Abstract
In quantum systems, signatures of multifractality are rare. They have been found only in the multiscaling of eigenfunctions at critical points. Here we demonstrate multifractality in the magnetic fieldinduced universal conductance fluctuations of the conductance in a quantum condensed matter system, namely, highmobility singlelayer graphene fieldeffect transistors. This multifractality decreases as the temperature increases or as doping moves the system away from the Dirac point. Our measurements and analysis present evidence for an incipient Andersonlocalization near the Dirac point as the most plausible cause for this multifractality. Our experiments suggest that multifractality in the scaling behavior of local eigenfunctions are reflected in macroscopic transport coefficients. We conjecture that an incipient Andersonlocalization transition may be the origin of this multifractality. It is possible that multifractality is ubiquitous in transport properties of lowdimensional systems. Indeed, our work suggests that we should look for multifractality in transport in other lowdimensional quantum condensedmatter systems.
Introduction
One of the unsolved problems in singlelayer graphene (SLG) is the nature of the electronic wavefunction near the chargeneutrality (Dirac) point. In principle, the charge carrier density of SLG should be continuously tunable, down to zero, leading to the largely unexplored regime of extremely weak interactions in a low carrier density system. The interaction parameter r_{s}, which parametrizes the ratio of the average interelectron Coulomb interaction energy to the Fermi energy, turns out to be independent of charge carrier density in the case of SLG where \(r_{\mathrm{s}} = e^2/(\kappa \hbar v_{\mathrm{F}})\). Here, κ is the dielectric constant of the surrounding medium and v_{F} is the Fermi velocity. In the case of SLG on an SiO_{2} substrate, r_{s} = 0.8. Thus SLG is a very weakly interacting system, when compared to other conventional twodimensional systems such as GaAs/AlGaAs and Si inversion layers, where the values of r_{s} are typically much higher. A naïve application of the scaling theory to such a system in this regime would predict Anderson localization—a disorderdriven quantum phase transition, leading to a complete localization of the charge carriers, and thence, to an insulator^{1,2,3,4,5}. Indeed, theoretical calculations for graphene indicate that intervalley scattering can lead to changes in the local and averaged electronic density of states with the creation of localized states^{6,7}. For example, grapheneterminated SiC (0001) surfaces undergo an Anderson localization transition upon dosing with small amounts of atomic hydrogen^{8}. This intervalley scatteringinduced localization is most effective near the Dirac point, where the screening of the impurity scatterers is negligible^{7}. Experiments, however, find the appearance of a minimum conductance value at the Dirac point with \(\sigma _{{\rm min}} \approx 2e^2/\pi \hbar\), instead of a diverging resistance which is the hallmark of a truly localized state; notable exceptions being carrier localization^{9} and an Anderson localization transition in bilayer graphene heterostructures^{10}.
By studying the scaling behavior of the universal conductance fluctuations (UCF), we look for signs of charge carrier localization near the Dirac point in ultrahighmobility SLG and uncover, as a result, an exotic multifractal behavior in the UCF. Multifractality, characterized by an infinite number of scaling exponents, is ubiquitous in classical systems. Since the pioneering work of Mandelbrot^{11}, the detection and analysis of multifractal scaling in such systems have enhanced our understanding of several complex phenomena, e.g., the dynamics of the human heart beat^{12}, the form of critical wavefunctions at the Anderson localization transition^{1}, the time series of the Sun’s magnetic field^{13}, in medical signal analysis (for instance, in pattern recognition, texture analysis and segmentation)^{14}, fully developed turbulence and in a variety of chaotic systems^{15,16}. In condensed matter systems, signatures of multifractality are usually sought in the scaling of eigenfunctions at critical points^{17,18,19,20,21,22,23,24}. Despite compelling theoretical predictions^{25,26,27,28,29,30}, there are no reports of the successful observation of multifractality in transport coefficients.
