Introduction

The 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 metal–semiconductor 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 contacts1,2,3,4 and strategies for encapsulation3 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 converters5. 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 contacts6

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 region7,8,9, whereas other investigations indicate a strong contribution from the contacts10,11,12. This distinction has remained elusive to existing studies7,8,9,10,11,12,13,14,15,16,17,18,19,20,21 because of the lack of a microscopic understanding of how processes characteristic to the metal–graphene junctions, in particular the current-crowding effect22,23,24,25,26,27, 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 LT22,28. Photocurrent measurements29,30,31 and Kelvin probe microscopy32,33 at the graphene–metal interface have already indicated the presence of current crowding with LT∼0.1–1 μm (refs 24, 25, 34). Restricting the effective current injection area leads to greater impact of local disorder kinetics, and hence larger 1/f noise35. 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 dispersion34,36. While it is clear that current crowding and the characteristics of the metal–graphene junction directly influence the contact resistance25,29,34,36,37,38,39,40,41,42, 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 , where Rc 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 noise43,44 and current-crowding effect in molybdenum disulphide (MoS2) and black phosphorus FETs26,27, 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) SiO2/p++–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 bandstructure45,46. This allows easy tuning of the doping, and correspondingly the resistance, of the contact region with backgate voltage (VBG). We measure the two-probe (R2P=R23,23) and four-probe (R4P=R23,14) resistance and noise as a function of VBG between the leads 2 and 3 (suffixes in RV+V−,I+I− indicate the voltage (V+, V−) and current (I+, I−) leads). The VBG dependence of R2P and R4P is shown in Fig. 1b. R4P shows a slightly asymmetric transfer characteristic, known to occur for asymmetric contact doping38, with a single Dirac point at VBG≈6 V. R2P, however, shows a second Dirac point at VBG≈31 V because of the combination of hole doping and weak pinning by Au at the contact region34,38,45,46, which divides the transfer behaviour in three parts (, n−p and ), 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−VBG characteristics confirms the structural integrity and gate tunability of the Fermi level of the graphene channel underneath the contact.

Figure 1: Contact resistance and current crowding in Au-contacted graphene.
figure 1

(a) Schematic of the device geometry and contact configuration. Inset shows the optical image; scale bar, 5 μm. (b) Resistance as a function of backgate voltage (VBG) measured in two-probe (R2P) and the four-probe (R4P) geometry. (c) The Fermi energy of graphene under the Au contacts can be tuned by applying a backgate voltage. (d) The difference of the R2P and R4P gives the measured contact resistance Rc (blue). The resistance RTcalc calculated for transport across the potential step (black). (e) Current injection into graphene occurs within a small length ∼LT for a contact length of Lc (top). Most of the potential drop occurs at the edge of the contact, shown at three VBG values (bottom), marked in d. Inset shows signification variation in LT with VBG (data near the main Dirac peak excluded).

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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 Rc, 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 VBG dependence of the experimental contact resistance Rc=R2P−R4P (corrected for the resistance of the small region of the probe arms) and that calculated assuming the Dirac-like dispersion and level broadening ≈80 meV underneath the contact and ≈57 meV in the channel (estimated from the experimental transfer characteristics, see Supplementary Note 2 for the full details of calculations). The agreement, both in VBG dependence and absolute magnitude (within 50% for all VBG), indicates that the contact resistance is primarily composed of the resistance RT of the graphene layer over the charge transfer length (LT) underneath the contacts. Owing to mismatch between the resistivities of the metal and graphene, LT is significantly smaller than the geometric width Lc (∼1–1.5 μm) 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 ρT (schematic in Fig. 1e). The potential profile in graphene under the metal is then given by22,

where I is the current flowing, ρc≈200 Ω μm2 (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 RT as the experimentally observed contact resistance Rc (Fig. 1d), we calculated the potential drop underneath the contact, normalized to its value at the edge x=0, for three gate voltages marked by the arrows in Fig. 1d. The potential drops exponentially over the gate voltage-dependent scale LT, being minimum (∼200 nm) for the second Dirac point at +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 ρT=RTW/LT, it is evident that the contact noise is essentially the resistance fluctuations in the graphene layer underneath the contact, that is, , where and nT are the phenomenological Hooge parameter and carrier density in the charge transfer region, respectively. is independent of nT 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 mobility46, the contact noise can be expressed as

and implies a scaling relation 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 Rc is ≈1 for interface-type contacts or ≈3 for constriction-type contacts35. (2) Since nT is the only gate-tunable parameter34,38,39, 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 R2P and R4P at all VBG consists of random time-dependent fluctuations with power spectral density (Fig. 2a), where 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’ , by integrating SR(f) numerically over the experimental bandwidth.

Figure 2: Contact noise in Au-contacted graphene.
figure 2

(a) Typical measured time series (left) and corresponding power spectral density SR (right) as a function of frequency. (b) Noise (variance) as a function of gate voltage in two-probe and four-probe geometry. (c) as a function of contact resistance Rc shows dependence. This dependence is valid for temperatures down to ∼80 K (inset). (d) Noise (4p) normalized by the graphene resistance as a function of number density.

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Figure 2b shows the ΔVBG dependence of and , where the maxima in both quantities align well with the Dirac points in R2P and Rc (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 channel–substrate interface7,8,15,16. Here peaks in the n–p region close to ΔVBG≈25–30 V, where the density of states in the charge transfer region is low34,38, indicating contact noise that originates because of poorly screened fluctuations in the local Coulomb disorder. In fact, the noise magnitude at the second peak (ΔVBG≈30 V) is ∼10 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, 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).

