A fast and sensitive room-temperature graphene nanomechanical bolometer

Bolometers are a powerful means of detecting light. Emerging applications demand that bolometers work at room temperature, while maintaining high speed and sensitivity, properties which are inherently limited by the heat capacity of the detector. To this end, graphene has generated interest, because it has the lowest mass per unit area of any material, while also possessing extreme thermal stability and an unmatched spectral absorbance. Yet, due to its weakly temperature-dependent electrical resistivity, graphene has failed to challenge the state-of-the-art at room temperature. Here, in a departure from conventional bolometry, we use a graphene nanoelectromechanical system to detect light via resonant sensing. In our approach, absorbed light heats and thermally tensions a suspended graphene resonator, thereby shifting its resonant frequency. Using the resonant frequency as a readout for photodetection, we achieve a room-temperature noise-equivalent power (2 pW Hz−1/2) and bandwidth (from 10 kHz up to 1.3 MHz), challenging the state-of-the-art.

fluctuations. This frequency time series data was then fit to a sine function with a frequency matching the AOM modulated frequency of the heating laser, with a modulated frequency between 40-100 Hz. To prevent any frequency drift due to substrate heating from interfering with the result, this fit was performed over many independent 100 ms time intervals and averaged. The amplitude obtained from the fit was used to calculate the frequency shift responsivity. Measurements of frequency noise were also performed with the PLL in the absence of heating radiation. This frequency noise time series data was used to calculate the Allan deviation 4 , defined as $ . = 1 2( − 1) 7 . 8 ( 9 − 9:; ) . < 9=.
where 9 is the average frequency measured over the th time interval of length .

