Quantum Noise and Asymmetric Scattering of Electrons and Holes in Graphene Atikur Rahman, Janice Wynn Guikema, and Nina Marković* Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, United States S Supporting Information *
ABSTRACT: We present measurements of quantum interference noise in double-gated single layer graphene devices at low temperatures. The noise characteristics show a nonmonotonic dependence on carrier density, which is related to the interplay between charge inhomogeneity and diﬀerent scattering mechanisms. Linearly increasing 1/f noise at low carrier densities coincides with the observation of weak localization, suggesting the importance of short-range disorder in this regime. Using perpendicular and parallel p−n junctions, we ﬁnd that the observed asymmetry of the noise with respect to the Dirac point can be related to asymmetric scattering of electrons and holes on the disorder potential. KEYWORDS: Graphene, 1/f noise, quantum transport, conductance ﬂuctuations
Low-frequency noise measurements were carried out using the ac noise measurement technique.17 A low-frequency ac bias current was applied to the sample, and a lock-in ampliﬁer was used to measure the voltage as a function of time. These data were then Fourier transformed to calculate the power spectral density SV, which represents the mean square voltage per unit frequency (see Supporting Information for the details on the noise measurement and data analysis). We ﬁnd that SV shows 1/fα dependence at all VTg, with values of α close to 1 (Supporting Information, Figure S2). In Figure 2a we show both the resistance and normalized noise as a function of carrier density (n) for VBg = 0. A minimum in the noise is found at low n (close to, but not coinciding with the maximum in the resistance), followed by an increase and then a decrease at larger n. The noise measurements were highly reproducible over time at all temperatures and did not depend on the direction or scan step of the gate voltage (Supporting Information, Figures S3−S5). The amplitude of the noise is found to increase with decreasing temperature and to decrease by half in magnetic ﬁeld (Supporting Information, Figures S6 and S7), suggesting a substantial contribution from QIN.2,7,15 Nonmonotonic QIN has been recently predicted due to nonuniversal conductance ﬂuctuations at a crossover between a pseudodiﬀusive and symplectic regime in graphene with longrange disorder.14 The nonmonotonic behavior was observed for low values of the impurity concentration, but it disappeared as the impurity concentration was increased. For the device shown in Figure 1, we found the mobility to be μ = 6052 cm2/(V s) and the residual carrier concentration n0 = 4.3 × 1011/cm2
uantum interference noise (QIN) arises from timedependent ﬂuctuations in the quantum interference between trajectories of charge carriers that are scattered on ﬂuctuating disorder.1−3 Closely related to universal conductance ﬂuctuations (UCF)4,5 and weak localization (WL),6 it is only observed at low temperatures, when the phase coherence length is larger than the mean free path. QIN shows a generic 1/f dependence on frequency, but it reduces by half in magnetic ﬁeld2,7 and increases with decreasing temperature,2 unlike the classical 1/f noise.8 In graphene, relativistic nature of the charge carriers modiﬁes their interaction with impurities,9 and quantum transport is aﬀected by pseudospin and valley degrees of freedom.10,11 WL and UCF have been studied extensively in graphene.9,12,13 In contrast, while QIN has been considered theoretically,14 experimental measurements have only recently been reported.15 In this work, we show measurements of QIN in disordered dual-gated graphene devices as a function of carrier density at low temperatures. We relate the nonmonotonic noise characteristics to the nature of the impurity scattering and ﬁnd that the asymmetry with respect to the gate voltage is related to asymmetric scattering of electrons and holes by disorder. The devices were fabricated by mechanical exfoliation of natural graphite (see Supporting Information for the details of the sample fabrication). Scanning electron microscope image of a typical device (500 nm wide graphene channel with a 500 nm wide top gate) is shown in Figure 1a. All the measurements were done in a 4-probe geometry using noninvasive voltage probes16 (schematic shown in Figure 1b). In Figure 1c, we show the resistance (measured at 250 mK) as a function of both back gate (VBg) and top gate voltage (VTg) for a locally gated device (SL1). All four regions (marked by p−p*−p, p− n*−p, n−n*−n, and n−p*−n) are clearly visible. © 2014 American Chemical Society
Received: August 26, 2014 Revised: October 10, 2014 Published: October 24, 2014 6621
