Eur. Phys. J. B (2014) 87: 16DOI: 10.1140/epjb/e2013-40756-0

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Edge states versus diffusion in disordered graphene flakes

Ioannis Kleftogiannis and Ilias Amanatidisa,b

Department of Physics, University of Ioannina, 45110 Ioannina, Greece

Received 12 August 2013 / Received in final form 6 November 2013Published online 20 January 2014 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2014

Abstract. We study the localization properties of the wavefunctions in graphene flakes with short rangedisorder, via the numerical calculation of the inverse participation ratio (IPR) and its scaling which providesthe fractal dimension D2. We show that the edge states which exist at the Dirac point of ballistic graphene(no disorder) with zig-zag edges survive in the presence of weak disorder with wavefunctions localized atthe boundaries of the flakes. We argue, that there is a strong interplay between the underlying destructiveinterference mechanism of the honeycomb lattice of graphene leading to edge states and the diffusiveinterference mechanism introduced by the short-range disorder. This interplay results in a highly abnormalbehavior, wavefunctions are becoming progressively less localized as the disorder is increased, indicated bythe decrease of the average 〈IPR〉 and the increase of D2. We verify, that this abnormal behavior is absentfor graphene flakes with armchair edges which do not provide edge states.

1 Introduction

Graphene [1,2] the first 2d metal ever made is a oneatom-thick layer of carbon atoms arranged in a honey-comb lattice structure offering large practical advantagesover conventional semiconductors, which make it an ex-cellent candidate for replacing silicon in future nanoelec-tronics. Most importantly graphene offers a vast field forfundamental theoretical work revealing phenomena suchas, relativistic behavior of the electrons at the Fermi level(E = 0), known as the Dirac point and topological phe-nomena like the edge states where the electron currentflows along the sample boundaries like in topological in-sulators. Edge states have been studied in [3–5] throughthe theoretical investigation of long stripes of grapheneknown as nanoribbons [6–9] while they have been experi-mentally observed in [10] through scanning tunneling mi-croscopy (STM) and spectroscopy (STS) techniques. Inreference [3], it is shown that nanoribbons with the so-called zigzag edges exhibit zero energy edge states withwavefunctions concentrated at the borders of the ribbons.These states are absent for nanoribbons with the otherpossible type of edge morphology, the armchair edges.So, zigzag nanoribbons exhibit vastly different electronicproperties compared to the armchair nanoribbons, at theFermi level. In principle, the type of edge (zig-zag orarmchair) at the boundaries of a graphene system playsa crucial role in its fundamental electronic properties.This argument becomes also apparent when studying con-fined graphene structures known as flakes [11–15], which

a e-mail: eliasamanatidis1@hotmail.comb Present address: Department of Electrophysics, National

Chiao Tung University, 1001 University Road, 30010 Hsinchu,Taiwan

have been experimentally fabricated in [1,16–20]. In ref-erences [12,14,15] it is shown that graphene flakes withzigzag edges exhibit also edge states which are absent forflakes with armchair edges. For instance, trigonal flakeswith zigzag edges provide edge states at the Fermi levelwhile hexagonal flakes with zig-zag edges give edge statesnear the Fermi level instead.

Apart from the detailed edge morphology of graphenesystems, another important factor that should be takeninto account when studying their electronic properties isthe presence of disorder which is an inevitable factor as inany mesoscopic material. The main sources of disorder ingraphene are the production method (synthesis) and theinteraction with the supporting substrate [21,22]. Disordercan appear as lattice distortions like wrinkles, rippling, orimpurities with various degrees of concentration comingfrom strains in the lattice or charge traps. Disorder is notalways an undesirable factor, it can be useful also in ap-plications, for example in spintronic devices through theinteraction with the spin [23,24] allowing the manipula-tion of the magnetic properties of disordered graphene sys-tems. Moreover, macroscopic graphene like lattice struc-tures (honeycomb) with controllable disorder have beenshown to be achievable in [25] through microwave simula-tion of the electronic waves. The theoretical treatment ofdisordered graphene requires the introduction of differentmodels like short or long range (smooth) disorder. Thetype of disorder plays a crucial role on the localizationproperties of the wavefunctions in graphene. Long rangedisorder retains the separation between the two so-calledvalleys, centered at the two non equivalent Dirac pointsof pure graphene (E = 0) at the corners of its hexagonalBrillouin zone, where the relativistic nature of electronsis revealed [21,22,26]. The separation between the two

