Low-Frequency Noise in Graphene Tunnel Junctions

: Graphene tunnel junctions are a promising experimental platform for single molecule electronics and biosensing. Ultimately their noise properties will play a critical role in developing these applications. Here we report a study of electrical noise in graphene tunnel junctions fabricated through feedback-controlled electroburning. We observe random telegraph signals characterized by a Lorentzian noise spectrum at cryogenic temperatures (77 K) and a 1/ f noise spectrum at room temperature. To gain insight into the origin of these noise features, we introduce a theoretical model that couples a quantum mechanical tunnel barrier to one or more classical ﬂ uctuators. The ﬂ uctuators are identi ﬁ ed as charge traps in the underlying dielectric, which through random ﬂ uctuations in their occupation introduce time-dependent modulations in the electrostatic environment that shift the potential barrier of the junction. Analysis of the experimental results and the tight-binding model indicate that the random trap occupation is governed by Poisson statistics. In the 35 devices measured at room temperature, we observe a 20 − 60% time-dependent variance of the current, which can be attributed to a relative potential barrier shift of between 6% and 10%. In 10 devices measured at 77 K, we observe a 10% time-dependent variance of the current, which can be attributed to a relative potential barrier shift of between 3% and 4%. Our measurements reveal a high sensitivity of the graphene tunnel junctions to their local electrostatic environment, with observable features of intertrap Coulomb interactions in the distribution of current switching amplitudes. Details of the classical ﬂ uctuator model used in the tight binding models; voltage and current scaling behavior of tight binding model I; results for the tight binding model II; electroburning and tunneling I − V traces for graphene tunnel junctions; statistical distribution of width of electroburnt devices and estimation of the gap width ﬁ tting error; simulated ( Δ I n , Δ I n +1 ) cluster asymmetry; long-term stability of RTSs; further details on the electrostatic shift of tunneling I − V due to charge traps; distribution of 1/ f noise γ slopes as a function of tunneling current; noise characterization of the measurement system, including open-circuit and thermal noise measurements (PDF)

G raphene tunnel junctions provide a two-dimensional platform for probing individual molecules. Recent experiments have demonstrated charge transport through single molecules that were firmly anchored between a pair of graphene electrodes via π−π stacking 1−3 or covalent bonding. 4−8 Moreover, graphene tunnel junctions have been proposed as candidate systems for molecular sensing, in particular for sequencing DNA molecules as they translocate through the gap. 9 These devices rely on the unique material properties of graphene: its two-dimensional nature, zeroenergy bandgap, and semimetallic type conductance. 10 The same properties also make graphene unique in the context of low-frequency noise, 11 with both carrier fluctuations and mobility fluctuations 12−29 playing an important role. 30 Whether graphene retains its favorable noise properties when structured into an ∼1 nm wide nanogap becomes particularly pertinent for applications that require a large signal-to-noise ratio, such as DNA sequencing. 31−34 Low-frequency 1/f noise or "flicker" noise is ubiquitous in nanoscale electronic systems, leading to prominent current fluctuations in semiconductor devices, 35−39 tunnel junctions, 40−43 and nanopores. 44−49 While the physical mecha-nisms that generate these fluctuations may vary and are often not known, it is generally accepted that 1/f noise is the result of a distribution of nonidentical random telegraph signals (RTSs). 11,35,36,39,50 These RTSs each have a Lorentzian noise power spectral density, the superposition of which results in a 1/f power spectral density. The emergence of 1/f noise from a distribution of nonidentical fluctuators was first described by McWhorter 35,51 in the context of interface traps in metaloxide-semiconductor field-effect transistors (MOSFETs), where trapping and detrapping of charge results in fluctuations in the number of charge carriers in the semiconductor channel. 36,37,39 RTSs have been observed experimentally in carbon nanotubes and have been predicted in graphene nanoribbons. These RTSs originate from the sensitivity of carbon nanotubes and graphene nanoribbons to a limited number of fluctuators in a small contact area. 52,53 In micrometer-scale graphene channels, relatively low noise amplitudes have been reported comparable to those found in state-of-the-art silicon transistors. 19 When the width of a graphene nanoribbon is reduced below 100 nm, the noise can increase by 2−3 orders of magnitude. 