Screening and collective modes in disordered graphene antidot lattices

12 Pages·2013·1.51 MB·English

PHYSICALREVIEWB88,195401(2013)
Screeningandcollectivemodesindisorderedgrapheneantidotlattices
ShengjunYuan,1,* FengpingJin,2RafaelRolda´n,3,† Antti-PekkaJauho,4 andM.I.Katsnelson1
1RadboudUniversityofNijmegen,InstituteforMoleculesandMaterials,Heijendaalseweg135,6525AJNijmegen,TheNetherlands
2InstituteforAdvancedSimulation,Ju¨lichSupercomputingCentre,ResearchCentreJu¨lich,D-52425Ju¨lich,Germany
3InstitutodeCienciadeMaterialesdeMadrid,CSIC,CantoblancoE28049Madrid,Spain
4CenterforNanostructuredGraphene(CNG),DTUNanotech,DepartmentofMicro-andNanotechnology,TechnicalUniversityofDenmark,
DK-2800KongensLyngby,Denmark
(Received23September2013;published5November2013)
Theexcitationspectrumandthecollectivemodesofgrapheneantidotlattices(GALs)arestudiedinthecontext
ofaπ-bandtight-bindingmodel.Thedynamicalpolarizabilityanddielectricfunctionarecalculatedwithinthe
random-phaseapproximation.Theeffectofdifferentkindsofdisorder,suchasgeometricandchemicaldisorder,
areincludedinourcalculations.WehighlightthemaindifferencesofGALswithrespecttosingle-layergraphene
(SLG). Our results show that, in addition to the well-understood bulk plasmon in doped samples, interband
plasmons appear in GALs. We further show that the static screening properties of undoped and doped GALs
quantitativelydifferfromSLG.
DOI:10.1103/PhysRevB.88.195401 PACSnumber(s): 73.21.La,72.80.Vp,73.22.Pr
I. INTRODUCTION graphene within the random-phase approximation (RPA),
see, e.g., Refs. 26–29. In fact, understanding the screening
Graphene antidot lattices (GAL), regular nanoscale per-
properties of this material is essential in order to exploit its
forations of the pristine graphene sheet, offer a way of
unique properties for plasmonic devices.23 However, we are
creating a band gap in graphene. Earliest investigations1 of
aware of only one paper (of which one of us is a coauthor
these structures date in fact back to the dark ages before the
of) of screening in GALs;30 in that paper the polarization
grapheneera,2butnewmomentumwasgainedwhenPetersen
function and plasmons of perfect GALs were studied in the
etal.3speculatedin2008thatthesestructuresmayhavemany
qa →0limit(qisthewavevectorandaisthegraphenelattice
interesting applications, even as a platform for spin-based
constant).Inthepresentpaperwepresentextensive,andcom-
quantum computation. Subsequently, scores of theoretical
plementary, results for the finite qa case: Using a numerical
papers have addressed various properties of GALs using a
methodwestudythescreeningpropertiesofdisorderedGALs.
varietyoftheoreticaltools(e.g.,Diracconeapproximationfor
The density of states (DOS) is obtained from a numerical
theunderlyinggraphenespectrum,3densityfunctionaltheory,4
solutionofthetime-dependentSchro¨dingerequation,31andthe
or within the tight-binding model5). Rather than attempting
polarizationfunctioniscalculatedfromtheKuboformula.31,32
to review this vast literature, we merely state that, in our
ThedielectricfunctionisobtainedwithintheRPA,usingthe
opinion, the electronic structure and its dependence on the
numerically computed polarization functions. The presence
underlying lattice symmetry and shape of the antidots,6,7 as
of disorder is unavoidable in experimental realizations of
well as transport and optical properties of perfect GALs, are
GALs. Here, we consider the generic kinds of disorder
fairly well understood,4 and what remains to be investigated
found in these systems, such as a random deviation of the
concerns the role of interactions, disorder, and extension of
periodicityandfluctuationsoftheradiiofthenanoholesfrom
the present theoretical methods to systems with large unit
the perfect array, as well as the effect of resonant scatterers
cells, such as the ones encountered in the laboratory. What
in the sample (such as vacancies, adatoms, etc.). We find
really has made GALs interesting is the rapid development
thatgappedandalmostdispersionlessplasmonsmayexistin
in fabrication techniques, and today several methods exist to
GALs,duetointerbandtransitionsbetweenthenarrowbands
create(reasonably)regularstructureswithperiodsinlowtens
characteristic of the GAL spectrum. However, these modes
of nanometers—a length scale at which the created gaps are
are expected to be highly damped. For doped samples, the
predicted to be in hundreds of millivolts, i.e., approaching
classicalplasmonmodewithadispersionrelationproportional
thetechnologicallyrelevantnumbers.Ashortandincomplete √
to q is also present. However, we find that the dispersion
catalog of fabrication methods includes block-copolymer
masks,8–10 nanoimprint lithography,11 e-beam lithography of this mode differs substantially from the corresponding
(either using conventional masks12–15 or focused e-beam dispersion of the plasmon mode in SLG. Finally, we study
direct writing of holes16), ion beam etching,17 nanosphere the main characteristics of the static screening in GALs,
masks,18,19 or nanoporous alumina membranes as etch paying special attention to their differences with respect
masks.20–22 toSLG.
