Optical conductivity of disordered graphene beyond the Dirac cone approximation
Shengjun Yuan, Rafael Roldán, Hans De Raedt, Mikhail I. Katsnelson
OOptical conductivity of disordered graphene beyond the Dirac cone approximation
Shengjun Yuan , Rafael Rold´an , , Hans De Raedt , and Mikhail I. Katsnelson Institute for Molecules and Materials, Radboud University of Nijmegen, NL-6525AJ Nijmegen, The Netherlands Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco E28049 Madrid, Spain Department of Applied Physics, Zernike Institute for Advanced Materials,University of Groningen, Nijenborgh 4, NL-9747AG Groningen, The Netherlands (Dated: October 29, 2018)In this paper we systemically study the optical conductivity and density of states of disordedgraphene beyond the Dirac cone approximation. The optical conductivity of graphene is computedby using the Kubo formula, within the framework of a full π -band tight-binding model. Differenttypes of non-correlated and correlated disorders are considered, such as random or Gaussian poten-tials, random or Gaussian nearest-neighbor hopping parameters, randomly distributed vacancies ortheir clusters, and random adsorbed hydrogen atoms or their clusters. For a large enough concentra-tion of resonant impurities, a new peak in the optical conductivity is found, associated to transitionsbetween the midgap states and the Van Hove singularities of the main π -band. We further discussthe effect of doping on the spectrum, and find that small amounts of resonant impurities are enoughto obtain a background contribution to the conductivity in the infra-red part of the spectrum, inagreement with recent experiments. PACS numbers: 72.80.Vp, 73.22.Pr, 78.67.Wj
I. INTRODUCTION
An important part of our knowledge on the elec-tronic properties of graphene, which consist of a two-dimensional (2D) lattice of carbon atoms, can be de-duced from optical spectroscopy measurements (for re-cent reviews see Refs. 2 and 3). Infrared spectroscopy ex-periments allows for the control of interband excitationsby means of electrical gating, similarly as electricaltransport in field effect transistors. Within the simplestDirac cone approximation, only vertical in wave-vectorspace transitions across the Dirac point are optically ac-tive, leading to a constant value for the optical conduc-tivity of undoped graphene of σ = πe / h . This leadsto a frequency-independent absorption of πα ≈ . α = e / (cid:126) c ≈ /
137 is the fine structure constant.This fact was observed for suspended graphene in ex-periments in the visible range of the spectrum and itwas later confirmed by further experiments in suspendedgraphene and epitaxial graphene on SiC substrate. For doped graphene with nonzero chemical potential µ ,at zero temperature, in the absence of disorder and with-out considering many body effects, the allowed excita-tions are only those between particle-hole pairs with anenergy difference larger than 2 µ , due to Pauli’s exclusionprinciple. This would lead to a zero infrared conductivitybelow the energy ω = 2 µ , and the optical conductivityshould be simply a step function σ ( ω ) = σ Θ ( ω − µ ).However, a background contribution to the optical con-ductivity between 0 < ω < µ was observed in Refs. 5and 7, pointing out the relevance of disorder and manybody effects. Another characteristic of the optical spec-trum is the Drude peak, which is built from a transferof spectral weight from the low-energy interband conduc-tance to the ω → although a strong suppression of the Drude peak at infrared ener-gies has recently been observed. Furthermore, the flat-tening of the π -bands at energies away from the Diracpoint is responsible for the strong peak in the spectrumat higher energies (of the order of 5eV) which is asso-ciated to optical transitions between states of the VanHove singularities. Finally, a method to control theintermediate excited states in inelastic light scatteringexperiments has been also reported, revealing the impor-tant role of quantum interference in Raman scattering. This intense experimental work has been accompaniedby a series of theoretical studies which have treated theproblem of the optical conductivity at different levels ofapproximation.
