We present results and analysis of the generation of sound in air from monolayer graphene field-effect transistors (FETs). A schematic of one of our devices is shown in Fig. 1(top inset). To explore a wide range of electrical parameters, we measured a total of 16 back-gated and top-gated FETs, the resistances of which varied in the overall range 10 Ω to 20 k Ω. For all devices, the graphene was etched to a square shape of side L = 6 mm. The back-gated FETs were graphene on SiO 2 (300 nm)/p+Si substrates. The p+Si formed the back gate electrode, separated from the graphene by the SiO 2 layer. The top-gated FETs were graphene on quartz substrates. The top gate was formed by a lithium perchlorate-based electrolyte29,30,31. (For full details, see Methods and SS2, 3). Sound pressure above the graphene was measured with a calibrated condenser microphone. We used a lock-in technique to resolve both the magnitude and phase (relative to the source) of the sound (SS4 and Figure S1). This technique gave a sufficiently high signal-to-noise ratio that acoustic isolation of the system was not required.

To facilitate ease of comparison later, it will often be useful to normalise the sound pressure by one or more of the parameters in equation (1). To indicate this, the parameter(s) of normalisation will appear as a superscript to δp. For instance, normalisation by the distance δp r ≡ δpr, frequency, δp f ≡ δp/f and power δp P ≡ δp/P allow ease of comparison of data taken at different distances, frequencies and powers without further analysis. Bias- and resistance-normalisation will also be used when full normalisation by the power is not appropriate.

Second harmonic generation

Second harmonic sound generation results from a source driven by a single frequency ac current. An example of this is shown in Fig. 1b. We used this to investigate the individual components of equation (1). Figure 2a shows the sound pressure spectra resulting from an ac bias voltage for both back- and top-gated FETs. The predicted sound pressure spectra from equation (1) are shown as dashed lines. The parameters used in the equation were either experimentally determined or established material properties (SS5 and Table S1). The thermal properties contained within E dictate the absolute magnitude of the sound. The back-gated FETs have relatively large substrate effusivity, ∼104 JK−1m−2s−0.5 (compared to values of ∼10 and 103 for air and quartz, respectively; Figure S2), and produced the quietest response; the loudest response was produced by graphene on quartz substrates (the top-gated FETs before the electrolyte was deposited). The overall linear dependence of δp 2 on frequency verified the thermoacoustic nature of the signal and allowed us to calculate δp f. The fine structure in the sound spectra was fully reproducible and found to be the result of phase variation as the sound wavefronts pass the microphone surface (SS6 and Figure S3). Figure 2b shows δp 2 as a function of inverse-separation. The linear increase with 1/r is that expected for a point source. By verifying this, it was possible to calculate δp r from measurements made at any separation within the experimental range.

Figure 2 Second harmonic generation. (a) Sound pressure spectra from a range of devices (measured at r = 50 mm). The bottom blue group are spectra from four back-gated FETs. The top red group are spectra from four top-gated devices before the electrolyte was deposited. The middle green group are spectra from two top-gated FETs after electrolyte deposition. Different line types designate device data within groups. In all cases, V g = 0. The expected dependences from equation (1) are shown as dashed lines of the same colour as the data. Indicative sound pressure levels, 20log(δp rf/δp ref ), where δp ref = 20 μ Pa m kHz−1, are shown as horizontal dotted lines. (b) Sound pressure from a back-gated FET with a source power of 3.4 W as a function of inverse device-microphone separation. The sound pressure has been averaged over the frequency range 20–50 kHz. (c) (Top) Device temperature measured in two back-gated FETs (differentiated by colour) and calculated (line) as a function of source power. (Bottom) Sound pressure at f 2 = 40 kHz (measured at r = 50 mm) for the same devices. The solid line is the expected dependence from equation (1). Full size image

