Introduction

Mid-infrared (mid-IR) photodetectors hold immense importance across diverse fields. Capturing and visualizing thermal radiation, they enable the study of celestial objects and their evolution1,2, diagnostics and therapeutics through non-invasive imaging3, nondestructive testing of components and detecting defects4. Mid-IR light carries information about molecular vibrations, providing valuable information about the chemical composition of materials for monitoring environmental pollutants5.

Tunneling devices based on van der Waals heterostructures are attractive for infrared detection due to the presence of strong phonon-polariton absorption lines in many layered dielectrics6,7,8,9, the most prominent example being the hyperbolic modes of hexagonal boron nitride (hBN)10,11,12. The intrinsic response time of tunneling detectors should be very short and limited only by vertical transport between few-nanometer-thick barriers13,14. As the tunneling probability is exponentially sensitive to the electron energy and barrier height, the tunnel-type photodetectors may have additional photocurrent gain mechanisms. They can be associated either with carrier heating or light-induced charge accumulation in the layers. The latter effect is especially pronounced in multilayer structures, such as those used in quantum well infrared photodetectors15,16,17. Strong non-linearity of the tunneling current–voltage characteristic should give rise to a pronounced radiation rectification, at least at ‘classical’ electromagnetic frequencies, when photon energy is below the barrier height18,19. At ‘quantum’ frequencies, a more adequate picture of photocurrent is the photon-assisted tunneling20. The photon-aided tunneling should benefit from singularities in the joint density of states between tunnel-coupled layers. Such a situation is realized in superlattice-based photodetectors21,22 and quantum cascade lasers23 and is further anticipated for tunnel-coupled graphene layers24,25.

A variety of light-induced physical effects in tunnel-coupled 2D layers makes it challenging to reveal the dominant photodetection mechanism in such structures. The experimental studies of such structures were concentrated on the visible range, where both heating-induced photocurrents26,27 and direct photon-aided tunneling were detected28. A reverse process of tunneling accompanied by visible light emission was also observed in coupled graphene layers29. The measurements of van der Waals tunneling photodetectors at lower electromagnetic frequencies are scarce. In the THz range, the measured photocurrents were speculated to originate from photon-aided tunneling30, yet no experimental proofs of this scenario were provided. As a result, the mechanism of photocurrent generation in graphene-based vertical tunnel structures in the infrared range remains unresolved.

In this work, we report on the photocurrent measurements of graphene/hBN/graphene tunnel structure under IR illumination with a photon energy of 144–207 meV. The graphene layers in our structures are twisted by a large angle ≳ 5° 31,32. In such a situation, the direct current can be dominated by electron resonant hopping via impurity levels inside the hBN barrier33,34, where the Ib–Vb-characteristic becomes ladder-type with sharp slopes. We find that the photocurrent Iph is proportional to the \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}\) and is maximum under the condition of tunneling through the impurity level in hBN. At a non-zero bias, the photocurrent appears proportional to the derivative of the DC tunnel current with respect to the base cryostat temperature dIb/dT. The proportionality between these three quantities (the photocurrent, the curvature of Ib–Vb-characteristic, and the temperature derivative of DC current) is accurately reproduced by a theoretical model where the incident radiation causes electron heating in coupled graphene layers. At finite bias, the origin of photocurrent can be termed as bolometric effect across the tunnel junction. At zero bias, the photocurrent emerges only upon the asymmetric heating of electrons in the two layers. The phenomenon is analogous to the photo-thermoelectric effect, albeit it is developed across the tunnel barrier. As the main reason for tunneling photocurrent lies in the radiation-induced electron heating, it becomes possible to perform the inverse reconstruction of electron temperature from the measured photocurrent. Such a technique can become a simple alternative to the Johnson noise thermometry35, especially at cryogenic temperatures36,37,38,39.

Results

Electrical characterization of tunneling infrared detector

Our tunnel IR detector was fabricated using dry transfer technique40 (see the “Methods” section for details). The structure consists of two flakes of single-layer graphene (SLG) separated by approximately three layers of hBN (thickness ~1 nm). The structure is located on an hBN/graphite stack, the latter serving as a back gate. The intersection area of the top and bottom graphene is ~2.3 μm2, which determines the area of the tunnel junction. A scheme of the stack and measurement configuration are presented in Fig. 1a. A photograph of the heterostructure with marked contacts used for measurements is shown in Fig. 1b. During measurements, the device was held in a cryostat at a base temperature of 7 K, unless otherwise is indicated.

