Efficient Linear Scaling Approach for Computing the Kubo Hall Conductivity
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Efficient Linear Scaling Approach for Computing the Kubo Hall Conductivity
Frank Ortmann , Nicolas Leconte , and Stephan Roche , Institute for Materials Science and Dresden Center for Computational Materials Science,Technische Universit¨at Dresden, 01062 Dresden, Germany ICN2Institut Catala de Nanociencia i Nanotecnologia,Campus UAB, 08193 Bellaterra (Barcelona), Spain and ICREA, Instituci´o Catalana de Recerca i Estudis Avan¸cats, 08070 Barcelona, Spain
We report an order- N approach to compute the Kubo Hall conductivity for disorderd two-dimensional systems reaching tens of millions of orbitals, and realistic values of the applied externalmagnetic fields (as low as a few Tesla). A time-evolution scheme is employed to evaluate the Hallconductivity σ xy using a wavepacket propagation method and a continued fraction expansion for thecomputation of diagonal and off-diagonal matrix elements of the Green functions. The validity ofthe method is demonstrated by comparison of results with brute-force diagonalization of the Kuboformula, using (disordered) graphene as system of study. This approach to mesoscopic system sizesis opening an unprecedented perspective for so-called reverse engineering in which the availableexperimental transport data are used to get a deeper understanding of the microscopic structureof the samples. Besides, this will not only allow addressing subtle issues in terms of resistancestandardization of large scale materials (such as wafer scale polycrystalline graphene), but will alsoenable the discovery of new quantum transport phenomena in complex two-dimensional materials,out of reach with classical methods. PACS numbers: 73.43.-f,72.80.Vp, 73.22.Pr
I. INTRODUCTION
The Hall effect is one of the central phenomenon inCondensed Matter physics with great importance formeasuring carrier density and mobility in semiconduc-tors. The long history started almost 150 years ago whenE. H. Hall observed a transverse voltage drop over a con-ducting bar driven by a longitudinal current and it expe-rienced an important turn in 1980 when K. von Klitzingobserved a quantized version of this effect, the so-calledQuantum Hall Effect (QHE) . This had an immediateimpact and triggered huge amount of work which led tothe definition of a resistance standard. The effect of disorder and related localization phe-nomenon is central to understand the integer QHE from abulk perspective, where σ xy plateaus develop by varyingthe charge density, while the plateau region coincide witha region of vanishing σ xx . This is explained by Andersonlocalization of the wave functions in the Landau levels in-duced by disorder. The most standard method to treatbulk conductivities microscopically is the linear-responsetheory, or the Kubo formula, which was first applied byAoki and Ando to the QHE problem . Such quantitycan be connected to measurements in the Corbino geom-etry for which electrodes are attached to the inner andouter perimeters of an annular sample, which allow toinvestigate bulk transport physics in the high magneticfield regime.Recently, the QHE has played a crucial role for thefirst unambigous proof of the existence of single atomiclayers of graphene . Indeed, in graphene, the pecu-liar Berry phase related to the pseudospin quantum de-gree of freedom results in a unique spectrum with four-fold degenerate Landau levels (due to the spin and valley degeneracies), and half-integer QHE with Hall plateausemerging at σ xy = e h ( n + 1 / n . QHErelated research in graphene is a vivid field, study-ing electron-electron interaction , superlattices ,graphene functionalization and topological currents. Ad-ditional transverse transport phenomena are being stud-ied such as (quantum-) anomaleous Hall effect, , spinHall effect and quantum spin Hall effect. The role of disorder in graphene is also debatedand the QHE features, with very robust behavior tolarge disorder , also present additional