The power dependence of the sound is more subtle than the frequency and separation dependences. Figure 2c shows this dependence for the equilibrium device temperature, T eq , and the sound pressure. The back-gated FETs were found to reach high temperatures at high ac biases. This allowed us to explore the mechanisms of heat loss other than sound from the devices. The temperature was found using a calibrated thermal camera, Fig. 2c(top). To a good approximation for powers up to 20 W, T eq = aP+T 0 , where a ≈ 6 K/W and T 0 = 293 K. This temperature is a result of the Joule heat produced in the device balanced by the convective and radiative heat losses to the surrounding air. The expected dependence (SS7) is shown in the figure (and Figure S4). A fit of the theory to the sound pressure is shown in Fig. 2c(bottom). The sublinearity at high powers is accounted for through the effect of the equilibrium temperature on the speed of sound in air immediately above the graphene (SS5). For the following experiments, we kept the power low (<10 W) to be in the linear regime and to calculate δp P.

First harmonic generation

By adding a dc current to the ac current, sound can be generated at the same frequency as the source. This first harmonic generation can be seen in Fig. 1a to result from the combination of the sum and difference heterodynes (where f B = 0). The second harmonic from the ac current remains; the dc current feeds into the dc power loss, P 0 . Therefore, the device generates both second and first harmonic sound simultaneously. If the magnitudes of the ac and dc currents are equal, the first harmonic sound, δp 1 , is four times larger than the second harmonic. This is not of particular note in itself, but its linear dependence on the dc current is. To explore this, we recast the expression for the power as P 1 = 2V dc V ac /R to clearly define the dc and ac components.

The expression for the power at the first harmonic indicates that the sound at this harmonic can be amplified by increasing the dc bias. Figure 3a shows the sound as a function of dc bias at a fixed ac bias of 20 V. We see a linear increase of δp 1 and a flat response of δp 2 . For sound reproduction, this is ideal as the second harmonic can be kept below the threshold of hearing (20 μPa) and ‘volume’ of the first harmonic can be tuned by the dc bias.

Figure 3 Bias control of the first harmonic generation. (a) Sound pressure measured at the first (blue) and second (green) harmonics of the source frequency as a function of dc bias across a back-gated FET. The sound pressure values are averaged over the frequency ranges f i = 38–42 kHz, i = 1, 2. The solid line is a linear fit to the first harmonic data; the dashed line is 〈δp〉 = 8μPa. P 2 was fixed at 0.1 W for all measurements. (b) The first harmonic sound as a function of both ac and dc bias at f = 12 kHz. The second harmonic sound increases quadratically with V ac : the yellow symbols indicate where, experimentally, δp 1 = δp 2 . (c) The phase of the first (black at f 1 = 12 kHz; red at f 1 = 15 kHz) and second (green) harmonic as a function of dc bias. (d) Example of a flat first harmonic sound spectrum (solid lines) created by decreasing the dc bias as V dc = V ref (f ref /f), where V ref = 10 V at f ref = 1 kHz. The ac bias was fixed at 10 V. The applied dc biases are shown (in Volts) as dotted lines in colours corresponding to the sound spectra, which have been normalised to V ref for comparison. Full size image

The balance between the two harmonic sounds can be achieved by varying the experimental parameters. Figure 3b shows δp 1 (V dc , V ac ). The magnitude of the first harmonic increases linearly with both V ac and |V dc | and is roughly symmetric about V dc = 0. In contrast, δp 2 has no dependence on V dc , Fig. 3a, but increases quadratically with V ac . (In fact, as will be shown later, δp 2 does have a small dependence on V dc ). As a result, for a given {V dc , V ac } either the first or second harmonic can dominate. The regions are delineated in the figure. The boundary δp 1 = δp 2 is linear, as expected from equations (1) and (2).

The boundary between first and second harmonic sound dominance is frequency dependent. We investigated this dependence explicitly for the phase of both harmonics. Figure 3c shows the phase of the first and second harmonics as a function of dc bias. The second harmonic behaves as we expect: like the magnitude, the phase is independent of dc bias. In contrast, irrespective of frequency, the phase of the first harmonic switches by half a cycle around V dc = 0. This can be seen to result from the linear dependence of δp 1 on this bias. The direction of the switch does depend on frequency: by changing the frequency through one period, Δ(1/f) = r/v a , (SS6) its direction about V dc = 0 is reversed.