For example, it has been suggestedthat the presence of spectral weight in the forbidden re-gion of the optical spectrum of doped graphene (below ω = 2 µ ) can be associated to disorder, electron-electron interaction or excitonic effects. In particular,the effect of electron interaction in the spectrum has beenconsidered in Refs. 27–35. Furthermore, understandingthe role played by the different kinds of disorder thatcan be present in this material is essential to increase themobility of the samples. Besides the long-range chargedimpurities, other possible scattering sources suchas ripples, strong random on-site potentials, largeconcentration of hydrogen adatoms, strain or ran-dom deformations of the honeycomb lattice have beenconsidered. In this paper, we perform a systemic study of the op-tical spectrum of graphene with different kinds of disor-der for both doped and undoped graphene, such as therandomness of the on-site potentials and fluctuation ofthe nearest-neighbor hopping. Special attention is paidto the presence of resonant impurities, e.g., vacanciesand hydrogen adatoms, which have been proposed as themain factor limiting the carrier mobility in graphene. a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Furthermore, depending on the way how the defects aredistributed over the lattice sites, each kind of disordercan be either non-correlated or correlated. The non-correlated one corresponds to the case with uniformlyrandom distributed disorder sources, i.e., the potential orhopping are randomly changed within a certain range, orthe resonant impurities (vacancies or hydrogen adatoms)are randomly positioned over the whole lattice; the cor-related one means that the distribution of the disorderfollow particular topological structures, such as Gaus-sian potentials or Gaussian hopping parameters, resonantclusters with groups of vacancies or hydrogen adatoms.In the present paper, we consider a noninteracting π -band tight-binding model on a honeycomb lattice andsolve its time dependent Sch¨odinger equation (TDSE) tocalculate the density of states (DOS). From this, the op-tical conductivity is calculated numerically by means ofthe Kubo formula.The paper is organized as follows. In Sec. II we givedetails about the method. In Secs. III and IV we showresults for the optical conductivity of undoped graphenein the presence of non-correlated and correlated disorder,respectively. In Sec. V we calculate the optical spectrumof doped graphene. Our main conclusions are summa-rized in Sec. VI. II. MODEL AND METHOD
The tight-binding Hamiltonian of a disordered single-layer graphene is given by H = − (cid:88) ( t ij a † i b j + h . c) + (cid:88) i v i c † i c i , + H imp , (1)where where a † i ( b i ) creates (annihilates) an electron onsublattice A (B), t ij is the nearest neighbor hopping pa-rameter, v i is the on-site potential, and H imp describesthe hydrogen-like resonant impurities: H imp = ε d (cid:88) i d † i d i + V (cid:88) i (cid:16) d † i c i + h . c (cid:17) , (2)where ε d is the on-site potential on the “hydrogen” impu-rity (to be specific, we will use this terminology althoughit can be more complicated chemical species, such as var-ious organic groups ) and V is the hopping between car-bon and hydrogen atoms. For discussions of the last termsee, e.g. Refs. 39, 44, and 47. The spin degree of free-dom contributes only through a degeneracy factor andis omitted for simplicity in Eq. (1). A vacancy can beregarded as an atom (lattice point) with and on-site en-ergy v i → ∞ or with its hopping parameters to othersites being zero. In the numerical simulation, the sim-plest way to implement a vacancy is to remove the atomat the vacancy site.The numerical calculations of the optical conductivityand DOS are performed based on the numerical solution of the TDSE for the non-interacting particles. In gen-eral, the real part of the optical conductivity contains twoparts, the Drude weight D ( ω = 0) and the regular part( ω (cid:54) = 0). We omit the calculation of the Drude weight,and focus on the regular part. For non-interacting elec-trons, the regular part is σ αβ ( ω ) = lim ε → + e − βω − ω Ω (cid:90) ∞ e − εt sin ωt × (cid:104) ϕ | f ( H ) J α ( t ) [1 − f ( H )] J β | ϕ (cid:105) dt, (3)(we put (cid:126) = 1) where β = 1 /k B T is the in-verse temperature, Ω is the sample area, f ( H ) =1 / (cid:2) e β ( H − µ ) + 1 (cid:3) is the Fermi-Dirac distribution opera-tor, J α ( t ) = e iHt J α e − iHt is the time-dependent currentoperator in the α (= x or y ) direction, and | ϕ (cid:105) is a ran-dom superposition of all the basis states in the real space,i.e., | ϕ (cid:105) = (cid:88) i a i c † i | (cid:105) , (4)where a i are random complex numbers normalized as (cid:80) i | a i | = 1. The time evolution operator e − iHt and theFermi-Dirac distribution operator f ( H ) can be obtainedby the standard Chebyshev polynomial representation. The density of states is calculated by the Fourier trans-form of the time-dependent correlation functions ρ ( ε ) = 12 π (cid:90) ∞−∞ e iεt (cid:104) ϕ | e − iHt | ϕ (cid:105) dt, (5)with the same initial state | ϕ (cid:105) defined in Eq. (4). For amore detailed description and discussion of our numeri-cal method we refer to Ref. 39. In this paper, we fix thetemperature to T = 300K. We use periodic boundaryconditions in the calculations for both the optical con-ductivity and the density of states, and the size of thesystem is 8192 × × III. NON-CORRELATED DISORDERA. Random on-Site Potentials or Nearest-NeighborHopping Parameters
