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Effects of large disorder on the Hofstadter butterfly Chenggang Zhou,1 Mona Berciu,2 and R. N. Bhatt1 1Department of Electrical Engineering, Princeton University, Princeton NJ 08544, USA 2Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada (Dated: February 2, 2008) 4 Motivatedbytherecentexperimentsonperiodicallymodulated,twodimensionalelectronsystems 0 placedinlargetransversalmagneticfields,weinvestigatetheinterplaybetweentheeffectsofdisorder 0 andperiodic potentials in theinteger quantumHallregime. Inparticular, westudythecase where 2 disorderislargerthantheperiodicmodulation,butbotharesmallenoughthatLandaulevelmixing n isnegligible. Inthislimit,theself-consistentBornapproximationisinadequate. Wecarryextensive a numerical calculations to understand the relevant physics in the lowest Landau level, such as the J spectrumandnature(localizedorextended)ofthewavefunctions. Basedonourresults,wepropose 2 a qualitativeexplanation of thenew features uncoveredrecently in transport measurements. ] PACSnumbers: 73.43.Cd l l a h I. INTRODUCTION modulationshavebeencreatedusinglithography7,8,9and - s holographic illumination.10 The lattice constants of the e resulting square lattices are of order 100 nm. As a re- m Two-dimensionalelectron systems (2DES) placed in a sult,the conditionφ/φ 2(forinstance)is satisfiedfor uniform perpendicular magnetic field exhibit a rich va- 0 ≈ t. riety of phenomena, such as the integer1 and fractional2 B 0.8 T. This is a very low value, in the Shubnikov- a quantum Hall effects.3 Another well-studied problem is de≈Haas (SdH) regime, not the high-B quantum regime. m Significant Landau level mixing and complications from thatofa2DESinauniformperpendicularmagneticfield - the fact that the Fermi level is inside one of the higher d subjectedtoaperiodicpotential. Evenbeforethediscov- n eryofthequantumHalleffects,Hofstadter4 showedthat Landau levels for such small B-values make the identifi- cation of the Hofstadter structure difficult. o in this case, the electronic bands split into a remarkable c fractalstructureofsubbandsandgaps,theso-calledHof- Recently,anewmethodforlateralperiodicmodulation [ stadter butterfly. Two “asymptotic” regimes are usually has been developed using a self-organized ordered phase 1 considered: (i) if the magnitude of the periodic poten- of a diblock copolymer deposited on a GaAs/AlGaAs v tial is very large compared to the cyclotron energy and heterostructure.11 The polymer spheres create a 2D tri- 7 the Zeeman splitting, then one can use lattice models to angular lattice with a lattice constant of about 39 nm. 0 describe the hopping of electrons between Wannier-like The corresponding unit cell area is almost an order of 0 states localized at the minima of the periodic potential, magnitude smaller than those achieved in previous ex- 1 0 whereas (ii) if the magnitude of the periodic potential is periments, implying that the condition φ/φ0 ≈ 2 is now 4 smallcomparedto the cyclotronenergy,the periodic po- satisfiedforverystrongmagneticfields,B 6T.Atsuch ≈ 0 tentiallifts the degeneracyofeachLandaulevel. Inboth highmagneticfieldsthe systemis inthe strongquantum / cases, the resulting butterfly structure is a function only regime, and Landau level mixing can be safely ignored. t a of the ratio between the flux φ = B of the magnetic For the experimental 2DES electron concentrations, the m field through the unit cell of the peAriodic lattice, and Fermilevelisinthespin-downlowestLandaulevel.11 As - the elementary magnetic flux φ =hc/e. Remarkably, if aresult,thisexperimentalsetupappearsmorepromising 0 d φ/φ of the first asymptotic case is equal to φ /φ of the for the successful observation of the butterfly. n 0 0 second case, their electronic structures are solutions of Nevertheless, one must take into account the disorder o c the same Harper’s equation.5 If the periodic potential is whichispresentinthesystem(withoutdisorder,thereis : comparabletothecyclotronenergy,Landaulevelmixing nointegerQuantumHallEffect–IQHE–tobeginwith). v must be taken into account; although Landau levels still Ifthedisorderisverysmallcomparedtotheperiodicpo- i X split into subbands, the structure is no longer universal, tential amplitude, one expects that the subbands of the r but depends also on the ratio of the periodic potential Hofstadterstructureare“smeared”onascale¯h/τ,where a amplitude and the cyclotron energy.6 τ is the scatteringtime, andτ asdisorderbecomes →∞ Experimentally, the case with a small periodic modu- vanishingly small. As a result, the larger gaps in the lation can be realized more easily. This is because the Hofstadterstructureshouldremainopenatthepositions periodic potential is usually imprinted at some distance predicted in the absence of disorder, and one expects a from the 2DES layer; as a result, its magnitude in the series of minima in the longitudinal conductivity as the 2DES is considerably attenuated. The interesting cases Fermi level traverses such gaps. The experiment indeed to study experimentally also correspond to small values shows a very non-trivial modification of the longitudi- of φ/φ (of order unity), where the butterfly structure nal resistivity, with many peaks and valleys appearing 0 shows a small number of subbands separated by large in what is (in the absence of the periodic modulation) a gaps,and shouldtherefore be easierto identify. Periodic smoothLorentz-likepeak.11 However,thepositionofthe 2 minimainρxxdonottrackthepositionsofthemaingaps y in the corresponding Hofstadter butterfly structure. In- stead, the data suggests that in this experimental setup, disorderisnotsmall,butratherlargecomparedtothees- x timatedamplitudeoftheperiodicpotential. Thisisnota consequenceofpoorsamples,sincethese2DEShavehigh