# Noise diagnostics of graphene interconnects for atomic-scale electronics

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### System preparation by gradual electrical breakdown

Our units encompass bowtie-shaped CVD graphene sheets25 terminated by two steel electrodes (see schematic in Fig. 1a). The constriction on the narrowest cross-section (Fig. 1b) is damaged by a feedback-controlled electrobreakdown process in air at room temperature. The ensuing nanometer-sized hole is seen as a darkish grey line within the SEM picture proven in Fig. 1c. Electrical breakdown is induced by periodically utilized 200 μs lengthy voltage pulses, whose amplitude (VPulse) is consecutively elevated by 1 mV. In the meantime the present (IPulse) is monitored (see Fig. 1d). To keep away from the abrupt breakdown, we cease the voltage ramp when the present decreases by a sure share in comparison with the utmost present inside the previous 200 mV voltage interval (see Supplementary Observe 1 for additional particulars on this protocol). Repeating the voltage ramp cycles after every suggestions occasion we will steadily improve the resistance till the specified nanogap is established. If the suggestions occasion just isn’t accompanied by a transparent junction resistance improve, the brink suggestions share is elevated within the subsequent voltage ramp cycle.

For instance the gradual narrowing of the contact we plot the constriction resistance (RC = (VBias/I) − RLead) as a perform of the voltage ramp cycle quantity in Fig. 1e (see additional examples for the evolution of RC alongside the EB cycles within the Supporting Info). Because the preliminary resistance of the constriction is negligible, the lead resistance (RLead) is estimated by the bottom resistance worth throughout the EB process. Sometimes the bottom resistance doesn’t belong to the primary cycle because of the present annealing of the graphene sheet22 (see Supplementary Observe 2 for extra particulars on this phenomenon). Surprisingly, after a properly controllable gradual narrowing interval, the constriction resistance displays a plateau within the regime of the G0 = 2e2/h conductance quantum unit. We declare that this plateau just isn’t a conductance quantization function, somewhat the resistance noise of the junction quickly will increase and triggers false suggestions occasions because the quantum conductance is approached and surpassed. Certainly, the magnified remaining sections of consultant voltage ramp cycles display the numerous noise improve (see the darkish pink, pink and yellow curves in comparison with the brown hint in Fig. 1f). Accordingly, the brink suggestions share must be considerably elevated to proceed with {the electrical} breakdown (see Fig. 1e high panel).

### Noise traits of graphene nanojunctions and nanogaps

The elevated noise ranges demonstrated in Fig. 1f are per prior research reporting excessive present fluctuations because the width of the graphene constriction is diminished under 100 nm.26,27,28 Vital noise ranges are additional maintained in graphene tunnel junctions29 which was attributed to tunnel barrier-height fluctuations within the nanogap area. Within the following, we research this noise enhancement intimately with the aim of understanding the underlying noise producing atomic processes. Thereby we purpose at proposing an optimized protocol for {the electrical} breakdown. First, (i) we carry out noise measurements on a statistically related variety of junctions to map the overall evolution of the ΔI/I relative present fluctuations because the constriction resistance is elevated alongside the EB steps. Subsequent, (ii) we resolve the more and more dominant single atomic fluctuators alongside the EB course of by decomposing the noise spectra to the 1/f-type background of many fluctuators and the Lorentzian contribution arising from single close by fluctuators. As well as, (iii) we determine the distinct, properly separable regimes of the EB course of by additionally evaluating the gate response and I(V) nonlinearity. Afterward, (iv) we quantitatively analyze the noise evolution when it comes to easy fashions considering single-atomic junction-width fluctuations within the nanometer-sized junction regime, and subatomic gap-size fluctuations within the nanogap regime. Lastly, (v) we apply nonlinear noise spectroscopy on the nanogap regime to underpin our gap-size modulation mannequin and to exclude the barrier-height fluctuation mannequin.

To realize a complete electrical characterization of the junction evolution alongside the EB protocol we frequently cease the EB course of to report the SI present noise energy spectral density (Fig. 2a–d), the present–voltage traits (Fig. 2e–h) and the gate response traits (Fig. 2i–l) of the particular junction (see Supplementary Observe 3 for extra particulars on the noise measurement protocol). We now have analyzed a statistical ensemble of EB processes. Throughout every course of we have now recorded the above traits at ≈10 distinct constriction resistance values.