Fig. 1: Tunneling detector scheme and electrical characterization.
figure 1

a The stack scheme of the tunneling detector. b Optical photograph of the sample with false-color images of the top, bottom graphene, and barrier hBN superimposed on it. Contacts to the top and bottom graphene used in measurements are designated by the letters “T” and “B” accordingly. AFM scan of barrier hBN edge is shown in the inset. There is a cutout in the stack made to avoid possible shorting of the top and bottom graphene. Scale bar is 10 μm. c Ib−Vb device characteristics at different gate voltages, showing ladder-type behavior. d Map of differential conductance as a function of bias and gate voltages at 7 K. The curves of maximum conductance are marked with i1t, i2t, i1b, i2b, and arrows of minimum conductance—cnpt, cnpb and blue dotted lines. The black dotted lines correspond to the theory. e Illustration of the tunneling process when the 1st impurity is aligned with the bottom graphene Fermi level, which corresponds to the i1b-curve on (d). f Same illustration, but the impurity is aligned with the bottom graphene CNP (cnpb-curve on (d)).

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Electrical measurements of the tunnel structure (Fig. 1c) confirm the rotational misalignment of graphene layers and the presence of resonant states in the dielectric. First of all, no traces of negative differential resistance are observed in the Ib–Vb-curves, which implies strong misorientation of graphene crystal structures. Second, the current-voltage characteristic has pronounced “steps”, which indicate the opening of new tunneling channels with an increase in the bias voltage. The detailed mapping of differential conductance dIb/dVb vs. gate and bias voltages shown in Fig. 1d confirms that these tunneling paths are associated with a resonant passage of electrons through the impurity levels. We identify four characteristic spike lines in the differential conductance map, marked as i1t, i2t, i1b, and i2b. The spike positions depend on both bias and gate voltage, the latter controlling the carrier density. This excludes the phonon-assisted origin of conduction. Instead, each spike can be attributed to the alignment of the Fermi level in either graphene layer μt,b with the level of impurity inside the band gap of boron nitride (Fig. 1e). More precisely, the resonant condition can be formulated as

$${\mu }_{{{{\rm{t,b}}}}}({V}_{{{{\rm{b}}}}},{V}_{{{{\rm{gate}}}}})={E}_{i,n}+eF({V}_{{{{\rm{b}}}}},{V}_{{{{\rm{gate}}}}}){x}_{i,n},$$
(1)

where Ei,n is the energy level of nth impurity in the absence of bias, the second term represents the bias-induced shift of impurity level, F is the electric field in the hBN barrier, and xi,n is the position of the nth impurity.

We managed to reproduce the experimentally measured positions of conduction spikes with the model (1) assuming two resonant levels within the barrier. The fitting procedure yields the defect levels Ei1 = 100 meV and Ei2 = −70 meV reckoned from the Dirac points of unbiased layers. Further fitting enables the determination of impurity positions: impurity #1 is located in the middle of the barrier, while impurity #2 is between the second and third hBN layers, counted from the bottom graphene. We also observe many other fainter resonant lines (curves near i1 and parallel to it) that emerge from other impurities, probably having a smaller overlap with the tunnel barrier.

Another prominent feature of the conduction map is the area with nearly zero differential conductivity. It forms two dark blue curves in Fig. 1d, labeled cnpt and cnpb. It corresponds to the neutrality point of either graphene layer and, hence, to zero tunneling density of states (DOS) (Fig. 1f). Both cnp-curves cross at nearly zero gate and bias voltages, which implies the absence of initial doping in both graphene layers. It can be noted that tunneling through impurity does not occur on these curves; lines i1 and i2 are interrupted.