Page 2 of 9 Eur. Phys. J. B (2014) 87: 16

valleys in disordered graphene results in many interestingphenomena like anti-localization [27] or minimum conduc-tance [28–30]. On the other hand, short range disordermixes the two valleys (inter-valley scattering), suppress-ing the relativistic effects at the Fermi level and resultingin Anderson localization [31–34]. In references [35–39] arough estimation of the localization properties of the wave-functions can be derived through the study of energy levelstatistics of disordered graphene flakes, verifying for ex-ample the Anderson localization for short range disorder.In general there has been an extensive study of Andersonlocalization phenomena in graphene.

However, the localization properties of the wavefunc-tions in the diffusive regime [40–42] has been much lessstudied in graphene, especially concerning the cases wherethe edge states are present. The diffusive regime is definedin systems with short range disorder when the system’slength is smaller than the localization length, i.e. beforethe onset to localization, where diffusive interference ef-fects are known to dominate the behavior of the wavefunc-tions. In this regime, the wavefunctions show a chaoticform with the amplitude randomly fluctuating coveringthe whole system area. Moreover, the random fluctuationsof the amplitude follow a multifractal form [41–50], a phe-nomenon that is absent in the localized regime, that is forlarge scales where Anderson localization is revealed. Inessence, this behavior owns its existence on the finite sizeof the system, so it is reasonable to use confined structureslike flakes for its investigation. Specifically, for graphenethe study of multifractality when edge states are presenthas shown interesting effects [50]. So, our main goal inthis paper is to investigate the edge states at the pres-ence of disorder in the diffusive regime, through the studyof disordered graphene flakes that provide edge states atthe zero disorder limit. Our analysis involves the numer-ical calculation of the inverse participation ratio (IPR)and its scaling behavior which gives the fractal dimen-sion D2 characterizing roughly the volume of a wavefunc-tion. Both measures combined provide a rough picture ofthe wavefunction form. Our calculations show evidence ofthe interplay between two mechanisms: the interferencemechanism of the honeycomb lattice of graphene, lead-ing to concentrated wavefunctions at the borders namelythe edge states, and the interference effects leading todiffusion of the wavefunctions in conventional disorderedsystems. The interaction between these two interferencemechanisms has a large impact on the localization prop-erties of the wavefunctions. When edge states are present,we observe a decrease of the average 〈IPR〉 and an in-crease of D2 with increasing disorder, implying that thewavefunctions become progressively less localized. This isa highly abnormal behavior compared to the conventional2d disordered systems, where the wavefunctions becomenaturally more localized with increasing disorder. We ver-ify that the normal behavior is reproduced for disorderedgraphene flakes without edge states.

The remainder of the paper is organized as follows.In Section 2, we introduce our numerical model based onthe tight-binding framework along with the short-range

disorder. In Sections 3 and 4, the numerical results forthe various graphene flake shapes at the presence of short-range disorder are presented. We study two different kindsof shapes, triangular and hexagonal, with zigzag andarmchair edges, respectively. We discuss our results andconclude in Section 5.

2 Model

For our analysis, we use the standard tight binding modelfor graphene with first nearest neighbor hopping andshort-range disorder simulated by a random on-site po-tential on each lattice site. The model is described by thefollowing Hamiltonian

H =∑

n

εnc†ncn +∑

〈n,m〉t(c†ncm + c†mcn), (1)

where c†n and cn are the creation and annihilation oper-ators for spinless fermions, 〈n, m〉 denotes nearest neigh-bors connected with constant hopping element t and εn

the random on-site potential, following the box distribu-tion P (ε) = 1/w, in the range [−w/2, w/2] with w de-noting the strength of the disorder. Additionally, all en-ergies E are measured in units of the hopping energy t,namely E ≡ E/t. This type of disorder simulates the exis-tence of impurities in the honeycomb lattice of graphene.It also mixes the two valleys resulting in inter-valley scat-tering in our problem, which suppresses the relativisticeffects at the Fermi energy while it breaks also the chi-ral symmetry of the graphene lattice. We focus our studyon graphene flakes with specific shapes that are knownto exhibit edge states at the limit of zero disorder. Theseinclude shapes studied in [12,14] like the trigonal and thehexagonal, with zig-zag edges. Also, we extend our studyon the same shape types but with armchair edges, inves-tigated also in [12,14] where the edge states are absent.We consider flakes that consist between a few hundred toa few thousand atoms, as the ones that have been studiedexperimentally in [12,16].

