Introduction

One of the unsolved problems in single-layer graphene (SLG) is the nature of the electronic wave-function near the charge-neutrality (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 rs, which parametrizes the ratio of the average inter-electron 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 vF is the Fermi velocity. In the case of SLG on an SiO2 substrate, rs = 0.8. Thus SLG is a very weakly interacting system, when compared to other conventional two-dimensional systems such as GaAs/AlGaAs and Si inversion layers, where the values of rs are typically much higher. A naïve application of the scaling theory to such a system in this regime would predict Anderson localization—a disorder-driven quantum phase transition, leading to a complete localization of the charge carriers, and thence, to an insulator1,2,3,4,5. 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 states6,7. For example, graphene-terminated SiC (0001) surfaces undergo an Anderson localization transition upon dosing with small amounts of atomic hydrogen8. This intervalley scattering-induced localization is most effective near the Dirac point, where the screening of the impurity scatterers is negligible7. 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 localization9 and an Anderson localization transition in bilayer graphene heterostructures10.

By studying the scaling behavior of the universal conductance fluctuations (UCF), we look for signs of charge carrier localization near the Dirac point in ultra-high-mobility 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 Mandelbrot11, 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 beat12, the form of critical wave-functions at the Anderson localization transition1, the time series of the Sun’s magnetic field13, in medical signal analysis (for instance, in pattern recognition, texture analysis and segmentation)14, fully developed turbulence and in a variety of chaotic systems15,16. In condensed matter systems, signatures of multifractality are usually sought in the scaling of eigenfunctions at critical points17,18,19,20,21,22,23,24. Despite compelling theoretical predictions25,26,27,28,29,30, 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 systems31,32,33,34,35. These arise through semi-classical electron wave interference processes whenever a system has mixed chaotic regular dynamics31,34,36,37. In all these systems, a primary prerequisite for the observation of fractal transport is that the electron dynamics be amenable to semi-classical 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 mixed-phase semi-classical 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 ≤ DF ≤ 2.

A semi-classical 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 high-mobility 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 high-mobility 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 cm2 V−1 s−1 were fabricated on SiO2 substrates by mechanical exfoliation from natural graphite, followed by conventional, electron-beam lithography38 (Fig. 1a, b). We begin with the dependence of the resistance (R) on the gate voltage (VG), for the device G28M6. In Fig. 1c we show plots of R versus ΔVG = VG − VD, where VD is the Dirac point, measured at different temperatures (T). The high mobility and the position of the charge neutrality point very close to VG = 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, non-periodic, but magnetic-field-symmetric, oscillations in G. The measurements were performed on multiple devices, over a wide range of VG and T. We find our UCF data to be in excellent agreement with previous studies of magnetoresistance and conductance fluctuations in SLG39,40,41,42,43. Figure 2a shows illustrative plots of G(B) with the Fermi energy (EF) 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, weak-localization 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 Lϕ, 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 Lϕ). The temperature dependence of the intervalley scattering length and the intravalley scattering length, extracted from weak localization measurements at T = 20 mK and ΔVG = 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 SLG42,44. We also observe an increase in Lϕ with increasing |ΔVG| = |VG − VD| (Fig. 2c), which we attribute to the increase in the screening of impurities by charge carriers39. We note an apparant saturation of Lϕ below a temperature of \(\simeq 100\) mK. The saturation of the phase-coherence length (Lϕ) with decreasing temperature is an issue that has been at the forefront of research in several other semiconducting materials including doped Si45,46,47. There are many effects, e.g., the presence of magnetic impurities47 or finite size, which can lead to a saturation of Lϕ. We note here that a decoupling of the electron and lattice temperature leading to a saturation of the Lϕ 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 Lϕ 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 VG 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 predictions48.

