Resonant terahertz detection using graphene plasmons

Plasmons, collective oscillations of electron systems, can efficiently couple light and electric current, and thus can be used to create sub-wavelength photodetectors, radiation mixers, and on-chip spectrometers. Despite considerable effort, it has proven challenging to implement plasmonic devices operating at terahertz frequencies. The material capable to meet this challenge is graphene as it supports long-lived electrically tunable plasmons. Here we demonstrate plasmon-assisted resonant detection of terahertz radiation by antenna-coupled graphene transistors that act as both plasmonic Fabry-Perot cavities and rectifying elements. By varying the plasmon velocity using gate voltage, we tune our detectors between multiple resonant modes and exploit this functionality to measure plasmon wavelength and lifetime in bilayer graphene as well as to probe collective modes in its moiré minibands. Our devices offer a convenient tool for further plasmonic research that is often exceedingly difficult under non-ambient conditions (e.g. cryogenic temperatures) and promise a viable route for various photonic applications.

As discussed in the main text, the responsivity of the THz detectors made of the field effect transistors is proportional to the sensitivity of the FET conductivity to the gate voltage variation. To improve the performance of our detectors, we took advantage of BLG's gate-tunable band structure and fabricated a dual-gated device (top and bottom insets of Supplementary Fig. 2a). The idea is that when an electric field is applied perpendicular to the channel it induces a band gap in BLG that leads to a steeper dependence of the FET resistance R 2pt on the gate voltage. Supplementary  Fig. 2a shows examples of R 2pt (V tg ) dependences measured at few V bg demonstrating the expected increase of R 2pt with increasing the average displacement field D = ε 2 (V tg /d bg − V bg /d tg ) applied to BLG, where d bg (d bg ) is the thickness of the bottom (top) hBN crystal and ε is its dielectric constant. Supplementary Fig. 2b shows the top gate voltage dependence of R a measured in response to 0.13 THz radiation in the dual-gated detector. In the case of zero back gate voltage (black curve), R a (V tg ) repeats the behaviour of another detector reported in Fig. 2a of the main text. Namely, the responsivity reaches its maximum of about 200 V/W near the NP where it flips its sign because of the change in the charge carrier type. Note, the absolute value of the maximum responsivity is very close to that reported in the main text (Fig. 2b) highlighting the reproducibility of our detectors' performance. When the back gate voltage is applied, the responsivity increases drastically (red and blue curves in Supplementary Fig. 2b). Already for a moderate D ∼ 0.1 V/nm the responsivity increases by more than an order of magnitude and exceeds 3 kV/W. The corresponding noise equivalent power, estimated using the Johnson-Nyquist noise spectral density for the same D, reaches 0.2 pW/Hz 1/2 . This makes our detector competitive not only with other graphene-based THz detectors 1 but also with some commercial semiconductor and superconductor bolometers (Table 1). To illustrate that the observed resonant photoresponse is reproducible for different electronic systems embedded in FETs of various lengths L and coupled to different antennas, Supplementary Fig. 3c shows another example of the photovoltage ∆U (V g ) emerging when the incoming 2 THz radiation is coupled to the broadband logarithmic spiral antenna connected to another FET. The latter is made of BLG having its crystallographic axis aligned with those of hBN, that reveals itself in peculiar three-peaks R(V g ) structure, shown in Supplementary Fig. 3e. The photoresponse curves are rather similar to those shown in Supplementary Fig. 2b of the main text, namely they follow the envelope trend set by the FET-factor F = − 1 σ dσ dVg ( Supplementary Fig. 3d) superimposed with the resonant peaks. The resonances are periodic in V −1/2 g (inset to Supplementary Fig. 3c) and are clearly seen for both electron and hole doping. Importantly, on the contrary to Fig. 2b, the photoresponse now changes sign multiple times following nontrivial F (V g ) evolution.
