## Introduction

Thermoacoustic sound is produced by a material without physical movement. In terms of applications, this has been its principle distinction from other generation mechanisms. It has been observed in many thin film conductors1,2,3,4,5 including graphene6,7,8,9,10, reduced graphene oxide11, 12 and graphene foams13, 14. The exceptionally high thermal conductivity15 and low heat capacity16 of graphene make it the quintessential material for investigating thermoacoustics. Up to now, the thermoacoustic process has been considered too uncontrollable and inefficient to be of scientific or practical interest. The focus has been to improve the efficiency of the transduction. Joule heat in graphene is lost via several routes17, 18. How it is lost is not only of fundamental importance but also key to thermal management applications19, 20. Loss in the form of light emission has been observed21,22,23. However, the most significant loss is to the substrate supporting the graphene. This is why the drive to greater efficiency has led to attempts to optimise the substrate, the ultimate being to remove it altogether. Practical applications will generally require a substrate, limiting the material choices and resulting efficiency. What has been largely overlooked so far is the role of the electronic properties. This role is the focus of our work.

The Joule heating mechanism acts as an ideal mixer for heterodyne generation. As we will show, heterodynes at the sum and difference of the source frequencies are generated most efficiently when the conductance is linear. (We will see later what happens in the case of nonlinearity). This is in stark contrast to traditional heterodyning systems that require a nonlinear element to mix the input signals (for example, diodes in electronics and nonlinear crystals in optics). Heterodyning is used extensively in telecommunications and signal processing, and has been suggested to be involved in bat echolocation24. It is finding new uses in acoustic metrology25, gas sensing26 and chemical spectroscopy27. As such, controllable heterodyning via a linear element will make a mark in a wide variety of fields.

The sound in air resulting from Joule heating can be modelled relatively simply2. The sound pressure

$$\delta p= {\mathcal E} \frac{fP}{r}\,,$$
(1)

where P is the power. The linear dependence of this sound pressure on frequency, f, is a hallmark of thermoacoustic generation. We assume here that r, the distance from the source, is large compared to the lateral size, L, of the graphene and that the sound is measured only along the surface normal. Therefore, sound is measured in the far field, r > L 2 f/4v a, and the graphene acts as a point source, Lv a/f28. The parameter, E contains the thermal properties: $${\mathcal E} =3{e}_{{\rm{r}}}\mathrm{/4}\pi {v}_{{\rm{a}}}^{2}$$, where e r is the thermal effusivity of the air relative to the whole system and v a is the velocity of sound in air. (Full details of the derivation can be found in Supplementary Information Section (SS) 1).

The power dependence of the sound pressure is the key to unlocking the potential of thermoacoustic generation. A current, I, driven through a conductor causes a voltage, V, to drop across it. The power dissipation, P = IV, is a result of the Joule heating mechanism. However, the power that appears in equation (1) is not the total power but only the power at frequency f. A single ac current source, Icos(ωt), will result in power IV(1 + cos(2ωt))/2. This has components at zero frequency (dc) and the second harmonic (2ω) of the source. The dc component only heats the device. The 2ω component is the source of power for the sound. This is the second hallmark of thermoacoustic generation: an input signal at f will cause sound generation only at 2f.

What if the source was driven by a current of more than one frequency? Consider two currents, A and B, with amplitudes I A,B and frequencies f A,B, respectively, as shown in Fig. 1(bottom inset). The power has several components:

$$\begin{array}{rcl}P & = & \tfrac{1}{2}R\{({I}_{{\rm{A}}}^{2}+{I}_{{\rm{B}}}^{2})+{I}_{{\rm{A}}}^{2}\,\cos \,\mathrm{(2}{\omega }_{{\rm{A}}}t)+{I}_{{\rm{B}}}^{2}\,\cos \,\mathrm{(2}{\omega }_{{\rm{B}}}t)\\ & & +2{I}_{{\rm{A}}}{I}_{{\rm{B}}}(\cos ({\omega }_{{\rm{A}}-{\rm{B}}}t)+\,\cos ({\omega }_{{\rm{A}}+{\rm{B}}}t))\}\\ & \equiv & {P}_{0}+{P}_{{\rm{2A}}}+{P}_{{\rm{2B}}}+{P}_{{\rm{A}}-{\rm{B}}}+{P}_{{\rm{A}}+{\rm{B}}}\,,\end{array}$$
(2)

where ω A±B ≡ ω A ± ω B and R is the resistance. This result is shown schematically in Fig. 1a. The powers P i generate sound δp i at frequency f i . The first three terms are what we expect from a linear superposition of two independent sources. The last two terms are heterodynes: the sum and difference frequencies of the sources.

