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Electronic Branched Flow in Graphene with Random Potential:

Theory and Machine Learning Prediction

Marios Mattheakis,1, ∗ G. P. Tsironis,1, 2 and Efthimios Kaxiras1, 3

1School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA2Department of Physics, University of Crete, Heraklion 71003, Greece

3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

(Dated: Draft of January 26, 2018)

We investigate the ultra-relativistic electronic flow in a two-dimensional random potential relevantto charge carrier dynamics in Dirac solids. As an example, the random potential in graphene, theprototypical Dirac solid, arises from inhomogeneous charge impurities in the substrate, a commonfeature in experimental systems. An additional bias voltage is introduce to tune the electronicpropagation. We show that the onset of electronic branched flow is determined by the statisticalproperties of the potential and we provide a scaling-type relationship that describes the emergenceof branches. We also show that, despite the statistical nature of branching, reservoir computingprovides an accurate detection mechanism for the caustics. The onset of branching is predictedthrough a deep learning algorithm, which may be implemented experimentally to improve materialsproperties of graphene substrates.

Keywords: Dirac solids; graphene; random potential; caustics; machine learning; reservoir computing

Wave focusing due to refractive index variation is acommon occurrence in many physical systems. In seawaves, the effective refractive index variation arises fromfluctuating depth [1–4]; in optical [5–10] or other media[11, 12] the index of refraction changes in a statisticalway due to small imperfections or distributions of de-fects. Random spatial variability of the index leads tolocal focusing and defocusing of the waves and the for-mation of caustics or wave branches with substantiallyincreased local wave intensity. Under general circum-stances the branching flow develops a stochastic web withstatistical patterns of persisting enhanced intensity wavemotion. Since electrons have also wave properties dueto their quantum nature, similar phenomena appear inthe quantum realm. Injected electrons in disordered twodimensional (2D) electron gas form coalescing trajecto-ries and manifest phenomena similar to wave motion inrandom media [13–16]. One aspect of the electronic mo-tion that has not yet been explored is the relativisticand in particular the ultra-relativistic one; the latter oc-curs in materials referred to as Dirac solids (DS), suchas graphene [17–21]. In the ultra-relativistic limit, themagnitude of the Dirac fermion velocity cannot be af-fected by external fields as it is already at its maximumvalue, leading to significant differences from the conven-tional non-relativistic flow. This is directly reflected inthe electronic branching properties and gives rise to dis-cernible differences, as we show here.

Pristine graphene is the prototypical 2D-DS, character-ized by linear dispersion in the electronic band structurenear the Fermi level [17, 18],

ǫk = vF ~|k|, (1)

where ǫk is the single particle energy, vF is the Fermivelocity and k is the wave-vector. Electron flow in these

band segments is ultra-relativistic with maximal propa-gation velocity vF [17, 21–24]. This electron flow is mod-ified by the presence of a bias potential applied along aspecific direction. The relativistic electronic dispersioncouples the motion of electrons along the direction ofbias and the one perpendicular to it [22, 23]. Electrondynamics is also subject to the presence of substitutionalor other type of weak disorder; the effects of such disorderare observed in graphene in the form of electronic pud-dles [24–28]. The combined presence of disorder and biasalters the electronic flow and thus the ultra-relativistictrajectories coalesce into branches of substantial localdensity. In this letter we show that the weak surfacedisorder produces a lensing mechanism for the electronicwaves that is clearly manifested in the form of caustics.Dirac fermions cannot be accelerated by a bias poten-tial, in contrast to the corresponding behavior of non-relativistic electrons in 2D metals with parabolic bands.As a result, a simple relationship derived here betweenthe caustic location and statistical properties of the in-trinsic potential remains valid even in the presence of ex-ternal fields; we derive this scaling-type formula analyti-cally and verify it by numerical simulations. In order topredict and monitor the stochastic caustic events we uti-lize a machine learning method implemented through thedeep learning algorithm of reservoir computing [29, 30].Scanning probe microscope (STM) techniques have beenused extensively to measure the electron density [20] andin turn, the branching flow in 2D electron gases [13, 15].We show that through the implementation of reservoircomputing, the range of STM measurements needed canbe drastically reduced, accelerating the image processingof electronic caustic events and introducing efficient datacompression.

Theory: The basic feature of a 2D-DS is the linear dis-persion relation of the energy with wave-vector, Eq. (1).