peculiarities de-pending on the symmetry breaking aspects conveyedby defects. Recently, we have studied the case ofoxidized graphene, and have shown the formation ofdisorder-induced resonant critical states appearing in thezero-energy Landau level with finite σ xx and suggestinga zero-energy σ xy quantized plateau . The confirmationof such a plateau will demand to revise the description oftopological invariant in disordered graphene, an excitingdirection of work. However a real space calculation of σ xy is needed and demand for the development of newmethodology.Indeed, the importance of an efficient computationalmethodology for the Hall conductivity σ xy stands insharp contrast to the limited possibilities to simulate thisquantity with usual approaches. One of them starts withthe Kubo formula and requires the knowledge of the fulleigenstates of a given system. . Such formulation allowsfor advanced analysis but suffers from unfavourablecubic scaling with system size which restricts to smallsystems because diagonalization is necessary. All eigen-states are needed and have to be combined to calculate σ xy . This computational limitation makes the analysisof realistic models of disordered samples a priori im-possible. Additionally, such approach is restricted tounreasonably strong magnetic fields that exceed by farwhat is achievable experimentally, and forces the mag-netic length scale to dominate over all other length scalesof the problem (mean free path, average distance betweenimpurities,...) limiting the exploration of complex mag-netotransport in intermediate or even low-magnetic fieldregimes, thus calling for approaches with improved nu-merical performance. In this paper, we present a highly efficient order- N algorithm for the computation of the Kubo Hall con-ductivity that allows us to circumvent aforementionedlimitations, and offer fascinating perspectives for explor-ing transverse transport phenomena in disordered two-dimensional materials with almost arbitrary complexity.A validation of the method on several models of clean ordisordered graphene is made by comparing this new timeevolution method with brute force diagonalization tech-nique. Some preliminary illustration of the method havebeen published in a prior work . The method is inspiredby the real space implementation of the dissipative Hallconductivity at zero frequency, which has proven undis-puted computational efficiency and predictive power (seeRef. for computational details, and Ref. for someillustration on realistic models of disordered graphene). II. METHODOLOGY
Kubo formalism –
The general framework is providedby Kubo’s linear response theory in which the dcconductivity is given as σ xy ( ω = 0) = 1 V Z ∞ dt Z β dλ Tr [ ρ j y j x ( t + i ~ λ )] (1)Starting from Eq. (1), the evaluation of the conductiv-ity σ xy proceeds by introducing an orthonormal many-particle basis with h n | H | n i = E n which allows to rewriteEq. (1) into σ xy = − V Z ∞ dt lim η → + X n Re h n | ρ j y E n − H + iη j x ( t ) | n i (2)where η is a small parameter necessary to converge thisexpression.In the case of non-interacting electrons one can proceedfurther towards the single-particle picture and obtain σ xy = − V Z ∞ dt lim η → + X k f ( ε k − ζ )Re (cid:20) h k | j y ε k − H + iη j x ( t ) | k i (cid:21) (3)where the Fermi-Dirac distribution is described by f ( ε k − ζ ) and the chemical potential ζ has been introduced.The single-particle eigenvalues ε k can in principle be ob-tained by matrix diagonalization of finite systems which has been widely employed in the literature for studyingmodel systems with phenomenological description of dis-order. Unfortunately such numerical operations quicklymakes the methodology computationally prohibitive, inparticular for simulating complex and large system sizescloser to the experimental ones and at moderate mag-netic fields. Clearly, the computational problem thenbecomes untreatable with diagonalization-based meth-ods, i.e. it becomes impossible to calculate (and store)all eigenvalues and eigenvectors, so