Simple fractal conductance fluctuations, on the other hand, have been observed in several condensed matter systems^{31,32,33,34,35}. These arise through semiclassical electron wave interference processes whenever a system has mixed chaotic regular dynamics^{31,34,36,37}. In all these systems, a primary prerequisite for the observation of fractal transport is that the electron dynamics be amenable to semiclassical analysis—the fractal nature of conductance fluctuations seen in these systems disappears as the system is driven deep into the quantum limit. For such a mixedphase semiclassical system, the graph of conductance (G) versus an externally applied magnetic field (B) has the same statistical properties as a Gaussian random process with increments of mean zero and variance (ΔB)^{γ}. These processes are known as fractional Brownian motion and have the property that their graph is a fractal of dimension \(D_{\mathrm{F}} = 2  \frac{\gamma }{2}\)^{31,34}, with 1 ≤ D_{F} ≤ 2.
A semiclassical description, however, breaks down in the case of materials where the charge carriers obey a Dirac dispersion relation, e.g., in graphene and topological insulators. In this report, we ask: can we find signatures of multifractality in a quantum system through transport measurements, specifically through the statistical properties of the graph of conductance fluctuations versus an external parameter like the magnetic field? We address this question by studying in detail the statistics of conductance fluctuations in highmobility SLG–FET devices as a function of the perpendicular magnetic field over a wide range of temperature and doping levels. We report, for the first time, the occurrence of multifractality in the UCF in highmobility graphene devices deep in the quantum limit. Our measurements and analysis suggest at an incipient Anderson localization transition in graphene near the Dirac point.
Results
Measurement of UCF
SLG–FET devices with mobilities in the range 20,000–30,000 cm^{2} V^{−1} s^{−1} were fabricated on SiO_{2} substrates by mechanical exfoliation from natural graphite, followed by conventional, electronbeam lithography^{38} (Fig. 1a, b). We begin with the dependence of the resistance (R) on the gate voltage (V_{G}), for the device G28M6. In Fig. 1c we show plots of R versus ΔV_{G} = V_{G} − V_{D}, where V_{D} is the Dirac point, measured at different temperatures (T). The high mobility and the position of the charge neutrality point very close to V_{G} = 0 V attests to the high quality of the devices. We measured the magnetoconductance (G) as a function of the magnetic field (B = (0, 0, B)), applied perpendicular to the plane of the device, in the range −0.2 T ≤ B ≤ 0.2 T. The presence of UCF was confirmed by the appearance of reproducible, nonperiodic, but magneticfieldsymmetric, oscillations in G. The measurements were performed on multiple devices, over a wide range of V_{G} and T. We find our UCF data to be in excellent agreement with previous studies of magnetoresistance and conductance fluctuations in SLG^{39,40,41,42,43}. Figure 2a shows illustrative plots of G(B) with the Fermi energy (E_{F}) maintained very close to the Dirac point (\(\Delta V_{\mathrm{G}} \simeq 0\)) for the device G28M6. The data for other devices are similar and are shown in the Supplementary Figure 1 (see Supplementary Note 1). At low values of B, near the minimum of G at B = 0, weaklocalization corrections are visible (Fig. 2a). As we move away from B = 0, the conductance fluctuations become prominent. The amplitudes of the UCF peaks, and the values of the charge carrier phase coherence length L_{ϕ}, obtained from the variance of the UCF (Fig. 2b), decrease with increasing temperature because of thermal dephasing (see Supplementary Note 2, Supplementary Figures 2, 3 and Supplementary Table 1 for a discussion on methods to obtain L_{ϕ}). The temperature dependence of the intervalley scattering length and the intravalley scattering length, extracted from weak localization measurements at T = 20 mK and ΔV_{G} = 0.2 V, are shown in the Supplementary Figure 4 (see Supplementary Note 3). We find them to be in excellent agreement with previous studies of localization in SLG^{42,44}. We also observe an increase in L_{ϕ} with increasing ΔV_{G} = V_{G} − V_{D} (Fig. 2c), which we attribute to the increase in the screening of impurities