Absolutely continuous energy bands and extended electronic states in an aperiodic comb-shaped nanostructure

0.64 MB·

by  Biplab Pal

Absolutely continuous energy bands and extended electronic states in
an aperiodic comb-shaped nanostructure
Biplab Pal∗
Department of Physics, University of Kalyani, Kalyani, West Bengal-741235, India
The nature of electronic eigenstates and quantum transport in a comb-shaped Fibonacci nanos-
tructure model is investigated within a tight-binding framework. Periodic linear chains are side-
attached to a Fibonacci chain, giving it the shape of an aperiodic comb. The effect of the side-
4 attachments on the usual Cantor set energy spectrum of a Fibonacci chain is analyzed using the
1 Greens function technique. A special correlation between the coupling of the side-attached chain
0 withtheFibonaccichainandtheinter-atomiccouplingoftheFibonaccichainresultsinadramatic
2 triggeringofthefragmentedCantorsetenergyspectrumintomultiplesetsofcontinuoussub-bands
of extended eigenstates. The result is valid even for a disordered comb and turns out to be a rare
n
exception of theconventional Anderson localization problem. The electronic transport thuscan be
a
madeselectively ballistic withindesired energyregimes. Thenumberandthewidthofsuchcontin-
J
uous sub-bands can be easily controlled by tuning the number of atomic sites in the side-coupled
7
periodiclinearchains. Thisgivesusascopeofproposingsuchaperiodicnanostructuresaspotential
1
candidates for prospective energy selective nanoscale filtering devices.

]
l PACSnumbers: 61.44.-n,73.20.Jc,73.22.Dj,73.63.-b
al Keywords: Aperiodicnanostructure, Singleelectronstates,Quantum transport,Tight-bindingmodel
h
-
s I. INTRODUCTION We intend to work with a golden mean Fibonacci
e
chainwherefinitearraysofQDsareside-graftedtoevery
m
atomic site, giving the lattice an aperiodic comb struc-
Fano-Anderson effect is an interesting phenomenon
.
t ture. The model has particular relevance in the con-
a exhibited by excitonic states in a tight binding lattice
textofrecentstudiesonspectralpropertiesofexcitonsin
m when a discrete, bound state ‘interacts’ with a contin-
nanoparticle array [21–25]. The comb-structure enables
uum[1–12]. Theeffectisexplicitlyseenwhenoneconsid-
- ustoexaminetheinterferenceofFanolineshapesandthe
d ers transmission spectrum of one dimensional quantum
n wire (QW) systems with a single quantum dot (QD), effect of the side coupled dots on the energy spectrum
o and transportproperties of sucha system. The presence
or a cluster of them is attached to the QW from one
c of multiple, closely spaced gaps and a multifractal spec-
side [7, 8, 10, 11]. Needless to say, the studies so far
[
trum, when coupled to discrete, bound states from one
has gone well beyond mere theoretical interest, thanks
1 to the present advanced stage of lithographic techniques side, offers a possibility of controlling the gaps,and may
v be even the basic character of the electronic states. In
and nanotechnology,and has incorporatedthe studies of
4 the presentdayofnanotechnologyandrevolutioninfab-
nonequilibriumdynamicsinopticaltransition[13],quan-
8 ricating tailor made lattices this can even open up the
2 tum simulation [14] and tunneling in a Kondo hole sys-
possibility of designing novel QD devices. These aspects
4 tem [15] to name a few.
prompt us to undertake the present study.
