In e play be ween c ys al-size and diso de e ec s in he high-ene gy op ical esponse
o pho onic c ys al slabs
Luis A. Do ado and Rica do A. Depine*
G upo de Elec omagne ismo Aplicado, Depa amen o de Física, Facul ad de Ciencias Exac as y Na u ales, Uni e sidad de Buenos
Ai es, Buenos Ai es C1428EGA, A gen ina
Gab iel Lozano and He nán Míguez
Ins i u o de Ciencia de Ma e iales de Se illa, Consejo Supe io de In es igaciones Cien í icas, Se illa 41092, Spain
共Recei ed 20 June 2007; e ised manusc ip ecei ed 13 Augus 2007; published 5 Decembe 2007兲
Expe imen al e lec ance spec a ha e been ob ained o colloidal c ys als whose wid hs anged om one o
se e al sphe e monolaye s, and hei ea u es in he highe o de band ene gy ange ha e been ep oduced
heo e ically. In o de o i he measu ed da a, op ical ex inc ion has been in oduced in he heo e ical model,
which accoun s o s uc u e impe ec ions and diso de , he main sou ces o losses in an ac ual measu emen .
A complex spec um in he high equency egion is obse ed e en o one o de ed monolaye , being his peak
s uc u e g adually modi ied as mo e laye s a e piled up. This allowed us o iden i y which peaks a e eminis-
cen o he op ical e lec ance ea u es o a single close-packed laye and which a e he esul o building up a
h ee dimensional pe iodici y. A clea co ela ion be ween he amoun o ex inc ion in oduced in he i ing and
he slab wid h has been ound, which demons a es ha wide eal c ys als p oduce less di usely sca e ed
ligh . A he same ime, we ind ha he op ical esponse o hinne c ys als is mo e obus agains he
in oduc ion o ex inc ion han ha o hicke ones, o which he e ec is d ama ic.
DOI: 10.1103/PhysRe B.76.245103 PACS numbe 共s兲: 78.20.Bh, 78.40.⫺q, 42.70.Qs, 41.20.Jb
Imp o emen s in he p ocesses o ab ica ion o sel -
assembled h ee dimensional 共3D兲pho onic c ys als 共PhCs兲,
which a e ma e ials wi h a spa ially pe iodic dielec ic unc-
ion in all h ee dimensions, ha e made possible he obse -
a ion o high quali y op ical spec a in highe o de band
equencies.1–3Also, he appea ance o localized modes in-
side band gaps in he pho onic band s uc u e due o he
addi ion o local de ec s is a well-known e ec , which is
analogous o he doping o semiconduc o s.4Poin , line, o
plane de ec s a e c ea ed wi hin he c ys al by locally adding
o emo ing ma e ial in a con olled manne ,5,6so hey a e
di e en om in insic de ec s, which a e unin en ional dis-
up ions o he spa ial pe iodici y o he dielec ic unc ion
ha a ise du ing he ab ica ion p ocess. Despi e new ab i-
ca ion echniques ha ha e alle ia ed he e ec o diso de
in he so-called low-ene gy ange, whe e he la ice cons an
is less han he wa eleng h o ligh , e y ecen ly, i has been
demons a ed ha ex inc ion due o in insic de ec s in 3D
PhCs s ongly a ec s he shape o he expe imen al spec a
in he high-ene gy ange, whe e he la ice cons an is g ea e
han he wa eleng h.7Also, calcula ions o he pho onic band
s uc u e, pe o med conside ing ex inc ion, ha e shown a
clea co ela ion be ween he beha io o he imagina y pa
o he wa e ec o and he op ical ea u es obse ed in he
spec a. This means ha diso de , while ha ing a small in-
luence on he measu ed spec a in he low-ene gy ange,
ampli ies i s e ec s a highe ene gies and, consequen ly, u -
he imp o emen o he ab ica ion echniques is equi ed o
achie e op ical quali y a hose pho on ene gies.8
In o de o pe o m a alid compa ison be ween he op i-
cal spec a o ini e c ys al slabs and he pho onic band s uc-
u e o an in ini e c ys al, PhC slabs o se e al laye s we e
analyzed in a p e ious wo k; an excellen ag eemen be ween
he expe imen al and he calcula ed op ical esponse a high
ene gies has been ound. He e, we p esen an analysis o he
beha io o he op ical spec a ea u es in he high-ene gy
ange as he wid h o he c ys al slab is g adually inc eased.