that time-evolutionbased approaches appear promising. By introducing R ∞−∞ dEδ ( E − H ) we arrive at an alternative represen-tation of the Hall conductivity σ xy = − N s V Z ∞ dt Z ∞−∞ dEf ( E − ζ )lim η → + Re (cid:20) h Ψ | δ ( E − H ) j y E − H + iη j x ( t ) | Ψ i (cid:21) (4)which proves useful for the numerical implementation.In Eq. (4), the trace is replaced by an average overa random-phase state | Ψ i according to P k h k | ... | k i → N s h Ψ | ... | Ψ i . Such states are normalized by using thenumber of sites N s . We further introduce the projectionoperator P N R j =1 | Ψ j ih Ψ j | , as well as the time-dependent σ ( t ) fulfilling σ xy = lim t →∞ σ xy ( t ) with σ xy ( t ′ ) = 2 πV s Z t ′ dt Z ∞−∞ dEf ( E − ζ )lim η → + N R X j =1 Im [ κ j ( E )]Re (cid:20) h Ψ j | j y U † ( t ) 1 E − H + iη j x U ( t ) | Ψ i (cid:21) , (5)where κ j ( E ) = h Ψ j | z − H | Ψ i (6)are the the matrix elements of the Green function (with z = E + iη ) and V s = V /N s is the volume per site.Equation (5) is the basis for the numerical evaluation ofthe Hall conductivity for which one eventually takes thelimit t ′ → ∞ . One notes, however, that the result doesnot converge in all cases when taking this limit and an os-cillatory behaviour of the integral is generally observed,particularly for the case of clean systems in high mag-netic fields (and well separate Landau levels), which isthe usual test reference for Kubo Hall transport simula-tions.The reason is that the Kubo formula Eq. (1) assumesthat at “infinite past” times, the system is in equilib-rium, i.e. at zero electric field, the current should vanish.In other words, the current-current correlation functionshould decay with time difference t . This should be ful-filled also for the final result. In pristine systems this isnot the case and must be imposed a posteriori . In orderto take this into account an additional small parameter δ > σ xy ( t ′ ) = 2 πV s Z t ′ dt Z ∞−∞ dEf ( E − ζ )lim η → + N R X j =1 Im [ κ j ( E )]Re (cid:20) h Ψ j | j y U † ( t ) 1 E − H + iη j x U ( t ) | Ψ i (cid:21) e − δt ′ . (7)converges.We note that the same parameter has to be included infull analogy in the standard expression for the calculationof the Hall conductivity based upon the knowledge of theeigensystem. We here display such formula σ xy ( t ′ ) = ~ V lim η → + lim δ → + X k k f ( ε k − ζ ) (cid:20) h k | j x | k ih k | j y | k i ( ε k − ε k + iη )( ε k − ε k − iδ ) − c.c. (cid:21) (8)which is identical to Ref. for the reasonable choice that η = δ . These two equations (7) and (8) are comparableand will be evaluated for a couple of examples in Sect.III as numerical checks. Numerical implementation –
The calculation proceedsby splitting the time-dependent and time independentparts ( κ j ( E )) while the sum over j in (7) controls the en-ergy dependent convergence (see details below). Firstly,in order to evaluate Eq. (6), we make use of the Lanc-zos tridiagonalization of the Hamiltonian H which allowsus to use the continued fraction expansion for the diag-onal Green function. From the diagonal elements onecan obtain the necessary off-diagonal elements in a sec-ond iterative step. We find a simple recursion relationconnecting both which reads for n > κ n +1 = 1 b n [ − b n − κ n − − ( z − a n ) κ n ] (9)while the first step in the iteration is given as κ = b [1 − ( z − a ) κ ].Secondly, the time evolutions of the quantities | Ψ i and | j y Ψ j i can be performed efficiently through a Chebychevexpansion of the time-evolution operator U ( t ) which usu-ally converges quite rapidly. For the final calculation ofthe brackets in Eq. (7) we have to calculate off diago-nal matrix elements with the advanced Green operator( z − H ) − . For the calculation of this second set of off-diagonal Green functions in Eq. (7), we use the continuedfraction expansion. All operations are linear in samplesize and this holds also for the total expression since thescalar product in j is restricted to a cutoff value N R .We use δ = 0 .
002 if not stated otherwise throughout thepaper.
