Current crowding mediated large contact noise in graphene field-effect transistors

The impact of the intrinsic time-dependent fluctuations in the electrical resistance at the graphene–metal interface or the contact noise, on the performance of graphene field-effect transistors, can be as adverse as the contact resistance itself, but remains largely unexplored. Here we have investigated the contact noise in graphene field-effect transistors of varying device geometry and contact configuration, with carrier mobility ranging from 5,000 to 80,000 cm2 V−1 s−1. Our phenomenological model for contact noise because of current crowding in purely two-dimensional conductors confirms that the contacts dominate the measured resistance noise in all graphene field-effect transistors in the two-probe or invasive four-probe configurations, and surprisingly, also in nearly noninvasive four-probe (Hall bar) configuration in the high-mobility devices. The microscopic origin of contact noise is directly linked to the fluctuating electrostatic environment of the metal–channel interface, which could be generic to two-dimensional material-based electronic devices.

T he wide spectrum of layered two-dimensional (2D) materials provides the opportunity to create ultimately thin devices with functionalities that cannot be achieved with standard semiconductors. The simplest of such devices is the field-effect transistor (FET). There are several factors that determine the performance of an FET, the key among them being the dielectric environment, quality of the metalsemiconductor contact and the level of low-frequency 1/f noise. Over the past few years, there has been tremendous progress in creating high-mobility, atomically thin FETs through a combination of low-resistance ohmic contacts [1][2][3][4] and strategies for encapsulation 3 of the active channel. However, there exists no consensus on the factors that determine the magnitude of the 1/f noise, which is known to degrade the performance of amplifiers, or introduce phase noise/jitter in high-frequency oscillators and converters 5 . Noise is especially detrimental to the performance of nanoscale devices and may cause variability even in ballistic transistor channels, where it has been suggested to arise from slow fluctuations in the electrostatic environment of the metal-semiconductor contacts 6 Even for the widely studied graphene FET, it is still unclear what the dominant contribution is to the 1/f noise. Conflicting claims exist, where some studies attribute the 1/f noise in graphene transistors primarily to noise generated within the channel region [7][8][9] , whereas other investigations indicate a strong contribution from the contacts [10][11][12] . This distinction has remained elusive to existing studies 7-21 because of the lack of a microscopic understanding of how processes characteristic to the metal-graphene junctions, in particular the current-crowding effect [22][23][24][25][26][27] , has an impact on the nature and magnitude of 1/f noise.
Fundamentally, current crowding is an unavoidable consequence of resistivity mismatch at the metal-semiconductor junction, where the injection and/or scattering of charge carriers between the semiconductor and the metal contact is restricted only close to the edge of the contact, over the charge transfer length L T 22,28 . Photocurrent measurements [29][30][31] and Kelvin probe microscopy 32,33 at the graphene-metal interface have already indicated the presence of current crowding with L T B0. 1-1 mm (refs 24,25,34). Restricting the effective current injection area leads to greater impact of local disorder kinetics, and hence larger 1/f noise 35 . For graphene-metal interfaces, the scenario is more complex than a typical metal-semiconductor junction, since it is known that metals such as Cr, Pd and Ti react to form metal carbides with graphene, altering the structural properties and causing strong modifications in its energy band dispersion 34,36 . While it is clear that current crowding and the characteristics of the metal-graphene junction directly influence the contact resistance 25,29,34,[36][37][38][39][40][41][42] , how these factors have an impact on the noise originating at the contacts (contact noise) is still not known.
In this work we study a series of graphene FETs with different mobilities, substrates and contacting configurations to demonstrate that electrical noise at the metal-graphene junction can be the dominant source of 1/f noise in graphene FETs, especially for invasive contacting geometry, where the probe contacts lie directly in the path of the current flow. The contact noise was found to scale as R 4 c , where R c is the contact resistance, in all devices and at all temperatures. While the noise magnitude is determined by the fluctuating charge trap potential at the oxide substrate underneath the metal contacts, a simple phenomenological model unambiguously attributes the scaling to the current-crowding effect at the metal-graphene junction. In view of the recent observations of contact noise 43,44 and currentcrowding effect in molybdenum disulphide (MoS 2 ) and black phosphorus FETs 26,27 , many of the results and concepts developed in this paper can be extended to other members of 2D semiconductor family as well.

