Metallic-to-insulating transition in disordered graphene monolayers
aa r X i v : . [ c ond - m a t . d i s - nn ] D ec Metallic-to-insulating transition in disordered graphene monolayers
M. Hilke
Dpt. of Physics, McGill University, Montr´eal, Canada H3A 2T8
We show that when graphene monolayers are disordered, the conductance exhibits a metallic-to-insulating transition, which opens the door to new electronic devices. The transition can beobserved by driving the density or Fermi energy through the mobility edge. At the Dirac pointthe system is localized, whereas at higher densities there is a region of metallic behavior before thesystem becomes insulating again at higher densities. The region of metallic behavior depends onthe disorder strength and eventually vanishes at high disorder. This result is quite unexpected sincein square lattices, scaling theory predicts that this metallic region does not exist in two dimensions,in contrast to graphene, where the lattice is a honeycomb.
In most active electronic devices, the conductivity canbe tuned from conducting to insulating by using a gate.With the recent discovery of graphene monolayers [1],and their potential for electronic devices [2], it is impor-tant for the conductivity to change substantially with thegate voltage. However, while in clean graphene monolay-ers, the conductivity changes almost linearly with gatevoltage, the off or minimum conductivity is still rela-tively large (of the order of the unit of conductance e /h )[4, 5, 6].Here we consider the situation of graphene nanorib-bons (with a honeycomb structure) of width W andlength L contacted by two large normal metals, assumedto have a square lattice. This leads to a typical two-terminal configuration as shown in figure 1, where thecontacts are assumed to be perfect. The disorder poten-tial is taken to be only onsite and uncorrelated. This isidentical to the large body of work on Anderson localiza-tion in tight binding models [7], which show a metal-insulator transition in dimensions strictly higher than2 [8, 9, 10, 11]. In two dimensional systems with asquare lattice no metal-insulator transition exists for non-interacting electrons [8, 12], except if correlations in thedisorder are present [13].In graphene, with its honeycomb lattice, the bandstructure was first studied theoretically by Wallace [14]using a tight binding Hamiltonian, where t ≃ V n the onsite energy. At the band edges, the band structureis very similar to the band edge of square lattices. At thecenter of band, however, and for V n = 0 Wallace showedthat there is a linear dispersion at the band center (Diracpoint) at two points in the reciprocal space leading to atwo-valley degeneracy. This has important implicationson scattering properties, such as a suppression of intra-valley backscattering. The linear dispersion close to theDirac point leads to dramatic new physics, such as ananomalous quantum hall effect and Dirac fermions [3]. Itis the absence of a gap at the Dirac point, which causesthe conductivity to not vanish at that point [4, 5, 6],which is a potential roadblock for applications in activeelectronics. Length −< l og ( G ) > E=−0.05W=68E=−0.1W=23 E=−0.15W=68−2(log
5. The solid lines represent linear fits of the data for L ≫ L C , the slope of which gives the inverse localizationlength L C . The horizontal line illustrates the value of L H . L and W are given in units of graphene atoms. Disorder plays an important role in graphene devicesand can be due to ripples [15, 16], defects in the substrate[17] and surface effects, such as partial hydrogenization[18]. To reduce disorder scattering, suspended graphenedevices [19] have been considered as well as the use ofother substrates [20].
