Conductance Peak Density in Disordered Graphene Topological Insulators
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Conductance Peak Density in Disordered Graphene Topological Insulators
Louis G. C. S. S´a , A. L. R. Barbosa , and J. G. G. S. Ramos Departamento de F´ısica, Universidade Federal da Para´ıba, 58297-000 Jo˜ao Pessoa, Para´ıba, Brazil Departamento de F´ısica, Universidade Federal Rural de Pernambuco, 52171-900 Recife, Pernambuco, Brazil (Dated: 7 de setembro de 2020)We investigate the universal properties of quantum transport in graphene nanowires that engen-der subtle universal conductance fluctuations. We present results for three of the main microscopicmodels that describe the sublattice of graphene and generate, as we shall show, all the chiral uni-versal symmetries. The results are robust and demonstrate the widely sought sign of chirality evenin the regime of many open channels. The fingerprints paves the way to distinguish systems withsublattice symmetry such as topological insulators from ordinary ones by an order of magnitude.The experimental realization requires a single measurement of the chaotic fluctuations of the asso-ciated valleytronics conductante. Through the phase coherence length, our theoretical predictionsare confirmed with the data from traditional measurements in the literature concerning quantummagnetotransport.
PACS numbers: 73.23.-b,73.21.La,05.45.Mt
I. INTRODUCTION
The transport phenomena in disordered mesoscopicsystems is strongly affected by the wave behavior of theelectron [1–6]. The wave scattering in nanostructures gi-ves rise to the fundamental phenomena of universal con-ductance fluctuation (UCF), which depends only on thedimensionality and the symmetries of the correspondingcoherent state [4,7–10]. An extraordinary characteristicof the UCF is the nature selection of just few ensembles todescribe its emerging properties. Despite the complexityof the mesoscopic device, atomic details are irrelevantand the transport properties depend only on fundamen-tal symmetries. According to Random Matrix Theory(RMT) [1,11,12], there are three ensembles: (1) circularorthogonal ensemble (COE) ( β = 1), if the Hamiltoniansupports time-reversal and spin-rotation symmetries, i.e.,if none magnetic field is applied B = 0 and the spin-orbitinteraction (SOI) is neglected; (2) circular unitary ensem-ble (CUE) ( β = 2), if time-reversal symmetry is brokenby a magnetic field, B = 0; (3) circular symplectic ensem-ble (CSE) ( β = 4), if spin-rotation symmetry is broken,while the time-reversal symmetry is preserved, i.e., theSOI is non-null.In bulk state, at the thermodynamic limit, the conduc-tance assumes fixed values in the same material. Howe-ver, in the mesoscopic regime, fluctuations that seem ran-dom appear as a function of some field or external energythat vary from sample-to-sample [13–17]. Interestingly,these fluctuations are, in fact, chaotic properties catego-rized through their amplitudes in any of the universal en-sembles previously mentioned, namely they depend onlyon fundamental symmetries of nature. One way to me-asure the correlation of such chaotic events is to run anaverage on the ensemble of achievements from various di-sordered devices [13,15]. This exhaustive process of makeand measure samples provides the important correlationwidth scale associated with chaos. Several experimental and theoretical results indicate the universality of thisscale, which act as a “chaotic number” [18]. For para-metric variations in energy, for instance, this average onsamples allows ones to find the electron dwell time asthe inverse of the corresponding autocorrelation width[19–22]. The measurements as a function of the exter-nal magnetic field, on the other hand, have the phase-coherence length as the associated physical measurable[10], a relevant parameter of the quantum scattering.Recent advances in nanotechnology have allowed theproduction and control of graphene monolayers withcarbon atoms distributed in honeycomb lattice [23,24].Graphene has received both experimental and theoreticalattention due to its special electronic transport proper-ties [8,10,14,23,25–28]. Another studies demonstrate theexistence of universality in graphene beyond the Wigner-Dyson classes previously mentioned [14,29–36]. In thechaotic