The dc bias can be used to arbitrarily shape the sound spectrum. For example, to create a white (flat) sound spectrum, the dc bias must be inversely proportional to the frequency. Experimentally, by applying such a bias we observe this white spectrum from 1 to 50 kHz, Fig. 3d. By minimising the fine structure (an experimental artefact), this example alone could easily find use as a calibration sound source.

Heterodyne generation

By sourcing ac currents at two frequencies, acoustic heterodynes can be generated at the sum and difference of these frequencies. The power at the heterodynes,

$${P}_{{\rm{A}}\pm {\rm{B}}}={I}_{{\rm{A}}}{I}_{{\rm{B}}}R,$$ (3)

is a simple combination of the current amplitudes and resistance. We can use equation (3) to test the ideal mixing expected from the Joule heating mechanism. An example of the acoustic sum heterodyne is shown in Fig. 1c. Figure 4a shows that if the frequency difference between the sources is maintained then the sound pressure at the heterodyne is independent of the absolute values of the source frequencies and increases linearly with the bias. (The slight suppression with increasing f A is due to capacitive loss). The only observed deviation from this behaviour occurs when the heterodyne coincides with the second harmonic frequency of one of the sources. In this instance, a quadratically increasing envelope of the pressure occurs with increasing bias (see Methods).

Figure 4 Heterodyning. (a) The acoustic difference heterodyne, |f A −f B | = 16 kHz, as a function of the ac bias magnitude of source A. Source B had a fixed magnitude equal to the maximum of A. Different colours correspond to different source A frequencies: f A = 1 kHz (black), 10 kHz (red) and 100 kHz (blue). The green, dashed curve shows f A = 8 kHz. (b) Homodyne sound generation as a function of the phase shift, θ, between A and B: f A,B = 5 kHz (black) and 10 kHz (blue). The phase shift between A and B is shown schematically below. Full size image

Homodyning occurs when the two source frequencies are equal. Homodynes are sensitive to the phase shift between the sources. As such, they are commonly used in optical and acoustic detection systems. In thermoacoustics, the two sources generate sound only at the second harmonic. This can be seen from Fig. 1a. The sum heterodyne combines with the second harmonics of the individual sources; the difference heterodyne adds to the dc power loss. Figure 4b shows the second harmonic sound as a function of the phase difference between the sources. Although the contribution of this phase in equation (2) appears rather complex, the effect on the magnitude of the sound is simple: if the phase difference is zero, the sound is maximised; if it is half a cycle, the sound is turned off. The sensitivity of the homodyne sound to this phase would make it useful as a detector of electronic phase changes in one of the sources.

Sound gating

Beyond its thermal properties, graphene plays two further roles in the sound generation. First, its electrical properties can be tuned by the transistor gate. This tuning could be used to switch or modulate the sound output. Second, it allows us to invert our original question: could we use the sound generation to reveal something about the conduction in graphene? Gate control of the sound output is possible in a field-effect transistor. The resistance and sound were measured by applying a gate voltage, V g , between the graphene and the gate electrode. Figure 5a,b shows measurements of the conductance, G = 1/R, and sound pressure as a function of V g for a back-gated FET (see Methods). As V g increases, the conductance decreases, approaching a minimum at V g = V D . In the limit P → 0, this minimum occurs at ∼140 V, which is coincident with the Dirac point (the energy at which the conduction and valence bands meet). In order to observe and compare the explicit dependence of δp 1,2 on R(V g ), they were normalised by the applied biases (see equation (2)): \(\delta {p}_{1}^{V}\equiv \delta {p}_{1}\mathrm{/(2}{V}_{{\rm{dc}}}{V}_{{\rm{ac}}})\); \(\delta {p}_{2}^{V}\equiv \delta {p}_{2}/({V}_{{\rm{ac}}}^{2}\mathrm{/2)}\). It can be seen that both harmonics are indeed proportional to 1/R, as predicted. As a result, the magnitudes and relative phases of the sounds generated at f 1 and f 2 can be completely specified by the set {V dc , V ac , V g }.

