Frequency response of electrolyte-gated graphene electrodes and transistors

The interface between graphene and aqueous electrolytes is of high importance for applications of graphene in the field of biosensors and bioelectronics. The graphene/electrolyte interface is governed by the low density of states of graphene that limits the capacitance near the Dirac point in graphene and the sheet resistance. While several reports have focused on studying the capacitance of graphene as a function of the gate voltage, the frequency response of graphene electrodes and electrolyte-gated transistors has not been discussed so far. Here, we report on the impedance characterization of single layer graphene electrodes and transistors, showing that due to the relatively high sheet resistance of graphene, the frequency response is governed by the distribution of resistive and capacitive circuit elements along the graphene/electrolyte interface. Based on an analytical solution for the impedance of the distributed circuit elements, we model the graphene/electrolyte interface both for the electrode and the transistor configurations. Using this model, we can extract the relevant material and device parameters such as the voltage-dependent intrinsic sheet and series resistances as well as the interfacial capacitance. The model also provides information about the frequency threshold of electrolyte-gated graphene transistors, above which the device exhibits a non-resistive response, offering an important insight into the suitable frequency range of operation of electrolyte-gated graphene devices.

the density of states near the Dirac point is strongly limited for graphene [12], which explains the low quantum capacitance of the material compared to metals or semiconductors. Therefore, in order to understand the graphene/electrolyte interface, the contribution of quantum capacitance has to be properly taken into account [13]. Considering the series arrangement of the electrolyte's double layer capacitance and the graphene's quantum capacitance, the inverse of the total capacitance is the sum of the inverses of the quantum capacitance and the double layer capacitance [14]. Thus, the total capacitance is limited by the capacitor with the smallest value, which is the quantum capacitance near the Dirac point. Many reports have recently focused on studying the total interfacial capacitance and the quantum capacitance in graphene in various electrolytes, like ionic liquids [13], ion-gels [15], and aqueous electrolytes [16][17][18], using electrochemical impedance spectroscopy [13,[15][16][17][18]. They show the voltage dependence of the interfacial capacitance in graphene, with a measured capacitance near the Dirac point ranging from 2 μF cm −2 [16] to 5 μF cm −2 [13]. However, these reports only consider experiments at low frequency, thus failing to display the complexity of the graphene/electrolyte interface. Here, we discuss the frequency dependence of the complex impedance of single-layer graphene electrodes and transistors as a function of the potential applied to the electrolyte. Our experiments reveal that the frequency response of devices based on the graphene/electrolyte interface is governed by the distribution of elements, resistors and capacitors along the graphene surface, resulting from the limited conductivity of graphene. Using both an analytical and a finite-element-method-based solution to study the behavior of the distributed elements, we successfully model the frequency-dependent impedance of electrolyte-gated single-layer graphene field-effect transistors and electrodes. Using this approach, we derive the intrinsic sheet resistance of graphene, the graphene/electrolyte interfacial capacitance, and the series resistance of electrolytegated graphene devices as a function of the applied electrolyte potential. Furthermore, our analysis identifies the threshold frequency that marks the transition between a resistive and non-resistive response along the transistor channel, or in the case of the electrode, between a purely capacitive and mixed response (resistive and capacitive distributed elements). It is shown that this threshold strongly depends on the dimensions of the device active area-i.e. the electrode diameter or the transistor gate area-as well as on the value of the interfacial capacitance and the intrinsic resistance of the graphene sheet. We demonstrate that for typical device dimensions in the mm range, the threshold frequency can be in the lower kHz range, which becomes of relevance for applications based on electrolyte-gated graphene devices.

