A real-space study of random extended defects in solids : application to disordered Stone-Wales defects in graphene

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by  Suman Chowdhury

A real-space study of random extended defects in solids : application to
disordered Stone-Wales defects in graphene.
SumanChowdhurya,SantuBaidyab,DhaniNafdayb,SoumyajyotiHaldard,MukulKabirc,Biplab
Sanyald,TanusriSaha-Dasguptab,DebnarayanJanaa,AbhijitMookerjeeb
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1 aDepartmentofPhysics,UniversityofCalcutta,92AcharyaPrafullaChandraRoad,Kolkata700009,India
0 bDepartmentofCondensedMatterandMaterialsScience,S.N.BoseNationalCentreforBasicSciences,BlockJD,SectorIII,SaltLake,
Kolkata700098,India
2
cDepartmentofPhysics,IndianInstituteofScienceEducationandResearch,PuneSaiTrinityBuilding,Pashan,Pune411021,India

n dDepartmentofPhysicsandAstronomy,UniversityofUppsala,Uppsala,Sweden
a
J

9

]
i Abstract
c
s Weproposehereafirst-principles,parameterfree,realspacemethodforthestudyofdisorderedextendeddefects
-
l in solids. We shall illustrate the power of the technique with an application to graphene sheets with randomly
r
t placedStone-Walesdefectsandshallexaminethe signatureof suchrandomdefectsonthe densityofstates as a
m
functionof their concentration. The techniqueis generalenoughto be appliedto a whole class of systems with
. latticetranslationalsymmetrybrokennotonlylocallybutbyextendeddefectsanddefectclusters. Therealspace
t
a approachwillallowustodistinguishsignaturesofspecificdefectsanddefectclusters.
m
Keywords: Extendeddisordereddefects,realspacerecursionmethod
-
d PACS:73.22.Pr;63.22.Rc;61.72.-y;46.35.+z
n
o
c
[ 1. Introduction

2
Theeffectofrandomdefectsonthepropertiesofsolidsisanimportantareainthestudyofmaterials. Defects
v
areubiquitousin solids[1]. Eitherformednaturallyduringtheirpreparationorartificiallyengineered,theymay
0
9 profoundlyaffect their physicalproperties. Chemical reactions, phase transitionsor plastic deformationsduring
2 theformativestagemayleavetheirimprintsasdefects. Thesedefectsmaybelocal,likevacancies[2,3],substi-
0 tutionalimpurities[4] oradatoms[5]. Defectscanalso beextended,likedislocations,stackingfaults, twinsand
.
1 grainboundaries. ExtendeddefectshavebeenvisualizedbySTM images[6]. Itis importantthereforeto setup
0 a first-principles,essentially parameterfree,theoreticalmethodologyforthe signatureofrandomdefects, andto
4
distinguish,in particular,the roleofdisorder. Localrandomdefectshasbeenthoroughlystudiedusingsophisti-
1
catedmean-fieldapproacheslike theitinerantcoherentpotentialapproximation(ICPA)[7], thetravellingcluster
:
v approximation(TCA)[8],bothderivedfromtheparentformalism: theaugmentedspaceformalism(ASF)[9,10].
i
X However,mostoftheworkonextendeddefectshaveinvolvedsuper-cellmethods.Theseworkslookatessentially
periodicarraysofdefectswithdisorderstretchingonlyoverthefinitesuper-cell[11–13]. Longrangeddisorderis
r
a notaccessibletothesereciprocalspacebasedmethods. Noneofthesupercellmethodscanaccuratelycapturethe
disorderinducedsmearingofthedensityofstatesduetotheself-energyarisingoutofscatteringofBlochwaves
by configurationfluctuations. This ‘life-time’effect is experimentallyaccessible throughneutronscatteringand
hasbeenfoundtobestronglydependentbothontheenergyEandthewave-vector~k. Inthisworkweshallpropose
afullyrealspacetechnique,wherethestructureandHamiltonianoftherelaxeddefectedlattice goesasaninput
intoarecursivealgorithmwhichgivesustheelectronicstructurecarryingthesignaturesofdisorder.Althoughthe
PreprintsubmittedtoPhysicaE January10,2014
applicationwillbetoaspecificproblem,themethodisgeneralenoughtobeappliedinanysolidcarryingrandom
extendeddefects.