mesoscopic regime, RMT predicts the existenceof ten symmetries classes according to the Cartan’s clas-sification [12], with the three of Wigner being the mostestablished. The honeycomb graphene lattice is dividedinto two sub-lattices, which givie rise to chiral symme-tries, allowing the emulation and control of the otherCartan classes in artificial atoms (quantum dots). TheChiral symmetry is an achievement of more general sys-tems also known as topological insulators [14–17,37,38]and the phenomenological counterpart, the relativisticchaos [2–4]. However, experimental detection of otherssymmetries is a hard task given that the chirality seemsto disappear according the number of open channels (le-ads widths) increases subtly. For two or more channels,this signal tends to disappear quickly.Faced with this scenario, two questions of experimen-tal and theoretical interest naturally arise. The first con-cerns the extraction of the magnetic correlation widthconsidering the requirement of several experimental de-signs and, therefore, the synthesis of a very large ensem-ble of nanowire samples: Is there a measurable capableof extract the correlation width through a single experi-mental design? And the second one deals with the cha-racteristic values associated with universality in topolo-gical insulators such as graphene: Does the autocorrela-tion width and consequently the phase-coherence lengthcarry peculiar information of topological insulators? Inthis work, we give a positive answer to both questions.The observable in question is the density of maxima (lo-cal maximum per magnetic field interval) already testedin different systems [13,18–22]. To extend the validity ofour result, we also investigate the graphene monolayerin different scenarios and find different numbers associa-ted with the chaos and universality that can be extractedfrom a single realization. Our results are confirmed byexperimental data available in the literature [39,40]. II. METHOD
In this work we investigate the three main models forgraphene nanowires. As we shall show, all of them exhi-bits the UCF. The first model (model I) supports spin-rotation symmetry with neglected SOI terms. On thesecond (model II) and third (model III) models, we im-plement the effect of the SOI in the electronic structure,proposed by Kane and Mele [41,42], as a spin-rotationsymmetry breaking mechanism.Disordered graphene in a tight-binding representationhas the Hamiltonian for honeycomb lattice defined as [14,41–43] H = X i ε i c † i c i − t X h i,j i e iφ ij c † i c j (1)for model I, H = H − i √ λ KM X hh i,j ii e iφ ij (cid:16) ˆ d in × ˆ d jn (cid:17) z s z c † i c j (2)for model II and H = H − iλ R X h i,j i e iφ ij (cid:16) s × ˆ d ij (cid:17) z c † i c j (3)for model III, where h ... i and hh ... ii denote the nearest-neighbor and next-nearest-neighbor interactions, respec-tively. On the model I, the first term introduce short-range disorder with ε i randomly chosen in the range( − W/ < ε i < W/ W the measure of the di-sorder strength and c i ( c † i ) is the annihilation (creation)operator on the i th lattice site. The second term repre-sents an usual nearest-neighbor interaction, with t de-noting the hopping between C atoms. Here we choosethe value t = 2.6 eV, following DFT calculations [44].The time-reversal symmetry breaking is generated by anexternal magnetic field B accounting the magnetic flux φ ij = e/ ¯ h R r j r i A · d l . In this work, we use the gauge A = ( − By, ,
0) as the vector potential for perpendicu-lar magnetic field ( z -direction) to graphene sheet. The π /a 2 π /a -3-2-10123 E ( e V ) π /a 2 π /a -3-2-10123 E ( e V ) π /a 2 π /a k x -3-2-10123 E ( e V ) a)b)c) λ KM = 0.1 λ R = 0.15 Figura 1: Band structures of ZGNR samples with 84 atoms.Blue line indicates the edge states. Band structure of (a)model I, preserved the time-reversal and spin-rotation sym-metries, characterizing by COE. The time-reversal symmetryis breaking for (b) model II and preserved for (c) model III,while spin-rotation symmetry is breaking in both models bySOI λ KM = 0.1 and λ R = 0.15. The model II and III arecharacterizing by CUE and CSE, respectively. second term contemplated on the model II is the mirrorsymmetric SOI that involves next nearest sites of indices i , j with n being the common nearest neighbor of i and j ,and, consequently, ˆ d in describes a vector pointing from n to i . The second term on the model III is a nearestneighbor Rashba term. The symbol s denotes the Paulimatrix that describes the electron spin.We perform tight-binding simulations through the -8 -7 -6 -5 -4 -3 -2 -1 0 E (eV) G ( e / h ) Model IModel IIModel III -1 -0.8 -0.6 -0.4 -0.2 0