## Results and Discussion

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 SiO2 (300 nm)/p+Si substrates. The p+Si formed the back gate electrode, separated from the graphene by the SiO2 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.

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.

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).

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}.

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 D0.6 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 = R1 ≡ 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.

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.

## Methods

### Device fabrication

Back-gated graphene FETs were created from CVD-grown monolayer graphene transferred to a degenerately doped silicon substrate coated by 300 nm silicon dioxide (SS2). The electrodes were formed of 50 nm thick Au thermally evaporated directly onto the graphene. A thin layer of Cr was used to anchor the extensions of these electrodes on the silicon dioxide surface. The graphene–gold interfacial contact resistance, determined to be 0.45 Smm−2, was sufficiently low (<3% of the total resistance) that it could be neglected (SS3). Four devices had 12 electrodes spaced around the edge of the graphene; other devices had two interdigitated electrodes separated by gaps of either 200 μm or 100 μm (Figure S1). Silver paint was used to fix the device to the base of a ceramic chip carrier. The paint allowed electrical connection to the Si and, therefore, to measure the device as a field-effect transistor by applying a gate voltage between the silicon and the graphene.

Top-gated FETs were created from CVD-grown monolayer graphene transferred to quartz substrates. A shadow mask technique was used to evaporate four Au/Cr electrodes on the corners of the graphene (a Van der Pauw geometry) and an additional electrode for the gate. All devices were nominally identical. The gate was formed by drop casting a lithium-based electrolyte layer onto the surface of the graphene. The electrolyte source material was created by mixing Poly(Ethyl Oxide) with Lithium Perchlorate (in an 8:1 ratio) in methanol, ultrasonicating the solution for 1 hour, then centrifuging at 10,000 rpm for 5 min. The supernatant was then drop cast by pipette onto the graphene.

### Thermoacoustic measurements

The ac voltage source was that of a lock-in amplifier (Signal Recovery 7265), amplified by a high voltage amplifier. The sound generation was measured using a calibrated condenser microphone (Earthworks M50), which had a constant sensitivity across the full spectral range of the experiment, at distances from 50 mm to 2 m away from the device (SS4 and Figure S1). The output signal from the microphone was amplified (Earthworks ZDT1022) and measured by lock-in amplifiers referenced to the source frequency. For the dc bias measurements, a floating dc voltage source (IOtech DAC488) was connected in series with the ac source. The full spectral response was additionally measured using a spectrum analyser (Stanford Research Systems SR785). For heterodyning, sources were added together (prior to the high voltage amplification) using a differential amplifier (Femto DLPVA-100). To measure the magnitude of the heterodyne signal, the lock-in amplifier was locked to the signal from the microphone at the heterodyne frequency (i.e. it uses the measured signal frequency as a reference as there is no real source at this frequency). The apparent periodicity seen in Fig. 4a is a temporal aliasing artefact caused by this different method of measurement.

The sound gating of the back-gated FETs required large voltages to be applied. To avoid breakdown of the gate dielectric in air, which would normally occur above |V g|  30 V, a steady stream of helium gas was directed across the device throughout. With this stream, the breakdown voltage was extended to 180 V. In top-gated FETs, the electrolyte forming the gate fails at V g 2 V31. As the resistances of the top-gated devices ranged from 2–200 kΩ, biases much greater in magnitude than this voltage were required to source sufficient power to generate sound. To apply such biases, we used the fact that the ions in the electrolyte are slow to respond to changes in the applied gate voltage: the maximum slew rate, ΔV gt, calculated from transient response measurements was found to be 10−2 V s−1. The rate of source signal change (104–106 V s−1) is much higher than this slew rate and, as a result, ac source–drain biases well over an order of magnitude larger than the gate voltage range could be applied to the channel without any gating artefacts.

### Electrical and thermal measurements

The differential resistance was determined by measuring the voltage drop across the device together with the voltage drop across a ballast resistor (1 kΩ, 50 W) in series with the device. These were measured simultaneously by lock-in amplifiers referenced from the source. Surface temperatures were measured using a thermal imaging camera (FLIR E6). Calibration of the camera was performed by measuring the surface temperature of a device in a calibrated heating stage (Linkam THM600) at a number of fixed stage temperatures.