Electronicstructureoflow-dimensionalsolidswithextendeddefectshavebeenaddressedearlier[11–13]. Let
usquoteShirodkarandWaghmare[13]:
“Ourwork,alongwithotherwork[11,12]hasinvolvedaperiodicarrayofSWdefects,whereasSW
defectsarerandomlydistributedinarealsample.”
Theauthorsgoontoremarkthatalthoughthisdisordermaynotaffectthevibrationalspectrumsignificantly,the
electronic structure obtained from a periodic array of SW defects needs to be interpreted with care. Our real
space recursion approach [42] makes no appeal to lattice translational symmetry and is thus ideal for studying
topologicaldisorderand its effecton electronicstructure. The illustrationof this pointis the justification of our
proposedmethodologyandthemainfocusofthiscommunication.
2. Therealspacerecursivealgorithm
Early in the seventies Heine and co-workersintroduced [41, 42] the recursion method based on a fully real
spacetechniqueto dealwithlattices withoutanytranslationsymmetry. Theessentialinputsarethe geometryof
thelatticeandatight-binding,sparseHamiltonian.
As anexample,andinview ofourlater applications,letuslookata honeycomblattice andmapitsvertices
ontothesetofpositiveintegers. Themappingisnotunique,butnecessarilyone-to-one(Fig.1). Thegeometryof
thelatticeisuniquelydescribedbyaconnectivitymatrixC(n,m)whichgivesusthem-thneighbourofthevertex
labelled as n. The central panel of Fig.1 shows a model in which up to third neighbouroverlapsare taken into
account.Theconnectivitymatrixis:
1 2 3 4 5 6 7 8 9 10 11 12 13
2 9 10 1 3 4 12 11 ... ... 5 8 ...
 
C(n,m)= 34 ..67....58... 11 131 142 22 1133 ...... ...... 76 190 ......  (1)
Thetight-bindingHamiltoniancanbewrittenas:
H = ǫ P + t (n−m)T (2)
α nα αβ nα,mβ
Xn Xα Xn,mXαβ
13 6 3 5 11 2 1 2 1 2 1 2
1 2 1 Ct’’ tC’ 1 2 1
10 a 2 A B 2
7 4 1 2 1 t’ t’ t’ 1
1
2 2 D’ D 2 1 2
8 9 nseecaorensdt nneeaigrh nbeoigrsh b 0o.r5s7 7 aa 1 2 1 1 2 1 1 2 1
12 third near neighbors 1.154a
Figure1:(Colouronline)(leftpanel)Themappingoflatticesitesonahoneycomblatticeontoadenumerablesetofpositiveintegers.(center)
Thenearest,next-nearestandthirdnearestneighbourvectorsonahoneycomblattice.(rightpanel)LatticedistortionsandchangeinHamiltonian
elementsaroundastructuraldefect.
2
Here, PandT areprojectionandtransferoperatorsonthetight-bindingbasislabelledbyn,α,wherenisthe
vertexlabelandαallotherpossible degreesoffreedomassociated with it. Similarly,we describea honeycomb
latticewithastructuraldefectshownintherightmostpanelofFig.1. Welabelthedistortedsitessaywithalabel
γ,thenwheneverα,β,γ,t (n−m) = t(n−m)andotherwiset (n−m) = t′(n−m).
αβ αβ
With this recursion method, we shall study the representationsof the Green operator or the resolventof the
Hamiltonian,
G(z)=(zI−H)−1
wherezisacomplexvariable.Mostphysicallymeasurablequantitiesdescribingtheelectronicpropertiesofasolid
arerelatedtodifferentmatrixelementsofG(z). Inparticular,thelocal(atomprojected)densityofstates(LDOS)
n(E)andthetotaldensitiesofstates(TDOS)n(E):
i
1
− lim ℑmG (z) = n(E)
πz→E−ı0+ ii i
1 1
− lim ℑmTrG(z) = n(E) (3)
πz→E−ı0+ N
Where N is the number of atoms in the system. It would be interesting to note that although the super-cell
basedmethodscaneasilyaccessboththebandprojectedpartialdensityofstates(PDOS)andthetotaldensityof
states(TDOS),itisonlythereal-spacebasedmethodsthatcangiveisasingleatomoranatomicclusterprojected
localdensitiesofstates(LDOS).