We first consider two different kinds of disorder: ran-dom local change of on-site potentials and random renor-malization of the hopping, which correspond to the diag-onal and off-digonal disorders in the single-layer Hamil-tonian Eq. (1), respectively. The former acts as a localshift of the chemical potential of the Dirac fermions, i.e.,shifts locally the Dirac point, and the latter arises fromthe changes of distance or angles between the p z orbitals.In order to introduce the non-correlated disorders in theon-site potentials, we consider that the on-site potential v i is random and uniformly distributed (independently ofeach site i ) between the values − v r and + v r . Similarly, -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 (a) Clean v r = 0.2t v r = 0.5t v r = t DO S ( /t ) E/t
Clean v r =0.2t v r =0.5t v r =t xx / /t (b) xx / /t -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 DO S ( /t ) E/t
Clean t r = 0.1t t r = 0.3t t r = 0.5t (c) Clean t r =0.1t t r =0.3t t r =0.5t (d) xx / /t xx / /t -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 DO S ( /t ) E/t
Clean n x =1% n x =5% n x =10% (e) Clean n x =1% n x =5% n x =10% (f) xx / /t xx / /t -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 (g) Clean n i =1% n i =5% n i =10% DO S ( /t ) E/t (h)
Clean n i =1% n i =5% n i =10% xx / /t xx / /t FIG. 1. (Color online) Numerical results for the density of states (left panels) and optical conductivity (right panels) of undoppedgraphene with different kinds of non-correlated disorders: (a,b) random on-site potentials, (c,d) random hopping parameters,(e,f) random distributed vacancies, and (g,h) random distributed hydrogen adatoms. Size of the samples is 4096 × × the non-correlated disorder in the nearest-neighbor hop-ping is introduced by letting t ij be random and uniformlydistributed (independently of couple of neighboring sites (cid:104) i, j (cid:105) ) between t − t r and t + t r . The presence of eachtype of disorder has quite similar effect to the density ofstates [see the numerical results with different magnitudeof disorders in Fig. 1 (a) and (c) for the random on-sitepotentials ( v r /t = 0 .
2, 0 . t r /t = 0 .
1, 0 . .
5) respectively]. The spectrumis smeared starting from the Van Hove singularities at | E | = t , and the smeared region expands around theirvicinal areas as the strength of the disorder is increased,whereas the spectrum around the vicinal region of theneutrality point keeps unaffected unless the disorder istoo strong. As the optical conductivity is proportionalto the density of states of the occupied and unoccupiedstates, one expects a peak in the spectrum of the opti-cal conductivity at the energy ω ≈ t , which correspondsto particle-hole excitations between states of the valenceband with energy E ≈ − t and states of the conductionband with energy E ≈ t . These processes contributeto the optical conductivity with a strong spectral weightdue to the enhanced density of states at the Van Hovesingularities of the π -bands. Because we are consideringa full π -band tight-binding model for our calculations,this peak is also present in our results for the opticalconductivity, as it is evident in Figs. 1(b) and (d) at ω/t ≈
2, in qualitative agreement with recent experi-mental results. Notice that the height of the peak issensitive to the presence of disorder, getting more andmore smeared as the strength of disorder is increased.On the other hand, for this kind of disorder, for whichthere is no big change in the DOS around the Dirac point,one expects that the low energy spectrum of the opticalconductivity should be robust for small disorder, i.e., theoptical conductivity should follow the same spectrum asthe clean sample without any disorder. These expec-tations are exactly what we observed in the numericalresults of σ ( ω ) shown in the insets of Fig. 1 (b) and(d). This is indeed the part of the spectrum that canbe accounted for within the continuum (Dirac cone) ap-proximation. We can conclude that the non-correlatedrandom disorder in the on-site potentials or hopping in-tegrals have almost no effect on the electronic properties(density of states and AC conductivity) in the low energypart of the spectrum unless the disorder is too large. Onthe other hand, the high energy inter-band processes be-tween states belonging to the Van Hove singularities ofthe valence and conduction bands are quite sensitive tothe strength of these two kinds of disorder. B. Random Distributed Vacancies or HydrogenImpurities
Next, we consider the influence of two other types ofdefects on graphene, namely, vacancies and hydrogen im-purities. Introducing vacancies in a graphene sheet will create a zero energy mode (midgap state), effect thathas been anticipated in many theoretical works, and which has been recently observed experimentallyby means of scanning tunneling spectroscopy (STM)measurements. It is shown that the number of midgapstates increases with the concentration of the vacancies ,and the inclusion of vacancies brings an increase of spec-tral weight to the surrounding of the Dirac point ( E = 0)and smears the van Hove singularities. This is infact the behavior found in Fig. 1 (e) for the DOS ofgraphene with different concentrations of vacancies n x ,where the numerical results with n x = 1%, 5%, 10% arerepresented and compared to the density of states of cleangraphene.The presence of hydrogen impurities, which are in-troduced by the formation of a chemical bond betweena carbon atom from the graphene sheet and a car-bon/oxygen/hydrogen atom from an adsorbed organicmolecule (CH , C H , CH OH, as well as H and OHgroups) have quite similar effect to the electronic struc-ture and transport properties of graphene.
