in a similar structure.¹⁷ Wu and Zhong parametrically enhanced $Q_{e f f}$ of a high $Q$ carbon nanotube resonator more than tenfold.³⁶⁷

TABLE VI.

Reports of $Q_{e f f}$ tuning via parametric pumping in chronological order. Papers that do not mention their device $Q$ are denoted by “N.M.”. Reports of self-oscillations are denoted by “self-osc.”.

Reference	$Q_{e f f}$ type	$Q$	$Q_{e f f}$
Rugar and Grütter¹⁵	Both	10 k	7.1 k, 250 k
Turner et al.⁹⁷	Enhance	3 k	Self-osc.
Dana et al.³²⁰	Enhance	3 k	Self-osc.
Carr et al.³⁵²	Both	810	580, 4.9 k
Kraus et al.³⁰³	Enhance	1.4 k	Self-osc.
Zalalutdinov et al.³⁵⁸	Enhance	11 k	7.8 k, Self-osc.
Yu et al.³⁶⁴	Enhance	2.9 k	Self-osc.
Zhang et al.³³⁶	Enhance	N.M.	Self-osc.
Napoli et al.³²¹	Enhance	2.2 k	Self-osc.
Patil and Dharmadhikari³²²	Enhance	N.M.	Self-osc.
Ono et al.³²³	Both	150	110, self-osc.
Ouisse et al.³²⁴	Enhance	300	410
Requa and Turner³²⁵	Enhance	1 k	Self-osc.
Moreno-Moreno et al.³²⁷	Enhance	520	Self-osc.
Koskenvuori and Tittonen³³⁸	Enhance	5.4 k	Self-osc.
DeMartini et al.³³⁷	Enhance	N.M.	Self-osc.
Requa and Turner³²⁶	Enhance	4 k	Self-osc.
Chan and Stambaugh³⁵³	Enhance	7.5 k	Self-osc.
Chan et al.⁹⁶	Enhance	10 k	Self-osc.
Chan et al.³⁵⁴	Enhance	10 k	Self-osc.
Mahboob and Yamaguchi⁹⁸	Both	110 k	78 k, 250 k
Mahboob and Yamaguchi³⁰⁴	Enhance	110 k	Self-osc.
Mahboob and Yamaguchi³⁰⁵	Enhance	110 k	Self-osc.
Harish et al.²⁹⁹	Enhance	45 k	Self-osc.
Nichol et al.³⁶⁵	Both	3 k	2.2 k, 24 k
Oropeza-Ramos et al.³⁴⁷	Enhance	7 k	Self-osc.
Karabalin et al.⁹⁹	Enhance	2.3 k	6.9 k
Krylov et al.³²⁸	Enhance	100	Self-osc.
Arslan et al.³⁵⁵	Enhance	N.M.	Self-osc.
Hu et al.³⁴⁸	Both	50 k	36 k, 2 M
Suh et al.³⁰⁶	Both	38 k	27 k, self-osc.
Karabalin et al.³⁰⁷	Enhance	2.7 k	Self-osc.
Collin et al.³²⁹	Both	5 k	3.6 k, self-osc.
Yie et al.³⁰⁸	Enhance	73	Self-osc.