In this basis the representation of the Hamiltonian is a matrix of infinite rank. The solution of the Kohn-
ShamequationofanelectroninthissystemcanbesimplifiedenormouslyiftheHamiltonianhaslatticetranslation
symmetry. InthatcasetheBlochtheoremintroducesthequantumlabel~kandreducestheeffectivelyinfiniterank
matrix representationof the Hamiltonianto a manageablefinite rank equalto the numberof bands(in this case
two).Consequently,crystallinesystemsarealwaysstudiedinthisBlochrepresentation.Butindisorderedsystems,
particularlywherethedisorderisatopologicaldistortionofthelatticeandthesedefectsaredistributedrandomly
throughoutit,suchperiodicityisabsentandthereciprocalspacerepresentationisnolongerstrictlyvalid. Insuch
situationsweneedtolookforreal-spacebasedtechniques.
Calculation of the resolvent requires inversion of a matrix of infinite rank. Haydock et.al. [41] proposed a
techniquetodoso. Westartfromasuitablevertexat|1i. Itessentiallyinvolvesgenerationofanewbasisthrough
athreetermrecurrencerelation:
‘
|1} = |nαi
|2} = (H−α )|1}
1
|n+1} = (H−α )|n}−β2 |n−1} (4)
n n+1
{n|H|n} {n|n}
α = and β2 = (5)
n {n|n} n {n−1|n−1}
Haydocket.al.[41]showedthatthematrixelementoftheresolventmaybewrittenasacontinuedfraction.
3
1
hnα|G(z)|nαi = {1|G(z)|1}=
β2
z−α − 2
1 β2
z−α − 3
2 β2
z−α − 4
3
...z−α −β2T(z)
N N
(6)
Rightatthestartweemphasizedthatwehavechosenarealspacealgorithmovermean-fieldandsupercellap-
proaches,becausewedonotwishtointroduceartificialperiodicityandconfinerandomnessoverafinitevolume.
But the problem with any numericalcalculation is that we can deal with only a finite number of operations. In
therecursionalgorithm,we cangoupto afinitenumberofstepsleadingtoexactlywhatwewishtoavoid. The
analysisoftheasymptoticpartofthecontinuedfractionthenisofprimeinteresttous. Thisisthe“termination”
procedureinwhichtheasymptoticbehaviourisassessedfromthewaythecoefficients{α ,β }behaveasn → ∞.
n n
Different terminators have been discussed in detail by Haydock and Nex [50], Luchini and Nex [51], Beer and
Pettifor[52]andinconsiderabledepthbyViswanathandMuller[53]. Theterminatorwhichdescribestheasymp-
toticallyfarenvironmentmustsatisfycertainbasicproperties:Thisterminationprocessisadelicatemathematical
approximation. We wouldlike tomaintainthe Herglotzanalyticpropertiesof theresolventaftertermination. A
functionT(z)iscalledHerglotzif:
(i) AllsingularitiesofT(z)lieontherealz-axis.
(ii) SingularitiesofT(z)formaboundedset.
(iii) ImT(z)≥0ifImz<0;ImT(z)≤0ifImz>0.
(iv) ReT(z)→0asRez→ ±∞
Thenextstepistoanalyzeourresolventtolocatesingularitiesonitscompactspectrum.Majorityofresolvents
with bounded spectra have singularities at the band edges. The termination of continued fractions describing
spectraldensitieswithcompactsupportandsingularitiesexceptatthebandedgeshavebeendescribedindetailin
earlierworks[50–53].
Thebestillustrationofthistechniqueistoapplyittoaninterestingmaterialwherealternativemethodsalways
leadstoproblemsofonesortoranother.WehavechosentostudygraphenewithrandomStone-Walesdefects.
3. TherelaxedgraphenelatticewithrandomStone-Walesdefects.
Inthiswork,weshallbeinterestedinintrinsicextendedstructuraldefectslikeStone-Wales(SW)defects[14]
ingraphene.Thestudyofgraphene,thetwo-dimensionalallotropeofCarbonisanareaofintenseinteresttoboth
theoreticalandexperimentalresearchersforvariousreasons.Thefirstisthepossibletechnologicalapplicationsof
graphene[15–19]. Second,grapheneseemstodefythetheoremofMerminandWagner[20]andformastabletwo-
dimensionalstructure. Finally, the electrons in graphene behave like massless charged particles, something not
encounteredinourthreedimensionalworld[21]. Therehasevenbeenfancifulapplicationsofgeneralrelativistic
ideasingraphene. TheSWdefectsareformedby90◦ rotationofaC-Cbond. Aftersucharotation,subsequent
re-bondingleadstotheformationoftwopentagonalandtwoheptagonalcarbonrings. ThisisillustratedinFig.2.