1. Inspite of the considerable volume of work existing in
0 this field, a practically unaddressed issue, to the best We find extremely interesting results. With a golden
4 of our knowledge, is how seriously does the presence of meanFibonaccichainhavingsidecoupledQDarrays,the
1 bound states caused by the attachment (from one side) presence of an infinite number of grafted chains enables
v: of a QD or an assembly of them, influence a singular one to control the transmission properties at all scales
i continuous spectrum. A singular continuous spectrum of energy by suitably changing the number of atoms in
X
is the hallmark of what are known as quasiperiodic sys- the side coupled array. The most important result is
r tems [16–20] such as the Fibonacci class of substitution- that, under certain special conditions the entire energy
a
ally generated atomic chains or, the array of Harper po- spectrum of the infinite system turns out to have abso-
tentials (Aubry-Andre chains). Due to the fragmented, lutely continuous subbands of energy. The occurrence of
Cantor like character of the energy spectra offered by these continuous subbands is triggered by a certain spe-
thesesystems,theboundstatescan,inprinciple,occupy cialrelationshipbetweenthe off-diagonalelementsofthe
any local gap inside the spectrum and may de-stabilize Hamiltonian. Surprisingly, this is something that is not
thefundamentalcharacteroftheentireenergyspectrum. expected in any conventional localization scheme, such
A central motivation of the present work is to look into as the Andersonlocalization[26], where the single parti-
such an issue. clestatesgetlocalizedinadisorderedarrayofpotentials
irrespective of the numerical values of the parameters of
the Hamiltonian. Our results therefore present a strong
case of a violation of this universality of Anderson local-
∗Electronicaddress: [email protected] ization.
2
The system to be described by a tight-binding Hamil- and so on. The number of L bonds and the number of
tonian, is depicted in Fig. 1. We have used real space S bonds bear the golden ratio τ = (√5 +1)/2 in the
renormalization group (RSRG) decimation scheme [27] thermodynamic limit.
and Green’s function technique to analyze the energy Dependingonthenearestneighborbondenvironment,
spectrum and the nature of eigenstates of the system. weidentifythreekindsofsites,viz.,α(reddots),β (dark
The side-attached QD arrays play a crucial role in gen- blue dots), andγ (lightblue dots)– flankedby L-L,L-S
erating and controlling the number of continuous energy and S-L bonds respectively as shown in Fig. 1. Periodic
bands. Extended character of the electron states popu- finite QD arrays of identical size (having N +1 QDs in
lated in the bands are reflected in a prominent enhance- each array) are side-attached to every α, β, and γ sites,
mentoftwo-terminalquantumtransportthroughthesys- while an extra QD arrayof size N +2 is hung from each
tem. α site, which couples to it througha coupling strength λ
Inwhatfollows,wedescribetheresultsobtainedbyus. as shown in Fig. 1, N being any positive integer. We
InsectionIIwepresentourmodelandthe mathematical adopt a tight-binding formalism and incorporate only
methods used for handling the problem. The effect of nearestneighborhopping. Withintheframeworkofnon-
the side-attachment on the energy spectrum of the sys- interacting electron picture, the tight-binding Hamilto-
temandhowtoobtaintheconditionforgeneratingabso- nian of the system can be expressed as,
lutelycontinuousbandsofextendedstatesisdiscussedin
section III. Section IV presents the two-terminal trans- H =Hbackbone+Hupperchains+Hlowerchains
mission characteristics of the system, and finally in sec- +Hbackbone-upperchains+Hbackbone-lowerchains (1)
tionVwedrawourconclusionwithpossibleapplications
for future study. where,
H = ǫ c†c + t c†c +h.c.
backbone i i i ij i j
II. DESCRIPTION OF THE MODEL AND THE Xi Xhiji(cid:0) (cid:1)
METHOD N+1
H = ǫ d†d + t d†d +h.c.
upperchains (cid:20) µ µ µ µν µ ν (cid:21)
A. The system and the Hamiltonian α,βX,γ sites µX=1 hXµνi(cid:0) (cid:1)
N+2
H = ǫ d†d + t d†d +h.c.