The extraordinary electronic and optical properties of The paper is organized as follows. In Sec. II we present
graphene have recently brought a lot of attention to this the details of the method. The main characteristics of the
material as an ideal candidate for plasmonics applications.23 excitation spectrum are discussed in Sec. III. The static
Plasmons, which are collective density oscillations of an dielectric screening is discussed in Sec. IV, and our main
electron liquid,24,25 have been extensively studied in pristine conclusionsaresummarizedinSec.V.
1098-0121/2013/88(19)/195401(12) 195401-1 ©2013AmericanPhysicalSociety
YUAN,JIN,ROLDA´N,JAUHO,ANDKATSNELSON PHYSICALREVIEWB88,195401(2013)
II. MODELANDMETHOD
WemodeladisorderedGALbytherealspacetight-binding
Hamiltonian,5
(cid:2) (cid:2)
H=− (t c†c +H.c.)+ v c†c +H , (1)
ij i j i i i imp
(cid:4)i,j(cid:5) i
†
where c (c ) creates (annihilates) an electron on site i of
i i
thehoneycomblatticeofgraphene,t isthenearest-neighbor
ij
hopping integral, and v is the on-site potential. The effect
i
of isolated vacancies can be modeled by setting the hopping
amplitudes to other sites to zero or, alternatively, with an
on-site energy v →∞. Further, additional resonant impu-
i
rities, such as hydrogen adatoms, can be accounted by the
termH inEq.(1):
imp
(cid:2) (cid:2) FIG.2. BandstructureofSLG(a)andofa{10,6}GAL(b).
H =ε d†d +V (d†c +H.c.), (2)
imp d i i i i
i i √
lattice constant a = 3a˜ ≈2.46 A˚, where a˜ ≈1.42 A˚ is
where ε is the on-site potential on the “hydrogen” impurity
d
the interatomic separation. It should be noted that there are
andV isthehoppingamplitudebetweencarbonandhydrogen
atoms.31,33,34 In our calculations, we fix the temperature to many other possible realizations of a GAL: Both the antidot
T =300 K and use periodic boundary conditions for both shapeanditsedgestructure,aswellastheunderlyinglattice
symmetry can be varied in a number of ways. The present
the polarization function and the density of states. The size
ofthesystemusedinoursimulationsis6600×6600atoms. model is chosen for several reasons: It is a generic model
usedinmanypreviousstudies,anditdisplaysagappedband
Throughout this paper we ignore the effects due to spin (in
our previous work5 we give a short discussion of this point), structure,6,7,35 which is of special interest for the present
study (see Fig. 2). Further, to the best of our knowledge, no
and thus the spin degree of freedom merely contributes a
experiments on plasmons in GALs have yet been reported,
degeneracyfactor2andisomittedforsimplicityinEq.(1).
and in a proof-of-principle study (such as the present one) a
Following Refs. 3 and 5, we model a GAL by creating a
thoroughexplorationofthevastparameterspacedoesnotseem
hexagonallatticeof(approxim√ately)circularholesofagiven
radiusRandaseparationP = 3Lbetweenthecentersoftwo warranted. On the other hand, when experiments emerge we
caneasilyextendourcalculationstoanyparticulargeometry.
consecutiveholes,whereListhesidelengthofthehexagonal
Weconsidertwokindsofgeometricdisordersinoursystems,
unit cell (see Fig. 1). GALs in this symmetry class can be
labeled by the parameters {L,R}, in units of the graphene whichleadtodeviationsoftheGALsfromperfectperiodicity.
First, we allow the center of the holes to float with respect
to their position in the perfect periodic lattice (x,y) around
(x±l ,y±l ). Second, we allow the radius of the holes to
C C
randomlyshrinkorexpandwithintherange[R−r ,R+r ].
R R
Throughoutthispaper,weexpressl andr inunitsofa.
C R
Thealgorithmusedinournumericalcalculationsisbased
onanefficientevaluationofthetime-evolutionoperatore−iHt
L (we use units such thath¯ =1) and Fermi-Dirac distribution
operator n (H)=1/[eβ(H−μ)+1] (β =1/k T, where T is
F B
the temperature, k is the Boltzmann constant, and μ is the
P B
chemical potential) in terms of the Chebyshev polynomial
R representation.31,37 The initialstate|ϕ(cid:5) isa random superpo-
sitionofallthebasisstatesintherealspace,i.e.,31,38
(cid:2)
|ϕ(cid:5)= a c†|0(cid:5), (3)
i i
i
where |0(cid:5) is the electron vacuum (cid:3)state and ai are random
complex numbers normalized as |a |2 =1. Since one
i i
initial state in our calculation contains all the eigenstates of
the whole spectrum, averaging over different initial states is
not required in our numerical computations.31,38 Moreover,
sincethesystemusedcontainsmillionsofcarbonatoms,and
onespecificdisorderconfigurationcontainsalargenumberof
differentlocalconfigurations,thereisnoneedtoaverageover
FIG.1. (Coloronline)SketchofaGAL,withthesetofgeomet- different realizations of the disorder.31 As shown in Refs. 38
ricalparametersthatareusedtodefineitasexplainedinthetext. and 31, the density of states (DOS) of the system can be
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SCREENINGANDCOLLECTIVEMODESINDISORDERED... PHYSICALREVIEWB88,195401(2013)
calculatedfromtheFouriertransformoftheoverlapbetween Brilloiun zone. In this paper, for simplicity, we fix the wave
the time-evolved state |ϕ(t)(cid:5)=e−iHt|ϕ(cid:5) and the initial state vectorqalongtheK-(cid:10)direction.