Figure 5 Sound gating with a field-effect transistor. (a,b) A back-gated FET. The conductance (a), and the first (solid green) and second (solid blue) harmonic sound (b) as a function of the back-gate voltage. The power-normalised sound pressures are shown as dotted lines: the units for δp rfP are μ Pa m kHz−1W−1. (c,d) A top-gated FET. Conductance (c) and second harmonic sound (d) as a function of top-gate voltage measured at f 1 = 18 kHz for different source powers from 0.05 W (black/green) to 0.5 W (red/blue). The ‘on’/‘off’ state of the sound, described in the main text, is indicated by the blue/red shaded regions, respectively. Full size image

The gate control could be used to switch the sound on and off. Figure 5c shows G(V g ) for a top-gated device for powers from 0.05 W to 0.5 W (see Methods). The conductance minimum is seen to occur at V D ∼0.6 V and this shifts with increasing power by ∼0.05 V over the power range considered. The two branches about V D do not vary up to 0.2 W; at higher powers they typically become less conductive. As with the back-gated FETs, the sound pressure varies in a similar way to G(V g ), so for a fixed bias the gate can be used to effectively switch the sound on and off by toggling its voltage between V g = V D and V g = V D − 2 V, Fig. 5d. The ‘on’ and ‘off’ state we define at P = 1 W as being at sound pressure levels, 20log(δp rf/δp ref ), of 0 dB and −20 dB, respectively. These correspond to ‘at’ and (an order of magnitude) ‘below’ the limit of human hearing at f = 1 kHz and r = 1 m, where δp ref = 20 μ Pa m kHz−1.

Nonlinear conduction in graphene

The sound generation and gate control can be used to investigate the conduction mechanisms in the graphene. If the bias-normalised sound pressure in Fig. 5b is further normalised by the resistance, it should be constant as a function of gate voltage. However, δp is found to be enhanced as the gate voltage approaches V D . This is possible, if the charge transport in graphene has a nonlinear component. To second order in the current, the voltage

$$V(t)={R}_{0}{I}_{{\rm{dc}}}+{R}_{1}{I}_{{\rm{ac}}}\,\cos (\omega t)+\tfrac{1}{2}{R}_{2}{I}_{{\rm{ac}}}^{2}{\cos }^{2}(\omega t)\,,$$ (4)

where R 0 = V dc /I dc is the dc resistance, R 1 = dV/dI is the differential resistance, and R 2 = d2 V/dI 2 is the second-differential resistance. Up to this point, we have assumed R 0 = R 1 ≡ R: experimentally, as R 2 I ac < R 1 we continue with this assumption. As a result, the source power has additional terms of 3R 2 (I ac /2)3 and R 2 I dc (I ac /2)2 for P 1 and P 2 , respectively (SS8). These power components can account for the differences seen in the dependences of δp 1 and δp 2 on V g , if |R 2 | increases with V g .

If the second-differential resistance is the origin of the enhancement then sound will be generated at the third harmonic. The power at this harmonic,

$${P}_{3}={R}_{2}{({I}_{{\rm{ac}}}\mathrm{/2})}^{3},$$ (5)