Diagonalizing the Hamiltonians for each graphene flakewe derive the wavefunction amplitudes Ψi at each latticesite i for energy E, which allow us to calculate the inverseparticipation ratio (IPR) given by IPR(E) =

∑i |Ψi(E)|4.

The participation ratio gives the information about the de-gree in which every site of the lattice is participating in thewavefunction. With its inverse we can get a rough estima-tion of the localization properties. In general, wavefunc-tions with densely distributed amplitude for example adiffusive or a ballistic wavefunction, will give values of IPRclose to zero while localized wavefunctions will give IPRvalues close to one. In addition, the scaling of IPR providesthe fractal dimension D2 of a wavefunction through the re-lation IPR ∼ L−D2 and is a characteristic measure of thevolume it roughly occupies. For instance, when analyzinga two dimensional system, D2 = 2 when the wavefunctionamplitude is equally distributed on the whole lattice whileD2 = 0 for a localized wavefunction. In general, D2 be-longs to a spectrum of fractal dimensions characterizing

Eur. Phys. J. B (2014) 87: 16 Page 3 of 9

a) b)

c) d)

α

Fig. 1. The flake shapes we study, triangular and hexagonal.(a) Equilateral triangular graphene flake with zigzag edges and(b) armchair edges. (c) Hexagonal flake with zigzag and (d)armchair edges. We choose the side length L of each shape,measured in units of the lattice constant α, to characterize theflake size (linear length scale). The flakes with zigzag edgesprovide edge states at the zero disorder limit in contrast to theflakes with armchair edges.

objects known as multifractals. Non integer values of D2

imply fractal wavefunctions [44–46,49,50] that are not ei-ther extended nor localized.

The value of D2 gives a good estimation of the vol-ume a wavefunction is occupying although it does notuniquely identify its form. For this reason, we illustratethe wavefunction probability amplitude |Ψi(E)|2 for eachcase that we are studying. With these three measuresthe IPR, D2 and the wavefunction probability, we canget a fairly complete picture of the wavefunctions indisordered graphene flakes.

3 Triangular graphene flakes

In this section, we examine the wavefunction localizationproperties of disordered graphene flakes of trigonal shapewith two different types of edges, zigzag and armchair(Fig. 1), via the analysis of IPR and its scaling whichgives the fractal dimension D2. One example of a trigonalgraphene flake with zigzag edges can be seen in Figure 1a.We characterize the size of the flake by the base lengthof the triangular shape L in units of the lattice constantα = 2.42 A, for example the length is L = 6α in Figure 1a.In the case of zero disorder the zig-zag triangle exhibitszero energy edge states as discussed in [12,14], with wave-functions concentrated on the zigzag edges (see Fig. 3a).In Figure 2, we plot the average value 〈IPR〉 versus theenergy E near the Fermi energy (E = 0) for a flake withL = 44 consisting of 2113 atoms and for different disorderstrengths w = 0.5, 1, 1.5. The number of disorder realiza-tions is 5000. Orange color in the background representsthe individual values of IPR for w = 0.5 used to obtain thecorresponding curve of 〈IPR〉. The 〈IPR〉 for w = 0 canbe also seen as large individual green colored dots, sepa-rated by large gaps. The green dot corresponding to the

Fig. 2. The average value of the inverse participation ratio(〈IPR〉) versus the energy E for a trigonal graphene flake withzigzag edges with L = 44 (2113 sites), for disorder strengthsw = 0.5, 1, 1.5 and 5000 realizations, along with the w = 0 case(green dots). The orange points in the background representthe individual values of IPR for w = 0.5 while the cases for theother disorder strengths are shown in the inset. 〈IPR〉 decreaseswith increasing disorder, implying less localized wavefunctions,visible also in the inset, where IPR concentrates on lower valueswith increasing disorder. There is an abrupt change of 〈IPR〉at E ∼ 0.1 evident also in the background (orange points),coming from the transition from edge states concentrated atlow energies to extended states for higher energies.