We compa e he expe imen al and heo e ical e olu ions o
he specula e lec ance spec a as we inc ease he c ys al
hickness om 1 o 18 sphe e laye s. We ound complex
spec a in he high equency egion e en o he ini ial
monolaye s uc u e,9being his peak s uc u e g adually
modi ied as mo e laye s a e piled up. This allowed us o
iden i y which peaks a e eminiscen o he op ical e lec-
ance ea u es o a single close-packed laye and which a e
he esul o building up a h ee dimensional pe iodici y. A
he same ime, we ind a clea co ela ion be ween he c ys al
size and he amoun o ex inc ion needed o a ain he bes i .
The in e play o c ys al size and diso de e ec s on he high-
ene gy op ical esponse is ho oughly analyzed, inding a
much mo e d ama ic e ec o he o me on he e lec ance
spec a o la ge c ys als.
Pho onic c ys al ilms we e made by e apo a ion induced
sel -assembly10 on glass subs a es o polys y ene sphe e
共IKERLAT, a e age diame e o 700 nm, polydispe si y be-
low 3%, densi y
␳
=1.1 g/cm3, e ac i e index n=1.59兲sus-
pended in wa e wi h pa icle olume ac ion anging om
0.05% o 0.20% and e apo a ed a empe a u es anging
om 30 o 60 °C. Se e al independen measu emen s o he
high-ene gy op ical esponse o simila s uc u es can be
ound in he li e a u e.1,11,12 Thus, he echnique is su i-
cien ly ma u e and eliable o p o ide ep oducible esul s in
di e en labo a o ies. The ypical c ys al g ow h di ec ion is
he 关111兴, and we will ocus on he op ical p ope ies when
ligh impinges in ha pa icula di ec ion since hey a e bes
known.13 Thus, in his case, a monolaye consis s o a ian-
gula la ice o close-packed sphe es and he slab wid h
is inc eased by piling monolaye s up in he sequence
PHYSICAL REVIEW B 76, 245103 共2007兲
1098-0121/2007/76共24兲/245103共4兲©2007 The Ame ican Physical Socie y245103-1
ABCABC¯. As i is usual, he pho on ene gy is exp essed in
educed uni s a/␭, whe e ais he la ice cons an o he
classical cubic cell, whose alue is a=冑2
␾
in an cc close-
packed s uc u e,
␾
being he sphe e diame e and ␭ he
wa eleng h o he inciden ligh .
Re lec ance measu emen s we e pe o med using a Fou-
ie ans o m in a ed spec opho ome e 共BRUKER IFS-
66兲a ached o a mic oscope. A 4⫻objec i e wi h a nume i-
cal ape u e o 0.1 共ligh cone angle o ±5.7°兲was used o
i adia e he la ices and o collec he e lec ed ligh a qua-
sino mal incidence wi h espec o i s su ace. A spa ial il e
was used o selec i ely de ec ligh om squa e spo s o
1000⫻1000
␮
m2.
Simula ed e lec ance spec a we e calcula ed by means o
he ec o Ko inga-Kohn-Ros oke me hod.14,15 To ensu e
con e gence in he high-ene gy ange, we used 41 wo di-
mensional 共2D兲 ecip ocal la ice ec o s in he plane wa e
expansions and sphe ical wa es wi h angula momen um up
o lmax=9. These alues a e he minimum ones ha p o ide
good con e gence in ha ange. Lowe alues o lmax and
less ecip ocal la ice ec o s s ill gi e a co ec esul o
lowe -ene gy bands, bu i is no su icien o p o ide us ul
in o ma ion on he egion we a e in e es ed in. Measu e-
men s we e pe o med wi h he PhC slab deposi ed on a
glass subs a e o e ac i e index 1.53, so his subs a e is
also included in he heo e ical model. The incoming me-
dium is ai 共 e ac i e index=1兲. The sphe es a e embedded
in ai and ha e a complex dielec ic cons an ␧s=2.5+i␧i,
which co esponds o la ex sphe es 共 e ac i e index=冑2.5
⬵1.58兲. An imagina y pa ␧iis added o he dielec ic con-
s an o he sphe es in o de o ake in o accoun all possible
sou ces o losses in ac ual measu emen s. Then, he alue o
␧iis chosen o i he expe imen al da a.