Photocurrent under mid-infrared illumination

We proceed to the characterization of the tunneling structure in the photodetector mode. The device is illuminated with mid-infrared radiation λ = 8.6 μm fed from a quantum cascade laser with output power P ≈ 7.2 mW. The photocurrent is measured using a lock-in amplifier (see Fig. 2a and the “Methods” section for details). The photocurrent map recorded by moving the laser beam across the device represents a single bright spot (Fig. 2b, c). This excludes the role of contact effects in the photocurrent generation process and shows that the tunnel structure itself acts as a photocurrent generator.

Fig. 2: Photocurrent measurements under λ = 8.6 μm illumination at T = 7 K.
figure 2

a Scheme of the photocurrent and \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}\) measurements. b Spatial photocurrent map. Deviation of the spot from the symmetrical Gaussian shape is aberration due to a slightly oblique incidence of light on the lens but not the sample features. c Slice of the map at y = 30 μm and spot size extracted by fitting to Gaussian distribution. d Photocurrent as a function of bias and gate voltages. e Second derivative of Ib(Vb) as a function of bias and gate voltages. It’s clearly seen how the photocurrent repeats all \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}\) features in details. f \({d}^{2}{I}_{{{{\rm{b}}}}}/d{V}_{{{{\rm{b}}}}}^{2}\) averaged over all gate voltages. g–i Slices of maps (d) and (e) at two different gate voltages: g Vgate = 0 V, h Vgate = −4 V and i at small bias Vb = 5 mV. Features at the energies of the phonon modes are marked by arrows on (d, e) and by vertical dashed lines on (f).

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Figure 2 d shows the gate- and bias-resolved map of the photocurrent. First, the correlation of the photocurrent extrema with the position of the impurities on the differential conductance map is striking. The photocurrent extrema very closely follows the curves i1 and i2 of the differential conductance map, i.e. are observed when the Fermi levels of impurities and graphene layers are aligned. The next interesting feature is that when Vb changes, and the impurity level passes through the Fermi level of one of the graphene layers, the photocurrent has two extrema of opposite signs, which are observed at ∣Vb∣ < 0.25 V, ∣Vgate∣ < 3 V. At large values of the bias and gate, these features persist, yet the photocurrent acquires a positive ‘background’ growing with the absolute value of bias.

The correlation between optoelectronic and electrical properties becomes even more apparent upon comparison of the photocurrent map with the \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}\) map shown in Fig. 2e. They match in the smallest details. The photocurrent extrema follows the \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}\) extrema for all values of the bias and gate, repeating the position dependencies on Vgate and Vb not only for the two main impurities but also for all others. For example, with negative gate and offset values, there are several impurities nearby. The photocurrent is amplified at each. At small values of Vb and Vgate (namely, at ∣Vb∣ < 0.25 V, ∣Vgate∣ < 3 V) there is a direct proportionality between the photocurrent and \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}\) clearly visible on the map slices in Fig. 2g–i. At larger gate and bias voltages, the presence of the background in photocurrent weakens this similarity, as indicated in Fig. 2h. In this regime, \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}\) has a strong sign-changing feature upon crossing the impurity level. The photocurrent does not change sign at these bias voltages, yet it demonstrates a spike at these points.

In addition, at Vb ≈ ±18–20 mV and Vb ≈ 175–200 mV the photocurrent map shows features that do not depend on Vgate, expressed in a sudden increase in Iph with increasing Vb. The latter energy is greater than the energy of the incident photon (144 meV) and coincides with the energy of the optical phonon modes of graphene. These features are also visible on the \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}\) map as local extrema. They are marked with small black arrows in Fig. 2d, e. The resonances are visible even better if one average the \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}\) dependences over all values of Vgate (Fig. 2f). This coincides with previously observed phonon modes of graphene obtained from transport measurements33,41. Moreover, while low-energy phonon modes have already been observed as phonon-assisted photocurrent under illumination with visible light, high-energy optical modes have been demonstrated. While in ref. 41 other phonon modes were also observed; they are less pronounced compared to the ones mentioned above and, for this reason, are not visible in our device.

All the key features of the dependencies of the photocurrent on the gate and bias, namely the proportionality of the photocurrent to the 2nd derivative of Ib(Vb), the presence of two extrema of the opposite sign when the levels of graphene and impurity are aligned, were repeated in the measurements of the device #2 (Supplementary Note 2). The device was a graphene/hBN/graphene stack encapsulated in protective hBN layers with the same hBN barrier thickness of 1 nm. The stack was different in that it was made of multilayer graphene (2L on top and 3L on the bottom), had a large tunnel junction area and a silicon gate, and was illuminated at a different wavelength of 6.0 μm.