mobilities. Itisduetothefactthattheperiodicmodula- − _L_x _L_x tionisconsiderablyattenuatedinthe 2DES,leadingtoa 2 2 small energy scale for the Hofstadter butterfly spectrum ascomparedto¯h/τ. Asaresult,theHofstadterstructure FIG. 1: The two-lead geometry considered: the finite-size predictedin the absenceof disorderis oflittle use forin- 2DES has periodic boundary conditions in the y-direction, terpretingtheexperimentaldata. Onemightexpectthat and is attached to metallic leads at the x=±Lx/2 ends. in this case the periodic potential should have basically no effect on the disorder-broadened Landau level. This hibits very interesting and non-trivial physics. is indeed true for the strongly localized states at the top The paper is organized as follows: in Section II we and bottom of the Landau level. However, states in the briefly review the computation of the Hofstadter struc- center of the Landau level extend over many unit cells ture for a small-amplitude periodic potential. In Sec- of the periodic potential, and, as we demonstrate in the tion III we describe the type of disorder potentials con- following, are non-trivially modified by its presence. sidered. Section IV describes the numerical methods In this paper, we investigate numerically the behavior used to analyze the spectrum andthe nature of the elec- of a 2DES subject to a perpendicular magnetic field, a tronicstates,withbothsemi-classicalandfullyquantum- periodic potential and a disorder potential, under con- mechanical formalisms. Results are presented in Sec- ditions applicable to the experimental system. The ef- tionV, while SectionVI containsdiscussionsandasum- fective electron mass in GaAs is 0.067me while the mag- mary of our conclusions. netic fields of interest are on the order of 10 T. Under these conditions, the cyclotron energy h¯ω , of the order c of 200 K,is the largestenergy scale in the problem. The II. PERIODIC POTENTIAL Zeeman energy g∗µ B for these fields is roughly 3 K, B but electroninteractioneffects leadto aconsiderableen- To clarify our notation, we briefly review the problem hancement of the spin splitting between the (spin po- of a free electron of charge e moving in a 2D plane larized) Landau levels, which has been measured to be − (from now on, the xy-plane, of dimension L L ) in a 20K.12 The amplitude ofthe periodic potential’s largest x× y magnetic field B = Be perpendicular to the plane, as z Fourier components is estimated to be of the order of described by 1 K, and the scattering rate from the known zero field mobility is estimated to be h¯/τ 8 K.13 As a result of 1 e 2 1 ∼ = p+ A gµ ~σ B this ordering of energy scales, we neglect Landau level H 2m c − 2 B · inter-mixingand study non-perturbativelythe combined (cid:16) (cid:17) effects of a periodic and a large smooth disorder poten- In the Landau gauge A = (0,Bx,0), the eigenfunctions tialontheelectronicstructureofthelowestLandaulevel. of the Schrödinger equation n,ky,σ = En,σ n,ky,σ H| i | i Previously, the effects of small disorder on a Hofstadter are: butterflyhavebeenperturbativelyinvestigatedusingthe self-consistent Born approximation (SCBA),14 and the rn,ky,σ = e−ikyye−12(xl−lky)2Hn xl −lky χσ, (1) combinedeffectofwhite-noisedisorderandperiodicmod- h | i Ly 2(cid:0)nn!√πl(cid:1) ulationonHall resistancewasstudied following the scal- ing theory of IQHE.15 Our results reveal details of the with eigenenergieps p electronic structure not investigated previously. 1 1 E =h¯ω n+ gµ Bσ. (2) The two-lead geometry we consider is schematically n,σ c B 2 − 2 showninFig.1: thefinite2DESisassumedtohaveperi- (cid:18) (cid:19) odicboundaryconditionsinthey-direction(alongwhich Herel = ¯hc/eB isthe magneticlength,ω =eB/mcis c theHallcurrentsflow),andisconnectedtometallicleads the cyclotron frequency, H (x) are the Hermite polyno- n at the x = Lx/2 and x = +Lx/2 edges. In particular, mials andpχT = (1 0), respectively χT = (0 1) are the − +1 −1 in this paper we study the effects of the periodic poten- eigenspinors of σ : σ χ =σχ . z z σ σ tialontheextendedstatescarryinglongitudinalcurrents ThedegeneracyofaLandaulevelisgivenbythenum- betweenthetwoleads,andidentifyanumberofinterest- berofdistinctk valuesallowed. Imposingcyclicbound- y ing properties, in qualitative agreement with simple ar- ary conditions in the y-direction, we find guments provided by a semi-classical picture. Our main conclusion is that while the beautiful Hofstadter struc- 2π k = j, (3) ture is destroyed by large disorder, the system still ex- y L y 3 where j is aninteger. The allowedvalues for j are found III. DISORDER POTENTIAL from the condition that the electron wave-functions, which are centered at positions x = l2k = l22πj/L j y y Real samples always have disorder. The current con- [see Eq.(1)] are within the boundary along the x-axis, i. sensus is that high-quality GaAs/AlGaAs samples ex- e. L /2 < x L /2. It follows that the degeneracy x j x hibit a slowly varying, smooth disorder potential. In a − ≤ of eachLandaulevel is N =L L B/φ , with φ =hc/e. x y 0 0 semi-classical picture, the allowed electron trajectories Considernowtheadditionofaperiodicpotential,with in the presence of such disorder follow its equipoten- a lattice defined by two non-collinear vectors a1 and a2, tial lines.3,19 Closed trajectories imply localized electron suchthatV(r)=V(r+na +ma )foranyn,m . The 1 2 states, while extended trajectories connecting opposite ∈Z periodicpotentialhasnon-vanishingFouriercomponents edges of the sample are essential for current transport onlyatthereciprocallatticevectorsg=hg +kg ,where 1 2 through the sample (for more details, see Sec. IVA). g a =2πδ and h,k are integers. Thus: i· j ij In typical experimental setups,11 dopant Si impurities V(r)= V eir·g. (4) with a concentration of 1013 cm−2 are introduced in g ∼ a thin layer of 6 nm in thickness, located 