Fig. 1 illustrates our system of interest. We have con- lowerchains (cid:20) µ µ µ µν µ ν (cid:21)
sidered a model quasiperiodic nanostructure, in which αXsites µX=1 hXµνi(cid:0) (cid:1)
atomic sites arearrangedfollowinga Fibonaccisequence Hbackbone-upperchains = ξ c†id1+d†1ci
onalinearchain(thebackbone)andperiodic,finitesized α,βX,γ sites (cid:0) (cid:1)
QD arraysare side-attachedto the backbone forming an
aperiodic comb-shaped system. The Fibonacci chain is Hbackbone-lowerchains = λ c†id1+d†1ci
αXsites (cid:0) (cid:1)
In the above, c†(c ) and d†(d ) represent the creation
i i µ µ
(annihilation) operators for the backbone and dangling
QD arrays respectively. ǫ is the on-site potential of an
i
i-thatomic site along the Fibonaccibackbone of the lat-
tice, which may take three different values, viz., ǫ , ǫ ,
α β
and ǫ corresponding to three different kinds of sites as
γ
mentionedabove. Thenearestneighborhoppingintegral
t takes two values t and t for electronhopping along
ij L S
the L and S bond respectively. ǫ is the value of on-site
FIG. 1: (Color online) Schematic diagram of a model µ
energy at an atomic site in the dangling chains, which
quasiperiodicnanostructure. Sitesinthebackbonearenamed
and colored differently to distinguish between nearest neigh- will assume a constant value equal to ǫ0 for all the QDs
bor bond environments. α, β, and γ sites are side-attached in an array. The hopping integral along a dangling QD
with finite size periodic QD arrays, while each α site has an arrayalsoassumesaconstantvalue tµν =ξ, andλis the
extra dangling QD chain coupled to it through a coupling couplingstrengthoftheextraQDarrayscoupledtoeach
strength λ. α-site.
grown recursively by repeated application of the infla-
tion rule [16, 18] L LS and S L, where L and S B. Renormalization into an effective system
→ →
stand for two “bonds”, viz., long bond and short bond
respectively. The first few generations are, The dangling chains are “folded back” into the back-
bone using a RSRGdecimationscheme [28]. As aresult,
G1:L, G2:LS, G3:LSL, G4:LSLLS the on-site energy of the renormalized atomic sites gets
3
dressed up, which we may call a kind of “self energy” of renormalized values of the on-site energy and hopping
the effective atomic sites. This is demonstrated pictori- integrals can be expressed by the following relations:
t2(n)+t2(n)
ǫ˜ (n+1)=ǫ˜ (n)+ L S
α γ
E ǫ˜ (n)
β
−
t2(n)
ǫ˜ (n+1)=ǫ˜ (n)+ S
β γ
E ǫ˜ (n)
FIG.2: (Coloronline)Schematicviewofrenormalizedversion − β
oftheoriginalsystem. Theside-attacheddanglingQDarrays t2(n) (4)
ǫ˜ (n+1)=˜ǫ (n)+ L
arefolded intoeffectiveatomic sitesshown byencircled dots. γ α E ǫ˜ (n)
β
−
t (n)t (n)
L S
ally in Fig. 2 and the renormalized on-site potentials of tL(n+1)=
E ˜ǫ (n)
the effective sites are given by, − β
t (n+1)=t (n)
S L
Σ =ǫ +∆
α α
where, n indicates the stage of renormalization. These
Σ =ǫ +∆
β β (2) recursion relations are used to obtain the energy spec-
Σ =ǫ +∆
γ γ trums of the system and are presented in the next the
Σ0 =ǫ0+∆ section.
Inthe aboveequations,∆isthetermwhichcontainsthe
entire information about the folded periodic finite size III. ENERGY SPECTRUMS OF THE SYSTEM
QD arrays and is given by,
A. Condition for obtaining the absolutely
ξ(E ǫ0)UN−1(x) ξ2UN−2(x) continuous bands
∆= − − , for N 1 (3)
(E ǫ0)UN(x) ξUN−1(x) ≥
− −
The renormalized effective 1-d Fibonacci chain is de-
wthheeCreh,eUbyNs(hxe)v=pol2yxnUoNm−ia1l(oxf)s−ecUonNd−k2i(nxd),,wfoirthNU−≥1 =1 0is, sMcrib,eadnbdyMase.toTfthhereeexptrraensssifoenrsmfaotrritchese[t1h8r],eeviztr.,anMsfαer,
U0 =1,andx=1/2[Tr(M)], M beingthe2×2transfer maβtrices are,γ
matrix of the form,
(E ǫ˜ )/t 1
M =(E−ǫ0)/ξ −1 Mα = −1α L −0 