Origins of photocurrent: bolometric and thermoelectric effects at the tunnel barrier

The two competing mechanisms contributing to the photocurrent in tunnel-coupled low-dimensional systems are photon-assisted tunneling and the tunneling of hot carriers. The direct rectification by nonlinearity of the tunneling Ib–Vb-characteristic should be considered as a low-frequency limit of the photon-aided tunneling and does not require a separate consideration. Were the photon-aided tunneling a dominant light detection mechanism, the photocurrent should peak when the Fermi level μt,b plus the photon energy ℏω reaches the level of impurity Ei42,43,44,45. Such energy constraint corresponds to the onset of tunneling by photoexcited carriers along the resonant levels. The photocurrent map for this detection mechanism should possess extrema curves parallel to the impurity lines i1t, i2t and shifted by ℏω = 144 meV on the bias scale. We observe no peculiarities of Iph(Vb, Vgate) at these energies and, therefore, exclude photon-assisted tunneling from relevant photodetection mechanisms.

An alternative to the photon-assisted tunneling realized in the case of fast inter-carrier energy exchange is the tunneling of electrons heated up by absorbed radiation. The light-induced change in electron temperature leads to the broadening of their Fermi distributions and, hence, to the change in average tunneling probability. Further physics depends essentially on whether the layers are biased or not and on the relative position of Fermi levels and impurity levels.

For biased structures, the photocurrent represents the change in total tunneling current induced by carrier heating. If the Fermi level of a particular graphene layer is biased slightly below the impurity level in the dark, the light-induced heating would push the electrons toward the resonant level. This would increase the total current and lead to a positive spike in the photocurrent (Fig. 3a). If the same layer is biased slightly above the impurity level in the dark, the light-induced heating would deplete the Fermi distribution in the vicinity of resonant energy. In such a situation, the negative spike in the photocurrent would be observed (Fig. 3b). Such a double-spike structure of photocurrent is indeed observed each time the Fermi level μt,b crosses the impurity level at not very large bias voltages (Fig. 2g). This fact can already be considered as a proof of hot-carrier origin of the photocurrent.

Fig. 3: Illustration and theory of the photocurrent generation.
figure 3

a and b Illustration of the thermal mechanism of the photocurrent generation when the top graphene layer is biased slightly (a) below or (b) above the impurity level (which corresponds to i1b, i2b-curves on Fig. 1d). c Illustration of the photocurrent generation at zero bias at 2 gate voltages near impurity alignment. d Theoretically calculated photocurrent and e \({d}^{2}{I}_{{{{\rm{b}}}}}/d{V}_{{{{\rm{b}}}}}^{2}\) map as a function of gate and bias voltages, well reproducing experimental results. f and g Slices of maps d and e at three different bias voltages: f Vb = 0.25 V; g Vb = 5 mV, along dashed line on (d); h Vb = 0. The red dashed and dotted lines on g, h demonstrate the contribution of the top and bottom layers, respectively, to the total photocurrent. The heating of the layers is assumed to be slightly different, δTt = 1.2δTb, which gives a non-zero photocurrent at Vb = 0.

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At near-zero bias, the Fermi levels of both graphene layers are very close. When the impurity level is aligned with them, the two effects described above are superimposed on each other, resulting in a picture with three spikes, which we observed at Vb = 5 mV (Fig. 2i).

At zero bias, the origin of photocurrent is distinct and can be termed as Seebeck effect across the tunnel barrier. Heating of both top and bottom electronic subsystems by the same amount cannot result in any current due to their partial thermal equilibrium. However, asymmetric heating of electrons in the layers would result in an imbalance between tunneling currents, measured as photocurrent. The photocurrent would be, therefore, proportional to the temperature difference between the layers, Iph ∝ Tt−Tb.