20 nm above g X the GaAs/AlGaAsinterface. Typically,upto10%ofthe Further, since V(r) is real, it follows that Vg =V−∗g. Si atoms are ionized. A small fraction of the ionized In the absence of Landau level mixing, the Hofstadter electrons migrate to the GaAs/AlGaAs interface where spectrum for both square4 they form the 2D electron gas. The electrostatic po- 2π 2π tential created by the ionized impurities left behind is Vs(x,y)=2A cos x+cos y , (5) the major source of disorder in the 2DES layer. On the a a (cid:20) (cid:21) length-scaleweareinterestedin,thereare104to105such and triangular16 ionizedSiimpuritiesperµm2. Theresultingdisorderpo- tential must be viewed as a collective effect of the den- 4π 2π Vt(x,y)= 2A cos x+cos x y√3 sity fluctuation of the ionized impurities20 rather than a − √3a √3a − (cid:20) (cid:16) (cid:17) simple summationof the Coulombpotentialof a few impurities. The electrostatic potential from Si impurities 2π +cos x+y√3 (6) is compensated and partially screened by other mobile √3a(cid:16) (cid:17)(cid:21) negative charges in the system such as, for example, the periodic potentials, with nonzero Fourier components surface screening effect by mirror charges considered by only for the shortest reciprocal lattice vectors, have been Nixonand Davies.20 An exacttreatmentof this problem studiedextensivelyinthe literature.4,16,17,18 Theparam- is difficult, since one should consider the spatial corre- eter defining the spectrum is the ratio between the flux lation of the ionized impurities.21,22 One model used to φ = B (a a ) of the magnetic field through a unit describesuchdisorderconsistsofrandomlyplacedGaus- 1 2 cell and·the ×elementary flux φ . For φ/φ = q/p, where sian scatterers.23 This model captures the main feature 0 0 pandq aremutuallyprimeintegers,the originalLandau ofasmoothdisorderpotentialandsupportsclassicaltra- level is split into q sub-bands. jectoriesonequipotentialcontours,but it hasno natural Wewouldliketoemphasizeaqualitativedifferencebe- energy/lengthscalesassociatedwithit. As a result,here tween the two types of potentials: the square potential wechoosetoalsoinvestigateadifferentmodelofthe dis- in Eq. (5) is particle-hole symmetric, since V (x,y) = order, which incorporates the smooth character of the s V (x+ a,y+ a). As a result, the sign of its amplitude Coulomb potential in real space. − s 2 2 isirrelevant. Onthe otherhand,thetriangularpotential We generate a realization of the disorder potential in doesnothavethissymmetry. WiththesignchoseninEq. the following way: positive and negative charges, cor- (6) and A > 0, V has deep local minima at the sites of responding to a total concentration of 103µm−3 are t the triangularlattice, whereasthe maxima are relatively randomly distributed within a volume [ L /2,L /2] x x − × flat and located on a (displaced) honeycomb lattice. As [ L /2,L /2] [20nm + d,26nm + d] above the elec- y y − × aresult,thesignofV ishighlyrelevant. Thesecondfact tron gas which is located in the z = 0 plane. Here, t that must be mentioned is that the choice made in Eqs. we choose d = 4 nm as an extra spacer since the elec- (5) and (6) is rather simple, since it aligns the periodic tronic wave-functions are centered about 3-5 nm below potential with the edges of the sample in a very specific the GaAs/AlGaAs interface. Since we are not simu- way. In general, however, one could consider the case lating single impurities but density fluctuations, these where the periodic lattice is rotated by some finite angle charges are not required to be elementary charges. In- withrespecttothesampleedges;studyofsuchcaseswill stead, we use a uniform distribution in the range [ e,e] − be discussed in future work. Finally, it may seem that for convenience (a Gaussian distribution would also be this choice of periodic potentials is very restrictive also a valid choice), and sum up all Coulomb potentials from because only the shortest lattice vectors have been kept thesecharges,usingthestaticdielectricconstantinGaAs in the Fourier expansion. In fact, the methods we em- ǫ = 12.91.24 The resulting disorder potential has energy ploycanbedirectlyusedforpotentialswithmoreFourier andlengthscalescharacteristicoftherealsamples. Typ- components, but their inclusion leads to no new physics. ical contours for such potentials are shown in Sec. V. 4 In an infinite system, in the quantum Hall regime, the existence of quantum Hall steps implies the exis- 10−4 Coulomb model tence of critical energies at which the localizationlength Gaussian model diverges.25 Thisisthequantumanalogofthetwodimen- sional percolation problem in a smooth random land- scape, for which there exists a single critical energy.19 eV In the case of potentials with electron-hole symmetry 1/2> V(r) = 0, the critical energy lies in the middle of the 2q)| 10−5 h i V( band(E =0),leadingtopercolatingpathathalffilling. <| c For a finite mesoscopic sample, however, not only does the percolating path (critical energy) E deviate from c this value, but in samples without a periodic boundary conditiononeneednothaveapercolatingpathtraversing 10−6 the system in the desired direction. This arises from the 0 5 10 15 20 fluctuations near the edge of a mesoscopic system with q (µ m−1) free boundary conditions. We circumvent such a possibility by adding an extra FIG. 2: Averaged Fourier amplitudes of two types of disor- smooth potential V′(x,y) to the impurity-induced dis- der potential as a function of wavevector q = |q|. For both order potential V (x,y), such that the total potential CoulombandGaussianmodel,V(q)2isaveragedover116dis- i V =V +V′ is zero on the opposite edges x= L /2 of order realizations. The relation between V(q) and V(r) and i x the sample where the metallic leads are attach±ed. The relevant parameters are discussed in the text. The standard