2

When the electronic density is low we may use the in-dependent electron model to describe the quasi-classicaldynamics of charge carriers [22, 23] through the quasi-classical ultra-relativistic Hamiltonian

H = ±vF

√

p2x + p2y + V (x, y), (2)

where p = (px, py) is the 2D momentum of the chargecarriers. The Hamiltonian (2) is the classical limit ofDirac equation [22] that describes dynamics of masslesselectron/hole quasi-particles in graphene and other DS[18]. Branched flow is an effect of ray fields associatedwith waves [1, 4, 11] and thus, we may use the Hamilto-nian (2) to give a ray description of the quantum flow ofDirac electrons. We note that STM measurements haveshown that classical ray-tracing simulations describe ac-curately the electron flow in graphene [20].We focus on 2D ultra-relativistic dynamics of particles

in a medium with potential V (x, y) = Vr(x, y) + Vd(x),where Vr(x, y) is a random δ-correlated potential withenergy scale much smaller than that of the electronicflow; this term comes from random charged impuri-ties in the substrate or other sources of disorder in thegraphene sheet [24–28]. The deterministic “control” po-tential Vd(x) = −αx is due to an externally applied volt-age in the x-direction with α fixed by experimental condi-tions. Electrons are injected in the Dirac sheet with ini-tial momentum p0 along the x-direction; due to the ballis-tic electronic motion along the x-axis we may ignore theeffects of the random potential in this direction [14, 16].We use plane wave initial conditions for the electrons,px(0) = p0 and py(0) = 0, and write the correspondingsolution of Hamilton’s equations as px(t) = p0 + αt andpy(t) = −

∫

∂yVr(x, y)dt, with px ≫ py. We expand theHamiltonian (2) up to second order in py/px to obtain

H = px +p2y2px

+ V (x, y), (3)

and set vF = p0 = 1 for simplicity. In order to studywave-like electronic flow through the Hamilton-Jacobiequation (HJE) we introduce the classical action S withpx = ∂xS and py = ∂yS, and express the HJE as

∂tS + ∂xS +(∂yS)

2

2∂xS+ V (x, y) = 0. (4)

Employing previous non-relativistic approaches to thebranching problem we derive an equation for the lo-cal curvature of the electronic flow determined throughu = ∂yyS [5, 8, 31]. To this effect we apply the operator

T ≡ (∂xx + ∂yy + 2∂xy) on Eq. (4) and obtain

∂tu+u2

∂xS+

∂yS

∂xS∂yu+ T V (x, y) = 0. (5)

Using the effective Hamiltonian (3) we calculate the equa-tions of motion for the x, px conjugate variables keeping

the lowest order terms in py/px, which leads to the simplerelation x(t) = t. Thus, under the approximation of thedominance of the momentum in the forward x-direction,we find that space and time variables are identical. Weuse this fact to simplify the HJE ray dynamics and turnEq. (5) into a quasi-2D version [1, 5], that is, we replacethe space coordinate x with t and use T V = ∂yyVr(t, y);this simplification is valid only in the ultra-relativisticlimit since non-relativistic electrons accelerate in thepresence of non-zero values for the parameter α. More-over, the convectional derivative of an arbitrary functionf(H), where H is the Hamiltonian (3), in the quasi-2Dapproximation is df/dt = [∂t + ∂yS/∂xS ∂y] f (see Sup-plemental Material at [32]), leading to the following ap-proximate ordinary nonlinear differential equation for thecurvature

du

dt+

u2

1 + αt+ ∂yyVr(t, y) = 0. (6)

The dynamics of Eq. (6) determines the onset of theregime for caustics; this occurs at times, or equivalentlylocations along the x-axis, where the curvature u becomessingular [5, 14, 16]. The first time when a singularity inu occurs determines the precise point for the onset ofray coalescence. Given that the term ∂yyVr(t, y) is fluc-tuating, we solve the first passage time problem for thecurvature to reach |u(tc)| → ∞, where tc is the time forthe occurrence of the first caustic. Ignoring the stochasticpotential term of Eq. (6) we find that

tc =eα/|u0| − 1

α,

where we set u0 = −|u0| since negative initial curvatureleads to positive tc. In the strong external potential limit(α → ∞) the caustic needs infinite time to develop (tc →∞), while in the weak limit of (α → 0) tc is finite andincreases linearly with α,

tc =1

|u0|

(

1 +α

2|u0|

)

. (7)

This behavior follows from the effective elimination ofthe nonlinear term in u of Eq. (6), in the large α limit,which is responsible for the creation of caustic events.