Westra et al.³⁰⁹	Both	58	43, 98
Midtvedt et al.³⁶⁶	Enhance	400	Self-osc.
Eichler et al.¹⁷	Enhance	1 k	Self-osc.
Karabalin et al.³¹⁰	Enhance	1.7 k	Self-osc.
Wu and Zhong³⁶⁷	Enhance	700	7 k
Villanueva et al.⁵¹	Enhance	1.2 k	Self-osc.
Hu et al.²⁹⁸	Enhance	28 k	Self-osc.
Thompson and Horsley²⁹⁷	Enhance	49	4 k
Mahboob et al.³¹¹	Enhance	140 k	Self-osc.
Sharma et al.³⁰⁰	Both	N.M.	Self-osc.
Cho et al.³¹³	Both	23 k	17 k, 220 k
Prakash et al.³³⁰	Enhance	350	Self-osc.
Droogendijk et al.³⁵⁶	Enhance	N.M.	Self-osc.
Suh et al.¹⁵³	Enhance	50 k	Self-osc.
Yie et al.³³¹	Enhance	8.6 k	19 k
Guo and Fedder³³⁹	Enhance	51	Self-osc.
Linzon et al.³³⁴	Enhance	120	Self-osc.
Thomas et al.³¹³	Both	500	360, 7 k
Szorkovszky et al.³³²	Enhance	480	Self-osc.
Vinante and Falferi³³³	Enhance	77 k	Self-osc.
Ahn et al.⁴	Enhance	110 k	880 k
Li et al.³¹⁴	Enhance	N.M.	Self-osc.
Poot et al.³⁴¹	Both	62 k	44 k, self-osc.
Lee et al.³⁴⁰	Enhance	1.9 k	2.2 k
Mahboob et al.³¹⁵	Enhance	200 k	Self-osc.
Pontin et al.³⁵⁹	Enhance	16 k	10 k
Shmulevich et al.³⁴³	Enhance	3.5 k	Self-osc.
Zega et al.³⁴⁹	Both	85 k	61 k, self-osc.
Wang et al.³³⁵	Both	3 k	2.2 k, 35 k
Nitzan et al.³⁵⁰	Both	80 k	57 k, 160 k
Poot et al.³⁴²	Both	60 k	43 k, self-osc.
Chowdhury et al.³⁶⁰	Enhance	3.1 k	Self-osc.
Mouro et al.³¹⁷	Enhance	1.6 k	Self-osc.
Kawai et al.³⁵⁷	Enhance	7 k	12 k
Ramini et al.³¹⁶	Enhance	N.M.	Self-osc.
Jia et al.³⁰¹	Enhance	17	Self-osc.
Mahboob et al.³¹⁸	Enhance	1.3 k	Self-osc.
Shmulevich and Elata³⁴⁴	Enhance	4.5 k	Self-osc.
Pallay and Towfighian³⁴⁶	Enhance	150	Self-osc.
Ozdogan et al.³⁶¹	Enhance	5 k	Self-osc.
Ganesan et al.³⁴⁵	Enhance	1.3 k	Self-osc.
Seitner et al.³¹⁹	Enhance	500 k	Self-osc.
Prasad et al.³⁶²	Enhance	500	Self-osc.
Jia et al.³⁰²	Enhance	N.M.	Self-osc.
Dolleman et al.³⁶³	Enhance	140	Self-osc.