54 Until now, RTSs have not been reported in graphene nanogaps. In the case of tunnel junctions, fluctuations in the electrostatic environment 55−57 and mechanical 58−61 instabilities will lead to noise in the tunnel current through modulation of the transmission function. 40,41,62 Here, we investigate the noise properties of nanometer-sized graphene tunnel junctions and present a theoretical description of RTSs and the emergence of 1/f noise, resulting from a quantum mechanical system coupled to either a single fluctuator or a distribution of classical fluctuators, respectively. Graphene tunnel junctions are fabricated using feedbackcontrolled electroburning (see Methods) and measured at room temperature and at 77 K. The current is sampled at 100 kHz with a low-pass filter with a cutoff frequency of 1 or 10 kHz. The mean current depends exponentially on the applied bias voltage and is well described by the Simmons model. 63 Fitting the I−V curves to the Simmons model yields an average gap size of ∼1.5 ± 0.2 nm (See Methods and the Supporting Information (SI) for further details concerning statistics of gap sizes and the method of their measurement), consistent with electroburnt gaps reported in earlier studies. 1,64,65 RESULTS AND DISCUSSION Current Fluctuations in Graphene Tunnel Junctions. Our devices consist of a graphene ribbon patterned on top of a pair of gold electrodes (see Figure 1A). The graphene ribbon has a 200 nm constriction, which allows for the localized electroburning of a tunnel junction between two parts of the graphene ribbon (see Figure 1B). Figure 1C and E shows typical current−time (I−t) traces measured for a graphene tunnel junction at room temperature and at 77 K, respectively. The room temperature I−t trace ( Figure 1C) shows characteristic flicker noise behavior, where, like the light of a flickering candle, the signal has a wandering baseline as the high frequency noise rides on a low frequency component. By contrast, the 77 K I−t trace ( Figure 1E) predominantly fluctuates between two levels, indicating that a single two-level fluctuator dominates the noise. The observed current fluctuations are also evident from the bimodal Gaussian distribution of current values ( Figure 1F) and can be measured for up to 6 h. (see the SI) A histogram of the room temperature current in graphene tunnel junctions ( Figure 1D) reveals a distinct log-normal distribution of the current values and gives a first hint at the physical mechanism behind the 1/f noise. A simplified formulation of the Simmons model gives the tunnel current 63 where n(E) is the carrier density and the probability that an electron can cross a tunnel barrier with width d and height φ is given by the WKB-approximation: If the number of charge carriers were to fluctuate according to a normal distribution, this would result in a normal distribution of the current values. However, if the barrier height or width fluctuates according to a normal distribution this results in the observed log-normal distribution of the current, due to the exponential dependence of the transmission function E ( ) . Noise Power Spectral Densities. By comparing the noise power spectral density (PSD) S I (f) of the tunnel junction at room temperature and at 77 K ( Figure 1G), we find that S I (f) at T = 293 K is well described by A f γ , whereas S I (f) at T = 77 K shows a distinct corner at f = 7.4 Hz superimposed onto a linear slope A f γ . Since the density of thermally activated fluctuators is typically not constant in space and activation energy, fluctuations can be dominated by a single fluctuator

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Article within a given spectral window when the temperature is sufficiently reduced. 36−38 The noise PSD of a single two-level fluctuator is given by 66,67 S f I f ( ) 2 4 (2 ) where ΔI is the change in the current induced by the fluctuator and τ the mean dwell time of the fluctuator. In the case of simple RTSs between up and down states, τ is an average value of the dwell time of the up (τ up ) and down (τ down ) level,  Figure 2C) shows a good separation between the current levels, which are centered at the mean values and can be fitted with a Gaussian distribution. If the fluctuations are thermally activated, the process follows an Arrhenius law τ −1 = τ 0 −1 e −E a /k B T , and reducing the temperature will decrease the corner frequency τ −1 . 35,36,39,69 By changing the temperature we therefore sample a different subset of the collection of nonidentical RTSs. The fact that we observe a single dominant RTS at 77 K indicates that at this temperature we are sampling a smaller number of RTSs. Similar temperature dependent behavior has previously been reported in metal-oxide-semiconductor devices, where it is attributed to the energy-dependent interface trap density in the oxide layer. [36][37][38][39]70 The dependence of the amplitude and dwell time of the RTS on applied voltage and mean current is presented in Figure 3.