Further proofs of the thermal origin of the photocurrent can be obtained by direct calculation of temperature-dependent current–voltage characteristics Ib(Tt, Tb). We have obtained the latter with the Bardeen transfer Hamiltonian approach

$${I}_{{{{\rm{b}}}}}({T}_{{{{\rm{t}}}}},{T}_{{{{\rm{b}}}}})=\frac{e}{\hslash }\int\limits_{-\infty }^{\infty }dE\left[{f}_{{{{\rm{t}}}}}(E)-{f}_{{{{\rm{b}}}}}(E)\right]{{{\mathcal{D}}}}(E),$$
(2)

where \({f}_{{{{\rm{t,b}}}}}={[1+\exp ((E-{\mu }_{{{{\rm{t,b}}}}})/{k}_{{{{\rm{B}}}}}{T}_{{{{\rm{t,b}}}}})]}^{-1}\) are the Fermi distribution functions in top and bottom layers with generally different temperatures (Tt and Tb) and Fermi levels. The function \({{{\mathcal{D}}}}(E)\) is the energy-dependent tunneling probability possessing sharp resonances at impurity levels E = Ei,n (see34 and the Supplementary Note 3 for explicit form). The model (2) is suitable both for calculations of DC source-drain current (with all its derivatives) and the photocurrent. For DC current, one sets the temperatures of both layers to the base cryostat temperature Tt = Tb = T0. Evaluating the photocurrent, one sets Tt,b = T0 + δTt,b, where δTt,b are the light-induced overheating of top and bottom layer. In explicit form

$${I}_{{{{\rm{ph}}}}}=I({T}_{0}+\delta {T}_{{{{\rm{t}}}}},{T}_{0}+\delta {T}_{{{{\rm{b}}}}})-I({T}_{0}).$$
(3)

The results of photocurrent calculations are presented in Fig. 3d–h and fully confirm the above intuitive picture on light-induced heating effects. Under finite bias Vb, the theoretically predicted photocurrent indeed possesses upward and downward spikes at both sides of impurity levels. Moreover, the theory reproduces the observed proportionality between photocurrent and \({d}^{2}{I}_{{{{\rm{b}}}}}/d{V}_{{{{\rm{b}}}}}^{2}\), which can be derived analytically in a fashion similar to the derivation of Wiedemann-Franz law (see Supplementary Note 3 for details).

At a bias close to zero, the photocurrent is a superposition of the photocurrent profiles from two graphene layers. At a bias of exactly zero, their sum gives the dependence of the photocurrent on the gate in the form of two spikes of opposite signs, the amplitudes of which are proportional to the temperature difference of the graphene layers Iph ∝ δTt − δTb (Fig. 3h).

However, applying even a small bias Vb = 5 mV spoils this ideal picture. The photocurrent profiles from the two layers are summed up into a pattern with three spikes: two upward (downward) spikes and a single downward (upward). A strong central spike mostly corresponds to the sum of the co-directional photocurrents from both graphene layers and represents the average heating of the graphene layers. While the two side spikes represent the sum of the opposing photocurrents from the two layers and contain information about the temperature difference between the layers (Fig. 3g). This is exactly what we see in the experiment at Vb ≈ 0 mV (Fig. 2i). The accuracy of setting and measuring the voltage did not allow us to clearly catch the case of zero bias (Supplementary Fig. 3).

Once the origin of photocurrent is thermal, its functional dependence Iph(Vb, Vgate) should be similar to that of temperature conductance coefficient dIb/dT0. Independent measurements of photocurrent and tunnel current Ib(T0) with variable base cryostat temperature confirm this idea, as shown in Fig. 4a, b.

Fig. 4: Estimation of the temperature of the electron gas heated by radiation from dIb/dT and photocurrent measurements.
figure 4

a dIb/dT and photocurrent as a function of bias voltage measured at Vgate = 0, b as a function of gate voltage at small bias Vb = 5 mV. Electron temperature rise is estimated to be about 8 K. c and d Theoretically calculated photocurrent and dIb/dT for c Vgate = 0 and d Vb = 5 mV for δTt = 1.2δTb. e Photocurrent dependence on Vb at elevated temperatures.