The ad-sorbates are described by the Hamiltonian H imp inEq. (1). The band parameters V ≈ t and (cid:15) d ≈ − t/ ab initio density functional theory(DFT) calculations. Following Refs. 39 and 44, we callthese impurities as adsorbates hydrogen atoms but ac-tually, the parameters for organic groups are almost thesame. As we can see from Fig. 1 (g), small concentra-tions of hydrogen impurities have similar effects as thesame concentration of vacancies to the density of statesof graphene. Hydrogen adatoms also lead to zero modesand the quasilocalization of the low-energy eigenstates,as well as to a smearing of the Van Hove singularities.The shift of the central peak of the density of states withrespect to the Dirac point in the case of hydrogen impu-rities is due to the nonzero (negative) on-site potentials (cid:15) d .The similarity in the density of states leads to simi-lar optical spectra for graphene with random vacanciesor hydrogen adatoms, as it can be seen in Fig. 1 (f)and (h). In the high and intermediate energy part ofthe spectrum it is noticeable, apart from the smearing ofthe ω ≈ t peak due to the renormalization of the VanHove singularities, the appearance of a new peak at anenergy ω ≈ t . This peak is associated to optical tran-sitions between the newly formed midgap states (withenergy E ≈
0) and the states of the Van Hove singular-ities (with energy E ≈ t ). Notice that, contrary to the ω ≈ t peak, the height of this ω ≈ t peak grows with thestrength of disorder, due to the enhancement of the DOSat the Dirac point. Therefore, we expect that this peakshould be observed in optical spectroscopy measurementsof graphene samples with sufficient amount of resonantscatterers.In the low energy part of the spectra, the new struc-ture of the DOS around the Dirac point leads to a mod-ulation of the infrared conductivity, as it can be seenin the insets of Figs. 1 (f) and (h). The lower peaks, -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 DO S ( /t ) E/t
Potential Gaussian-Cluster
Clean v =t,d=5a P v =0.1% v =3t,d=0.65a P v =1% (a) E/tE/t (b) xx / /t Potential Gaussian-Cluster
Clean v =t,d=5a P v =0.1% v =3t,d=0.65a P v =1% xx / /t -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 (c) DO S ( /t ) E/t
Hopping Gaussian-Cluster
Clean t =0.5t,d=5a P t =0.25% t =t,d=0.65a P t =2.5% E/tE/t (d) xx / /t Hopping Gaussian-Cluster
Clean t =0.5t,d=5a P t =0.25% t =t,d=0.65a P t =2.5% xx / /t -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 Clean III: R=5a II: 0< R <5a I: R=0 (e) DO S ( /t ) E/t
Vacancy Cluster n x ~1% E/tE/t
Clean III: R=5a II: 0< R <5a I: R=0 (f) xx / /t Vacancy Cluster n x ~1% xx / /t -3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 Clean III: R=5a II: 0< R <5a I: R=0 (g) DO S ( /t ) E/t
Hydrogen Cluster n i ~1% E/tE/t
Clean III: R=5a II: 0< R <5a I: R=0 (h) xx / /t Hydrogen Cluster n i ~1% xx / /t FIG. 2. (Color online) Numerical results for the DOS (left panels) and optical conductivity (right panels) of undopped graphenewith different kinds of correlated disorders: (a,b) Gaussian potentials, (c,d) Gaussian hoppings, (e,f) vacancy clusters, and (g,h)hydrogen clusters. The distribution of the clusters of impurities used for the results (e)-(h) are sketched in Fig. 3. which in Figs. 1 (f) and (h) corresponds to a conductiv-ity σ ≈ . σ for different concentration of impurities,might have their origin from excitations involving statessurrounding the zero modes (central high peak in thedensity of states). At slightly higher energies there isa new set of peaks that can be associated to processesinvolving states at the boundaries of the midgap states.The optical conductivities in the region between thesetwo peaks are in general smaller compared to those inclean graphene, which can be due to the fact that themidgap states are quasilocalized states. IV. CORRELATED DISORDERSA. Gaussian Potentials and Gaussian Hoppings
