Electrical Low-Frequency 1/fγ Noise Due to Surface Diffusion of Scatterers on an Ultra-low-Noise Graphene Platform

Low-frequency 1/f γ noise is ubiquitous, even in high-end electronic devices. Recently, it was found that adsorbed O2 molecules provide the dominant contribution to flux noise in superconducting quantum interference devices. To clarify the basic principles of such adsorbate noise, we have investigated low-frequency noise, while the mobility of surface adsorbates is varied by temperature. We measured low-frequency current noise in suspended monolayer graphene Corbino samples under the influence of adsorbed Ne atoms. Owing to the extremely small intrinsic noise of suspended graphene, we could resolve a combination of 1/f γ and Lorentzian noise induced by the presence of Ne. We find that the 1/f γ noise is caused by surface diffusion of Ne atoms and by temporary formation of few-Ne-atom clusters. Our results support the idea that clustering dynamics of defects is relevant for understanding of 1/f noise in metallic systems.


I Experimental procedures
In our experiments, we measured graphene Corbino disks, freely suspended by dissolving away a LOR sacrificial layer, onto which natural graphite had been exfoliated. Our sample fabrication techniques and illustrations of samples can be found in Refs. 1,2. Our most extensive data set was taken using a Corbino disk with distance L = 1.3 µm between the electrodes (inner and outer diameters of 1.8 and 4.5 µm, respectively). The value of the gate capacitance C g = 1.5 × 10 −5 F/m 2 was determined using the spread of a Landau level fan diagram. 1 A close-up from a scanning electron microscope (SEM) image of a graphene Corbino sample is illustrated within the measurement schematics in Fig. S1.
Following the initial characterization at room temperature, the samples were mounted on a Bluefors LD250 dry dilution refrigerator and cooled down to T = 10 mK. Prior to lowfrequency noise measurements, the graphene devices were current annealed at 4 K, at which cryopumping guaranteed a UHV level cryogenic vacuum. Cryopumping by the cryostat worked also at higher temperatures against impurity gases such as O 2 and N 2 . In addition, current annealing was employed at regular intervals to clean neon away from the graphene surface, thereby verifying that the results indeed were caused by Ne atoms. The applied gate voltage V g was employed to determine the charge carrier density according to n = (V g − V D g )C g e, where V D g denotes the gate voltage value of the Dirac point. Measurements of zero-bias resistance R 0 (V g ) yielded the field-effect mobility µ F E ∼ 10 5 cm 2 /Vs using µ F E = (σ − σ 0 ) ne, where the minimum conductivity σ 0 corresponds to the maximum of measured resistivity at the Dirac point. According to µ F E determination, adsorption of Ne increased the mobility of graphene by ∼ 30% at small charge densities n ≲ 2 × 10 14 m −2 . This indicates additional screening of Coulomb impurities by adsorbed Ne. The effect of Ne-induced charge screening/charge modulation could be removed by thermal annealing, which was repeatedly done during our experiments.
The measurement schematics is depicted in Fig. S1. The graphene disk with its gold electrodes, seen in the false color SEM image, is located on top of a 500-nm-thick lift off resist (LOR) layer. Strongly doped substrate Si ++ acted as the gate for charge density control, coupled by capacitance C g to the sample via a 500-nm vacuum gap and 280-nm layer of SiO 2 .
Bias-T components (a combination of inductance L and capacitance C in the schematics) were employed to separate the radio frequency (rf) section from the low frequency parts. The Bias-T components were employed to separate the radio frequency section used for mechanical vibrations from the low frequency parts employed to measure zero-bias resistance, IV characteristics, and noise. For details, see text.
rf-circuitry was needed to excite (using V rf ) and to record the mechanical resonance of the graphene membrane. Zero-bias resistance R 0 of the sample (consisting of a sum of graphene resistance R gr and contact resistance R c between graphene and gold) was measured at ∼ 35