Reference	$Q_{e f f}$ type	$Q$	$Q_{e f f}$
Rugar and Grütter¹⁵	Both	10 k	7.1 k, 250 k
Turner et al.⁹⁷	Enhance	3 k	Self-osc.
Dana et al.³²⁰	Enhance	3 k	Self-osc.
Carr et al.³⁵²	Both	810	580, 4.9 k
Kraus et al.³⁰³	Enhance	1.4 k	Self-osc.
Zalalutdinov et al.³⁵⁸	Enhance	11 k	7.8 k, Self-osc.
Yu et al.³⁶⁴	Enhance	2.9 k	Self-osc.
Zhang et al.³³⁶	Enhance	N.M.	Self-osc.
Napoli et al.³²¹	Enhance	2.2 k	Self-osc.
Patil and Dharmadhikari³²²	Enhance	N.M.	Self-osc.
Ono et al.³²³	Both	150	110, self-osc.
Ouisse et al.³²⁴	Enhance	300	410
Requa and Turner³²⁵	Enhance	1 k	Self-osc.
Moreno-Moreno et al.³²⁷	Enhance	520	Self-osc.
Koskenvuori and Tittonen³³⁸	Enhance	5.4 k	Self-osc.
DeMartini et al.³³⁷	Enhance	N.M.	Self-osc.
Requa and Turner³²⁶	Enhance	4 k	Self-osc.
Chan and Stambaugh³⁵³	Enhance	7.5 k	Self-osc.
Chan et al.⁹⁶	Enhance	10 k	Self-osc.
Chan et al.³⁵⁴	Enhance	10 k	Self-osc.
Mahboob and Yamaguchi⁹⁸	Both	110 k	78 k, 250 k
Mahboob and Yamaguchi³⁰⁴	Enhance	110 k	Self-osc.
Mahboob and Yamaguchi³⁰⁵	Enhance	110 k	Self-osc.
Harish et al.²⁹⁹	Enhance	45 k	Self-osc.
Nichol et al.³⁶⁵	Both	3 k	2.2 k, 24 k
Oropeza-Ramos et al.³⁴⁷	Enhance	7 k	Self-osc.
Karabalin et al.⁹⁹	Enhance	2.3 k	6.9 k
Krylov et al.³²⁸	Enhance	100	Self-osc.
Arslan et al.³⁵⁵	Enhance	N.M.	Self-osc.
Hu et al.³⁴⁸	Both	50 k	36 k, 2 M
Suh et al.³⁰⁶	Both	38 k	27 k, self-osc.
Karabalin et al.³⁰⁷	Enhance	2.7 k	Self-osc.
Collin et al.³²⁹	Both	5 k	3.6 k, self-osc.
Yie et al.³⁰⁸	Enhance	73	Self-osc.
Westra et al.³⁰⁹	Both	58	43, 98
Midtvedt et al.³⁶⁶	Enhance	400	Self-osc.
Eichler et al.¹⁷	Enhance	1 k	Self-osc.
Karabalin et al.³¹⁰	Enhance	1.7 k	Self-osc.
Wu and Zhong³⁶⁷	Enhance	700	7 k
Villanueva et al.⁵¹	Enhance	1.2 k	Self-osc.
Hu et al.²⁹⁸	Enhance	28 k	Self-osc.
Thompson and Horsley²⁹⁷	Enhance	49	4 k
Mahboob et al.³¹¹	Enhance	140 k	Self-osc.
Sharma et al.³⁰⁰	Both	N.M.	Self-osc.
Cho et al.³¹³	Both	23 k	17 k, 220 k
Prakash et al.³³⁰	Enhance	350	Self-osc.
Droogendijk et al.³⁵⁶	Enhance	N.M.	Self-osc.
Suh et al.¹⁵³	Enhance	50 k	Self-osc.
Yie et al.³³¹	Enhance	8.6 k	19 k
Guo and Fedder³³⁹	Enhance	51	Self-osc.
Linzon et al.³³⁴	Enhance	120	Self-osc.
Thomas et al.³¹³	Both	500	360, 7 k
Szorkovszky et al.³³²	Enhance	480	Self-osc.
Vinante and Falferi³³³	Enhance	77 k	Self-osc.
Ahn et al.⁴	Enhance	110 k	880 k
Li et al.³¹⁴	Enhance	N.M.	Self-osc.
Poot et al.³⁴¹	Both	62 k	44 k, self-osc.
Lee et al.³⁴⁰	Enhance	1.9 k	2.2 k
Mahboob et al.³¹⁵	Enhance	200 k	Self-osc.
Pontin et al.³⁵⁹	Enhance	16 k	10 k
Shmulevich et al.³⁴³	Enhance	3.5 k	Self-osc.
Zega et al.³⁴⁹	Both	85 k	61 k, self-osc.
Wang et al.³³⁵	Both	3 k	2.2 k, 35 k
Nitzan et al.³⁵⁰	Both	80 k	57 k, 160 k
Poot et al.³⁴²	Both	60 k	43 k, self-osc.
Chowdhury et al.³⁶⁰	Enhance	3.1 k	Self-osc.
Mouro et al.³¹⁷	Enhance	1.6 k	Self-osc.
Kawai et al.³⁵⁷	Enhance	7 k	12 k
Ramini et al.³¹⁶	Enhance	N.M.	Self-osc.
Jia et al.³⁰¹	Enhance	17	Self-osc.
Mahboob et al.³¹⁸	Enhance	1.3 k	Self-osc.
Shmulevich and Elata³⁴⁴	Enhance	4.5 k	Self-osc.
Pallay and Towfighian³⁴⁶	Enhance	150	Self-osc.
Ozdogan et al.³⁶¹	Enhance	5 k	Self-osc.
Ganesan et al.³⁴⁵	Enhance	1.3 k	Self-osc.
Seitner et al.³¹⁹	Enhance	500 k	Self-osc.
Prasad et al.³⁶²	Enhance	500	Self-osc.
Jia et al.³⁰²	Enhance	N.M.	Self-osc.
Dolleman et al.³⁶³	Enhance	140	Self-osc.