1 0
 
 
(E ǫ˜ )/t t /t
for the atomic sites in the QD arrays. Mβ = − β S − L S  (5)
Now we have a renormalized system with effective 1 0
 
atomic sites arranged in a Fibonacci order along the
banacekffbeocnteiveanddanegalcinhgeQffeDcttivheroαug-shitaescoisupnloinwgcsotruepnlgedthtλo Mγ =(E−ǫ˜γ)/tL −tS/tL 
1 0
(Fig.2). We caneasilydecimate outthe dangling“effec-  
tive”QD,whichwillmakethesystemintoaneffective1-d and, M and M = M .M follow an arrangement
α γβ γ β
Fibonacci array of renormalizedatomic sites with renor-
in the Fibonacci sequence. It can be easily verified that,
malized on-site potentials, viz., ǫ˜α =Σα+λ2/(E−Σ0), if we choose ǫα = ǫβ = ǫγ = ǫ0 and tL = tS, then the
ǫ˜β =Σβ,andǫ˜γ =Σγ asdepictedinFig.3. Actuallythe commutator [M ,M ] vanishes irrespec6 tive of the en-
α γβ
ergy E, if we set λ = t2 t2. So, under this condi-
S − L
tion,whenthetransferpmatricescorrespondingtoαclus-
tersandthe β-γ clusterscommute,theycanbearranged
FIG.3: (Coloronline)Effective1-dFibonacciarrayofatomic in any desired fashion, say, for example, in a perfectly
sites obtained by decimating the dangling renormalized QDs periodic pattern. The spectrum offered by a periodic
at theα-sites in figure 2. arrangement of building blocks is an absolutely contin-
uous one. Once the commutator [M ,M ] is made
α γβ
on-site potentials at the β and γ sites do not get mod- to vanish, the entire Fibonacci lattice can be thought
ified, but the to make the notations consistent we have to be equivalent with two periodic infinite sub-lattices,
renamed Σ and Σ as ǫ˜ and ǫ˜ respectively. onecomprisingthe“renormalized”α-sitesalone,andthe
β γ β γ
Using the Fibonacci growth-rule in the reverse direc- other formed by a periodically repeating cluster of βγ
tion it is simple to rescale any generationof the effective pairs. Each subsystem has its own continuous energy
1-dFibonaccisystemintoitsearliergeneration[27]. The spectrum with a band-multiplicity that arises out of the
4
FIG. 4: (Color online) LDOS-E spectrum at an α-site of the system. The upper panel shows the LDOS-E spectrum under
thesuitable resonance condition,while thelower panelrepresentstheLDOS-E spectrum for a50% deviation from thedesired
resonance condition. In (a) and (c) numberof atomic sites in each lower chain is 5 and in each upper chain is 4, while for (b)
and (d) those are set equal to 7 and 6 respectively. The values of other parameters are ǫα =ǫβ =ǫγ =ǫ0 =0, tL =1, tS =2,
andξ =1. Forupperpanelvalueofλ=√3,andforlowerpanelλ=√3/2whichisa50%deviationfrom theresonancevalue.
side-coupled QD array. Most interestingly, the separate by,
band structure arising in these two cases map onto each
1
other, giving rise to a unique, absolutely continuous en- ρ (E)= lim Im Gi (6)
i 00
ergyspectrumasverifiedbyanextensivenumericalwork. δ→0 (cid:20)−π (cid:21)
(cid:0) (cid:1)
This conclusively leads to the existence of a dense, gap-
The results are presented in Fig. 4. We have verified
less set of extended eigenfunctions when the resonance
that the LDOS at an α, or β, or γ site turns out to
conditionissatisfied. Thefragmented,multifractalchar-
be the same. So to save space we have presented the
acter of a typical Fibonacci quasicrystal disappears and
LDOSatanα-site only. In the upper panelofFig.4, we
all the eigenstates populating the continuous subbands
have shown the LDOS at an α-site under the resonance
are of extended character.
condition. We have set ǫα = ǫβ = ǫγ = ǫ0 = 0, tL = 1,
t = 2, ξ = 1, and λ = √3. As we set λ = √3, which
S
satisfiesthedesiredconditionformatrixcommutationas
described in Sec. IIIA, it triggers absolutely continuous
subbands in the energy spectra (Fig. 4(a) and (b)).
