By solving a master equation in the Sierpinski lattice and in a planar
random-resistor network, we determine the scaling with size L of the shot noise
power P due to elastic scattering in a fractal conductor. We find a power-law
scaling P ~ L^(d_f-2-alpha), with an exponent depending on the fractal
dimension d_f and the anomalous diffusion exponent alpha. This is the same
scaling as the time-averaged current I, which implies that the Fano factor
F=P/2eI is scale independent. We obtain a value F=1/3 for anomalous diffusion
that is the same as for normal diffusion, even if there is no smallest length
scale below which the normal diffusion equation holds. The fact that F remains
fixed at 1/3 as one crosses the percolation threshold in a random-resistor
network may explain recent measurements of a doping-independent Fano factor in
a graphene flake.Comment: 6 pages, 3 figure

B. I. Shklovskii

C. W. Groth

C. W. J. Beenakker

D. Stauffer

G. M. Schütz

J. Tworzydło

W. Sierpiński

English

arXiv

Crossref

Electronic Shot Noise in Fractal Conductors

Groth, C.W.

Tworzydlo, J.

Beenakker, C.W.J.

Leiden University Scholary Publications

Electronic Shot Noise in Fractal ConductorsC. W. Groth,1 J. Tworzydło,2 and C. W. J. Beenakker11Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands2Institute of Theoretical Physics, University of Warsaw, Hoża 69, 00-681 Warsaw, Poland(Received 13 February 2008; published 30 April 2008)By solving a master equation in the Sierpiński lattice and in a planar random-resistor network, wedetermine the scaling with size L of the shot noise power P due to elastic scattering in a fractal conductor.We find a power-law scaling P / Ldf2, with an exponent depending on the fractal dimension df andthe anomalous diffusion exponent . This is the same scaling as the time-averaged current I, whichimplies that the Fano factor F  P=2e I is scale-independent. We obtain a value of F  1=3 foranomalous diffusion that is the same as for normal diffusion, even if there is no smallest length scalebelow which the normal diffusion equation holds. The fact that F remains fixed at 1=3 as one crosses thepercolation threshold in a random-resistor network may explain recent measurements of a doping-independent Fano factor in a graphene flake.DOI: 10.1103/PhysRevLett.100.176804 PACS numbers: 73.50.Td, 05.40.Ca, 64.60.ah, 64.60.alDiffusion in a medium with a fractal dimension is char-acterized by an anomalous scaling with time t of the root-mean-squared displacement . The usual scaling for inte-ger dimensionality d is  / t1=2, independent of d. If thedimensionality df is noninteger, however, an anomalousscaling  / t1=2, with > 0, may appear. This anom-aly was discovered in the early 1980s [1–5] and has sincebeen studied extensively (see Refs. [6,7] for reviews).Intuitively, the slowing down of the diffusion can be under-stood as arising from the presence of obstacles at all lengthscales—characteristic of a self-similar fractal geometry.A celebrated application of the theory of fractal diffu-sion is to the scaling of electrical conduction in random-resistor networks (reviewed in Refs. [8,9]). According toOhm’s law, the conductance G should scale with the linearsize L of a d-dimensional network asG / Ld2. In a fractaldimension, the scaling is modified to G / Ldf2, de-pending both on the fractal dimensionality df and on theanomalous diffusion exponent. At the percolation thresh-old, the known [6] values for d  2 are df  91=48 and  0:87, leading to a scaling G / L0:97. This almostinverse-linear scaling of the conductance of a planarrandom-resistor network contrasts with the L-independentconductance G / L0 predicted by Ohm’s law in twodimensions.All of this body of knowledge applies to classical resis-tors, with applications to disordered semiconductors andgranular metals [10,11]. The quantum Hall effect providesone quantum mechanical realization of a random-resistornetwork [12], in a rather special way because time-reversalsymmetry is broken by the magnetic field. Very recently[13], Cheianov et al. announced an altogether differentquantum realization in zero magnetic field. Followingexperimental [14] and theoretical [15] evidence for elec-tron and hole puddles in undoped graphene [16], Cheianovet al. modeled this system by a degenerate electron gas [17]in a random-resistor network. They analyzed both the high-temperature classical resistance as well as the low-temperature quantum corrections by using the anomalousscaling laws in a fractal geometry.These very recent experimental and theoretical develop-ments open up new possibilities to study quantum me-chanical aspects of fractal diffusion, both with respect tothe Pauli exclusion principle and with respect to quantuminterference (which are operative in distinct temperatureregimes). To access the effect of the Pauli principle, oneneeds to go beyond the time-averaged current I (studied byCheianov et al. [13]) and consider the time-dependentfluctuations It of the current in response to a time-independent applied voltage V. These fluctuations existbecause of the granularity of the electron charge, hencetheir name ‘‘shot noise’’ (for reviews, see