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Modeling electronic structure and transport properties of graphene with resonant

scattering centers

Shengjun Yuan,1, ∗ Hans De Raedt,2, † and Mikhail I. Katsnelson1, ‡

1Institute for Molecules and Materials, Radboud University of Nijmegen, NL-6525ED Nijmegen, The Netherlands2Department of Applied Physics, Zernike Institute for Advanced Materials,

University of Groningen, Nijenborgh 4, NL-9747AG Groningen, The Netherlands

(Dated: October 17, 2018)

We present a detailed numerical study of the electronic properties of single-layer graphene withresonant (“hydrogen”) impurities and vacancies within a framework of noninteracting tight-bindingmodel on a honeycomb lattice. The algorithms are based on the numerical solution of the time-dependent Schrödinger equation and applied to calculate the density of states, quasieigenstates, ACand DC conductivities of large samples containing millions of atoms. Our results give a consistentpicture of evolution of electronic structure and transport properties of functionalized graphene in abroad range of concentration of impurities (from graphene to graphane), and show that the formationof impurity band is the main factor determining electrical and optical properties at intermediateimpurity concentrations, together with a gap opening when approaching the graphane limit.

PACS numbers: 72.80.Vp, 73.22.Pr, 78.67.Wj

I. INTRODUCTION

The experimental realization of a single layer of carbonatoms arranged in a honeycomb lattice (graphene) hasprompted huge activity in both experimental and theo-retical physics communities (for reviews, see Refs. 1–10).Graphene in real experiments always has different kindsof disorder or impurities, such as ripples, adatoms, ad-molecules, etc. One of the most important problems ingraphene physics, especially, keeping in mind potentialapplications of graphene in electronics, is understandingthe effect of these imperfections on the electronic struc-ture and transport properties.

Being massless Dirac fermions with the wavelengthmuch larger than the interatomic distance, charge car-riers in graphene scatter rather weakly by generic short-range scattering centers, similar to weak light scatteringfrom obstacles with sizes much smaller that the wave-length. The scattering theory for Dirac electrons in twodimensions is discussed in Refs. 11,13–15. Long-rangescattering centers are of special importance for trans-port properties, such as charge impurities6,16–18, ripplescreated long-range elastic deformations7,19, and resonantscattering centers12,13,19–22. In the latter case, the diver-gence of the scattering length provides a long-range scat-tering and a very slow, logarithmic, decay of the scatter-ing phase near the Dirac (neutrality) point. Earlier theresonant scattering of Dirac fermions was studied in acontext of d-wave high-temperature superconductivity23.For the case of graphene, vacancies are prototype exam-ples of the resonant scatterers21,24. Numerous adatomsand admolecules (including the important case of hydro-gen atoms covalently bonded with carbon atoms) provideother examples25–27. Recently, some experimental28 andtheoretical29 evidence appeared that, probably, the res-onant scattering due to carbon-carbon bonds betweenorganic admolecules and graphene is the main restrict-

ing factor for electron mobility in graphene on a sub-strate. Resonant scattering also plays an important rolein interatomic interactions and ordering of adatoms ongraphene30. This all makes the theoretical study ofgraphene with resonant scattering centers an importantproblem.

In the present paper, we study this issue by directnumerical simulations of electrons on a honeycomb lat-tice in the framework of the tight-binding model. Nu-merical calculations based on exact diagonalization canonly treat samples with relative small number of sites,for example, to study the quasilocalization of eigenstateclose to the neutrality point around the vacancy12,31

and the splitting of zero-energy Landau levels in thepresence of random nearest neighbor hoping32. Forlarge graphene sheet with millions of atoms, the nu-merical calculation of an important property, the den-sity of states (DOS), is mainly performed by the recur-sion method31,33,34 and time-evolution method29,35. Thetime-evolution method is based on numerical solution oftime-dependent Schrödinger equation with additional av-eraging over random superposition of basis states. In thispaper, we extend the method of Ref. 36 to compute theeigenvalue distribution of very large matrices to the cal-culation of transport coefficients. It allows us to carryout calculations for rather large systems, up to hundredsof millions of sites, with a computational effort that in-creases only linearly with the system size. Furthermore,another extension of the time-evolution method yieldsthe quasieigenstate, a random superposition of degener-ate energy eigenstates, as well as the AC and DC29 con-ductivities.

The numerical calculation of the conductivity is basedon the Kubo formula of noninteracting electrons. Thedetails of these algorithm will be given in this paper.Our numerical results are consistent with the resultson hydrogenated graphene37 and graphene with vacan-cies38, which are based on the numerical calculation of

http://arxiv.org/abs/1007.3930v2

2

the Kubo-Greenwood formula39. Another widely usedmethod of the numerical study of electronic transport ingraphene is the recursive Green’s function method40–49,which is generally applied to relatively small samples fol-lowed by averaging of many different configurations. Therecursive Green’s function method is a powerful tool tocalculate the electronic transport in small system such asgraphene ribbons, while the method that we employ inthis paper is more suitable for large systems having mil-lions of atoms and therefore does not involve averagingover different realizations.The paper is organized as follows. Section II gives a de-

scription of the tight-binding Hamiltonian of single layergraphene including different types of disorders or impu-rities, in the absence and presence of a perpendicularmagnetic field. In section III, we first discuss briefly thenumerical method used to calculate the DOS, and showthe accuracy of this algorithm by comparing the analyti-cal and numerical results for clean graphene. Then, basedon the calculation the DOS, we discuss the effects of va-cancies or resonant impurities to the electronic structureof graphene, including the broadening of the Landau lev-els and the split of zero Landau levels. In section IV, weintroduce the concept of a quasieigenstates, and use it toshow the quasilocalization of the states around the vacan-cies or resonant impurities. Sections V and VI give dis-cussions of the AC and DC conductivities, respectively.The details of numerical methods and various examplesare discussed in detail in each section. Finally a briefgeneral discussion is given in section VII.

II. TIGHT-BINDING MODEL

The tight-binding Hamiltonian of a single-layergraphene is given by

H = H0 +H1 +Hv +Himp, (1)

where H0 derives from the nearest neighbor interactionsof the carbon atoms:

H0 = −∑

tijc+i cj , (2)

H1 represents the next-nearest neighbor interactions ofthe carbon atoms:

H1 = −∑

t′ijc+i cj , (3)

Hv denotes the on-site potential of the carbon atoms:

Hv =∑

i

vic+i ci, (4)

and Himp describes the resonant impurities:

Himp = εd∑

i

d+i di + V∑

i

(d+i ci +H.c.

). (5)

x

y

A

B

n

m m+1m -1

n+1

a

n-1B

Ba a

FIG. 1: (Color online) The lattice structure of a graphenesheet. Each carbon is labeled by an coordinate (m,n), wherem is along the zigzag edge and n is along the armchair edge.Each carbon (red) has three nearest neighbors (yellow) andsix next-nearest neighbors (blue).