by charge carriers^{39}. We note an apparant saturation of L_{ϕ} below a temperature of \(\simeq 100\) mK. The saturation of the phasecoherence length (L_{ϕ}) with decreasing temperature is an issue that has been at the forefront of research in several other semiconducting materials including doped Si^{45,46,47}. There are many effects, e.g., the presence of magnetic impurities^{47} or finite size, which can lead to a saturation of L_{ϕ}. We note here that a decoupling of the electron and lattice temperature leading to a saturation of the L_{ϕ} is also possible. However, the data shown in Figs. 1c and 2a show a continuous evolution of both the conductance and conductance fluctuations down to 20 mK. This rules out any saturation of the electron temperature down to 20 mK, and hence, the observed saturation of L_{ϕ} below a certain temperature is not an experimental artifact (see Supplementary Note 3). In a separate set of measurements, we obtain the magnitude of the UCF, at a given magnetic field, by sweeping over V_{G} and calculating the rms value of the fluctuations. We found that this quantity decreased sharply with increasing magnetic field (see Supplementary Note 4, Supplementary Figure 5) in conformity with theoretical predictions^{48}.
Analysis of fractal scaling of the UCF
UCF represents quantum correction to Drude conductivity arising from the interference of electronic wavefunctions; it is a fingerprint of the disorder configuration in the conducting channel. Besides information about the phasecoherence of the charge carriers, it can also provide crucial insights into the electron dynamics and distribution of eigenstates through a scalingdimension analysis of the magnetoconductance traces^{31,32,34}. We first compute the simple fractal dimensions D_{F} of the UCF curves via the Ketzmerick variance method (see Supplementary Note 5, and Supplementary Figure 6 for details). Figure 3a shows plots of D_{F} versus T for the device G28M6. At very low T and small  ΔV_{G}, we find 1 < D_{F} < 2. With increasing temperature, D_{F}→1 monotonically. In this highT regime, the thermaldiffusion length \(L_{\mathrm{T}} = \sqrt {\hbar D/k_{\mathrm{B}}T} \ll L_\phi\), with D the thermal diffusion coefficient of the charge carriers; therefore, quasiparticle phase decoherence, induced by inelastic thermal scattering, suppresses quantum interference. For large ΔV_{G}, the magnitude of the UCF is comparable to, or smaller than the background electrical noise, so D_{F}→2, the value for Gaussian white noise. In Fig. 3b, we plot D_{F} versus L_{ϕ} for two different devices: G28M6 and G30M4; remarkably, all the data points from these two devices cluster in the vicinity of a curve, with \(D_{\mathrm{F}} \propto \ln(L_\phi )\). In the limits \(L_\phi \ll L\), the UCF is nonfractal (\(D_{\mathrm{F}} \simeq 1\)), whereas for large L_{ϕ}, the UCF is a fractal, so 1 < D_{F} < 2.
Analysis of multifractal scaling of the UCF
We build upon the predictions of multifractal scaling of conductance fluctuations in quantum systems^{25} by carrying out a multifractal detrended fluctuation analysis of our UCF (see Supplementary Note 6, Supplementary Figures 7 and 8 for details). The multifractality can be represented in the following two ways: (1) by the generalized Hurst exponent h(q), defined using the orderq moment of the UCF as \(\langle {\mathrm{rms}}[\Delta G(\Delta B)]^q\rangle ^{1/q} \sim [\Delta B]^{h(q)}\), and (2) by the multifractal spectrum f(α), obtained from the Legendre transform of h(q). For a monofractal function, h(q) has a single, qindependent value. For each one of our UCF plots, we obtain h(q) in the range −4 ≤ q ≤ 4. An illustrative plot of h(q) versus q that we obtain from our magnetoconductance data at 20 mK (Fig. 4a) is shown in Fig. 4b; h(q) goes smoothly from \(\simeq 1.9\), at q = −4, to \(\simeq 0.85\), at q = 4; the corresponding f(α) spectrum is plotted in Fig. 4c. The singularity spectra have a definite maximum value of 1 (which is the dimension of the support graph). The width of the multifractal spectrum is defined as Δα ≡ h(q)_{max} − h(q)_{min}. The wide range of h(q), or, equivalently, the wide spectrum (Δα = 1.05) quantifies the multifractality of the UCF. This is the first observation of multifractality of a conductance in any quantumcondensed matter system and is the central result of our work.