|ϕ(cid:5)asfollows: ThescreeningpropertiesoftheGALaredeterminedbythe
(cid:4)
∞ dielectricfunctionε(q,ω),whichweconsiderherewithinthe
1
d(ε)= eiεt(cid:4)ϕ|ϕ(t)(cid:5)dt. (4) RPA:
2π −∞
ε(q,ω)=1−V (q)(cid:6)(q,ω), (10)
Furthermore, the dynamical polarization can be obtained
fromtheKuboformula39 as where
(cid:4)
i ∞ 2πe2
(cid:6)(q,ω)= dτeiωτ (cid:4)[ρ(q,τ),ρ(−q,0)](cid:5), (5) V (q)= (11)
A κq
0
w(cid:3)here A denotes the area of the unit cell, ρ(q)= is the 2D Fourier transformation of the Coulomb interaction
c†c exp(iq·r ) is the density operator, and the average and κ is the dielectric constant of the embedding medium
i i i i
is taken over the canonical ensemble. For the case of the (we take in all our plots κ =1). Although self-energy and
single-particleHamiltonian,thepolarizationfunction(5)can vertex corrections are not included within the RPA, this
bewrittenas31 approximationissufficienttocapturethemainfeaturesofthe
(cid:4)
∞ plasmon modes, which is the main focus of this work. From
2
(cid:6)(q,ω)=− dτeiωτIm(cid:4)ϕ|n (H)eiHτρ(q) thedielectricfunction,wecanstudythecollectiveexcitations
F
A
0 ofthesystem.Thedispersionrelationforthecollectivemodes
×e−iHτ [1−n (H)]ρ(−q)|ϕ(cid:5). (6) isobtainedfromthesolutionof
F
Byintroducingthetimeevolutionoftwowavefunctions, Reε(q,ω (q))=0, (12)
pl
|ϕ (τ)(cid:5)=e−iHτ [1−n (H)]ρ(−q)|ϕ(cid:5), (7) where ω is the energy of the collective (plasmon) mode.
1 F pl
The condition for those modes to be long-lived is that
|ϕ (τ)(cid:5)=e−iHτn (H)|ϕ(cid:5), (8) Im(cid:6)(q,ω )=0, such that the plasmon cannot decay into
2 F pl
electron-holepairs.Otherwise,therewillbeafinitedamping
we obtain the real and imaginary part of the dynamical ofthemodeγ ∝Im(cid:6)(q,ω ).
polarizationas40 pl
(cid:4)
∞
2
Re(cid:6)(q,ω)=− dτcos(ωτ)Im(cid:4)ϕ (τ)|ρ(q)|ϕ (τ)(cid:5), III. DIELECTRICFUNCTIONANDCOLLECTIVEMODES
2 1
A
(cid:4)0 Inthissectionwepresenttheresultsforthepolarizationand
∞
2
Im(cid:6)(q,ω)=− dτsin(ωτ)Im(cid:4)ϕ (τ)|ρ(q)|ϕ (τ)(cid:5). dielectricfunctionofGALs,analyzingtheirmaindifferences
2 1
A with respect to SLG. For completeness, we summarize the
0
(9) salientresultsfortheeffectoftheantidotlatticeontheDOS
(both pristine and disordered systems), as discussed in our
In the numerical calculation, the integral of Eqs. (5) and (9) earlier papers.3,5 The DOS of a {10,6} GAL follows from
isreplacedbythesumoftheintegratedfunctionatfinitetime
Eq. (4), and the results are shown in Fig. 3. The lattice of
steps. The value of the time step is set to be π/Emax, which antidotssplitsthebroadπ andπ∗bandsofSLGintoanumber
is small enough to cover the whole spectrum. Here E is
max ofgapped,narrow,andflatbands,modifyingtheDOSofSLG
the maximum absolute value of the energy eigenvalues, and
[givenbytheredlineinFigs.3(a)and3(b)]intoasetofpeaks
for pristine graphene and GALs itis 3t.The number of time
associatedtotheVanHovesingularitiesofthesubbandsofthe
steps determines the energy resolution and typically we use
antidotlattice[blacklinesinFigs.3(a)and3(b)].Figures3(c)
1024timesteps.WealsouseaGaussianwindowtoalleviate
and 3(d) show the modification in the DOS of GALs due to
the effect of the finite time used in the numerical integration
geometricaldisorder,associatedtoirregularitiesintheantidot
inEqs.(5)and(9).
lattice,suchaschangesinthecenter-to-centerdistanceofthe
Wenoticeherethatthepresenceoftheantidotarraybreaks
etchedholesortovariationsinthesizeoftheholes.Ingeneral,
the translational invariance of the graphene layer. Hence, in
the gaps are rather robust against geometrical disorder, and
the most general case, the polarization function (cid:6) should be
onlyafteralargedeviationoftheGALarrayfromtheperfect
a matrix in the G, G(cid:8) space, where G and G(cid:8) are reciprocal
periodicity do the gaps close. We further show the effects
latticevectorsassociatedtothenewperiodicityimposedinthe
of resonant impurities, such as vacancies or adatoms, in the
system by the GAL.30 Therefore, in writing the polarization
DOSofGALs[Figs.3(e)and3(f)].Thiskindofdefectsleave
function in the form of Eq. (5), we are neglecting local field
thestructureoftheDOSpracticallyunchanged,apartfromthe
effects,whichisequivalenttoassumingthattherelevantwave
creationofamidgapbandassociatedtolocalizedstatesaround
vectorsq aresmallcomparedtothereciprocallatticevectors theimpurities,whichleadstothepeakatE ≈0intheDOS.
