Novel circuit design for high-impedance and non-local electrical measurements of two-dimensional materials

Two-dimensional materials offer a novel platform for the development of future quantum technologies. However, the electrical characterisation of topological insulating states, non-local resistance and bandgap tuning in atomically-thin materials, can be strongly affected by spurious signals arising from the measuring electronics. Common-mode voltages, dielectric leakage in the coaxial cables and the limited input impedance of alternate-current amplifiers can mask the true nature of such high-impedance states. Here, we present an optical isolator circuit which grants access to such states by electrically decoupling the current-injection from the voltage-sensing circuitry. We benchmark our apparatus against two state-of-the-art measurements: the non-local resistance of a graphene Hall bar and the transfer characteristic of a WS2 field-effect transistor. Our system allows the quick characterisation of novel insulating states in two-dimensional materials with potential applications in future quantum technologies.

A widely and long-accepted classification of the materials and states of matter relies on the electrical conductivity of systems which are metallic, insulating or semiconducting.
However, their characterization has often proven to be very challenging. Indeed, the difficulty of measuring a superconducting zero-state resistance faced in the early days by Kamerling Onnes 1 is nowadays being emulated by the difficulty of measuring highly resistive states under extreme conditions (e.g. high magnetic field and low temperature), in systems with extremely low dimensionality such as atomically-thin materials [2][3][4] . The recent observations of insulating states of topological origin 5 or induced by many-body interactions 6 , a nonsaturating linear magneto-resistance of quantum mechanical origin 7 , the observation of giant non-locality 8 , viscous electron flow 9 and tunable bandgap 10 in single-and few-layer graphene devices are examples of recently discovered high-impedance (Hi-Z) states and phenomena of high interest for their potential applications in the field of quantum technologies. However, contrasting experimental reports in high quality devices and materials suggest that the electronics used in the characterization of these states needs to be reconsidered to eliminate spurious signals.
There is a wide consensus that alternate-current (AC) lock-in measurements offer higher resolution and greater noise rejection than direct-current (DC) techniques and, for this reason, they are widely used to characterize the response of electronic devices as well as in many practical applications, including medical diagnostics 11 , classical and quantum metrology 12 .
Careful consideration needs to be given to the characterization of insulating states with this technique. For example, a spurious negative value of resistance can appear in the non-local transport of a graphene Hall bar 13,14 due to the incomplete rejection of the common-mode voltage (CMV) present at the input of lock-in amplifiers. Such spurious voltage is the direct consequence of the finite input impedance of the measuring amplifier and the dielectric leakage of the transmission lines, usually BNC-terminated coaxial cables. DC amplifiers and sources might offer a straightforward solution since they typically have a higher input impedance than their AC counterparts. However, they are characterized by a larger noise floor than AC electronics making the characterization of insulating states with narrow energy gaps, such as in 2D materials 10,15 , challenging.
In this letter, we present an experimental apparatus designed specifically to overcome the aforementioned limitations, allowing us to eliminate spurious artefacts and grant direct access to the aforementioned Hi-Z states. The key to achieving this goal lies in the ability to decouple electrically (float) the circuits connected to the current-injection probes from that of the voltage-sensing contacts. To this end, we developed a battery-powered optical isolator (or optocoupler) circuit, together with a custom-designed measuring chamber which is able to float both core and shell of the coaxial cables used to interface the device to the instruments.
In this manuscript we present two versions of this circuit: low-power and low-noise. The low-noise circuit is suitable for high-precision measurements as it is characterised by a noise level well below the intrinsic noise of the commercial measuring instrument used in these experiments (< 10 −5 V/ √ Hz at frequencies below 100 Hz). On the other hand, the low-power circuit demonstrates an increase in the period between battery charges from 4 to more than 30 days with a noise floor of ∼ 10 −4 V/ √ Hz, making it ideal for remote, long-term, operation.
We benchmark the use of these circuits in two state-of-the-art experiments: measurement of the non-local resistance in a graphene Hall bar in perpendicular magnetic field 8 and AC electrical characterisation of an atomically thin WS 2 field-effect transistor. Finally, the reduced number of components and the possibility of miniaturisation using surfacemount-devices makes our approach of interest for applications with stringent requirements on integration such as particle accelerators and space applications.

I. OPERATING PRINCIPLES AND INSTRUMENTATION
To allow the measurement of Hi-Z devices in AC and prevent common-mode voltage (CMV), inductive coupling, and dielectric leakage currents from affecting the measuring instruments, it is necessary to effectively float the voltage probes of such instruments from the current-injection leads of the device under test. To achieve this goal, we incorporate into our apparatus an optical isolator or optocoupler. This is a device which converts an electric signal into light using a light-emitting-diode (LED) and converts it back into an electrical signal through one, or more, photodiodes. An ideal optocoupler has a linear response with a gain of 1. Whilst optical coupling is a well-known technique to isolate electric signals, crucially in the specific case of AC lock-in measurements, the optical isolation of the core signal of the wire is not sufficient to eliminate the emergence of spurious electrical signals.
Indeed, the dielectric leakage in the coaxial cables (including connectors and terminations) between the core wire and the ground shielding causes the signal in one section of the circuit to be coupled to the ground, known as ground potential difference 16 , and therefore to be measured elsewhere in the circuit. This coupling is one kind of CMV present at the input of a differential amplifier. For this reason, in AC measurements, it is also necessary to decouple the elements of the circuit used in defining the electrical ground. We realise such decoupling by using a carefully designed measuring probe which is able to isolate the cores from the grounding shells of the BNC wires with a resistance in excess of 1TΩ.
The core electronic circuit is shown in Figure 1. This consists of a bipolar optocoupler with integrated dual supply based on two HCNR201 high-linearity analog optocoupler (OC1 and OC2) integrated circuits (ICs). Such ICs comprise a LED and two photodiodes. A primary photodiode is used to transfer the input signal to the output while the secondary photodiode is used as a feedback to ensure optimal linearity and stability. This feature makes the performance of the HCNR201 very close to that of an ideal optocoupler. Indeed, the typical The noise as a function of frequency has been measured using an Ametek Model 7270 DSP Lock-in amplifier in Noise-measurement mode. Figure 3b shows the noise from the lock-in (black line) and the two versions of the optocoupler. We can see that the low-power version has a noise level of > 10 −5 V/ √ Hz with a pronounced peak around 100 Hz. At higher frequencies the noise level is between one and two orders of magnitude higher than the lockin intrinsic level (< 10 −7 V/ √ Hz), which is mostly due to the noise performance of the Op-Amps used. The sharp peak observed at 6 kHz corresponds to the frequency at which the ICL7660A operates to produce the negative power supply voltage. On the contrary, the low-noise version shows a noise level at low frequency identical to the lock-in amplifier (< 10 −5 V/ √ Hz), indicating that its intrinsic noise level is at least one order of magnitude smaller than that of the lock-in. The same behaviour of the low-power version is observed at higher frequencies, albeit with slightly higher noise level above 1 kHz. In both versions, the frequency of the ICL7660A is easily adjustable in a wide range by adding a simple capacitor to the circuit (between pin 7 and GND in figure 1), allowing for the customisation of the high-frequency noise level of the optocoupler. days.

