Effect of disorder on the optical properties of colloidal crystals

Rajesh Rengarajan and Daniel Mittleman*Department of Electrical and Computer Engineering, Rice University, MS-366, P.O. Box 1892, Houston, Texas 77251, USA

Christopher Rich and Vicki ColvinDepartment of Chemistry, Rice University, MS-60, P.O. Box 1892, Houston, Texas 77251, USA

sReceived 21 April 2004; published 25 January 2005d

Colloidal crystals offer a promising route for the formation of three-dimensional photonic crystals. Theprimary constraint in working with these materials is the disorder present in these self-assembled materials.Sphere vacancies, line dislocations, and random position errors all lead to a degradation in the optical prop-erties. It is important to characterize these effects so as to guide further developments in colloidal crystaloptics. Here, we report a systematic and quantitative study of disorder in colloidal crystals with visiblediffractive properties. Using optical spectroscopy and digital imaging we have correlated several measures ofstructural disorder with variations in the transmissive and reflective optical properties. We observe a criticalsize distribution above which rapid deterioration of the lowest stop band is observed. Below this critical sizedistribution, we observe excellent optical quality, nearly independent of the size distribution. Single spherevacancies also increase in crystals formed from more polydisperse spheres. The primary effect of this type ofdefect is to increase the broadband diffuse scattering.

DOI: 10.1103/PhysRevE.71.016615 PACS numberssd: 42.70.Qs, 61.72.Dd, 81.16.Dn

I. INTRODUCTION

For many years photonic band gap materials have pro-vided scientists with diffractive systems rich in complexphysics and chemistryf1–3g. These materials exhibit dielec-tric periodicity in one, two or three dimensions. As a result ofthis periodicity, electromagnetic waves of certain frequencieshave no propagating modes within the material. This resultsin a gap in the photonic band structure. Three-dimensionalphotonic crystals are especially valuable for applications thatexploit the complete confinement of light, such as the inhi-bition of spontaneous emissionf1g. Lithography provides anideal tool for forming high quality one and two-dimensionalcrystals for applications such as wave-guiding, wavelengthmultiplexing, cavity quantum electrodynamics, filtering andoptical switchingf4–14g. However, such structures, whichrely on waveguide effects for confinement in the remainingdimensions, can suffer from excessive leakage lossesf15g. Inaddition, lithographic or “top-down” approaches are rela-tively expensive and time-consuming processes; the genera-tion of arbitrary three-dimensional structures is especiallyproblematic for a top-down fabrication strategyf16g.

In contrast, chemical assembly of ordered dielectrics pro-vides a simpler and less costly approach to generating large-scale photonic crystalsf17–26g. These techniques make useof the spontaneous assembly of submicron particles into spe-cific well-ordered lattice structures. These colloidal latticeshave been used as templates to create periodic structures us-ing a wide variety of materials, including semiconductorsf19,27–32g, insulatorsf33,34g, and even metalsf35–39g. Re-cently techniques in substrate patterning and soft lithographyhave been combined with colloidal crystal fabrication. This

has permitted the control of lithographic approaches to becombined with the ease of preparation of the self-assemblymethodologyf40–45g.

In this paper we fabricate colloidal crystals using con-trolled drying, a widely employed method for crystal forma-tion also known as convective self-assemblyf23,26,44,46g.This technique, which has largely supplanted gravity sedi-mentation as the preferred method for colloidal crystal for-mation, produces single-crystal planar films of colloidalcrystals over large areas, with precise control over the filmthickness. Colloids in a liquid suspension are drawn up into asolvent meniscus where they order and pack into a face-centered cubicsfccd array under the influence of surface ten-sion forcesf23g. A vertical glass slide acts as a substrate onwhich this photonic crystal lattice is deposited. As the sol-vent evaporates and the meniscus drops with respect to theglass slide, the colloidal crystal is continuously deposited onthe substrate.

In the controlled drying method, there are several factorsthat are critical to the formation of a uniform, well orderedcrystal. It has been shown that the evaporation rate, the par-ticle flux, the wetting of the substrate and the meniscus prop-erties are important control parametersf47g. Though the dy-namics of the process are not completely understood, it isclear that self-assembly has the potential for inefficient pack-ing leading to imperfect crystal formation.

