Contact gating at GHz frequency in graphene

The paradigm of graphene transistors is based on the gate modulation of the channel carrier density by means of a local channel gate. This standard architecture is subject to the scaling limit of the channel length and further restrictions due to access and contact resistances impeding the device performance. We propose a novel design, overcoming these issues by implementing additional local gates underneath the contact region which allow a full control of the Klein barrier taking place at the contact edge. In particular, our work demonstrates the GHz operation of transistors driven by independent contact gates. We benchmark the standard channel and novel contact gating and report for the later dynamical transconductance levels at the state of the art. Our finding may find applications in electronics and optoelectronics whenever there is need to control independently the Fermi level and the electrostatic potential of electronic sources or to get rid of cumbersome local channel gates.

In general, contacts behave as passive elements impeding the performance of electronic devices. In high mobility graphene transistors the contact resistance competes with the channel resistance already for channel lengths in the hundred nanometer range 1 . A standard route to minimize this spurious contribution is to match contact doping with channel doping using an overall back gate. Introduced in silicon and carbon nanotube Schottky barrier transistors 2,3 , back gating proves particularly efficient in graphene [4][5][6] and other 2D materials 7 , thanks to the weak screening at 2D. Going one step further, one can think of turning contacts to active elements by controlling independently and dynamically their electric and chemical potentials using individual contact gates; the non linear element is then the Klein tunneling barrier that develops at the contact edge due to the work function mismatch between metal and graphene 8 . Such a contact gating is challenging as it requires a set of local gates to engineer the doping profile all along the graphene sheet with the Fermi wave length resolution. In the present work we have realized dual gate Klein barrier transistors (KBTs) achieving these conditions using a nano-patterned bottom gate array, thin hexagonal boron nitride (hBN) dielectric, and palladium (Pd) contacts. The contact gate controls the doping of contacted graphene locally, including its polarity, which allows operating the KBT in a fully tunable bipolar regime. Its low bias DC properties are accurately mapped using a ballistic Klein tunneling junction model 9 from which we extract the relevant contact parameters. Its dynamical response is investigated at GHz frequency, where contact gating is benchmarked against conventional channel gating. The realization of gated contacts opens the way to new types of devices such as (channel) gate-free transistors for microwave or photo-detectors 10,11 , tunable nanosecond electron sources for quantum electronics 12,13 or spintronics 14 , as well as new architectures based on Dirac Fermion optics [15][16][17] .
A graphene contact is a composite element made of two junctions in series: a vertical metal-graphene junction and an in-plane contact junction. The former stems from the momentum mismatch between zone-center metallic and zone-edge graphene electrons and the latter results from doping gradients at the contact edge. According to Giovanetti et al. 8 , a charge transfer takes place at the metal-graphene junction as a result of work function imbalance Δ W between graphene and metal; its equilibrium density is governed by Δ W and the screening energy ϵ c of graphene (see Eq. (3) below). However the electrostatic problem is not fully closed in general as the doping may be further controlled by remote electrostatic influence from the back side of the contact. Therefore Eq. (3) incorporates contact back gating by the voltage V cont , which fully sets the Fermi energy of the graphene area located beneath the metal. In-plane junctions have been investigated theoretically (see e.g. ref. 9) and experimentally 1,4,6 . As a difference with gate defined channel junctions [18][19][20] , contact junctions are generally sharp and ballistic; their resistance is determined by the junction length d and the doping (polarity and concentration) on both sides. With k F d/2π ~ 1, where k F is the electronic wave number in the least doped side, sharp p-n junctions have a high transparency  .
T 0 5 9 . In this work we demonstrate a Klein barrier transistor, equipped with a bottom channel gate (bias V ch ) and two interconnected source and drain contact gates (bias V cont ) as sketched in Fig. 1a. The independent dual gate control of Klein barriers is highlighted in the low-bias resistance curve R(V ch ), where R is strongly modulated by V cont at large channel p-and n-doping (Fig. 1b)