G∝1/L.Thisapproximationisjustifiedforthesmallvalue
of L considered here, L=10a, although local field effects
A. Electron-holecontinuumandcollectivemodes
shouldbetakenintoaccountforlargevaluesofL.Thiseffect
is beyond the scope of this paper, and it will be discussed ThestructureoftheDOSdiscussedabovewillbeusefulto
in a future work.41 Furthermore, the dynamical polarization understand the polarization and dielectric functions of GAL.
functionisanisotropicanddependsonthedirectionofqinthe We first consider the case of a clean GAL. Our results for
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YUAN,JIN,ROLDA´N,JAUHO,ANDKATSNELSON PHYSICALREVIEWB88,195401(2013)
3 1
(a) (b)
2.5
GALs {10,6} 0.8 GALs {10,6}
Graphene Graphene
2
1/t) 1/t) 0.6
S( 1.5 S(
O O
D D 0.4
1
0.2
0.5
0 0
-3 -2 -1 0 1 2 3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
E(t) E(t)
1 1
(c) (d)
0.8 lc=0.0 0.8 rR=0.00
l =0.5 r =0.25
c R
l =1.0 r =0.50
c R
1/t) 0.6 1/t) 0.6
S( S(
O O
D 0.4 D 0.4
0.2 0.2
0 0
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
E(t) E(t)
1 1
(e) (f)
0.8 nx=0.00% 0.8 ni=0.00%
n =0.05% n=0.05%
x i
n =0.10% n=0.10%
x i
n =0.20% n=0.20%
x i
1/t) 0.6 1/t) 0.6
S( S(
O O
D 0.4 D 0.4
0.2 0.2
0 0
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
E(t) E(t)
FIG.3. (Color online) (a) DOS of a {10,6} GAL (black lines) and of SLG (red lines). (b) Same as (a) but zoomed in at low energies.
(c)Randomshiftofthecenteroftheantidotswithintherange(x±l ,y±l ).(d)Randomvariationoftheantidotradiuswithintherange
C C
[R−r ,R+r ].(e)Randomvacanciesforfourconcentrations.(f)Randomhydgrogenadsorbantsforfourcoverages.Thedottedvertical
R R
linesinpanel(b)correspondtothetwodifferentFermienergiesconsideredinthiswork:undopedGALwithμ=0anddopedGALwith
μ=0.115t.
(cid:6)(q,ω)andε(q,ω)asafunctionoffrequencyωareshownin wealsoshowthepolarizationfunctionforundopedSLGfrom
Fig.4.Intheundopedregime[Figs.4(a)and4(b)]weobtain a π-band tight-binding model40 (blue line) and within the
an Im(cid:6) which consists of a series of peaks [the black line Dirac cone approximation (red line). In both cases, undoped
in Fig. 4(a)], which indicate the electron-hole continuum in graphene and undoped GAL, only interband transitions are
GALs, defined as the region of the energy-momentum plane possible. However, there are strong differences in the two
whereparticle-holeexcitationsarepossible.Forcomparison, spectra.Ontheonehand,theinterbandelectron-holeexcitation
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SCREENINGANDCOLLECTIVEMODESINDISORDERED... PHYSICALREVIEWB88,195401(2013)
0.015
(a) 0.01 (b)
0.008 μ=0 20 GALs {10,6}
0.012 0.006 q=0.05a-1 Graphene
Graphene Dirac Cone
0.004
15 Approximation
2) 0.009 0.002
-a 0
-1Π(t 0 0.5 1 1.5 2 2.5 3 εRe 10
m 0.006 GALs {10,6}
-I Graphene μ=0
Graphene Dirac Cone 5
Approximation q=0.05a-1
0.003
0
0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ω(t) ω(t)
0.06
(c) 20 (d)
μ=0.115 GALs {10,6}
0.05 0.01 q=0.05a-1 Graphene
15 Graphene Dirac Cone
Approximation
0.04 10
2)
-a 0
-1Π(t 0.03 0 0.5 1 1.5 2 2.5 3 εRe 5
m 0
-I 0.02 GALs {10,6}
Graphene μ=0.115
-5
Graphene Dirac Cone q=0.05a-1
0.01 Approximation
-10
0 -15
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ω(t) ω(t)
FIG.4. (Coloronline)ComparisonbetweenSLGandGALs.Polarizationfunction[(a)and(c)]anddielectricfunction[(b)and(d)]fora
{10,6}GAL.BlacklinescorrespondtoGAL,whereasthebluelinescorrespondtoSLGwithintheπ-bandtight-bindingmodel,andthered
linescorrespondtoSLGusingtheDiracconeapproximation.Panels(a)and(b)areforundopedsamples(μ=0),andpanels(c)and(d)are
fordopedsampleswithμ=0.115t.Theinsetsin(a)and(c)givethespectrainabroaderrangeofenergies.