Ene gy losses due o powe dissipa ion a e negligible in
la ex sphe es, so s uc u e impe ec ions and diso de a e e-
sponsible o he expe imen al de ia ion o he spec um
shape wi h espec o he heo e ical op ical esponse wi hou
ex inc ion.7In he high-ene gy ange, he a e age size o
hese s uc u al impe ec ions, measu ed in wa eleng hs, be-
comes mo e impo an han ha in he low-ene gy bands. Fo
his eason, hese in insic de ec s16 cause ligh o be sca -
e ed di usely, emo ing pa o he ene gy om he specu-
la ly e lec ed o ballis ically ansmi ed beam. In his wo k,
we will ocus on he analysis o specula e lec ance esul s,
since hey can be eadily a ained by mic ospec oscopy om
single domains o he c ys alline ilm using a low nume ical
ape u e objec i e, ensu ing he hickness uni o mi y o he
es ed spo and he no mal incidence o he beam. This is a
much mo e complica ed ask o be pe o med in ansmission
mode, since he inciden beam is ypically ocused wi h a
la ge nume ical ape u e objec i e.
In Figs. 1and 2, he measu ed and calcula ed specula
e lec ances 共o R0,0兲can be seen as a unc ion o he la ice
cons an a/␭ o di e en c ys al slab wid hs. Ve y good
ag eemen be ween he expe imen al and heo e ical e lec-
ance spec a has been ob ained by in oducing ex inc ion in
he heo e ical model, as we ha e demons a ed o much
hicke c ys als be o e. Figu e 1shows he expe imen al and
heo e ical specula e lec i i y spec a o PhC slabs o a
ew laye s 共N=numbe o laye s兲. The alue o ␧iin he
calcula ed spec a was chosen o ob ain he bes i . No e ha
all oscilla ions p esen a simila heigh wi h espec o he
backg ound in bo h he expe imen al 共black solid lines兲and
he simula ed 共 ed dashed lines兲spec a, which was ac ually
he c i e ion ollowed o choose he bes i . Di e ences in
he baseline a e due o he ac ha only one in e ace o he
glass subs a e is conside ed o he calcula ions in o de o
a oid he p esence o e y sho equency lobes in he spec-
a. These a ise om he in e e ence be ween beams e-
lec ed a he uppe and lowe aces o he hick subs a e
共1mm兲unde conside a ion. Also, some dispe sion in he
eal pa o he dielec ic cons an was added, which accoun s
o a small e ac i e index inc ease o ma e ials as we mo e
owa d highe ene gies.
In Fig. 1共a兲, wo main peaks o he monolaye 共N=1兲can
FIG. 1. 共Colo online兲Measu ed 共solid lines兲and calcula ed
共dashed lines兲specula e lec ance spec a o glass-suppo ed slabs
o Nlaye s composed o sphe es o dielec ic cons an ␧s=2.5+i␧i
in ai . 共a兲N=1, ␧i=0.30. 共b兲N=2, ␧i=0.25. 共c兲N=3, ␧i=0.15. 共d兲
N=4, ␧i=0.13.
FIG. 2. 共Colo online兲Measu ed 共solid lines兲and calcula ed
共dashed lines兲specula e lec ance spec a o glass-suppo ed slabs
o Nlaye s composed o sphe es o dielec ic cons an ␧s=2.5+i␧i
in ai . 共a兲N=6, ␧i=0.08. 共b兲N=13, ␧i=0.06. 共c兲N=18, ␧i=0.06.
DORADO e al. PHYSICAL REVIEW B 76, 245103 共2007兲
245103-2
be seen: one nea a/␭=0.55 and he o he a a/␭=1.0. These
peaks mo e g adually owa d highe ene gies as he slab
wid h is inc eased, as Figs. 1共a兲–1共c兲show. Also, seconda y
peaks ela ed o he Fab y-Pe o oscilla ions appea be ween
hese main peaks o slabs o N=2, 3, and 4 laye s. Since all
ou measu emen s we e aken a om he edges o he c ys-
al, in which di e en ypes o s acking sequences ha e been
epo ed,17 he ai ness o he i ings we a ain conside ing
only he ABCABC¯sequence indica es ha an cc la ice is
buil as mo e monolaye s a e piled up.