Localization in graphene nanoribbons : Early numericalstudies in disordered honeycomb lattices, were limited tothe Dirac point and showed that the states are local-ized at this point [12]. These results were obtained byevaluating the localization length ( L C ) from the small-est Lyapounov exponent of a finite width ( W ) ribbonusing a transfer matrix approach. L C is then studiedas a function of W . Using scaling arguments [11], lo-calized states are identified when L C /W decreases with W , whereas extended states are characterized by L C /W increasing with W . The point where there is no de-pendence is then inferred to as the critical point, wherethe localization-delocalization transition occurs. In twodimensional square lattices, only localized states werefound for all energies [21].Here we use a very similar approach, but insteadof considering a transfer matrix, we evaluate the two-terminal conductance of the system, which allows usto directly evaluate the transport properties of the sys-tem. For a given width of the graphene nanoribbon,we evaluate the zero-temperature two-terminal conduc-tance ( G ) of the disordered graphene attached to metallic(square lattice and non-disordered) leads using an itera-tive Green’s function technique. Since the conductancedepends on the given disorder configuration we considera configurational average h·i by averaging over many dis-order configurations. We used two disorder distributions,either uniform with − V / < V n < V /
2, or binary ( V n = ± V / √
12) both characterized by h V n V m i = δ n,m V / −h log( G ) i for different values of the Fermi energy inunits of t . Because of the symmetric band structurearound the Dirac point, all results are symmetric around E = 0. Two regimes can be identified: (i) the ballis-tic regime, when L ≪ L C and (ii) the localized regime,where −h log( G ) i ∼ L/L C for L ≫ L C . In the bal-listic regime ( L ≪ L C ), the conductance is dominatedby mesoscopic conductance fluctuations [23, 24], where δG ≪ G , hence − h log( G ) i ≃ − log h G i ≃ log h R i . (1) R = 1 /G is the two terminal resistance of the device and δG is the standard deviation of G . In the localized regime( L ≫ L C ), on the other hand, where the conductancevanishes exponentially with the length of the system, wehave δG ≫ G . Because of the statistical properties ofthe conductance of a quasi-one dimensional system, thisyields [25] − h log( G ) i ≃ − h G i ≃ log h R i / ≃ L/L C − α. (2) α is a parameter, which is close to unity for G in unitsof 2 e /h as shown in figure 1. Relation (2) becomes ex-act in the limit where L → ∞ . Hence, the localizationlength can be extracted using any of the average trans- −0.6 −0.4 −0.2 Width L C / W E=−0.1E=−0.3E=0W E=−0.8 −1 −0.8 −0.6 −0.4 −0.2 00.20.40.60.811.2
E [t] L C / W E C W=252W=72Insulating Metallic
FIG. 2: Left: The dependence on width of the ratio L C /W for different values of the energy. Right: The dependence ofthe ratio L C /W on energy for different values of the width.The energy E C labels the critical energy at which the widthdependence of the ratio changes sign, indicative of a metallic-to-insulating transition. port quantities ( h log( G ) i , h G i or h R i ), but the conver-gence of h log( G ) i is much faster as shown in figure 1.While equation (2) requires L ≫ L C in order to ex-tract L C accurately, an approximate L C can also be ob-tained by looking at the crossover from ballistic to lo-calized, which will happen when equations (1) and (2)both hold, i.e., when each term is zero. This correspondsto δG ≃ G , since δG is close to one in units of 2 e /h [23]. Hence, defining L H as the length which minimizesmin {h log( G ( L )) i + (log h G ( L ) i ) + (log h R ( L ) i ) } ⇒ L H ,we obtain a length, which characterizes the crossoverfrom ballistic to localized. It turns out that for all valuesof interest, L H ≃ L C within 10%. Scaling behavior:
In order to determine if a state isexponentially localized in the two dimensional limit, weneed to evaluate L C /W as a function of W . L C is alwaysfinite in the quasi one-dimensional limit, since all statesare localized. For the two-dimensional case, L C /W is therelevant quantity, since ultimately we are interested inthe average conductivity σ = h G i L/W for infinite widthand length. By leaving the aspect ratio constant andequal to one, i.e., L = W when taking the limit to largesizes, we obtain σ ∼ e − W/ L C , since h G i ∼ e − L/ L C fromequation (2). Hence, if with increasing width L C /W → σ → L C /W → ∞ , for increasing width, thisimplies that there are no exponentially localized statesand we refer to this state as metallic.We now apply this analysis to our graphene device andshow the results for L C /W in figure 2. We observe thatat the Dirac point, the ratio L C /W monotonously de-creases with the width, which implies that the systemis insulating at the Dirac point. This is consistent withearlier results [12, 26, 27]. More interestingly, away fromthe Dirac point, the dependence of L C /W is more com-plicated and the dependence becomes non-monotonous.Indeed, at small widths the ratio decreases at first, beforeincreasing again at larger values of the width, which is asignature for metallic behavior. This increase of L C /W log(Width) E [t] L C /W log(Width) E [t] Lc/W log(Width) E [t] Lc/W
FIG. 3: Contour plots of the ratio L c /W as a function of theenergy and width. Top graph is for V = 1 .