B. Local density of states
We have minutely examined each of these continuum
sub-bands for a very fine scanning of the energy inter-
The local density of states (LDOS) at an α, β or γ val, and the continua still persist. This confirms the
is obtained by calculating the respective local Green’s robustness of these continuum subbands under the reso-
function Gi = (E +iδ ǫ∗)−1, where, i = α, β or γ. nance condition. The values of the hopping integrals t
00 − i L
δ is a very small imaginary part added to the energy E, andt remainsnon-zerooverarbitrarilylargenumberof
S
and ǫ∗ represents the fixed point value of the respective RSRG iteration steps for any energy inside eachof these
i
on-site potential as the hopping integrals flows to zero continuous subbands. This proves the extended nature
under RSRG iterations of Eq. (4). The LDOS is given of the electronic eigenstates in each of these subbands.
5
This fact is corroborated by an prominent enhancement of√3,absolutelycontinuoussub-bandsstartdominating
of two-terminal transport through the system under the the spectrum, and multifractality disappears.
resonanceconditionwhichwillbedescribedinsectionIV.
Thenumberandthepositionofsuchsubbandsaresensi-
tivetonumberofatomicsitesintheside-attachedchains. IV. TWO TERMINAL TRANSMISSION
This fact is elucidated in Fig. 4(a) and (b). In Fig. 4(a), CHARACTERISTICS
number of sites in each lower chain is set equal to 5 and
in each upper chain is set equal to 4 respectively, and
Toobtainthethetwo-terminaltransmissioncharacter-
for Fig. 4(b) these are 7 and 6 respectively. Clearly, the
istics of the system, we clamp the system between two
numberandpositionofsubbandsinthetwocaseschange.
semi-infiniteorderedleads,viz.,sourceanddrain(Fig.6).
Asweturnawayfromtheresonancecondition,thecon-
The leads are characterized by uniform on-site potential
tinua in the energy spectra get destroyed, and the typ-
ǫ and nearest neighbor hopping integral τ . We renor-
L L
ical Cantor set singular continuous energy spectra of a
quasiperiodic system returns. This is shownin the lower
panelpanelofFig.4,wherewehavesetλ=√3/2which
isa50%deviationfromitsresonancevalue. Allotherpa-
rameters remain same as in upper panel of Fig. 4. How-
ever, isolated extended states still exists in such cases.
These can be evaluated exactly.
C. The energy eigenvalue distribution
FIG.6: (Coloronline)Schematicviewofthesystemclamped
between two semi-infinite ordered leads, viz., source and
We have explored the energy eigenvalue distribution drain.
(Fig. 5) as a function of the coupling strength λ of the
dangling chains with the α-sites. Fig. 5 is representative malize the system clamped between the two leads into
an effective dimer by decimating out the internal sites
selectively. To get unique recursion relations at the two
boundary sites of the system, we consider system with
odd generation index only. This should be appreciated
that, as we interested to look at the behavior of trans-
mission characteristics of a large finite size system, by
choosing system with odd generation index only we do
not lose any physics whatsoever. Instead of deflation
rule used in Sec. IIB, we now use an alternative defla-
tion rule LSL L′ and LS S′ [10]. By using this
→ →
deflation rule, staring with an (2n+1)-th generationFi-
bonaccisystem,decimationofnsteps convertsitintoan
effectivedimerwithtworenormalizedatomicsiteshaving
effective on-site potentials ǫ˜∗ and ǫ˜∗ (for the “left” and
l r
“right” edge sites) and are connected to each other by
an effective hopping integral t∗. The recursion relations
L
arises out of this decimation procedure are given by,
FIG. 5: (Color online) Distribution of energy eigenvalues
spectrum of the proposed system as a function of the cou-
n−1 n−1
pling strength λ. We have chosen ǫα = ǫβ = ǫγ = ǫ0 = 0, ˜ǫα,n =ǫ˜α,n−1+ X + Y
tcLou=ple1d,ltoSwe=r c2h,aainndisξse=t to1.5Nanudmibnereaocfhsuitpepseirncheaacinhissidseet- Zt2Ln,−n−11 Zn−n1−1
to 4. ˜ǫβ,n =˜ǫα,n−1+ Eβ,n−1 + Yn−1
Z
t2
of an infinite system. In Fig. 5 we observe formation of ˜ǫγ,n =˜ǫγ,n−1+ S,n−1 + Xn−1
multiple bands and gaps in the energy eigenvalue spec- Eβ,n−1 Zn−1 (7)
trum as we vary the value of λ. For values of λ away
n−1
from√3,thespectrumshowsthefragmentationwithrel- ˜ǫl,n =ǫ˜l,n−1+ X
n−1
ativelydensepackingofenergyeigenvaluearoundcertain Z
n−1
energy intervals which basically are due to isolated ex- ˜ǫr,n =ǫ˜r,n−1+ Y
tendedstates(or,atleaststateswithalargelocalization Zn−1
length) clustered together. Band-crossing is, in general t2L,n−1tS,n−1 tL,n−1tS,n−1
t = , t =
avoidedinsuchcases. Withλclosetotheresonancevalue L,n n−1 S,n Eβ,n−1
Z
6
FIG. 7: (Color online) Transmission characteristics of a 9-th generation system as a function of the energy E. (a) represents
the transmission characteristics under the resonance condition (λ= √3) and (b) is for the 50% deviation from the resonance
condition. Number of sites in each side-coupled lowerchain is 5 and in each upper chain is 4. The values of other parameters
are ǫα =ǫβ =ǫγ =ǫ0 =0, ǫl =ǫr =0, tL=1, tS =2, and ξ=1. The lead parameters chosen ǫL =0 and tL =2.