We note that there are two distinct properties of UCF that can give rise to its multifractal behavior^{49,50}: (i) a fattailed, nonGaussian distribution of the UCF differences (as a function of δB) or (ii) longrange, in the magnetic field B, correlations of the fluctuations of δg(B). We have verified that the distribution of our measured δg(B) is not lognormal. Having ruled out (i), we now give a convenient test for (ii): we check for multifractality in a data set obtained from a random shuffling (see Supplementary Note 7) of the original sequence of δg(B). If the longrange correlations (ii) exist, then signatures of multifractality must be absent in the reshuffled data. Indeed, in our experimental data, we observe a nearcomplete suppression of multifractality in the shuffled data set with \(h_{\mathrm{shuf}}(q) \simeq 0.5\) for all values of q and Δα = 0.05 (Supplementary Figure 9). Hence it is reasonable to infer that the multifractality in our UCF can be traced back to longranged correlations, that are otherwise difficult to measure.
Figure 4d shows plots of f(α) for different values of T and ΔV_{G} = 0.2 V. The symmetry of f(α) about α_{0} (Fig. 4c) reflects the distribution of fluctuations, about the mean of the UCF. The largefluctuation (smallfluctuation) segments contribute predominantly to the q > 0 (q < 0) part of h(q). The q > 0 (q < 0) part of h(q) maps onto the α < α_{0} (α > α_{0}) region of f(α), which is moreorless symmetric about α_{0} at low T (Fig. 4d). As we increase T, this symmetry is lost. The magnitude of the skewness \(\langle [\delta \alpha ]^3\rangle /\langle [\delta \alpha ]^2\rangle ^{3/2}\), where δα = (α−α_{0}), increases with T. Therefore, as T increases, largeamplitude conductance fluctuations become rarer than smallamplitude fluctuations. This is consistent with our plots of the UCF (Fig. 2a).
Discussion
A naïve characterization of the fractal property of a curve, say by the measurement of one fractal dimension, does not rule out multifractality of this curve, which requires the calculation of an infinite number of dimensions^{51} (related to h(q)). One dimension suffices for monofractal scaling (as, e.g., in the scaling of velocity structure functions in the inverse cascade region of forced, twodimensional fluid turbulence^{52,53}). Our measurements of the UCF in SLG show that it is multifractal only if (i) the temperature is low and (ii) \(\Delta V_{\mathrm{G}} \simeq 0\). If either one of these conditions is not met, the plot of δg versus B is a monofractal (see Supplementary Note 8, and Supplementary Figure 10); at sufficiently large T, D_{F}→1 and the plots are nonfractal.
What can be the possible origin of our multifractal UCF? We list three potential causes: (1) scarring of wave functions (e.g., because of classicallychaotic billiards^{54}); (2) quasiperiodicity in the Hamiltonian induced by a magnetic field^{55,56,57} and its analog for graphene^{58}; (3) Anderson localizationinduced multifractality^{1}. We critically examine each one of these possibilities below and conclude that our results are most compatible with the last of these mechanisms.