5, bottom left for V = 2 and bottom right for V = 2 . only occurs in a small window of energies, between E = 0and E = E C as shown in figure 2. This is typical formetal-insulator transitions as seen in three dimensions[11]. However, this is quite unexpected in two dimensionswithout a magnetic field. Indeed, numerical studies ondisordered systems with square lattices show no metallicbehavior [12], i.e., L C /W always decreases with size.Interestingly, the continuous increase of L C /W overa wide range of widths close to the Dirac point, whichis a signature for a metallic behavior, is correlated withthe existence of additional channels. Indeed, for a non-disordered system close to the Dirac point, there is onlyone quantized transport channel, leading to a quantizedconductance of 2 e /h [28] if the width of the nanorib-bon is smaller than W D = 2 π ~ v F /E F , where v F is theFermi velocity and E F the Fermi energy. The value of W D is shown in figure 2 for E F = − . t and is corre-lated to a jump in L c /W . This is quite different from thesquare lattice case, where the addition of another chan-nel is always correlated with a trough of L C [29] andsuggests the two situation to be very different in natureand indicates that the reason for this delocalization isintimately related to the linear dispersion, which leadsto a suppression of intra-valley back-scattering. In termsof the beta function, Nomura and coworkers argued forthe topological delocalization of two-dimensional mass-less Dirac fermions [30].A more detailed analysis of the dependence of L C /W is provided by the contour plots shown in figure 3. For V = 1 . L C /W increases moves towards the Dirac point at E = 0,yielding an increase in L C /W as a function of W for an −1−0.500 100 200 30001234 E [t]Length σ [ / h ] FIG. 4: The dependence of the conductivity on energy andlength of a nanoribbon 272 atoms wide for V = 1 . energy close to the Dirac point. In contrast, the contourplot for V = 2 . L C /W de-creases with the width of the system. This implies thatall states are exponentially localized for V = 2 .
2. Thecase of V = 2 is interesting since it corresponds to thecrossover between the two behaviors. Indeed, for V = 2and close to the Dirac point, the contour lines of con-stant L C /W are almost constant in energy, indicative ofa critical behavior, in contrast to the V = 1 . W ,which is opposite to the V = 2 . V ≃ on/off ratio as a function of density (gate voltage),which is infinite when taking disorder into account. Inorder to illustrate this point, we evaluated the averageresistivity ρ = h R i√ W/ L for V = 1 . σ ) in figure 4 as a function of energy and lengthof the nanoribbon for a fixed width of W=272 atoms. Thenumerical coefficient in determining ρ is the geometricalfactor associated with the way we defined the atoms forour numerical implementation. Under our scheme, thetotal number of graphene atoms is given by W × L . Weused h R i and not h G i , because in most experiments ρ ismeasured as a function of density using a fixed current.For the maximum length (L=300), the conductivityvanishes at E = 0 due to localization at the Dirac point.Moving away from the Dirac point the conductivity in-creases with energy due to the metallic behavior, be-fore reaching a maximum, which leads to a diverging σ max /σ min as a function of energy (density) for a givengeometry and disorder strength. At even higher ener-gies, the conductivity decreases again due to localiza-tion. This behavior is consistent with some recent ex-periments, where a similar behavior has been observedby photo-emission, consistent with a metal to insulatortransition [31]. In epitaxial graphene, a metal to insula-tor transition was also observed by molecular doping [32].Several authors have computed the inverse participationratio to show the existence of a gap close to the Diracpoint [33]. In a related work, the density of states wascomputed numerically and found to be consistent witha localization-delocalization transition close to the Diracpoint [34, 35].Summarizing, we have shown that there is a metallic-to-insulating transition in disordered graphene monolay-ers, which happens close to the Dirac point and leadsto a window of energy, where a metallic behavior exists.This metallic behavior exists only for small enough dis-order ( V .
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