For discussions of the last term see, e.g. Refs. 27,50.The spin degree of freedom contributes only through a

degeneracy factor and is omitted for simplicity in Eq. (1).Vacancies are introduced by simply removing the corre-sponding carbon atoms from the sample.If a magnetic field is applied to the graphene layer, the

hopping integrals are replaced by a Peierls substitution51,that is, the hopping parameter becomes

tmn → tmneie∫

n

mA·dl = tmne

i(2π/Φ0)∫

n

mA·dl, (6)

where∫ nmA ·dl is the line integral of the vector potential

from site m to site n, and the flux quantum Φ0 = ch/e.Consider a single graphene layer with a perpendicular

magnetic fieldB = (0, 0, B). Let the zigzag edge be alongthe x axis, and use the Landau gauge, that is, the vectorpotential A = (−By, 0, 0) Then H0 changes into

H0 =∑

m,n

t(m,n),(m,n−1)a+m,nbm,n−1

+t(m,n),(m−1,n)eiπn(Φ/Φ0)a+m,nbm−1,n

+t(m,n),(m+1,n)e−iπn(Φ/Φ0)a+m,nbm+1,n

+H.c. (7)

where

Φ ≡ 3√3

2Ba2, (8)

a is the nearest-neighbor interatomic distance.

III. DENSITY OF STATES

The density of states describes the number of states ateach energy level. An algorithm based on the evolution

3

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.25

0.50

0.75

1.00

1.25

Analytical 512X512 4096X4096

DOS

E/t

FIG. 2: (Color online) Comparison of the analytical DOS(in units of 1/t, black solid) with the numerical results of asample contains 512 × 512 (red dash) or 4096 × 4096 (greendot) carbon atoms.

of time-dependent Schrödinger equation (TDSE) to findthe eigenvalue distribution of very large matrices was de-scribed in Ref. 36. The main idea is to use a randomsuperposition of all basis states as an initial state |ϕ (0)〉:

|ϕ (0)〉 =∑

i

ai |i〉 , (9)

where {|i〉} are the basis states and {ai} are random com-plex numbers, solve the TDSE at equal time intervals,calculate the correlation function

〈ϕ (0)| e−iHt |ϕ (0)〉 , (10)

for each time step (we use units with ~ = 1): and thenapply the Fourier transform to these correlation functionsto get the local density of states (LDOS) on the initialstate:

d (ε) =1

2π

∫ ∞

−∞

eiεt 〈ϕ (0)| e−iHt |ϕ (0)〉 dt. (11)

In practice the Fourier transform in Eq. (11) is per-formed by fast Fourier transformation (FFT). We use aGaussian window to alleviate the effects of the finite timeused in the numerical time integration of the TDSE. Thenumber of time integration steps determines the energyresolution: Distinct eigenvalues that differ more than thisresolution appear as separate peaks in the spectrum. Ifthe eigenvalue is isolated from the rest of the spectrum,the width of the peak is determined by the number oftime integration steps.By averaging over different samples (random initial

states) we obtain the density of states:

D (ε) = limS→∞

1

S

S∑

p=1

dp (ε) . (12)

For a large enough system, for example, graphene crys-tallite consisting of 4096 × 4096 ≈ 1.6 × 107 atoms, oneinitial random superposition state (RSS) is already suf-ficient to contain all the eigenstates, thus, its LDOS isapproximately equal to the DOS of an infinite system,i.e.,

D (ε) ≈ d (ε) . (13)

For the proof of this results and a detailed analysis ofthis method we refer to Ref. 36. To validate the method,we will compare the analytical and numerical results forclean graphene.The numerical solution of the TDSE is carried out

by using the Chebyshev polynomial algorithm, which isbased on the polynomial representation of the operatorU (t) = e−itH (see Appendix A). The Chebyshev poly-nomial algorithm is very efficient for the simulation ofquantum systems and conserves the energy of the wholesystem to machine precision. In order to reduce the ef-fects of the graphene edges on the electronic properties(see, e.g., Ref. 35), we use periodic boundary conditionsfor all the numerical results presented in this paper.

A. DOS of Clean Graphene

The analytical expression of the density of states of aclean graphene (ignoring the next-nearest neighbor in-teraction t′ and the on-site energy) was given in Ref. 52as

ρ (E) = {2Et2π2

1√F (E/t)

K

(4E/t

F (E/t)

), 0 < E < t,

2Et2π2

1√4E/t

K

(F (E/t)4E/t

), t < E < 3t,

(14)

where F (x) is given by

F (x) = (1 + x)2 −

(x2 − 1

)2

4, (15)

and K (m) is the elliptic integrals of first kind:

K (m) =

∫ 1

0

dx[(1− x2

) (1−mx2

)]−1/2. (16)

In Fig. 2, we compare the analytical expressionEq. (14) with the numerical results of the density of statesfor a clean graphene. One can clearly see that these nu-merical results fit very well the analytical expression, andthe difference between the numerical and analytical re-sults becomes smaller when using larger sample size (seethe difference of a sample with 512× 512 or 4096× 4096in Fig. 2). In fact, the local density of states of a samplecontaining 4096× 4096 is approximately the same as thedensity of states of infinite clean graphene, which indi-cates the high accuracy of the algorithm.

4

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0.75

1.00

1.25

-3 -2 -1 0 1 2 30.00

0.25

0.50

0.75

1.00

1.25

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.30.0

0.1

0.2

0.3

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.30.0

0.1

0.2

0.3

ni=0% ni=0.1% ni=0.2% ni=0.3% ni=0.4% ni=0.5% ni=1% ni=5%

DOS

(b)

(a) (c)

(d)

DOS

E(t)

nx=0% nx=0.1% nx=0.2% nx=0.3% nx=0.4% nx=0.5% nx=1% nx=5%

E(t)

FIG. 3: (Color online) Density of states (in units of 1/t) as a function of energy E (in units of t) for different resonant impurity(εd = −t/16, V = 2t) or vacancy concentrations: ni(nx) = 0.1%, 0.2%, 0.3%, 0.4%, 0.5%, 1%, 5%. Sample size is 4096× 4096.

B. DOS of Graphene with Impurities

Next, we consider the influence of two types of defectson the DOS of graphene, namely, vacancies and reso-nant impurities. A vacancy can be regarded as an atom(lattice point) with and on-site energy v → ∞ or withits hopping parameters to other sites being zero. In thenumerical simulation, the simplest way to implement avacancy it to remove the atom at the vacancy site. Intro-ducing vacancies in a graphene sheet will create a zeroenergy modes (midgap state)12,31,33. The exact analyti-cal wave function associated with the zero mode inducedby a single vacancy in a graphene sheet was obtained inRef.33, showing a quasilocalized character with the am-plitude of the wave function decaying as inverse distanceto the vacancy. Graphene with a finite concentration ofvacancies was studied numerically in Ref. 31. The num-ber of the midgap states increases with the concentrationof the vacancies. The inclusion of vacancies brings an in-crease of spectral weight to the surrounding of the Diracpoint (E = 0) and smears the van Hove singularities12,31.Our numerical results (see Fig. 3) confirm all these find-

ings.