The dwell time distribution shows no meaningful trend within the experimental error bars with increasing voltage ( Figure  3A). There is an approximately linear increase of the RTS amplitude ΔI with increasing mean tunneling current ( Figure  3B). This indicates that the tunneling current does not drive the observed fluctuations in conductance, but that these fluctuations exist independently of the current and the current is merely a readout method of the independent fluctuations. 39 The same approximately linear relationship for low voltages is obtained in the tight binding model presented below, where the environmental fluctuators driving the tunnel barrier are independent of the current or applied voltage (Figures 3B and  SI2).
To characterize the 1/f noise amplitude, we compare the normalized noise power spectral density S I ( f)/I 2 for 35 devices in Figure 4. The noise spectra recorded for several voltage values show that the 1/f noise profile is present independent of the applied voltage and increasing voltage does not induce Lorentzian noise spectrum at room temperature ( Figure 4A). We find that the exponent γ = 1 ± 0.2 ( Figure 4B) does not

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Article depend on the tunneling current ( Figure SI14). Deviations from a 1/f noise profile are typically attributed to variations in the distribution of the RTSs, 35,36,39,50 and the γ values obtained in our graphene tunnel junctions are in the same range as values obtained for silicon devices, 35−39 tunnel junctions, 40−43 and nanopores. 44−49 We also find that S I (f)/I 2 measured for the same device at different bias voltages remains unchanged, indicating that the noise is not driven by the current and that ΔI ∝ I.
More surprising are the values for the normalized noise amplitude, or pseudo-Hooge parameter, α = f S I (f)/I 2 , which ranges from log α = −3 to 0 ( Figure 4C). These values are 7−9 orders of magnitude larger than those reported in micrometersized graphene channels, 13,16,18,19,71,72 and 2 to 3 orders of magnitude higher than the normalized noise amplitude measured in graphene nanopores of comparable size to our tunnel junctions. 48,73 This may be attributed to the extreme sensitivity of the tunnel current (compared to for example the ionic current in nanopores) to environmental fluctuations. When we compare the noise characteristics of our devices to those reported for MOSFET-type device of similar dimensions we find that pseudo-Hooge parameters in silicon devices are at least 2 orders of magnitude lower, 74−78 which is likely due to the highly optimized semiconductor fabrication processes that minimize the number of interface traps in the oxide. 77,79 When we compare our devices to CNT transistors on thermally grown SiO 2 , 80−84 we find similar noise values to our devices. In the remainder of this work we shall present a theoretical model explaining the sensitivity of graphene tunnel junctions to fluctuations in their electrostatic environment, and identify the potential mechanisms for causing these fluctuations.
Tight Binding Model of a Tunnel Junction. One possible origin of the observed RTS and Lorentzian noise spectrum is the presence of charge traps distributed in the substrate underlying graphene tunnel junctions. By changing their charge state between empty and occupied, traps alter the electrostatic environment of the junction, which may lead to the shift of the potential barrier in the junction with respect to the Fermi level of graphene electrodes. To gauge the effect of fluctuations in the charge trap occupation on the current through the tunnel junction we employ a simple onedimensional Huckel tight binding model. The model consists of a quantum tunnel barrier driven by the classical environment. The tunnel barrier is modeled as a scattering region containing N quantum levels, connected to two semi-infinite electrodes ( Figure 5). The barrier is coupled to the classical fluctuating environment, which is represented by one or more generalized coordinates x i corresponding to charge traps. The modeled coupling between the quantum system and environmental classical system yields a simple linear ε ∼ x relationship.
The aim of the model is to understand how different parameters describing the classical environment affect the changes in tunneling signal and in particular to estimate the magnitude of potential barrier fluctuations which can give rise to the observed current features. We investigate two models representing four limiting cases. Model I describes the case where five quantum levels in the scattering region are driven synchronously {ε 1 = ε 2 = ... = ε 5 = ε} by the collective effect of Fluctuations in the Tunnel Barrier. In the tight binding model, the height of the resulting tunnel barrier u between two leads is the difference between the Fermi level (black dashed line in Figure 6A) and the mean value of the lowest eigenvalue of the scattering region, corresponding to the nearest

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Article transmission resonance (at 0.25 eV in Figure 6A). For the model Ia, because of the influence of the generalized environmental coordinate x 1 , the lowest eigenvalue E 1 fluctuates over time with a mean value u (blue dashed line in Figure 6B) and mean upper and lower values (black dashed lines in Figure 6B). Current Fluctuations in the Tight Binding Model. The I−t traces for models Ia ( Figure 7C) and IIa ( Figure SI5C) show a distinct RTS, in contrast to the I−t ftraces for the case of Ib ( Figure 7A) and IIb ( Figure SI5A), which have the characteristic wandering baseline associated with flicker noise. The current histograms for models Ia ( Figure 7D) and IIa ( Figure SI5D) contain two Gaussian peaks, while the histograms for models Ib ( Figure 7B) and IIb ( Figure SI5B) have the log-normal distribution that was observed in our room temperature experiments. The noise spectra for the single-trap models (Ia and IIa) have a Lorentzian frequency dependence and as more environmental fluctuators are activated in the models Ib and IIb, a 1/f noise spectrum emerges, corresponding to the thermal activation of multiple RTSs at room temperature. We find that the slope varies between 0.9−1.3, when tuning the tunnel barrier height u shown in blue dashed line in Figure 6B, which agrees with measured sample to sample variations (see more details in Figures SI3 and SI6).