spectruminSLGisacontinuum which(neartheDiracpoint) excitations,whichareobtainedfromEq.(12),weobservethat
correspondstotheregionintheω-q planeinwhichω>v q, Reε vanishesforundopedGALsatleastonce,whereasthere
F
where v is the Fermi velocity of graphene near the Dirac isnosolutionofEq.(12)forundopedgraphene[theblueand
F
point.Infact,Im(cid:6)(q,ω>v q)isamonotonicfunctionfora redlinesinFig.4(b)].Thisisthewell-knownresultaboutthe
F
broadrangeofenergies,Fig.4(a).Athighenergies,oftheorder absence of plasmons in undoped SLG within the RPA.26–29
of ω≈2t ∼5 eV, the spectral function shows a peak which The situation differs somewhat for GALs, whose peculiar
isassociatedtointerbandtransitionsbetweenstatesoftheVan band structure and DOS lead to a collective mode which
Hovesingularitiesofthevalenceandoftheconductionbands is associated to interband electron-hole transitions between
[seeinsetofFig.4(a)].Noticethatthiseffectisnotcapturedby the valence and conduction bands, with energies (for the
theDiracconeapproximation(redline),whichisvalidonlyfor {10,6}structure)E ≈±0.12t,asshowninFig.3(b).However,
transitionsinthevicinityoftheDiracpoint.Theelectron-hole this mode cannot be considered a fully coherent plasmon
excitation spectrum qualitatively differs for GALs [the black since its dispersion lies in a region of the spectrum where
line of Fig. 4(a)]. Most saliently, there are regions of zero Im(cid:6)(q,ω )(cid:12)=0. Therefore, such a mode will be damped,
pl
spectral weight in Im(cid:6)(q,ω) which are associated to the decaying into electron-hole pairs. It is interesting to notice
interbandgapsopenedintheGALbandstructure,ascompared thatthismodehasasimilaroriginastheso-calledπ-plasmon
to the continuum spectrum associated to the broad π and mode in graphene,40 which is a damped mode in single-
π∗ bands in graphene [see, e.g., Figs. 3(a) and 3(b) for the layer and multilayer graphene originated from particle-hole
correspondingDOS]. transitions between the Van Hove singularity of the valence
In addition, the dielectric function of undoped GALs bandatE ≈−t ∼−2.7eVandtheVanHovesingularityof
and SLG qualitatively differ, as shown in Fig. 4(b). The the conduction band at E ≈+t ∼+2.7 eV and which has
oscillatoryDOSoftheGALleadstoadielectricfunctionwith been observed by electron energy loss spectroscopy (EELS)
an oscillatory behavior, too. Second, focusing on collective experiments.42
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YUAN,JIN,ROLDA´N,JAUHO,ANDKATSNELSON PHYSICALREVIEWB88,195401(2013)
The results for doped GALs are shown by the black lines the spectrum, as seen by the zeros of Reε(q,ω) in Fig. 4(d).
in Figs. 4(c) and 4(d), which are compared to the results This solution of Eq. (12) corresponds to the usual plasmon
of SLG from a π-band tight-binding model (blue lines) and mode which, in this case, is long-lived due to the fact that
fromtheDiracconeapproximation(redlines).Thechemical it disperses (for a certain range of energies) in the region of
potentialμ≈0.115t issuchthattheFermienergylieswithin the (q,ω) space where Im(cid:6)(q,ω)=0. In fact, such a mode
the first peak in the DOS, as shown by the corresponding is present also in SLG, as seen by the zero-energy cut of the
dotted black line of Fig. 3(b). Here we define the chemical blue line in Fig. 4(d). However, for a given wave vector, the
potential as measured from the E =0 Dirac point energy in energyatwhichReε(q,ω)=0differsforGALandforSLG,
graphene. In GAL, since it is a semiconductor, it is useful and,hence,thedispersionoftheplasmondiffersmarkedlyin
to define μ≈(cid:13)/2+E , where (cid:13) is the gap energy and the two cases. In particular, from our results one can expect
F
E is the Fermi energy measured, as usual, from the bottom that the velocity of the mode (given by the slope of the band
F
of the conduction band. One first notices that Im(cid:6)(q,ω) has dispersion)willbelowerinGALthaninSLG.Consideringthat
twodifferentcontributions:oneatlowenergies,whichisdue in2Dthedispersionrelationoftheplasmonsatlowenergies
tointrabandexcitations,andwhichisverynarrowduetothe canbeapproximatedby25
(cid:5)
smallbandwidthWofthedopedbandinGAL.Thisbandwidth
2πne2q
W imposesalimitfortheenergyofintrabandtransitions.The ω (q)≈ , (13)
secondcontributionemerges(forq =0)atω≈(cid:13)+2EF and pl mb
isduetointerbandexcitationsacrossthebandgap.Thisisthe where n is the carrier density and m is the effective mass
b
counterpartoftheinterbandelectron-holecontinuuminSLG, of the band, our numerical result for the existence of slow
anditistheonlycontributionintheμ=0case,asdiscussed plasmons in GALs is expected due to the flatness of their
above.Interestingly,dopingleadstoanewplasmonmodein bands [see Fig. 2(b)], which typically implies large effective
FIG.5. (Coloronline)DensityplotsofReε(q,ω).(a)Undoped(μ=0)GALwith{10,6}periodicity.(b)DopedGALwithμ=0.115t.