Supplementary We have also studied the performance of our detectors at frequencies intermediate to those reported in Figs. 2a and b of the main text and found that the resonant operation onsets already in the middle of the sub-THz domain. Figure  4 shows the gate voltage dependence of R a recorded in response to 460 GHz radiation. In the vicinity of the charge neutrality point (NP) the responsivity peaks and changes its sign in agreement with the evolution of the FET-factor with the gate voltage (black curve in Supplementary Fig. 4) as discussed in the main text. However, away from the NP the responsivity peaks for both electron and hole doping (stars in Supplementary Fig. 4) despite the fact that F (V g ) is featureless. These peaks stem from the plasmon resonances in the FET channel as it follows from the comparison of the experimental data with theory (inset of Supplementary Fig. 4). In good agreement with theory, at lower frequencies the number of resonant modes, which can be observed for the same gate voltage span, is smaller compared to that found at 2 THz (Fig. 2b of the main text). In addition, the resonances appear much broader than those observed at 2 THz (Fig. 2b) which is consistent with the reduced quality factor at sub-THz frequencies. Supplementary where ω and q are the plasmon wavelength and wave vector, respectively, τ is the momentum relaxation time, n 0 is the carrier density, m * is the effective mass of charge carriers, d it the distance to the gate, ε is the dielectric permittivity, and ε 0 is the vacuum permittivity. Confinement of a 2d channel by source and drain contacts quantizes the wave vector q and leads to emergence of discrete plasmon frequencies. The quantization conditions can be obtained by requiring the oscillating quantity (e.g. voltage V ω ) to return to its original value after the channel round trip: where r s and r d are the complex-valued reflection coefficients at the source and drain terminals, respectively. Therefore, eigen frequencies of bounded plasmons can be found from To see that the latter dispersion relation indeed appears in the nonlinear response functions, we model the FET channel as a transmission line (TL) fed by antenna voltage U 1ω = V a cos ωt at the source side 3,4 . The antenna may have finite impedance Z a which will be taken into account at the end of this section. The TL is terminated by load impedance Z gd at the drain side, and by impedance Z gs at the source side. In real device, these impedances are due L is the kinetic inductance of electrons, ρ is the channel resistivity, C is the effective gate-to-channel capacitance, and Z gd is the load resistance at the drain side. All quantities are measured per unit length of the channel. Z gd → ∞ corresponds to Dyakonov-Shur boundary condition to the capacitive coupling between the respective electrodes. The TL model is justified by the formal coincidence of TL equations (Telegrapher's equation) with transport equations in a gated FET channel. The TL elements are specific inductance capacitance per unit length and resistance where W is the channel width. It is readily seen that the dispersion relation for waves in an infinite transmission line 5 coincides with plasma wave dispersion (1) with proper values of line parameters (4-6). The characteristic (wave) impedance of transmission line is A well-known result for current reflection coefficient from a loaded (drain) end of transmission line reads while for source end with fixed voltage When the reflection coefficients and conditions at the ends of cavity are specified, it is straightforward to write down the solution for voltage across the TL (which is the gate-to-channel voltage in the actual FET): here r d and r s are given by Eqs. (9) and (10), respectively. The longitudinal electric field in the channel is given by As the nonlinear response of the FET is proportional to the properly averaged square of ac electric field in the channel (12), it becomes apparent that responsivity would possess a plasma resonant factor |1 − r s r d e 2iqL | −2 , independent of the detection mechanism.
The account of finite antenna impedance results in a a simple "renormalization" of input voltage in Eqs. (11) and (12): where stands for parallel connection of impedances, and Z tl,in = Z tl (1−r d e 2iqL )/(1+r d e 2iqL ) is the input impedance of the transmission line (we have used r s = 1). It is straightforward to show that the modification of input voltage can be translated in the modification of "resonant denominator" The effect of Z a in the square bracket is the reduction of input voltage due to the drop at internal antenna resistance. Finite value of Z a in the second term leads to extra broadening or resonances, as analyzed below.
Supplementary Note 6: Gate tuning of graphene plasmons: monolayer vs bilayer We briefly review the density dependences of plasmon frequencies in single layer graphene (SLG) and bilayer graphene (BLG). The general dispersion relation for gated plasmons in two-dimensional electron system with sheet conductivity σ reads 2 The study of plasmon dispersions in various two-dimensional systems is therefore reduced to evaluation of their frequency-dependent conductivity. In the classical ( ω ε F ) long-wavelength (q ω/v F ) limit, the latter is found from the Boltzmann equation where ρ(ε) is the density of states, v p is the velocity of carrier with momentum p, and f 0 is the equilibrium distribution function. In case of BLG, ρ(ε) = 2m/π 2 , v p = p/m, which results in ordinary Drude conductivity In case of SLG, ρ(ε) = 2ε/π 2 v 2 F , v p = v F , and the conductivity reads The latter equality is valid at low temperatures. Using the low-temperature relation between density and Fermi energy in SLG n = ε 2 F /p 2 v 2 F , we readily observe that classical conductivity of SLG is still given by the Drude formula (17) with density-dependent mass m → ε F /v 2 F ∝ n 1/2 . Combining Eqs. (15), (17) and 18, we observe that plasma frequency in BLG scales as ω ∝ n 1/2 , while in SLG ω ∝ n 1/4 .