Resonant impurities are introduced by the formationof a chemical bond between a carbon atom from graphenesheet and a carbon/oxygen/hydrogen atom from an ad-sorbed organic molecule (CH3, C2H5, CH2OH, as well asH and OH groups)29. To be specific, we will call adsor-bates hydrogen atoms but actually, the parameters fororganic groups are almost the same29. The adsorbatesare described by the Hamiltonian Himp in Eq. (1). Theband parameters V ≈ 2t and ǫd ≈ −t/16 are obtainedfrom the ab initio density functional theory (DFT) calcu-lations29. As we can see from Fig. 3, small concentrationsof vacancies or hydrogen impurities have similar effectsto the DOS of graphene. Hydrogen adatoms also lead tozero modes and the quasilocalization of the low-energyeigenstates, as well as to smearing of the van Hove sin-gularities. The shift of the central peak of the DOS withrespect to the Dirac point in the case of hydrogen impu-rities is due to the nonzero (negative) on-site potentialsǫd.

Now we consider the electronic structure of graphenewith a higher concentration of defects. Large con-centration of vacancies in graphene leads to well pro-

5

-3 -2 -1 0 1 2 30

2

4

60

10

200

20

40

0

50

100

0

200

400

E/t

nx=10%

nx=20%

nx=30%

nx=50%

nx=90%DOS

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.00

0.25

0.50

0.75

1.00

nx=5% nx=10% nx=20% nx=30%

E/t

FIG. 4: Density of states (in units of 1/t) as a function ofenergy E (in units of t) for the vacancies with large concen-trations: nx = 5%, 10%, 20%, 30%, 50%, 90%. Sample sizeis 4096 × 4096 for nx ≤ 50% and 8192× 8192 for nx = 90%.

nounced symmetric peaks in the DOS: a very high cen-tral peak at the Dirac point, two small peaks at theVan Hove singularities, and tiny peaks at |E| /t =0.618, 0.766, 1.414, 1.618, 1.732, 1.848 (see Fig. 4). Theseresults indicate the emergence of small pieces of iso-lated carbon groups, shown in Fig. 5. The positions

E/t

0

1

2½

3½

( 1 5½)/2

(2 2½)½

0, 1, 3½

P

(1- nx)nx3

(1- nx)2nx

4

(1- nx)3nx

5

(1- nx)4nx

6

(1- nx)4nx

6

(1- nx)5nx

7

(1- nx)5nx

7

FIG. 5: Typical atomic structures of most favourable isolatedcarbon groups in graphene with large concentration of vacan-cies. The energy eigenvalues of each group are listed in thecentral column (in units of t), and P is the probability of aparticular group to be found in a graphene sample.

of the peaks in the DOS match very well with the en-ergy eigenvalues of these small subgroups. For example,non-interacting carbon atoms contribute to the peak atDirac point, and isolated pairs contribute to the peaksat Van Hove singularities. Graphene with very high va-cancy concentration, e.g., nx = 90%, is mainly a sheetof non-interacting carbon atoms, with small amount ofisolated pairs, and tiny amounts of isolated triples. Onlythe peaks corresponding to these groups appear in thecalculated DOS of nx = 90% in Fig. 4.

Graphene with 100% concentration of hydrogen impu-rities is not graphene, but pure graphane53. Graphaneis shown to be an insulator because of the existence ofa band gap (in our model, 2t), see the bottom panel inFig. 6. Graphene with large concentration (ni) of hy-drogen impurities corresponds to graphane with smallconcentrations (1 − ni) of vacancies of hydrogen atoms,which leads, again, to appearance of localized midgapstates (shifted from zero due to nonzero ǫd) on the car-bon atoms which have no hopping integrals to any hy-drogen, see these central peaks in Fig. 6. Despite thefact that our model is oversimplified for dealing with fi-nite concentrations of hydrogen (in general, parametersof impurities should be concentration dependent, directhopping between hydrogens should be taken into account,etc.), this conclusion is in an agreement with first prin-ciple calculations54.

6

0

2

4

6

0.0

0.5

1.0

0.0

0.5

1.0

-4 -3 -2 -1 0 1 2 3 40.0

0.5

1.0

0

2

4

6

8

ni= 90%

ni= 99%

ni= 99.5%

ni=100%

E/t

ni=50%DOS

FIG. 6: Density of states (in units of 1/t) as a function ofenergy E (in units of t) for the resonant impurities (εd =−t/16, V = 2t) with large concentrations: ni = 50%, 90%,99%, 99.5%, 100%. Sample size is 2048 × 2048.

C. DOS of Graphene with Impurities in the

Magnetic Field

A magnetic field perpendicular to a graphene layerleads to discrete Landau energy levels. The energy ofthe Landau levels of clean graphene is given by2,3

EN = sgn(N)√2e~v2FB |N |, (17)

where in the nearest-neighbor tight binding model

vF /t = 3a/2~. (18)

Our numerical calculations reproduces the positions ofthe Landau levels. Introducing impurities or disordersin graphene will broaden the Landau levels. Fig. 7

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.200.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.088 0.089 0.090 0.091 0.092 0.0930.0

0.2

0.4

0.6

0.8

-0.003 -0.002 -0.001 0.000 0.001 0.002 0.0030.0

0.2

0.4

0.6

0.8

0.021 0.022 0.023 0.024 0.0250.01

0.02

0.03

0.04

0.05DOS

=7.03x10-4

DOS

DOS

=7.09x10-4

=7.87x10-4

B=60Tnx=0.01%

8192x8192

DOS

E/t

FIG. 7: (Color online) Density of states (in units of 1/t) asa function of energy E (in units of t) in the presence of auniform perpendicular magnetic field (B = 60T ) with vacancyconcentration nx = 0.01%. The red curves are Gaussian fits ofEq. (19) centered about each Landau levels, with w = 7.09×10−4 for EN = 0 (N = 0), w = 7.03× 10

−4 for EN = 0.0909t(N = 1), and w = 7.87× 10−4 for E = 0.0232t (between zeroand first Landau levels). Sample size is 8192 × 8192.

presents the numerical results for a uniform perpendic-ular magnetic field (B = 60T ) applied to a 8192× 8192graphene sample with a small concentration of vacancies(nx = 0.01%). The spectral distribution near each Lan-dau level fits well to the Gaussian function

ρ (E) = A exp

[− (E − EN )

2

2w2

], (19)

with w ≈ 7×10−4t. Between two Laudau levels, there areextra peaks which also fit to a Gaussian distribution withw ≈ 8×10−4t. These additional localized states were alsofound in other numerical simulations34 of much smaller96 × 60 samples with a stronger magnetic field (B ≈400T ) and larger concentration of vacancies (nx = 0.21%and 0.42%).Increasing the concentration of the vacancies will

smear and suppress the Landau levels except the one at

7

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0.05

0.10

0.15

0.20

0.25

0.30

0.35

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0.45

4096X4096B=20T

B=0 n

x=0%

nx=0.01%

nx=0.05%

nx=0.1%

nx=0.5%

DOS

E/t

FIG. 8: (Color online) Density of states (in units of 1/t) as afunction of energy E (in units of t) in the presence of a uni-form perpendicular magnetic field (B = 20T ) with differentvacancy concentrations: nx = 0%, 0.01%, 0.05%, 0.1%, 0.5%.Sample size is 4096× 4096.