Results
Characterization of Au-contacted graphene. We first focus on a single-layer graphene channel on conventional (300 nm) SiO 2 /p þ þ -Si substrate, etched into a Hall bar shape with surface-contacted Au (99.999%) leads (Fig. 1a). Here we used pure gold contact (without a wetting underlayer of, for example, Cr or Pd) because gold (hole) dopes the graphene underneath without pinning the Fermi energy, or causing substantial modification in the bandstructure 45,46 . This allows easy tuning of the doping, and correspondingly the resistance, of the contact region with backgate voltage (V BG ). We measure the two-probe (R 2P ¼ R 23,23 ) and four-probe (R 4P ¼ R 23,14 ) resistance and noise as a function of V BG between the leads 2 and 3 (suffixes in R V þ V À ,I þ I À indicate the voltage (V þ , V À ) and current (I þ , I À ) leads). The V BG dependence of R 2P and R 4P is shown in Fig. 1b. R 4P shows a slightly asymmetric transfer characteristic, known to occur for asymmetric contact doping 38 , with a single Dirac point at V BG E6 V. R 2P , however, shows a second Dirac point at V BG E31 V because of the combination of hole doping and weak pinning by Au at the contact region 34,38,45,46 , which divides the transfer behaviour in three parts (p À p 0 , n À p and n À n 0 ), based on the sign of doping in the channel and contact regions. The position of the Fermi level at the two Dirac points is shown in the schematic of Fig. 1c. The observation of double Dirac point in R À V BG characteristics confirms the structural integrity and gate tunability of the Fermi level of the graphene channel underneath the contact.
Contact resistance with Au contacts and current crowding. The shift in the local chemical potential by metal contacts results in a change in resistance that contributes to contact resistance, and determines the extent of current crowding at the gold-graphene interface. To compute the contact resistance R c , we follow the Landauer approach where the net transmission probability T across the contact is determined by the interplay of the number of propagating modes in the channel and metal regions ( Fig. 1c) 38 . Figure 1d shows the V BG dependence of the experimental contact resistance R c ¼ R 2P À R 4P (corrected for the resistance of the small region of the probe arms) and that calculated assuming the Dirac-like dispersion and level broadening E80 meV underneath the contact and E57 meV in the channel (estimated from the experimental transfer characteristics, see Supplementary Note 2 for the full details of calculations). The agreement, both in V BG dependence and absolute magnitude (within 50% for all V BG ), indicates that the contact resistance is primarily composed of the resistance R T of the graphene layer over the charge transfer length (L T ) underneath the contacts. Owing to mismatch between the resistivities of the metal and graphene, L T is significantly smaller than the geometric width L c (B1-1.5 mm) of the metal lead, resulting in the current-crowding effect.
To visualize this quantitatively, we consider the transmission line model where the graphene layer below the contacts is represented with a network of resistors characterized by sheet resistivity r T (schematic in Fig. 1e). The potential profile in graphene under the metal is then given by 22 , where I is the current flowing, r c E200 O mm 2 (ref. 45) is the specific contact resistivity and is charge transfer length from the edge by which 1/e of the current is transferred to the metal contact (W is the contact width). Taking R T as the experimentally observed contact resistance R c (Fig. 1d), we calculated the potential drop underneath the contact, normalized to its value at the edge x ¼ 0, for three gate voltages marked by the arrows in Fig. 1d. The potential drops exponentially over the gate voltage-dependent scale L T , being minimum (B200 nm) for the second Dirac point at þ 31 V where the mismatch between the resistivity of the metal and that of the graphene layer underneath is maximum.