As discussed in the previous section, the change of on-site potential can be regarded as a local chemical poten-tial shift for the Dirac fermions. If the random potentialsare too large, characteristics of the graphene band struc-ture such as the Dirac points or the Van Hove singular-ities can disappear completely, and the whole spectrumbecomes relatively flat over the whole energy range. Therefore in order to introduce large values of randompotentials but keep a relatively similar spectrum, in thissection we use small concentrations of correlated Gaus-sian potentials, defined as v i = N vimp (cid:88) k =1 U k exp (cid:32) − | r i − r k | d (cid:33) , (6)where N vimp is the number of the Gaussian centers, whichare chosen to be randomly distributed over the car-bon atoms ( r k ), U k is uniformly random in the range[ − ∆ v , ∆ v ] and d is interpreted as the effective potentialradius. The typical values of d used in our model are d = 0 . a and 5 a for short- and long-range Gaussianpotential, respectively. Here a ≈ . N vimp is characterized by the ratio P v = N vimp /N , where N is the total number of carbon atoms of the sample.As one can see from Fig. 2(a), in the presence of locallystrong disorders (∆ v = 3 t and t for short and long rangeGaussian potentials, respectively) the whole spectrum ofDOS is quite similar to the case of clean graphene, butwith the emergence of states in the vicinal area aroundthe Dirac point, and also a smearing of the Van Hovesingularities. This kind of disorder leads to regions ofthe graphene membrane where the Dirac point is locallyshifted to the electron ( U k <
0) or to the hole ( U k > ω ≈ . t . The optical conductivityin the region ω < . t is larger than in clean graphenebut becomes smaller for ω > . t . The increase of theconductivity might have its origin in the possible exci-tations between electron and hole puddles. Indeed, therenormalization of the spectrum obtained by consider-ing long-range Gaussian potentials leads to a larger opti-cal contribution than for short-range Gaussian potentials,which yield infra-red spectra much more close to that ofa clean graphene membrane.The local strong disorder in the hopping between car-bon atoms is introduced in a similar way as the corre-lated potentials, i.e., with a distribution of the nearest-neighbor hopping parameter given by t ij = t + N timp (cid:88) k =1 T k exp (cid:32) − | r i + r j − r k | d t (cid:33) , (7)where N timp is the number of the Gaussian centers ( r k ), T k is uniformly random in the range [ − ∆ t , ∆ t ] and d t is interpreted as the effective screening length. Simi-larly, the typical values of d t are the same as for theGaussian potential, i.e., d t = 0 . a and 5 a for short-and long-range Gaussian random hopping, respectively,and the values of N timp are characterized by the ratio P t = N timp /N .Numerical results for the DOS and optical conductiv-ity of graphene with short- (∆ t = 3 t, d t = 0 . a ) andlong-range (∆ t = 1 t, d t = 5 a ) Gaussian hoppings areshown in Fig. 2(c-d). This kind of disorder accountsfor the effect of substitutional impurities like B or N in-stead of C, or local distortions of the membrane. Con-cerning the physics around the neutrality point, in thiscase the Dirac point remains unchanged although thereis a local renormalization of the slope of the band. As aconsequence, the Fermi velocity around Dirac is locallyincreased (when t k >
0) or decreased (when t k < E ≈ B. Vacancy Clusters and Hydrogen Clusters
Correlated resonant impurities are introduced by theformation of groups of vacancies or adsorbed hydrogenatoms (see Fig. 3). The center of the formed vacancy
FIG. 3. (Color online) Sketch of a gaphene sheet with vacan-cies (left pannels) or hydrogen adatoms (right pannels). Thevacancies are presented as missing carbon atoms, whereas thehydrogen adatoms are highlighted in red. From top to bot-tom: Resonant impurites are distributed as the formation I( R = 0), II (0 ≤ R ≤ a ) and III ( R = 3 a ) as described in thetext. For illustrative purposes, the size of the sample shownin this sketch is 60 ×
40, and the concentration of impuritiesis approximately equal to 2%. or hydrogen cluster ( r c ) is randomly distributed over thehoneycomb lattice sites, with equal probability on bothsublattices A and B. Each site ( i ) whose distance to one ofthe centers ( R ≡ | r − r c | ) is smaller than a certain value( R c ), is assumed to be part of the cluster, i.e., being avacancy or adsorbing a hydrogen atom. We further in-troduce another freedom of the resonant clusters namelythat their radius can change within the sample, allowingfor a graphene layer with cluster of impurities of differentsize. This means that the value of R c for each resonantcluster can either be different and randomly distributedto a maximum value, or can be kept fixed for all the clus- ters in the sample. We want to emphasize that as thecenter of the cluster is located on a particular sublatticeA or B, the formation of the cluster does not preservethe