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.200.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

4096X4096B=50T

B=0 ni=0% ni=0.01% ni=0.05% ni=0.1% ni=0.5% ni=1%

DOS

E/t

FIG. 9: (Color online) Density of states (in units of 1/t) as afunction of energy E (in units of t) in the presence of a uniformperpendicular magnetic field (B = 50T ) with different hydro-gen concentrations ni = 0%, 0.01%, 0.05%, 0.1%, 0.5%, 1%.Sample size is 4096× 4096.

zero energy12, see Fig. 8. The zero-energy Landau levelseems to be robust with respect to resonant impuritiessince the latter form their own midgap states.The presence of hydrogen impurities has similar effects

on the spectrum as in the case of vacancies (compare Fig.8 and 9) except that, because of the non-zero on-site en-ergy (ǫd) of hydrogen sites, the zero-energy Landau levelsplits into two for a certain range of hydrogen concen-trations (for example, see ni = 0.05% in Fig. 9). Thepeak at the neutrality point corresponds to the originalzero-energy Landau level whereas the other one origi-nates mainly from hybridization with hydrogen atoms.

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

0.0000

0.0005

0.0010

0.0015

0.0020

/t

nx=0.1%

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

0.000

0.001

0.002

0.003

/t

ni=0.1%

FIG. 10: (Color online) The error (δ and σ) of the approx-imation of |Ψ(ε)〉 of a quasieigenstate in a graphene sam-ple (4096 × 4096) with vacancies or hydrogen ((εd = −t/16,V = 2t) impurities. The concentration of the defeats is 0.1%.

The splitting of zero-energy Landau level by other kindsof disorder is also observed, for example, with randomnearest-neighbor hopping between carbon atoms as re-ported in Ref. 32.For small concentration of hydrogen impurities (ni =

0.01% in Fig. 9), there are also extra peaks between zeroand first Landau levels, similar as in the case for lowconcentration of vacancies. The difference is that thesetwo extra peaks are not symmetric around the neutralitypoint, because of non-zero on-site energy (ǫd).

IV. QUASIEIGENSTATES

For the general Hamiltonian (1) and for samples con-taining millions of carbon atoms, in practice, the eigen-states cannot be obtained directly from matrix diagonal-ization. An approximation of these eigenstates, or a su-perposition of degenerate eigenstates can be obtained byusing the spectrum method55. Let |ϕ (0)〉 = ∑n An |n〉be the initial state of the system, and {|n〉} are the com-plete set of energy eigenstates. The state at time t is

|ϕ (t)〉 = e−iHt |ϕ (0)〉 . (20)

Performing the Fourier transform of |ϕ (t)〉 one obtainsthe expression

∣∣∣Ψ̃ (ε)〉

=1

2π

∫ ∞

−∞

dteiεt |ϕ (t)〉

=1

2π

∑

n

An

∫ ∞

−∞

dtei(ε−En)t |n〉

=∑

n

Anδ (ε− En) |n〉 , (21)

which can be normalized as

|Ψ(ε)〉 = 1√∑n |An|

2δ (ε− En)

∑

n

Anδ (ε− En) |n〉 .

(22)

It is clear that |Ψ(ε)〉 is an eigenstate if it is a single (non-degenerate) state, and some superposition of degenerateeigenstates with the energy ε, otherwise.

8

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-30

-20

-10

0

10

20

30

40

Y(a)

X(a)-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40E/t=-0.0626

X(a)

Y(a)

1E-11

2E-10

3E-09

4E-08

6E-07

1E-05

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-30

-20

-10

0

10

20

30

40

X(a)

Y(a)

1E-11

2E-10

3E-09

4E-08

6E-07

1E-05

E/t=-1.5x10-4

-40 -30 -20 -10 0 10 20 30 40-40

-30

-20

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0

10

20

30

40E/t=-0.09376

X(a)

Y(a)

1E-11

2E-10

3E-09

4E-08

6E-07

1E-05

-40 -30 -20 -10 0 10 20 30 40-40

-30

-20

-10

0

10

20

30

40E/t=-0.0310

X(a)

Y(a)

1E-11

2E-10

3E-09

4E-08

6E-07

1E-05

-40 -30 -20 -10 0 10 20 30 40-40

-30

-20

-10

0

10

20

30

40E/t=-0.25

X(a)

Y(a)

1E-11

8E-11

6E-10

5E-09

4E-08

3E-07

3E-06

1E-05

FIG. 11: (Color online) Position of hydrogen impurities (black dots in the top left panel) and contour plot of the amplitudesof the quasieigenstates in the central part of a graphene sample (4096× 4096) with different energies. The concentration of thehydrogen impurities (εd = −t/16, V = 2t) is 0.1%.

In general, |Ψ(ε)〉 will not be an eigenstate but maybe close to one and therefore we call it quasieigenstate.Although |Ψ(ε)〉 is written in the energy basis, the ac-tual basis used to represent the state |ϕ (t)〉 can be any

orthogonal and complete basis. It is convenient to intro-duce two variables δ (ε) and σ (ε) to measure the differ-ence between a true eigenstate and the quasieigenstate

9

-40 -30 -20 -10 0 10 20 30 40-40

-30

-20

-10

0

10

20

30

40E/t=-5.6x10-6

X(a)

Y(a)

1E-12

1E-11

1E-10

1E-09

2E-08

2E-07

2E-06

1E-05

-40 -30 -20 -10 0 10 20 30 40-40

-30

-20

-10

0

10

20

30

40E/t=-0.0982

X(a)

Y(a)

1E-12

1E-11

1E-10

1E-09

2E-08

2E-07

2E-06

1E-05

-40 -30 -20 -10 0 10 20 30 40-40

-30

-20

-10

0

10

20

30

40E/t=-0.0491

X(a)

Y(a)