Supplementary Note 1.
We characterized 12 devices for this work. Supplementary Figure 2 shows a gallery of SEM images of all the graphene nanomechanical bolometers (GNB) characterized. Supplementary Table 1 shows the mechanical and bolometric properties of these devices. The most sensitive noise equivalent power (device a) was calculated by measuring the Allan deviation at 1 ms, which we measured to be ? = 1.9 × 10 :D .
(pW Hz -1/2 ) Shows the detector sensitivity ( ) at a 100 Hz bandwidth, the bandwidth estimated from the thermal response time (BW H ), the mechanical bandwidth (BW 9 ), frequency responsivity ( L ) to absorbed power, Allan Deviation ( $ ) over a 10 ms integration time, initial resonance frequency ( 7 ), mechanical quality factor ( ), gate voltage used for the measurements ( "# ), tether width ( ), and initial diameter ( ), for the 12 bolometers characterized in this work. Source data are provided as a Source Data file. Devices are indicated by a-l and correspond to the images in Supplementary Figure 2.
Thermal Circuit. The response bandwidth and the overall response spectrum ( Figure 3c in the main text) of our GNB are well described by a thermal RC circuit model, shown schematically in Supplementary Figure 3a. In this model, H is the combined thermal resistance of all four tether supports, is the thermal capacitance given by the heat capacity of the suspended graphene, and is the amplitude of absorbed power that is modulated at frequency . The surrounding support has relatively large thermal mass and is therefore assumed to be a thermal sink.
The effective thermal impedance of the circuit is  The characteristic time of the response is given by the thermal RC time constant: Frequency-shift Responsivity. This will lead to a shift in the resonance frequency of the membrane. We relate ΔT to the frequency shift responsivity L by examining the change in resonance frequency (Δ 7 ) that results from thermal contraction of the graphene sheet. The equation for the mechanical resonance for a thin circular membrane is given by where is the radius, is the in-plane stress, and is the 2D mass density. When the temperature of the suspended membrane increases, the stress changes according to the stressstrain relation, Δ = −( Δ ) i ;:j , where is the Young's modulus, 7 is the initial stress, is the Poisson ratio, and is the thermal expansion coefficient 5 , which is negative for graphene for the temperature range used in our experiments. To first order, We find excellent agreement between the model and the data for L ( ) (for example, see Fig. 3c in the main text). Moreover, the responsivity should be independent of incident power, in accord our measurements (see Fig. 2b in the main text).
Taking the thermal resistance as H ~z { | } , where H is the 2D thermal resistivity, and and are the tether length and width, respectively, and in the limit ≪ The general L ∝ ; } prediction agrees well with the data (see Fig. 2c in main text.) We note that L is independent of the device area.
For a given device, L will decrease with added stress in the graphene. To check this prediction, we apply electrostatic stress with a back-gate bias "# , which pulls the graphene structure toward the silicon back-gate, while simultaneously measuring L . As predicted, we see that L decreases monotonically with increasing bias, as shown in Supplementary Figure 3b.
Noise-equivalent Power. The noise-equivalent power is defined as could be a function of tuv , but in our system it is a constant. Thus, where we have substituted the Allan deviation for fractional noise, $ = L / 7 , and used the definition of the frequency-shift responsivity, L ≡ Δ 7 /( 7 tuv ). Using Supplementary Equation 3, is given by The predicted tether-width dependence ∝ agrees well across all devices tested in this work ( Figure 2f main text), despite some variations in the Allan deviation. Thus, lower stress devices with a narrower tether width will have a lower noise-equivalent power (i.e. will be more sensitive to light.) As with L , is independent of the device area.
Through narrowing the tether width and increasing the tether length , we expect a H ~ 10 ;; K W -1 (near the blackbody radiation limit 6 ) to be within feasible experimental reach (the most sensitive devices characterized in this work have H ~ 10 Š K W -1 see Supplementary Note 7). For our most sensitive GNB (device a), we would expect being able to access ~1 fW Hz -1/2 , while still preserving a bandwidth of ~ 10 Hz. As a comparison, a 100 nm thick, 5 μm diameter bulk microbolometer with a heat capacity of ~ 10 :;7 J K -1 and H ~ 10 ;; K W -1 would have a bandwidth of ~10 mHz (see the following subsection Response Bandwidth), which would be over a 1000 × slower than our GNB and impractical for most applications. The measured bandwidth data agrees well with model prediction BW ∝ (see Fig. 3e in the main text). The BW we measure is likely lower than what we would expect for pristine graphene, as the mass density inferred from the resonance frequency gate dependence (see Supplementary Note 7) is about ~7.5 × greater than pristine graphene. Our experiments did not broadly sample the device area , so we could not robustly test the prediction BW ∝ :; . However, our limited data do agree with the area prediction. If the area dependence holds, the BW for a 1 μm diameter drumhead would reach ~40 MHz, and together with pristine graphene, the BW could reach ~200 MHz.

Finally, combining Supplementary Equation 3 with
' š (;:j) is defined as the frequency coefficient of temperature. The prediction ∝ agrees well with our data, as shown in Figure 3f in the main text.
Mechanical Linewidth and Quality Factor. The mechanical damping time constant can be extracted from an amplitude frequency response curve (see Fig. 1d in main text). We use a model for a damped driven oscillator 7 where is the 1/e decay rate, is the driving force, = 2 , and 7 = 2 7 . We use this decay time to estimate the "mechanical bandwidth" from the resonance linewidth in the same way as from the thermal time constant, BW ¡ = √OE .
• . The quality factor is calculated according to =