sublattice symmetry and therefore can lead to theappearance of midgap states.Firts, in Fig. 2(e) and (g), we compare the density ofstates with the same total number of resonant impurities(vacancies or hydrogen adatoms) but with different kindsof formations. We consider three different situations, i.e.,randomly distributed uncorrelated single impurities (for-mation I), randomly distributed correlated clusters withvaried radius of clusters (formation II) or with fixed ra-dius of clusters (formation III). The different structuresare sketched in Fig. 3. Notice that the formation I isa limiting case of the formation III with all the radiusof clusters being zero. As we can see from the resultsof the simulations, the number of midgap states is largerin the cases of uncorrelated single resonant impurities,and smaller for the case of fixed radius of resonant clus-ters. This is expected since the midgap states are stateswhich are quasilocalized around the vacancies or carbonatoms which adsorb hydrogen atoms. Therefore,for the same concentration of impurities, the number ofmidgap states will grow with the isolation of the impu-rities in small clusters. Something similar happens forthe case of hydrogen clusters. This can understood bylooking at Fig. 4, where we present contour plots of theamplitudes of quasieigenstates at the Dirac point or out-side the midgap region. The quasieigenstate | Ψ ( ε ) (cid:105) isa superposition of the degenerated eigenstates with thesame eigenenergy ε , obtained by the Fourier transforma-tion of the wave functions at different times | Ψ ( ε ) (cid:105) = 12 π (cid:90) ∞−∞ dte iεt | ϕ ( t ) (cid:105) , (8)where | ϕ ( t ) (cid:105) = e − iHt | ϕ (cid:105) is the time evolution of the ini-tial state | ϕ (cid:105) defined in Eq. (4). Although the quasieigen-state is not exactly the energy eigenstate unless the cor-responding eigenstate is not degenerated at energy ε ,we can still use the distribution of the amplitude inthe real space to verify the quasilocalization of the zeromodes in the presence of random impurities, or ob-tain the DC conductivity at certain energies or carrierdensities. As we can see from Fig. 4, the con-tour plots of the quasieigenstates of graphene with va-cancy and hydrogen clusters are quite similar, i.e., theamplitudes on the carbon atoms which adsorb an hy-drogen atom are almost zero, just like if they are va-cancies. Furthermore, at the Dirac point (left panels ofFig. 4, corresponding to E = 0) the quasieigenstatesare semi-localized around the edge of the clusters (seethe red color in the regions around the cluster). On theother hand, for energies above the impurity band, thestates are not localized around the resonant cluster, andthe amplitudes of the quasieigenstates are more or lessuniformly distributed over the sample except within theclusters, where the amplitudes are zero. Therefore, aswe have discussed above, for a given concentration of im- FIG. 4. (Color online) Contour plot of the amplitudes of quasieigenstates at energy E = 0 or E = 0 . t . The radius of theresonant clusters is fixed at R c = 5 a . purities, the number of carbon atoms which are locatedaround an impurity will be larger in the formation I thanin the formation III. Then, the number of zero modesis also larger in I than in III, leading to spectra for theDOS and optical conductivity similar to the ones of cleangraphene for samples in which disorder is concentratedin a small number of big clusters (formation III) thanspread into a large number of small clusters (formationI), as it can be seen in Figs. 2(e)-(h). Finally, notice thatthe possibility for new excitations between the impurityand the carrier bands, leads to a modulation of the op-tical conductivity (as compared to the clean membrane)whose peak structure depends on the renormalized DOSand band dispersion of each case. V. OPTICAL CONDUCTIVITY OF DOPEDGRAPHENE
