Description of the GNB fabrication and mechanical measurements

In our GNB, light is detected by tracking changes to the fundamental mode frequency of a graphene nanomechanical resonator (see Supplementary Methods). The graphene structures are made by transferring graphene23 onto a silicon/silicon oxide support substrate with patterned holes, resulting in circular drumhead resonators (Fig. 1b). Some drumheads are patterned into trampoline geometries using a focused ion beam (FIB) technique20 (see Methods for details), as shown in Fig. 1c. We drive motion of the graphene resonators24 by applying an a.c. voltage between the graphene and the backgate (Fig. 1a), and we measure the motion with a scanning laser interferometer25 operated with a low-power, power-locked laser. By sweeping the a.c. drive frequency, we obtain amplitude and phase spectra, as seen in Fig. 1d for the first fundamental mode of a trampoline. The resonance frequency can be inferred from either the phase or the amplitude spectrum, which from Fig. 1d is ~10.7 MHz. We obtain the resonance gate dependence by applying a d.c. bias to the graphene, while measuring the amplitude spectrum, as shown in Fig. 1e. By using an electromechanical model (see Supplementary Note 7), the gate dependence (Supplementary Fig. 7) reveals the graphene membrane mass density (ρ), Young’s modulus (Y), and initial stress (σ 0 ). We track the frequency during light illumination with frequency modulation detection26, which uses a phase-locked loop (PLL) with the phase locked on resonance. A key advantage of using frequency modulation is that the GNB response BW is not determined by the resonance linewidth, as it is with amplitude modulation detection. The PLL BW allows tracking up to ~50 kHz. For frequency-shift measurements, we maximize the signal-to-noise ratio in several ways. First, we use the scanning interferometer to obtain a 2D spatial map of the vibrational amplitude of the resonator. A map for a trampoline (Fig. 1f) shows 90° rotational symmetry in agreement with the trampoline geometry and goes to zero near the clamping of the tethers, indicating they are the only point of contact to the substrate. Using these maps, we position the interferometer laser to maximize the amplitude signal. Moreover, we adjusted the a.c. voltage level to just below the onset of bistability to maximize the resonator amplitude and to avoid nonlinear effects, such as phase instability, which can disrupt the phase locking.

Measurement of the noise-equivalent power

The noise-equivalent power (pW Hz−1/2) of the GNB is calculated with the expression \(\eta = \sigma _f\sqrt t /(f_0\;R_f)\), where σ f is the frequency noise, t is the measurement time, and R f is the frequency-shift responsivity (i.e., the fractional change in resonance frequency per unit of absorbed power), defined as \(R_f \equiv \frac{1}{{f_0}}\frac{{{\mathrm{d}}f_0}}{{dP_{{\mathrm{abs}}}}}\). To determine R f , we illuminate the GNB membrane with an amplitude-modulated heating laser (532 nm) and measure f 0 with a PLL. A time recording of f 0 when the GNB is exposed to sinusoidally modulated light is shown in Fig. 2a, in which P abs = 4.4 nW. Here we assume the absorption is 2.3% of the incident power6,10,12. The shift Δf 0 is inferred from a sine fit (Fig. 2a black curve) as the peak-to-peak amplitude. For the data shown in Fig. 2a, Δf 0 = 8.5 kHz, corresponding to ~72% of the resonator linewidth. The power dependence of Δf 0 for a trampoline GNB (Fig. 2b) shows that Δf 0 is linear with P abs (in the range of 1–100 nW) and therefore \(\frac{{df_0}}{{dP_{{\mathrm{abs}}}}} = \frac{{{\mathrm{\Delta }}f_0}}{{P_{{\mathrm{abs}}}}}\) is a constant. The linear power dependence of f 0 was observed in all GNB devices. In Fig. 2c, we plot \(R_f = \frac{1}{{f_0}}\frac{{{\mathrm{\Delta }}f_0}}{{P_{{\mathrm{abs}}}}}\) vs. tether width (w) for nine different trampolines and three different drumheads; for drumheads, w is given by one-fourth the circumference. The trampoline width (w) is indicated in Fig. 3b. We tested trampoline GNBs with w ranging from 200 nm to 1.4 µm and GNBs with a d of 6 and 8 μm. In general, the drumheads had R f values about 1% that of trampolines. Our most sensitive device, a 6 µm diameter trampoline with 200 nm wide tethers, had R f ~ 300,000 W−1, a factor 100 greater than state-of-the-art nanomechanical bolometers15. As seen from Fig. 2c, R f increases with smaller w for trampolines.