Supplementary Note 3.
Bandwidth Measurement using Resonant Frequency-Shift Response. We can measure the bandwidth of the GNB by increasing the modulation frequency of the heating laser and monitoring the response, thus directly measuring L ( ) and ( ), and thus the bandwidth . While tracking the resonant frequency shifts with the PLL, we output a voltage proportional to the frequency shift from the PLL and input this into a second lock-in amplifier channel in the Zurich HFLI2. This signal is referenced to input of the AOM. By sweeping the frequency of the AOM drive signal we could quickly extract how the resonant frequency shift amplitude drops as the modulation frequency of the heating laser increases. Representative data for this measurement is shown in Fig. 3c in the main text. The 3dB point provides a direct measure of BW.
Bandwidth Measurement using Off-Resonant Thermomechanical Response. We were unable to measure the bandwidth by looking at the change in frequency shift with the phase locked loop (PLL) when the device bandwidth exceeded that of the PLL. Instead, we infer bandwidth from the off-resonant frequency response of thermomechanical 8,9 displacement of the graphene membrane. The out-of-plane displacement of a curved membrane occurs when thermal stress tightens and flattens the membrane (Supplementary Figure 4). In the limit of small displacement and first-order thermal expansion, the mechanical displacement amplitude will be proportional to the change in temperature, ∝ Δ .
The displacement amplitude is a direct response to thermomechanical tensioning of the membrane, just as with the frequency-shift response of the bolometer. For the off-resonant where H = H is the thermal response time fit parameter and is the angular frequency of the heating source.
In off-resonant experiments, we use the 532 nm heating laser to create an AC heat source and a 633 nm interferometer to measure the deflection of the graphene. The real and imaginary amplitudes (defined by the phase difference between the mechanical amplitude and the heating laser intensity) are shown in Supplementary Figure 5 for two different GNBs. The black traces are the model fits. The bandwidth can be obtained from the fitted thermal response time H according to: We compared the response bandwidth obtained from the off-resonant and resonant approaches, denoted BW H and BW, respectively. The selected results are shown in Supplementary Table 2 for devices e, f, and i (see Supplementary Figure 2). These results provide evidence that the BW obtained from the two approaches are equivalent, and that thermomechanical tensioning is responsible for both off-resonant deflection and resonant frequency shifts. We also compared the BW to the equivalent bandwidth obtained from the mechanical resonance linewidth (denoted BW ¡ ). A full comparison is provided in Supplementary Table 1. The results show that the response bandwidth ascertained from either the resonant or off-resonant approach is not determined or limited by the mechanical linewidth, as is expected 3 . For example, the bandwidth, BW H , for device l differs from the its linewidth bandwidth by over a factor of 100.

Supplementary Note 4.
The figure of merit commonly used to compare different bolometers 10 is calculated using FOM = NETD × H × © where NETD is the noise equivalent temperature difference and © is the detector area 11 . The NETD is proportional to the noise-equivalent power through the relation 11 where is the optical aperture (typically = 1), Δ« ΔZ = 0.84 W m -2 sr -1 K -1 is the luminance variation with scene temperature around 300 K, © is the noise equivalent power to incident radiation, and is the measurement time. Increasing the thermal resistance improves © at the expensive of bandwidth and increasing the detector area improves the NETD at the expensive of pixel pitch. Therefore, the FOM removes geometric considerations when comparing bolometer technologies because both the thermal resistance and the pixel area can usually be tuned by changing the geometry. This is true for our GNB, as it is common to fabricate suspended graphene sheets with diameters ranging from 1 − 25 µm (ref. 12 ) and we have demonstrated that the thermal resistance can be tuned by varying the trampoline tether width. Doing this calculation for the most sensitive trampoline, with © = 300 pW Hz -1/2 , = 10 ms, H = 26 µs, we obtain FOM = 1.18 × 10 D mK ms µm 2 . The lowest reported FOM for room-temperature microbolometers 10,11,13,14 is of order 10 D mK ms µm 2 . Thus, despite not yet being optimized and a low optical absorption (2.3%), our GNB has already matched these record-low FOM values.