Methods
Following ref. 21 we use h-BN flakes as a mobility preserving back-gate dielectric (dielectric constant κ .  4 2). A 16 nm-thin h-BN flake is deposited on a 20 nm-thick tungsten film that is nanostructured with 30 nm gaps to realize a back gate array as shown in Fig. 1a (inset). The 30 nanometer long intergate trenches are achieved by e-beam lithography and dry etching. Tungsten was used here as a refractory metal preserving the gate pattern resolution during annealing processes. In addition, the h-BN dielectrics shows superior dynamical properties, as compared with conventional thin oxides, due to the absence of spurious charge traps. The exfoliated graphene and h-BN flakes are transferred with a dry technique 22 . Palladium contacts are deposited and aligned with the trenches to set independent channel and contact gating (inset of Fig. 1a). The channel length and width are The transistor is embedded in a three port coplanar wave guide (CPW) (Fig. 5a) used for DC and RF characterization; the transfer curves and scattering parameters are measured identically for contact and/or channel gating. Experiments are carried out in a 40 GHz variable temperature probe station, with a measuring window limited to 3 GHz by the finite resistivity (~300 Ω square) of the thin tungsten metallizations. The experimental data below refer to room temperature measurements, but we have checked that device properties are preserved, and eventually enriched by Fabry-Pérot oscillations, at cryogenic temperatures 1,24,25 . Figure 1c shows the resistance transfer curve R(V ch ) for contact gate voltages V cont = − 1.2→ 1.2 V. As seen in the figure, contact gating efficiently modulates the device resistance away from charge neutrality point (CNP), R(V ch ) being modulated by a factor 2.3 for V ch < 0 and 1.3 for V ch > 0. By contrast, the CNP conductance remains unchanged indicating a mobility limited conductance (µ − −  6000 cm V s 2 1 1 as explained below). Note that, in contrast with previous investigations using a combination of back and top gates 6 , the CNP-peak position is not The contact area is represented by a resistance R mg in parallel with an electrical double layer capacitance C dl that stand for electrochemical balance. The doping of graphene underneath the contact is modulated through the capacitance C cont . The "K" symbol represents the Klein tunneling junction between contact area and channel area with its associated resistance

DC Results
is the diffusive resistance of the channel modulated by the channel gate through C ch . Panel (b) Electrical potential map of the device at the contact junction from finite element simulation for 2 typical channel and contact gate voltages, V cont = − 1 V and V ch = + 2 V, the contact being at V source = 0 V. Dimensions are labeled in nanometers, the graphene layer being located at z = 0. Panel (c) Horizontal cut of the potential through the junction together with a Fermi like function. The 10-90% extension is  d nm 30 .
shifted by the contact gate. The independence of the channel and contact gating not only facilitates the control of contact junctions at DC but also paves the way for new RF operation. For a quantitative analysis of the combined effects of channel and contact gating we consider below the full set of resistance transfer curves.
A color plot representation of the R(V ch , V cont ) is presented in Fig. 2a. As seen in the figure, the experimental data map the four polarities of channel and contact doping.  Fig. 2c and calculations below) by a small amount, Ω 110 , attributed to the metalgraphene resistance (  R 55 Ohms mg per contact). Our overall contact resistance compares favorably with the best achievements of 100 Ω .μm 4,26 . The asymmetry between p-doped and n-doped regimes, , is itself consistent with the ballistic junction model below. Figure 2f,g are R(V cont ) cuts for p-and n-channel doping for contact junction dominated regimes. The curves show a maximum and an asymmetry which are typical of ordinary transfer curves, confirming our ability to drive the contact doping through neutrality. These curves are more direct evidence of contact doping reversal than the conventional ones obtained with remote back gates where polarity reversal is signaled by a smeared second peak 6,26 . The positions of the maxima are independent of R(V ch ) (given a channel polarity) confirming the small electrostatic coupling between contact and channel gates, the R(V cont ) curves being mainly shifted upward on decreasing channel doping due the finite carrier mobility contribution of the channel. The main features of the R(V ch , V cont ) data are well reproduced in the plot of Fig. 2b, obtained using the simple model detailed below.