II. Neon on graphite
Atomic neon films on graphite provide a good starting point for understanding our submonolayer Ne films on graphene. An excellent overview of neon films on graphite is provided in Ref. 4. In general, quantum effects are a way smaller for neon atoms than for helium or hydrogen but they are still significant. For example, diffusion of Ne atoms is governed by quantum tunnelling at sub-Kelvin temperatures but, in our work at T > 4 K, thermally activated diffusion still dominates. Our adsorbed neon films were prepared around 20 K with pressure p ∼ 10 −4 mbar. On the basis of the empirical phase diagram for thin Ne films on graphite, 4,9 we conclude that no registered phases will be generated, and the behavior of the adsorbed atoms at T = 10 − 37 K will display gas or fluid like behavior. At temperatures T < 10 K, coexistence of solid phase and vapor becomes possible. 9 An immobile solid phase will be irrelevant for the generation of low frequency noise. Fluctuations of the shape of the solid, however, will contribute to the noise in a similar fashion as fluctuations in the clustered atom regions seen in our simulations.
The addition of neon changed the dependence of graphene sample resistance, R, on the gate-induced carrier density n as illustrated in Fig. S2a. Clearly, R(n) is more peaked near the Dirac point while the saturation conductance (inverse resistance) value becomes larger with Ne adsorption; this saturation value at V g ≫ V D g is taken to correspond to the effective contact resistance R c . Consequently, the data in Fig. S2a yields R c = 500 Ω and 405 Ω for clean and Ne-adsorbed states, respectively. With addition of Ne, the Dirac point of the sample also moved upwards nearly by 0.8 V in V g , which has been taken into account when calculating the charge density. is the range where most of our noise data have been measured. This increase in mobility is quite close to the quoted 40% increase in µ 0 found from the magnetoresistance.
The asymptotic resistance value at large negative gate voltages in Fig. S2a is reduced by ∼ 95 Ω by the addition of Ne. This indicates that resistance for hole transport near the gold contact is influenced by the adsorption of Ne and the hole contact resistance R h c (resistance not influenced by V g ) is lowered. Reduction in R h c can be assigned to Ne-induced strain that causes a change in the scalar and vector potentials in graphene in the neon-covered region.
The strain-induced scalar potential modifies charge density near the contact and R h c becomes lowered as seen in Fig. S2a. For electronic transport at V g >> V D g , the modification in the electronic contact resistance R e c is nearly zero.
We recently investigated magnetoresistance of the very same suspended Corbino disk. 10 Those experiments indicated that Coulomb scatterers at small carrier densities close to the Magnetoresistance investigations also allowed us to determine the contact resistance. 10 We employed the B 2 dependence of geometric magnetoresistance of graphene in Corbino geometry. The magnetoresistance yields mobility vs. gate voltage V g that can be fitted accurately using scattering by short ranged and Coulombic impurities. These scattering contributions together govern conductance G(V g ) of bulk graphene. By subtracting the bulk contribution out, we obtain for the contact resistance R c ≃ 350 − 400 Ω, nearly independent of the gate voltage. At large charge density the total resistance is dominated by R c as is common in good quality samples. The actual value of R c depended on the cleanliness of the sample and the presence of Ne on the sample. Note that the contact resistance is smaller with the presence of Ne on the sample.

III. Thermal activation and potentials
Our basic model for atomic Ne fluxes and trapping of atoms is illustrated in Fig. S3. The joint dynamics of gaseous neon and adsorbed atoms is governed by three different energy scales. The largest is the Ne-graphite adsorption energy E a k B which is on the order 350 K. 6 For the atoms at the electrodes of the Corbino geometry there is an additional Ne-boundary adsorption energy E b k B which amounts approximately to a few hundred Kelvin which is a typical noble gas-metal adsorption energy. 12 The smallest scale is related to the corrugation of the graphite potential E d that provides the diffusion barrier of the neon atoms moving along graphene. We employ E d k B = 32 K which has been reported for graphite. 13,14 In our simple modeling, we neglect the Ne-Ne interaction, but our Monte Carlo simulations do take this energy into account (see Sect. V).
When Ne atoms land on clean graphene, there are very few sticking sites for the atoms to get attached before they reach the boundary of the sample. Consequently, in our modeling we assume that neon atoms, becoming adsorbed from the gas phase, will become attached first to the wall potential marked by E b in Fig. S3. We are interested in the time development of the number of mobile Ne atoms N on the graphene surface. Since neon atoms diffusing on graphene are supplied by N b trapped atoms at the walls, we have two rate equations governing the dynamics: where the upper equation describes the behavior at the wall and the latter one on graphene.
N b τ s denotes the rate at which atoms are released from the wall whileṄ  In the steady state, dN b dt = 0 and dN dt = 0, which yields Using the definitions in Fig. S3, the time scales can be written as All time scales involve exponential activation type of behavior, but with quite distinct energy scales. The prefactors are related to inverse attempt frequencies f −1 0 ; these frequencies are expected to be in the range of f th 0 ≃ 10 8 − 10 12 s −1 , but they vary depending on the actual curvature of the underlying potential and the temperature of the environment. However, since effective attempt frequencies have been observed to be even smaller than f th 0 with noble gases on metal surfaces, 12 we regard τ ds as fit parameters. Diffusion of Ne atoms along the graphene is also thermally activated 12 and depends exponentially on temperature according to . At high temperatures T > 25 K, desorption of atoms is faster than their diffusion across the sample, and we have τ ds ≪ τ d , while the opposite limit τ ds ≫ τ d is realized at low temperatures. In the former case, we have because τ ds andṄ In the low-temperature limit with τ ds ≫ τ d , all the atoms from the gas phase have been adsorbed to surfaces, and we expect that the trapping states at the boundary are basically fully occupied, i.e. N b = N S , where N S denotes the saturation amount at the wall.
, there should be a strong T -dependence in N , and the number becomes reduced with lowering temperature. This will lead to more infrequent encounters between neon atoms and clustering of atoms becomes reduced at low T . This would then favor random walk type of noise which is detailed in Sect. IV next.
Finally, let us make a remark concerning the trapping potential E b , which we have regarded as a constant. However, the potential at the boundary will have several energy levels for Ne atoms and the atoms on higher levels will experience a smaller trapping potential.
Consequently, the effective trapping barrier E ef f b for release of Ne atoms at low T may be smaller than E b = 200 K, which would lead to a reduced decrease of N with lowering T than obtained from Eq. 10.