Supplementary Table 2.
This table shows the bandwidth estimated from the thermal response time (BW H ), the mechanical bandwidth measured from the quality factor (BW 9 ), and the 3dB bandwidth measured directly (BW). The direct bandwidth measurement agrees with the bandwidth limited by the thermal response time in agreement with frequency modulation.
Optical Absorption Estimate from Cavity Effects. The GNB device architecture used in our studies forms a Fabry-Perot cavity. Optical cavity effects due to reflections at interfaces will lead to an optical absorption that is a function of the device dimensions and the wavelength of the absorbed light. We calculate the complex amplitude of the reflected electromagnetic wave from the Si and SiO2 system at the location of the suspended graphene. Summing over all reflections according to the wave transfer matrix method for calculating transmission and reflection through multilayer media 15 gives the wave-transfer matrix where describes a dielectric boundary reflection and describes the propagation through a homogeneous medium. We calculate by using standard values for the refractive index of SiO2 and Si ( ¹ = 1.5 and v = 4.14) and by measuring the thickness of the oxide layer and the distance between the graphene and the oxide with atomic force microscopy (see Supplementary Figure 6 for schematic displaying ¹ and AE ), from which we obtain ¹ = 353 nm and AE = 552 nm, respectively, and use = 532 nm.
In terms of the wave-transfer matrix components and , the amplitude of the complex reflected wave is É = − • Ê = (−0.55 + 0.26 ). The electric field intensity at the graphene can be written as the sum of the incident and reflected wave | | . = |1 + É | . . The power absorbed by the graphene can be written as a function of electric field intensity at the graphene because the reflection coefficient of graphene is small (~0.01) 16 , = |1 + É | . = 0.6% Figure 6. Sketch of the GNB where Ì is the distance between the suspended graphene and the SiO2 and º is the oxide layer thickness protecting the silicon back gate.

Supplementary
where is the absorption coefficient. We note that an engineered cavity could be used to enhance the absorption to = |1 + 1| . = 9.2%.

Supplementary Note 6.
Photothermal Back-action Frequency Responsivity. Here we estimate the frequency shift due to photothermal backaction 16 . Depending on the location of the graphene membrane in the optical field of the cavity, this photothermal backaction could either enhance or weaken the GNB thermomechanical responsivity. Our modeling and calculations show that photothermal backaction does not cause a significant frequency shift in a GNB when compared to photothermal tensioning.
The effective frequency ÍÎÎ and damping Γ ÍÎÎ due to photothermal backaction can be written as 16

∇ -
where is the thermal response time, ∇ = "Ò ÓÔÕ "Ö is the derivative of the photothermal force as a function of displacement (ref. 16  In this work, we did not have a back-reflecting mirror to enhance cavity effects. Therefore, we use the measured results from Ref. 16 and assume an optimized photothermal backaction setup to estimate the upper limit of the frequency responsivity due to photothermal back-action. We use Û ÙÚÚ /Û:; mW -1 , ~500, tuv~ÜÝÞ × 0.023, 7 = 2 × 5 MHz, and we estimate ~300 ns instead of the theoretical estimate provided in Ref. 16 to account for the slower than theoretically predicted thermal response time in suspended graphene 8 . The ~300 ns estimate used here is consistent with the thermal response time from our measurements and that of Dolleman et al. 8 Altogether, we estimate L,×$~1 0 . W -1 for a 10-micron drumhead with a back-reflecting mirror. Therefore, the change in frequency due to photothermal back-action, even assuming a perfect reflecting back mirror, is a factor of 10 . − 10 ß lower than direct photothermal tensioning (see Supplementary Table 1 for L .) Moreover, because for a given absorbed power and displacement the device heats up to a higher temperature resulting in a larger photothermal force-and thus ∇ ∝ H -any enhancement to ∇ due to an increased H would be canceled by an increase in thermal response time, where ∝ H . Consequently, the trampolines, which possess a larger H , would see little if any enhancement in photothermal backaction when compared to the drumheads.