These defects are responsible not only for bringing about fractures and embrittlement [22, 24–26], but also for
alteringthechemicalreactivityandtransportproperties[27–29]andforcausingtheripplingbehaviorofgraphene
sheets [30]. SW defects can lead to out-of-planedisplacementof carbon atoms in graphene[31–34] and induce
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Figure2:(Colouronline)SWdefectformationduetoaπ/2rotationofaC-Cbond
curvature[35]. Recently, based on these three-dimensionalrippleson the graphenesheets with SW defects, the
assumptionthatSWdefectsareperfecttwodimensionalhasbeenquestioned[27,28,36–40].
For setting up the lattice model, we have started with a flat graphene sheet cluster, containing 3200 atoms.
Different concentrations of Stone-Wales defects were then created by randomly choosing bonds on the lattice,
rotatingbyπ/2andrebondingasshowninFig. 2. Thefirststepofrelaxationsalllayonthegraphenesheetand
involved mainly on-plane relaxation of the C-atoms. The left panel in Fig.3 shows a vertical SW defect in the
unrelaxedlatticewhiletherightpanelshowstheeffectofrelaxationonthedefect.
Latticerelaxationwascarriedoutusingthedensityfunctionaltightbindingmethodasimplementedinasparse-
matrix based DFTB+ code [43]. In this method, a first-principlesformof density functionalis used and hence,
largesystemscouldbehandledwithreasonableaccuracy.Wehaveusedtheconjugategradientmethodtorelaxthe
internalcoordinateswith 10−2 eV/Aforceconvergenceforthe ionicrelaxationand10−4 eV energyconvergence
fortheelectronicrelaxation. GeometryoptimizationsweredoneusingcalculationsattheΓpointintheBrilluoin
zone(BZ).TheFermismearingmethodhasbeenusedwithanelectrontemperatureof100K.
ThisrelaxationfirstcausestherotatedC-Cbondlengthtocompressto1.32Afromitsunrelaxedvalueof1.44A.
Asaconsequenceofthisshorteningofbondlength,thebondangleattheapexofthepentagoncompressesfrom
140degreeto115degree,whichis 18.5%. Becauseoftheselargechangesthesystemexperiencesacompressive
stressalongthedirectionofunrotatedC-Cbonds,andtensilestressalongtherotatedC-Cbond.Thegraphenesheet
thenreducesthisinplanestressbybucklinginthez-direction,forminganon-planar,rippledstructure.Onstructure
relaxationthesystemenergyislowered.
Once the in-plane relaxation was completed we went a step further and relaxed the lattice in full three di-
mensions,allowingmovementofatomsperpendiculartotheoriginalgrapheneplane. ThetopleftpanelofFig. 4
showstheripplingofthegraphenesheetonasingleSWdefect.Theotherthreepanelsshowripplinginasheetwith
higherconcentrationsofSW-defect. Fulllatticerelaxationisthusextremelyimportantforthestructuralchanges
1.43 1.32
140 140 115 115
Figure3:(Left)AverticalSWdefectintheunrelaxedlattice(Right)Thesamedefectafterplanarrelaxation.
5
Figure4:(Top,Left)RipplingofthegraphenelatticeonasingleSWdefect.(Top,RightandBottom)Ripplingofagraphenesheetwithvarying
concentrationsofSW-defects.
onrelaxationmayhaveveryimportanteffectontheelectronicstructureofthe materials. We shalldealwith the
effectsofstructuralchangeswithHarrisonscaling[45,46].
4. FirstPrinciplesderivationoftheHamiltonian.
OncewehaveconstructedtherelaxedgraphenesheetswithSWdefects,thenextstepistoobtainaHamiltonian
forthebandduetotheπ-bonded p electronsfromfirst-principles,asfarfreefromfittedparametersaspossible.
z
Thecalculationsbeginwithaself-consistentgroundstatecalculationforthesinglelayergrapheneusingthetight-
binding, linear muffin-tin orbitals (TBLMTO) method within the atomic sphere approximation. Three empty
spheres were needed to achieve space filling. The muffin tin radii used for carbon (C) and the empty spheres
were1.56a.u. and2.76a.u. respectively. Theminimalbasissetfortheself-consistentcalculationconsistedofC
s,p ,p ,p andemptyspheresstates. Fig.4leftpanelshowsthefullTB-LMTObands.
x y z
Theactiveorbitalswererecognizedasthe p . Theremainingdegreesoffreedomwereintegratedout[44,47]
z
usinga downfoldingprocedure. Formally,we partitiontheHilbertspaceH inwhichtheHamiltonianisdefined
Figure5: (Colouronline)(Leftpanel)TheTB-LMTObands forpristinegraphene (Rightpanel) TheNMTOdown-folded bandsduetoπ
bondingbetweenthepzstates.