supplementarycontributionV′(x,y)canbethoughtofas deviation, (LxLy)−1 drV2(r) for the Coulomb model is simulating the effect of the leads on the disorder poten- 3.2×10−7eV2, and 2.1×10−7eV2 for the Gaussian model. (cid:10)R (cid:11) tial, since the metallic leads hold the potential on each edge constant by accumulating extra charges near the interface. Therefore, physically we expect that the ex- IV. NUMERICAL CALCULATIONS tra potential V′ decays exponentially over the screening length λ inside the sample. This implies: In this section we discuss the numerical methods we use, including derivations of some relevant formulas. As V ( L /2,y)+V (L /2,y) cosh(x/λ) V′(x,y)= i − x i x already stated, we focus on the case where the ampli- − 2 cosh(Lx/2λ) tudes of the periodic and disorder potentials are very small comparedto the cyclotronenergy and the Zeeman splitting, and therefore inter-level mixing is ignored. V ( L /2,y) V (L /2,y) sinh(x/λ) i x i x + − − 2 sinh(L /2λ) x where λ is taken to be 100 nm in our calculation. A. Semi-classical treatment In Fig. 2, we plot the average of Fourier transform of the magnitude of the random potential V(q)2 The semi-classical approach is valid19 for the integer h| | i versus q = q for the Coulomb model and the Gaus- quantum Hall effect in the presence of a slowly varying, | | p sian model. The Gaussian model is generated by adding smoothdisorderpotentialandlargemagneticfields(such 100 randomly placed Gaussian scatterers on an area of as we consider), so that the magnetic length l which de- 3µm 3µm, each contributing A e−r2/d2, where A is terminesthespatialextentoftheelectronwave-functions d d × uniformly distributed in [ 2,2] meV, and d is uniformly is much smaller than the length scale of variation of − distributed in [0,0.2] µm. V(q) is related to V(r) by the smooth disorder potential, V(r) ¯hω /l. Then, c |∇ | ≪ V(r) = V(q)eiq·r, where the summation is over all semi-classically the electron moves along the equipoten- q the wavevectors involved in the fast Fourier transforma- tial contours of the disorder potential V(r), in the di- tion. ThPe Gaussian model has an arbitrary energy scale rection parallel to V(r) B. Since the kinetic energy ∇ × which is fixed by the maximum value of the distribu- is quenched in the lowest Landau level, the total energy tion A . Here A = 2 meV. As can be seen, V(q) of of the electron simply equals the value of the disorder m m both models are decreasing functions of q. The trends potential on the equipotential line on which its trajec- of decay are exponential at large q. At small q, the two tory is located. As a result, the density of states in the models behave differently. Despite the difference, both semi-classical approach is directly given by the proba- models lead to the same qualitative results, although, bility distribution for the disorder potential, which can as expected, minor quantitative differences are present. becalculatedbyrandomlysamplingthepotentialenergy This shows that the physics we uncover is independent and plotting a histogram of the obtained values.19,26 oftheparticulartypeofslowly-varyingdisorderpotential In Sec. V we compare the results obtained within this considered, and therefore should be relevant for the real semi-classical approach with fully quantum mechanical samples. results. As expected, the agreement is good if only the 5 disorder potential is present. However, if the periodic either case, there are Fourier components corresponding modulation is also included, the lattice constant a pro- to q = 2π/a and q = 0. Since only basis vectors y y ± videsanewlength-scalewhichiscomparabletothemag- for which the difference k k′ =q give non-vanishing y − y y neticlengthl,andthesemi-classicalpicturebreaksdown. matrix elements, we must choose the length L of the y Quantum mechanical calculations are absolutely neces- sampletobe amultipleintegerofa,the latticeconstant. sary to quantitatively treat this case. Thematrixelementsofthedisorderpotentialarecom- puted ina similarway. We use a gridof dimensionN x × N tocoverthesampleandgeneratethevaluesofthedis- y B. Quantum Mechanical Treatment orderpotentialonthisgrid. Then,fastFouriertransform (FFT)27 is used to find the long wavelengthcomponents As showninSec. II, for a finite sample of size L L of the disorder potential corresponding to the allowed x y × atagivenmagneticfieldB,thedegeneracyoftheunper- values q = 0, 2π ,..., Nx,y 2π (proper care is turbed Landau level is N = L L B/φ = L L /(2πl2). x,y ±Lx,y ± 2 Lx,y Sincethe disordervariesverysxlowyly,we0needxtoyconsider taken to define Fourier comphonentis so that Vq = V−∗q). The matrix elements of this discretized disorder poten- systems with L ,L l to properly account for its ef- x y ≫ tial are then computed using Eq. (7). In principle, finer fects. As aresult,the numberofstatesinaLandaulevel grids (increased values for N and N ) will improve ac- can be as large as 104 in our calculations. Storage of x y curacy. However, they also result in longer computation the Hamiltonian as a dense matrix requires considerable times,sincetheyaddextramatrixelementsinthesparse amount of computer memory and its direct diagonaliza- matrix, corresponding to large wave-vectors. We have tion is prohibitively time-consuming. Sparse matrix di- verifiedthatagridsizeofdimensionN =N =72isal- agonalizationtechniquescouldbeemployed,buttheyare x y readylargeenoughtoaccuratelycapturethelandscapeof less efficient when all eigenvectors are needed, and also a 3µm 3µm sample and the computed quantities have have some stability issues. × already converged, with larger grids leading to hardly Here we introduce the numerical methods we use to noticeable changes. This procedure is also justified on computedensitiesofstatesandinferthenature(localized a physical basis. First, the neglected