1E-12

1E-11

1E-10

1E-09

2E-08

2E-07

2E-06

1E-05

-40 -30 -20 -10 0 10 20 30 40-40

-30

-20

-10

0

10

20

30

40E/t=-0.245

X(a)

Y(a)

1E-12

1E-11

1E-10

1E-09

2E-08

2E-07

2E-06

1E-05

FIG. 12: (Color online) Contour plot of the amplitudes of the quasieigenstates in the central part of a graphene sample(4096 × 4096) with different energies. The concentration of the vacancy impurities (indicated by black dots) is 0.1%.

|Ψ(ε)〉:δ (ε) = 〈Ψ(ε) |H |Ψ(ε)〉 − ε, (23)

σ (ε) =

√〈Ψ(ε) |H2|Ψ(ε)〉 − 〈Ψ(ε) |H |Ψ(ε)〉2.(24)

As δ (ε) is a measure of the energy shift and σ (ε) isthe variance of the approximation, both variables shouldbe zero if |Ψ(ε)〉 is a quasieigenstate with the energyε. From numerical experiments (results not shown),we have found two ways to improve the accuracy ofthe quasieigenstates. One is that the Fourier transformshould be performed on the states from both positive andnegative times, and the other is that the wave function|ϕ (t)〉 should be multiplied by a window function (Han-ning window56) (1 + cos(πt/T ))/2 before performing theFourier transform, T being the final time of the propa-gation. The propagation in both positive and negativetime is necessary to keep the original form of the inte-gral in Eq. (21), and the use of a window improves theapproximation to the integrals.In Fig. 10 we show δ (ε) and σ (ε) of the calculated

quasieigenstates in graphene with vacancy or hydrogenimpurities. The time step used in the propagation ofthe wave function is τ = 1 in the case of vacancies andτ = 0.6 in the case of hydrogen impurity. The totalnumber of time steps is Nt = 2048 in both cases. Onecan see that the errors in the energy of |Ψ(ε)〉 are quitesmall (|δ (ε)| < 5 × 10−4), and the standard deviationσ (ε) is less than 2 × 10−3 and 3 × 10−3 for vacanciesand hydrogen impurities, respectively. The value σ (ε)is smaller in the case of the vacancies due to the largertime step and larger propagation time used. The fluctu-ations of δ (ε) in the region close to the neutrality point(ε = 0) are due to the error introduced by the finitediscrete Fourier transform in Eq. (21), because near theneutrality point, the finite discrete Fourier transformmaymix components from the eigenstates in the opposite sideof the spectrum. In fact, it would be more accurate todirectly use 〈Ψ(ε) |H |Ψ(ε)〉 instead of ε as the energy ofthe quasieigenstate. Notice that the error of σ (ε) withε = 0 is smaller than in the case of nonzero ε, since forε = 0 there is no error due the combination of the factor

10

eiεt(= 1) with the state |ϕ (t)〉. All the errors of δ (ε) andσ (ε) as well as these fluctuations around δ (ε) can be re-duced by increasing the time step τ and/or total numberof time steps Nt.Although quasieigenstates are not exact eigenstates,

they can be used to calculate the electronic propertiesof the sample, such as the DC conductivity (as will beshown later). The contour plot of the amplitudes ofthe quasieigenstates directly reveals the structure of theeigenstates with certain eigenenergy, for example, thequasilocalization of the low-energy states around the va-cancy or hydrogen impurity, see Fig. 11 and Fig. 12. Thequasilocalization of the states around the impurities oc-curs not only for zero energy, but also for quasieigen-states with the energies close to the neutrality point.This quasilocalization leads to an increase of the spec-tral weight in the vicinity of the Dirac point (E = 0), seeFig. 3. The states with larger eigenenergy are extendedand robust to small concentration of impurities, and theirspectral weight is close to that in clean graphene. Onecan see that in the case of hydrogen impurities, thequasieigenstates that are close enough to the impuritystates, i.e., E/t = −0.0626 ≈ εd in Fig. 11, are dis-tributed in the whole region around hydrogen atoms.The carbon atoms coupled to hydrogens look like “va-cancies”, with very small probability amplitudes, whichexplains why hydrogen impurities and vacancies producesimilar effects on the electronic properties of graphene.

V. OPTICAL CONDUCTIVITY

Kubo’s formula for the optical conductivity can be ex-pressed as57

σαβ (ω) = limε→0+

1

(ω + iε)Ω{ − i 〈[Pα, Jβ ]〉

+

∫ ∞

0

ei(ω+iε)tdt 〈[Jα (t) , Jβ ]〉 }, (25)

where P is the polarization operator

P = e∑

i

ric+i ci, (26)

and J is the current operator

J =·

P = e∑

i

·ric

+i ci =

i

~[H,P ] . (27)

For a generic tight binding Hamiltonian, the current op-erator can be written as

J = − ie~

∑

i,j

tij (rj − ri) c+i cj , (28)

and

[Pα,Jβ ] = −ie2

~

∑

i,j

tij

[(ri − rj)α (rj − ri)β

]c+i cj .

(29)

The ensemble average in Eq. (25) is over the Gibbsdistribution, and the electric field is given by E (t) =E0 exp (iω + ε) t (ε is a small parameter introduced inorder that E (t) → 0 for t → −∞). In graphene, P andJ are two-dimensional vectors, and Ω is replaced by thearea of the sample S.In general, the real part of the optical conductivity

contains two parts, the Drude weight D (ω = 0) andthe regular part (ω 6= 0). We omit the calculation ofthe Drude weight, and focus on the regular part. Fornon-interacting electrons, the regular part is58

Reσαβ (ω) = limε→0+

e−β~ω − 1~ωΩ

∫ ∞

0

e−εt sinωt

×2Im 〈f (H)Jα (t) [1− f (H)] Jβ〉 dt,(30)

where β = 1/kBT, µ is the chemical potential, and theFermi-Dirac distribution operator

f (H) =1

eβ(H−µ) + 1. (31)

In the numerical calculations, the average in Eq. (30)is performed over a random phase superposition of allthe basis states in the real space, i.e., the same initialstate |ϕ (0)〉 in calculation of DOS. The Fermi distribu-tion operator f (H) and 1 − f (H) can be obtained bythe standard Chebyshev polynomial decomposition (seeAppendix B).By introducing the three wave functions59

|ϕ1 (t)〉x = e−iHt

~ [1− f (H)]Jx |ϕ〉 , (32)|ϕ1 (t)〉y = e−

iHt

~ [1− f (H)]Jy |ϕ〉 , (33)|ϕ2 (t)〉 = e−

iHt

~ f (H) |ϕ〉 , (34)

we get all elements of the regular part of Reσαβ (ω):

Reσαβ (ω) = limε→0+

e−β~ω − 1~ωΩ

∫ ∞

0

e−εt sinωt

×[2Im 〈ϕ2 (t) |Jα|ϕ1 (t)〉β

]dt. (35)