Refs. [18,19]).Shot noise is quantified by the noise power P  2Z 11dthI0Iti (1)and by the Fano factor F  P=2e I. The Pauli principleenforces F < 1, meaning that the noise power is smallerthan the Poisson value 2e I—which is the expected valuefor independent particles (Poisson statistics).The investigation of shot noise in a fractal conductor isparticularly timely in view of two different experimentalresults [20,21] that have been reported recently. Both ex-periments measure the shot noise power in a graphene flakeand find F < 1. A calculation [22] of the effect of the Pauliprinciple on the shot noise of undoped graphene predictedF  1=3 in the absence of disorder, with a rapid suppres-sion upon either p-type or n-type doping. This prediction isconsistent with the experiment of Danneau et al. [21], butthe experiment of DiCarlo et al. [20] gives instead anapproximately doping-independent F near 1=3. Computersimulations [23,24] suggest that disorder in the samples ofDiCarlo et al. might cause the difference.PRL 100, 176804 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 MAY 20080031-9007=08=100(17)=176804(4) 176804-1 © 2008 The American Physical SocietyMotivated by this specific example, we study here thefundamental problem of shot noise due to anomalousdiffusion in a fractal conductor. While equilibrium thermalnoise in a fractal has been studied previously [25–27], itremains unknown how anomalous diffusion might affectthe nonequilibrium shot noise. Existing studies [28–30] ofshot noise in a percolating network were in the high-voltage regime where inelastic scattering dominates, lead-ing to hopping conduction [31], while in the low-voltageregime of diffusive conduction we have predominantlyelastic scattering.We demonstrate that anomalous diffusion affects P and Iin such a way that the Fano factor (their ratio) becomesscale-independent as well as independent of df and .Anomalous diffusion, therefore, produces the same Fanofactor F  1=3 as is known [32,33] for normal diffusion.This is a remarkable property of diffusive conduction,given that hopping conduction does not produce a scale-independent Fano factor [28–30]. Our general findings areconsistent with the doping independence of the Fano factorin disordered graphene observed by DiCarlo et al. [20].To arrive at these conclusions, we work in the experi-mentally relevant regime where the temperature T is suffi-ciently high that the phase coherence length is  L andsufficiently low that the inelastic length is L. Quantuminterference effects can then be neglected, as well asinelastic scattering events. The Pauli principle remainsoperative if the thermal energy kT remains well belowthe Fermi energy, so that the electron gas remains degen-erate. We neglect Coulomb interactions between the elec-trons. In the application to graphene, this requires that theelectron and hole puddles have a conductance large com-pared to e2=h (no Coulomb blockade).We first briefly consider the case that the anomalousdiffusion on long length scales is preceded by normaldiffusion on short length scales. This would apply, forexample, to a percolating cluster of electron and holepuddles with a mean free path l which is short comparedto the typical size a of a puddle. We can then rely on thefact that F  1=3 for a conductor of any shape, providedthat the normal diffusion equation holds locally [34,35], toconclude that the transition to anomalous diffusion on longlength scales must preserve the one-third Fano factor.This simple argument cannot be applied to the moretypical class of fractal conductors in which the normaldiffusion equation does not hold on short length scales.As representative for this class, we consider fractal latticesof sites connected by tunnel barriers. The local tunnelingdynamics then crosses over into global anomalous diffu-sion, without an intermediate regime of normal diffusion.A classic example is the Sierpiński lattice [36] shown inFig. 1 (inset). Each site is connected to four neighbors bybonds that represent the tunnel barriers, with equal tunnelrate  through each barrier. (The tunnel rate is voltage-independent in the low-voltage limit considered here.) Thefractal dimension is df  log23, and the anomalous diffu-sion exponent is [6]   log25=4. The Pauli exclusionprinciple can be incorporated as in Ref. [37], by demandingthat each site is either empty or occupied by a singleelectron. Tunneling is therefore allowed only between anoccupied site and an adjacent empty site. A current ispassed through the lattice by connecting the lower-leftcorner to a source (injecting electrons so that the siteremains occupied) and the lower-right corner to a drain(extracting electrons so that the site remains empty). Theresulting stochastic sequence of current pulses is the ‘‘tun-nel exclusion process’’ of Ref. [38].The statistics of the current pulses can be obtainedexactly (albeit not in closed form) by solving a masterequation [39]. We have calculated the first two cumulantsby extending to a two-dimensional lattice the one-dimensional calculation of Ref. [38]. To manage the addedcomplexity of an extra dimension, we found it convenientto use the Hamiltonian formulation of