Fig. 2 Frequency responsivity to absorbed light and frequency noise measurements of graphene resonators. a Mechanical resonance frequency vs. time for a 8 µm diameter trampoline with 500 nm wide tethers. The device is subject to 190 nW of incident radiation modulated at 40 Hz. Assuming 2.3% absorption, the absorbed power is \(P_{{\mathrm{abs}}} = 4.4\) nW, which causes a frequency shift of Δf 0 = 8.5 kHz. b Measured resonance shift vs. absorbed power. A best-fit line to this data yields a 2.3 kHz nW−1 resonance shift per incident power. c Frequency responsivity, R f , vs. tether width, w, for nine different trampolines and three different drumheads. For the drumheads, the tether width is taken to be 1/4 of the drumhead circumference. d Resonance frequency vs. time for a trampoline GNB device. The device is not exposed with heating laser light other than that needed for the measurement. e Allan deviation, σ A , of the frequency noise vs. measurement time in a log–log plot. The resonance frequency was tracked with the PLL to obtain temporal frequency data. f Sensitivity, η, vs. tether width for nine different trampolines and three different drumheads. Symbol legend is shared between c and f. Circles indicate a trampoline with a 6 µm diameter, turquoise triangles indicate a trampoline with an 8 µm diameter, and magenta triangles indicate a drumhead resonator of either 6 or 8 µm diameter. Source data are provided as a Source Data file Full size image

Fig. 3 Modeling and bandwidth measurements of graphene resonators. a Thermal circuit model. b False-colored scanning electron microscope image of a trampoline of tether width w = 200 nm and diameter d = 6 µm. Black scale bar is 2 µm. c Normalized frequency-shift responsivity \({R}_{f}^ \ast\) as the amplitude of the heating laser is modulated from 100 Hz to 50 kHz. The total resonance shift was found to be constant for low modulation frequency and reached half its maximum value at BW = 13.8 kHz. A thermal circuit model was used to fit the thermal response time of the trampoline; the fitted curve using Eq. 2 is shown in black. d Real and imaginary amplitude of thermal expansion induced displacement for a trampoline (w = 1.2 μm, d = 6 μm). The black curve is a fit to the thermal circuit model. From this fit, we extract the thermal response time, \({\tau }_{\mathrm{T}} = 2.4\) μs. e Bandwidth vs. tether width for nine different trampolines and three different drumheads. For the drumheads, the tether width is taken to be 1/4 of the drumhead circumference. f Sensitivity vs. bandwidth for nine different trampolines and three different drumheads. The black line is the linear fit, \({\mathrm{BW}} \propto {\eta }\) (R-value of 0.97). All bandwidth values in e and f were inferred from the off-resonant thermomechanical method. Symbol legend and vertical axis is shared between e and f. Circles indicate a trampoline with a 6 µm diameter, turquoise triangles indicate a trampoline with an 8 µm diameter, and magenta triangles indicate a drumhead resonator of either 6 or 8 µm diameter. Source data are provided as a Source Data file Full size image

As a measure of the fractional noise, σ f /f 0 , we used the Allan deviation27, which we calculate (see Supplementary Methods) from temporal recordings of the frequency while the heating laser is turned off (Fig. 2d). Representative Allan deviation data for varying measurement intervals are presented in Fig. 2e. Across the sampling range and for all devices, the Allan deviation was flat, taking on a value of ~10−5, indicating that σ f is dominated by flicker noise (1/f) and not by thermomechanical noise28. In this case, the frequency noise is not generally reduced with a larger quality factor29,30.