A. Optical Conductivity of Clean Graphene

In Fig. 13, we compare our numerical results to theanalytical results obtained in Refs. 60–62, where the realpart of the conductivity in the visible region has theform60

Reσxx = σ0

[πt2a2

8Ac~ωρ

(~ω

2

)(18− ~

2ω2

t2

)+

~2ω2

4!24t2

]

(tanh

~ω + 2µ

4kBT+ tanh

~ω − 2µ4kBT

),

(36)

11

0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

T=300K=0

xx/

0

/t

Simulation Analytical I Analytical II

0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

T=300K=0.2eV

xx/

0

/t

Simulation Analytical I Analytical II

0.0 0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.8

1.0

1.2

xx/

0

/t

Simulation Analytical I Analytical II

T=300K=0

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.2

0.4

0.6

0.8

1.0

1.2

T=300K=0.2eV

xx/

0

/t

Simulation Analytical I Analytical II

FIG. 13: (Color online) Comparison of the numerically calculated optical conductivity (µ = 0 or 0.2eV, T = 300K) withEq. (36) (analytical I) and Eq. (37) (analytical II). The size of the system is M = N = 8192.

with the minimum conductivity σ0 = πe2/2h. Around

ω = 0 the real part of the conductivity can be simplifiedas60–62

Reσxx = σ0

(1

2+

1

72

~2ω2

t2

)

(tanh

~ω + 2µ

4kBT+ tanh

~ω − 2µ4kBT

). (37)

As we can see from Fig. 13, the numerical and analyt-ical results match very well in the low frequency region,but not in the high frequence region. This is because theanalytical expressions are partially based on the Dirac-cone approximation, i.e., the graphene energy bands arelinearly dependent on the amplitude of the wave vector.It is exact for the calculations of the low-frequence op-tical conductivity, but not for high-frequence. Our nu-merical method does not use such approximation and hasthe same accuracy in the whole spectrum. Furthermore,our numerical results also show that the conductivity ofReσxx with µ = 0 in the limit of ω = 0 converges to theminimum conductivity σ0 when the temperature T → 0.

B. Optical Conductivity of Graphene with

Random on-Site Potentials

The on-site potential disorder can change the elec-tronic properties of graphene dramatically. For example,if the potentials on sublattices A and B are not symmet-ric, a band gap will appear. If we set vd and −vd asthe on-site potential on sublattice A and B, respectively,then a band gap of size 2vd is observed in the centralpart of DOS and the optical conductivity in the region0 < ω < 2vd becomes zero, see the red dashed lines(vd = t) in Fig. (14). If the potentials on sublattice Aand B are both uniformly random in a range [−vr, vr],then the spectrum is broaden symmetrically around theneutrality point (because of the random character of thepotentials on sublattice A and B), and there is no bandgap, see the colored lines (except the red one) in Fig. (14).It softens the singularities in the DOS, the smearing be-ing larger for a larger degree of disorder. The smearingof the DOS leads to the smearing of the optical conduc-tivity, see σxx in Fig. 14.

12

-4 -2 0 2 40.0

0.2

0.4

0.6

0.8

1.0

1.2 vr=0 vd=1 vr=1 vr=2 vr=2.5 vr=3D

OS

E/t

0 1 2 3 4 5 6 7 8 90

1

2

3

4

4096X4096=0,T=300K

xx/

0

/t

vr=0 vd=1 vr=1 vr=2 vr=2.5 vr=3

FIG. 14: (Color online) Comparison of DOS (in units of 1/t)and optical conductivity (µ = 0, T = 300K) with symmetri-cal random (vr) or antisymmetrical fixed (±vd) potential onsublattices A and B. The size of the system is M = N = 4096.

C. Optical Conductivity of Graphene with

Resonant Impurities

In Fig. 15 we present the optical conductivity ofgraphene with various concentrations of hydrogen im-purities. Small concentrations of the impurities havea small effect on the optical conductivity, but higherconcentrations change the optical properties dramati-cally, especially when the concentration reaches the max-imum (100%), i.e., when graphene becomes graphane.Graphane has a band gap (2t), see bottom panel in Fig. 6,which leads to the zero optical conductivities within theregion |ω| ∈ [0, 2t], see Fig. 15) for ni = 100%. At in-termediate concentrations, one can clearly see additionalfeatures in the optical conductivity related with the for-mation of impurity band. The Van Hove singularity ofclean graphene is smeared out completely for concentra-tions as small as 1%.

0 1 2 3 4 5 6 70

1

2

3

4

xx/

0

/t

4096X4096=0,T=300K

ni=0% ni=0.1% ni=1% ni=50% ni=90% ni=100%

FIG. 15: (Color online) Comparison of optical conductivity(µ = 0, T = 300K) with different concentration of hydrogenimpurities. The size of the system is M = N = 4096, exceptfor the clean graphene (M = N = 8192).

VI. DC CONDUCTIVITY

The DC conductivity can be obtained by taking ω → 0in Eq. (25) yielding58

σ = − 1VTr

{∂f

∂H

∫ ∞

0

dt1

2[JJ (t) + J (t)J ]

}. (38)

We can use the same algorithm as we used for the opticalconductivity to perform the integration in Eq. (38), but itis not the best practical way since it only leads to the DCconductivity with one chemical potential each time, andthe number of non-zero terms in Chebyshev polynomialrepresentation growth exponentially when the tempera-ture tends to zero. In fact, at zero temperature ∂f∂H canbe simplified as

− ∂f∂H

= δ (EF −H) , (39)

and therefore Eq. (25) can be simplified as

σT=0 =π

NVRe

N∑

m,n=1

〈n| J |m〉 〈m| J |n〉

×δ (EF − Em) δ (EF − En) . (40)By using the quasieigenstates |Ψ(ε)〉 obtained from thespectrum method in Eq. (21), we can prove that (seeAppendix C)

σ =ρ (ε)

V

∫ ∞

0

dtRe[e−iεt 〈ϕ| JeiHtJ |ε〉

]≈ σT=0,

(41)

where |ϕ〉 is the same initial random superposition stateas in Eq. (20) and

|ε〉 = 1|〈ϕ|Ψ(ε)〉| |Ψ(ε)〉 . (42)

13

-0.006 -0.004 -0.002 0.000 0.002 0.004 0.0060

5

10

15

20

25

30

35

40-0.003 -0.002 -0.001 0.000 0.001 0.002 0.00305

1015202530354045

-0.02 -0.01 0.00 0.01 0.0202468

1012141618

-0.009 -0.006 -0.003 0.000 0.003 0.006 0.00905

1015202530354045

-0.010 -0.005 0.000 0.005 0.0100

5

10

15

20

25

30

35

-0.04 -0.02 0.00 0.02 0.040.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

ni=0.2% nx=0.2%

(e2 /h

)

ni=0.1% nx=0.1%

(e2 /h

)

ni=1% nx=1%

ni=0.3% nx=0.3%

(e2 /h

)

ne

ni=0.5% nx=0.5%

(f)

(e)

ni=5% nx=5%

ne

(a)

(b)

(d)

(c)

FIG. 16: (Color online) Conductivity σ (in units of e2/h) as a function of charge carrier concentration ne (in units of electronsper atom) for different resonant impurity (εd = −t/16, V = 2t) or vacancy concentrations (nx) : (a) ni = nx = 0.1%, (b) 0.2%,(c) 0.3%, (d) 0.5%, (e) 1%, (f) 5%. Numerical calculations are performed on samples containing (a) 8192 × 8192 and (b-f)4096 × 4096 carbon atoms. The charge carrier concentrations ne are obtained by the integral of the corresponding density ofstates represented in Fig. 3.