Results and discussion
The top-view schematics of the investigated electrode and transistor configurations are depicted in figure 1(a). Singlelayer graphene (SLG) films, as confirmed by Raman spectroscopy (see the supporting information (SI) (stacks.iop. org/JPhysD/50/095304/mmedia)), were employed in all the experiments in this work. The SLG was grown by chemical vapor deposition (CVD) on Cu foils as described elsewhere [19], transferred onto the substrate, and electrically connected using metal contacts. The precise insulation of the contacts with a silicone passivating film reduces the ungated graphene area-i.e. the graphene that is not in contact with the electrolyte-and therefore, the so-called access resistance. In the electrode configuration (figure 1(a) left), a circular area of 0.90 cm 2 is exposed to the electrolyte and a circular metal ring is used to minimize the contact resistance. In the transistor configuration, a rectangular area of × 0.96 0.96 cm 2 is left uncovered and two metal contacts on each side serve as the source and drain of the transistor. The potential in the electrolyte is changed using a Ag/AgCl reference electrode (see figure 1(b)). More details about the fabrication of the device can be found in the methods section.
The interface between SLG and the electrolyte is illustrated in figure 1(c). In the equivalent circuit model, the interfacial capacitance C d takes into account both the quantum capacitance and the electric double layer capacitance. In order to account for the heterogeneity of the electrode/electrolyte interfaces, the capacitance is typically replaced by a constant phase element (CPE) [14]. The Faradaic impedance is represented by Z f and the intrinsic sheet resistance of graphene by R n [14]. In the absence of charge transfer induced by Faradaic reactions at the graphene/electrolyte interface, Z f can be removed leading to the distribution of R-C or R-CPE elements along the graphene's surface. In graphene, the values of R n and C d strongly depend on the gate voltage and the applied potential between the electrolyte and graphene. As the lowest charge carrier density is observed at potentials near the Dirac point or the so-called 'charge neutrality point' [3], a high sheet resistance and a low capacitance can be expected at these potentials. Increasing the gate potential with respect to the Dirac voltage leads to an increase of the charge carrier density and, therefore, to lower R n and higher C d values.
Figures 2(a) and (b) show the Bode plot-i.e. the impedance and the phase angle of SLG measured in an electrode configuration (figure 1(a) left)-for a DC potential near the charge neutrality point (solid symbols). At frequencies lower than approximately 1 kHz, an almost purely capacitive regime with a phase shift near − 90 can be observed. At higher frequencies (1-100 kHz), the electrode shows a mixed response (capacitive and resistive) with a phase angle between − 45 and 0°. A simple equivalent circuit model consisting of a resistor, representing the series combination of the contact, access, and electrolyte resistances, and a CPE (see inset in figure 2(b)) can be used to reproduce the data at low frequencies (<100 Hz). At higher frequencies (>100 Hz), however, this circuit is not sufficient to fit the measured impedance, and more complex modeling is required. As schematically explained in figure 2(c), the graphene/electrolyte interface is governed by the distribution of elements, resistors and capacitors along the graphene surface, resulting from the limited conductivity of graphene. Charge carriers can flow through different paths along the graphene sheet before charging the distributed capacitor elements; the impedance of each current path corresponds to the impedance of the capacitive element plus the resistance of the graphene from the contact to the corresponding capacitive element. At low frequencies, the impedance of the current paths is dominated by the capacitive elements; thus, all paths contribute equally to the total current (figure 2(c) top). Increasing the frequency leads to a reduction of the impedance of the capacitance elements with respect to the resistive paths. As a result, the current paths do not contribute equally to the current (figure 2(c) center). The higher the frequency is, the more relevant this effect gets, and the current will preferentially flow through the paths that are very close to the contact (figure 2(c) bottom and figure S10). However, for graphene-related materials with a low sheet resistance, as is the case of multilayer graphene (MLG), the influence of the distributed elements can be neglected, and the impedance of the electrode can be modeled using a simple equivalent circuit consisting of a resistor and a CPE (see SI).