E (eV) G ( e / h ) Figura 2: Conductance of a graphene nanowire as a functionof Fermi energy in absent of both disorder and magnetic fieldfor ZGNR sample with 84 atoms and 100 nm. Inset: the con-ductance for energy above − . Kwant code [45]. We calculated the conductance usingthe Landauer-B¨uttiker formulation, G = e /h T r ( tt † ) , where t is the transmission matrix block of the scatteringmatrix, written in terms of Green’s function. The systemis coupled to two semi-infinite ideal leads and the sample-to-sample fluctuation behavior can be characterized bythe conductance deviation rms[ G ] = p h G i − h G i . III. RESULTS AND DISCUSSIONS
For pedagogical reasons, we divide this section in thefollowing four subsections: A) We show the effects of SOIon the graphene band structure and in the correspon-ding conductance without disorder; B) We incorporateeffects of disorder on the graphene conductance and alsoin its UCF; C) We describe the conductance peak densityand analyze the corresponding numerical data using, as amethod, results from the principle of maximum entropy;D) The analysis of the conductance peak density will beapplied to UCF experimental data from Refs. [39,40].
A. The Graphene Wire in the Absence of Disorder
We begin the investigation obtaining known resultsand analyzing the graphene band structure. We explorea zigzag graphene nanribbon (ZGNR) with 84 atoms, inabsence of disorder and magnetic field. The results aredepicted in the Fig.(1). For the model I, Fig.(1.a) showsthat the bands connected at Fermi energy ( E =0) are po-pulated by the edge states and the other ones are thebulk bands, unveiling degenerate copies for each band.For the model II, according to the results shown in theFig(1.b), the effect of the spin-rotation symmetry brea-king with the SOI is to open the edge bands and the gap undergoes an increment of 1.0 eV to 1.5 eV, which isin accordance with the Kane-Mele model [41]. The latestmodel provides results explaining that the edge states arenot chiral since each edge has propagating states in bothdirections. The model III contemplates the Rashba termwhich violates z −→ − z mirror symmetry [46], shiftingsome bands as depicted in the Fig.(1.c).We investigate the ZGNR conductance in a samplewith 84 atoms and 100 nm of length, in absence of di-sorder and magnetic field. Results for the three modelsare shown in the Fig.(2). Without SOI (model I), theconductance is quantized, as expected, and it is null forany energy out of the range | E | > . | E | = 2 . λ KM = 0 . − . − . λ R = 0 .
15) also inducesconductance fluctuation.
B. Disordered Graphene Wire
The main purpose of this present investigation is tosimulate samples whose relevant properties are manifestin the UCF whenever the electron transport is diffusive.Within this general purpose, we analyze the conductanceaverage and its deviation as a function of the disorderstrength W , as shown in the Fig.(3), for which we takethe typical values λ KM = 0 .
15 and λ R = 0 .
15 on mo-dels II and III, respectively. Although the absence ofdisorder, W = 0, can induce the system to behave asideal, the conductance decreases according the disorder ismagnified, as depicted in the Fig.(3.a), and it is also thedisorder that induces the sample-to-sample fluctuation,Fig.(3.b). Therefore, with moderate values of W , the dif-fusive regime is activated, and indicates that the conduc-tance deviations Fig.(3.b) support an expected characte-ristic value of a quasi-one-dimension nanowire, describedin the framework of RMT [1]. For large values of W ,the conductance performs a conductor/insulator transi-tion occasioned by the Anderson localization [6], i.e., theconductance and its deviation tends to zero, as expected.Another perspective of the UCF can be enlightenedthrough the conductance average and its deviation as afunction of the Fermi energy, Fig.(4), with λ KM = 0 . λ R = 0 .
15 for models II and III, respectively, and adisorder value W = 0 .