System description
The electronic properties ofgraphene can be described by the standard orthogonal j | κ j | ( a r b . un it s ) j(a) (b) FIG. 1: (color online) Absolute values of the off-diagonal ma-trix elements Eq. (6) for pristine graphene (a) and disorderedgraphene (b). Different curves correspond to energies aroundthe lowest-energy Landau level. tight-binding model H = X α V α | α ih α | − γ X h α,β i e − iϕ αβ | α ih β | , (10)where γ = 2 . eV is the nearest neighbor transfer inte-gral and V α the on-site energy, which can be chosen todescribe various disorder models. Spatially uncorrelatedAnderson disorder is introduced through on-site energiestaken at random from [ − W γ / , W γ /
2] where W givesthe disorder strength. This is a commonly used disor-der model for exploring the metal-insulator transitionin low-dimensional systems , which, in average, doesnot break electron-hole symmetry or inversion symmetry.The inversion symmetry of the system related to the sym-metry of AB sublattices can be artificially broken by astaggered potential of the form V α = +( − ) V AB for A(B)sublattices (including all atoms). A less artificial vari-ant of such potential can be introduced by diluting suchsublattice terms for a weaker and random distribution,which mimicks some correlation in the potential beyondthe Anderson model.The magnetic field is implemented through a Peierlsphase added to γ which determines the magnetic fluxto φ = H A · d l = h/e P hexagon ϕ αβ per plaquette. Spin degeneracy is assumed throughout the paper andan anti-symmetrization procedure for σ xy ( E ) has beenperformed consistent with electron-hole symmetry for allconsidered disorder potentials. III. RESULTS
Before we enter into the discussion of the physical re-sults, we illustrate the convergence behaviour of the nu-merical algorithm which can be controled by the expres-sion (6). A key quantity for the convergence is κ j ( E ) -0.1 -0.05 0 0.05 0.1 0.15E ( γ )-12-8-404812 σ xy ( e / h ) B=45.8TB=22.9 TB=11.4T -0.8 -0.4 0 0.4 0.8-808
Diag: Φ =2 π *0.006Diag: Φ =2 π *0.012Diag: Φ =2 π *0.024TEK: B=1925TTEK: B=458T high fieldsextremelymedium fields FIG. 2: (color online) Hall conductivity for varying magneticfield strength. Main frame: moderately high fields; lowerinset: extremely fields. Dashed curves in all frames indicatesimulations with time-evolution Kubo (TEK) approach. Notethe magnetic-field scaling of Landau level energies E ∝ √ B which is reflected in varying energy scales of main frame andinset ( γ units). which can be studied prior to the time evolution ofwavepackets and allows to estimate the required numberof recursion steps which depend on the specific physicaldetails (e.g. disorder, magnetic field), on the selectedenergy, and on the broadening η . Fig. 1 presents themodulus of κ plotted versus j for some selected ener-gies in the case of pristine graphene (a) and disorderedgraphene (b). As a general trend, Fig. 1 shows that incase of pristine systems, the convergence of the Lanczosrecursion can be reached with a low number of recursionsteps ( N R ), while increasing disorder requires N R to beincreased typically to a few thousands.Turning to the simulations of the Hall conductivity,we first discuss the results for pristine graphene in Fig.2 which shows the comparison of diagonalization methodand time-evolution Kubo approach. Figure 1 (inset)shows the case of extremely high magnetic fields (fluxes).We recall that, in order to fulfill the boundary conditionsof the periodic system, the limit of very high fields is usu-ally considered by diagonalization-based studies, becauseonly small sample sizes can be treated by matrix diag-onalization. The magnetic field is simultaneously givenin the inset of Fig. 2 in terms of the total flux pene-trating the sample. The corresponding results from thetime-evolution Kubo method (TEK) is plotted as dashedlines for the same magnetic fields (values indicated inthe inset legend). For the selected energies, the curvescoincide visually. The square root scaling of their posi-tion with magnetic field is evident and Hall-conductancesteps occur at the expected positions and with the ex-pected height corresponding to the half-integer sequence σ xy = ± (1 / n )4 e /h .Next, we consider the interesting case of moderately -1 -0.5 0 0.5 1E ( γ ) -30-20-100102030 σ xy ( e / h )
458 T963 T1926 T-0.4 -0.2 0-6-4-202 W = 0.25W = 0.5 -0.2 -0.1 0-18-12-60 W = 0.25W = 0.5W = 1.0- - - DIAGTEK
W = 0.5 963 T 45 T