6
(eV) ε t t t
1 2 3
-0.291 -2.544 1.668 -1.586
Table1:Tight-bindingparametersgeneratedbyNMTO.
intoonewhichisspannedbytheactiveorbitalsH andtherest,whichweshallcallthe‘bath’: H =H ⊗H
act act B
sothat
H H′
H = act
H′ HB !
ThepartitionordownfoldingtheoremgivestheeffectiveHamiltonianinthesubspaceH :
act
H = H +H′†G (ε )H′
eff act B n
G iscalculatedbyinverting(ε I−H )inthebathsubspaceH . Forthepresentstudy,onlytheC-p orbitals
B n B B z
werekeptactive.AllothersincludingC-s,p ,p andemptyspheres-orbitalsweredownfolded.Theintegratedout
x y
orbitalsrenormalizedtheactiveC-p orbital. TherenormalizedC-p NMTOswereconstructedusingfourenergy
z z
meshpointsε -ε whichgives3rdordermuffintinorbitals.TheenergymeshusedfortheconstructionofNMTOs
0 3
were chosen to be in the energy window spanning the C-p bands. The minimal set of C-p NMTOs serve the
z z
purposeofeffectivelocalizedC-p Wannierfunctions.
z
(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) miltonian element (eV)----21100.....05050 t2 t3 t4 t5 t6 t7
Ha
-2.5 t1
-3.00.5 1.0 1.5 2.0
Distance (units of a)
Figure6:(Colouronline)(Left)pzorbitalsperpendiculartothegrapheneplane.(Right)Hamiltonianparameters
TheeffectiveC-p -C-p hoppinginteraction:t(R,R′),connectingC-p NMTO-sχ(R)centeredatthesiteRto
z z z
theneighbouringC-p NMTOχ(R′)centeredattheR′,wereobtainedbyFouriertransformationofthelowenergy
z
C-p HamiltonianH (~k)[47,48]. TheseareshowninFig.6andTable1.
z pz
The downfolded bands are shown on the top right panel of Fig.5. The corresponding t(R) is shown in the
bottomrightpanelofFig.5andTable1. Pleasenotethattheoverlapisreasonablylongranged. Thussomeofthe
earliercalculationswithasimplenearestneighbouroverlap(Reichet.al.[49])maynotshowtheaccuratepicture.
Longerrangedoverlapmodelshavealsobeenattempted. However,allofthemarefittingprocedurestothesingle
C-p bands. Our procedure of getting the bands from first-principles DFT calculations and then systematically
z
down-foldingawaythenon-activebandsisbothphysicallymoreappealingandmathematicallymorerigorous.
5. TheTerminator
HavingobtainedtheonebandHamiltonian(arisingfromtheπbonded p orbitals),wecarryouttherecursion
z
algorithm up to a finite number of steps. As discussed earlier, from the asymptotic behaviour of the recursion
7
coefficientsweestimatetheterminator. Weanalyzeourresolventtolocatesingularitiesonitscompactspectrum.
Majority of resolventswith boundedspectra have singularities at the band edges. The terminationof continued
fractionsdescribingspectraldensitieswith compactsupportandsingularitieson ithave beendescribedin detail
inearlierworks[50–53]. Forgraphene,weexpecttheπbandstohaveaspectraldensitywhichhasanadditional
singularityattheDiracpoint. TerminatorsappropriatetosuchproblemshavebeendiscussedbyMagnus[54]and
ViswanathandMu¨ller[53]. Theyproposeaterminatoroftheform:
2π(E )(α+2β+1)/2)
T(z)= m |z−E |α{(z−E )(E −z)}β (7)
0 1 2
B α+1,1+β
2
(cid:16) (cid:17)
25 0.2
20 0.18
2βn 0.16
15 -1V)0.14
100 2 4 6 8 10 n 12 14 16 18 20 States (e0.01.21
T(E) in scaled units Density of 0000....00002468
0-8 -6 -4 -2 0 2 4 6 8 10
-8 -6 -4 -2Ene0rgy (eV2) 4 6 8 Energy (eV)
Figure7: (Left,Top)Theterminatorwithaninternalsingularityappropriateforgraphene. (Left,bottom)Asymptoticpartsofthecalculated
pristinegrapheneGreenfunctioncontinuedfractioncoefficientsobtainedbyrecursion. (Right)TheTDOSofpristinegraphenetakingupto
thirdnearestneighbouroverlaps.