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scattering.22,24 Some of these mechanisms are also likely to aﬀect QIN, as discussed below. We consider the following contributions to the noise: carrier density ﬂuctuations, disorder ﬂuctuations, electron−hole puddles, transport through p−n junction interfaces and dependence of the phase coherence length on n. Carrier Density Fluctuations. If we assume that the noise is determined entirely by the ﬂuctuations in the carrier density25 or mobility,26 then the normalized noise power, SV/ V2, and the contribution to the resistance ﬂuctuations due to the carrier density, (dR/dVTg)2/R2, should closely follow each other.20,24 Although the two curves have a similar qualitative dependence, they do not coincide, as shown in Figure 2b, indicating that the noise is not dominated by the ﬂuctuations in the carrier density (see Supporting Information for data on other devices). Disorder Fluctuations. Two main types of disorder are known to play an important role in graphene: short-range (SR) and long-range (LR) disorder.12,13 SR disorder arises from lattice defects or edges that lift the valley degeneracy, while the LR disorder is typically due to charged impurities with Coulomb potentials that vary slowly on the length scale of the lattice constant. If scattering on SR disorder were the dominant source of resistance ﬂuctuations, then the normalized noise would increase monotonically with increasing carrier concentration (n). However, if LR disorder dominated the noise, then the noise magnitude would decrease with increasing n. 27 A crossover from SR to LR disorder-dominated ﬂuctuations could therefore account for the observed nonmonotonic behavior.24 To see whether that makes sense physically, one should clearly distinguish the average resistance from the timedependent f luctuations: while the average resistance is determined by combined contributions from all scatterers, the time-dependent ﬂuctuations will be dominated by the scatterers that ﬂuctuate on the relevant time scales of the measurement (the slow ones in this case since we are looking at low frequencies). Therefore, we should not assume that the same mechanism will dominate both the average resistance and the time-dependent ﬂuctuations. Naively, one may expect lattice defects, vacancies, and edges to have a higher activation energy than reconﬁgurations of charges trapped at the substrate. A possible evidence of SR disorder in our samples is presented in Figure 3. The resistance increases with decreasing temperature in the vicinity of the Dirac point but does not otherwise show signiﬁcant temperature dependence (Figure 3a). As shown in Figure 3b, the temperature-dependent regime coincides with the range of VTg for which the noise increases. Negative magnetoresistance MR is observed in the same regime, as shown in Figure 3c,d. Fitting the MR data using the WL theory,10 we estimate the phase coherence length to be lϕ ≈ 300 nm (Supporting Information Figure S9a). The logarithmic temperature dependence of the average resistance is also consistent with WL10 (see Supporting Information Figure S9b). Another likely contribution to the MR may come from the transmission through the p−n interfaces formed by electron and hole puddles at low carrier densities, although it is not clear that this mechanism would result in negative MR.28 The observation of WL would imply the presence of SR disorder in our samples10,13 (possibly due to the top gate dielectric29 or patterning of graphene using oxygen plasma). A crossover from SR to LR disorder-dominated ﬂuctuations could occur at higher carrier densities, when eﬃcient screening
Figure 1. Sample schematic and resistance as a function of gate voltages. (a) Colored SEM micrograph of a dual-gated single-layer graphene device. The scale bar is 1 μm. The width of the graphene channel (blue) is 500 nm, and the distance between voltage leads (yellow) ranges from 150 nm to 3 μm for diﬀerent devices (1.5 μm for the device shown). The width of the top gate (brown) ranges from 150 to 500 nm (500 nm for the device shown). (b) Schematic of the 4probe measurement conﬁguration for a device shown in panel a. (c) Color plot of the resistance as a function of top gate voltage and the back gate voltage (device SL1). The carrier type and density are controlled independently by the two gates to create p−p*−p, p−n*− p, n−n*−n, and n−p*−n regions (here * denotes the carrier type under the top gate). For VTg = 0, the maximum in the resistance (VD) is located at VBg = −6 V. From the shift of (VD) as a function of VTg and VBg, we found the capacitive coupling ratio between the top-gate and back-gate dielectrics (CTg/CBg) = 30 (see also Supporting Information Figure S1). This provides estimated top gate capacitance value CTg = 345 × 10 −9 Farad/cm2.
Figure 2. Noise and resistance as a function of carrier density. (a) Normalized noise power (left axis) and resistance (right axis) as a function of carrier density for zero back gate voltage (see Supporting Information for additional data from this device and from other devices). (b) SV/V2 and (dR/dVTg)2/R2 dr/dVTg as a function of carrier density. It is evident that the two curves do not coincide. The measurements were done at 250 mK.