75. As expected, the conductancedeviation goes to the COE ( β = 4) value 0.74 e /h for mo-del I, as depicted in the Fig.(4.b). For model II, Fig.(4.a),the edge states ( E > -0.4 eV) are unaffected by disorderand indicates a ballistic behaviour of the electron trans-port. The robustness of the topological edge states [37],although the conductance fluctuation, Fig.(4.b), exhibitsthe universal behavior with the UCF value of 0.52 e /h in the diffusive regime ( E < -0.4 eV), a value that corres-pond to the CUE ( β = 2). Notice that the model II cor- W (Disorder strength) 〈 G 〉 ( e / h ) Model IModel IIModel III
W (Disorder strength) r m s ( G ) ( e / h ) β = 1 β = 2 β = 4a)b) Figura 3: Disorder effects in graphene. (a) Conductance ave-rage and (b) its deviation as a function of disorder strength W at an energy of − . β = 1), CUE( β = 2), and CSE ( β = 4). respond to the CSE ( β = 4) in the framework of RMT. Asdiscussed by Choe and Chang [14], the distinct UCF va-lue in the Kane-Mele model is attributed to the particularform of H which can be written as a sum of two HaldaneHamiltonians [37], H = H + Haldane ⊕ H − Haldane , a directsum of spin-up and spin-down Haldane terms with eachcomponent supporting opposite sign. The Haldane mo-del is categorized as the circular unitary ensemble ( β =2) since the phase acquired by the next-nearest-neighborhopping term breaks the time-reversal symmetry. Hence,the model II exhibits a UCF value 0.52 e /h whereasits Hamiltonian is founded in the Haldane model. Forthe model III, the edge states are affected by disorderas indicates the Fig.(4.b) and the conductance deviationconverges to the GSE value 0.37 e /h , as expected. -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 E (eV) 〈 G 〉 ( e / h ) Model IModel IIModel III -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
E (eV) r m s ( G ) ( e / h ) β = 1 β = 2 β = 4 b)a) Figura 4: (a) Conductance average and (b) its dviation as afunction of Fermi energy in presence of disorder W = 0.75.The lines in (b) represent the deviation values predicted bythe RMT for COE ( β = 1), CUE ( β = 2) and CSE ( β = 4). G ( e / h ) Square Φ / Φ G ( e / h ) Model IModel IIModel III
Graphene
Square lattice
Figura 5: Conductance as a function of perpendicular magne-tic flux, with a disorder strength of W = 0.75 at an energy of E = -1.2 eV. For model II and III, the values used are λ KM = 0.15 and λ R = 0.15, respectively. C. Conductance Peak Density
In order to investigate the connection between the con-ductance peak density and its correlation function, weanalyze the conductance behavior as a function of a per-pendicular magnetic field. Effectively, in all the simula-tions we use a disorder strength W = 0 .
75 and a Fermienergy tuned in − . N and the range of dimen-sionless perpendicular magnetic flux, ρ Φ = N/ (∆Φ / Φ )[19]. Hence, we build the central sector of the Table Ifrom the Fig.(5), which shows the maxima number andthe CPD for the three models. Notice we use in allgraphene models the magnetic flux range ∆Φ / Φ = 0 . / Φ = 0 . Φ , and the density of ma-xima, ρ Φ , as proposed in the reference [19]. The methodwas applied in a variety of scenarios, yielding applicationson different systems [20,21]. The experimental obtentionof the correlation width requires an average under thedata derived from the synthesis and measurement on aensemble of samples. Therefore, it is a costly procedure,although it results in this important characteristic num-ber of chaos. The requirement of a large amount of datain order to extract the average in the ensemble can bereplaced by a simple and unique measurement throughthe maxima density observable. The method determinessuch relation through the formula ρ Φ = 3 π √ Φ ≈ . Φ . (4)To confirm the results obtaining previously, present inthe central sector of the Table I, we calculate the auto-correlation function C (∆(Φ / Φ )) by simulating severalsamples through subtly modifications in the boundaryconditions of the wire in each sample in order to esta-blish a connection with the chaos. Once with the data,we extract the Γ Φ . The former was calculated throughits usual definition C (∆Φ / Φ ) = h G (∆Φ / Φ ) G (0) i − h G (∆Φ / Φ ) ih G (0) i , yielding the results displayed in the Fig.