large wave-vector or extended) as well as the spatial distribution of the components describe very short-range spatial features, wave-functions, while avoiding direct diagonalization. which are probably not very accurately captured by our disorder models to begin with, and which are certainly notbelievedtoinfluencethebasicphysics. Secondly,this 1. Matrix elements procedure insures that the actual disorder potential we useisperiodicinthey-direction,sinceeachFouriercom- Since inter-level mixing is ignored, the Hilbert sub- ponentretainedhasthisproperty. Thisisconsistentwith spaces corresponding to different spin-polarized Landau our use of a basis of wave-functions which are periodic levelsdonothybridize. EachHilbertsubspace(n,σ)has along y. a basis described by Eq. (1), containing N orthonormal The matrix elements of the Hamiltonian within a vectors indexed by different ky values. given Landau level (n,σ) are then n,k ,σ n,k′,σ = In order to compute matrix elements of the total E + n,k n,k′ , where E hareygive|Hn|by Eyq.i(2) Hamiltonianinsuchabasis,weusethefollowingidentity n,σ h y|V| yi n,σ and the matrix elements of both the periodic and the derived in Ref. 18 (notice their different sign convention disorderpartofthe potential arecomputed as already for k . If σ =σ′, the overlapis zero): V y discussed. Thisproducesasparsematrix,whichisstored 6 efficiently in a column compressed format. n′,ky′ eiq·r|n,kyi=δky′,ky−qyLn′,n(q)eil22qx(ky′+ky), (7) (cid:10) (cid:12) where (cid:12) 2. Densities of States and Filling Factors n−n′ 1 A quantity that canbe computed without directdiag- Ln′,n(q)=(cid:18)Mm!!(cid:19)2 i|n′−n| qxqx2++iqqyy2 Foneramliziaetnioenrgiys.tThheefifilllilninggfafcatcotrorνnis,σd(eEfiFn)edasasa:function of ×e−12QQ12|n′−qn|L(m|n′−n|)(Q), ν (E )= 1 Θ(E E ), (8) n,σ F F n,α,σ N − with Q = 12l2(qx2+qy2), m and M the minimum and the Xα maximum of n′ and n respectively, and L(|n′−n|)(Q) the where Θ(x) is the Heaviside function and N is the total m associated Laguerre polynomial. When band-mixing is number of states in the (n,σ) Landau level. (Since we neglected n = n′ and n,n(q) = e−21QLn(Q). For the neglect Landau-levelmixing, we can define this quantity L first Landau level, L (x)=1. forindividuallevels.) Thefillingfactortellsuswhatfrac- 0 Eq. (7) gives us the matrix elements for the square tion of the states in the given Landau level are occupied [Eq. (5)] or triangular [Eq. (6)] periodic potentials. In at T = 0, for a given value of the Fermi energy. This 6 corresponds to the averagefilling factor measured in ex- (no Landau level mixing), it follows: periment and is also proportionalto the integrated total (as opposed to local) density of states. n,α,σ n,α,σ GˆR,A(ω)= | ih | = GˆR,A(ω). (10) The filling factor is straightforward to compute if the ¯hω E iδ n,σ n,α,σ n,α,σ − ± n,σ eigenenergies E are known. However, we want to X X n,α,σ avoid the time-consuming task of numerical brute force The exact eigenstates can be expanded in terms of the diagonalization. The strategy we follow is a generaliza- basis states n,k ,σ as tion to Hermitian matrices of the method used in Ref. | y i 28. We restate the problem in the following way: as- sume we have a Hermitian matrix of size N N (no n,α,σ = cn,α(ky)n,ky,σ . (11) × | i | i Landau level mixing), given by the matrix elements of Xky M = E 1 in the basis n,k ,σ (1 is the unit ma- F y trix).HTh−en, ν (E ) is pro|portionial to the number of Since the states n,k ,σ arelocalizednearx=k l2 [see n,σ F y y | i negative eigenvalues of the matrix M. We now generate Eq.(1)],thecoefficientsc (k )describetheprobability n,α y the quadratic form = N ζ ζ∗M , and transform amplitude for an electron in the state n,α,σ . Knowl- itintoits standardfMorm i=,j=1Ni jd χij 2 using theJa- edge of these coefficients allows us to in|fer wheither such MP i=1 i| i| states are extended or localized in the x-direction, i.e. cobian method described below. Here, d ’s are all real P i whether they can carry currents between the leads. numbers,andtheχ ’sarelinearcombinationsoftheζ ’s. i i However, as already stated, we wish to avoid direct This is a similarity transformationwhich retains the sig- diagonalization. We can still infer whether the Hamil- nature of the matrix. As a result, even though the num- tonian has extended or localized wave-functions near a bers d arenoteigenvaluesofM,the numberof negative i given energy h¯ω in the following way. We introduce the eigenvalues equals the number of negative d values. It i matrix elements: followsthatν (E )isobtainedbysimplycountingthe n,σ F number of negative d values for the given E . i F GR,A(k ,k′;ω)= n,k ,σ GˆR,A(ω)n,k′,σ The Jacobian method is iterative in nature. First, all n,σ y y h y | | y i terms containing ζ and ζ∗ are collected and the needed 1 1 complementary terms are added to form the first total squared1|χ1|2,sothatζ1 andζ1∗ areeliminatedfromthe = cn,α(ky)c∗n,α(ky′). (12) rest of the quadratic form . The procedure is then ¯hω E iδ repeated for all ζ and ζ∗ terMms (producing d ) etc., un- Xα − n,α,σ± 2 2 2 til all N values d are found. Computationally, this can i IfLandaulevelmixingisneglected,Eq.