(Supporting Information, Figure S8),18 corresponding to an impurity concentration nimp = 1.3 × 1012/cm2.19 This value is slightly larger, but of the same order as nimp that was predicted to result in a nonmonotonic behavior.14 While such a crossover resembles our data, we cannot claim that our samples are in the correct regime: in disordered samples that also include shortrange impurities (such as ours seem to be, as we show below), one would generally not expect to ﬁnd either pseudodiﬀusive regime (expected in clean samples) or symplectic regime (expected in samples with preserved valley symmetry). Similarly M-shaped classical 1/f noise characteristics have also been reported in graphene at higher temperatures, away from the quantum transport regime.20−24 The nonmonotonic behavior as a function of gate voltage has been attributed to charge noise,20 interplay between majority and minority carriers, 21 and competition between diﬀerent types of 6622
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Figure 3. Temperature and magnetic ﬁeld dependence of the resistance. (a) Resistance of a single layer graphene device at two diﬀerent temperatures as a function of top gate voltage at zero back gate voltage (corresponding carrier density shown in the top axis). (b) The noise SV/V2 (left axis) and R0.25K − R2.5K (right axis) as a function of top gate voltage at zero back gate voltage. (c) Color plot of the resistance as a function of VTg − VD and the magnetic ﬁeld at zero back gate voltage. Negative magnetoresistance is observed in the vicinity of the Dirac point, with the largest magnitude near the Dirac point (VTg − VD =0, with VD = −1.1 V), and decreasing away from it. In this sample, magnetoresistance remains negative for negative VTg − VD, but it becomes positive at relatively large magnetic ﬁelds for positive values of VTg − VD. (d) Magnetoresistance as a function of magnetic ﬁeld for three diﬀerent values of top gate voltage (−1.1, −5, and 5 V) lines (corresponding to VTg − VD = 0, −3.9, and 6.1 V) in panel c.
Figure 4. Parallel p−n junction device. (a) Scanning electron microscope image of a single-layer parallel p−n junction device (SL4). Position of the graphene ﬂake is indicated by the blue shaded region. The top gate only partially covers the graphene channel. The scale bar is 1 is μm long. (b) Schematic of the four-probe measurement for the parallel p−n junction. (c) The noise (left axis) and the resistance (right axis) as a function of back gate voltage for zero top gate voltage. The noise minimum is located at VTg = 8 V. (d) The schematic and the noise data as a function of top gate voltage at VBg = 8 V. The blue solid line represents the noise signal from the portion of the sample under the top gate, the straight black line represents the noise signal from the portion of the sample that is not covered by the top gate (this part is independent of the top gate voltage), and the red dashed line indicates the total noise signal that would be measured, to be compared with the data below the schematic. Similar schematics and the data are shown for (e) VBg = −4 V and (f) VBg = −26 V.
eﬀectively slows down the ﬂuctuations due to charged impurities. Electron−Hole Puddles. At low carrier densities, the transport is aﬀected by electron and hole puddles.30 Assuming that the noise is inversely proportional to the number of carriers, it has been suggested before that the contribution from the minority carriers can dominate the noise, resulting in a net increase.21 The exact shape of the noise characteristics was found to depend on the relative strengths of the SR and LR disorder.22 p−n Junction Interfaces. To investigate the possibility that the measured noise is generated at the p−n junction interfaces, we have also studied parallel p−n junctions, as shown in Figure 4a. The resistance of the sample (see Figure 4b for a schematic) is determined by the parallel combination of the resistance of the two segments (portions with and without top gate) and lowest resistive path will dominate. In contrast, the noise of the entire system will be dominated by the maximum noise in either of the two segments. In Figure 4c, the minimum in the noise is found at VBg = +8 V with VTg = 0. As a function of VTg, two maxima appear in the noise (Figure 4d). As the back gate is set at the value at which the noise shows a minimum, we can safely assume that the maxima in the noise arise from the part of the sample under the top gate. The combined contributions from both segments result in diﬀerent shapes of the noise characteristics (Figures 4e,f). This implies that the observed nonmonotonic noise is not generated at the p−n junction interfaces because then we would not have observed it in the case of a parallel p−n junction (or a globally gated graphene sample, as shown in Supporting Information, Figure S10).