The frequency-dependent response of the distribution of elements in SLG can be modeled as follows. Considering a circular graphene sheet (figure 1(a) left) with r being the radial distance to the center, the following differential equations can be written to correlate the measured current I with the applied potential U: [20] where Z i is the interfacial impedance represented by C d and Z f in figure 1(c), and ω is the angular frequency. Solving these equations (see SI) yields for the total impedance Z of the graphene electrode: with L being the radius of the circular electrode and J 0 and J 1 the hyperbolic Bessel functions of the first kind and of the zeroth and first order, respectively. In the coefficient with the parameters Q 0 and β (see SI) and the imaginary number i, since the Faradaic reactions (represented by Z f in figure 1(c)) can be excluded.
At high frequencies, equation (3) predicts that the real and imaginary parts are the same, and thus the phase should reach a value of − 45 [20], which is not observed experimentally (see figure 2(b)). The discrepancy is attributed to the contribution of the resistance R c in series with Z ( figure 2(c)). R c is the sum of contributions from the electrolyte resistance, the contact resistance, and the so-called access resistance of the device, i.e. the resistance of the region of graphene not exposed to the electrolyte. The relation between the electrode impedance Z and R c will determine the response at high frequencies (see SI for a detailed discussion of the effect of R c ). In the case of a non-zero R c value, the phase at high frequencies can reach 0° because R c is purely real and Z decreases with increasing frequency.
Equation (3) modified with the series contribution of R c has been used to fit the impedance measured as a function of the electrolyte potential in an aqueous electrolyte; R n , R c , and the parameters of the CPE (Q 0 and β) were used as the fitting parameters. In the case of the electrolyte-gated graphene devices, the expected position of the charge neutrality point is at −0.1 V, which is the difference of the work function of graphene (4.6 eV) and the Ag/AgCl reference electrode (4.7 eV) [23]. However, as will be shown below, the charge neutrality point was found to be at = U 0.05 D V versus Ag/AgCl (see SI), which is generally attributed to the p-doping introduced by the polymer used during the transfer (see methods section). Figures 3(a) and (b) show the measured module and phase angle (symbols) of the impedance, respectively, near to ( + U 0.05 D V) and away from ( + U 0.35 D V) the charge neutrality point. In both cases, an almost ideal capacitive regime can be observed at frequencies lower than 1 kHz. However, at frequencies higher than 1 kHz, the frequency response departs from the one predicted by the Randles circuit depending on the applied electrolyte potential. At potentials away from the charge neutrality point (red symbols), where the sheet resistance is lower, the electrode impedance is more dominated by R c , as the phase shift approaching 0° at the high frequency shows. Near the charge neutrality point, where the sheet resistance is higher, the contribution of the distributed elements discussed above becomes dominant in the high frequency regime. The solid lines represent the fit according to equation (3) modified with the series contribution of R c , showing an excellent agreement with the measured data. The dashed line in figure 3(b) represents the case R c = 0. A fit using the finite element method (FEM) is presented in the SI. Figure 3(c) depicts the capacitance of the SLG electrode (green symbols) as a function of the gate voltage U G with respect to the charge neutrality point voltage U D . Here, the capacitance has been obtained from the fitted Q 0 parameter of the CPE, and the parameter β was found to be 0.96. The experimental data is in good agreement with the calculated capacitance (orange line), which takes into account the series combination of the electric double layer capacitance and the quantum capacitance (purple line), calculated using an extended Poisson-Boltzmann model [21,22]. The lowest measured capacitance is 1.15 μF cm −2 , which is slightly higher than the calculated value. We tentatively attribute this discrepancy to the contribution of impurities in the graphene sheet, like defects and doping, which are known to be introduced during the graphene transfer [24] (see methods section and SI). Away from the charge neutrality point, the capacitance increases and starts to saturate at values above 2.0 μF cm −2 , which has been previously discussed in terms of the dominant contribution of the electrical double layer capacitance over the quantum capacitance [13]. The intrinsic sheet resistance, extracted by fitting the impedance spectra at different electrolyte voltages, has its maximum (4 kΩ) at the charge neutrality point (0.05 V versus Ag/AgCl) and decreases away from the charge neutrality point (see figure 3(d)). At very high frequencies (>100 kHz), the module of the graphene electrode impedance asymptotically approaches a resistive term which represents the combined contribution of the contact, access, and electrolyte resistance.