Combining R f and the Allan deviation (measured at 100 Hz), we calculate the noise-equivalent power η for each device and plot η vs. w, shown in Fig. 2f. This data illustrates that η decreases with narrower tether width. A trampoline with a tether width of 200 nm exhibited the best power sensitivity, η = 2 pW Hz−1/2 (at 1 kHz BW), which is also the lowest reported value of noise-equivalent power for a room-temperature graphene bolometer to date12. The η is much larger for drumheads; the largest value (η ~ 1 nW Hz−1/2) is over 200 times greater than the most sensitive trampoline. From these trends, it is clear that reducing the tether width provides a straightforward means to lower, and thus improve, the GNB’s η.

Our measurement of η assumes 2.3% absorption. However, cavity effects and surface contaminants could lead to large deviations from 2.3%. Our cavity modeling (see Supplementary Note 5 and Supplementary Fig. 6) predicts that variations in the absorption are dominated by interference, which changes the overall intensity at the surface of the graphene membrane. For the device geometry used in this work, the intensity, and thus the effective absorption, is reduced to ~0.6%. Moreover, photothermal back-action cavity effects have a negligible effect on R f in this configuration (see Supplementary Note 6). For the most sensitive device, cavity effects indicate that the absorbed power could be lower than predicted by the 2.3% absorption estimate and therefore the NEP could be as sensitive as η = 500 fW Hz−1/2. However, by combining the measured frequency shift and resonance frequency gate dependence (Fig. 1e) with predictions from mechanical modeling for R f (see Supplementary Note 7), we calculate an experimental value for the optical absorption of 2.0%. Surface contaminants on the graphene, which the measured mass density indicates are present, likely increases the total absorption from that predicted from cavity modeling. For the sake of comparison with previous work10,12, we use the standard absorption estimate6 of 2.3%.

Modeling of the frequency responsivity

The observations of R f and η can be understood through a thermomechanical model that combines a thermal circuit with membrane mechanics. The circuit (shown schematically in Fig. 3a) treats the GNB as a thermal capacitance C, given by the membrane heat capacity, in parallel with a thermal resistance R T , governed largely by the tethers (or boundary circumference for drumheads). The absorbed power, I = P abs , obeys Fourier’s heat law, ΔT = P abs R T , where ΔT is the temperature difference between the graphene and the surrounding substrate (assumed to be a room-temperature thermal ground.) By using first-order thermal expansion, we relate ΔT to the mechanical strain in the GNB membrane to calculate Δf 0 . For an absorbed power modulated at angular frequency ω, the model provides an expression for the frequency-shift responsivity

$$R_f\left( \omega \right) = - \frac{{{\mathrm{\alpha Y}}}}{{2\sigma _0\left( {1 -

u } \right)}}\frac{{R_{\mathrm{T}}}}{{\sqrt {1 + \omega ^2R_{\mathrm{T}}^2C^2} }}$$ (2)

again where α is the thermal expansion coefficient, v is the Poisson ratio, σ 0 is the initial in-plane stress, and Y is the 2D elastic modulus. The full details of the thermomechanical model are provided in Supplementary Note 2 and Supplementary Fig. 3. We note Eq. 2 predicts R f is independent of incident power, in accord with the measurements given in Fig. 2b. In the low-frequency limit (i.e., \(\omega \ll \frac{1}{{R_TC}}\)) and with tether resistance \(R_{\mathrm{T}} = \frac{{\rho _{\mathrm{T}}l}}{w}\), where ρ T is the 2D thermal resistivity of graphene, and l and w are the tether length and width, respectively, Eq. 2 becomes

$$R_f = \frac{{\alpha Y\rho _{\mathrm{T}}}}{{2\;\sigma _0\left( {1 -

u } \right)}}\frac{l}{w}$$ (3)

Measurements of R f vs. w for trampolines given in Fig. 2c agree well with Eq. 3; a fit to \(R_f \propto w^{ - 1}\) for trampolines has a statistical R-value of 0.74. Moreover, the model predicts η ∝ w, which is also in agreement with our measurements (Fig. 2f; R-value of 0.70). In both cases, the agreement is good, despite some variations in σ 0 and l.