(c)Sameas(b)forareducedregionofthespectrumcorrespondingtosmallwavevectorsandfrequencies.(d)DopedSLGwithμ=0.115t.
Noticethedifferentscalesoffrequenciesandmomentaineachplot.Theplotsstartatq =0.05/a.43
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SCREENINGANDCOLLECTIVEMODESINDISORDERED... PHYSICALREVIEWB88,195401(2013)
masses.Furthermore,wenoticethatthetheoreticalstudiesfor the band structure, (cid:13)≈0.24t for the present case, and it is
thebandstructureofGALsshowthattheyhaveadirectgap, almost dispersionless.43 The reason for this weak dispersion
locatedinthe(cid:10)pointoftheBrillouinzone.3Theminimumof of the gapped mode is the narrowness and flatness of the
theconductionbandandthemaximumofthevalencebandcan bands involved in the interband transitions. Apart from this
be well approximated, in leading order, by a parabolic band well-definedmodewithagap(cid:13)≈0.24t,Fig.5(a)alsoshows
dispersion,whoseeffectivemassissimplyrelatedtotheDOS, several resonances at higher energies. They are signatures of
d(μ),bytheusualrelationm =2πh¯2d(μ)/g ,whereg =2 electron-hole transitionsofhigher energies,associatedtothe
b σ σ
isthespindegeneracy.Therefore,takingintoaccountthatthe bandswhichleadtothedifferentpeaksintheDOSofFig.3.
nanoperforationofthegraphenelatticeallowsustomanipulate Similarly as in SLG,42 this structure should be accessible
theDOSofthesystem,thecorrespondingdispersionrelation by means of EELS experiments, which could be useful to
oftheplasmonmodewillbespecificanddifferforeachpair determinethemaingap,aswellasthecharacteristicenergies
{L,R}thatcharacterizesagivenGAL. oftheotherflatbands,whichwouldleadtofurtherresonances
These features are more clearly seen in Fig. 5, where we intheEELSspectrum.
showadensityplotofReε(q,ω)intheω-qplane.Considerfirst The situation differs for doped GALs [Fig. 5(b)]. Apart
the undoped μ=0 case, Fig. 5(a). In the absence of charge fromthegappedmode(with(cid:13)≈0.24t)discussedabove,we
carriers, the GAL does not exhibit any low-energy plasmon observeastrongfeatureatlowenergieswhichcorrespondsto
mode, and the first line of zeros of the dielectric function thegaplessclassicalplasmon,withdispersionrelationgivenby
corresponds to the gapped plasmon which is associated, as Eq.(13).Thelow-energyandsmall-q regionofthespectrum
discussed above, to interband transitions between states of of doped GAL is shown for clarity in Fig. 5(c), where we
the bands adjacent to the Fermi level μ=0. The energy observetheexistenceofacollectivemodeinaω-q regionof
of the mode at q →0 coincides with the gap opened in the spectrum which, for the corresponding undoped case of
FIG.6. (Coloronline)DensityplotofReε(q,ω)forfourdifferenttypesofdisorder.(a)Randomcenter;(b)randomradius;(c)random
vacancies;(d)randomhydrogenadsorbants.Theparameterscharacterizingthedisorderareindicatedinthefigures.
195401-7
YUAN,JIN,ROLDA´N,JAUHO,ANDKATSNELSON PHYSICALREVIEWB88,195401(2013)
Fig. 5(a), is entirely featureless. This collective mode is the which accounts for deviations of the GAL from the perfect
counterpartofthewell-knownplasmonmodecharacteristicof periodicity,andresonantimpurities,whichcanbeassociated
a 2DEG24,44 or doped SLG.26–29 For comparison, we show to vacancies in the graphene lattice or to adatoms which
a density plot of Reε(q,ω) for doped SLG in Fig. 5(d), can be adsorbed by the surface. The results are shown in
where one ca√n clearly observe the classical plasmon mode Fig.6.ThemodificationofthebandstructureandDOSdueto
with ω(q)∝ q at small wave-vectors q.43 By comparing geometrical disorder, as shown by Figs. 3(c) and 3(d), leads
Fig.5(d)toFig.5(c),weobservethattheslopeoftheplasmon toadramaticeffectonthestructureofthespectrum,ascanbe
mode in SLG is much higher than the corresponding slope seen by comparing Figs. 6(a) and 6(b) to Fig. 5(a). Changes
of the low-energy plasmon in doped GAL. The qualitatively in the center-to-center separation of the antidots leads to a
different band structures of SLG and GAL explain the blurring of the resonance peaks in the spectrum, as seen in
different behaviors of the plasmon m√ode in the two systems. Fig.6(a),wherethefeatureassociatedtothegappedplasmon
Furthermore, notice that the ω(q)∼ q plasmon is the only withenergy(cid:13)≈0.24t haspracticallydisappeared.Infact,no
activemodeindopedSLG[Fig.5(d)],whichdoesnotpresent zeros are present in the dielectric function after this kind of
additional optical resonances at higher energies, as the ones disorder is considered, which means that the plasmon mode
presentinthespectrumofGALs[Fig.5(b)]. hasdisappeared.