IV. Random walk on Corbino disk geometry
Random walk simulations were performed with a Corbino disk geometry of two concentric circles of 1.8 µm (inner) and 4.5 µm (outer) forming the contacts (see Fig. S4a). The particle starts at the outer contact and then moves on the graphene with a step size of 2.5 nm. Here a move is allowed with equal probability in the up, down, left or right direction. When the particle reaches the inner or outer contact it will get adsorbed. The amount of steps, which is proportional to the time diffusing on the graphene, is then recorded. Additionally, it is observed that 99.8% of the random walks that started at the outer contact also end there.
In Fig. S4b 1000000 of such random walks are compared and a linear fit is applied to the first 1000 double logarithmic data points. The linear fit of ax + b yields a slope of k = −1.5 which is in accordance with the one-dimensional calculation of Yakimov. 15 According to Ref.
15, the exponent for the spectrum is obtained from k as γ = 3 + k = 1.5.  corresponding to the electrodes of an infinitely large Corbino disk. We assume that all lattice sites are equivalent for impurity atoms, i.e. we neglect fully ripples on the graphene membrane, although such deformations might lead to clustering of impurities as such. We also neglect the Ne-Ne repulsion at short distances (see Sect. II) and allow the particles to occupy the nearest neighbour sites for computational simplicity.
In our starting configuration, 25 defects were placed on randomly selected lattice sites.
Assuming vacancy diffusion type dynamics 16 -equivalent to the 2-state Ising model performing Kawasaki dynamics, where only one defect can occupy one lattice site at a time and the defects are allowed to move via thermally activated diffusional hops to any of the eight nearest non-defect lattice sites with the rates governed by the following equations: where r is the average rate of a hop to one of the neighboring sites, f 0 is the attempt frequency, ∆E is the change in the system energy and E d is the activation energy for the diffusional hop, and T is the temperature. ∆E is determined by the coordination number between the neighboring defects assuming that when the coordination number increases by one, ∆E = −2, and when it decreases by one, ∆E = 2, and so on. Thus, according to Eqs. 11 the rate of cluster formation is somewhat higher than that of the dissociation and the rates depend on temperature. In our present simulations, simple energy relations were applied also for the other parameters: f 0 = 1, E d = 4 and k B T = 1.2 or k B T = 2.
After the kMC simulations, the produced time series of the positions of the moving defects were used as an input to calculate the corresponding time series of the fluctuating resistance of the system. A minimal model for impurity scattering was employed to estimate the induced resistance change due to the diffusing particles: The resistance of a defect site was taken to be much lower, or alternatively much larger, 1 than that of the background lattice. 18 In our finite element method (FEM) calculations, 19 we assigned a 10 5 times smaller

VI. Graphene noise vs. contact noise
In the main paper, we considered the 1 f γ noise as coming from graphene without trying to separate the exact origin of the noise, whether it comes purely from graphene or whether it is also related to electrical contacts. As is well known, the resistance in high quality graphene samples originates mostly from the contacts, and the same could happen with the 1 f γ noise. The separation of noise contribution from contacts has been discussed in clean graphene in Ref. 20 in presence of incoherent noise sources. Specific features of the measured S I (V g ) could be related to contact noise S c I and to the graphene noise S gr I . In particular, a M-shaped S I (V g ) curve with leveling off at large charge densities could be explained using the incoherent noise source model, in which the dip in the noise is related to S c I and to the coexistence of electrons and holes. 20 Moreover, the contact noise governs the leveling off of S I (V g ) at large carrier densities. and T = 20 K. The data display a well-defined minimum of noise at the Dirac point, followed by a maximum in S I (V g ) at gate voltage V g ≃ ±10 V, and finally a clear decrease of noise at V g > 10 V which tends to saturation when V g → 50 V. The behavior thus follows exactly the V g dependence outlined in the analysis of Ref. 20. Consequently, we can conclude that the division between contact noise S c I and graphene noise S gr I is qualitatively similar in the presence of adsorbed Ne as in clean graphene. From the data at T = 4 K, however, we may infer that the contact noise becomes asymmetric with respect to electrons and holes: electron conduction is seen to have smaller noise than the hole conduction at V g < 0. Comparing with Fig. S2, we see that the change in contact resistance due to Ne atoms is stronger in the hole carrier regime, i.e. under similar conditions as S c I . Hence, we conclude that both contact noise and graphene noise are influenced by adsorbed Ne. Apart from local pseudomagnetic fields due to individual adsorbed atoms, the noise in both cases is due to changes in the local doping of graphene due to variation in Ne-induced scalar potential. Our kMC analysis with Corbino like boundary conditions includes significant clustering of atoms at the contacts and thereby noise due to changes in the boundary layer are an integral part of our numerical noise analysis. On broader scale, the origin of the contact noise is the same as that of graphene, even though the presence of the Au side wall changes the basic conditions for clustering.