So far we have discussed the effects of disorder on theoptical response of undoped graphene. In this section,we study the optical conductivity of graphene for finitevalues of the chemical potential, taking into account theeffect of disorder. At zero temperature, a clean sheet ofgated (doped) graphene has a zero optical conductivity in the region ω < µ , and an universal conductivity of σ ( ω ) = σ , due to optically active inter-band excitationsthrough the Dirac point, for energies above the threshold ω > µ . In the presence of the disorder, thebroadening of the bands as well as the appearance of pos-sible midgap states leads to a more complicated selectionrule for the optical transitions, making possible to haveexcitations in the forbidden region < ω < µ , as ob-served experimentally. In this section, we are interestedin studying the effect on the optical spectrum of dopedgraphene of the different kinds of disorder, considered inthe previous section.In Fig. 5 we compare the numerical results forthe optical conductivity of doped graphene, consider-ing four different types of non-correlated disorder (ran-dom potentials, random hoppings, vacancies and hydro-gen adatoms) as well as clean graphene. First, one noticethat the effect of doping is not relevant in the high energypart of the spectrum ( ω (cid:29) µ ), and σ ( ω ) follows the samebehavior discussed in Secs. III and IV, with a peak corre-sponding to particle-hole inter-band transitions betweenstates of the Van Hove singularities at ω ≈ t . However,the spectrum changes dramatically in the infra-red re-gion, as shown in the insets of Fig. 5. Therefore, fromnow on we will focus our interest on the effect of disor- Clean v r =0.2t t r =0.1t n x =1% n i =1% xx / /t =0.1t xx / /t =0.2t Clean v r =0.2t t r =0.1t n x =1% n i =1% xx / /t xx / /t FIG. 5. (Color online) Simulation results of the optical con-ductivity of doped graphene with diffenrent kinds of non-correlated disorders. The chemical potential is µ = 0 . t in(a) and 0 . t in (b). der on this low energy part of the spectrum. First, onenotices that for all kinds of disorder, there is a peak in σ ( ω ) close to ω = 0, whereas at slightly higher energies, σ ( ω ) drops to almost zero for the case of non-resonantscatterers (red and green curves), while there is still anon-zero background contribution when resonant scat-terers are considered (light and dark blue curves). Thiscan be understood as follows: for all the cases, disor-der leads to a broadening of the bands, which allows forintra-band transitions between surrounding states of theFermi level. However, we have seen that resonant impu-rities create an impurity band at the Dirac point, withthe corresponding peak in the DOS at E = 0, whereasnon-resonant impurities are not so effective in creatingmidgap states. Therefore, the background contributionthat we find in Fig. 5(a)-(b) between 0 < ω < µ forsamples with resonant scatterers are due to transitionsbetween the newly formed impurity band and the con-duction band. Taking into account that resonant impu-rities are believed to be the main source of scattering ingraphene, our results suggest that this kind of im-purities could be behind the background contribution tothe optical conductivity observed experimentally. Finally, notice that the peak observed in σ ( ω ) for thecase of resonant impurities at the energy ω ≈ µ is associ- ated to transitions between the above discussed impurityband and states at the Fermi level.To gain more insight about the effect of disorder in theoptical conductivity of doped graphene, in Fig. 6 we show σ ( ω ) for different values of µ at fixed concentration of im-purities (upper panels), and σ ( ω ) for different concentra-tions of impurities and fixed µ . In the first case, the mainfeature is that the conductivity increases as the dopingdecreases, in a qualitative agreement with the experimen-tal results. When the chemical potential is fixed and theconcentration of impurities changes (bottom panels), oneobserve that the conductivity in the region 0 < ω < µ grows with n x ( i ) from σ ( ω ) = 0 for a clean sample to σ ( ω ) ≈ . σ for the larger concentration of impuritiesconsider ( n x ( i ) = 0 . .
25% of resonant impuritieswould lead to a background contribution similar to theone reported by Li et al. for graphene on SiO , whereasonly a ∼ .
1% of resonant impurities would be neces-sary to quantitatively reproduce the results of Chen et al. for graphene doped with a high-capacitance ion-gel gatedielectric. Finally, we can see that similar results areobtained when a sample with correlated on-site potentialdisorder distributed in the form of Gaussian clusters, asshown in Fig. 7. Therefore, we conclude that there areseveral kinds of disorder (resonant scatterers and corre-lated impurities) that can induce a finite conductivity inthe infra-red region of the spectrum, as observed exper-imentally. It is the whole set of data on DC and ACtransport from which one may infer the dominant typeof defects in real graphene.