InFig.7weshowresultsforthepolarizationanddielectric
function of GAL, obtained from Eqs. (5) and (10), in the
B. Effectofdisorder presence of geometrical disorder for the wave vector q =
We next address the effects due to disorder. We have 0.05a−1. The presence of geometrical disorder can lead to
consideredtwomainsourcesofdisorder:geometricaldisorder, the disappearance of the plasmon mode. This can be seen in
0.015 10
(a) 0.01 (b)
0.008 μ=0 l =0.0
0.012 0.006 q=0.05a-1 8 lcc=0.5
l =1.0
0.004 c
6
2) 0.009 0.002
-1-Π(ta 0 0 0.5 1 1.5 2 2.5 3 εRe 4 μq==00.05a-1
l =0.0
-Im 0.006 lcc=0.5 2
l =1.0
c
0
0.003
-2
0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ω(t) ω(t)
0.015 10
(c) 0.01 (d)
0.008 μ=0 r =0.00
0.012 0.006 q=0.05a-1 8 rRR=0.25
r =0.50
0.004 R
6
2) 0.009 0.002
-1-Π(ta 0 0 0.5 1 1.5 2 2.5 3 εRe 4 μq==00.05a-1
r =0.00
-Im 0.006 rRR=0.25 2
r =0.50
R
0
0.003
-2
0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ω(t) ω(t)
FIG.7. (Coloronline)SameasshowninFig.4butinthepresenceofgeometricaldisorder.Panels(a)and(b)correspondstoIm(cid:6)(q,ω)
andReε(q,ω),respectively,foraGALinwhichthecenteroftheholesisshiftedrandomlywithrespecttotheoriginalpositionintheperfect
periodic array, within the range (x±l ,y±l ). The different colors correspond to different values of l (in units of a), as denoted in the
C C C
insetofthefigures.Panels(c)and(d)correspondstoaGALwheretheradiusoftheholesisrandomlyshrunkorenlargedwithintherange
[R−r ,R+r ].Differentcolorscorrespondtodifferentvaluesofr (inunitsofa).Theinsetsinpanels(a)and(c)showthepolarizationin
R R R
abroaderrangeofenergies.
195401-8
SCREENINGANDCOLLECTIVEMODESINDISORDERED... PHYSICALREVIEWB88,195401(2013)
Figs.7(b)and7(d),wherethereisanabsenceofzerosforthe in Fig. 6(b), where a pronounced resonance is still observed
dielectric function of disordered GALs (blue and red lines), atanenergy(cid:13)≈0.1t.Thereasonforthelow-energyfeature
whereas Reε for clean GAL clearly shows a solution for the is clearly understood by looking at the green dashed line of
plasmonequation(12).Furthermore,thewell-separatedbands Fig.3(d),whichcorrespondstotheDOSofaGALsamplewith
of clean GAL leads to a discretization of the electron-hole this kind of geometrical disorder and where we see that two
continuum at low energies. This can be seen by looking at newpeakshaveemergedintheDOSatenergiesE ≈±0.05t.
theblacklineinFigs.7(a)and7(b),whichshowsanabsence Therefore,theresonanceinFig.6(b)atω≈0.1t hasitsorigin
ofspectralweightbetweenthewell-definedpeaksofIm(cid:6)at inelectron-holetransitionsbetweenthosebandscreatedinthe
low energies. However, the presence of geometrical disorder spectrumbytheeffectofrandomradiusgeometricaldisorder.
strongly modifies the band structure, and, as a consequence, We emphasize that this is not a plasmon, since it does not
thereisatransferofspectralweighttothegappedregionsof correspondtoazeroofthedielectricfunction.
thespectrum. Wehavealsoconsideredtheeffectofresonantimpuritiesin
Therefore, the impurity bands which emerge in the spec- thespectrum.Thiskindofdisorderhasalessdramaticeffect
trum due to the presence of geometrical disorder, leads to ontheexcitationspectrumofthesystem,asshowninFig.6.
a significant modification of the electron-hole continuum as Bothrandomdistributionofvacanciesandrandomdistribution
well as the dielectric function, and the results for GAL with ofhydrogenimpuritiesinthesamplehaveasimilareffecton
thiskindofdisorderdoesnotshowanysignatureofplasmons. thedielectricfunctionofGAL,leadingtoabroadeningofthe
Nevertheless, notice that a resonance is still visible in the plasmonmodes,whicharemoreefficientlydampedduetothe
spectrumofaGALinwhichthesizeoftheholesvarywithin presenceofthiskindofimpurities.However,thesefeaturesstill
somerange,atanenergywhichissmallerthantheenergyof correspondtoaplasmonmode:Thisisprovedbylookingatthe
the plasmon mode in the clean GAL sample. This is shown Reε plots of Figs. 8(b) and 8(d), which present well-defined
0.015 7
(a) 0.01 (b)
0.008 μ=0 6
0.012 0.006 q=0.05a-1 5 nx=0.00%
n =0.05%
x
0.004 n =0.10%
4 x
2) 0.009 0.002 nx=0.20%
-a 0 3
-1Π(t 0 0.5 1 1.5 2 2.5 3 εRe 2
n =0.00%
m 0.006 x
-I nx=0.05% 1
n =0.10%
x
nx=0.20% 0 μ=0
0.003 -1 q=0.05a-1
-2
0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ω(t) ω(t)
0.015 7
(c) 0.01 (d)
0.008 μ=0 6
0.012 0.006 q=0.05a-1 5 ni=0.00%
n=0.05%
i
0.004 n=0.10%
4 i
2) 0.009 0.002 ni=0.20%
-a 0 3
-1Π(t 0 0.5 1 1.5 2 2.5 3 εRe 2
n=0.00%
m 0.006 i
-I ni=0.05% 1
n=0.10%
i
ni=0.20% 0 μ=0
0.003 -1 q=0.05a-1
-2
0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ω(t) ω(t)
FIG.8. (Coloronline)Same asshown inFig. 4butinthepresence ofresonantimpurities.Panels (a)and (b)correspondtoIm(cid:6)(q,ω)
andReε(q,ω),respectively,fora{10,6}GALwitharandomdistributionofvacancies.Thedifferentcolorscorrespondtodifferentamounts
ofmissingdanglingbonds,asdenotedintheinsetofthefigures.Panels(c)and(d)correspondstoaGALwithhydrogenadatoms.Different
colorscorrespondtodifferentpercentageofadatomsinthesample.Theinsetsinpanels(a)and(c)showthepolarizationinabroaderrangeof
energies.