depends exclusively on R 2 . An explicit measurement of R 2 was made along with this predicted harmonic component. Figure 6a shows R 2 (V g ) of a back-gated FET for different source powers. In this experiment only, the device was immersed in liquid helium to distinguish the effect of bias from the effect of an increase in T eq (Figure S4). That it is R 2 and not an artefact of the resistance is shown in Fig. 6b, where the voltage drop across the channel of the FET at the second harmonic of the source frequency (V 2 ) is shown to have the quadratic dependence on the source current (equation (4)). The origin of R 2 (V g ) is not important for the present discussion. (Various types of nonlinear behaviour have been predicted32,33,34 and observed17, 35,36,37). The significance here is that R 2 is finite and increases in magnitude as the gate voltage approaches the Dirac point. Figure 6c–e shows that this does indeed account for the observed enhancement of the sound pressure: Fig. 6c shows the direct correlation between R 2 and δp 3 ; Fig. 6d shows the dependence of δp 2 on V dc ; and Fig. 6e shows the dependence of each harmonic on the ac bias. For Fig. 6e, the wide range of dc biases over which the data are averaged, and the fact that \({R}_{{\rm{2}}}{I}_{\mathrm{dc},\mathrm{ac}}\ll {R}_{1}\) means that the predicted power-law dependence on \({V}_{{\rm{ac}}}^{n}\) is n = 1, 2, 3 for δp 1,2,3 , respectively. The δp 2 (V dc ) dependence shown in Fig. 6d can be used to estimate a value for R 2 of ∼+100 ΩA−1 at V g = 0, which is comparable to that measured directly in the charge transport. For sufficiently large values of R 2 beyond our current experimental range, the second harmonic acoustic response could be turned off altogether.

Figure 6 Third harmonic generation. (a) Second-differential resistance as a function of gate voltage of a back-gated FET immersed in liquid helium (T = 4.2 K). The curve colour ranges from green at P = 0.007 W to blue at P = 0.7 W (not in equal steps). (b) The voltage drop across the device at f 2 as a function of ac bias. The data (symbols) are taken at the gate voltages indicated by identically coloured arrows in a; the dashed lines are the expected \({I}_{{\rm{ac}}}^{2}\) dependence. (c) R 2 (V g ) in ambient conditions at P = 0.7 W shown together with the simultaneously measured current-normalised third harmonic sound pressure, \(\delta {p}_{3}^{I}=\delta {p}_{3}/{({I}_{{\rm{ac}}}\mathrm{/2)}}^{3}\) (measured at f 3 = 42 kHz and r = 25 mm). \(\alpha \equiv \mu {\rm{Pa}}\,{\rm{m}}\,{{\rm{kHz}}}^{-{\rm{1}}}\,{{\rm{A}}}^{-{\rm{3}}}\). (d) δp 2 as a function of dc bias, normalised to its value at V dc = 0. Curves at four ac biases: 10 V (black), 20 V (red), 30 V (green) and 40 V (blue). (e) First, second and third harmonic sound pressures as a function of ac bias. Each datum point is an average over a range of frequencies 10 < f < 14 kHz and dc biases −34 < V dc < +34 V. The dashed lines are the predicted \({I}_{{\rm{ac}}}^{n}\) dependences: n = 1 (blue), n = 2 (green) and n = 3 (red). \(\beta \equiv \mu {\rm{Pa}}\,{\rm{m}}\,{{\rm{kHz}}}^{-{\rm{1}}}\,{{\rm{\Omega }}}^{-{\rm{1}}}\,{{\rm{A}}}^{{\rm{2}}-{\rm{n}}}\). Full size image

In summary, we have demonstrated a highly versatile thermoacoustic sound generator ranging from audible to ultrasonic frequencies. The most significant result of our work was to show that the Joule heating mechanism in graphene controllably mixes frequency components of a current source together. This not only has applications in acoustics but also in signal processing where it could be used to create an acoustically-coupled, linear electronic mixer. We further showed the simplicity of this mixing in heterodyning, homodyning, amplification and equalisation. In addition to modulation achieved using a transistor gate, this afforded full control over the sound output. Such a generator has a wide range of potential applications, from multiplexing in telecommunications to calibrated sound sources for metrology and sensing. One of our most intriguing results was that this generation can be used to quantitatively measure the conduction properties of graphene. Nonlinearity in the conduction has important consequences for optical, electronic and thermal applications of this material so our acoustic probe will provide fresh insights in these areas.