lowest energy for w = 0 is the average 〈IPR〉 over 43 edgestates with energy E = 10−16. The number of these edgestates is significantly larger than the number of extendedstates (6 states) that lie higher in the energy spectrum. Atthe presence of disorder both the edge states and the ex-tended states disperse creating two separated energy areascharacterized by vastly different values of IPR, somethingthat is especially evident for the weaker disorder w = 0.5.For the low energy regime with the larger IPR values,approximately until E ∼ 0.1 clearly the IPR is in aver-age decreasing with increasing disorder strength. This isalso evident in the inset of Figure 2 where we show thecorresponding values of IPR. The points concentrate pro-gressively lower with increasing disorder resulting in lowervalues of 〈IPR〉 as we have seen in the main figure. Keepingin mind the two trivial limits of IPR, IPR = 0 for extendedwavefunctions (ballistic) and IPR = 1 for localized wave-functions, the overall behavior of IPR implies that the lowenergy wavefunctions of the zigzag triangle with disorderare becoming in average less localized as the disorder isincreased. This behavior is highly abnormal and is absentin normal disordered materials, for instance a square lat-tice or a chain, where the wavefunctions become naturallymore localized with increasing disorder, with the IPR in-creasing in average. We have verified that the abnormalbehavior starts approximately at w = 0.25. From w = 0to w = 0.25 a normal behavior occurs with IPR increasing.In Figure 2, we can also observe an abrupt change of IPRstarting at E ∼ 0.1 resulting in large fluctuations un-til E ∼ 0.17. This is especially evident for w = 0.5, inboth the curve of 〈IPR〉 and the individual values of IPR

Page 4 of 9 Eur. Phys. J. B (2014) 87: 16

a) b)

c) d)

Fig. 3. The wavefunction probability amplitude |Ψ |2 for a flakewith L = 44 (2113 sites) for different disorder strengths w =0.5, 1.5, 5 at energy E ∼ 0.07 (one disorder realization) alongwith a wavefunction for w = 0 at E = 10−6. (a) For w = 0, theamplitude is concentrated on the edges of the flake (edge state).(b) For w = 0.5, the wavefunction becomes localized along theborder. (c) For w = 1.5, the amplitude penetrates slightly theflake, despite being mostly concentrated on the border, show-ing also abrupt fluctuations. (d) The wavefunction for largedisorder strength w = 5 concentrates in the flake’s bulk. Thereis no sign of the edge states in this case, the wavefunctionbecomes localized inside the flake instead.

(orange color). Below E ∼ 0.1, we can distinguish a wholearea of points with values of IPR in average much higherthan the values corresponding to energies in the inter-val E ∼ 0.17−0.25.

The energy area E ∼ 0−0.1 consists mainly of wave-functions that have their amplitude concentrated at theedges of the trigonal flake as shown in Figures 3b and 3cwhere the wavefunction probability is plotted for E ∼ 0.07and different strengths of disorder. Comparing the edgestate in Figure 3a for w = 0 with the wavefunction in Fig-ure 3b we can see that weak disorder (w = 0.5) localizesthe wavefunction along a random area on the border ofthe flake acting in this way as a pertubation on the zerodisorder limit studied in [14], where the edge state am-plitude spreads almost periodically along the whole bor-der. So, we can say that the edge states survive for weakdisorder in the sense that the amplitude remains mostlyconcentrated at the border of the flake. The abnormal be-havior that we distinguished through the analysis of IPRin the energy area E ∼ 0−0.1 can be understood by look-ing at Figures 3c and 3d, where we plot the wavefunctionprobability for stronger disorder w = 1.5 and w = 5. InFigure 3c the amplitude although still mostly localizedalong the border, has started extending across it whileit also penetrates slightly the flake, resulting in less lo-calized wavefunctions and in lower IPR values as seen inFigure 2. Even larger disorder (w = 5) in Figure 3d tendsto localize the wavefunction inside the flake instead of theedges. For sufficiently strong disorder, the destructive in-terference mechanism of the honeycomb lattice that leadsto edge states in graphene is completely destroyed by the

3.0 3.2 3.4 3.6 3.8 4.0ln(L)