The promise in using controlled drying methods to fabri-cate photonic crystals lies in the economy and ease associ-ated with the manufacturing process, as well as the diversearray of material systems that can be formed from colloidalcrystal templates. An important challenge for this method isthe reduction in the number of crystal defects generated dur-ing thin film growth. Typical defects may include line de-fects, point defects, drying cracks, stacking faults and ran-dom variations in the sphere positions. These defects perturbthe structural periodicity of the photonic crystal, and can*Corresponding author. Email address: daniel@rice.edu

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therefore alter their optical performance. Several theoreticaltreatments suggest that though the full photonic band gapsbetween the 8th and 9th bandsd is extremely sensitive to anydisruption in the periodicity of the photonic crystal lattice,the lower energy pseudogapsbetween bands 2 and 3d is onlymoderately sensitive to crystal defectsf48,49g. A quantitativeexperimental evaluation of the sensitivity of the optical prop-erties to these defects, however, is still lacking. Such infor-mation is vital to developing colloidal crystals as optical de-vices, as well as for determining the level of perfectionrequired for particular applications.

There have been several studies, both theoreticalf48–51gand experimentalf52–57g, which have investigated specificdefects in gravity-sedimented, shear alignedf25g and dis-persedf58–60g colloidal crystals, and their connection to theoptical properties. As expected, the presence of defects cor-relates with a diminished reflectivity inside the stop band.However, in those cases it was not possible to systematicallyevaluate the role of defects in the optical properties. In par-ticular, crystals grown from gravity sedimentation form withmany grain boundaries, and are in effect polycrystalline. Bycontrast, the thin films described here are single crystals withfew grain boundariesf23g.

Clearly, the origin of many of the defects in colloidalcrystals is the nonuniformity in the size of the spherical col-loids. This fact suggests an interesting method for experi-mentally controlling the quality of the colloidal crystal films.By using colloids with a range of well-characterized sizedistributions, we can fabricate films with a range of crystal-line order. It has been theoretically demonstrated that theband gap, as determined from the density of states, is moresensitive to size randomness in the colloidal particles than tosite randomnessf49g. This suggests that the monodispersityof the colloids is the critical factor in determining the qualityof a photonic crystal. Also, the size dispersion of the colloi-dal particles has a direct effect on the structural quality of thecrystal; a crystal made from colloids having a large size dis-tribution, for example, will not pack into a well-orderedclose-packed lattice. Just as it is hard to stack billiard balls ofuneven size, it is difficult to obtain a perfect close-packedlattice with spheres having a large size distribution. Even onelarge sphere in a uniform lattice can cause local changes inthe positions of neighboring colloidal sitesf46g.

The aim of this work is to evaluate the role of disorder indetermining the quality of a photonic crystal. We describequantitative methods to characterize disorder in colloidalcrystals and correlate the structural quality with the opticalquality. This data permits us to determine how much disordercan be tolerated in a sample before significant departuresfrom the properties of the ideal crystal are observed. Thegoals are to determine what level of structural order can beachieved with near perfect colloids and how this structuralorder degrades when imperfect colloids are used for filmformation. Optical methods provide the ideal tools for moni-toring the quality of the crystals and thus enable a correlationbetween colloidal sphere monodispersity and photonic be-havior to be established.

II. EXPERIMENT

In this work we studied assembled crystals of silica col-loids. Figure 1 shows the photonic band structuref61g for a

face-centered cubic close-packed crystal of silica colloidswith refractive indexn=1.38 f62g. Colloids arranged in aface-centered cubicsfccd lattice possess a pseudogap at ascaled frequencyva/2pc=0.48, wherea is the lattice pa-rameter of the primitive cell, equal to the sphere diameter.This is attributed primarily to scattering from thef111gplanes, oriented parallel to the substrate. Although it is wellknown that silica colloidal crystals have no full photonicband gap, they are nonetheless the starting point from whichhigh index contrast macroporous photonic structuresswhichcan have a full gapd are fabricatedf63–65g. Samples withhigher index contrast are formed using templating tech-niques, which involve intercalating materials into the inter-stitial regions of the close-packed silica sphere arrays, andthen removing the spheres by chemical etchingf19,21,34,66–71g. This permits the fabrication of inversestructures which retain the precise crystalline geometry ofthe starting film. Since templating preserves not only thecrystal structure but also all of the lattice defects of the tem-plate, the structural properties of the template dictate theproperties present in the final architectures. For this reason,we have exclusively studied films consisting of close-packedarrays of silica colloids, which consist of silica spheres sur-rounded by air.