195401-9
YUAN,JIN,ROLDA´N,JAUHO,ANDKATSNELSON PHYSICALREVIEWB88,195401(2013)
0.14 8
(a) μ=0 (b)
0.12 7
6
0.1
2)
-a 5
-1(t0 0.08 =0) 4
= ω
Πω 0.06 ε( μ=0
Re Graphene 3 Graphene
- 0.04 pristine GALs pristine GALs
l =0.5 2 l =0.5
c c
r =0.25 r =0.25
0.02 n =R0.20% 1 n =R0.20%
x x
n=0.20% n=0.20%
i i
0 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
q(a-1) q(a-1)
0.14 30
(c) μ=0.115 (d) Graphene
pristine GALs
0.12
25
0.1
2) 20 μ=0.115
-a
-1(t0 0.08 =0) 15
= ω
Πω 0.06 ε(
e
R 10
- 0.04
5
0.02
Graphene
pristine GALs
0 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
q(a-1) q(a-1)
FIG.9. (Coloronline)Staticpolarization(cid:6)(q,ω=0)andstaticdielectricfunctionε(q,ω=0)ofGALs.Plots(a)and(b)areforundoped
samples,μ=0,whereasplots(c)and(d)correspondtoafinitedopingμ=0.115t.Differentkindsofdisorderareconsideredinpanels(a)
and(b),asdescribedintheinset.Forcomparison,weincludethecorrespondingpolarizationanddielectricfunctionforSLG,asgivenbythe
fullblacklinesineachplot.
zeros for both clean and disordered samples. Furthermore, transitions.Suchacontributionisshownbythesolidblackline
the energies of the gapped modes remain almost unchanged inFig.9(a).FordopedSLG,thereisanextracontributiondue
as compared to the spectrum of a perfect and clean GAL tointrabandexcitations,whichcanbewrittenas
[Fig.5(a)].However,thereisanadditionaltransferofspectral
weighttothelow-energyregionofthespectrum,associatedto (cid:6) (q (cid:2)2k )≈−d(E )[1−q/(2k )], (14)
intra F F F
theimpuritybandswhichhavebeencreatedandwhichaccount
for localized states around the impurities.5 This feature in where d(EF)=EF/(2πvF2) is the DOS at the Fermi level.
thespectrumisespeciallyvisibleinFigs.8(a)and8(c),where Both contributions lead to a constant static polarization
we observe a contribution to the polarization function in functionatsmallwavevectors,(cid:6)(q (cid:2)2kF)=−d(EF),asis
the low-energy region of the spectrum which increases with showninFig.9(c),typicalofmetallicscreening,25whereasthe
theconcentrationofimpuritiesinthesample. interbandlineartermleadstoaninsulatinglikescreening.The
abovepolarizationfunctionleadstoastaticdielectricfunction
for SLG as shown by the solid black lines of Figs. 9(b) and
IV. STATICSCREENING
9(d).UndopedSLGhasaconstantdielectricfunction,ε(q)=
Inthissectionwefocusonthestaticdielectricscreeningof 1+πe2/(2κv )=ε ≈4.9 for the parameters used in
F SLG
a GAL: We calculate the polarization and dielectric function this work, and it corresponds to the black horizontal line
in the ω→0 limit, using the Kubo formula [Eq. (5)]. The of Fig. 9(b). Doping a SLG leads, as we have mentioned
results for clean and disordered GALs, as compared to the above, to a metallic like screening with the corresponding
correspondingpolarizationanddielectricfunctionforSLG,are 1/q divergence as q →0, as it is shown by the continuous
showninFig.9.Westartbyreviewingthemaincharacteristics blacklineinFig.9(d).
of the static screening in SLG. As is well known,26–28,45,46 ThesituationdiffersinGALs.First,Fig.9(a)showsastatic
undopedSLGshowsalinearincreaseofthestaticpolarization polarization function that grows with q, as in undoped SLG,
functionwithq,(cid:6) (q)=−q/(4v ),associatedtointerband signaling a semiconducting type of screening. However, by
inter F
195401-10