where, Xn−1 = t2L,n−1Eγ,n−1, Yn−1 = t2L,n−1Eβ,n−1, V. CONCLUSION
Zn−1 =Eβ,n−1Eγ,n−1−t2S,n−1,andEj,n−1 =(E−ǫj,n−1)
for j = α, β, γ. The transmission coefficient across the In conclusion, we have studied the electronic energy
effective di-atomic system is given by [29], spectrumandtransmissioncharacteristicsofamodelFi-
bonacci quasiperiodic system with side-coupled periodic
4sin2ka QD arrays within a tight-binding framework. The lo-
T =
2+ 2 cal density of states and the transmission coefficient are
A B computed using an RSRG analysis. The major result
with, =[(P12 P21)+(P11 P22)coska]
A − − is that, we are able to find a certain relationship be-
and B =[(P11+P22)sinka] (8) tween the parameters of the Hamiltonian of the system,
whichtriggersabsolutelycontinuoussubbandsintheen-
where, P11 = [(E −ǫ˜∗l)(E −˜ǫ∗r)]/t∗LτL −t∗L/τL, P12 = ergyspectrumofthesystem. Theextendednatureofthe
−(E−ǫ˜∗r)/t∗L, P21 =(E−ǫ˜∗l)/t∗L, P22 =−τL/t∗L are the electronicstatesunderthesuitableresonanceconditionis
matrixelementsofthetransfermatrixfortheeffectivedi- affirmedbyaprominentenhancementinthetransmission
atomic system, and coska=(E ǫL)/2τL, ‘a’ being the coefficient. The number and the position of the contin-
−
latticeconstantoftheleadsandissettounitythroughout uous subbands can be controlled by tuning the number
the calculation. of atomic sites in the side-attached chains. This opens
In Fig. 7 we have presented the transmission charac- up a possibility of devising nanoscale electronic filtering
teristics of a 9-th generation system. As we tune the devices with such systems. We have investigated the ef-
value of λ to √3, which is the resonance value of λ, the fect of side-attachments on a Fibonacci type backbone –
transmittance of the system gets enhanced significantly this study can be carried over to some other quasiperi-
as apparent from Fig. 7(a). This result is at par with odic structures like copper mean, Thue-Morse etc. with
the results discussed in Sec. III and confirms the ex- side-attached periodic chains.
tended character of the corresponding electronic eigen- Before we end, it is to be appreciated that the central
states. The oscillation in the T-E profile is due to the methodpresentedinthis communicationis applicableto
finite dangling chains, where a standing wave pattern is avarietyofbuilding blocksarrangedinacompletely dis-
formed. InFig.7(a)weobserveseveralhigh-transmission ordered fashion. The results indicate a unique delocal-
zonesseparatedbyzero-transmissionzones. Thisfeature ization of electronic state contrary to the canonical case
can be useful in designing small scale novel filtering de- of Anderson localization.
viceswithsuchsystems. Theeffectofdeviationfromthe
resonance condition on the transmission coefficient T is
shown in Fig. 7(b), where we have chosen λ = √3/2,
Acknowledgments
which is the 50% deviated value of λ from its resonance
value. Thetransmissioncharacteristicsunderthiscondi-
tionexhibitahighlyfluctuatingbehaviorindicatingpoor The author is thankful to Prof. Arunava Chakrabarti
conducting nature of a quasiperiodic system in general. forhisilluminatingcommentsandsuggestionsduringthe
Sudden bursts of transmission peaks are due to the for- preparation of the manuscript and acknowledges DST,
mation of isolated extended states as mentioned earlier. India for providing funding through an INSPIRE fellow-
7
ship.