(9)canberewrit- be done by scanning the lower or upper triangle of the ten in the basis n,k ,σ as: y Hermitian matrix M only once. The total number of | i operations is proportional to the number of nonzero elements of the matrix, meaning that for a dense matrix it (h¯ω±iδ)δky,ky′′ −hn,ky,σ|H|n,ky′′,σi scales with N2 (sparse matrices require much fewer op- Xky′′ h i erations). Asaresult,thisprocedureismuchfasterthan brute forcediagonalizationwhichscales with N3 (for us, lNike∼fu1n0c4t)i.onTs,hweitfihllisntgepfsaclotocratνend,σa(tEt)heiseaigesnuvmaluofess.teBpy- ×GRn,,σA(ky′′,ky′;ω)=δky,ky′. (13) scanningE andidentifyingthepositionofthesestepswe We use the popular numerical library SuperLU,29 canalsofindthetrueeigenvaluesE ,withthedesired n,α,σ basedonLUdecompositionandGaussianreductionalgo- accuracy. Finally, the total density of states is given by rithmforsparsematrices,tosolvetheselinearequations. ρ (E)=dν (E)/dE. nσ n,σ Consider now the matrix element GR,A(k ,k ;ω) n,σ min max corresponding to the smallest k = k and the largest y min k = k values. If all wave-functions with energies y max 3. Green’s functions: extended vs. localized states close to ¯hω are localized in the x-direction, it follows that GR,A(k ,k ;ω) is a very small number, of the | n,σ min max | The advanced/retarded Green’s functions are the so- order e−Lx/ξ(ω), where ξ(ω) is the localization length at lutions of the operator equation the given energy. On the other hand, we expect to see a sharp peak in the value of GR,A(k ,k ;ω) if h¯ω is (h¯ω iδ)GˆR,A(ω)=1, (9) in the vicinity of an extende|dns,tσatemeiingenmvaalxue, s|ince [see −H± Eqs. (11,12)] both c (k ) and c (k ) are non- n,α min n,α max whereδ 0+. (Inpracticeweuse asetofsmallpositive vanishing for an extended wave-function with significant → numbers,andusethedependenceonδ toobtainresults.) weight near both the L /2 and the L /2 edges. More- x x − If the exact eigenstates and eigenvalues of the total over,theheightofthispeakscaleslike1/δ,sobyvarying Hamiltonian are known, n,α,σ = E n,α,σ , δ wecaneasilylocatetheenergiesoftheextendedstates. n,α,σ H H| i | i 7 4. Green’s functions: local densities of states have the added advantage that they can be easily stored as sparse vectors, because of their Gaussian profiles [see We canalsouse Green’sfunctions techniquestoimage Eq. (16)]. Moreover, in the limit l 0 (B ) where → →∞ the local density of states at a given energy E. By def- rx0,y0 δ(x x0)δ(y y0), the projected density |h | i| → − − inition (and neglecting Landau level mixing), the local of states ρP(x0,y0;E) ρ0,σ(r;E). Therefore, for the → density of states in the level (n,σ) is: large B values that we consider here, the projected density of states ρ should provide a faithful copy of the P local density of states. ρ (r;E)= rn,α,σ 2δ(E E ) n,σ |h | i| − n,α,σ We compute the projected local density of states fol- α X lowing the method of Ref. 30. Let u be the vector with 0 elements k x ,y obtained from the representation of y 0 0 = π1Imhr|GÂn,σ(E)|ri, (14) |mxa0,tryi0xioinfhtthhe|eH|kaymiiibltaosnisia[nsee Eiqn.t(h1e5)]k, anbdalseist.HWbeegtehne- y H | i erate the series of orthonormalvectors u ,u ,... using: where the second equality follows from Eq. (10). This 0 1 function traces the contours of probability |φn,α,σ(r)|2 v1 = Hu0, for electrons with the given energy E. Its direct compu- a = u†v , tation, however,is difficult and very time-consuming. 0 0 1 v a u For the rest of this subsection, the discussion is re- u = 1− 0 1 , 1 stricted to the Lowest Landau level n = 0 (the value v†v a2 of σ is irrelevant). We know that in the lowest Landau 1 1− 0 q level,electronicwave-functionscannotbelocalizedinany and for n 2, ≥ direction over a length-scale shorter that the magnetic v = Hu , length l. As a result, it suffices to compute a projected n n−1 local density of states on a grid with l l (or larger) an−1 = u†n−1vn, × spacings. Theprojectionismadeonmaximallylocalized b = u† v , wave-function, defined as follows. Let r =(x ,y ) be a n−2 n−2 n 0 0 0 v a u b u point on the grid. We associate it with a vector: u = n− n−1 n−1− n−2 n−2. n v†v a2 b2 n n− n−1− n−2 x ,y = k k x ,y , (15) 0 0 y y 0 0 q | i | ih | i The numbers a and b can be shown to be real. We Xky donothavea“tnerminatnor”30 toendthisrecursiveseries. whereweusethesimplifiednotation k n=0,k ,σ Instead,ourprocedureendswhentheorthonormalsetof y y | i≡| i for the basis states ofthe first Landaulevel (see Eq.(1)) vectors u0,u1,... exhausts a subspace of the lowest Lan- and we take dau level containing all states coupled through the dis- orderand/or periodic potential to the state x ,y (i.e., 0 0 hky|x0,y0i=s2Llπy21e−2xl202−ky22l2+ky(x0+iy0). (16) aplolinstta).teIsntthhaetpcroensternibceutoef tdoisothrdeepr,rothjeicstuedsuDa|lOlySinacitlutdheiss the entire lowest Landau level. It is then straightforwardto show that Then, the projected density of states is given by Eq. (18), where the matrix element of the Green’s func- rx0,y0 = 1 e−(x−4lx20)2−(y−4ly20)2e−2li2(x+x0)(y−y0). tion is the continued fraction: h | i √2πl (17) x0,y0 GA(E)x0,y0 = h | | i In other words, x ,y is an eigenstate of the first Lan- 0 0 daulevelstrongl|ypeakiedatr=r0. (The phase factoris E iδ a b2 E iδ a b2[...]−1 −1 −1 due to the proper magnetic translation). We then define − − 0− 0 − − 1− 1 the projected density of states [compare with Eq. (14)]: (cid:20) h i (cid:21) (19) Because the Hamiltonian is a sparse matrix, the genera- 1 ρ (x ,y ;E)= Im x ,y GÂ(E)x ,y , (18) tionoftheseorthonormalsets andcomputationofρ (E) P 0 0 0 0 0 0 p π h | | i forallthegridpointsisarelativelyfastprocedure. More- and use it to study the spatial distribution of the elec- over,thiscomputationisideallysuitedforparallelization, tron wave-functions at different energies. Strictly speak- with different grid points assigned to different CPUs. ing, the local density of states defined in Eq. (14) cannot be projected exactly on the lowest Landau level, because the lowest Landau level does not support a δ- V. NUMERICAL RESULTS function ( rn,k ,σ = 0, n). However, the coher- y h | i 6 ∀ ent states x ,y we select are the maximally spatially- In this section we present numerical results obtained 0 0 | i localized wave functions in the lowest Landau level, and usingthese methods. We haveanalyzedover20different 8 ν(E) ν(E) 0.8 (a) Disorder only 0.8 (b) A = 0.05meV 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 -0.5 -0.25 0 0.25 0.5 0.75 -0.5 -0.25 0 0.25 0.5 0.75 E (meV) E (meV) ν (E) ν (E) 0.8 (c) A = 0.5meV 0.8 (d) A = 5meV 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 -2 -1 0 1 2 -10 -5 0 5 10 E (meV) E (meV) FIG.4: Semi-classical(dashedline)andquantum(solidline) filling factors for the disorder potential shown in Fig. 3, but different amplitudes of the triangular periodic potential (a) A=0;(b)A=0.05meV; (c)A=0.5meV and(d)A=5meV.As expected,agreement exists only in thelimit A→0. 