Supplementary Note 7.
Resonance Frequency Gate Dependence. The resonance frequency gate dependence ( ( "# ))-as seen in Supplementary Figure 7-can be used to infer 17 the graphene membrane mass density ( ), Young's modulus ( ), and initial stress ( 7 ). In turn, these quantities provide independent ways to experimentally calculate , H , L , and the absorption . The relevant equations are derived from the capacitive potential energy of the graphene sheet and the Si ++ back-gate.
Referring to the schematic given in Supplementary Figure 6, the series capacitance of the vacuum and oxide is where is the area of the drumhead and É = 3.9 is the relative permittivity of SiO2, AE is the distance between the suspended graphene and oxide (which is in vacuum), and ¹ is the thickness of the oxide between the Si ++ and vacuum, and = AE + " Â á â . The capacitive potential energy is where is the displacement of the suspended membrane, and "# is the applied DC gate bias. We then use # in the Lagrangian procedure outlined in Ref. 17 where is the radius of the drumhead, 7 is the permittivity of free space, and = AE + " Â á ae is the effective capacitive distance, which we measure from atomic force microscopy to be = 642 nm.
Fitting the experimental data of ( "# ) for a given membrane device using Supplementary Equations 4-5 yields the parameters , , and 7 . Using the data for the drumhead device l shown in Supplementary Figure 6, we extract 7 = 0.1 N m -1 , = 7.5 × ç , and = 110 N m -1 , where ç is the intrinsic mass density of monolayer graphene (~7.7 × 10 :è kg m -2 ). The amount of contaminating mass observed in this device is consistent with other graphene nanomechanical systems that used a PMMA transfer technique to suspended graphene sheets 16,18 . We expect that the mass density and modulus of all trampoline and drumhead devices on the chip containing device l will be the same, namely = 7.5 × ç and = 110 N m -1 .
Membrane Heat Capacity. The heat capacity can be estimated from the mass density = 7.5 × ç . The heat capacity for the GNB is where . is the device area (typically ~ 25 -50 µm 2 ), ç = 700 J kg -1 K -1 is the specific heat of graphene, and © ~ 1500 J kg -1 K -1 is the specific heat of PMMA. Together, the heat capacity for device l is ~ 4 × 10 :;OE J K -1 .
Thermal Resistance. We can determine the thermal resistance H experimentally from the measured H and with the expression: For the data shown in Figure 3c Figure 1). To extract the resonance frequencies, amplitude-frequency response curve data was fit using damped driven oscillator at varying gate voltages. Using these values, the mass density, initial stress, and elastic modulus were extracted. Source data are provided as a Source Data file.
Frequency-shift Responsivity. Experimental values for , 7 , obtained from electromechanical measurements together with H obtained from off-resonant bandwidth measurements can be used to calculate the frequency-shift responsivity, which for a circular drumhead is given by For the drumhead device l, H = 230 ns and ~4 × 10 :;OE J K -1 . Using = −7.4 × 10 :ê K -1 (ref. 18 ) and the Poisson ratio for graphene = 0.16, we calculate a frequency responsivity of L~ 2900 W -1 , which agrees within 10% of the independently measured value from frequency shifts measurements L~2 600 W -1 , also for Device l.
For the trampoline device e, we use the heat capacity and mass density measured from the gate dependence of device l. However, we expect that the initial stress on the device was changed during the focused ion beam shaping. To estimate the initial stress more accurately, we use finite element analysis to determine the amount of initial stress required in a trampoline geometry to reproduce the measured 7 = 10.7 MHz, which we find to be 7 = 0.19 N m -1 . Using these values along with measured H = 20 µs and ~2 × 10 :;OE J K -1 (and assuming this model can be applied to trampolines), Supplementary Equation 7 gives L~ 250000 W -1 , which reasonably close to the directly measured value of L~1 80000 W -1 .
Optical Absorption Estimated from Mechanical Properties. Using the above frequency responsivity (Supplementary Equation 7) together with measured values of Δ 7 , 7 , and the incident power ( ÜÝÞ ) allows for a calculation of the optical absorption coefficient ≡ Δ 7

L ÜÝÞ
For the drumhead device l and trampoline device e, we find = 2% and = 1.6%, respectively. These values agree well with expected absorption for monolayer graphene 19,20 = 2.3%. The absorption obtained from cavity modeling was = 0.6% but the mass density shows that the graphene is likely coated with contaminants, perhaps PMMA residue or amorphous carbon. This residue will absorb some light, which would raise above 0.6%, closer to the measured 2.0%.
To quote the most conservative value for the noise equivalent power, we use the highest estimate for the absorption at = 2.3% for our calculation in the main text.