We discovered that the noise spectra both observe a transparent 1/f-type frequency dependence (see Fig. 2a, b), or a kink is superimposed on the general 1/f-type background (see Fig. 2c, d). We attribute the emergence of such kinks to a single dominant atomic fluctuator situated within the junction area,30 whereas an ensemble of distant fluctuators consequence within the extra generic 1/f-type background.31 A single fluctuator with a well-defined τ time fixed yields to a Lorentzian-type noise spectrum which is fixed at low frequency and follows a 1/f2 dependence at fτ−1. In distinction, the Lorentzian spectra of an ensemble of distant fluctuators sum as much as a 1/f-type envelope. Accordingly, we match all noise spectra with the

$${S}_{I}(f)=beta cdot {left(f/{f}_{0}proper)}^{gamma }+frac{alpha tau }{1+{(2pi ftau )}^{2}}$$

(1)

system, the place f0=1 Hz is the reference frequency, β is the magnitude and γ ≈−1 is the exponent of the 1/f-type noise, α is the low-frequency amplitude of the Lorentzian noise and τ is its time fixed. Determine 2c exemplifies the fitted 1/f-type (blue) and Lorentzian (yellow) noise contributions along with their superposition (black line). From the fitted spectra we calculate the built-in imply squared present deviation, ({({{Delta }}{I}_{{rm{Complete}}})}^{2}=int {S}_{I}(f)df) for the 50 Hz–30 kHz frequency band (see the blue traces in Fig. 2a), from which the relative ΔIComplete/I present fluctuation is obtained. Alternatively, we will combine the ({({{Delta }}{I}_{{rm{Lorentzian}}})}^{2}) imply squared present deviation arising from the Lorentzian noise contribution, and quantify the relative noise contribution of the only dominant fluctuator as ({({{Delta }}{I}_{{rm{Lorentzian}}})}^{2}/{({{Delta }}{I}_{{rm{Complete}}})}^{2}).

The I(V) curves are properly fitted with an ICubic = ILin + bV3 cubic perform, the place ILin = RLinV is the linear a part of the I(V) curve as decided by its low-bias slope, RLin. We measure the diploma of nonlinearity with (ICubic − ILin)/ILin, which we consider at VBias = 300 mV (see Fig. 2h). The gate response curves measured within the VGate= ± 40 V gate voltage window are characterised by their (({R}_{{rm{Max}}}-{R}_{{rm{Min}}})/{R}_{{rm{Min}}}) values (see Fig. 2j).

Determine 3 reveals the evolution of the 4, above outlined attribute portions, the relative present fluctuation (a), the relative Lorentzian noise contribution (b), the present–voltage non-linearity (c) and the relative gate response amplitude (d) as a perform of the constriction resistance. The statistics are created from the electroburning of 9 graphene nanocontacts fabricated on SiO2 substrates (grey symbols) and an equivalent management machine using a Si3N4 substrate (blue symbols). The evolution of those 4 attribute portions highlights three clearly separated regimes which might be established throughout the EB process.

Within the first, low resistance regime (RC < 3  103 Ω, pink background) the relative present fluctuation stays at its low preliminary worth of 10−3 (Fig. 3a), whereas the Lorentzian kind noise has negligible contribution (Fig. 3b). The present–voltage traces present ohmic habits with negligible nonlinearity (Fig. 3c). The relative gate response amplitudes have lowering tendency, which will be attributed to the shift of the gate response curves to larger resistance, whereas ({R}_{{rm{Max}}}-{R}_{{rm{Min}}}) along with the general form of the gate response curve stays roughly unchanged (see the person curves in Fig. 2i, j). Afterward, on the identical resistance regime the place {the electrical} breakdown turns into much less controllable (3  103 Ω < RC < 6  105 Ω, orange background) the whole present noise quickly will increase by greater than two orders of magnitude. In the meantime, the ratio of the Lorentzian noise considerably enhances, in lots of junctions it exceeds 50%. In accordance with the noise measurements, each the I(V) non-linearity and the relative gate response amplitude improve greater than two orders of magnitude. Lastly, within the excessive resistance regime (RC > 6  105 Ω, yellow background) the relative present fluctuation stays at excessive saturated degree (ΔI/I ≈ 0.1−1) exhibiting excessive machine to machine variation. The Lorentzian noise reduces in common, but it surely nonetheless has appreciable contribution (see the black line for the imply Lorentzian contribution in Fig. 3b). The I(V) traces present a saturated nonlinearity, whereas the relative gate response amplitude decreases to a degree, which is per the general resistance noise within the yellow regime. Whatever the substrate materials all graphene samples present the identical options throughout the electrical characterizations (grey and blue factors). This discovering demonstrates that the substrate has a major impact neither on the breakdown course of24 nor on {the electrical} noise.26