G. Other pumping techniques

There are many potential sources of back-action within a resonator, and so, there is no doubt that there will be many future reports of $Q_{e f f}$ tuning using different coupling or internal feedback approaches. One such technique is quantum back-action, which was first experimentally demonstrated by Naik et al.³⁶⁸ The Heisenberg force exerted on a resonator during its measurement induces a tiny velocity-proportional feedback. This feedback is always present when transducing the motion of a resonator, but it is exceedingly difficult to measure in all but the smallest resonators with the most sensitive measurement readouts. References 368–383 cannot be easily classified into Sections V A–V F. Brown et al. passively suppressed $Q_{e f f}$ of a microcantilever by coupling it to an electrical circuit with a phase lag.³⁶⁹ Ayari et al. demonstrated self-oscillations of a silicon carbide nanowire arising from electron field emission to an adjacent electrode.³⁷⁰ Lassagne et al. demonstrated $Q_{e f f}$ and frequency tuning of a single-walled carbon nanotube due to the coupling between the mechanical motion and electron transport in the Coulomb blockade regime.³⁷¹ Weldon et al. studied self-oscillations of a carbon nanotube arising from electron field emission to an adjacent electrode.³⁷² Fairbairn et al. used a piezoelectric shunt to passively tune $Q_{e f f}$ of an AFM cantilever without the need for an explicit feedback loop.³⁷³ Rieger et al. tuned the resonant frequency and effective quality factor of a silicon nitride clamped-clamped beam by more than sixfold using dielectric electric-field-gradient forcing.³⁷⁴ Barois et al. demonstrated pW power consumption self-oscillations in two contacting nanowires³⁷⁵ and then theoretically and experimentally studied the threshold for self-oscillations due to electron field emission from a nanowire.³⁷⁶ Sarkar and Mansour demonstrated current-controlled $Q_{e f f}$ tuning of a piezoresistive AFM in a matched Wheatstone bridge.³⁷⁷ Nigues et al. observed $Q_{e f f}$ suppression and $Q_{e f f}$ enhancement up to the self-oscillation threshold in a silicon carbide nanowire using a focused electron beam.³⁷⁸ Barois et al. demonstrated self-oscillations in a silicon carbide nanowire by applying a capacitive drive at a frequency above the resonant frequency and the electrical cutoff frequency.³⁷⁹ Yasuda et al. observed $Q_{e f f}$ enhancement in a carbon nanotube due to electrostatic interactions with charges on a nearby insulator.³⁸⁰ McAuslan et al. cooled a mechanical mode of a microtoroid down to an effective temperature of 100 mK using superfluid evaporation-induced feedback.³⁸¹ Okazaki et al. studied back-action in a piezoelectric doubly clamped resonator due to interactions with an embedded quantum dot.³⁸² Macquarrie et al. demonstrated back-action cooling of a bulk acoustic resonator via coupling to a nitrogen vacancy center.³⁸³

VI. UTILIZING EFFECTIVE QUALITY FACTOR TUNING

By understanding how $Q_{e f f}$ tuning affects the dynamics and noise of a mode and understanding the ways to tune $Q_{e f f}$ in MEM/NEM resonators, these techniques can be selected to improve MEM/NEM sensors and oscillators for specific applications. The application area, the transduction method, the total power consumption, the fabrication constraints, and the desired size of the MEM/NEM device and supporting components can all influence the selection of a particular $Q_{e f f}$ tuning method.

An important consideration is the motional readout. Capacitive, piezoelectric, and piezoresistive measurement readouts are not likely to monitor a MEM/NEM resonator with a thermomechanical-noise-limited resolution, while optical readouts are likely to easily resolve the thermomechanical motion. The lower sensitivity measurement readouts can be implemented into handheld electronics, while bulky laboratory lasers preclude this possibility for the higher sensitivity readouts. When using measurement readouts that are amplifier-noise-limited, $Q_{e f f}$ tuning will improve the SNR of the corresponding amplitude modulated sensor if the harmonic force to be detected is applied within roughly one linewidth of mechanical resonance. Any signals at frequencies away from resonance will not be amplified by the $Q_{e f f}$ enhancement, irrespective of whether amplifier noise or thermomechanical noise dominates. Since the linewidth of the resonance narrows with increasing $Q_{e f f}$ ⁠, the range of frequencies that signals will be amplified in decreases as $Q_{e f f}$ increases. When using a thermomechanical-noise-limited motional readout, the SNR does not improve with $Q_{e f f}$ enhancement because $Q_{e f f}$ enhancement amplifies the signal and thermomechanical noise at resonance equally.