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More precisely, in a linear approximation of Eq. (3) on temperature, the photocurrent is given by

$${I}_{{{{\rm{ph}}}}}=\frac{{\rm {d}}{I}_{{{{\rm{b}}}}}}{{\rm {d}}{T}_{{{{\rm{t}}}}}}\delta {T}_{{{{\rm{t}}}}}+\frac{{\rm {d}}{I}_{{{{\rm{b}}}}}}{{\rm {d}}{T}_{{{{\rm{b}}}}}}\delta {T}_{{{{\rm{b}}}}}.$$
(4)

When eVb ≫ kBT0 ≈ 0.7 meV and away from the i1t, i1b, i2t, i2b lines crossing areas determined by the energy width of the impurity levels (≈4 meV), photocurrent depends only on dIb/dTδT of one of the graphene layers, which Fermi level is aligned with the impurity level. The Fermi level of the other graphene layer is far away and does not significantly contribute to the photocurrent. It becomes possible to calculate the increase in the electron temperature of each layer by dividing the measured photocurrent value by dIb/dT at curves i1t, i2t for the top layer and i1b, i2b for the bottom.

We calculated the derivative of the current with respect to temperature as dIb/dT = (Ib(T2)−Ib(T1))/(T2−T1), where T1, T2 are 10 and 20 K for Fig. 4a, and 7 and 20 K for Fig. 4b. Comparing its profile with the photocurrent measured at 7 K at the positions of spikes i1t, i2t, i1b, i2b we estimate the average heating of electrons in both graphene layers at our incident light power 7.2 mW to be δT ≈ 8 K. The spread of the top and bottom layer heating values does not allow us to estimate their difference.

In the unbiased case, we did not observe ideal photocurrent profiles with two spikes, only a pattern corresponding to a near-zero bias. This still allows us to evaluate the difference in heating of the graphene layers. Fitting calculated Iph(Vgate) curves at Vb = 5 mV to the experimental data, we estimate the temperature difference between the layers to be ≲ 1.5 K.

As the temperature increases, the Fermi distribution blurs, which, according to the theory, leads to a decrease in dIb/dT inversely proportional to temperature, and hence to a decreasing in the photocurrent. In Fig. 4e the photocurrent dependence on Vb is presented at different temperatures from 31 to 100 K, demonstrating a decline with increasing the temperature, as expected.

Discussion

In summary, we have elucidated the mechanism of photocurrent generation in graphene/hBN/graphene tunnel structures with localized defect states in a barrier upon mid-infrared illumination. The photocurrent appears upon electron heating, resulting either in a Seebeck-type effect at zero bias or in a bolometric effect at finite bias. Both effects differ from their classical counterparts observed upon longitudinal Ohmic transport, as here they develop across the tunneling barrier. Both effects are maximized at the gate and bias voltages corresponding to the I(V)-curve steps. At these steps, the Fermi level of either graphene layer is aligned with a resonant-tunneling defect state and even tiny changes in electron temperature result in strong variations of the energy-averaged electron tunneling probability.

We have developed a theoretical model of such photo-thermal tunneling current and reproduced the experimental data in detail at all bias and gate voltages. At cryogenic temperatures T ~ 9 K corresponding to the actual experiment, the photothermoelectric effect across the barrier is observed only at very low bias voltages Vb ≲ kT/e ~ 1 mV and surpasses the bolometric effect otherwise. At nearly room temperature, the thermoelectric effect is observed in a much broader range of biases.

Absence of a direct photon-assisted tunneling component in the observed photocurrent may lie in the fast thermalization of photoexcited electrons, as compared to their tunneling exchange between layers. The scattering time of electron above the Fermi surface scales quadratically with its elevation46, thus τth ≈ 4πεF/ℏω2. The corresponding thermalization time at a realistic Fermi energy at εF = 200 meV estimates to τth ≈ 80 fs. The tunneling time can be estimated \({\tau }_{{{{\rm{tun}}}}}=(\hslash /{E}_{{{{\rm{bar}}}}}){{\rm {e}}}^{2\kappa {d}_{{{{\rm{bar}}}}}}\), where Ebar ≈ 1.5 eV is the barrier height at the graphene/hBN interface and κ ≈ 6 nm−1 is the electron wave function decay rate47. For barrier thickness d ≈ 1 nm, the tunneling time τtun ≈ 40 ps appears three orders of magnitude longer than the thermalization time. At lower excitation frequencies ω/2π ≲ 1 THz, the thermalization becomes slower than tunneling transfer, and direct photon-assisted current may be observed.