The tight binding model also reproduces scaling features of the experimental data showing an exponential increase of the amplitude ΔI of RTSs as a function of bias voltage ( Figure  SI2A). This feature arises, because the Fermi level is located in the exponential tail of the transmission coefficient E ( ) , which is controlled by the lowest eigenvalue. The model also shows the linear increase of the amplitude ΔI as a function of the mean current in agreement with the experimental data ( Figure  3B). The slope of the ΔI ∼ I dependence is 0.1 which is also in qualitative agreement with the experimental results. This qualitative agreement corresponds to a relative barrier-height fluctuation of 0.028 u u = Δ for the model Ia and 0.035 u u = Δ for the model IIa. Experimentally, potential shifts of this order can be induced by switching of an electron from a charge trap located at distance of a few nm from the junction to another one that is a few nm further away (details in the SI, Figure  SI13). The tight binding model also shows that five traps controlling transport through the tunnel junction are sufficient to produce 1/f noise over a four-decade frequency range, consistent with other reports. 50 The room temperature models Ib and IIb also show that the normal distribution of the potential shifts Δu results in the log-normal current distribution. The width Δu = ⟨(E 1 − u) 2 ⟩ 1/2 of the modeled potential distributions is equal to 0.057 . In Figure 8B, we plot, for example, the cumulative net potential as a function of radius R for nine randomly chosen dipole lattice distributions (with different random orientations Θ i of dipoles at a given lattice node). Only the dipoles nearest to the junction significantly affect the potential. Charge traps at large distances R > 400 nm do not induce large changes in the net potential, due to the decreasing contribution from each dipole and the increasing number of randomly oriented dipoles. Therefore, the potential value summed for all traps with r ≤ 1000 nm is taken as the final potential value.
In order to simulate the dynamic behavior of the charge traps, we simulated an ensemble of 2000 independent charge trap dipole lattices, such as the one presented in Figure 8A, assuming that differences between the obtained net voltage, resulting from all the traps at distance r ≤ 1000 nm, correspond to variability in potential barrier measured in experiments. 86 In Figure 8C we show the resulting distribution of the potential values at the center of the graphene tunnel junction. The distribution can be fitted with a Gaussian function with the standard deviation σ, σe = Δφ = 30 meV, and The estimated potential shifts are calculated assuming that there is a single point of junction sensitive to the electrostatic environment. Although tight binding models I and II are both capable of reproducing the main characteristics of current measurements at both cryogenic and room temperature (Figure 7), comparing small tunneling distance (1−2 nm) to relatively large intertrap spacing (∼10 nm), we regard model I as more realistic.
Charge traps are distributed also over the entire graphene− substrate interface, but only those traps located in the vicinity of the junction exert a sizable shift of the tunneling barrier. Traps located away from the junction, under the graphene leads, can still influence the conductance of the device by locally changing the density of states of carriers or their mobility. 86,87 However, the effect of traps located under wider regions of graphene electrodes is limited, because these traps are not synchronized and switching of each of them gates only a small fragment of the graphene electrode, while there are many more parallel conduction paths. 88 The same argument holds for fluctuations resulting from the electromigration of metal atoms at the gold-graphene interface: 16,89 the contact resistance is only a fraction of resistance of the tunnel junction, such that the contribution of contact resistance fluctuations will be negligible. The large distance from the metal contacts to the tunnel junction (2 μm) will also prevent metal atoms from migrating to the junction. Therefore, we conclude that the tunnel barrier in the junction remains the area of the device that is most sensitive to changes in the electrostatic environment. This highly localized sensitivity can be harnessed for molecular sensing applications. One example of high sensitivity of the investigated devices is the analysis of charge trap interactions in the vicinity of tunnel junction.