Dependence of the Phase Coherence Length on the Carrier Density. Speciﬁcally for QIN, the increasing noise could also be due to an increase in the phase coherence length as a function of gate voltage.31 However, this eﬀect is unlikely to dominate since we have also observed similar nonmonotonic noise in devices that were shorter than the phase coherence length (see Supporting Information, Figure S10). Asymmetry. Both the noise and MR are asymmetric with respect to VD. If VD is positive, larger negative MR is observed on the electron side and vice versa (Figure 3). The position of VD also aﬀects the noise amplitude away from VD. When VD is negative, then the noise peak is larger on the hole side (Figures 2a, 3b, and 5b). Similarly, if VD is positive, then the noise peak is larger on the electron side (Figure 4d). For the sample shown in Figure 4, in which VD is positive, we ﬁnd that the noise peak is larger when the majority carriers under the top gate are electrons. In Figure 4f, we see that for VTg = 0 V the majority carriers are holes (VBg = −26 V). If we add more holes by applying a negative VTg, we ﬁnd that the noise increases slightly. However, for positive VTg, the majority carriers under the top gate are electrons, and a sharp increase in noise is observed. This excess noise on one side suggests that 6623
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noise20−22,24 and UCF.31,35 We ﬁnd that (a) asymmetric peak heights are not due to the transport through p−n junctions, (b) they are not due to contacts, (c) they are correlated with asymmetric negative MR, and (d) they depend on the location of the Dirac point, which is determined by the sign of the charged impurities.33 This suggests that the peak heights are related to the scattering on charged impurities, and such scattering is asymmetric with respect to the sign of the charge carriers. This observation is consistent with the theoretical prediction of asymmetric scattering of massless carriers by charged impurities36 and with STM observation of quasiparticle interference on localized scattering centers, observed only in the electron puddles in the sample in which VD was positive.37 We conclude by noting that QIN is a phenomenon that is sensitive to underlying symmetries and degrees of freedom and is complementary to conductance, STM measurements, and even UCF measurements: it has been reported that conductance ﬂuctuations in graphene are nonergodic35,38 (Supporting Information, Figures S11 and S12). This is not surprising, given that the time-dependent ﬂuctuations reﬂect a natural average over impurity conﬁgurations, while the magnetic ﬁeld-dependent ﬂuctuations probe one particular impurity conﬁguration. Further theoretical and experimental work on QIN should explore its potential as a useful probe for scattering mechanisms.
Figure 5. Asymmetric shape. The upper panels are schematics of the regions in which SR-type and LR-type noise is observed. Representative experimental data are shown for the cases in which the VD was located at +8 V (lower left panel, device SL4) and −10 V (lower right panel, device SL6).
scattering by the disorder present in the system has a stronger eﬀect on one type of a carrier as compared to the other. We note that an asymmetric contribution to 1/f noise could also arise due to ﬂuctuating gate leakage current.24 However, this contribution can be ruled out in our experiment (see Supporting Information for details). Another possible source of asymmetry could come from Klein tunneling in the case of the p−n−p or n−p−n junctions,32 but this is quite unlikely given the relatively large width of our top gates (150−500 nm). Discussion. The shape of the noise characteristics and their asymmetry can be summarized as shown in Figure 5. The noise shows a minimum near the Dirac point and increases monotonically with increasing gate voltage. This increase, along with the negative MR and the temperature dependence of the resistance observed in the same regime, can all be explained by SR-type disorder scattering. We note that this would imply that the vanishing carrier density coincides with the minimum in the noise power, not with the minimum in the conductance (which is conventionally associated with the Dirac point33). This regime is likely to be aﬀected by the puddles, and the noise could also be increasing due to ﬂuctuations in the minority puddles. At higher carrier densities, where screening increases, the noise becomes dominated by LR-type scattering. The gate voltage dependence of the noise contains both the classical and QIN contributions, while the temperature and magnetic ﬁeld dependence is dominated by QIN. Further theoretical considerations that include the eﬀect of SR disorder are needed to see whether the observed nonmonotonic noise could be reﬂecting a crossover between a pseudodiﬀusive and diﬀusive regime, as discussed by Rossi et al.14 One would intuitively assume that our disordered samples could not be in the pseudodiﬀusive regime. However, it has been suggested that the evanescent modes arising from SR disorder could bring about an intrinsic pseudodiﬀusive regime, which is analogous to the pseudodiﬀusive transport in clean graphene.34 Some asymmetry with respect to gate voltage is observed in almost all experiments reported so far, including classical 1/f
S Supporting Information *
Methods and materials for graphene device fabrication; measurement details; Figures S1 to S12. This material is available free of charge via the Internet at http://pubs.acs.org.