In Fig. 2, he e lec i i y spec a o PhC slabs o N=6,
13, and 18 laye s can be app ecia ed. The main peak in he
low-ene gy band ends o i s inal posi ion, which is a a/␭
=0.61, as can be p edic ed om he pho onic band s uc u e
o he sphe e cc la ice unde conside a ion, because he e is
a pseudoband gap in he ⌫Ldi ec ion cen e ed a his pa -
icula ene gy.7The i s peak in he high-ene gy egion has
mo ed o a/␭=1.07, and a second double degene a ed peak
nea a/␭=1.2 is p esen . I is in e es ing o no e ha his
second peak becomes double degene a ed in slabs wi h a
leas six laye s. In o he wo ds, we mus comple e a leas
wo ABC sequences in he slab in o de o ob ain he peak a
a/␭=1.2. Figu e 2also shows ha his peak is di icul o see
in he measu ed da a o N=6 and 13 laye s, bu Fig. 2共c兲 o
N=18 laye s con i ms i s exis ence. The e is a second peak
s uc u e in he ange o 1.5⬍a/␭⬍1.8, which is composed
o wo peaks o simila ampli udes. They can be obse ed in
Figs. 1共c兲,1共d兲, and 2, so he numbe o laye s needed mus
be Nⱖ3, which co esponds o one o mo e comple e se-
quences o ABC. The g adual appea ance o hese high-
ene gy peaks as he numbe o sphe e laye s is inc eased can
be p edic ed om he pho onic band s uc u e calcula ed
wi h ex inc ion.7Howe e , o a/␭⬎1.63, di ac ed spo s
emo e pa o he ene gy om he specula ly e lec ed
beam. Then, abo e ha ene gy, bo h diso de and di ac ion
con ibu e o he educ ion o in ensi y o he specula ly e-
lec ed ligh .
The a io be ween he powe ca ied by each di ac ed
mode and he inciden powe is gi en by a e lec ion coe i-
cien Rp,q, whe e he pai o in ege s 共p,q兲indica e he o de
o di ac ion, and 共p,q兲=共0,0兲co esponds o he specula ly
e lec ed ligh , he quan i y ha has been measu ed o his
wo k. The onse o di ac ion in ai a a/␭=1.63 can be
p edic ed by equi ing he conse a ion o he angen ial
componen o he wa e ec o s o a 2D iangula la ice,
ha is,
a
␭艌冑2
n冑p2+共2q+p兲2
3,共1兲
whe e nis he e ac i e index o he di ac ion medium
共n=1 o ai 兲. This same exp ession explains he appea -
ance o di ac ed modes only wi hin he glass subs a e
共n=1.53兲and no in he incoming medium a a/␭=1.07 o
he suppo ed la ice.7In Fig. 3, we compa e he calcula ed
spec um o R0,0 and ha o 兺共p,q兲⫽共0,0兲Rp,q共 he sum o he
in ensi ies o all di ac ed modes in he homogeneous inci-
den medium o e ac i e index n=1兲 o he slab composed
o 18 laye s. The o al e lec i i y R=兺共p,q兲Rp,qis also plo ed
in Fig. 3共a兲, whe e he in ensi y educ ion o R0,0 compa ed
o Rcan be app ecia ed o pho on ene gies a/␭⬎1.63.
The impac o diso de in he op ical esponse o he
glass-suppo ed PhC slab can be obse ed in Fig. 4, whe e
he specula e lec ance spec a o N=1 and 5 laye s is
shown o di e en alues o he imagina y dielec ic con-
s an ␧i. The a enua ion o he e lec ance alues compa ed
o he case ␧i=0.0001, which co esponds o a slab wi h al-
mos no diso de , can clea ly be seen in bo h cases. Howe e ,
his e ec is much mo e d ama ic in he case o he i e laye
la ices, whose maximum a a/␭=1.15 is dec eased by a ac-
o abo e 10, while o he monolaye , he co esponding
maximum a a/␭=1.0 is educed by ac o smalle han 2.