-4.0

-3.5

-3.0

-2.5

ln(<

IPR

>)

w=0.5,D2=0.17

w=1.0,D2=0.36

w=1.5,D2=0.59

Fig. 4. The scaling of IPR for a trigonal zig-zag flake for dif-ferent disorder strengths, averaged over energies in the interval[0, 0.1] and over 5000 realizations of the disorder. The slope ofln(〈IPR〉) versus ln(L) gives the fractal dimension D2 whichallows the estimation of the wavefunction volume. It is clearlyincreased with increasing disorder while the points for IPR forsmall disorder strengths lie above the corresponding points forlarger disorder in agreement with the behavior demonstratedin Figure 2 considering the average behavior of IPR.

interference mechanism coming from the short-range dis-order, leading to Anderson localization with the wavefunc-tions becoming completely localized in the interior (bulk)of the flake instead of the edges. In essence, the on-siteshort-range disorder destroys the special topology of thehoneycomb lattice that favors the creation of edge states.However,the effects of the edge states are not immedi-ately washed out but only in the large disorder limit. Forweaker disorder the abnormal behavior we analyzed oc-curs, coming from the interplay between the interferencemechanisms of the diffusion and the edge states result-ing in less localized wavefunctions with increasing disor-der. Apart from this interplay another factor that plays arole in the abnormal behavior we obtain is the progressivemixing of the edge states with the extended states as thedisorder is increased, which becomes more important forstrong disorder. We have considered low disorder strengthvalues (w < 2) in the study of IPR in order to minimizethe effect of this mechanism. For energies above E ∼ 0.17in Figure 2 the localization properties of the wavefunc-tions change drastically, spreading along the whole flake,indicated by IPR obtaining much lower values than forE ∼ 0−0.1. The transition from edge states to these ex-tended states creates the large fluctuations in the energyinterval E ∼ 0.1−0.17.

We continue our analysis by studying the scaling ofIPR from which we can derive the fractal dimension D2

providing a rough estimation of the wavefunction’s vol-ume. In Figure 4, we show D2 for different strengths ofdisorder with the values of the average 〈IPR〉 for eachsize being over 5000 disorder realizations and energiesinside the window [0, 0.1], where the edge states lie ap-proximately according to our previous analysis. SinceIPR ∼ L−D2 , we plot ln(〈IPR〉) versus ln(L) in order toget the exact value of D2. We can observe that the slope

Eur. Phys. J. B (2014) 87: 16 Page 5 of 9

Fig. 5. 〈IPR〉 versus the energy for a trigonal graphene flakewith armchair edges of size L = 44.5 (2106 sites) for disor-der strengths w = 0.5, 1, 1.5 and 5000 realizations, along withthe w = 0 case (green dots). The orange points in the back-ground are the individual values of IPR for w = 0.5, withthe other cases shown in the inset. IPR increases in aver-age with increasing disorder, implying more localized wave-functions, a behavior observed in normal disordered systemslike the square lattice. The gaps appearing at the distributionof IPR for w = 0.5 (orange points) come from the respectivegaps present for w = 0. As the disorder is increased the gapsare disappearing, as shown in the inset.

of each curve representing D2 increases with increasingdisorder strength, implying that the volume occupied bythe corresponding wavefunctions increases also. Moreover,ln(〈IPR〉) and consequently 〈IPR〉 for each individual sizeaveraged over the energies and realizations decreases, inagreement with the results obtained in Figure 2 for thecurves of 〈IPR〉 versus E. Additionally, D2 obtains noninteger values implying multifractality inside the chosenenergy window, evident from the abrupt fluctuations ofthe amplitude in Figure 3. The values of D2 below one arereasonable considering the wavefunctions for w = 0.5, 1.5in Figures 3b and 3c being mostly concentrated along theborder of the trigonal flake extending slightly inside. Theoverall behavior of D2 versus the disorder strength forthe graphene triangle with zigzag edges can be seen inFigure 14. We should clarify that we have restricted ouranalysis of D2 on the diffussive regime, for larger flakesizes Anderson localization takes place in all cases givingzero D2. So, we have found that although weak disorderpreserves the edge states in trigonal flakes with zig-zagedges, its increase results in a highly abnormal behav-ior, the edge states become progressively less localized,extending inside the flakes.

