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

Abstract

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 cm²V⁻¹s⁻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.

The performance of graphene field effect transistors is adversely affected by fluctuations in the electrical resistance at the graphene/metal interface. Here, the authors unveil the microscopic origin of such contact noise, highlighting the role of current crowding.

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 L_T²²,²⁸. Photocurrent measurements²⁹,³⁰,³¹ and Kelvin probe microscopy³²,³³ at the graphene–metal interface have already indicated the presence of current crowding with L_T∼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 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 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 (V_BG). We measure the two-probe (R_2P=R_23,23) and four-probe (R_4P=R_23,14) resistance and noise as a function of V_BG between the leads 2 and 3 (suffixes in R_{V+V−,I+I−} indicate the voltage (V⁺, V⁻) and current (I⁺, I⁻) 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 doping³⁸, with a single Dirac point at V_BG≈6 V. R_2P, however, shows a second Dirac point at V_BG≈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−V_BG 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.

(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 (V_BG) measured in two-probe (R_2P) and the four-probe (R_4P) geometry. (c) The Fermi energy of graphene under the Au contacts can be tuned by applying a backgate voltage. (d) The difference of the R_2P and R_4P gives the measured contact resistance R_c (blue). The resistance R_Tcalc calculated for transport across the potential step (black). (e) Current injection into graphene occurs within a small length ∼L_T for a contact length of L_c (top). Most of the potential drop occurs at the edge of the contact, shown at three V_BG values (bottom), marked in d. Inset shows signification variation in L_T with V_BG (data near the main Dirac peak excluded).

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=R_2P−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 ≈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 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 (∼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 ρ_T (schematic in Fig. 1e). The potential profile in graphene under the metal is then given by²²,

where I is the current flowing, ρ_c≈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 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=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 (∼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 ρ_T=R_TW/L_T, it is evident that the contact noise is essentially the resistance fluctuations in the graphene layer underneath the contact, that is, , where and n_T are the phenomenological Hooge parameter and carrier density in the charge transfer region, respectively. 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 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 R_c is ≈1 for interface-type contacts or ≈3 for constriction-type contacts³⁵. (2) Since n_T 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 R_2P and R_4P at all V_BG 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 S_R(f) numerically over the experimental bandwidth.

Figure 2. Contact noise in Au-contacted graphene.

(a) Typical measured time series (left) and corresponding power spectral density S_R (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 R_c 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.

Figure 2b shows the ΔV_BG dependence of and , 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 channel–substrate interface⁷,⁸,¹⁵,¹⁶. Here peaks in the n–p region close to ΔV_BG≈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 (ΔV_BG≈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.

(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 (R_2P, R_4P and R_c) as a function of backgate voltage. (c) Noise variance as a function of contact resistance R_c 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–V_G characteristics for the device. (f) Noise shows R⁴ dependence (dotted line) at higher gate voltages indicating the dominant contact contribution.

To verify the contact origin of noise, we have plotted as a function of contact resistance R_c in Fig. 2c. Remarkably, for all V_BG 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 R_c 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.

(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 R_4P 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).

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 (V_D) for both R_2P and R_4P (Fig. 3b inset), as expected for a Cr underlayer³⁹. Both and decrease with increasing |V_G−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 and exhibit the scaling over three decades (Fig. 3c). The scaling of is unexpected, although the suppression is close to the nonlocal factor ∝ exp[−2πL_T/W] for realistic L_T 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 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≈2R_MG+2R_T+R_g and R_4P≈2R_T+R_g, respectively (see schematics in Fig. 3d), where R_MG (∼300 Ω) and R_g are the metal–graphene interface resistance and graphene channel resistance, respectively. Owing to the inseparability of R_T(=R_c) and R_g within this contacting scheme, we plot as a function of R_4P in Fig. 3f. It is evident that for R_4P≲150–200 Ω, where R_g is small because of heavy electrostatic doping of the channel. However, for R_4P≳200 Ω, the deviation from the scaling is likely due to finite R_g that causes R_4P to overestimate the true R_c. 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 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 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 R_cW, 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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Supplementary Material

Supplementary Informations

Supplementary Figures 1-5, Supplementary Notes 1-3 and Supplementary References.