This is a heo e ical con i ma ion ha he hinne he c ys als,
FIG. 3. 共Colo online兲共a兲Calcula ed specula 共solid line兲and
o al 共dashed line兲 e lec ance spec a, R0,0 and R, o a glass-
suppo ed slab o 18 laye s and sphe es o dielec ic cons an ␧s
=2.4964+0.06iin ai . 共b兲Re lec ed di ac ed ligh , 兺共p,q兲⫽共0,0兲Rp,q,
spec a. The e ical dashed line indica es he onse o di ac ion.
FIG. 4. 共Colo online兲Calcula ed specula e lec ance spec a
o 共a兲a glass-suppo ed monolaye 共N=1兲and 共b兲a glass-
suppo ed slab o N=5 laye s, bo h wi h sphe es o dielec ic con-
s an ␧s=2.5+i␧iin ai o di e en alues o ␧i.
INTERPLAY BETWEEN CRYSTAL-SIZE AND DISORDER…PHYSICAL REVIEW B 76, 245103 共2007兲
245103-3
he mo e obus i s op ical esponse agains he in oduc ion
o ex inc ion.
In he expe imen al spec a o Figs. 1and 2, we can ap-
p ecia e a g adual inc ease o he e lec ance ampli udes as
he numbe o laye s is inc eased. Also, he imagina y pa ␧i
o he dielec ic cons an o he sphe es dec eases as he
numbe o laye s Nis inc eased in he calcula ed spec a.
This endency is plo ed in Fig. 5, whe e ␧ican be seen as a
unc ion o N. Hence, he e exis s a clea co ela ion be ween
he ex inc ion in oduced and he wid h o he slab, which
indica es ha di usely sca e ed ligh inc eases o hinne
c ys als. In he case o c ys als o a ew laye s, besides he
e ec o diso de , in e ac ions wi h he glass subs a e p o-
duce an addi ional a enua ion o he e lec i i y alues.
The e o e, he inc ease o ␧ias he c ys al wid h is dec eased
means ha mo e di usely sca e ed ligh is p oduced, he
combina ion o diso de and glass-c ys al in e ac ion being
esponsible o his e ec . Ne e heless, o c ys als wi h six
o mo e laye s, ␧ibecomes p ac ically a cons an , as Fig. 5
shows, which means ha any su ace e ec and glass-c ys al
in e ac ion a e negligible, he alue o ␧ibeing a measu e o
he deg ee o diso de in hese cases. Thus, we ound a
me hod o quan i ying diso de in a eal c ys al because we
can use a single pa ame e such as ␧i, which is de e mined by
i ing he heo e ical model o he expe imen al da a, as a
measu e o he s uc u al quali y o he PhC slab. A simila
me hod has al eady been p oposed o he same ype o PhC
ilms bu based on he i ing o he low-ene gy op ical e-
sponse o he la ices.18 In e es ingly, he same end o he
ex inc ion needed o i he expe imen al cu e e sus he
numbe o close-packed laye s in he ilm is ound.
In conclusion, we ha e ealized an in eg al app oach o
he op ical esponse o 3D PhC slabs as a unc ion o he slab
wid h in he highe o de band ene gy ange. By analyzing
he g adual modi ica ion o ha esponse om one o se e al
monolaye s, we could iden i y which peaks a e eminiscen
o he op ical e lec ance ea u es o a single close-packed
laye and which a e he esul o building up a h ee dimen-
sional pe iodici y. Fu he mo e, we ha e ound a clea co e-
la ion be ween he ex inc ion in oduced in ou heo e ical
model and he wid h o he eal slab, which indica es ha
wide c ys al slabs p oduce less di usely sca e ed ligh and
p o ides a me hod o quan i ying diso de in hese cases.
This wo k has been ealized in he amewo k o a join
Spanish-A gen inian coope a ion p ojec CSIC-CONICET
共G an No. 2005AR0070兲. R.A.D. and L.A.D. acknowledge
suppo om Consejo Nacional de In es igaciones Cien í i-
cas y Técnicas 共CONICET兲, Uni e sidad de Buenos Ai es
共UBA兲, and Agencia Nacional de P omoción Cien í ica y
Tecnológica 共ANPCYT-BID 802/OC-AR03-14099兲. H.M. is
g a e ul o inancial suppo om he Ramón A eces Foun-
da ion and he Spanish Minis y o Science and Educa ion
unde G an No. MAT2005-03028.
*[email p o ec ed]
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