Similar to the case of the electrode configuration, the distribution of elements has a great influence on the frequency response of SLG electrolyte-gated field-effect transistors. Figure 4 shows the impedance of SLG measured in a transistor configuration ( figure 1(a) (right)), i.e. it represents the impedance measured between the drain and the source contacts. To this end, an AC voltage was applied between the drain and source while an external voltage U G was applied to the electrolyte to change the position of the Fermi level in the graphene. At high frequencies (≈100 kHz), the transistor shows nonresistive behavior, with a phase angle which is significantly more negative than 0° for a gate bias near ( − U 0.03 D V, blue symbols) and away ( − U 0.5 D V, red symbols) from the charge neutrality point ( figure 4(b)). With a decreasing frequency, the phase angle reaches a minimum close to − 30 at around 2 kHz, and increases to 0° at frequencies lower than 10 Hz. Accordingly, the module of the impedance (figure 4(a)) does not exhibit any frequency dependence at frequencies below 10 Hz; in this frequency regime, the measured impedance corresponds to the resistance of the graphene sheet from source to drain. As expected, the graphene sheet resistance depends on the applied gate voltage and varies from 1.8 kΩ near the charge neutrality point to 0.42 kΩ away from the charge neutrality point ( figure 4(c)). At low frequencies (below 10 Hz for the measured device), the impedance of the distributed capacitive elements representing the graphene/electrolyte interface is significantly larger than the impedance of the resistive elements representing the graphene sheet resistance; as a result, it can be assumed that the only equivalent circuit elements contributing to the current density are the resistors along the graphene sheet (see illustration in figures 4(e) and S13). With an increasing frequency, however, the impedance of the interfacial capacitive elements decreases and its contribution cannot be neglected anymore. As a result, part of the injected signal at the drain or source contact 'leaks' through the graphene/electrolyte interface, offering an alternative conduction path through the capacitors and the electrolyte. This leads to a decrease in the phase angle towards values close to − 45 at high frequencies, characteristic of an interface with resistive and capacitive distributed elements as shown in the illustration of figure 4(e). At high frequencies (⩾2 kHz), the phase angle of the measured impedance approaches 0° again: in this frequency range, the flow of charge from source to drain across the electrolyte (the low impedance path due to the used high ionic strength) becomes dominant over the flow along the graphene sheet. As a result, the frequency response  [21,22] that takes the quantum capacitance C Q and the double layer capacitance into account, is also shown. (d) The calculated intrinsic sheet resistance as a function of the applied potential. is dominated by the series resistance R c , as already discussed for the case of the electrode configuration (a detailed discussion of the contrib ution of R c in the transistor configuration is provided in the SI). The analytical solution of the distribution of elements with a rectangular graphene area (the full derivation is presented in the SI) is used to fit the measured data and is represented by the solid lines in figures 4(a) and (b). The dashed line represents the case in which R c = 0. The fits show good agreement, except for the frequency range around 2 kHz where the maximum phase angle difference is observed. This deviation can be explained by the oversimplification of the analytical model relating to the negligible electrolyte resistance, and homogeneous graphene conductivity and capacitance along the channel. Simulations of the impedance using an FEM model, which is able to overcome the limitations of the analytical solution can be found in the SI. An important outcome of this work is the realization that the discussed analytical model can be used to extract the series resistance (figure 4(c)), which includes the contact resistance, and the interfacial capacitance (figure S6) using the frequency-dependent drain-to-source impedance. The voltage dependence of the extracted series resistance results from the ambipolar transport in graphene and the characteristic p-type doping of CVD graphene [24]. When the transistor is biased in the hole regime ( < U U G D ), the Fermi level in the channel area is in the graphene valence band, resulting in the formation of a pp-junction at the interface between the channel and ungateable p-doped graphene near the metal contacts. In the electron region ( > U U G D ), in contrast, a pn-junction is formed, which increases the contact resistance and thus the series resistance.