We contrast this result to the behavior of the IPRand D2 observed for trigonal flakes with armchair edgesshown in Figure 1b, for which edge states are absent forzero disorder according to [12,14]. 〈IPR〉 versus the en-ergy E for L = 44.5 (2106 atoms) and for different disor-der strengths can be seen in Figure 5 along with theirrespective distributions of IPR in the background (or-ange points–w = 0.5) and inside the inset (brown points-w = 1.0, grey points–w = 1.5). Also, the 〈IPR〉 for w = 0

Fig. 6. The wavefunction probability amplitude for an arm-chair trigonal flake of size L = 44.5 (2106 sites) for strengthof disorder w = 1.5 at E ∼ 0.07. The amplitude spreads ran-domly all over the flake, this is a common picture of a diffusivewavefunction encountered in the diffusive regime of normal 2dsystems.

3.0 3.2 3.4 3.6 3.8ln(L)

-7.0

-6.5

-6.0

-5.5

-5.0

ln(<

IPR

>)

w=0.5,D2=1.77

w=1.0,D2=1.69

w=1.5,D2=1.61

Fig. 7. The scaling of IPR for a trigonal armchair flake fordifferent disorder strengths, averaged over the energy inter-val [0, 0.2] and over 5000 realizations. The calculated slope D2

characterizing the wavefunction volume is clearly decreasedwith increasing disorder while the points along the curves in-crease their values, in agreement with the average behaviorof IPR observed in Figure 4.

can be seen as large green dots. Clearly, as the disorder isincreased, 〈IPR〉 increases inside the whole energy window(E = 0−0.26), which is evident also in the insets wherethe individual points of IPR concentrate progressively inhigher values with increasing disorder. The gaps at specificenergies for w = 0.5 are a consequence of the respectivegaps appearing in the energy spectrum for zero disorder asseen from the 〈IPR〉 for w = 0. Apart from these gaps inFigure 5, IPR behaves smoothly with the energy in con-trast to the trigonal flake (Fig. 2), where we observed tworegions with vastly different values of IPR. In Figure 6,we show a characteristic example of a wavefunction lyinginside the energy window of Figure 5 at E ∼ 0.07. The am-plitude fluctuates wildly, randomly spreading on the wholeflake, a common picture of a diffusive wavefunction.

The results for the fractal dimension D2 can be seenin Figure 7 for different strengths of disorder. The valueof IPR for each size is calculated as an average over

Page 6 of 9 Eur. Phys. J. B (2014) 87: 16

Fig. 8. 〈IPR〉 vs. E for a hexagonal flake with zigzag edgesof size L = 18 (1944 sites) for different disorder strengths and5000 realizations, along with the case w = 0 (green points)and the individual values of IPR in the background (orangepoints–w = 0.5) and the inset. 〈IPR〉 decreases with increasingdisorder in agreement with the behavior of the IPR in theinset, implying in overall less localized wavefunctions. AroundE ∼ 0.15 there is a transition from edge states with high IPRvalues to extended states with low IPR values. The overallbehavior is qualitatively similar to that of the trigonal flakewith zigzag edge.

different realizations of the disorder and over the energywindow [0, 0.2]. In contrast to the trigonal flakes withzigzag edges, D2 decreases with increasing disorder, as fornormal 2d disordered systems where the wavefunctionsoccupy progressively less volume, becoming less dense, asmore disorder is introduced. The overall behavior of D2

versus the disorder strength can be seen in Figure 14. Weshould also remark that for sufficiently large flakes D2

goes to zero because of Anderson localization. So, the trig-onal flakes with armchair edges at the presence of shortrange disorder do not exhibit the abnormal behavior weencountered in the disordered trigonal flakes with zigzagedges. Instead, they behave as normal disordered metalswith their corresponding wavefunctions becoming morelocalized with increasing disorder.