Figure 3: Noise in high-mobility graphene
figure 3

(a) Hall bar device with graphene encapsulated by hexagonal BN and contacted by etching the top BN; scale bar, 10 μm in the optical image (bottom). (b) Noise variance as a function of gate voltage in two-probe and four-probe geometry. Inset shows resistance (R2P, R4P and Rc) as a function of backgate voltage. (c) Noise variance as a function of contact resistance Rc shows dependence (dotted line) for both two-probe and four-probe measurements. (d) Electron microscope image of graphene on BN with invasive linear contacting geometry. Bottom schematic shows the effect of charge scattering, from the region under the contacts, on four-probe measurements as well. (e) measured in two-probe and four-probe geometry roughly coincide for all gate voltages, indicating that noise generated in the channel is negligible. Inset shows R–VG characteristics for the device. (f) Noise shows R4 dependence (dotted line) at higher gate voltages indicating the dominant contact contribution.

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To verify the contact origin of noise, we have plotted as a function of contact resistance Rc in Fig. 2c. Remarkably, for all VBG collapses on a single trace, and varies as 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 higher than the channel noise (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, is shown as a function of the carrier density n in Fig. 2d. For large hole doping (≳1012 cm−2), that is, p–p′ regime where Rc reduces to ≲1 kΩ, we observed (dashed line), suggesting Hooge-type mobility fluctuation noise in the graphene channel, with a Hooge parameter ∼10−3 (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 Au-contacted 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 SiO2 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.

Figure 4: Noise mechanism.
figure 4

(a) Schematic of the contact region with metal deposition, and possible effects on the band dispersion and structural properties of graphene underneath. (b) Device image and schematic showing a portion of graphene on BN and the rest on SiO2. Scale bar, 10 μm. (c) Resistance as a function of gate voltage on BN (blue) and on SiO2 (black). (d) Approximately 5–20 times lower noise in graphene on BN (black squares) than on SiO2 (blue circles). (e) as a function of R4P for graphene on SiO2 (squares) and for graphene on BN (circles); the dotted lines are R4. (f) Comparison of contact noise in various devices; noise depends strongly on the substrate (left) or the contact material used (right).

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To examine the generality of the 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 cm2V−1s−1 in the electron-doped and hole-doped regimes, respectively. The transfer characteristics show only one Dirac point (VD) for both R2P and R4P (Fig. 3b inset), as expected for a Cr underlayer39. Both and decrease with increasing |VG−VD| (Fig. 3b), except over a small region around VD where the the distribution of charge in graphene becomes inhomogeneous. Away from the inhomogeneous regime, both and exhibit the scaling over three decades (Fig. 3c). The scaling of is unexpected, although the suppression is close to the nonlocal factor ∝ exp[−2πLT/W] for realistic LT of ∼400 nm (refs 24, 49), suggesting that this could be a nonlocal effect due to finite dimensions of the voltage leads 2 and 3 (ref. 50; see Supplementary Note 3). This also explains the reduced, but perceptible signature of contact noise in in 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 noise8,15,16,18,51, 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 ∼35,000 cm2V−1s−1 (Fig. 3e inset). Strikingly, the magnitudes of and were found to be almost equal over the entire range of VBG (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 R2P≈2RMG+2RT+Rg and R4P≈2RT+Rg, respectively (see schematics in Fig. 3d), where RMG (∼300 Ω) and Rg are the metal–graphene interface resistance and graphene channel resistance, respectively. Owing to the inseparability of RT(=Rc) and Rg within this contacting scheme, we plot as a function of R4P in Fig. 3f. It is evident that for R4P≲150–200 Ω, where Rg is small because of heavy electrostatic doping of the channel. However, for R4P≳200 Ω, the deviation from the scaling is likely due to finite Rg that causes R4P to overestimate the true Rc. We have observed an R4 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 decades6,35,43,52,53,54,55,56,57, and except for a few early models based on kinetics of interface disorder such as adsorbate atoms53, the most common mechanism is based on time-dependent fluctuations in the characteristics of the Schottky barrier at metal–semiconductor junctions53,55,56,57. The linearity of I–V characteristics (not shown) and temperature independence of Rc (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 SiO2 surface44,58,59,60,61,62, which has been suggested to cause contact noise even in ballistic semiconducting carbon nanotube FETs6,63. 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 SiO2, as indeed shown recently for noise at grain boundaries in graphene64.

To verify this, we have fabricated an invasively Cr/Au-contacted device where a single graphene channel was placed partially on BN (thickness ∼10 nm), thus physically separating the channel from the oxide traps65, whereas the other part was directly in contact with SiO2 (Fig. 4b). The four-probe transfer characteristics (Fig. 4c) confirm that the region of graphene placed on SiO2 shows lower carrier mobility (7,500 and 4,000 cm2V−1s−1 for hole- and electron-doping, respectively) than the corresponding mobility (8,000 and 7,500 cm2V−1s−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 SiO2 surface. Although 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 SiO2 substrate is up to a factor of 10 larger than that on BN, similar to that reported recently19,20. The scaling (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 SiO2 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 as a function of specific contact resistance RcW, 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 SiO2, lower on devices with graphene on BN and lowest for suspended graphene devices where all SiO2 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 SiO2 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 graphene36,39, 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 SiO2. 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 SiO2 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 CHF3 and O2 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 ≲10−6 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 ∼100 nA to ensure linear transport regime67. 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.

Additional information

How to cite this article: Karnatak, P. et al. Current crowding mediated large contact noise in graphene field-effect transistors. Nat. Commun. 7, 13703 doi: 10.1038/ncomms13703 (2016).

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