Along the propagation axis t, the fluctuating term inEq. (6) acts as a δ-correlated noise ξ(t) with zero meanand standard deviation σ, ∂yyVr(t, y) = σ2ξ(t), and Eq.(6) becomes a stochastic Langevin equation [4, 14, 16].For α = 0 the curvature Eq. (6) reduces to the non-relativistic case with the average first caustic time 〈tc〉obeying the scaling relationship 〈tc〉 ∼ σ−2/3 [1, 5, 16].To quantify the scaling for the first relativistic causticevent including α, we use a self-consistent approach basedon the approximation that replaces the time variable inthe nonlinear term of Eq. (6) with the caustic time, t =

3

〈tc〉. The modified Eq. (6) becomes

du

dt= −γu2 − σ2ξ(t), (8)

with the constant coefficient

γ =1

1 + α〈tc〉. (9)

We solve the first passage problem of Eq. (8) as in [5]and arrive at the expression

〈tc〉 ∼ (σγ)−2/3. (10)

The use of Eqs. (9), (10) leads to the modified ultra-relativistic scaling-type expression

〈tc〉 ∼ σ−2/3

(

1 + 2α+ 3α2 +10

3a3)

, (11)

where α = 1.11ασ−2/3, is the relativistic correction termin the presence of a deterministic potential [32].Simulations: We now depart from the quasi-2D ap-

proximation and solve numerically the characteristicequations for the full Hamiltonian (2) while constructinga random potential based on experimental observations.In particular, impurities in the substrate of graphene cre-ate a smooth landscape of charged puddles of radiusR ≈ 4 nm [27, 28]. In our model, each puddle sizeis drawn from a two-dimensional Gaussian distributionwith standard deviation R. The location of each pud-dle is randomly chosen through a uniform distribution.We perform simulations for quasi-classical electron dy-namics in a graphene sheet of size 400× 400 nm, whereseveral caustics are observed; periodic boundary condi-tions are used to ensure that all the rays reach a caus-tic. The random potential consists of 2000 randomly dis-tributed Gaussian defects with R = 4 nm. A collec-tion of 1000 ultra-relativistic rays, initially distributeduniformly along y axis, are injected into the graphenesheet from the x = 0, with plane wave initial conditions,px(0) = p0 = 1 and py(0) = 0. We select a single causticevent out of many to show how the deterministic partof the potential affects the onset of this event, see Fig.1. The rays propagate in the disordered potential withσ = 0.1 and after time tc a caustic event occurs. Theray-tracing simulations are performed for three differentvalues of α = [0, 0.05, 0.1], to show that tc increases lin-early with α, a behavior expected from Eq. (7). We thusconfirm numerically the quasi-2D analytical predictionthat the presence of a small voltage in graphene shiftsthe location of the first caustic, a fact that can be testedexperimentally. The location and the shape of causticsare modified by the external potential Vd; in particular,as α increases the passage to branched flow is delayedand the caustics disperse slower, see Fig. 1.The classical electron trajectories can be used to de-

termine the onset of a caustic in the context of the

FIG. 1. Two-dimensional numerical ray-simulations deter-mine the onset of a caustic event in a disordered poten-tial with σ = 0.1 and for a deterministic potential withα = [0, 0.05, 0.1]. (Left) The lower panel shows the ran-dom potential. The remaining images represent the densityof rays I . The green dashed line shows that the first caustictime tc increases linearly with α. (Right) The ray density ofbranched flow in a graphene sheet for α = 0 and α = 0.1.

stability matrix M. The latter describes the evolu-tion in time of an infinitesimal volume of phase space,δxi(t) = Mδxi(0), where xi = (x, y, px, py) is the fourdimensional phase space vector, and the elements of Mread mij(t) = ∂xi(t)/∂xj(0) [12, 16]. The evolution of

M is given by M(t) = KM(t) with initial conditionmij(0) = δij and the ultra-relativistic symplectic matrix

K =

0 0 p2y/p3 −pxpy/p

3

0 0 −pxpy/p3 p2x/p

3

−Vxx −Vxy 0 0−Vxy −Vyy 0 0

, (12)

where p = |p| =√

p2x + p2y [32]. A caustic occurs when

the classical density of rays diverges giving rise to the con-dition for caustic emergence (−py, px, 0, 0)

T Mδxi(0) = 0[12, 31]. We use this condition to determine numericallythe time tc where the first caustic occurs by calculatingthe average time needed for the first caustic event. Tothis end we study a wide range of strength for the Gaus-sian defects in order to obtain many Vr with differentvariances σ2. In addition, we examine several bias po-tentials with α ranging between 0 and 0.1 while realizing30 disorder potentials for each pair of σ and α values.Furthermore, in these simulations we consider 104 ultra-relativistic electrons distributed uniformly along y andejected at x = 0. In Fig. 2 we present the relation be-tween 〈tc〉 and the potential parameters (σ, α). For α = 0we obtain 〈tc〉 ∼ σ−2/3, the scaling of conventional 2Dmetals in the absence of bias potential [14, 16]. When

4

FIG. 2. Simulation results for the mean first caustic time 〈tc〉for the occurence of a caustic event as a function of the ran-dom potential standard deviation σ and for bias determinedby α in the range [0, 0.1] (dashed lines conect data points).The color solid lines indicate the theoretical predicted rela-tionship between 〈tc〉 and σ in log space, in the range whereEq. (11) is valid.