(6) for thegraphene models (bottom horizontal axis) and for thesquare lattice (top horizontal axis). The autocorrelationwidth is also defined in the usual way, i.e, the ∆Φ / Φ N ρ Φ Γ Φ ρ Φ Model I 8 1333 4.4 × − × − × − × − ρ Φ / Φ = N/ ∆Φ / Φ with ∆Φ / Φ = 0 . ⊥ ) obtained from auto-correlation function Fig. (6); Fifth column: the conductancepeak density obtain from Eq. (4). There is a great agreementbetween the both methods of obtaining the conductance peakdensity. value at half height C (Γ Φ ) C (0) = 12 . The Γ Φ values obtained from Fig.(6) are presented inthe right side column of the Table I. Substituting the Γ Φ values in the Eqs.(4), we obtain the CPD, which is alsopresented on the right side of the table I. The resultsof third and firth columns are in great agreement anddemonstrate the efficiency of the CPD procedure for adisorder graphene device.On the one hand, as depicted in the Figs.(3.b)-(4.b),the conductance deviations of disordered graphene na-nowire follow the fundamental symmetries of Wigner-Dyson ensembles, that is, they do not provide any in-formation related to the graphene chyral symmetry. Onthe other hand, the results in the Fig.(5) show a signifi-cant change in the UCF, leaving clear the fingerprint ofchiral symmetry. These changes affect the CPD and canadequately characterize chiral fundamental symmetriesin transport measurements. We performed the same si-mulation for a square lattice for which there is no sublat-tice thus referring to the usual fundamental symmetriesof Wigner-Dyson ensembles. Our results are demonstra-ted in the Fig.(6) and also in table I, confirming the resultof Ref.[13].As exposed in the Table I, the CPD of disorderedgraphene nanowire ranges from 1333 to 3333, while fortypical nanowire the value is 371. Even more surprisingly,we show that there is a difference of an order of magni-tude in all conductance measurements of a topologicalinsulator (honeycomb lattice of graphene) compared toa typical nanowire (square lattice). Therefore, our resultdemonstrates that UCF clearly carry information aboutthe fundamental symmetry of the sub-lattice structure. D. Chirality Fingerprints Underlying ExperimentalSignals
Our results suggest that chirality can be supportedeven with a high number of open channels, leaving fin-gerprints on the conductance/UCF. Experimental data Φ / Φ -0.4-0.200.20.40.60.81 Model IModel IIModel III C / C SquareGrapheneSquare lattice
Figura 6: Conductance correlation in function of perpendicu-lar magnetic flux obtained from 10 realizations. For modelII and III were used λ KM = 0.15 and λ R = 0.15, respectively. H (T) -0.500.51 G - 〈 G 〉 ( e / h )
80 90 100 110 120 B ⊥ (mT) G ( e / h ) a)b) Figura 7: Experimental data of a monolayer graphene conduc-tance as a function of the magnetic field. The experimentaldata (green) is obtained (a) from the Ref.[39] and (b) fromthe Ref.[40]. The smooth of experimental conductance data,in black. on mesoscopic diffusive wires with a number of channelsin the order of a few dozen would, consequently, be sig-nificantly relevant to prove exactly the autocorrelationwidth length and other previously established observa-bles. However, the experimental data available is, untilour knowledge, for more than 100 channels. We followthe previous results in order to find, through a single re-alization, the fingerprints of chirality. Therefore, in thissection, we apply the developed methodology in experi-mental data found in the literature [39,40].Ojeda et. al. [39] and Lundeberg et. al. [40] developedexperimental measures of conductance as a function ofthe magnetic field in monolayer graphene, whose resultsare shown in the Fig. (7. a and b), respectively. In theformer, the monolayer graphene nanowire was depositedonto doped silicon and has dimensions of 2.7 µ m of widthand 0.8 µ m of length, while in the latter, it was depositedonto an SiO /Si wafer and has dimensional of 4.1 µ m ofwidth and 12.9 µ m of length. In spite of the experimentalsample lengths be one order greater than those used inour numerical simulations, Fig.