While describing a quantum system, whose classical analog is chaotic, one encounters scarred wavefunctions, whose intensity is enhanced along unstable, periodic orbits of the classical system. This nonuniform distribution of intensity of wavefunctions results from quasiparticle interference. Quantum scars can lead to pointer states^{59} with long trapping times, and, consequently, to large conductance fluctuations. Relativistic quantum scars have been predicted theoretically in geometrically confined graphene stadia, which exhibit classical chaos^{60}. However, our devices are not shaped like billiards that are classically chaotic; and the charge carriers are not in the ballistic regime. Therefore, quantum scars cannot be the underlying cause for our multifractal UCF.
Fractal energy spectra can arise in tightbinding problems with an external magnetic field, which can be mapped onto Schrödinger problems with quasiperiodic potentials^{55,56,57,61}. It has been argued^{58}, therefore, that a fractal conductance can also arise, via Hofstadterbutterflytype spectra, in Dirac systems at sufficiently high magnetic fields \(B_{\mathrm{H}} \simeq \phi _0{\mathrm{/}}A_0\) (\(\simeq 10^5\) T for our samples), where \(\phi _0 = 2\pi \hbar /e\) is the flux quantum and A_{0} is the unit cell area. Our measurements use very lowmagnitude magnetic fields (\(\lesssim 0.2\) T); this rules out a quasiperiodicityinduced multifractal UCF.
The most compelling explanation of the multifractal UCF we observe in our SLG samples is an incipient Anderson localization near the charge neutrality point. Multifractality of the local amplitudes of critical eigenstates near Anderson localization has been studied, theoretically, in several quantumcondensed matter systems^{19,22,23,24}. The multifractality of the eigenstates near the critical point directly affects the twoparticle correlation function through the generalized diffusion coefficient^{62,63}, which, in turn, affects the local current fluctuations in the system via the Kubo formula. It is not obvious that this must be reflected in the (macroscopic) conductance, or its moments; however, it is plausible that near the critical point, the UCF may inherit multifractal behavior from its counterpart in the eigenfunctions^{64}. Indeed, there are several theoretical predictions of multifractality in transport coefficients including conductance jumps near the percolation threshold in random resistor networks^{27,28}, conductance fluctuations in quantum Hall transitions^{29}, and the temperature dependence of the peak height of the conductance at the Anderson localization transition^{30}.
Spectroscopic studies on singlelayer onsubstrate graphene devices have revealed that the local potential fluctuations in this system are strongest when E_{F} is close to the Dirac point^{7,9}. This leads to electronic states that are quasilocalized^{65,66,67}. Such quasilocalized states have a high inverse participation ratio^{6} that can lead to the multifractality seen in our experiments. If this is true, then the multifractality in the UCF should be largest near the Dirac point; and then fall off on either side of it. In Fig. 5 we show the dependence of Δα on T and ΔV_{G} for the device G28M6 (data obtained for other devices are qualitatively similar). We observe that Δα is indeed largest near ΔV_{G} = 0 and at low T, where the conductance of the device is of the order of e^{2}/h, and it sharply decreases as either T or the magnitude of ΔV_{G} increases. Similarly, as T is increased, thermal scattering increases quasiparticle dephasing, and eventually at high T, when \(L_\phi \sim L_{\mathrm{T}} \ll L\), quantuminterference effects are masked. From our observation that a large multifractality arises only when quantum interferenceinduced charge carrier localization is significant, we propose that an incipient Anderson localization near the Dirac point is the most plausible origin of multifractal UCF in SLG.
This interpretation of the multifractality of the UCF in SLG devices is based on previous theoretical predictions and analysis. We summarize our argument below.
Conductance fluctuations, as a function of the magnetic field, have been shown to have a fractional fractal dimensions in some simple, onedimensional, quantum systems, e.g., the kicked rotor^{64,68}. In these systems, such a fractional fractal dimension arises if one of the following conditions holds: (1) The PDF P(t), of the charge carrier survival time t, has a powerlaw form \(P(t) \propto t^{  \gamma }\) at large t (as opposed to an exponential decay); (2) the energy correlation C(ΔE) of elements of the Smatrix exhibit power laws (i.e., C(ΔE) \(\propto\) (ΔE)^{−γ})^{31,34,68,69}. Such survival probabilities are related to the conductance^{64,68}; and their multifractal behavior has been explored^{25}.