Instead of using $Q_{e f f}$ enhancement to improve the SNR of amplitude modulated sensors, which is restricted to amplification in a very narrow frequency range around resonance, the MEM/NEM resonator can be embedded into a phase-locked loop and used to detect stimuli via a shift in the resonant frequency. Because phase-independent $Q_{e f f}$ enhancement and phase-dependent $Q_{e f f}$ suppression increase the phase slope at resonance, there will be a larger shift in the phase-locked loop phase for a given shift in the resonant frequency with pumping. This pumping will improve the sensor SNR with the increasing phase slope, until the phase shift is dominated by the frequency fluctuations.

$Q_{e f f}$ suppression can be implemented with any of the phase-independent $Q_{e f f}$ tuning mechanisms and can be useful for increasing the bandwidth of sensors based on high $Q$ mechanical modes without compromising the thermal SNR. Engineering a high $Q$ mode will improve the thermomechanical noise floor and associated SNR of a force sensor implemented with that mode, at the expense of increasing the ring-down time. This reduces the sampling bandwidth of the corresponding amplitude modulated force sensor. If a larger sampling bandwidth is desired, the sensor can either be operated in a frequency-modulated setup, so that the sensor bandwidth is decoupled from $Q$ ⁠, or $Q_{e f f}$ suppression can be applied to the mode.

While external feedback control is the most widely used technique for constructing oscillators with MEM/NEM resonators, there are opportunities for developing oscillator topologies with the other $Q_{e f f}$ tuning mechanisms.^51,194,281 These techniques may eventually enable commercial oscillators with lower phase noise, higher frequencies, and lower power consumption than the state of art.

VII. EXPERIMENTAL RESULTS

To compare $Q_{e f f}$ tuning using phase-dependent and phase-independent mechanisms, we study the device in Fig. 7(a) by pumping it via parametric pumping and thermal-piezoresistive pumping. The device consists of a released silicon proof mass supported by two anchored beams. The two supporting beams of the resonator have different widths to enable the thermal-piezoresistive effect in the fundamental in-plane mode.¹⁴ The wide “spring” beam is 12 μm wide, and the narrow “engine” beam is 3 μm wide. Ohmic contacts through the encapsulation to the two anchors enable a current flow through the device. We fabricated the resonators within a process that yields high quality factor, highly stable resonators within a vacuum-sealed hermetic environment.³⁸⁴ A cross-section of a cleaved device is shown in Fig. 7(b). The resonator material is n-type doped single crystal silicon with an antimony concentration of roughly 10¹³ cm⁻³.

FIG. 7.

View large Download slide

(a) The patterned device prior to encapsulation. (b) A cross-section of the device after encapsulation.

Figure 8 depicts the device operation. The electrodes on either side of the proof mass are used to capacitively drive and sense the resonator. The electrode on the engine beam side of the device enables the application of an external electric field on the device with a frequency near $ω_{0}$ for actuation or a frequency near ${2 ω}_{0}$ for parametric pumping. We bias the spring beam anchor and connect a current supply between the two anchors to induce thermal pumping with increasing current. We use a large-valued resistor at the output of the current supply to raise its output impedance.

FIG. 8.

View large Download slide

The measurement setup. One anchor of the resonator is biased at $V_{b}$ for the capacitive readout, while a current $I_{d c}$ flows through the device for thermal-piezoresistive pumping. One electrode is used to apply $V_{a c}$ ⁠, which can consist of a probe near resonance or a pump at twice the resonant frequency for parametric amplification. The resonator motion induces a motional current out of the adjacent electrode, which is transduced into a voltage with a transimpedance amplifier (TIA).

We read out the resonator displacement by measuring the changing capacitance across the gap on the spring beam side with a custom transimpedance amplifier. The amplifier is sensitive enough that the noise at resonance is dominated by the device thermomechanical motion (which is approximately threefold larger than the amplifier noise). The amplifier is linear over the entire device operating range, including displacements that exceed the advent of Duffing nonlinearity. We keep both the resonator and the amplifier in a temperature-controlled chamber at 25 °C.