The accuracy of the quasieigenstates in Eq. (21) aremainly determined by the time interval and total timesteps used in the Fourier transform. The main limitationof the numerical calculations using Eq. (21) is the sizeof the physical memory that can be used to store thequasieigenstates |Ψ(ε)〉.We used the algorithm presented above to calculate the

DC conductivity of single layer graphene with vacanciesor resonant impurities. The results are shown in Fig. 16.As we can see from the numerical results, there is plateau

of the order of the minimum conductivity63 4e2/πh in thevicinity of the neutrality point, in agreement with theo-retical expectations64. Finite concentrations of resonantimpurities lead to the formation of a low energy impu-rity band (see increased DOS at low energies in Fig. 3).At impurity concentrations of the order of a few percent(Fig. 16 e, f) this impurity band contributes to the con-ductivity and can lead to a maximum of σ in the midgapregion. The impurity band can host two electrons perimpurity. For impurity concentrations below ∼ 5%, this

14

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.150

2

4

6

8

10

12

14

16 ni=0.01%

fit with q0=0.01A-1, R=1.25A

1/2

KF

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.150

2

4

6

8

10

12

14

16 nx=0.01%

fit with q0=0A-1, R=1.28A

1/2

KF

FIG. 17: (Color online) Red dots: conductivity σ (in unitsof e2/h) as a function of KF (in units of Å

−1) for reso-nant impurity (top panel, εd = −t/16, V = 2t) or va-cancy (bottom panel). The concentration of the impuritiesis ni = nx = 0.01%. Numerical calculations are performed onsamples containing 4096 × 4096 carbon atoms. Black lines:

fit of Eq. (44) with q0 = 0.01Å−1

, R = 1.25Å for ni = 0.01%,and q0 = 0, R = 1.28Å for nx = 0.01%.

leads to a plateau-shaped minimum of width 2ni (or 2nx)in the conductivity vs. ne curves around the neutralitypoint. Analyzing experimental data of the plateau width(similar to the analysis for N2O4 acceptor states in Ref.25) can therefore yield an independent estimate of impu-rity concentration.Beyond the plateau around the neutrality point, the

conductivity is inversely proportional to the concentra-tion of the impurities, and approximately proportional tothe carrier concentration ne. This is consistent with theapproach based on the Boltzmann equation, which in thelimit of resonant impurities with V → ∞, yields for theconductivity13,19,29,50

σ ≈ (2e2/h) 2π

neni

ln2∣∣∣∣E

D

∣∣∣∣ , (43)

where ne = E2F /D

2 is the number of charge carriers per

carbon atom, andD is of order of the bandwidth. Equan-tion (43) yields the same behavior as for vacancies21.Note that for the case of the resonance shifted with re-spect to the neutrality point the consideration of Ref. 13leads to the dependence

σ ∝ (q0 ± kF ln kFR)2 , (44)

where ± corresponds to electron and hole doping, respec-tively, and R is the effective impurity radius. The Boltz-mann approach does not work near the neutrality pointwhere quantum corrections are dominant20,63,65. In therange of concentrations, where the Boltzmann approachis applicable the conductivity as a function of energy fitsvery well to the dependence given by Eq. (44), as for ex-

ample shown in Fig. (17), with q0 = 0.01Å−1

, R = 1.25Åfor ni = 0.01%, and q0 = 0, R = 1.28Å for nx = 0.01%.The relation of these results to experiment is discussedin Ref. 29.The advantage of the method used here for the calcu-

lation of the DC conductivity is that the results do notdepend on the upper time limit in the integration sincethe contributions to the integrand in Eq.(41) correspond-ing to different energies tends to zero fast enough whenthe time is large. The propagation time for the integra-tion depends on the concentration of the disorder, i.e.,larger concentration leads to faster decay of the correc-tions. The disadvantage of this method is that a lot ofmemory may be needed to store the coefficients of manyquasieigenstates. Furthermore, since |ε〉 in Eq. (42) con-tains the factor 1/ |〈ϕ|Ψ(ε)〉|, this may cause problemswhen |〈ϕ|Ψ(ε)〉| is very small. For example, when us-ing this method to calculate the Hall conductivity in thepresence of strong magnetic fields, tiny |〈ϕ|Ψ(ε)〉| (outthe Landau levels) will leads to large fluctuations of thecalculated conductivity. Nevertheless, the conductivitieswithout the presence on the magnetic filed in our paperare agreement with the results reported in Refs. 37 (hy-drogenated graphene) and 38 (graphene with vacancies),and both papers are based on the numerical calculationof the Kubo-Greenwood formula, as proposed in Ref.39.To calculate the Hall conductivity accurately our methodshould be developed further.

VII. SUMMARY

We have presented a detailed numerical study of theelectronic properties of single-layer graphene with res-onant (“hydrogen”) impurities and vacancies within aframework of noninteracting tight-binding model on thehoneycomb lattice. The algorithms developed in thispaper are based on the numerical solution of the time-dependent Schrödinger equation, the fundamental opera-tion being the action of the evolution operator on a gen-eral wave vector. We do not need to diagonalize theHamiltonian matrix to obtain the eigenstates and there-fore the method can be applied to very large crystallites

15

which contains millions of atoms. Furthermore since theoperation of the Hamiltonian matrix on a general wavevector does not require any special symmetry of the ma-trix elements, this flexibility can be exploited to studydifferent kinds of disorder and impurities in the nonin-teracting tight-binding model.The algorithms for the calculation of density of states,

quasieigenstates, AC and DC conductivities, are applica-ble to any 1D, 2D and 3D lattice structure, not only toa single layer of carbon atoms arranged in a honeycomblattice. The calculation for the electronic properties ofmultilayer graphene can be easily obtained by adding thehoping between the corresponding atoms of different lay-ers.Our computational results give a consistent picture of

behavior of the electronic structure and transport prop-erties of functionalized graphene in a broad range of con-centration of impurities (from graphene to graphane).Formation of impurity bands is the main factor deter-mining electrical and optical properties at intermediateimpurity concentrations, together with the appearance ofa gap near the graphane limit.