[1] A. E. Miroshnichenko and Y. S. Kivshar, Phys. Rev. E Lett. 50, 1870 (1983).
72, 056611 (2005). [17] J. B. Sokoloff, Phys. Rep.126, 189 (1985).
[2] A.E.Miroshnichenko,S.Flach,andY.S.Kivshar,Rev. [18] M.Kohmoto, B.Sutherland,andC.Tang, Phys.Rev.B
Mod. Phys. 82, 2257 (2010). 35, 1020 (1987).
[3] P. A.Orellana, G. A.Lara, and E. V. Anda,Phys. Rev. [19] E.Maci´aandF.Dom´ınguez-Adame,Electrons,Phonons
B 74, 193315 (2006). and Excitons in Low Dimensional Aperiodic Systems
[4] R. Franco, M. S. Figueira, and E. V. Anda, Phys. Rev. (Editorial Complutense, Madrid, 2000).
B 73, 195305 (2006). [20] G. G. Naumis, Phys. Rev.B 59, 11315 (1999).
[5] A.A.AligiaandL.A.Salguero,Phys.Rev.B70,075307 [21] C.Forestiere,G.Miano,G.Rubinacci,andL.DalNegro,
(2004). Phys. Rev.B 79, 085404 (2009).
[6] K. Kobayashi, H. Aikawa, A. Sano, S. Katsumoto, and [22] C.Forestiere, G.Miano, C.Serpico,M.dAquino,andL.
Y.Iye,Phys. Rev.B 70, 035319 (2004). Dal Negro, Phys.Rev. B 79, 214419 (2009).
[7] P.A.Orellana,F.Dom´ınguez-Adame,I.G´omez,andM. [23] L. Dal Negro and Ning-Ning Feng, Optics Express 15,
L. Ladr´on de Guevara, Phys.Rev. B 67, 085321 (2003). 14396 (2007).
[8] V. Pouthier and C. Girardet, Phys. Rev. B 66, 115322 [24] A. Gopinath, S. V. Boriskina, B. M. Reinhard, L. Dal
(2002). Negro, Optics Express17, 3741 (2009).
[9] Z.-Y.Zhang,S.-J.Xiong,andS.N.Evangelou,J.Phys.: [25] N. E. Kaputkina, Yu E. Lozovik, R. F. Muntyanu, and
Condens. Matter 10, 8049 (1998). YuKhVekilov,J. Phys.: Conf. Ser.226,012028 (2010).
[10] A.Chakrabarti, Phys.Rev. B 74, 205315 (2006). [26] P. W. Anderson, Phys. Rev. 109, 1492 (1958); P. A.
[11] A.Chakrabarti, Phys.Lett. A 366, 507 (2007). Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287
[12] U.Fano, Phys. Rev.124, 1866 (1961). (1985).
[13] F. Haupt, S. Smolka, M. Hanl, W. Wu¨ster, J. Miguel- [27] A.Chakrabarti,S.N.Karmakar,andR.K.Moitra,Phys.
Sanchez, A. Weichselbaum, J. von Delft, and A. Rev. B 39, 9730 (1989).
Imamoglu, Phys. Rev.B 88, 161304(R) (2013). [28] S. Jana and A. Chakrabarti, Phys. Rev. B 77, 155310
[14] N. Zuniga-Hansen, Y.-C. Chi, and M. S. Byrd, Phys. (2008).
Rev.A 86, 042335 (2012). [29] A.D.Stone,J.D.Joannopoulos, andD.J.Chadi,Phys.
[15] J.-X. Zhu, J.-P. Julien, Y. Dubi, and A. V. Balatsky, Rev. B 24, 5583 (1981).
Phys.Rev.Lett. 108, 186401 (2012).
[16] M. Kohmoto, L. P. Kadanoff, and C. Tang, Phys. Rev.