Device layout

The layout of a typical device investigated here is shown in Fig. 8.1. The stub tuner made out of superconducting niobium is based on two transmission lines TL1 and TL2 of lengths l and d, each close to λ/4 [263]. The working principle of the stub tuner can be understood in a simple analogy to optics.

The incoming microwave signal is split at the T-junction that acts as a beam splitter. The microwave signal then travels down both arms and reflects at the open and at the device end. Depending on the device load, different interference conditions are present at the T-junction that provide the resonance condition.

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8.2. Device layout

VG signal line ground

L

Figure 8.1. Sample layout: (a) An optical picture of the stub tuner with arm-lengths l and d. The central conductor and the gap widths of the transmission lines are 15 µm and 6 µm respectively. Light areas show the Nb film and darker areas are exposed SiO2 substrate after the Nb is etched away.

(b) An SEM image near the l end showing a narrow slit between the signal line and the ground plane. (c) An SEM image of a hBN/Gr/hBN stack for device B placed over the slit. Areas A1 and A2 correspond to the two parts of the graphene lying on the signal line and the ground plane, respectively. (d) A cross section schematic of the device near the slit. (e) An equivalent circuit with lumped capacitance and resistance elements. c 2018 American Physical Society

The circuit is patterned in a 100 nm thick Nb film employing standard e-beam lithography and a dry etching process employing Ar/Cl2. In order to minimize microwave loss, the sample is fabricated on an intrinsic, high resistive silicon wafer with a 170 nm thick SiO2layer. The signal line of the TL1features a narrow slit (450 nm) insulating it from the ground plane to the right, see also Fig.8.1(b) and (c). We placed a hBN/Gr/hBN heterostructure across this slit using a dry pick-up method described in more detail in section 2.1. The stack was positioned such that parts of it were lying on top of the signal line (area A1) and parts on the ground plane (area A2). E-beam lithography and reactive ion etching were then used to structure the stack into a well defined rectangular geometry. Details on the fabrication of the niobium resonator can be found in appendixA.

The lack of electrical contacts allowed us to change the shape of the device

8. Quantum capacitance and dissipation in graphene pn-junctions

while using the same RF circuit. We first fabricated a device with dimensions W × Lof 6.5 µm × 13 µm (device A), with W and L indicated in Fig.8.1(c).

After measuring device A, the stack was reshaped into the following dimensions of 6.5 µm × 9.6 µm (device B). Both devices consisted of an area A1 lying on top of the signal line (gate) of 6.5 µm × 3.4 µm. The graphene section lying above the ground plane (area A2) was 6.5 µm × 9.6 µm for device A and 6.5 µm

×3.8 µm for device B. Hence, device A was asymmetric whereas device B was quasi symmetric around the slit. This two devices, using the same resonant circuit, allowed for consistency checks and helped in understanding the gating of these devices. A third symmetric device C of dimension 5 µm × 12 µm fabricated on a separate resonator circuit and a different vdW-heterostructure was also investigated.

8.2.1. Measurement principle

The graphene properties were extracted by measuring the complex reflection coefficient from the stub tuner, which depends on the RF admittance of the load [264]. A vector network analyser is used to measure the reflected part of the probe RF signal fed into the launcher port of the circuit. In order to change the Fermi energy of the graphene a gate voltage VGis applied to the signal line with the help of a bias tee as shown in Fig. 8.1 (a). The gate voltage changes locally the density and hence the quantum capacitance and also the resistance of the graphene. Careful analysis of the RF response of the circuit at different gate voltages allowed us to extract differential capacitance, related to geometric and quantum capacitance, and dissipation, related to losses and charge relaxation resistance in the graphene. All measurements were performed at a temperature of 20 mK and with an input power of −110 dBm.

Even though the Nb resonator could also be operated at higher temperatures (still below Tc), 20 mK was chosen to maximize the quality factor and hence the sensitivity of the RF resonator to changes in C and R of the load impedance.

Since the graphene is not connected to any electrode the total charge on the graphene is conserved and therefore gating works a bit different than in conventional devices. A gate voltage on the signal line induces charge carriers on the graphene part lying on top of it (area A1). The only source of charge carriers is the graphene area lying on top of the ground plane (area A2).

Therefore, gating will always influence both areas simultaneously. In the case of a pristine graphene sheet (Fermi energy at the CNP without gating), any gate voltage applied to the signal line will lead to the formation of a pn-junction at the slit. However, if a finite offset doping is present, which is generally the case for any graphene device, an offset voltage has to be applied to the gate to drive the graphene charge neutral. The CNP is reached at two different gate voltages, once for each part of the graphene. At voltages larger than this offset voltages (in absolute values) a pn-junction will be present in the

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