THz-circuits driven by photo-thermoelectric, gate-tunable graphene-junctions

For future on-chip communication schemes, it is essential to integrate nanoscale materials with an ultrafast optoelectronic functionality into high-frequency circuits. The atomically thin graphene has been widely demonstrated to be suitable for photovoltaic and optoelectronic devices because of its broadband optical absorption and its high electron mobility. Moreover, the ultrafast relaxation of photogenerated charge carriers has been verified in graphene. Here, we show that dual-gated graphene junctions can be functional parts of THz-circuits. As the underlying optoelectronic process, we exploit ultrafast photo-thermoelectric currents. We describe an immediate photo-thermoelectric current of the unbiased device following a femtosecond laser excitation. For a picosecond time-scale after the optical excitation, an additional photo-thermoelectric contribution shows up, which exhibits the fingerprint of a spatially inverted temperature profile. The latter can be understood by the different time-constants and thermal coupling mechanisms of the electron and phonon baths within graphene to the substrate and the metal contacts. The interplay of the processes gives rise to ultrafast electromagnetic transients in high-frequency circuits, and it is equally important for a fundamental understanding of graphene-based ultrafast photodetectors and switches.

: Spectra of the pump laser pulse that is used to excite the graphene at the junction between both gates. Figure S2: Fit of total ( g1 , g2 ) for (a) charge neutrality at gate1 ( g1 = d1 = −1.9 ) and (b) charge neutrality at gate2 ( g2 = d2 = −2.8 ). Figure S3: Calculation of the thermopower from the derivative of the resistance. We fit total ( g1 , g2 ) for g1,2 ≥ −4 V to more precisely account for the feature that will determine the manifold sign change of the thermoelectric current for g1,2 ≥ 0 V. Generally, a numerical evaluation of the derivative total results in a noisy computed Seebeck coefficient. Figure S5a  that are calculated from the fit of total ( g1 , g2 ). The charge neutrality point is identical for both methods. Therefore, total ( g1 , g2 ) is fitted, and we use the resulting parameters to model the thermoelectric current according to equation eq.(3) of the main manuscript. Sketch of a split back-gate geometry with a lateral distance of 2 µm between the two individual gates. C1 is the capacitance between the two waveguides and C2 between one stripline and the back-gate neglecting the impact of graphene. C3 is the additional capacitance between the two gates of the split back-gate. Figure S6: Geometry for the finite element simulations in the time domain using the radio frequency package of Comsol. The 10 nm thin back-gate below the 30 nm thick ALD Al2O3-layer has a size of 120 µm times 80 µm. It is indicated by the red rectangle. Both coplanar striplines have a width of 5 µm. They are separated by a distance of 10 µm. The optical excitation of the graphene is simulated by a surface current element between the two waveguides. It has a length of 25 µm in the y-direction. For the evaluation of the resulting electromagnetic pulse, the dashed line along the inner side of the right waveguide is used.      Table S1: Simulation parameters. The relative permittivity εr of Al2O3 is taken as a mean value from (2)(3) and it is used for both the ALD layer and sapphire substrate. For the back-gate, εr varies between the given values for common metals (4)(5)(6). The conductivity for Au waveguides is taken from (7)  In this supplementary note, we present numerical calculations how a dual-gated graphene junction needs to be designed such that highfrequency photo-thermoelectric pulses from a graphene-junction can couple into on-chip THz-striplines. As is demonstrated below, the crucial point is the capacitive interaction of the back-gates with the striplines. To start with, a sketch of the cross section of the sample with a global back-gate below the graphene is shown in the supplementary Figure S5a. The split-gate-geometry used in the ultrafast experiments of the main manuscript is depicted in the supplementary Figure S5b. In both cases, we examine the influence of the backgate onto an electromagnetic pulse propagating along the coplanar stripline. In particular, we examine a simplified 3-dimensional model of the sample geometry (cf. supplementary Figure S6), which is implemented in the finite element software Comsol Multiphysics. The simulations are performed in the time domain using the so-called radio frequency physics model package (8)(9). Comsol solves the relevant partial differential equation for the vector potential A (S1) with εr the relative permittivity, µr the relative permeability, σ the conductivity, ε0 = 8. In a next simulation, a single back-gate is assumed which spans from one stripline to the other (cf. Supplementary Figure S7b), and which exhibits a metallic conductivity (σ = 10 7 Sm −1 ). The simulation shows that no pulse reaches the model boundaries (i.e. the pulse has an amplitude of only Ewith back-gate ≈ 4·10 −4 Vm −1 ). Instead, the pulse is reflected in-between the back-gate boundaries in the ydirection. To further analyze this issue, the back-gate conductivity σ is varied as listed in Table S1 using a parametric sweep in Comsol.