We emphasize, that the ample data of the overall noise evolution alongside the EB cycles permits the optimization of the EB protocol: by consequently setting the brink suggestions share in response to the typical noise degree on the precise junction resistance, false or uncontrolled suggestions occasions will be effectively eradicated. This adaptive management of {the electrical} breakdown course of is demonstrated in Supplementary Observe 4.

Earlier than decoding the above resistance regimes we estimate the related size scales within the system. To this finish, we analyze the gate response curve in Fig. 2i, which was measured after the present annealing, however the junction was nonetheless huge sufficient to estimate the λF ≈ 30 nm Fermi wavelength and the l ≈ 31 nm imply free path in response to widespread bulk concerns (see Supplementary Observe 5 for extra particulars). Because the junction narrows towards the mesoscopic regime bulk portions like conductivity and mobility fail to explain the system. As an alternative, the resistance of every junction configuration turns into delicate to the place of particular person scattering facilities in addition to the superb particulars of quantum interference phenomena27,32 and edge termination options.33,34,35 Our samples are thought-about as disordered junctions, the place neither the situation for ballistic transport nor the presence of a well-defined edge-termination is happy, i.e., the conductances are hardly described by easy theoretical concerns. As an alternative, we estimate the width of our junctions counting on transmission and scanning electron microscopy measurements34,36,37 delivering empirical relations between the measured junction width and conductance values. These research have proven that the resistance of graphene nanojunctions will increase inversely proportionally to the junction width all the best way from the 100 nm scale to contacts with 1−2 nm width. Consequently, the constriction resistance throughout the entire narrowing course of will be approximated as RC = ρW−1, the place ρ ≈ 172 kΩ  nm is an empirical fixed for brief single-layer graphene junctions36 just like our buildings, and W is the width in nanometers. Based mostly on this system, we estimate a constriction width of W ≈ 57 nm on the border between the pink and orange regimes, whereas W ≈ 0.29 nm is estimated on the border between the orange and yellow regimes. Observe, that the latter quantity is past the vary of the transmission electron microscopy measurements, i.e., it’s thought-about as a tough estimate reflecting subnanometer-sized, atomic-scale junctions.

Based mostly on these concerns, we argue that the crossover between the pink and orange regimes happens the place atomic-scale junction width fluctuations begin to dominate over the noise generated within the leads. Alongside the orange regime, the graphene junction narrows towards the last word atomic width, which is mirrored within the fast improve of the relative present fluctuations. Lastly, across the border between the orange and yellow regimes the atomic-scale graphene junction breaks, and a tunneling nanogap types. Within the latter (yellow) regime the noise is both associated to gap-size variations or barrier peak fluctuations. The corresponding states of the junction evolution are illustrated in Fig. 3e.

We argue that electronically clear, brief graphene junctions are solely out there as much as the quantum resistance, ({G}_{0}^{-1}approx 12.9)kΩ, which is said to a ≈13 nm huge junction in response to the above width estimate system. At larger resistances, the junctions begin to act as limitations, which results in the emergence of nonlinear options within the I(V) curve. Moreover, relying on the precise junction association, resonant transport options may come up within the gate response.26,32 Observe, that whereas the relative present fluctuations and the relative Lorentzian contributions already begin to improve within the okayΩ regime (see Fig. 3a, b), pronounced I(V) nonlinearity and intense gate response are hallmarks of the ({R}_{{rm{C}}}>{G}_{0}^{-1}) resistance regime (Fig. 3c, d).

### Mannequin evaluation of the noise traits

To validate our intuitive conceptions concerning the dominant noise sources, we flip to the mannequin evaluation of the above situations. For this goal we quantitatively analyze the tendencies of the relative present fluctuations for a selected machine (see black dots in Fig. 4a).