VI. CONCLUSION AND DISCUSSION
We have presented a detailed theoretical study of theoptical conductivity of graphene with different kinds ofdisorder, as resonant impurities, random distributionof on-site potential or random renormalization of thenearest-neighbor hopping parameter (which can accountfor the effect of substitutional defects). Furthermore, wehave consider the possibility for the impurities to be cor-related or non-correlated.For all types of disorder considered, the high energypeak at ω ≈ t , due to inter-band excitations betweenstates of the Van Hove singularities of the valence andthe conduction bands, are always sensitive to disorder,getting smeared out proportionally to the strength of dis-order. On the other hand, the low energy part of the op-tical spectrum is strongly dependent on the type of disor-der, as well as its strength and concentration. In general,for undoped graphene and in the presence of small dis-order of the on-site potentials or in the nearest-neighborhopping between the carbon atoms, the characteristics ofthe single-particle Dirac cone approximation are clearlypresent in the spectrum, and σ ( ω ) ≈ σ at energies forwhich the continuum approximation applies. This is alsotrue when we consider Gaussian hopping parameters. On0 n x =0.25%Random Vacancies =0.05t =0.1t =0.15t =0.2t =0.25t =0.3t =0.35t =0.4t xx / /t xx / /t xx / /t =0.05t =0.1t =0.15t =0.2t =0.25t =0.3t =0.35t =0.4t RandomHydrogen adatomsn i =0.25% xx / /t Random Vacancies n x =0 n x =0.125% n x =0.25% n x =0.5% xx / /t =0.2t xx / /t =0.2t RandomHydrogen adatoms n i =0 n i =0.125% n i =0.25% n i =0.5% xx / /t xx / /t FIG. 6. (Color online) Optical conductivity of doped graphene with resonant scatterers. Upper panels: Fixed concentrationof impurities, σ ( ω ) for different values of µ . Lower panels: Fixed chemical potential µ , σ ( ω ) for different concentration ofimpurities. the other hand, if there are long-range Gaussian poten-tials, the local shifts of the Dirac points leads to electron-hole puddles and to the emergence of states in the vic-inal region of the Dirac points. As a consequence, weobserve an enhancement of the optical conductivity inthe infra-red part of the spectrum. Interestingly, in thepresence of resonant impurities (vacancies or hydrogenadatoms) there appear midgap states which are quasilo-calized around the impurities, the number of which isproportional to the number of carbon atoms which arelocated around the impurities. Completely random dis-tributed (non-correlated) resonant impurities lead to thestrongest enhancement of zero modes (seen as a promi-nent peak in the DOS at zero energy) and also the largesteffect on the optical spectrum. In fact, for a large enoughamount of resonant impurities, we obtain a new peak inthe optical conductivity at an energy ω ≈ t , which is as-sociated to optical transitions between the midgap statesand states of the Van Hove singularities. When, for agiven concentration of impurities, they merge togetherforming clusters, instead of staying uncorrelated, the in-fluence of disorder on the electronic properties becomessmaller, especially if these clusters form large islands. Finally, we have considered the effect of doping on thespectrum. Whereas for clean graphene, only inter-bandprocesses with an energy larger than ω = 2 µ are opticallyactive, the presence of disorder leads to a low energy peakin σ ( ω ) (associated to transitions near the Fermi level)plus a possible spectral weight in the region 0 < ω < µ for disorders that can create an impurity band at zeroenergy. Most importantly, we have found that a smallamount of resonant impurities ∼ . − . σ ( ω ) between 0 < ω < µ in qualitative and quantitative agreement with recentspectroscopy measurements. VII. ACKNOWLEDGEMENT
The authors thank useful discussions with E. Cap-pelluti and F. Guinea. This research is supportedby the Stichting Fundamenteel Onderzoek der Materie(FOM), the Netherlands National Computing Facilitiesfoundation (NCF), the EU-India FP-7 collaboration un-der MONAMI and the grant CONSOLIDER CSD2007-00010.1 xx / /t =0.05t =0.1t =0.15t =0.2t =0.25t =0.3t =0.35t =0.4t Potential Gaussian-Cluster v =t,d=5aP v =0.1% xx / /t Potential Gaussian-Cluster v =3t,d=0.65aP v =1% =0.05t =0.1t =0.15t =0.2t =0.25t =0.3t =0.35t =0.4t xx / /t xx / /t Potential Gaussian-Cluster v =t,d=5a=0.2t P v =0 P v =0.05% P v =0.1% P v =0.2% xx / /t xx / /t xx / /t P v =0 P v =0.05% P v =0.1% P v =0.2% Potential Gaussian-Cluster v =3t,d=0.65a=0.2t xx / /t FIG. 7. (Color online) Same as Fig. 6 but for doped graphene with on-site potential disorder distributed in gaussian clusters. A. H. Castro-Neto, F. Guinea, N. M. R. Peres,K. Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109(2009). N. M. R. Peres, Rev. Mod. 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