For each simulation, the ratio of Ewith back-gate/Ewithout back-gate (at y = 200 µm for t = 2.32 ps) is plotted versus the corresponding conductivity (supplementary Figure S7d). Starting at σ ≈ 10 3 Sm −1 , the pulse damping is increased with a higher back-gate conductivity instead of propagating further along the coplanar stripline. The influence of the backgate's relative permittivity εr on the simulation results is examined by varying εr. Values for εr in the range of common metals are used as listed in the supplementary Table S1 for a fixed conductivity. Only a small variation of the pulse amplitudes is found, but no change in the overall temporal evolution. Also increasing the standard deviation b to 10 ps or 10 ns of the surface current (cf. supplementary Equation S3) and adjusting the simulation time gives the same result concerning the damping through the back-gate (cf. supplementary Figure S8). Only on a nanosecond timescale, the pulse amplitude outside the back-gate area rises. The latter explains why the stripline-circuits can be used also for a time-integrated measurement of the photocurrent (Iphoto in the main manuscript). However, supplementary Figure S7b clearly reveals that time-resolved measurements on a picosecond timescale are not possible using a single metal back-gate which spans from one stripline to the other (σ = 10 7 Sm −1 ).
The above simulated results can be further explained by describing the coplanar stripline with the well-known telegraph equations. By solving these differential equations for the elementary segment shown in supplementary Figure S9, a frequency dependent wave impedance of (S4) can be derived (10). The impedance Z(ω) depends on the resistance R, the inductance L, the conductance G, and the capacitance C per unit length as well as the angular frequency ω = 2π·f. Assuming an ideal stripline (R = 0, G = 0), supplementary equation (S4) simplifies to the frequency independent impedance Z = √ / . The capacitance per unit length is calculated to be C = εr ε0 ·l / d. For simplicity, one can assume that all capacitances can be calculated using the formula of a plate capacitor. For the global back-gate geometry (cf. supplementary Figure S5a), the capacitance between the striplines can be computed to be in the order of C1 ≈ 0.22 pFm −1 (εr = 1 for vacuum, l = 250 nm, d = 10 µm). The additional capacitance caused by the back-gate Cgate is a series connection of two times the capacitance between one stripline and the backgate C2 ≈ 0.14 nFm −1 (εr = 9.4 for ALD-grown Al2O3, l = 5 µm, d = 30 nm, and graphene is neglected). Therefore, the overall capacitance of the coplanar stripline and the back-gate Ctot is the sum (parallel connection) of C1 and Cgate = C2 /2 resulting in Ctot ≈ 0.07 nFm −1 . Thus, the coplanar striplines have a lower impedance at the position of the global back- The impedance mismatch at the boundaries of the metal back-gate explains the reflections of the pulse in the y-direction for the case when the back-gate spans from one stripline to the other (supplementary Figure S7b). To reduce the impedance mismatch at the back-gate boundaries along the striplines, Ctot should be comparable to C1. This can be reached by reducing Cgate. For this purpose, the back-gate is split in the middle of the striplines into two gates with a distance of 2 µm (supplementary Figure S5b). The new capacitance between the two gates can be estimated to be C3 = 0.42 pFm −1 (εr = 9.4, l = 10 nm, d = 2 µm). Now, Cgate consists of a series connection of C3 and two times C2 resulting in Cgate = (C2C3)/(2C3 + C2) ≈ C3, which is much smaller compared to the case of a global back-gate.
All performed simulations for the global back-gate are repeated using the new split back-gate sample geometry. Supplemenatry Figure   7c shows the corresponding temporal evolution of the pulse at three different times in the case of the mentioned split-gate geometry (σ = 10 7 Sm −1 ) (cf. supplementary Figure S5a). Now, a pulse reaches the model boundaries. Compared to the simulation without a gate (cf. supplementary Figure 7a), the outcoupling pulse has a lower amplitude. This difference is still caused by the gate boundaries in the ydirection, since Cgate is unequal zero. The ratio of Ewith split back-gate/Ewithout gate for the parametric sweep of the gate conductivity σ is shown in supplementary Figure 7d. An apparent result of the geometric adjustment of the sample geometry is an increased electric field ratio of about 62% for σ = 10 7 Sm −1 using a metal split back-gate. Hence, a metal split-gate is suitable for allowing time-resolved measurements on photo-thermoelectric graphene-junctions.
We note that instead of a geometric adjustment, a suitable global back-gate material with a conductivity σ below 10 3 Sm −1 might appear equally suitable (cf. supplementary Figure S7d). In a naïve picture, this can be understood that reducing the conductivity is equivalent to removing an electrode with a capacitance Cgate. We note that the utilized O + -ion-implanted silicon on the sapphire substrate has a conductivity of approx. σ = 10 −1 Sm −1 . In this sense, in principle, it could be used as a gate material. However, it has a bandgap which is smaller than the pump laser energy. Therefore, the back-gate would be photoactive. To simulate the impact of such a scenario, the surface current element is placed below the ALD-grown Al2O3-layer (see Comsol simulation in the supplementary Figure S10). The additionally generated photocurrent capacitively couples from the photo-active back-gate into the coplanar striplines. Hence, the photoresponse of graphene and a possible silicon back-gate would overlay in time-resolved measurements. Another issue with a low conductive gate is that as in the case of O + -ion-implanted silicon, such materials typically have a strongly temperature-dependent conductivity. In other words, cooling the sample with liquid nitrogen or even helium freezes-out the charge carriers and therefore, turns the conductivity towards zero. In turn, a gating of the graphene would no longer be possible.
To summarize, in this supplementary note, we present a design guideline for a split back-gate geometry that is suitable for time-resolved measurements on dual-gated graphene-junctions and in a broader sense, for the integration of photo-thermoelectric graphene-junctions into high-frequency circuits.