Fig. 1
figure 1

Device structure and characteristics. a False-colour SEM image and b schematic device configuration of our SLG–FET. The scale-bar in a is 2 μm. c Plot of R versus ΔVG, measured at different temperatures. The values of the temperatures at which the data were collected are mentioned in the legend. In the inset, we plot the same data, zoomed in near the Dirac point (|ΔVG| ≤ 2 V)

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Fig. 2
figure 2

UCF in graphene. a Illustrative plots of the magnetoconductance G versus the magnetic field B measured at 0.02, 0.05, 0.075, 0.10, 0.20, 0.50, 1.00, and 2.00 K (from top to bottom—curves for different temperatures have been shifted vertically for clarity). The data shown here are from the device G28M6 measured close to the Dirac point (ΔVG = 0.2 V). b Plot of the phase-coherence length Lϕ versus the temperature T extracted from the UCF measured at ΔVG = 0.2 V. c Plot of Lϕ versus ΔVG, extracted from the UCF measured at T = 20 mK. The error-bars in b, c represent the standard deviation in Lϕ

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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 phase-coherence of the charge carriers, it can also provide crucial insights into the electron dynamics and distribution of eigenstates through a scaling-dimension analysis of the magnetoconductance traces31,32,34. We first compute the simple fractal dimensions DF of the UCF curves via the Ketzmerick variance method (see Supplementary Note 5, and Supplementary Figure 6 for details). Figure 3a shows plots of DF versus T for the device G28M6. At very low T and small | ΔVG|, we find 1 < DF < 2. With increasing temperature, DF→1 monotonically. In this high-T regime, the thermal-diffusion 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 |ΔVG|, the magnitude of the UCF is comparable to, or smaller than the background electrical noise, so DF→2, the value for Gaussian white noise. In Fig. 3b, we plot DF versus Lϕ 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 non-fractal (\(D_{\mathrm{F}} \simeq 1\)), whereas for large Lϕ, the UCF is a fractal, so 1 < DF < 2.

Fig. 3
figure 3

Fractal dimension of UCF in graphene. a Plots of the fractal dimension DF versus the temperature T for different values of ΔVG. The error-bars represent the standard deviation in DF. b DF versus the charge carrier phase coherence length Lϕ on a semi-log scale; the data points, with error bars representing standard deviation in DF and Lϕ, are from two different devices (filled symbols: G28M6; open symbols: G30M4) and for the following parameter ranges: 20 mK < T < 10 K and −4.5 V < ΔVG < 4.5 V. The numbers in the legend refer to the (VG, T) parameter space. The red curve is \(D_{\mathrm{F}} \propto \ln(L_\phi )\) (see text)

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Analysis of multifractal scaling of the UCF

We build upon the predictions of multifractal scaling of conductance fluctuations in quantum systems25 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 order-q 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, q-independent 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 quantum-condensed matter system and is the central result of our work.

Fig. 4
figure 4

Multifractality of UCF in graphene. a A typical plot of the magnetoconductance at T = 20 mK, ΔVG = 0.2 V for our device G28M6. b Plot of the generalized Hurst exponent h(q) versus q, calculated from the UCF data in a. A few representative error bars, defining the standard deviation in the data, are shown in the plot. c Plot of the multifractal singularity spectrum f(α) versus α that follows from the h(q) versus q data plotted in b. The dotted vertical line marks the location of the maximum of f(α). d Plots illustrating the temperature dependence of f(α) at ΔVG = 0.2 V for the device G28M6

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We note that there are two distinct properties of UCF that can give rise to its multifractal behavior49,50: (i) a fat-tailed, non-Gaussian distribution of the UCF differences (as a function of δB) or (ii) long-range, 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 log-normal. 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 long-range correlations (ii) exist, then signatures of multifractality must be absent in the reshuffled data. Indeed, in our experimental data, we observe a near-complete 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 long-ranged correlations, that are otherwise difficult to measure.

Figure 4d shows plots of f(α) for different values of T and ΔVG = 0.2 V. The symmetry of f(α) about α0 (Fig. 4c) reflects the distribution of fluctuations, about the mean of the UCF. The large-fluctuation (small-fluctuation) 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 < α0 (α > α0) region of f(α), which is more-or-less symmetric about α0 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−α0), increases with T. Therefore, as T increases, large-amplitude conductance fluctuations become rarer than small-amplitude 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 dimensions51 (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, two-dimensional fluid turbulence52,53). 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, DF→1 and the plots are non-fractal.