*E-mail: [email protected]
The manuscript was written through contributions of all authors. All authors have given approval to the ﬁnal version of the manuscript. Notes
The authors declare no competing ﬁnancial interest.
ACKNOWLEDGMENTS This work was supported by NSF Grant No. DMR-1106167. N.M. would also like to thank the Aspen Center for Physics (through the NSF Grant No. PHYS-1066293) for hospitality.
(1) Feng, S.; Lee, P. A.; Stone, A. D. Sensitivity of the Conductance of a Disordered Metal to the Motion of a Single Atom: Implications for 1/f Noise. Phys. Rev. Lett. 1986, 56, 1960. (2) Birge, N. O.; Golding, B.; Haemmerle, W. H. Electron Quantum Interference and 1/f Noise in Bismuth. Phys. Rev. Lett. 1989, 62, 195. (3) Altshuler, B. L.; Spivak, B. Z. Change of Random Potential Realization and Conductivity of Small Size Sample. JETP Lett. 1985, 42, 447. (4) Lee, P. A.; Stone, A. D.; Fukuyama, H. Universal Conductance Fluctuations in Metals: Effects of Finite Temperature, Interactions, and Magnetic Field. Phys. Rev. B 1987, 35, 1039. (5) Altshuler, B. L. Fluctuations in the Extrinsic Conductivity of Disordered Conductors. JETP Lett. 1985, 41, 648. (6) Bergmann, G. Weak Localization in Thin Films, a Time-of-Flight Experiment with Conduction Electrons. Phys. Rep. 1984, 107, 1.
dx.doi.org/10.1021/nl503276s | Nano Lett. 2014, 14, 6621−6625
(7) Stone, A. D. Phys. Rev. B 1989, 39, 10736. (8) Balandin, A. A. Nat. Nanotechnol. 2013, 8, 549. (9) Kotov, V. N.; Uchoa, B.; Pereira, V. M.; Guinea, F.; Castro-Neto, A. H. Electron-Electron Interactions in Graphene: Current Status and Perspectives. Rev. Mod. Phys. 2012, 84, 1067. (10) McCann, E.; Kechedzhi, K.; Falko, V. I.; Suzuura, H.; Ando, T.; Altshuler, B. L. Weak-Localization Magnetoresistance and Valley Symmetry in Graphene. Phys. Rev. Lett. 2006, 97, 146805. (11) Tworzydlo, J.; Trauzettel, B.; Titov, M.; Rycerz, A.; Beenakker, C. W. J. Sub-Poissonian Shot Noise in Graphene. Phys. Rev. Lett. 2006, 96, 246802. (12) Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. The Electronic Properties of Graphene. Rev. Mod. Phys. 2009, 81, 109. (13) Das Sarma, S.; Adam, S.; Hwang, E. H.; Rossi, E. Electronic Transport in Two Dimensional Graphene. Rev. Mod. Phys. 2010, 83, 407. (14) Rossi, E.; Bardarson, J. H.; Fuhrer, M. S.; Das Sarma, S. Universal Conductance Fluctuations in Dirac Materials in the Presence of Long-Range Disorder. Phys. Rev. Lett. 2012, 109, 096801. (15) Rahman, A.; Guikema, J. W.; Markovic, N. Quantum Interference Noise near the Dirac Point in Graphene. Phys. Rev. B 2014, 89, 235407. (16) Huard, B.; Stander, N.; Sulpizio, J. A.; Goldhaber-Gordon, D. Evidence of the Role of Contacts on the Observed Electron-Hole Asymmetry in Graphene. Phys. Rev. B 2008, 78, 121402. (17) Scofield, J. H. ac Method for Measuring Low Frequency Resistance Fluctuation Spectra. Rev. Sci. Instrum. 1987, 58, 985. (18) Kim, S.; Na, J.; Jo, I.; Shahrjerdi, D.; Colombo, L.; Yao, Z.; Tutuc, E.; Banerjee, S. K. Realization of a High Mobility Dual-Gated Graphene Field-Effect Transistor with Al2O3 Dielectric. Appl. Phys. Lett. 2009, 94, 062107. (19) Adam, S.; Hwang, E. H.; Galitski, V. M.; Das Sarma, S. SelfConsistent Theory for Graphene Transport. