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Charge transport mechanism in networks of armchair graphene nanoribbons

In graphene nanoribbons (GNRs), the lateral confinement of charge carriers opens a band gap, the key feature that enables novel graphene-based electronics. Despite great progress, reliable and reproducible fabrication of single-ribbon field-effect transistors (FETs) is still a challenge, impeding the understanding of the charge transport. Here, we present reproducible fabrication of armchair GNR-FETs based on networks of nanoribbons and analyze the charge transport mechanism using nine-atom wide and, in particular, five-atom-wide GNRs with large conductivity. We show formation of reliable Ohmic contacts and a yield of functional FETs close to unity by lamination of GNRs to electrodes. Modeling the charge transport in the networks reveals that transport is governed by inter-ribbon hopping mediated by nuclear tunneling, with a hopping length comparable to the physical GNR length. Overcoming the challenge of low-yield single-ribbon transistors by the networks and identifying the corresponding charge transport mechanism is a key step forward for functionalization of GNRs.


S1. Raman spectroscopy of -AGNR films
In Fig. S1 we show Raman spectra taken on a 9-AGNR film as-grown on Au and after the transfer on SiO2 demonstrating the intactness of the film after the transfer. We find the RBLM peak at approximately 311 cm −1 , which is the expected value for 9-atom wide GNRs 1 . As no additional peaks appear we can conclude that the GNRs are not significantly altered by the transfer process. Figure S1. Raman spectrum of a 9-AGNR film on Au and SiO2 surfaces. The low-frequency line at 311 cm −1 can be attributed to the presence of intact 9-AGNRs (the ribbon width is denoted by , the number of carbon atoms across the ribbon).

S2. Gate-leakage and bias symmetry in 5-AGNR network FETs
As described in the main text, in FET devices parasitic leakage currents can occur through the gate barrier. At high gate voltages and low drain voltages, the leakage leads to small systematic errors in the measurement. In order to quantify the impact of leakage currents, we measure the current at the gate electrode = + , where is the current flowing between the source electrode and the gate electrode. As shown in Fig. S2, the gate current is small compared to the drain current and its dependence on is negligible at small voltages applied to the gate electrode. Only at large , the leakage starts to affect slightly. However, since does not change with , we correct for its influence by simply subtracting from the I-V curves, such that lim →0 = 0. Figure S2. Gate leakage current and bias symmetric channel current for a representative 5-AGNR device presented in a semi-logarithmic plot. The gate current is orders of magnitude below the channel current and does not depend on bias voltage. We plot the absolute value of the channel current in order to provide a facile comparison of positive and negative bias.
Although the current at negative drain voltages is systematically lower than the current at positive drain voltages, this difference is very small. Hence, the use of data from only positive bias is justified.

S3. GNR-network FET characterization
In a conventional FET, the ratio on/off , which compares the current in the saturation regime (on-state) with the current in the subthreshold regime (off-state), is widely used as a figure of merit to quantify the effect of the gate voltage. Since a current saturation regime is not reached in our devices, we define in analogy to on/off a current modulation ratio where we compute the ratio of channel current at specific gate voltages. Furthermore, the field-effect mobility can be extracted from the transfer curve via 2 where = ⁄ is the transconductance, is the channel length, is the channel width and Ox = 0 / Ox ≈ 1.15 × 10 −4 F/m 2 is the geometrical capacitance density assuming that the channel and the gate electrode form a parallel plate capacitor. Here, we use Ox = 300 nm, a relative permittivity of SiO2 = 3.9 and a vacuum permittivity 0 = 8.854 F/m.
Ideally, the transconductance is determined in the linear regime of the transfer curve.
However, with our GNR network FETs, a linear regime is not reached. Therefore, we use a linear approximation of the curve in the range ≤ −35 V. In the linear regime of transfer curves, the channel current is usually much larger than in the subthreshold regime and therefore the transconductance, which we extract in this way leads to a systematic underestimation of . Hence, the values given for are lower bounds. However, the contact resistance-free mobilities are of the same magnitude showing that the systematic error in the field-effect mobility is small and hence these values represent a good approximation of the charge carrier mobility in the devices.
With nuclear tunneling-assisted hopping as the dominant charge transport mechanism, we can further rationalize the values for the field-effect mobility. Although a degradation of the charge carrier mobility with larger band gaps is expected, using terahertz spectroscopy, values for the mobility in the order of 10 2 cm 2 V -1 s -1 have been experimentally observed in 9-AGNR samples 1 . Hence, the mobility of individual GNRs can be much larger than the field effect mobilities determined in our charge transport experiments. On the other hand, when hopping is the dominant charge transport mechanism, mobilities in the range of 10 −1 cm 2 V -1 s -1 to 10 −4 cm 2 V -1 s -1 are typically observed 3 .

S4. Gate voltage-dependence of the contact resistance
The gate voltage dependent contact resistances were determined for a series of 5-AGNR devices by measuring charge transport at room temperature for various channel lengths and using the transmission line method 3 . The total device resistance consists of the channel resistance and the contact resistance, on = channel + . The channel resistance is a function of gate voltage and is proportional to the geometrical aspect ratio of the channel / , with the channel length and , the channel width. We use the Ohmic part of the I-V curves at low drain voltages to determine the device resistance on as the slope of a linear model via a least squares fit. The data points for each gate voltage were again linearly fitted (least squares fit) to extract the total contact resistance (source + drain) from the intercept of the fit curve with the ordinate-axis at zero channel length. In Fig. 2 (d)  To corroborate this further, we exemplify a correction for the charge carrier mobility and compare the corrected value to the as-measured data. With the help of the width-normalized inverse channel resistance = ( ⁄ ) −1 ⁄ at different gate voltages (Fig. S3) the contact resistance-corrected charge carrier mobility 3 = 1 Ox ⁄ ( ⁄ ) −1 = (3.4 ± 0.0.2) × dependence of the field-effect mobility originating from contact resistance at the source and drain electrodes is eliminated. However, for this set of devices extracted from the transfer curves ranges from 0.01 cm 2 V -1 s -1 to 0.03 cm 2 V -1 s -1 ( = 15 V ) showing that the difference is small and thus corroborating that the influence of the contacts is negligible. Figure S3: Contact resistance of metal/GNR interfaces. Width-normalized reciprocal slopes of the total resistance depending on gate voltage allowing for the determination of a contact resistance-corrected charge carrier mobility.

S5. Charge carrier density as a function of temperature
For the determination of the charge carrier density as we combine I-V curves ( Fig. 3 (a) in the main text) and transfer curves (Fig. S4)  The systematic error in the mobility, which we discuss above (S3), of course propagates to the charge carrier density. Nevertheless, the temperature dependence of the transfer curves is captured correctly in the field-effect mobility and therefore our results are robust against these systematic uncertainties. Figure S4. Temperature dependence of charge carrier density 5-AGNR network FETs. In (a), we present the temperature evolution of the transfer curves. The field-effect mobility is extracted from these curves and used to estimate the charge carrier density. As shown in (b), the charge carrier density is constant over a wide temperature range between 100 K and 260 K. Lines in are guides for the eye.

S6. Additional temperature dependent measurements
Additional to the charge transport data from the I-V curves shown in the main text, we performed a temperature sweep down to 5.6 K at fixed = 10 V and = 0 as shown in Fig. S5. These data are also included in the universal scaling shown in Fig. 3 (c) of the main text. We also tested our data against a variable range hopping (VRH) model as this is a competing hopping mechanism. However, the model failed to describe the experimental observation thus ruling out this model. Figure S5. Temperature dependence of the channel current at = 10 V and = 0.