As the radiation-induced electron heating with subsequent thermoelectric/bolometric effects is the main physical reason for photocurrent generation, the correspondence between photocurrent Iph in vertical tunnel structures and electron temperature Te becomes one-to-one. This correspondence, treated reversely, can be used to measure Te in highly non-equilibrium conditions, e.g., at strong photoexcitation power. The knowledge of Te in photoexcited 2d materials can shed light on the mechanisms of carrier cooling and provide useful information on the efficiency of light–matter coupling. More precisely, to extract the electron temperature in photoexcited 2d material, one may equip it with a vertical tunnel contact and measure Iph at different light intensities. Such a technique can become a convenient alternative to Johnson noise38 and Coulomb blockade48 thermometry used previously for similar purposes.

It is worth noting that the magnitude of the photocurrent can be significantly increased with a large area of the tunnel region and a small barrier thickness. The device presented in the article demonstrated a photocurrent of up to 120 pA. And device #2, which has 10 times greater conductivity due to the size of the tunnel junction region, demonstrated 50 times greater photocurrent, up to 5 nA. The sensitivity of device #2 is 0.8 mA W−1. And the noise equivalent power NEP* = 830 pW Hz−1/2 (Supplementary Note 2). Using less resistive materials as a barrier layer, for example, WS249 will allow for an even more significant increase in sensitivity. Such tunnel micro-detectors demonstrating high photocurrent could be envisioned as a building block for multipixel mid-IR cameras.

Methods

Device fabrication

Devices were made using the dry transfer technique40. This involved a standard dry-peel technique to obtain graphene and hBN crystals. The flakes were stacked on top of each other (from top hBN to bottom graphite) using a stamp made of PolyBisphenol carbonate (PC) on polydimethylsiloxane (PDMS) and deposited on top of an oxidized (280 nm of SiO2) high-conductivity silicon wafer (KDB-0.001, ~0.001–0.005 Ω cm). The resulting thickness of the hBN layers was measured by atomic force microscopy. Then electron-beam lithography and reactive ion etching with SF6 (30 sccm, 125 W power) were employed to define contact regions in the obtained hBN/graphene/barrier hBN/graphene/hBN/graphite heterostructure. Metal contacts were made by electron-beam evaporating 3 nm of Ti and 70 nm of Au. The second lithography was done to make a cutout to avoid possible shorting of the top and bottom graphene due to the displacement of the thin hBN layer during transfer. It was followed by reactive ion etching using PMMA as the etching mask.

Measurements

The sample was held at 7 K inside a cold finger closed-cycle cryostat (Montana Instruments, s50). Ib–Vb characteristics were measured using Keithley 2636B sourcemeter. Differential conductance was calculated by numerical Ib(Vb) differentiating. \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}\) was measured using AC–DC mixing technique. Source-meter (Keithley Instruments, 2636B) DC voltage and lock-in amplifier (Stanford Research, SR860) output AC sine voltage at 4 Hz frequency was summed up by voltage divider resulting in DC bias with small VAC = 4.8 mV (rms) component applied to the sample. Second derivative of Ib–Vb was calculated from lock-in second harmonic readings I2ω at +90° phase as \({{\rm {d}}}^{2}{I}_{{{{\rm{b}}}}}/{\rm {d}}{V}_{{{{\rm{b}}}}}^{2}=-\sqrt{2}{I}_{2\omega }/{V}_{{{{\rm{AC}}}}}^{2}\). Photocurrent was also measured using lock-in amplifier. Linearly polarized light from a quantum cascade laser with a wavelength of 8.6 μm was modulated by a chopper at a frequency of 8 Hz. Light was focused by ZnSe lens through polypropylene film cryostat window to an almost diffraction-limited spot. Motorized XY stage allowed precise aligning of sample and laser spot. Binding of the chopper phase was done by comparing the photocurrent phase with that obtained in the case of laser current modulation and additionally controlled from amplified waveforms on the oscilloscope. Deeper details are presented in Supplementary Note 1.