At the Anderson localization transition, it is known that both the probability density function describing the diffusion of a wavepacket and the twoparticle correlation function decay algebraically with a fractional power^{62,63,70}. Hence, as in the case of the simple quantum systems mentioned above, we may expect multifractal fluctuations of the conductance at the Anderson localization transition. However, to the best of our knowledge, there are no exact, analytical results that yield a onetoone correspondence between the multifractality of a critical wavefunction and the multifractality of the magnetoconductance. Thus, we propose that the multifractality of the critical wavefunctions at the Anderson localization is the most plausible cause of the multifractality of the UCF we have observed.
In conclusion, we have uncovered and quantified the multifractal structure of mesoscopic conductance fluctuations in SLG devices. We speculate that our results are indicative of an incipient Anderson localization in this system. In particular, we quantify the multifractality of transport in a quantum condensed matter system. There may well be multifractality in transport properties in systems other than graphene, and that multifractality is not unique to graphene. Our work provides a natural framework for studying the multifractality of such transport properties.
Methods
Sample fabrication
The graphene flakes were exfoliated from natural graphite onto highly doped Si wafer with thermally grown, 285nm thick SiO_{2} on top. SLG flakes were identified by optical microscopy, and further, confirmed by Raman spectroscopy, or by measurement of integer quantumhall plateau positions. Electrical contacts to the SLG were made by standard electronbeam lithography followed by thermal evaporation of 5 ~ 7 nm chromium and 60 nm gold. The highly doped Si was used as the backgate electrode, and the SiO_{2} was used as the gate dielectric, which enabled us to tune the charge carrier density, and hence the Fermi level of the device, globally.
Measurement technique
The devices were pumped overnight, before cooling down, to remove moisture and other adsorbents from the graphene surface. The electrical transport characteristics of the devices were measured in a cryogenfree Oxford Instruments Triton 400 dilution refrigerator. Cryogenic filters were used to remove highfrequency noise. The conductance of the device were measured in standard lowfrequency ac lockin measurement technique, in a fourprobe configuration. The biasing current (typically 0.25–0.5 nA) was kept sufficiently small to avoid electron heating.
Data analysis
The computation of the fractal dimension of the data were carried out using the Ketzmeric variance method. The multifractal exponents were computed following Multifractal Detrended Fluctuation Analysis method. The details of these methods have been discussed in the Supplementary Note 5 and Supplementary Note 6, respectively.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
We acknowledge discussions with AD. Mirlin and S. Bhattacharjee. A.B. acknowledges financial support from Nanomission, DST, Govt. of India project SR/NM/NS35/2012; SERB, DST, Govt. of India and IndoFrench Centre for the Promotion of Advanced Recearch (CEFIPRA). K.R.A. thanks CSIR, MHRD, Govt. of India for financial support. S.S.R. acknowledge financial support from the AIRBUS Group Corporate Foundation Chair in Mathematics of Complex Systems established in ICTSTIFR, the support from DST, Govt. of India, project ECR/2015/000361, and the IndoFrench Center for Applied Mathematics (IFCAM). R.P. acknowledges DST, Govt. of India for support. N.P. thanks UGC, Govt. of India for financial support.
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K.R.A and A.B. designed the experiment, fabricated the devices and carried out the measurements. K.R.A., A.B., N.P., S.S.R. and R.P. carried out the data analysis. All authors discussed the results and wrote the paper.
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Amin, K.R., Ray, S.S., Pal, N. et al. Exotic multifractal conductance fluctuations in graphene. Commun Phys 1, 1 (2018). https://doi.org/10.1038/s4200501700014
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