We can define the threshold frequency f th above which the frequency response of the device's impedance is governed by the contribution of the distribution of the elements. For the particular case of the transistor, the threshold frequency separates the resistive and non-resistive response of a transistor. It can be shown (see SI) that = π f R C th 1 2 n int , i.e. it depends on the capacitance and resistance of the active channel and thus on the device dimension and the electronic properties of the used material. For instance, the threshold frequency of the transistor discussed above, with a dimension of × 0.96 0.96 cm 2 , is close to 1 kHz. While for a microtransistor ( µ × 10 20 m 2 ) the threshold frequency was measured above 10 MHz (see figure S14), the threshold frequency for a mm sized SLG transistor is expected to be in the higher kHz range. On the other hand, for a mm sized transistor based on reduced graphene oxide, with a typical sheet resistance of 43 kΩ ⋅sq [25] and an interfacial capacitance of 283 μF cm −2 [26], the threshold frequency is expected in the range of 1 Hz, suggesting that the operation frequency of the sensors based on reduced graphene oxide should be restricted to frequencies lower than 1 Hz in the case of mm sized devices.

Conclusion
In conclusion, we have investigated the frequency-dependent impedance of SLG electrolyte-gated field-effect transistors and electrodes. Our work shows that the frequency response of the graphene/electrolyte interface has to be described considering the distribution of resistive and capacitive elements along the graphene surface related to the graphene sheet resistance and the interfacial capacitance. Using an analytical solution to the problem of the distribution of elements, the frequency response of the SLG electrodes and transistors was reproduced in terms of the module and phase angle, and the intrinsic resistance, the series resistance, and the interfacial capacitance were obtained from the measurements in the electrode and transistor configurations. Our model is able to predict a threshold frequency above which the device response is dominated by the distribution of elements. In the particular case of the transistors, the threshold frequency separates the resistive and non-resistive regimes of the drain-source impedance. At frequencies higher than the frequency threshold, the capacitive impedance is low, therefore, the electrolyte provides a less resistive path for the signal injected to the drain or source contacts, which effectively short-circuits the drain-source conductive path along the electrolyte. Our results are of relevance especially for applications using electrolyte-gated graphene devicesfor instance biosensing and bioelectronics applications-in order to understand the frequency response of the devices and, even more importantly, to estimate at which frequency these devices can be operated.

Fabrication of graphene electrodes and transistors
The graphene was grown by chemical vapor deposition using copper as the metal catalyst and methane as the carbon source. Before growth, copper foil (25 μm thickness, Alfa Aesar) was electropolished [27] for 5 min at a current density of 12 mA cm −2 . After an annealing step at 1015 °C with an argon flow of 400 sccm and a hydrogen flow of 100 sccm at 100 mbar, a methane flow of 0.2 sccm for 30 min was introduced to start the graphene nucleation and growth at 15 mbar. In order to get full graphene coverage, the carbon supply was continuously increased by increasing the methane flow to 0.5 sccm. The copper was etched by a chemical solution containing 0.5 m FeCl 3 and 2 m HCl using PMMA 950K A2 as the supportive layer. The graphene sheet was fished onto a glass substrate, which was cleaned in oxygen plasma (200 W, 3 min), and dried on a hotplate at 50 °C. The polymer was removed in hot acetone for 1 h. An aluminum wire was connected to the graphene sheet via silver paste and insulated with a silicone layer (Scrintec 601).

Electrochemical measurements
All electrochemical measurements were recorded using a Parstat (Princeton Applied Technologies) system in a three electrode configuration with a Ag/AgCl electrode as a reference. In the electrode configuration, a platinum wire was used as the counter electrode and the electrolyte consisted of a 5 mm PBS buffer containing 500 mm KCl. The pH value was adjusted to = pH 3 by adding HCl. In the transistor configuration, the gate potential was changed using an external voltage source (Keithley 2400) in series with a 50 MΩ resistor.