Ref. [40]. The 0.3 0.40.001 0.01 0.1 1 1  10  100FIG. 1. Lower panel: Electrical conduction through aSierpiński lattice. This is a deterministic fractal, constructedby recursively removing a central triangular region from anequilateral triangle. The recursion level r quantifies the size L 2ra of the fractal in units of the elementary bond length a (theinset shows the third recursion). The conductance G  I=V(open dots, normalized by the tunneling conductance G0 of asingle bond) and shot noise power P (solid dots, normalized byP0  2eVG0) are calculated for a voltage difference V betweenthe lower-left and lower-right corners of the lattice. Both quan-tities scale as Ldf2  Llog23=5 (solid lines on the double-logarithmic plot). The Fano factor F  P=2e I  P=P0G0=Grapidly approaches 1=3, as shown in the upper panel.PRL 100, 176804 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 MAY 2008176804-2hierarchy of linear equations that we need to solve in orderto obtain I and P is derived in the supplement [41].The results in Fig. 1 demonstrate, first, that the shotnoise power P scales as a function of the size L of thelattice with the same exponent df  2   log23=5 asthe conductance and, second, that the Fano factor F ap-proaches 1=3 for large L. More precisely (see Fig. 2), wefind that F 1=3 / L1:5 scales to zero as a power law,with F 1=3< 104 for our largest L.Turning now to the application to graphene mentioned inthe introduction, we have repeated the calculation of shotnoise and Fano factor for the random-resistor network ofelectron and hole puddles introduced by Cheianov et al.[13]. The construction of the network is explained in Fig. 3.While this model was inspired by a specific application[13], it is generic and representative for any model of two-dimensional percolation in the low-voltage limit. The re-sults, shown in Fig. 3, demonstrate that the shot noisepower P scales with the same exponent L0:97 as theconductance G (solid lines in the lower panel) and thatthe Fano factor F approaches 1=3 for large networks(upper panel). This is a random, rather than a deterministic,fractal, so there remains some statistical scatter in the data,but the deviation of F from 1=3 for the largest lattices isstill <103 (see the circular data points in Fig. 2).In conclusion, we have found that the universality of theone-third Fano factor, previously established for normaldiffusion [32–35], extends to anomalous diffusion as well.This universality might have been expected with respect tothe fractal dimension df (since the Fano factor isdimension-independent), but we had not expected univer-sality with respect to the anomalous diffusion exponent .The experimental implication of the universality is that theFano factor remains fixed at 1=3 as one crosses the perco-lation threshold in a random-resistor network—therebycrossing over from anomalous diffusion to normal diffu-sion. This is consistent with the doping-independent Fanofactor measured in a graphene flake by DiCarlo et al. [20].A discussion with L. S. Levitov motivated this research,which was supported by the Netherlands ScienceFoundation NWO/FOM. We also acknowledge supportby the European Community’s Marie Curie ResearchTraining Network under Contract No. MRTN-CT-2003-504574, Fundamentals of Nanoelectronics.[1] I. Webman, Phys. Rev. Lett. 47, 1496 (1981).[2] S. Alexander and R. Orbach, J. Phys. Lett. 43, L625(1982).[3] D. Ben-Avraham and S. Havlin, J. Phys. A 15, L691(1982).[4] Y. Gefen, A. Aharony, and S. Alexander, Phys. Rev. Lett.50, 77 (1983).[5] R. Rammal and G. Toulouse, J. Phys. Lett. 44, L13 (1983). 0.001 0.01 0.1 1 1  10  100 – 5 10 – 4 10FIG. 2. The deviation of the Fano factor from 1=3 scales tozero as a power law for the Sierpiński lattice (triangles) and forthe random-resistor network (circles). 0.3 0.40.001 0.01 0.1 1 1  10  100FIG. 3. The same as Fig. 1, but now for the random-resistornetwork of disordered graphene introduced by Cheianov et al.[13]. The inset shows one realization of the network for L=a 10 (the data points are averaged over ’ 103 such realizations).The alternating solid and dashed lattice sites represent, respec-tively, the electron (n) and hole (p) puddles. Horizontal bonds(not drawn) are p-n junctions, with a negligibly small conduc-tance Gpn  0. Diagonal bonds (solid and dashed lines) eachhave the same tunnel conductance G0. Current flows from theleft edge of the square network to the right edge, while the upperand lower edges are connected by periodic boundary conditions.This plot is for undoped graphene, corresponding to an equalfraction of solid (n-n) and dashed (p-p) bonds.PRL 100, 176804 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 MAY 2008176804-3[6] S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695(1987).[7] M. B. Isichenko, Rev. Mod. Phys. 64, 961 (1992).[8] D. Stauffer and A. Aharony, Introduction to PercolationTheory (Taylor & Francis, London, 1994).[9] S. 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Electronic shot noise in fractal conductors

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Electronic shot noise in fractal conductors

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