α 6= 0, ln(〈tc〉) decays practically linearly with ln(σ) re-vealing that the ultra-relativistic nature of Dirac fermionsretains the scaling also in the presence of a bias potential,as predicted theoretically by Eq. (11). The color solidlines show the theoretical prediction of 〈tc〉 through theuse of Eq. (11) in the range that is valid, ασ−2/3 ≪ 1.

The spatiotemporal trajectories may be altered whenstructural defects are present in graphene. In this casea small energy gap appears in the electronic band struc-ture, which leads to a discernible electronic effective massm [18, 19]. In the vicinity of small mass, px >> m, wefind that the quasi-2D approach is still valid, with x = t,and the curvature equation is the same with Eq. (6) [32].Thus, the Dirac branching is robust and not affected bythe presence of few structural defects in graphene.

Machine Learning Prediction: The analysis of branch-ing flow in DS provides a theoretical framework for thephenomenon. In practice it would be useful if we couldpredict the onset of the branching flow, especially incases where available experimental or numerical dataare scarce, for instance, in order to reduce the STMmeasurements and speed up the electronic flow imag-ing process, and thus compress efficiently the experi-mental data. We next show that, although branchingis a complex nonlinear dynamical and stochastic phe-nomenon, the implementation of appropriate machinelearning methods can lead to accurate prediction of caus-tic events. We use a recurrent neural network (RNN)implemented through reservoir computing, an approachthat works well for time series prediction, including sys-tems that have chaotic behavior [29, 30]. The crucial ideafor caustic event prediction is that the longitudinal direc-tion x can be considered as time, and therefore, we can

FIG. 3. Machine learning prediction of caustic formation ina disorder potential with σ = 0.02 (α = 0). (Left) The actualdensity of rays I . The solid white line marks the end of neuralnetwork training time. During the electron propagation wefeed the network with input data in locations shown throughblack arrows. (Right) We compare the intensity profiles ofactual (filled color) and predicted data (lines) in two positionsindicated by magenta and blue dashed line. The intensitiesare plotted in the same scale of left image.

map the stationary 2D problem onto an 1D spatiotem-poral study. Thus, in the framework of the quasi-2Dapproach we transform the ray density matrix I into avector of time series u(t). The machine learning methodwe use consists of an input and output layer of nodesthat are modified through training as well as a random“reservoir” layer that incorporates recurrent links andgives additional flexibility to the network. Our aim isto predict the generation and location of random causticevents that lead to the production of branching in theelectronic flow. From the comparison of the actual flowto the predicted patterns in Fig. 3, we conclude that theRNN is capable of predicting not only the peak intensityof the caustic but also its first occurrence (see Supple-mental Material at [32] for further details of the RNNparameters that are used in Fig. 3).

In conclusion, we extended the investigation of branch-ing phenomena that appear in 2D electronic flows in weakdisorder potentials to ultra-relativistic dynamics relevantto 2D-DS such as graphene. We showed that whilebranching persists in these systems, an applied voltagein the form of an external potential affects the branch-ing dramatically by reducing the dispersion of causticsand increasing the location (or time) of the first causticevent. The presence of the bias field modifies the scal-ing relationship that connects the caustic location to thestatistical properties of the intrinsic electronic potential.This quantitative theoretical prediction, also verified nu-merically, opens a clear experimental possibility for thestudy of branching current patterns in “puddled” anddisordered 2D-DS. Although the branching scaling rela-tionship gives quantitative information on the onset ofbranches it is only a statistical measurement. In orderto improve the predictability of caustic events we imple-mented reservoir computing techniques to describe the

5

flow patterns. We found patterns that this specific ap-plication of deep learning methods produces clear, quan-titative information on the onset of flow caustics. Thisapproach may be implemented in cases of partial experi-mental information and can lead to accurate predictionsin the singularities of flow patterns. It would be inter-esting to study the applicability and usefulness of thespecific deep learning approach to the onset of singu-lar events in other complex systems, including photonicand plasmonic materials, radiation patterns, and waterwaves.

We acknowledge support by ARO MURI Award No.W911NF-14-0247 (M.M., E.K.), EFRI 2-DARE NSFGrant No. 1542807 (M.M.), European Union projectNHQWAVE MSCA-RISE 691209 (G.P.T.). We usedcomputational resources on the Odyssey cluster of theFAS Research Computing Group at Harvard University.M.M. and G.P.T acknowledge helpful discussions withDr. Ragnar Fleischmann and Dr. Jakob J. Metzger,and G.P.T. acknowledges useful discussions with Profes-sor Edward Ott.

∗ mariosmat@g.harvard.edu;http://scholar.harvard.edu/marios_matthaiakis

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