We first demonstrate parametric enhancement and suppression of our device using a ${2 ω}_{0}$ pump, applied to the drive electrode in addition to a constant 100 μV drive voltage at $ω_{0}$ ⁠. By repeatedly sweeping the drive frequency across $ω_{0}$ while sweeping the pump frequency at $2 ω$ for the same drive signal amplitude but increasing pump signal amplitude, we obtain Fig. 9. In Figs. 9(a) and 9(b), the pump is in phase with the drive signal, leading to parametric amplification of the motion and the corresponding increase in $Q_{e f f}$ and the amplitude. For the 880 mV pump, we observe the advent of Duffing nonlinearity due to electrostatic spring softening.²⁵ In Figs. 9(c) and 9(d), the pump is out of phase with the drive signal, leading to parametric suppression of the motion and a reduction in the amplitude. At large pump amplitudes, we observe phase-dependent parametric suppression. The phase between the pump and the drive is fixed, but it is the phase difference between the pump and the displacement that determines the parametric gain, and this changes with frequency due to the frequency-dependence of the resonator phase lag behind the driving force. This leads to the dimple in the response of Fig. 9(d). For the increasing parametric pump, the bandwidth method indicates $Q_{e f f}$ enhancement, while the phase slope method indicates $Q_{e f f}$ suppression. For parametric suppression, the bandwidth method indicates $Q_{e f f}$ suppression, while the phase slope method indicates $Q_{e f f}$ enhancement.

FIG. 9.

View large Download slide

The (a) amplitude and (b) phase of our driven micromechanical resonator subjected to parametric enhancement, showing more than two orders of magnitude increase in the amplitude at resonance and a reduction in the phase slope. The (c) amplitude and (d) phase of our device subjected to parametric suppression, showing a reduction in amplitude, an increase in phase slope, and parametric splitting due to the changing phase between the parametric pump and the resonator displacement with frequency.

We previously experimentally showed that for thermal pumping of a similar device, $Q_{e f f}$ enhancement results in a narrowing linewidth and a steepening of the phase slope,⁹² in contrast to the degenerate parametric pumping experiments here. This contrary behavior between the amplitude and the phase slope distinguishes phase-dependent $Q_{e f f}$ tuning techniques, such as parametric pumping, from phase-independent techniques, such as thermal pumping. Phase-independent techniques exhibit an increasing measured $Q_{e f f}$ with increasing pumping using either the bandwidth method or the phase slope method.

We next return to the assertion that $Q_{e f f}$ tuning mechanisms modify the resonator transfer function, $H (ω)$ ⁠, but not the thermomechanical noise force, $f_{t h} (ω)$ ⁠. The thermomechanical ASD of a resonator subjected to one of the $Q_{e f f}$ tuning mechanisms is described by Eq. (20) with constant $Q$ and changing $Q_{e f f}$ and can be rewritten as

V (ω) = \sqrt{R^{2} \frac{4 k_{B} T (\frac{ω_{0}}{m Q})}{{({ω_{0}}^{2} - ω^{2})}^{2} + {(\frac{ω ω_{0}}{Q_{e f f}})}^{2}} + N_{a}^{2}},

(64)

where $V (ω)$ is the ASD as measured using a spectrum analyzer, $N_{a}$ is the amplifier noise floor, and $R$ is the amplifier responsivity. Equation (64) assumes that the device thermal motion is uncorrelated with the amplifier noise.

We connect the output of our amplifier to a scalar spectrum analyzer and monitor the ASD of the motion in the proximity of the resonant frequency without applying an AC voltage to the drive electrode. Figure 10 shows the thermal ASD of our device for the progressive increase in direct current through the resonator. As we increase the current, thermal pumping increases the amplitude of resonator thermomechanical displacement ASD at resonance and narrows the linewidth. We visually fit Eq. (64) to the thermal noise ASD by manually adjusting $R$ ⁠, $N_{a}$ ⁠, and $Q$ ⁠. We also attempt to fit an erroneous model, $V_{*} (ω)$ ⁠, to the thermomechanical noise using $R$ and $N_{a}$ from the zero pumping case, while assuming that pumping increases a mechanical $Q_{*}$ parameter in both the numerator and the denominator of Eq. (64). We measure the resonant frequency, $ω_{0}$ ⁠, from the center of the peak, calculate a lumped mass of $m$ = 4.29 μg for our modeshape, and sense the electrode configuration using finite-element-analysis. We estimate $Q$ = 9 k to match the linewidth of the ASD in the absence of pumping.

FIG. 10.

View large Download slide

(a)–(f) The amplitude-spectral-density (ASD) of the thermomechanical noise at resonance for increasing current, $I_{d c}$ ⁠, and no drive voltage. Experimental data are plotted against a correct visual fit, $V$ ⁠, using Eq. (64) where only $Q_{e f f}$ increases, and an incorrect fit, $V_{*}$ ⁠, using Eq. (64) by falsely positing that thermal-piezoresistive pumping increases the mechanical quality factor.