Inourproblem,E istheDiracpointandE ,E arethespectralbounds,E2 =E E ,α = β = 1.
0 1 2 m 1 2
Magnus[54]hascitedaclosedformforthecontinuedfractioncoefficientsoftheterminator:
4E2n(n+1) E2(2n+2)(2n+4)
β2 = m β2 = m (8)
2n (4n+2)(4n+4) 2n+1 (4n+4)(4n+6)
Theparametersoftheterminatorareestimatedfromtheasymptoticpartofthecontinuedfractioncoefficients
calculatedforourproblem.ThetopleftpanelinFig.7showsthecontinuedfractioncoefficientsfortheViswanath-
Mullerterminatorshowninthebottomleftpanelofthisfigure.Thecalculatedcontinuedfractioncoefficientsfrom
recursionareshownintherightpanel. Theparametersoftheterminatorcoefficientsarefittedfromtheseresults.
The resultant density of states is shown in the right panel of Fig.7. Unlike the usual nearest neighbour models
the density is not symmetric round the Dirac point. If we write the Green function as contributions from non-
intersectingpaths(basedonFeenbergperturbation[51])theninthenearestneighbourmodelallnon-intersecting
pathsareofevenlengthandall{α }isaconstant,leadingtoasymmetricdensity. Assoonasweintroducelonger
n
rangedt,oddlengthnon-intersectingpathsappear.Thenα varywithnandthedensitybecomesnon-symmetric.
n
6. ResultsandDiscussion.
SincetheaxisoftheSWdefectsarealongthebondwhichisrotatedinitsformation,wenotethatthedefects
areeither verticalor atangles60o tiltedto the leftandrightofthe vertical. Closeupsoftwo suchisolated tilted
defectsareshownonthetoprowofFig.8. We alsoshowadoubledefectonetiltedtotheleftandanothertothe
rightsharinga commonhexagon. In the sampleswith higherconcentrationsof defectswe also have clustersof
connecteddefects. TwosuchexamplesofdefectclustersareshowninthebottomrowofFig.8.
8
Inordertoanalyzethesignatureofindividualdefectswefirstshowthetotaldensityofstatesforseveraldefect
concentrationsThisisshowninthetoppanelsofFig.9. Wenotethattheoccupiedpartofthespectrumbelowthe
Diracpointshowverylittle change. Theisolateddefectsshowadefectstatearound1eVabovetheDiracpoint.
For the double defect this structure widens. There is also a defect induced signature around 3-4 eV above the
Diracpoint.ThesesignatureswerealsoobservedbyShirodkarandWaghmare[13](comparewiththeirFig. 5).Of
course,inourrealspacepicturethereisnoconceptofdefect‘bands’,butthespectralsignaturesareverysimilar.
Thesupercellapproachesindicatethelocaldefectlevelsareverysimilartoourrealspaceapproach. However
it has been known that the effect of the far disordered environmentleads to a ’self-energy’ [1], the real part of
whichshiftsthedefectlevelsandtheimaginarypartcausesthemtowiden.Inourcontinuedfractionapproachthe
terminatorplaystheroleoftheself-energy. Asaresultweclearlyseethewideningofthedefectstructureswith
increasingdisorder. Noperiodicsupercelltechniquecangivethisdisorderbroadeningwithaccuracy. Oneadded
advantageofthe realspaceapproachisthatwe canfocusofthe atomorclusterprojectedlocaldensityofstates
andassigndifferentsignaturesinthespectrumtoparticularisolateddefectsordefectclusters.
We proposethattherealspacerecursionmethodisapowerfultechniqueforthestudyofextendeddefectsin
disorderedsolids. OurapplicationtorandomStone-Wellsdefectsingraphenejustifiesourproposal.
Acknowledgements
SCwouldliketothankDST,IndiaforfinancialsupportthroughtheInspireFellowship. Thisworkwasdone
undertheHYDRAcollaborationbetweenourinstitutes.
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9
0.15 0.15
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