Charge Trap Interactions. Until now we have treated the RTSs as a purely stochastic process, with the independent dwell time values for consecutive current levels governed by Poisson statistics and random values of the switching current amplitude distributed according to a Gaussian distribution. However, it is known from single molecule measurements that the analysis of correlations in current values can reveal more details of a transport mechanism than a simple analysis of current traces. 90,91 The correlation in RTSs in a graphene tunnel junction is evident from correlation diagrams showing the amplitude of n+1 transition as a function of n transition (ΔI n ,ΔI n+1 ). The RTS data takes the form of two main point clusters (Figure 9) corresponding to a down → up transition sequence (ΔI down ,ΔI up ) ( Figure 9A) and up → down  Experimentally measured distributions of pairs of (ΔI 1 ,ΔI 2 ) points with overlaid bivariate Gaussian distribution fits. There is higher asymmetry in the distribution of (C) (ΔI down ,ΔI up ) events than of (D) (ΔI up ,ΔI down ) events. (E) Diagram showing schematically how occupation of different empty traps (white dots) with a charge carrier (red dot) leads to the different current levels and results in the broadening of a current distribution for the low conductance state.

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Article (ΔI up ,ΔI down ) sequence ( Figure 9B). In the case of a single independent trap governing the transport the absolute values of the step amplitudes should be equal, |ΔI up | = |ΔI down | resulting in symmetric circular distributions of points. There is, however, a sizable asymmetry in the (ΔI down ,ΔI up ) distribution ( Figure 9C) compared to the (ΔI up ,ΔI down ) distribution ( Figure 9D), which can be explained assuming that charge traps experience Coulomb interactions from their environment, that is other traps. 40 If a trap is occupied, it prevents occupation of neighboring traps through Coulomb repulsion, however the neighboring traps might be energetically equivalent and thus any of them can be filled by a charge carrier. Occupation of different traps leads to a slightly different current level in the down state ( Figure 9E). In contrast there is only one configuration for the up state, corresponding to the narrower distribution of possible current values. The asymmetry of the (ΔI n ,ΔI n+1 ) distribution can be reproduced by assuming the Gaussian distribution of the possible current values for both up and down states with the higher standard deviation of the latter distribution (SI).

CONCLUSIONS
We have demonstrated the presence of RTSs and a Lorentzian noise spectrum in graphene devices. The switching process leading to RTSs is not generated by the tunneling current, which serves only as a readout mechanism, as is evident from the constant relative current step amplitude ΔI/I ∼ 0.1. The capability of detecting single switching events shows high sensitivity of the graphene tunnel junctions to the local environment, which allows us to envisage highly sensitive graphene tunnel junction biosensors. The high sensitivity leads however to high noise levels.
The observed switching features can be explained by the gating of the tunnel barrier by charge carriers switching between oxide charge traps. Correlations in the amplitude of switching event pairs (ΔI n ,ΔI n+1 ) suggest the presence of Coulomb repulsion between traps, allowing only a single trap in the vicinity of the junction to be occupied.
At cryogenic temperatures only single traps are available, whereas at elevated temperatures more thermally excited traps can take part in switching. I−t traces affected by these traps have a wandering line and log-normal current distribution due to the normal distribution of potential barrier heights. The superposition of Lorentzian spectra with different characteristic frequencies leads to the observation of 1/f noise spectrum.
Our tight binding model reproduces qualitatively all the features of observed RTSs at cryogenic temperature and 1/f noise at room temperature. The model assumes that the fluctuations are caused by the interaction of the quantum tunnel barrier with a classical environment. Our first model assumes that all quantum levels in the scattering region are driven collectively due to the averaged effect of all traps buried deeper in the oxide. Our second model assumes an individual interaction of the quantum levels in the scattering region with individual traps, which corresponds to the traps located close to the barrier. Both of the tight binding models lead to results which are consistent with the experimental measurements, indicating that in the measured graphene tunnel junctions both of the individual and collective models might be observed.
Our numerical model calculates the potential shift Δφ and resulting current fluctuations amplitude ΔI due to the net effect of the traps in the substrate, assuming their constant density and dipole-type interactions. Agreement between the parameters related to current and potential shift obtained from experimental data, tight binding model, and numerical model supports attributing the noise in graphene tunnel junctions to charge traps.