Supplementary Note 7: Resonance broadening and plasmon lifetime
Before discussing the physics beyond THz rectification, we specify mechanism-independent quantities, namely, the positions of plasma resonances and resonance width. Introducing the complex reflection phase we transform the "resonant denominator" in eq. (1) of the main text R(ω, V g ) ∝ |1 − r s r d e 2iqL | −2 = 1 2 e θ r +2q L cosh(θ r + 2q L) + cos(θ r + 2q L) , The maxima of responsivity correspond to wave vectors In the case of Dyakonov-Shur boundary conditions realized in our devices, θ r = 0, and the first resonance corresponds to L equal to the quarter of plasmon wavelength. Assuming reflection and scattering losses to be small, the lineshape (20) can be transformed to Lorentzian in the vicinity of each peak The full width at half-height is given by here we have introduced the plasmon lifetime τ p which is below the scattering time τ p due to resonator loss. It is also possible to take into account the effect of finite antenna resistance on plasmon linewidth. To this end, one should transform resonant denominator of the form (14) in the vicinity of resonance. This leads us to The above equation clearly demonstrates that inverse plasmon lifetime τ −1 p is the sum of electron momentum relaxation rate τ −1 , contact damping rate and damping rate due to antenna resistance The two latter contributions to damping rate are minimized in the vicinity of charge neutrality point. Examples of calculated plasmon lifetime including the contributions of contacts and antenna are shown in Supplementary Fig. 6. Effects of radiative contribution to plasmon damping on detector responsivity is shown in Supplementary Note 9, along with the discussion of detection mechanisms. As the wave vector at fixed frequency is inversely proportional to wave velocity, q = ω/s ∝ V Asymmteric feeding of THz radiation results in asymmetric heating of the device and emergence of thermoelectric effect. The resulting dc voltage is 7 where S ch is the Seebeck coefficient in the gated channel, and S cont -in the metal-doped graphene contact, T s is the local temperature at the source junction and T d is at the drain junction . From now on, we refer to the gated part of graphene as "channel" and ungated part -as "contact". The doping of ungated part does not depend on gate voltage, however, it can be non-uniform due to the effects of built-in field near metal contacts. The temperature difference T s − T d induced by non-uniform heating of the device can be found from the solution of heat transfer equation in the channel: where q(x) = 2Reσ ω |E xω | 2 is the Joule heating power, χ ch is the electron thermal conductivity in the channel, L T = (χ ch τ ε /C e ) 1/2 is the thermal relaxation length, τ ε is the energy relaxation time due to heat sink into substrate phonons 8 , and C e is the heat capacitance of the electronic system. Equation (29) is supplemented by the boundary conditions at the boundaries of gated domain these conditions follow from the continuity of heat flux at the interfaces. The sought-for temperature difference between source and drain can be obtained in the closed form under the following simplifying assumptions (1) the Joule heating occurs only in the channel (2) the temperature drop across the contacts is much less than maximum overheating in the channel. Both conditions are justified by the small length of the contacts L cont L. Under these assumptions, the expression for the photo-thermoelectric voltage acquires a physically appealing form The quantity χ cont /l cont is the thermal conductance of the contact, while the integral is the difference of heat fluxes traveling toward the source and toward the drain. The kernel of the integral is anti-symmetric with respect to the middle of the channel x = L/2, therefore, the PTE signal appears only due to asymmetric heating q(x). Final evaluation of PTE voltage is performed by substituting the solution for electric field (12) into (32): where α = θ r + qL/2.

Supplementary Note 9: Dyakonov-Shur rectification in Fabri-Perot cavity
The so-called Dyakonov-Shur rectification includes two physically different nonlinearities. One contribution to the rectified current appears due to simultaneous modulation of 2d channel conductivity and application of longitudinal field. This effect, also known as resistive self-mixing, results in the rectified voltage Here σ ω=0 = ne 2 τ /m is the dc conductivity of a 2D channel, n is the carrier density, e and m are the elementary charge and effective mass of charge carriers respectively, τ is the momentum relaxation time, and σ ω = σ ω=0 /(1−iωτ ) is the high-frequency conductivity. Evaluation of the integral leads us to the result Another contribution to rectified voltage stems from the difference of kinetic energies of electron fluid at the source and drain side (Bernoulli law). The underlying nonlinearity is manifested by convective term (u∇)u in the Euler equation for electron fluid 9 . The corresponding rectified voltage is where E xω is the complex amplitude of high-frequency longitudinal field in the channel given by (12). Using the result for electric field (12), we find Equations (35) The factor in square brackets peaks when the length of the FET channel matches odd multiples of the plasmon quarter-wavelength. We note that Eqs. (35), (37) and (38) diverge as the dc gate voltage V g tends to zero. In fact, this divergence stems from the gradual-channel approximation, relating carrier density and gate voltage CV g = en, that fails near the charge neutrality point. The account of ambipolar transport involving electrons and holes leads to a simple replacement in Eqs. (35), (37) and (38): where n and p are electron and hole densities, m n and m p are their effective masses, and s 2 = (n/m n + p/m p )e 2 /C is the plasma wave velocity in ambipolar system. In the main text, the experimental photoresponse was compared with the DS photovoltage from eq. (38) corrected by eq. (39). The full picture of calculated Dyakonov-Shur photovoltage vs gate voltage is shown in Supplementary  Fig. 7. Along with the result for perfect reflection from the drain and ideal voltage source (Z a = 0, red line), it also shows the effect of finite antenna impedance on resonance width (green and blue lines). In accordance with the discussed antenna-induced "renormalization" of input voltage, eq. 13, the resonances at high carrier density are highly broadened due to finite Z a . The width of resonances at low density, on the contrary, is mainly determined by momentum relaxation time.