FIG. 3: (color online). Hall conductivity for Anderson disor-der (dashed curves: exact diagonalization; solid curves: TEK)at different magnetic fields (main frame). Effect of ncreasingdisorder for high field 963T (upper inset) and intermediatefields of 45T (lower inset). high magnetic fields such as frequently observed in cur-rent experiments in the main frame of Fig. 2. TheKubo approach allows to reduce magnetic fields down toonly few Teslas where the first Landau levels appear atFermi energies of few tens of meV. Still the plateau se-quence is obtained with very good accuracy, which showsthat such approach can be of great practical use. Mostnotably, the figure demonstrates conductance quantiza-tion for fields as low as 11 Tesla. As an effect of finiteenergy resolution, we observe that the relative sharpnessof the steps is reduced with lowering the field in thisregime. The conductance slope starts to become noti-cable as Landau levels get closer (as seen for 11.4 T at σ xy = ± e /h ). This consequence of the considered en-ergy resolution (here η = 0 . η .As a second example to illustrate the performance ofthe algorithm, we focus on the case of homogeneouslydisordered graphene. The case of Anderson disorder ischosen for Fig. 3, because this disorder is expected toinduce a transition from the QHE regime to the con-ventional Anderson insulating state for sufficiently highvalue of W . The main-frame indicates very good agree-ment (within few percents) obtained between the Kuboalgorithm and exact diagonalization techniques at lowenergy for very high magnetic fields. At higher energy, η allows for fast convergence, at the expense of small loss ininformation compared to the exact method when Landaulevels are getting closer in energy. To illustrate the effectof increasing disorder, a zoom on the first two Hall stepsis provided for 963 T in the upper inset. The Landau lev-els are broadened with increasing W around the criticalstates, as expected from perturbation theory. Simulta-neously, the first and second plateaus remain at ± e /h -0.1 -0.05 0 0.05 0.1E ( γ )-4-2024 σ xy ( e / h ) B=9T B=25 T B=45T -0.01 0 0.01-4-202
FIG. 4: (color online) Hall conductivity for p=2.5% of AB-sublattice breaking defects (strenght V s = 0 . γ ) distributedat random in space (see main text). Vertical dotted line indi-cates nominal gap of a correspondingly homogenous AB po-tential of strength pV AB = 0 . γ . and ± e /h indicating robust QHE. In the lower inset, amore realistic magnetic field is chosen (45 T) to illustrate(i) the performance of the algorithm to probe low mag-netic fields and (ii) the possibility to destroy conductancequantization at higher energy for W = 1. For such dis-order, only the zero-energy Landau level fully develops,while all states become localized for E > E .As another challenging example, we present the anal-ysis of a weakly correlated potential. We chose a glob-ally sublattice-symmetry breaking potential that is lo-cally disordered. In contrast to the above Anderson dis-order model, which shifts all on-site energies at random,the present potential is only locally present, i.e. onlyfor a fraction of sites that are selected at random. Thedisorder is chosen such that it breaks globally the AB-sublattice symmetry. For the results shown in Fig. 4(main frame), we use a strength of V AB = 0 . γ with p = 2 .
5% of sites affected. Besides ordinary conductancequantization at the values ± e /h, ± e /h, ... , an addi-tional plateau is induced at zero energy with zero Hallconductivity. The plateau is clearly visible for fields be-tween 9 T and 45 T. We find the width of the zero-energyplateau equal for all calculated magnetic fields. The cor-responding energy value of the onset of the plateau at σ = 0 is determined by the strength of the potential which is given as pV AB = 0 . γ (indicated by dashedvertical line). The transition to higher plateaus (i.e. from ± e /h to ± e /h ) is rather influenced by the magneticfield-dependence of Landau level position.It is interesting to study the emergence of such plateauwhen gradually increasing the concentration of impuri-ties. Results are plotted for the case of 9T in Fig. 4 (in-set). The plateau-length scaling with p is evident fromthe figure. For the considered energy resolution in theinset ( η = 0 . pV AB = 0 . γ or 2.7 meV). IV. CONCLUSION
To conclude, an efficient real space algorithm to com-pute the Hall conductivity with the Kubo formula hasbeen presented and validated on simple graphene-basedsystems. Such approach should become a useful compu-tational tool to simulate QHE in very large size complexdisordered materials such as polycrystalline graphene ,graphene subjected to weak van der Waals interactionsuch as by a boron-nitride substrate , or other types ofdisordered two-dimensional materials. It should also al-low to corroborate the formation of zero-energy plateausof σ xy driven by disorder-induced critical states, and dis-connected from degeneracy lifting of Landau levels , anissue of genuine fundamental interest in the context oftopological interpretation of the quantized conductance. Acknowledgments
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