First, we concentrate on the orange nanojunction regime. Contemplating ρ and W fluctuations within the above empirical resistance estimate system, the relative present fluctuation is expressed as ((Delta I/I)^{2}=(Delta W {cdot} {R}_{{rm{C}}}/rho )^{2}+(Delta rho /rho)^{2}). The second time period is unbiased of the width, and due to this fact it may well solely contribute to the baseline of the noise degree (≈10−3). Due to this fact, within the orange regime, we attribute the noise improve to the primary time period which means a straight line with a unity slope on the log(ΔI/I)−log(RC) plot, whose intercept corresponds to ΔW/ρ. By becoming a straight line to our knowledge (see the blue line in Fig. 4a) we acquire a width fluctuation of ΔW = 0.27 nm. The dashed blue traces display the calculated noise degree utilizing a doubled/halved ΔW fluctuation. This evaluation confirms that the efficient width of the junction fluctuates on the size of the lattice fixed. This atomic-scale junction-width fluctuation accounts for the noticed two orders of magnitude improve of the relative noise because the junction width decreases from a couple of tens of nanometers to the atomic regime. The image of atomic-scale fluctuations can be supported by the dominance of the Lorentzian contribution, i.e., the dominance of single fluctuators within the spectrum (Fig. 3b). We observe, that this efficient width fluctuation just isn’t essentially associated to the direct width-fluctuation of the graphene lattice, however additional options, like edge termination fluctuations38 or cost trapping/de-trapping on the disordered graphene edges26,39 may yield an efficient width fluctuation.

Subsequent, we analyze the resistance dependence of ΔI/I within the yellow, nanogap regime. Prior research have proven13,21,22,24,29,32 that the transport by means of graphene nanogaps is properly described by the Simmons mannequin40 counting on two key parameters, the Φ peak and the d size of the tunneling barrier between the 2 sides of the nanogap. Right here, we additionally describe the I(V) traits when it comes to a numerically evaluated Simmons mannequin (see Supplementary Observe 6) contemplating a rounded rectangular potential barrier (see inset of Fig. 4c). The rounded barrier ensures the graceful transition from the direct tunneling to the sector electron emission regime, which is crucial in our later evaluation of nonlinear noise phenomena. We measure the I(V) attribute after every EB step within the yellow resistance regime and decide the d and Φ parameters by Simmons becoming (see inset of Fig. 4a). This evaluation reveals that Φ stays roughly fixed ((overline{{{Phi }}}) = 0.31 ± 0.1 eV) in the complete yellow regime, whereas the rise of the d barrier size is proportional to the logarithm of the resistance.

We emphasize, that within the framework of the Simmons mannequin, a very totally different resistance dependence of ΔI/I is obtained if solely barrier-length fluctuations or solely barrier-height fluctuations are thought-about because the dominant noise supply (see Supplementary Observe 6). That is demonstrated by the pink (brown) dots in Fig. 4a, the place ΔI/I is calculated by inserting the fitted d and Φ values of the particular junction to the Simmons mannequin, and contemplating a continuing Δd gap-size fluctuation (pink) or a continuing ΔΦ barrier-height fluctuation (brown) inside the complete yellow regime. The plotted pink (brown) dots signify the Δd = 0.012 nm (ΔΦ = 0.042 eV) values, for which the calculated resistance dependence matches the measured resistance dependence of ΔI/I the very best. The barrier-height fluctuation mannequin yields a comparable relative fluctuation (ΔΦ/Φ ≈ 0.13) as a beforehand utilized tight-binding mannequin29Φ/Φ ≈ 0.057). The latter mannequin efficiently accounted for the noticed temperature and frequency dependence of the noise in graphene nanogaps, nonetheless, our evaluation demonstrates that the measured resistance dependence of ΔI/I is totally inconsistent with the barrier-height fluctuation mannequin (brown dots). In distinction, a ten pm scale gap-size fluctuation (pink dots) i.e., subatomic fluctuations of the graphene nanogap edges properly describe the noticed resistance dependence of the relative present fluctuations.

Lastly, we apply nonlinear noise spectroscopy to additional assist the clear dominance of gap-size fluctuations over barrier-height fluctuations within the noise traits of graphene nanogaps. Up to now all offered noise measurements have been restricted to sufficiently low voltage driving, the place the linearity of the I(V) curve is happy. On this linear regime the relative resistance and present noise values are equal, ΔR/R = ΔI/I, i.e., steady-state resistance fluctuations yield voltage-independent ΔI/I values. Nevertheless, within the nonlinear transport regime ΔI/I additionally displays a definite voltage dependence. Moreover, the nonlinearity of the I(V) curve converts to a very totally different voltage dependence of ΔI/I, when purely gap-size fluctuations or purely barrier-height fluctuations are thought-about.