4 Hexagonal graphene flakes

We now extend our analysis on hexagonal graphene flakeswith zigzag edges which provide edge states at the zero dis-order limit (see Fig. 9a) as in the case of the correspondingtrigonal flakes [12,14] concentrated near the Fermi energy.Again, we are interested in the effect of the disorder on theedge states obtained through the scaling analysis of IPR.The overall behavior of IPR versus the energy can be seenin Figure 8 for a hexagonal flake with L = 18 consisting of1944 sites. The behavior is similar to that of the zig-zagtriangle. For low energies below E ∼ 0.1, where the edgestates are concentrated (see Figs. 9b and 9c), 〈IPR〉 andIPR are decreasing with increasing disorder meaning thatthe wavefunctions are becoming progressively less local-ized as indicated by the wavefunction form in Figure 9ccompared to Figure 9b. This abnormal behavior starts

a) b)

c) d)

Fig. 9. The wavefunction probability amplitude for a flake ofsize L = 184 (1944 sites) for different disorder strengths w =0.5, 1.5, 5 at E ∼ 0.07 along with the w = 0 case at E ∼ 0.0005.(a) The amplitude for w = 0 clearly extends along the border ofthe flake. (b) For w = 0.5, the amplitude remains concentratedon the border although localized in a specific area. (c) For w =1.5, the amplitude clearly starts penetrating the flake. (d) Forstrong disorder w = 5, the wavefunction becomes localized inthe bulk. The overall behavior is similar to the trigonal flakeswith zigzag edges.

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0ln(L)

-4.2

-4.0

-3.8

-3.6

-3.4

-3.2

ln(<

IPR

>)

w=0.5,D2=0.32

w=1.0,D2=0.56

w=1.5,D2=0.72

Fig. 10. The scaling of IPR for a hexagonal zig-zag flake forincreasing disorder strengths averaged over the energy interval[0,0.1] and 5000 realizations. The slope D2 is increased withincreasing disorder while the points for IPR for small disor-der lie above the corresponding points for larger disorder, inagreement with Figure 7.

approximately from w = 0.25 as in the zig-zag triangles.IPR changes drastically above E ∼ 0.1, where extendedstates start to appear, obtaining much lower values de-spite the fact that the transition from the edge states tothe extended states is smoother in this case than it isfor the zigzag triangle. This is because of the denser en-ergy spectrum compared to the triangular flakes, evidentfrom the comparison of 〈IPR〉 for w = 0 (green points)between Figures 2 and 8. Sufficiently strong disorder lo-calizes the wavefunction inside the flake (see Fig. 9d).As in the case of the zigzag trigonal flakes, D2 in Fig-ure 10 is increased as we increase the disorder (wavefunc-tion volume increases), while its non-integer values imply

Eur. Phys. J. B (2014) 87: 16 Page 7 of 9

Fig. 11. 〈IPR〉 for a hexagonal flake with armchair edges ofsize L = 20.2 (2382 sites) for different disorder strengths and5000 realizations along with IPR in the background and theinset including the w = 0 case. Overall, IPR increases withincreasing disorder like in the case of the trigonal armchairflake. This is the behavior observed in the diffusive regime ofnormal disordered systems.

multifractality in agreement with the abrupt fluctuationsof the amplitude in Figure 9. It is clear that for weak disor-der the edge states survive also in the case of the hexagonalflakes with zigzag edges and result in less localized statesas the disorder is increased. So, the abnormal behavior wepointed out for the trigonal flakes with zigzag edges existsalso in the case of hexagonal flakes with zigzag edges. Weconclude that this effect is independent of the overall flakeshape and is related with the existence of edge states, gov-erned by the detailed edge structure on the borders of thegraphene flakes.

To finalize our study, we consider the case of hexag-onal flakes with armchair edges shown in Figure 1d. Inthis case, there are no edge states in the limit of zero dis-order, in contrast to the flakes with zigzag edges. In Fig-ure 11, we show 〈IPR〉 for a hexagonal flake with armchairedges with L = 20.2 consisting of 2382 sites for disorderstrengths w = 0, 0.5, 1.5 and 5000 realizations along withthe corresponding IPR values in the background and theinset. Overall IPR increases as the disorder is increasedlike in the case of the trigonal shape with armchair edges.This is compatible with the behavior observed in normaldisordered systems like a square lattice or a linear chain,as we have already pointed out. Also in Figure 13, weobserve that D2 is clearly decreased. In Figure 12, thecorresponding wavefunction has the characteristic diffu-sive form with randomly fluctuating amplitude coveringthe whole lattice, compatible with the non integer valuesof D2, close to two. So, the hexagonal flakes with arm-chair edges at the presence of short range disorder, do notexhibit the abnormal behavior we encountered in flakeswith zigzag edges which provide edge states.