Model
To describe the contact properties we use an electrostatic model for the contact doping from ref. 8

and a
Fermi-function potential model for the contact junction resistance from ref. 9. In the electrostatic model the metal-graphene contact is described by a tunnel resistance R mg in parallel with a metal-graphene capacitance C dl (per unit area), the later accounting for the charge double layer formed at the metal-graphene interface as a result of work function mismatch Δ W. Whereas a metallic contact sets the electrochemical potential of the contacted graphene, both the electrostatic V gr and the chemical E F cont potentials can vary. Writing the electrochemical equality (1) and the electrostatic relationship (2),  one obtains the expression for the chemical potential of contacted graphene (3). In addition we plot in Fig. 4a The doping polarity is determined by the sign of the effective electrochemical potential mismatch ϵ W , which is tuned by the local bottom gate voltage V cont according to the capacitance level arm ratio C cont /C dl . Ab-initio calculations 8 predict small double layer thickness (~0.3-0.5 nm) and accordingly large C dl ~ 15-30 fF/μm 2 . However the metal graphene coupling is process dependent and experiments show large scatter in the contact resistance presumably due to doping variations. In addition, our measurements show that, for a given sample, the contact coupling eventually depends also on thermal cycling. The term ϵ c defines the screening energy scale of the gated contact assembly; at low doping   ( )  W c , the screening by graphene is weak and the contact gate controls the   the charge is carried by the graphene itself that screens the contacting metal. Eq. (3) accounts for the crossover between the two regimes and is plotted in Fig. 4a. In the channel, the doping is determined by V ch according to the usual parallel plate capacitor model.
The resistance is calculated using the lumped electrical elements description in Fig. 3a, where the contact junction is represented by a "K" symbol (by reference to Klein tunneling junctions). Several models have been proposed to calculate the transmission T(θ) of a potential step as function of the incident angle θ of carriers to the junction normal 9,28,29 . We assume ballistic junctions and rely on the Fermi-function model of ref. 9 where the potential step is described the function ( ) ∝ ( + ) x w 1 (this step shape is sketched in Fig. 1b) where w is the step length scale. As a benefit it provides analytic expressions for T(θ) for arbitrary junction length 9 . The transmission is given by k n F cont ch cont ch using the Snell-Descartes relation: Using numerical simulation tools (Fig. 3b), we have checked that the potential step at a contact edge in Fig. 3c does mimic a Fermi function step with a 10-90% extension   w d 4 30 nm. In this simulation we have neglected graphene screening, an assumption that is valid at low carrier density, and strictly relevant at p-n contact junctions. The resistance R K in Fig. 3a is then calculated using ch is the angular average of the junction transmission appropriate for diffusive leads (also plotted in Fig. 4b in two representative cases) and k F cont the Fermi momentum of contacted graphene 9 . Restricting ourself to linear conductance and assuming drain-source symmetry and incoherent transport in the channel, the drain and source junction resistances simply add up. Finally we add a channel resistance R ch in Fig. 3a, which is calculated taking an energy independent mobility μ = Const. Based on this simple model we can reproduce in Fig. 2b the main features of the experimental plot R(V ch , V cont ) in Fig. 2a. In the simulation we have imposed C ch = C cont = 2.3 fF/μm 2 and the junction length d = 30 nm from device geometry. The fitting parameters are R mg = 55 Ohm, μ = 6000 cm 2 V −1 s −1 , C dl = 4.5 fF/μm 2 (ϵ c = 58 meV) and Δ W = 50 meV. From these numbers we deduce the investigated contact doping range Note that the value of the mobility is consistent with a ballistic length l B = μE F /ev F ~ 60 nm (at E F ~ 0.1 eV) that is larger than the junction length. In addition we have accounted for a gaussian broadening of the density corresponding to an experimental minimum sheet carrier concentration of . × −  n 2 5 10 cm 0 11 2 . The above fitting process involves a priori five adjustable parameters, Δ W, C dl , R mg , μ and n 0 . The last two ones are determined as usual by the R(V ch ) dependence at DP while the three others control R(V ch ) away from DP. They can be determined thanks to the additional V cont dependence controlling the asymmetry of the R(V ch ) curves from which we deduce the contact parameters Δ W, C dl . As mentioned above, we finally deduce R mg from the (constant) symmetric part of the R(V ch ) curves. We found a work function mismatch between Pd and graphene of Δ W = 50 meV while the tabulated value is on the order of 1.1 eV 8 . However, according to Giovannetti et al. we should take into account a metal-graphene chemical interaction, that depends on the metal-graphene distance and scales down the work function mismatch. For a typical distance of 0.3 nm the correction is large, ~0.9 eV 8 , so that one expects Δ W th ~ 200 meV some factor above the measured value. This discrepancy can be attributed to a process induced chemical doping. A superiority of our dual gate transistor is that it provides a direct estimate of this effective work function mismatch as well as the screening energy.