What can be the possible origin of our multifractal UCF? We list three potential causes: (1) scarring of wave functions (e.g., because of classically-chaotic billiards54); (2) quasi-periodicity in the Hamiltonian induced by a magnetic field55,56,57 and its analog for graphene58; (3) Anderson localization-induced multifractality1. 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 non-uniform distribution of intensity of wavefunctions results from quasiparticle interference. Quantum scars can lead to pointer states59 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 chaos60. 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 tight-binding problems with an external magnetic field, which can be mapped onto Schrödinger problems with quasiperiodic potentials55,56,57,61. It has been argued58, therefore, that a fractal conductance can also arise, via Hofstadter-butterfly-type 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 A0 is the unit cell area. Our measurements use very low-magnitude magnetic fields (\(\lesssim 0.2\) T); this rules out a quasi-periodicity-induced 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 quantum-condensed matter systems19,22,23,24. The multifractality of the eigenstates near the critical point directly affects the two-particle correlation function through the generalized diffusion coefficient62,63, 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 eigenfunctions64. Indeed, there are several theoretical predictions of multifractality in transport coefficients including conductance jumps near the percolation threshold in random resistor networks27,28, conductance fluctuations in quantum Hall transitions29, and the temperature dependence of the peak height of the conductance at the Anderson localization transition30.

Spectroscopic studies on single-layer on-substrate graphene devices have revealed that the local potential fluctuations in this system are strongest when EF is close to the Dirac point7,9. This leads to electronic states that are quasi-localized65,66,67. Such quasi-localized states have a high inverse participation ratio6 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 ΔVG for the device G28M6 (data obtained for other devices are qualitatively similar). We observe that Δα is indeed largest near ΔVG = 0 and at low T, where the conductance of the device is of the order of e2/h, and it sharply decreases as either T or the magnitude of ΔVG increases. Similarly, as T is increased, thermal scattering increases quasiparticle dephasing, and eventually at high T, when \(L_\phi \sim L_{\mathrm{T}} \ll L\), quantum-interference effects are masked. From our observation that a large multifractality arises only when quantum interference-induced 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.

Fig. 5
figure 5

Width of the multifractal spectrum. a Dependence of the width Δα on temperature at ΔVG = 0.2 V; the data were obtained from the UCF plots shown in Fig. 2a. b Blue filled circles (left axis) show the dependence of the width of the multifractal spectrum, Δα on ΔVG; the data were obtained at 20 mK. The red line shows the dependence of R on VG, measured at 20 mK (right axis). It is clear from the plot that Δα is large only over a narrow range of gate voltage around ΔVG = 0 V, and sharply decreases as the system is driven away from the Dirac point. The error bars in a, b represent the standard deviation in Δα

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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, one-dimensional, quantum systems, e.g., the kicked rotor64,68. 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 power-law 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 S-matrix exhibit power laws (i.e., C(ΔE) \(\propto\) (ΔE)−γ)31,34,68,69. Such survival probabilities are related to the conductance64,68; and their multifractal behavior has been explored25.

At the Anderson localization transition, it is known that both the probability density function describing the diffusion of a wave-packet and the two-particle correlation function decay algebraically with a fractional power62,63,70. 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 one-to-one 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, 285-nm thick SiO2 on top. SLG flakes were identified by optical microscopy, and further, confirmed by Raman spectroscopy, or by measurement of integer quantum-hall plateau positions. Electrical contacts to the SLG were made by standard electron-beam lithography followed by thermal evaporation of 5 ~ 7 nm chromium and 60 nm gold. The highly doped Si was used as the back-gate electrode, and the SiO2 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 cryogen-free Oxford Instruments Triton 400 dilution refrigerator. Cryogenic filters were used to remove high-frequency noise. The conductance of the device were measured in standard low-frequency ac lock-in measurement technique, in a four-probe 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.