(5), the experimental datahave a similar behavior as depicted in the Fig.(7).We focus the investigation of the experimental dataon one of the most relevant experimental observable, thephase-coherence length L φ , which has a direct relationwith the autocorrelation width L φ = r he Γ ⊥ , (5)with h and e denoting the Planck constant and the elec-tronic charge, respectively. The substitution of the Eq.(4)in the Eq.(5) render L φ = r √ πh e ρ ⊥ , (6)which provides a direct relation between ρ ⊥ and L φ . Thisindicates that we can obtain the phase-coherence lengththrough a simple calculation of the conductance peak den-sity directly from a experimental data even without theinformation of correlation width.We first remove the random noise due to both the ther-mal interference and the experimental apparatus fromthe data. A simple and straightforward way to performthe extraction is through the B´ezier algorithm as usedand described in the Refs.[13,14]. The smooth conduc-tance as a function of a perpendicular magnetic field isdepicted by black color in the Fig.(7). The number ofmaxima contained in the data of the Fig.(7.a) is N = 16while the magnetic field range is ∆ B = 0 .
45 T, whichallows one to directly infer that ρ ⊥ = 35 . − . The-reafter, by replacing the value in the Eq.(6), we obtainthe phase-coherence length L φ ≈ . µ m, which is inagreement with literature of monolayer graphene withmobility µ ≈ cm /Vs, Refs.[25,47–50]. We make a di-rect comparison between the CPD results of a monolayergraphene and those from the Ref.[13], which investigateInAs nanowire samples (metallic regime). For such me-tallic samples (Wigner-Dyson ensembles) with mobilityalso of µ ≈ cm /Vs, Ref.[51], we found the CPD ρ ⊥ = 3 . − and, by replacing the value in the Eq.(6),we obtain L φ ≈ . µ m. Remarkable, in similar expe-rimental situations (small mobility values), the CPD ofgraphene is ten times greater than that of InAs nanowire,which confirm that UCF carry information about the fun-damental sublattice symmetry. Also, there is a peculiarcoherence length fingerprint in the Universal Chiral Sym-metries, confirming nicely our theoretical predictions.Additionally, the maxima number of the Fig.(7.b) is N = 32 and ∆ B = 0 .
045 T, generating the numbers ρ ⊥ = 711 . − and L φ ≈ µ m, which is in agreementwith literature of monolayer graphene with high mobility µ ≫ cm /Vs, Refs.[52–54]. Furthermore, not onlywe show that the CPD can be understood as an univer-sal sublattice characteristic number but we also obtaina law that relates the coherence phase length with thesquare root of the maxima density. The rapid oscillationof the conductance as a function of the field in grapheneexplains its strong quantum coherence behavior. IV. CONCLUSIONS
In conclusion, we investigated three widely used mo-dels to describe graphene nanowires. The three onesgenerate universal fluctuations in conductance, each be-longing to different classes of fundamental symmetries:orthogonal, unitary or symplectic ensembles. The studydemonstrates the connection between the typical spec-trum of systems with the sublattice symmetry, the for- mation of edge states and the classes of universal symme-tries. The manifestations in diffusive electron magneto-transport are evident in these different scenarios.Through the connection between the conductance sig-nals with the principle of maximum entropy, we identifieda measurable capable of extracting the correlation lengththrough a single experimental realization of a graphenemonolayer. Remarkable, we identified a clear fingerprintof the sublattice structure and, as a deployment, the sig-nal coming from topological insulators by simply coun-ting the maxima number even in the regime of manyopen channels.We obtained a law relating the phase coherence lengthto the square root of the maximum density, L φ ∝ √ ρ ⊥ Eq. (6), showing through experimental data, the sublat-tice signal by an order of magnitude when compared tothe magnetoconductance of usual semiconductor systems.Our study paves the way for the search for coherent quan-tum transport signals in chiral systems.
Acknowledgments
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