Since refs 22,24) and r T ¼ R T W/L T , it is evident that the contact noise is essentially the resistance fluctuations in the graphene layer underneath the contact, that where g T and n T are the phenomenological Hooge parameter and carrier density in the charge transfer region, respectively. g T is independent of n T and is determined by the kinetics of local disorder induced by trapped charges, chemical modifications and changes in the band dispersion due to hybridization. Assuming a diffusive transport in the charge transfer region with density-independent mobility 46 , the contact noise can be expressed as and implies a scaling relation DR c ð Þ 2 / R 4 c that can be readily verified experimentally. Note that (1) the scaling is different from that suggested for metal and three-dimensional (3D) semiconductors where the exponent of R c is E1 for interfacetype contacts or E3 for constriction-type contacts 35 . (2) Since n T is the only gate-tunable parameter 34,38,39 , the scaling of contact resistance and electrical noise can be dynamically monitored by varying the gate voltage, circumventing the necessity to examine multiple pairs of contacts to isolate the contact contribution to noise. (3) Although the absolute magnitude of the contact noise is device/contact-specific, the scaling of equation (2) is expected to hold irrespective of the geometry, material or chemical nature of the contact (wetting or non-wetting).
Noise measurement in Au-contacted graphene. Noise in both R 2P and R 4P at all V BG consists of random time-dependent fluctuations with power spectral density S R f ð Þ / 1=f a (Fig. 2a), where a % 1 indicates usual 1/f noise due to many independent fluctuators with wide distribution of characteristic switching rates. However, to estimate and compare the total noise magnitude, we have evaluated the 'variance' h DR ð Þ 2 i ¼ R S R f ð Þdf , by integrating S R (f) numerically over the experimental bandwidth. Figure 2b shows the DV BG dependence of h DR 2P ð Þ 2 i and h DR 4P ð Þ 2 i, where the maxima in both quantities align well with the Dirac points in R 2P and R c (Fig. 1b,e). The origin of the maximum in noise at the Dirac point is a debated topic, and has often been attributed to low screening ability of the graphene channel to fluctuating Coulomb potential at the channelsubstrate interface 7,8,15,16 . Here h DR 2P ð Þ 2 i peaks in the n-p region close to DV BG E25-30 V, where the density of states in the charge transfer region is low 34,38 , indicating contact noise that originates because of poorly screened fluctuations in the local Coulomb disorder. In fact, the noise magnitude at the second peak (DV BG E30 V) is B10 times larger than that at the main Dirac peak, indicating the significantly larger noise where the current crowding is most severe and contact resistance is the largest. Surprisingly, h DR 4P ð Þ 2 i shows a weak increase in this regime as well, suggesting a leakage of the noise at the contacts even in four-probe measurements (discussed in more detail in the context of Fig. 3c and in Supplementary Note 3).
To verify the contact origin of noise, we have plotted h DR 2P ð Þ 2 i as a function of contact resistance R c in Fig. 2c. Remarkably, h DR 2P ð Þ 2 i for all V BG collapses on a single trace, and varies as h DR 2P ð Þ 2 i / R 4 c over four decades of noise magnitude, suggesting that the measured noise in two-probe configuration originates almost entirely at the contacts, which is at least a factor of 10-100  ð Þ 2 (circles in Fig. 2b). Similar behaviour was observed over a wide temperature range as shown in the inset of Fig. 2c. In order to analyse the channel contribution to noise, DR 4P ð Þ 2 is shown as a function of the carrier density n in Fig. 2d. For large hole doping (\10 12 cm À 2 ), that is, p-p 0 regime where R c reduces to t1 kO, we observed DR 4P ð Þ 2 =R 2 4P / 1=n (dashed line), suggesting Hooge-type mobility fluctuation noise in the graphene channel, with a Hooge parameter B10 À 3 (refs 8,11,13,47,48). However, in the n-n regime, where the contact contribution is dominant, noise deviates from 1/n behaviour.
Noise in high-mobility graphene hybrids. The dependence of the noise magnitude with the contacting geometry in the Aucontacted device (Fig. 2) led us to explore the contact contribution to noise in three other device geometries: (1) graphene, encapsulated between two hexagonal boron nitride (BN) layers and etched into a Hall bar, contacted by etching only the top BN (see Methods and Supplementary Note 1), shown in Fig. 3a. (2) Graphene on SiO 2 and BN substrates ( Fig. 3d and Fig. 4b), in surface-contacted linear geometry, where the contacts extend on to the channel region (invasive contacts) and (3) suspended graphene devices that are intrinsically in two-probe contact configuration. A 5 nm Cr underlayer was used with 50 nm Au films as contact material in all these devices. The details of the device fabrication process are given in the Methods section and Supplementary Note 1. Similar to the Au-contacted device, the noise measurements were performed from 80 K to room temperature and no appreciable qualitative difference was observed.