2.964µm. With these choices, the Landau level contains N =10108 states. The disorder potential obtained with our scheme described in Sec. III is shown in Fig. 3, both with and without the correction V′(r). An extended equipotential line appears, as expected, at ν 0.5. ≈ InFigs. 4and5weplotthefillingfactorν(E)andthe correspondingtotaldensityofstatesρ(E)asafunctionof E (computationdetailsweregiveninSec.IVB2). These quantitiesareobtainedinthesemi-classicallimit(dashed line) and with the full, quantum-mechanical treatment (solid line). Results are shown for 4 different cases: (a) only disorder potential and (b, c, d) disorder plus a tri- angularperiodicpotentialwithamplitudesA=0.05,0.5 and 5 meV, respectively. We only plot a relatively small FIG. 3: Profile of the disorder potential obtained from our Coulomb model on a 3.11µm×2.96µm sample, without (up- perpanel)andwith (lowerpanel)theV′(r) correction at the ρ (E) ρ (E) x = ±Lx/2 edges. The disorder potential varies between 1000 (a) 1000 (b) −3 meV and 3 meV, on a spatial length-scale much larger 800 800 than l = 12.03 nm. The critical region containing extended 600 600 states is in the vicinity of E = 0.06 meV. The contours are 400 400 shown for E =0.0575 meV (dashed), 0.17 meV (thick solid) 200 200 and 0.31 meV (thin solid). These energy values correspond 0 0 -0.5 -0.25 0 0.25 0.5 0.75 -0.5 -0.25 0 0.25 0.5 0.75 to classical filling factors ν=0.47, 0.58 and 0.68 in the upper E (meV) E (meV) panel and ν=0.45, 0.56, 0.66 in the lower panel. The differ- ρ (E) ρ (E) ence is dueto thesupplementary smooth potential V′. (c) 200 (d) 600 150 400 100 disorder realizations for samples of different sizes, and 200 50 all exhibit the same qualitative physics. Here, we show 0 0 results for several typical samples. The lattice constant -2 -1 0 1 2 -10 -5 0 5 10 E (meV) E (meV) is always a = 39 nm if periodic potential is present, as defined by the experimental system.11 FIG. 5: Semi-classical (dashed line) and quantum (full line) For the first sample, we consider φ/φ = 3/2 (B = densityof states calculated from corresponding filling factors 0 4.71 T). The magnetic length is l = 12.03 nm, and we inFig. 4. Weshowonlythecenterofthedisorder-broadened choose a sample size L = 3.11µm and L = 76a = lowest Landau level, where the density of states is large. x y 9 energy interval where the DOS is significant, and ignore gies. Finally,inpanel(c)weshowthecomparisonwitha the asymptoticregionswithlongtailsoflocalizedstates. largerenergy interval. The value of the Green’s function While the agreement between the semi-classical and decreases exponentially fast on both sides of the critical quantum-mechanical treatment is excellent in the limit region, indicating strongly localized states. Here, data A 0, the two methods give more and more different for δ = 10−6 eV is a smooth curve, whose magnitude is res→ults as the periodic potential amplitude is increased. muchlessthanthat ofthe otherthree values evenfor lo- As already explained, this is a consequence of the fact calizedstates. Thisisdue tothe factthatthisδ islarger that the magnetic length l is comparable to the lattice thantypicallevelspacings. Asaresult,severallevelscon- constanta,leadingtoafailureofthesemi-classicaltreat- tribute significantlyto Green’s function at eachE value, ment when this extra length-scale is introduced. In par- and the destructive interference of the random phases ticular, in the case with the largest periodic potential of different eigenfunctions lead to the supplementary δ- [panel (d) of Figs. 4 and 5] we can clearly see the ap- dependence. Weconcludethatthedisorderpotentialhas pearance of the 3 subbands expected for the Hofstadter acriticalenergyregimeofapproximately0.3meVwidth, butterfly at φ/φ = 3/2, although the disorder leads to coveringlessthan5%(inenergy)and20%(innumberof 0 broadened and smooth peaks, and partially fills-in the states) of the disorder-broadened band with total width gapbetweenthelowertwosubbands. Thispicture[panel 6meV.Thepositionofthecriticalenergyintervalisin ∼ (d)] is quite similar to the density of states that Ref. 14 agreement with the semi-classical results which suggest calculated using the self-consistent Born approximation. an extended state in the vicinity of E = 0.06meV. By This is expected since the SCBA approach is valid in comparison with Fig. 4, we can also see that this criti- thelimitofstrongperiodicpotentialwithweakdisorder. cal regime corresponds to a roughly half-filled band, in However, the SCBA approach is not appropriate in the agreement with the experiment. limit of moderate or strong disorder, where the higher The effect of an additional triangular periodic poten- order terms neglected in SCBA are no longer small. For tial is shown in Fig. 7, where we plot the same quantity disorder varying on a much longer length-scale than the shown in Fig. 6 for a fixed δ = 10−7 eV and different periodic potential, one still expects that locally, on rel- amplitudes A = 0, 0.05, 0.5 and 5 meV, respectively. ative flat regions of disorder, the system exhibits the TheseresultscorrespondtoadifferentCoulombdisorder Hofstadter-type spectrum. However, these spectra are potential (not shown), as can be seen from the different shifted with respect to one another by the different local locationof its extended states. Here we see how the nar- disordervalues. Ifdisordervariationsaresmall,thenthe row