II. NON-LOCAL AND HIGH-IMPEDANCE MEASUREMENTS
In order to demonstrate the use and capabilities of our apparatus we present two stateof-the-art measurements which would be adversely affected by artefacts in a normal experimental setup: the non-local resistance in a graphene Hall bar and the characterisation of a The graphene is contacted using Cr/Au electrodes in a side-contact configuration (or 1D contacts) 22 . This structure and contact geometry allows the realisation of the ultra-high mobilities necessary to observe non-local effects in micron-scaled graphene devices 8,21 .
A typical room temperature non-local lock-in measurement of a graphene Hall bar is shown schematically in figure 4a. The current is injected between contacts 1 and 2, through a known ballast resistor R B and the voltage drop (V nl ) is measured between contacts 3 and 4.
Such arrangement gives the results shown in figure 4b, where the non-local resistance (R nl ) is plotted as a function of gate voltage (V bg ) in the presence, and absence, of a magnetic field (B) applied perpendicularly to the sample. We notice that the curve in figure 4b for B = 2 T shows a negative resistance peak at V bg = −10 V, which corresponds to the charge neutrality point of the graphene Hall bar. Although the appearance of a negative bending resistance has been observed in graphene, and it is a signature of room temperature ballistic transport 23,24 , such effect cannot be measured in our geometry because it requires a cross-shaped device, in which the current-injection (and voltage-sensing) is performed using pairs of orthogonal contacts. Therefore, the observed feature is an artefact introduced by the measurement configuration. This is caused by the incomplete rejection of the CMV (V cmv ) at the input of the amplifier. A mismatch in the resistance between the current-injection leads produces a voltage drop which is then coupled to the voltage-probes via ground coupling and through the device itself (between pins 1 − 3 and 2 − 4). The spurious non-local voltage can be calculated as 13 : where R in is the input impedance of the voltage amplifier, typically 1 − 10 MΩ, and R 1 (R 2 ) is the total resistance across the local part of the circuit, which include R B , the contact resistance of points 1 and 3 (or 2 and 4) and R in . The ground coupling is due to the dielectric leakage of the BNC cables towards the cable shielding, which effectively allows a signal generated between contacts 1 and 2 to be sensed between contacts 3 and 4 by-passing the device. The value of V cmv increases with the channel resistance and it is maximum at the charge-neutrality point. Notably, depending on the values of R 1 and R 2 , the value V nl can be larger than the actual non-local signal and it can also have opposite sign (see figure 4b). In order to suppress this spurious signal we adopted the configuration shown in characteristic (figure 5c, red line) is observed. In this case a sharp decrease of the channel resistance with applied bias voltage is expected in such material due to the large barrier formed at the contacts, a well-known issue in both exfoliated and CVD-grown transitionmetal dichalcogenides 26,27 . Such characteristic is correctly displayed in the DC measurement as well as in the AC configuration using the optical isolators.

III. CONCLUSION
In conclusion, we have presented an integrated opto-electrical circuit which allows the use of low-noise and low input impedance AC lock-in for the characterization of high-Z devices.
We show that this optocoupler eliminates large spurious electrical signals by effectively decoupling the voltage sensing instruments from the current-driving circuitry. We demonstrate its performance in terms of fidelity, noise level, which is as low as the intrinsic noise level of the measuring instrument, and power consumption, which results in a continuous operation up to 40 days using a small-size battery. The circuit can easily be miniaturized using surfacemount components and quickly up-scaled to fit multi-purposes applications when more than two contacts are needed in a small space, such as in modern experiments on quantum computing, large machines such as particle accelerators or in space applications. The developed optocouplers are highly suitable for the low-noise electrical characterization of narrow-gap states and semiconductors with potential applications in future quantum technologies such as computation, communication and sensing. Although in this work we focussed on the use of our circuitry to the characterization of 2D materials, the technique can be readily applied to the electrical measurement of a multitude of materials in which high-impedance states need to be accessed, such as polymers and composites, metalling thin-films, nanowires, nanotubes and quantum dots.

SUPPLEMENTARY MATERIAL
Supplementary material is submitted with the current manuscript. Additional data, including EAGLE files for the schematics and PCB broads, related to this paper may be requested from the authors. Correspondence and requests for materials should be addressed to S.R.