# High-Frequency Limits of Graphene Field-Effect Transistors with Velocity Saturation
## 2 Methods  
Samples used in this work are made of CVD graphene on a thin h-BN flake deposited on top of a metallic bottom gate. The substrate consists in a 280 nm thermal SiO\({}_{2}\) insulating barrier on top of high-resistivity Si suitable for RF measurements. Two types of gate electrodes are used: a thin tungsten electrode deposited on SiO\({}_{2}\) (type A samples) for low-frequency devices and DC characterization, and a thick gold dual gate buried in SiO\({}_{2}\) (type B samples) for RF devices. Several samples of both types have been measured presenting similar characteristics, but for simplicity, we focus on one illustrative sample of each type. Sample-A (Figure 1a) is made out of a 20-nm-thin tungsten film that is etched to define the gate whose dimensions are \(L=0.5\)\(\upmu\)m for the length and \(W=1.5\)\(\upmu\)m for the width. In addition, we have included local contact gates (also tungsten) controlling the carrier density of the contacted graphene to reduce the contact resistance (see [19]). An h-BN flake, of thickness \(t_{BN}=16\) nm, is deposited on top of the gate by a dry transfer technique [2]. The h-BN surface is cleaned by furnace annealing at 300 \({}^{\circ}\)C under an Ar/H\({}_{2}\) flow. The CVD graphene is grown on copper following the _pulsed CVD_ method at Institut Neel [3] and transferred with a wet technique on top of h-BN. Again, the polymer residues are removed by furnace annealing. Finally, 50 nm of Pd is evaporated for the electrical contacts, and devices are embedded in a 50 \(\Omega\)-matched coplanar waveguide (CPW) used for DC and RF characterization. A schematic of a type-A device can be seen in Figure 1b. In very-high mobility devices such as the Zener-Klein transistor of [11] radiative cooling by hyperbolic phonon polaritons in the can be achieved, which is optimized using thicker (\(t_{BN}\gtrsim 100\) nm) h-BN substrates. Note also that alternative WSe2 substrates can be used that provide even higher mobility at room temperature [20]. The fabrication steps of sample-B (see Figure 2a below) are identical, except for the gate electrode: First a trench of SiO\({}_{2}\) is realized with e-beam lithography and dry etching with a carefully controlled depth of 200 nm. Another e-beam lithography step defines a gate electrode within the SiO\({}_{2}\) trench, and gold is evaporated up to a nearly perfect alignment with the SiO\({}_{2}\) surface (an excess of a few nm of gold is preferable to avoid any air gap between the gate and h-BN). The gate dimensions are \(L=0.8\)\(\upmu\)m and \(W=22\)\(\upmu\)m, and h-BN thickness is 18 nm. The low contact resistance of sample-A (\(R_{c}\approx 125\)\(\Omega\upmu\)m) is minimized by contact gating implying that the total resistance is dominated by the channel. Fully ambipolar contact gating was demonstrated in Ref. [19]; in this work the channel length was small so that the transistor was controlled by the contact junction resistance. Here we work with long channels, longer than the optical phonon scattering length, to realize velocity saturation transistors with large voltage gain; contact gating is used instead to minimize the contact resistance. This geometry is well suited for the identification of such mechanisms and their modeling. The sample-B geometry is best suited for maximizing the oscillation frequency \(f_{max}\), as dual gates provide a better impedance match of the device and the thick gold allows a reduction of the gate electrode resistance, which is a limiting factor for \(f_{max}\). Our devices fulfill the filed-effect mobility requirement \(\mu\gtrsim 1\) m\({}^{2}\)V\({}^{-1}\)s\({}^{-1}\) at low temperature. Experiments are carried out in a Janis cryogenic probe station at 8 K in the band 0-40 GHz. A vector network analyzer (VNA) is used to measure the S-parameters of our transistors. Figure 1: Low frequency field effect transistors (GFETs). (**a**) SEM image of sample-A. Note the absence of ungated area. (**b**) Electrical sketch of samples used in this study.