critical interval of extended states grows gradually total spectrum shows somewhat shifted subbands with as the amplitude of periodic potential is increased and partially filled-in gaps, but overall the Hofstadter struc- finally exhibits the three well-separated extended sub- ture is still recognizable. On the other hand, for moder- bands expected for φ/φ = 3/2 in the limit of vanishing 0 ate andlargedisorder,the detailed structureofthe local disorder. Thethreesubbandscanalreadyberesolvedfor density of states from various flat regions are hidden in the moderate case A=0.5 meV, although they are very the total density of states. All one sees are some broad- wide and exhibit significant overlap. ened, weak peaks and gaps superimposed on a broad, Qualitatively similar behavior is obtained if we use continuous density of states. the Gaussian scatterers model for disorder. A typical We now analyze the nature of the electronic states for realization of this disorder is shown in Fig. 8. Results these configurations. We start with the case which has for the Green’s function’s values with such disorder are onlydisorder. InFig.6weplot GR(k ,k ;E)2 asa shown in Fig. 9, for cases with pure disorder, and also min max | | function of the energy E, for different values of δ (com- cases with either a triangular or a square periodic po- putation details were given in Sec. IVB3). As already tential. The magnetic field has been doubled, such that discussed, extended states are indicated by large values φ/φ = 3. Similar to the case shown in Fig. 7, the peri- 0 of this quantity, as well as a strong (roughly 1/δ2) de- odic potential leads to a widening of the critical regime. pendence on the value of the small parameter δ. For large periodic potentials, the expected Hofstadter- Figure 6 reveals that as δ is reduced, resonant be- likethree-subbandstructureemergesagain. Weconclude havior appears in a narrow energy interval E = 0.02 that Coulomb and Gaussian disorder models show qual- 0.36 meV. Panel (a) shows that results correspondin−g itatively similar behavior. to δ = 10−7 eV and δ = 10−8 eV indeed differ by We now analyze the projected local density of states roughly2ordersofmagnitude,withδ =10−8eVshowing ρ (E) discussed in Sec. IVB4, in order to understand P sharper resonance peaks. The difference between results the reason for this substantial widening of the critical for δ =10−8 eV and δ =10−9 eV shown in panel (b), is region by even small periodic potentials. We consider a no longer so definite. The reason is simply that for such smallersample,ofsizeapproximately1.6µm 1.6µm,and × smallδ, the denominator in the Green’s function expres- compute the projected density of states for 500 equally- sion is usually limited by E E and not by δ [see spaced energy values, on a 60 60 square grid and for a n,α,σ Eq. (12)], and the depend|enc−e on δ is| minimal. Only if valueδ =10−8 eV.Thisδ valu×eiscomparableorsmaller E is such that E E < δ can we expect to see a than the level spacing, so we expect to see sharp res- n,α,σ | − | δ dependence, and indeed this is observed at some ener- onances from the contribution of individual eigenfunc- 10 FIG. 6: Semi-log plot of the amplitude of Green’s function matrixelementbetweenthetwoedgestatesnearx=±Lx/2, as a function of energy. Only the disorder potential of Fig. 3 is present. (a) comparison between δ = 10−7 and FIG. 8: A disorder potential of Gaussian type on a roughly δ = 10−8 results; (b) comparison between δ = 10−8 and 3µm×3µm square. The three lines are equipotential con- δ=10−9 results; (c) comparison between resultscorrespond- tours close to the critical regime, with energies of -0.1 meV ing to δ = δ = 10−6,10−7,10−8 and 10−9. (the last three (dashed),0meV(thicksolid)and0.1meV(thinsolid). Cyclic curves are indistinguishable to the eye on this scale.) All δ boundary condition are applied in they direction. values are in eV units. 20 Disorder only Triangular, A/A = 0.2 m Square, A/A = 0.04 0 m Triangular, A/A = 0.04 m 2E)| ;max-20 k ,min k R( G -40 g|10 o l -60 -80 -1 -0.5 0 0.5 1 E (meV) FIG. 9: Green’s functions for asample with Gaussian disor- derandvariousperiodicpotentials. Thecalculation included FIG. 7: The effect of a triangular periodic potential on the 20216stateswithφ/φ0 =3. SimilartoresultsshowninFig.9, critical energy regime. The disorder potential used here (not we see that the periodic potentials widen thecritical region. shown)supportsanarrowintervalofextendedstatescentered at about −0.6 meV. As the amplitude A of the periodic potential increases, the range of extended states increases dra- Figures10and11showsomeofourtypicalresults. The matically. The left panel shows results for disorder-only and twofiguresarecalculatedforthesameCoulomb-disorder two relatively weak periodic potentials, while theright panel potential, for values of E = 0.504 meV (at the bottom showstwolargerperiodicpotentialswherethethree-subband of the band) and E = 0.1−24 meV (close to, but be- structureexpected for φ/φ0 =3/2 is clearly seen. low the band center) res−pectively. Each figure contains four panels, panel (a) shows the profile of the disorder potential as well as an equipotential line (solid black) tionsaswescantheenergyspectrum. Eachcomputation correspondingto the valueE considered;the otherthree generates a large amount of data (roughly 24M), corre- panelsshowtheprojecteddensityofstatesρ (E)for(b) P sponding to the 500 plots of the local density of states puredisorder;(c)disorderplustriangularperiodicpoten- at the 500 values of E. Since we cannot show all this tial with A=0.1 meV; (d) disorder plus square periodic data,weselectacouple ofrepresentativecasesandsome potential with A = 0.1 meV. In Fig. 10, this equipoten- statistical data to interpret the overallresults. tialline(whichtracesthesemi-classicaltrajectoryofelec-