Determine 4c reveals the voltage dependence of ΔI/I for a consultant nanogap junction (see the black/grey crosses similar to the upward/downward voltage sweep), whereas the I(V) curve of the identical junction is proven in Fig. 4b (black dots). From the latter, the Simmons becoming (pink curve) reveals the barrier parameters (Φ = 0.35 eV, d = 0.91 nm). Utilizing these parameters we decide the amplitude of pure gap-size fluctuations (Δd = 0.054 nm) and pure barrier-height fluctuations (ΔΦ = 0.057 eV) in such a approach that the relative present fluctuations calculated from the Simmons mannequin reproduce the measured fixed low-bias ΔI/I values. The measured noise knowledge display, that the voltage independence of ΔI/I holds as much as 300 mV. At larger voltage, the relative present fluctuations markedly lower with the voltage. In keeping with our earlier findings, the gap-size fluctuation mannequin (pink line) reproduces the measured noise dependence. In sharp distinction, the barrier-height fluctuation mannequin (brown curve) is absolutely inconsistent with the measured noise knowledge. On this case, the calculated voltage dependence of ΔI/I displays a peak within the voltage regime similar to the barrier peak. At this bias voltage, the profile of the potential barrier modifications from trapezoidal to triangular and the present turns into extra delicate to barrier peak fluctuations.

Within the above evaluation the d = 0.91 nm size of the tunneling barrier is taken into account as an efficient hole dimension. Because of the exponential dependence of the present on the size of the barrier we anticipate that this efficient size is near the precise dimension of the nanogap on the narrowest spot, and the ≈30% Lorentzian contribution to the noise (Fig. 3b) is said to a single fluctuator close to the narrowest spot. Nevertheless, {the electrical} breakdown is taken into account as a self-limiting course of, i.e., as soon as a phase is disconnected, it doesn’t widen additional. This implies, that the hole size stays within the vary of the few nanometers alongside the complete width of the graphene stripe.22 Accordingly, it’s conceivable that additional segments of the nanogap additionally contribute to the present, and the fluctuators round these segments contribute to the 1/f-type background noise. The latter background noise may relate to the diffusion of adatoms across the narrowest spot missing a well-defined time-scale, equally to the sub-Ångström gap-size fluctuations in atomic-sized gold tunnel junctions.41 In distinction, the thermal bond fluctuations in response to the Bose-Einstein distribution of the phonon modes should not thought-about as a supply of the 1/f-type background noise. The latter concerns are mentioned additional in Supporting Observe 6.

### Conclusions

In conclusion, we have now demonstrated that the superior evaluation of low-frequency noise measurements provides a wealthy supply of knowledge on the managed electrobreakdown means of graphene nanojunctions. Following the evolution of the relative noise amplitudes we have been capable of optimize {the electrical} breakdown protocol and to determine distinct regimes of the nanogap formation course of. The latter clearly highlighted the crossover between unbroken atomic-scale junctions and nanogap units. Analyzing the frequency dependence of the noise, we’re capable of observe how single atomic fluctuators begin to dominate over a broader 1/f-type noise background as the last word atomic junction dimensions are approached. The numerical evaluation of the markedly totally different resistance dependence of the noise within the nanojunction and nanogap regimes allowed us to determine the microscopic noise producing mechanisms. Within the former case atomic-scale junction-width fluctuations, whereas within the latter subatomic gap-size fluctuations are recognized as the key noise sources. Lastly, we utilized the method of nonlinear noise spectroscopy demonstrating that the conversion of present–voltage nonlinearity to nonlinear noise phenomena is delicate to the microscopic origin of the dominating fluctuations. Utilizing this distinction we supplied an unambiguous proof that gap-size fluctuations are extremely dominating over barrier-height fluctuations in graphene nanogap units. In keeping with pioneering outcomes on MoS2 nanojunctions42,43 and the research of {the electrical} breakdown in MoS2 field-effect transistors44 we consider that the offered adaptive breakdown protocol and the scheme of the noise measurements are properly relevant in different sufficiently conducting 2D supplies as properly.