To examine the generality of the R 4 c scaling in high-mobility graphene FETs, we first measured both two-probe and four-probe noise in the BN-encapsulated graphene hall bar device, which exhibited room temperature (four probe) carrier mobilities of 58,000 and 35,000 cm 2 V À 1 s À 1 in the electron-doped and holedoped regimes, respectively. The transfer characteristics show only one Dirac point (V D ) for both R 2P and R 4P (Fig. 3b inset), as expected for a Cr underlayer 39 . Both DR 2P ð Þ 2 and DR 4P ð Þ 2 decrease with increasing |V G À V D | (Fig. 3b), except over a small region around V D where the the distribution of charge in graphene becomes inhomogeneous. Away from the inhomogeneous regime, both h DR 2P ð Þ 2 i and h DR 4P ð Þ 2 i exhibit the R 4 c scaling over three decades (Fig. 3c) Fig. 2b,d. It is also interesting to note the drop in contact noise magnitude in the inhomogeneous regime, which could be due to the dominance of McWhorter-type number fluctuation noise 8,15,16,18,51 , rather than just mobility fluctuations in the charge transfer region.
The effect of contact noise becomes more severe for invasive surface contacts (leads extending to the current flow path), as demonstrated with a device that has graphene on BN (Fig. 3d). The transfer characteristics show a single Dirac point with carrier mobility B35,000 cm 2 V À 1 s À 1 (Fig. 3e inset). Strikingly, the magnitudes of h DR 2P ð Þ 2 i and h DR 4P ð Þ 2 i were found to be almost equal over the entire range of V BG (Fig. 3e), suggesting that the dominant contribution to noise arises from the charge transfer region underneath leads 2 and 3. To establish this quantitatively, we note that R 2P E2R MG þ 2R T þ R g and R 4P E2R T þ R g , respectively (see schematics in Fig. 3d), where R MG (B300 O) and R g are the metal-graphene interface resistance and graphene channel resistance, respectively. Owing to the inseparability of R T ( ¼ R c ) and R g within this contacting scheme, we plot h DR 4P ð Þ 2 i as a  function of R 4P in Fig. 3f. It is evident that h DR 4P ð Þ 2 i / R 4 4P for R 4P t150-200 O, where R g is small because of heavy electrostatic doping of the channel. However, for R 4P \200 O, the deviation from the R 4 4P scaling is likely due to finite R g that causes R 4P to overestimate the true R c . We have observed an R 4 scaling of noise for high-mobility suspended graphene devices as well (see Fig. 4f).

Discussion
Contact noise at the metal-semiconductor interface has been extensively researched over nearly seven decades 6,35,43,[52][53][54][55][56][57] , and except for a few early models based on kinetics of interface disorder such as adsorbate atoms 53 , the most common mechanism is based on time-dependent fluctuations in the characteristics of the Schottky barrier at metal-semiconductor junctions 53,[55][56][57] . The linearity of I-V characteristics (not shown) and temperature independence of R c (see Supplementary Fig. 4) in our devices, however, eliminate the possibility of Schottky barrier-limited transport. An alternative source of time-varying potential is the trapped charge at the SiO 2 surface 44,58-62 , which has been suggested to cause contact noise even in ballistic semiconducting carbon nanotube FETs 6,63 . The reaction of graphene with metals spontaneously leads to chemical modification (for example, carbide formation) and introduction of defects (see schematic in Fig. 4a). The chemical modification and defect formation can strongly influence the bandstructure of graphene underneath the metal, suppressing the screening of Coulomb impurities. This makes the charge transfer region susceptible to mobility fluctuations because of trapped charge fluctuations in SiO 2 , as indeed shown recently for noise at grain boundaries in graphene 64 .