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 18392. (20) Heller, I.; Chatoor, S.; Männik, J.; Zevenbergen, M. A. G.; Oostinga, J. B.; Morpurgo, A. F.; Dekker, C.; Lemay, S. G. Charge Noise in Graphene Transistors. Nano Lett. 2010, 10, 1563. (21) Xu, G.; Torres, C. M., Jr.; Zhang, Y.; Liu, F.; Song, E. B.; Wang, M.; Zhou, Y.; Zeng, C.; Wang, K. Effect of Spatial Charge Inhomogeneity on 1/f Noise Behavior in Graphene. Nano Lett. 2010, 10, 3312. (22) Zhang, Y.; Mendez, E. E.; Du, X. Mobility-Dependent LowFrequency Noise in Graphene Field-Effect Transistors. ACS Nano 2011, 5, 8124. (23) Pal, A. N.; Ghatak, S.; Kochat, V.; Sneha, E. S.; Sampathkumar, A.; Raghavan, S.; Ghosh, A. Microscopic Mechanism of 1/f Noise in Graphene: Role of Energy Band Dispersion. ACS Nano 2011, 5, 2075−2081. (24) Kaverzin, A. A.; Mayorov, A. S.; Shytov, A.; Horsell, D. W. Impurities as a Source of 1/f Noise in Graphene. Phys. Rev. B 2012, 85, 075435. (25) McWhorter, A. L. In Semiconductor Surface Physics; Kingston, R. H., Ed.; University of Pennsylvania: Philadelphia, PA, 2012. (26) Hooge, F. N. 1/f Noise Is No Surface Effect. Phys. Lett. A 1969, 29, 139. (27) Ando, T. Screening Effect and Impurity Scattering in Monolayer Graphene. J. Phys. Soc. Jpn. 2006, 75, 074716. (28) Cheianov, V. V.; Fal’ko, V. I. Selective Transmission of Dirac Electrons and Ballistic Magnetoresistance of n-p Junctions in Graphene. Phys. Rev. B 2006, 74, 041403. (29) Jang, C.; Adam, S.; Chen, J.-H.; Williams, E. D.; Das Sarma, S.; Fuhrer, M. S. Tuning the Effective Fine Structure Constant in Graphene: Opposing Effects of Dielectric Screening on Short- and Long-Range Potential Scattering. Phys. Rev. Lett. 2008, 101, 146805. (30) Cheianov, V. V.; Fal’ko, V. I.; Altshuler, B. L.; Aleiner, I. L. Random Resistor Network of Minimal Conductivity in Graphene. Phys. Rev. Lett. 2007, 99, 176801.
(31) Pal, A. N.; Kochat, V.; Ghosh, A. Direct Observation of Valley Hybridization and Universal Symmetry of Graphene with Mesoscopic Conductance Fluctuations. Phys. Rev. Lett. 2012, 109, 196601. (32) Young, A. F.; Kim, P. Quantum Interference and Klein Tunneling in Graphene Heterojunctions. Nat. Phys. 2009, 5, 222. (33) Chen, J. H.; Jang, C.; Adam, S.; Fuhrer, M. S.; Williams, E. D.; Ishigami, M. Charged Impurity Scattering in Graphene. Nat. Phys. 2008, 4, 377. (34) Dietl, P.; Metalidis, G.; Golubev, D.; San-Jose, P.; Prada, E.; Schomerus, H.; Schn, G. Disorder-Induced Pseudodiffusive Transport in Graphene Nanoribbons. Phys. Rev. 2009, B 79, 195413. (35) Ojeda-Aristizabal, C.; Monteverde, M.; Weil, R.; Ferrier, M.; Gueron, S.; Bouchiat, H. Conductance Fluctuations and Field Asymmetry of Rectification in Graphene. Phys. Rev. Lett. 2010, 104, 186802. (36) Novikov, D. S. Elastic Scattering Theory and Transport in Graphene. Phys. Rev. B 2007, 76, 245435. (37) Zhang, Y.; Brar, V. W.; Girit, C.; Zettl, A.; Crommie, M. F. Origin of Spatial Charge Inhomogeneity in Graphene. Nat. Phys. 2009, 5, 722. (38) Bohra, G.; Somphonsane, R.; Aoki, N.; Ochiai, Y.; Akis, R.; Ferry, D. K.; Bird, J. P. Nonergodicity and Microscopic Symmetry Breaking of the Conductance Fluctuations in Disordered Mesoscopic Graphene. Phys. Rev. B 2012, 86, 161405.
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