METHODS
Fabrication of Graphene Devices. CVD-grown graphene, whose synthesis procedure has been previously described in ref 92, is transferred into p-doped Si wafers with 300 nm SiO layer and patterned 10 nm Cr/70 nm Au electrodes. Graphene is patterned into 200 nm wide constrictions using a combination of electron-beam lithography (JEOL 5500FS) with a negative resist ma-N 2405 and oxygen plasma etching.
Electroburning of Tunnel Junctions. Devices are contacted using automated probe station. The formation of tunnel junctions is achieved by feedback-controlled electroburning of graphene constrictions. Electroburning relies on the application of bias to the constriction with the simultaneous measurement of current ( Figure  SI8A for electroburning traces). The bias is increased at low constant rate of 750 mV s −1 resulting in initial linear increase of the current; at some point further increase of the voltage leads to the decrease of the slope of I−V curve and consequent decrease of current. This point marks the onset of electroburning due to the removal of carbon atoms caused by the high temperature in the constriction due to the joule heating. Once the current drop is detected the feedback loop decreases the voltage to zero at a high rate of 225 V s −1 to prevent the uncontrolled breakdown of the constriction. This electroburning cycle is repeated multiple times for each device, with increased resistance after each iteration, verified by the I−V measurement. The process is stopped at 500 MΩ resistance, which corresponds to the formation of a tunnel junction. The tunneling regime is confirmed by the measurement of a nonlinear I−V ( Figure SI8B).
Determination of Tunnelling Distance. The nonlinear I−V curves obtained for successfully burned graphene devices are subsequently used to estimate the tunneling distance, which is achieved by fitting the I−V curves to a nonlinear Simmons model, assuming tunneling process through an asymmetric potential barrier. 1,63 The fitting model is implemented in a form of iterative script which calculates current values for given voltage range, using as fitting parameters the width, height, and asymmetry factor of the potential barrier, with tunneling barrier width corresponding to the size of tunnel gap. Details of the implementation of fitting with Simmons model are given in the SI, as well as statistical distribution of fitted tunneling gap widths and estimation of the fitting error. An example of measured I−V curve and fitted Simons curve is also presented in Figure SI8B.
Electric Measurements. Devices with features of tunneling current and the tunneling distance obtained from the Simmons fit on the order of 1−2 nm were used for further measurements. Devices were measured in a custom-built cryogenic liquid dipper, which was vacuum pumped to the pressure of 10 −4 mbar and dipped in liquid nitrogen to obtain temperature of 77 K. Devices at room temperature were measured both in vacuum and ambient atmosphere, without any difference in the current signal or noise. Room temperature measurements were also performed in the same dipper, which also screens external electric fields. All measured devices were connected to Axopatch 200B voltage clamp amplifier which offers unrivalled noise performance among other commercial discrete electronic measurement systems. 93 The graphene devices were connected through the Axopatch headstage preamplifier, which was kept in a Faraday box to minimize the external noise contributions. The length of wires connecting the head stage and dipper was kept to minimum (∼10−20 cm) to minimize the noise pick-up and capacitance of the wires. The Axopatch 200B was operated in a voltage clamp mode and was used to bias the devices. The measured current was recorded and applied voltage controlled through Digidata 1440A acquisition card. A Bessel filter with 1 or 10 kHz filter frequency was applied to the signal and current was sampled at 100 kHz frequency. Noise spectra were calculated on the basis of Fourier transform of I−t traces. Recorded

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Article traces were divided into 10 sections, and noise spectrum was calculated for each of the sections individually. The spectra shown in this Article are an average of 10 noise spectra. In order to characterize the intrinsic noise level of the measurement system and prevent any instrumentation artifact, we characterized also open circuit noise level and thermal noise recorded in resistors ( Figure SI15) ASSOCIATED CONTENT

* S Supporting Information
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.8b04713.
Details of the classical fluctuator model used in the tight binding models; voltage and current scaling behavior of tight binding model I; results for the tight binding model II; electroburning and tunneling I−V traces for graphene tunnel junctions; statistical distribution of width of electroburnt devices and estimation of the gap width fitting error; simulated (ΔI n ,ΔI n+1 ) cluster asymmetry; long-term stability of RTSs; further details on the electrostatic shift of tunneling I−V due to charge traps; distribution of 1/f noise γ slopes as a function of tunneling current; noise characterization of the measurement system, including open-circuit and thermal noise measurements (PDF)