Measurement of the bandwidth

Another important metric in a bolometer is the response BW, which determines its ability to detect transient signals and fast variations of the radiation intensity. We characterize the BW in two ways. First, we infer the BW from the 3 dB roll-off of R f (ω), which we get by sweeping the modulation frequency, ω, of the heating laser at fixed power and measuring Δf 0 with the PLL and a second lock-in (see Supplementary Note 3). We fit the measured R f (ω) with Eq. 2 to extract the fit parameter \(\tau _T = R_{\mathrm{T}}C\) (i.e., the characteristic time of the circuit), thereby obtaining \({\mathrm{BW}} = \sqrt 3 /(2\pi R_TC)\). An R f spectrum for a trampoline GNB is illustrated in Fig. 3c, where the black trace is the fit to Eq. 2. This spectrum has a nearly flat response, before falling off at BW ~ 13.8 kHz. As seen from the fit, the measured R f (ω) obeys the circuit model very well.

These spectra provide a direct measure of the BW of R f , but are limited by the measurement BW of the PLL. To overcome these speed limitations, our second approach infers the BW from the off-resonant thermomechanical out-of-plane displacement of the graphene membrane, which occurs when thermal stress tightens and locally flattens the membrane19,31. In the limit of small displacement and first-order thermal expansion, the mechanical displacement amplitude will be proportional to the change in temperature, \(A \propto {\mathrm{\Delta }}T\) (Supplementary Fig. 4). Therefore, the displacement amplitude due to a modulated heating laser of frequency ω will obey our thermal circuit model and will have the same frequency dependence as R f (ω), as given in Eq. 2. For these off-resonant measurements, we sweep the modulation frequency of the heating laser at frequencies well below the mechanical resonance (in the absence of any electrical actuation) and record the amplitude A(ω) (see Supplementary Note 3). By fitting our measurements of A(ω) to our model, we extract the thermal response time \(\tau _{\mathrm{T}} = R_{\mathrm{T}}C\) and thus BW. Figure 3d illustrates the real and imaginary parts of A(ω) along with the model fit (black traces) for a trampoline device with \(\tau _{\mathrm{T}} = 2.4\) μs or BW = 120 kHz (see Supplementary Fig. 5). Where possible, we compared the BW obtained from R f (ω) and A(ω), finding excellent agreement (see Supplementary Note 3 and Supplementary Table 2). Again, we note that the 3 dB BW is not limited by the mechanical linewidth of the resonator when using frequency moduluation26. In practice, the BW is limited by either the thermal circuit or PLL BW.

The response BW is strongly correlated with the tether width, where wider tethers produce a faster response. We plot BW vs. w in Fig. 3e. The BW of trampolines ranged between 10 and 100 kHz, while for drumheads the BW was as high as 1.3 MHz. For trampolines, our model predicts

$${\mathrm{BW}} = \frac{{\sqrt 3 }}{{2\pi c\rho \rho _Tl}}\frac{w}{A}$$ (4)

where c is the membrane-specific heat, ρ is the membrane mass density, and A is the membrane area. The measured BW data in Fig. 3e agrees well with the model prediction BW∝w; for 6 μm diameter trampolines, the linear fit R-value is 0.9. Although our experiments did not broadly sample the device area A, our limited data do agree with the prediction \({\mathrm{BW}} \propto A^{ - 1}\). 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 6) is about a factor of ~7.5 greater than pristine graphene.

The BW and the noise-equivalent power are expected to be directly proportional, regardless of the device geometry. Specifically, our model predicts

$${\mathrm{BW}} = \left( {\frac{{\sqrt 3 }}{{2\pi }}\frac{{\alpha _T}}{{\sigma _A\sqrt t c\rho }}\frac{1}{A}} \right) \cdot \eta$$ (5)