RF Results
Based on the understanding of the DC properties of our dual gate KBT, we proceed by investigating its dynamical properties. Especially, we show here that the performance of the transistor, driven from its contact gate, can be as good as when it is driven from the usual channel gate. Here we are restricted to the 0.1-3 GHz frequency range, due to the combined effects of the gate resistance ( . Ω)  k 1 5 and stray capacitance ( ) 50 fF . In this window the transconductance data g m in Fig. 5b,c are frequency independent. We have selected representative gate biasing conditions for both channel-and contact-gating excitations. The sample is biased with a drain-source voltage V ds < 0.3 V to achieve a finite g m while preserving linear conditions ∝ g V m ds . At higher biases, velocity saturation effects arise which are beyond the scope of the present work. In Fig. 5c we compare the RF transconductance for a channel gate excitation g ch RF (red squares) with that of a contact gate excitation g cont RF (blue dots). This benchmarking is performed in the case of neutral contacts (V cont = 0 V) where the device is most symmetric. We first note that channel and contact gating are equally efficient with a transconductance level that matches the current state of the art 21,30,31 . For a more quantitative account of the dynamics of contact gating we have also plotted in Fig. 5c the DC transconductances g ch DC and g cont DC (dashed lines). The drop from DC to RF (Fig. 4c,d) is more pronounced for contact gating, presumably due to a larger gate source capacitance. In Fig. 5d we analyze the influence of contact doping on the contact-gate transconductance; we observe that g cont RF is strongly enhanced when contacted graphene is driven deep in the hole doping range (V cont = − 0.9 V) reaching a value as large as 0.6 mS/μm/V. These transconductance levels are very promising for applications in RF transistors that would however require low-resistance contact gates.
Finally, we show in Fig. 4b the response of the dual gate KBT under pulsed contact gating. As seen in the figure the sign of output current is reversed upon changing the channel doping polarity (at fixed contact doping) in agreement with the transconductance measurements for similar bias conditions (panel c). A technical but important feature is the fact that the output current pulse is a faithful replica of the contact gate pulse. We have realized similar devices where hBN is replaced by a thin aluminium oxide layer, where the situation is very different with the drain current drooping at the nanosecond scale after a gate voltage step. We attribute this affect to the relaxation of spurious charges at the graphene-oxide interface. Our hBN devices are devoid of trapped charges and therefore highly suitable for pulsed electronic applications.

Conclusions
In conclusion we have demonstrated contact gating both at DC and RF, in a dual gate Klein barrier transistor using graphene on boron nitride with a set of local bottom gates. Our device operates in the fully bipolar regime with transport properties that are accurately mapped with a ballistic Klein tunneling junction model. Active contact gating is demonstrated at RF frequency, and compares favorably with conventional channel gating. This finding opens new routes for graphene electronics and optoelectronics by using low resistance bottom gates and/ or separate contact gates to tune independently the drain and source doping.