To verify this, we have fabricated an invasively Cr/Aucontacted device where a single graphene channel was placed partially on BN (thickness B10 nm), thus physically separating the channel from the oxide traps 65 , whereas the other part was directly in contact with SiO 2 (Fig. 4b). The four-probe transfer characteristics (Fig. 4c) confirm that the region of graphene placed on SiO 2 shows lower carrier mobility (7,500 and 4,000 cm 2 V À 1 s À 1 for hole-and electron-doping, respectively) than the corresponding mobility (8,000 and 7,500 cm 2 V À 1 s À 1 ) of the part on BN, as well as strong substrate-induced doping, both of which can be readily understood by the proximity to charge traps at the SiO 2 surface. Although h DR 4P ð Þ 2 i in both parts shows strong peaks at the respective Dirac points (Fig. 4d), it is evident that the normalized noise magnitude in the graphene on SiO 2 substrate is up to a factor of 10 larger than that on BN, similar to that reported recently 19,20 . The scaling h DR 4P ð Þ 2 i / R 4 4P (Fig. 4e) over three decades of noise magnitude, irrespective of the substrate, unambiguously indicates the dominance of contact noise, and that the contact noise in graphene FETs is primarily a result of mobility fluctuations in the charge transfer region due to fluctuating Coulomb potential from local charge traps (predominantly from the SiO 2 surface).
Finally, in order to outline a recipe to minimize the contact noise in graphene devices, we have compiled the normalized magnitude of specific contact noise h DR c ð Þ 2 iW 2 as a function of specific contact resistance R c W, from different classes of devices that were studied in this work. We identify two key factors that have an impact on the contact noise: first, as can be clearly seen in Fig. 4f (left), the specific contact noise is largest for graphene on SiO 2 , lower on devices with graphene on BN and lowest for suspended graphene devices where all SiO 2 has been etched away  from under the graphene channel as well as partially from below the contact region (see Supplementary Fig. 5). Moreover, noise data from all devices with BN as substrate collapse on top of each other, regardless of mobility values, indicating that the separation of contacts from the SiO 2 traps is the primary factor that determines the noise magnitude rather than the channel quality itself. Second, it can also be seen from Fig. 4f (right) that the device with Cr/Au contacts, which are known to chemically modify graphene 36,39 , exhibits higher noise than the device with Au contacts, which is expected to leave graphene intact, despite the fact that the former device has a BN substrate, whereas the later SiO 2 . This highlights the major role of defects under metal contacts in noise generation. Combining these factors leads to the conclusion that minimizing environmental electrostatic fluctuations and developing a contacting scheme that preserves the chemical/structural integrity of graphene will be necessary for ultralow noise graphene electronics.
In conclusion, we have studied electrical noise at the metal contacts in graphene devices with a large range of carrier mobility, on multiple substrates with various device and lead geometries. Using a phenomenological model of contact noise for purely 2D materials, we show that contact noise is often the dominant noise source in graphene devices. The influence of contact noise is most severe in high-mobility graphene transistors. Most surprisingly, we discover the ubiquity of contact noise, which is seen to affect even four-probe measurements in a Hall bar geometry. Our analysis suggests that contact noise is caused by strong mobility fluctuations in the charge transfer region under the metal contacts because of the fluctuating electrostatic environment. A microscopic understanding of contact noise may aid in the development of ultralow noise graphene electronics.

Methods
Device fabrication. Graphene and hexagonal BN were exfoliated on SiO 2 using the 3 M scotch (Magic) tape. The heterostructures were assembled using a method similar to that described in ref. 66 in a custom-built microscope and transfer assembly. For parameters similar to those described in ref. 3, we determined the etching rate of BN, in a CHF 3 and O 2 plasma, to be 23±2 nm per 60 s (see Supplementary Fig. 1). The device shown in Fig. 3a was fabricated by etching only the top BN (21 ± 3 nm, etched for 60 s). Two layers of PMMA (450 and 950 K) were spin-coated for electron beam lithography and act as masks for metal deposition and etching. Graphene was contacted by thermally evaporating Au (50 nm) or Cr/Au (5/50 nm) at t10 À 6 mbar.
Measurements. Both average resistance and time-dependent noise were measured in a standard low-frequency lock-in technique, with a small source-drain excitation current B100 nA to ensure linear transport regime 67 . Background noise was measured simultaneously and was subtracted from total noise to determine the sample noise.
Data availability. The data that support the findings of this study are available from the corresponding author upon request.