$V (ω)$ and $V_{*} (ω)$ agree in Fig. 10(a) because at zero current, the system is in thermal equilibrium. For increasing current in Figs. 10(b)–10(f), we increase $Q_{e f f}$ while holding $Q$ and all other parameters constant to touch $V (ω)$ to the top of the ASD and increase $Q_{*}$ while holding all other parameters constant to attempt to fit $V_{*} (ω)$ to the data with the incorrect assertion that thermal pumping increases the mechanical quality factor. $V_{*} (ω)$ diverges from the data, while $V (ω)$ matches the data with $Q_{e f f}$ enhancement: thermal-piezoresistive pumping modifies $Q_{e f f}$ ⁠.

In Fig. 11, we repeat the same procedure for parametric pumping as we did for Fig. 10. We observe that the starting $Q$ value is slightly higher in Fig. 11 than in Fig. 10, possibly because the heating of the device during strong thermal pumping partially annealed the resonator or cleaned adsorbents off the surface. We apply a signal to the drive electrode at twice the resonant frequency to parametrically amplify the thermomechanical noise. For zero parametric pump, $V (ω)$ and $V_{*} (ω)$ agree in Fig. 11(a). Because parametric pumping is a phase-sensitive $Q_{e f f}$ tuning mechanism, the thermal noise of our device is amplified in phase with the parametric pump and is suppressed out of phase. The ASD in Fig. 11 is the phase-average of the two quadratures of the displacement noise,⁶⁵ which is still described in Eq. (64). For Figs. 11(b)–11(f), $V_{*} (ω)$ again diverges from the data, which suggests that parametric pumping increases $Q_{e f f}$ of the phase-averaged ASD, not the mechanical quality factor.

FIG. 11.

View large Download slide

(a)–(f) The amplitude-spectral-density (ASD) of the thermomechanical noise at resonance for the increasing parametric pump, $V_{2 ω}$ ⁠, and no drive voltage. Experimental data are plotted against a correct visual fit, $V$ ⁠, using Eq. (64) where only $Q_{e f f}$ increases, and an incorrect fit, $V_{*}$ ⁠, using Eq. (64) by falsely positing that parametric pumping increases the mechanical quality factor.

VIII. CONCLUSION

There are many methods for tuning the effective quality factor of micro- or nano-electromechanical resonators, with important applications for sensors, oscillators, and ground state cooling. A change in the effective quality factor modifies the resonator transfer function, without modifying the thermomechanical noise force, while a changing mechanical quality factor modifies both the resonator transfer function and the thermomechanical noise force. Different phase-independent effective quality factor tuning mechanisms can be compared using their equivalent linear feedback parameters. We experimentally demonstrate that the resonator response to effective quality factor tuning can be distinguished from the mechanical quality factor in a micromechanical resonator by studying the phase-averaged thermomechanical displacement noise amplitude-spectral-density. We show that the phase slope of a micromechanical resonator steepens for parametric suppression and becomes less steep for parametric enhancement, in contrast to the phase-independent effective quality factor tuning mechanisms.

ACKNOWLEDGMENTS

J.M.L.M. and A.A. are grateful to Mark Dykman, Steven Shaw, Amy Duwel, and Ali Mohazab for proofreading this manuscript and Daniel Rugar, Ali Hajimiri, Yasunobu Nakamura, Hiroshi Yamaguchi, Imran Mahboob, John Teufel, José Aumentado, Daniel López, Kimberly Foster, David Horsley, Ashwin Seshia, Alireza Ramezany, and Nicholas Miller for helpful discussions. The authors also appreciate the feedback and suggestions provided by the anonymous reviewers.

J.M.L.M. was supported by the National Defense Science and Engineering Graduate (NDSEG) Fellowship and the E.K. Potter Stanford Graduate Fellowship. L.G.V. acknowledges financial support from the Swiss National Science Foundation (SNF) under Grant No. PP00P2-170590. This work was performed in part in the nano@Stanford labs, which are supported by the National Science Foundation (NSF) as part of the National Nanotechnology Coordinated Infrastructure under Award No. ECCS-1542152, with support from the Defense Advanced Research Projects Agency Precise Robust Inertial Guidance for Munitions (PRIGM) Program, managed by Robert Lutwak, and the NSF under Grant No. CMMI-1662464.

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