VIII. ACKNOWLEDGEMENT

The support by the Stichting Fundamenteel Onderzoekder Materie (FOM) and the Netherlands National Com-puting Facilities foundation (NCF) are acknowledged.

IX. APPENDIX A

Suppose x ∈ [−1, 1], then

e−izx = J0(z) + 2∞∑

m=1

(−i)m Jm (z)Tm (x) , (45)

where Jm(z) is the Bessel function of integer orderm, andTm (x) = cos [m arccos (x)] is the Chebyshev polynomialof the first kind. Tm (x) obeys the following recurrencerelation:

Tm+1 (x) + Tm−1 (x) = 2xTm (x) . (46)

Since the Hamiltonian H has a complete set of eigen-vectors |En〉 with real valued eigenvalues En, we can ex-pand the wave function |φ(0)〉 as a superposition of theeigenstates |n〉 of H

|φ(0)〉 =N∑

n=1

|n〉 〈n|φ(0)〉 , (47)

and therefore

|φ(t)〉 = e−itH |φ(0)〉 =N∑

n=1

e−itEn |n〉 〈n|φ(0)〉 . (48)

By using the inequality∥∥∥∑

Xn

∥∥∥ ≤∑

‖Xn‖ , (49)

with the Hamiltonian H of Eq.(1) we find

‖H‖b ≡ 3tmax + 6t′max + |v|max + |εd|+ |V |> max{En}. (50)

Introduce new variables t̂ ≡ t ‖H‖b and Ên ≡ En/ ‖H‖b,where Ên are the eigenvalues of a modified HamiltonianĤ ≡ H/ ‖H‖b, that is

Ĥ |En〉 = Ên |En〉 . (51)

By using Eq. (45), the time evolution of |φ(t)〉 can berepresented as

|φ(t)〉 =[J0(t̂)T̂0

(Ĥ)+ 2

∞∑

m=1

Jm(t̂)T̂m

(Ĥ)]

|φ(0)〉 ,

(52)

where the modified Chebyshev polynomial T̂m

(Ên

)is

T̂m

(Ên

)= (−i)m Tm

(Ên

), (53)

obeys the recurrence relation

T̂m+1

(Ĥ)|φ〉 = −2iĤT̂m

(Ĥ)|φ〉 + T̂m−1

(Ĥ)|φ〉 ,

T̂0

(Ĥ)|φ〉 = I |φ〉 , T̂1

(Ĥ)|φ〉 = −iĤ |φ〉 . (54)

X. APPENDIX B

In general, a function f(x) whose values are in therange [−1, 1] can be expressed as

f(x) =1

2c0T0 (x) +

∞∑

k=1

ckTk (x) , (55)

where Tk (x) = cos (k arccosx) and the coefficients ck are

ck =2

π

∫ 1

−1

dx√1− x2

f (x)Tk (x) . (56)

Let x = cos θ, then Tk (x) = Tk (cos θ) = cos kθ, and

ck =2

π

∫ π

0

f (cos θ) cos kθdθ

= Re

[2

N

N−1∑

n=0

f

(cos

2πn

N

)e

2πinkN

], (57)

which can be calculated by the fast Fourier transform.For the operators f = ze−βH/

(1 + ze−βH

), where z =

exp (βµ) is the fugacity, we normalize H such that H̃ =

16

H/ ||H || has eigenvalues in the range [−1, 1] and put β̃ =β ||H ||. Then

f(H̃)=

ze−β̃H̃

1 + ze−β̃H̃=

∞∑

k=0

ckTk

(H̃), (58)

where ck are the Chebyshev expansion coefficients of

f (x) =ze−β̃x

1 + ze−β̃x, (59)

and the Chebyshev polynomial Tk

(H̃)can be obtained

by the recursion relations

Tk+1

(H̃)− 2H̃Tk

(H̃)+ Tk−1

(H̃)= 0, (60)

with

T0

(H̃)= 1, T1

(H̃)= H̃. (61)

XI. APPENDIX C

The random superposition state (RSS) |ϕ〉 in the realspace can be represented in the energy eigenbases as

|ϕ〉 =∑

n

An |n〉 . (62)

By using the expression Eq. (21) of |Ψ(ε)〉 we obtain

|〈ϕ|Ψ(ε)〉| =√∑

n

|An|2 δ (E − En), (63)

and

|ε〉 = 1∑n |An|

2δ (ε− En)

∑

n

Anδ (ε− En) |n〉 . (64)

Therefore the conductivity in Eq. (41) becomes

σ =1

V

ρ (ε)∑

n |An|2δ (ε− En)

∫ ∞

0

dtRe[e−i(ε−Em)t

×∑

m,k

A∗k 〈k|J |m〉 〈m|J∑

n

Anδ (ε− En) |n〉]

=π

V

ρ (ε)∑

n |An|2δ (ε− En)

Re∑

m,k,n

AnA∗k

×〈k| J |m〉 〈m| J |n〉 δ (ε− Em) δ (ε− En) . (65)

Dividing∑

m,k,n into two parts with k = n and k 6= n,the conductivity reads

σ =π

V

ρ (ε)∑

n |An|2δ (ε− En)

Re∑

m,n

|An|2

×〈n| J |m〉 〈m| J |n〉 δ (ε− Em) δ (ε− En)

+π

V

ρ (ε)∑

n |An|2δ (ε− En)

Re∑

m,k 6=n

AnA∗k

×〈k| J |m〉 〈m| J |n〉 δ (ε− Em) δ (ε− En) ,(66)

When the sample size N → ∞, the RSS in real spaceis equivalent to a RSS in the energy basis, and we have|An|2 ≈ 1/N, ρ (ε) ≈

∑n |An|

2δ (ε− En). Then the

second terms in above expression is close to zero becauseof the cancellation of the random complex coefficientsAnA

∗k. Thus, we have proven that

σ =ρ (ε)

V

∫ ∞

0

dtRe[e−iεt 〈ϕ| JeiHtJ |ε〉

]

≈ πNV

Re∑

m,n

〈n| J |m〉 〈m| J |n〉 δ (ε− Em) δ (ε− En) ,

(67)

which is just Eq. (40).

Introducing

|ϕ1 (t)〉x = e−iHtJx |ϕ〉 , |ϕ1 (t)〉y = e−iHtJ |ϕ〉 , (68)

the DC conductivities at zero temperature is given by

σαβ (ε, T = 0) =1

V

∫ ∞

0

Re[e−iεt α 〈ϕ1 (t) |Jβ | ε〉

]dt.

(69)

The DC conductivity for temperature T > 0 is

σαβ =∑

ε

β [1− f (ε)] f (ε)σαβ (ε, T = 0) . (70)

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