5 Discussion and conclusions

In this paper, we have presented numerical results forgraphene flakes with short-range disorder that show a

Fig. 12. The wavefunction probability for a hexagonal arm-chair flake consisting of 2382 sites (L = 20.2) for strength ofdisorder w = 1.5 at energy E ∼ 0.07. The amplitude is spreadon the whole lattice, fluctuating randomly, a characteristicexample of a diffusive wavefunction.

2.40 2.60 2.80 3.00ln(L)

-6.75

-6.25

-5.75

-5.25ln

(<IP

R>

)

w=0.5,D2=1.85

w=1.0,D2=1.81

w=1.5,D2=1.74

Fig. 13. The scaling of IPR for a hexagonal flake with arm-chair edges for disorder strengths w = 0.5, 1, 1.5 averaged inthe energy interval [0, 0.2] and 5000 realizations. D2 behavesconventionally, decreasing with increasing disorder.

highly abnormal behavior for the localization propertiesof the wavefunctions when edge states are present. Weobserve a decrease of the inverse participation ratio 〈IPR〉and an increase of the fractal dimension D2 with in-creasing disorder, implying that the wavefunctions be-come roughly less localized as the disorder is increased.We argue that the underlying mechanism that causes thisbehavior is the interplay between the destructive interfer-ence mechanism that produces edge states (concentratedwavefunctions at the borders) and the diffusive interfer-ence mechanism, known to prevail in 2d mesoscopic sys-tems with short range disorder for scales below the lo-calization length. We have verified this behavior throughthe study of trigonal and hexagonal graphene flakes withzigzag edges where edge states are present. The abnormalbehavior is absent for flakes with armchair edges whichdo not result in edge states. Moreover the edge states sur-vive for weak disorder. On the other hand, for sufficientlystrong disorder the edge state mechanism is completely

Page 8 of 9 Eur. Phys. J. B (2014) 87: 16

0 0.5 1.0 1.5 2.0w

0

0.5

1.0

1.5

2.0

D2

Hexagonal-Armchair EdgesTrigonal-Armchair EdgesHexagonal-Zigzag EdgesTrigonal-Zigzag Edges

Fig. 14. The fractal dimension D2 vs. the disorder strength wfor trigonal and hexagonal flakes with zigzag and armchairedges. For the flakes with zigzag edges and w > 0.25, D2 is in-creased with increasing disorder in contrast to the flakes witharmchair edges for which D2 behaves conventionally, decreas-ing with the disorder. This means that when edge states arepresent the volume of the corresponding wavefunctions is inaverage increasing, they become less localized with increasingdisorder as we have shown through the analysis of IPR. At thelimit of zero disorder (w = 0) D2 = 1 for both the zigzag tri-angle and hexagon since both flakes exhibit zero energy edgestates extended along the flake’s border. For w = 0−0.25 lo-calization of these edge states results in a steep decrease of D2,followed by the abnormal behavior we described.

suppressed by the destructive interference mechanism ofthe short-range disorder (Anderson localization) result-ing in localization of the wavefunctions in the bulk of theflakes instead of the edges. The abnormal behavior we ob-tained exists in the intermediate regime between the weakand strong disorder limit.

So, we have shown that when edge states are present,the consideration of interference effects in graphene sys-tems with short-range disorder is very important and leadsto unexpected behavior. In our work we concentrated inthe diffusive regime while in future studies, we also intendto investigate the localized regime. Additionally, we wouldlike to extend our analysis in order to include effects likethe magnetization of the edges [23,24] in graphene systemswith disorder or to investigate the connection with topo-logical insulators which has been shown to carry resem-blance to graphene [51,52], due to the edge states mimick-ing the topological property of the electron current flowingfrom the boundary surfaces in these materials. We hopethat our work will motivate further experimental investi-gation of the edge states in graphene systems, and theirimpact on the electronic properties.

We would like to thank D. Katsanos, V.A. Gopar and S.N.Evangelou for useful discussions and careful reading of themanuscript. We also acknowledge the computer resources andassistance provided by the Institute for Biocomputation andPhysics of Complex Systems (BIFI) of the University ofZaragoza.

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