Using Stober’s method, numerous suspensions of silicacolloids were preparedf72g. Colloidal crystals from thesesuspensions were prepared using the controlled drying tech-nique discussed above, described in detail elsewheref23g.The crystals were grown in a plexiglass chamber containinga dehydrating agent in order to keep the humidity levels lows8–12 %d. We have found that colloidal crystals grown fromethanol suspensions are less uniform if there is any watercontamination. A flow of ultrapure nitrogen was also main-tained through the chamber at all times during the growthprocess. The flow rate of the nitrogen was adjusted to finetune the evaporation rate of the ethanol colloidal suspension.All crystals were grown at room temperatures22–23 °Cd.Typical sample growth rates were approximately 150mm/h.

The size and size distribution of the colloids used to makephotonic crystal films were measured using scanning elec-tron microscopysSEMd. The images were acquired from thesame location on the sample from which optical spectra werepreviously acquired. Typically, five SEM images were col-

FIG. 1. Photonic band structuref61g for an fcc crystal of close-packed silica colloids, with refractive indexn=1.38. The lowest 10energy bands are shown.

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lected within a circle of diameter 200 microns. Each imageused for sizing contained approximately 60 colloids. Sincethe colloids touch their nearest neighbors in a crystallinelattice, particle sizing by automated digital analysis softwaresImagePro, Digital Micrographd returned diameters smallerthan the true value by up to 15%. In order to minimize suchsystematic errors these measurements were performed manu-ally, reducing the errors to less than 1%. The average spherediameters of all the suspensions fell in the range of250–350 nm. The sphere size distributions,d, of the suspen-sions varied from 3%smonodispersed to 10%spolydispersed.Here, we define the parameterd as the ratio of the standarddeviation of the sphere diameters to the mean diameter.

The colloidal crystals thus prepared were characterized intwo ways. The first method involved measuring the reflectionand transmission spectra at specific points on the sample,using a microscope-enhanced uv-visible spectrometerf73g.An Olympus BX60 microscope was coupled to an ISA JobinYvon-Spex HR460 monochromator. The monochromator hada liquid nitrogen cooled CCD detector and could scan wave-lengths from 400 nm to 1050 nm. The system was capableof measuring spectra in both reflection and transmissionmodes. Three objectivess203 ,503 ,1003 d and an internalfield stop allowed the spot size to be varied from50 mm to 1100mm on the sample surface. Since the fieldstop is placed in the image plane, the collimation of thesystem in unchanged for any change in the field stop aper-ture. All measurements were performed at normal incidencealong thef111g direction of the fcc lattice. In order to mini-mize the cone angle of collection to under 5°sto avoid de-tecting light scattered from planes other thanf111gd an exter-nal aperture was used to reduce the effective numericalaperture to 0.1. The sample to be measured was placed on astage withx-y translation. The stage had a measurement ac-curacy of 100mm in both axis directions, though we havefound that the repeatability of translation fell within 40mm.The reference spectrum for the reflection and transmissionmeasurements were obtained using an aluminum-sputteredglass slide and a plain glass slide, respectively. Several posi-tions on each sample were selected for measurements. Thecoordinates of the positions were noted for subsequent SEMimaging. At each position the reflection spectrum was ac-quired for spot sizes of 50, 100, 150, 200, 220, 300, 440,500, 1000, and 1100mm and the transmission spectrum wasobtained for spot sizes of 220, 440, and 1100mm.

The second characterization procedure involved structuralanalysis from SEM imaging. Since the coordinates at whichthe optical spectra were taken were known, the same posi-tions could be located on the sample under the SEM towithin 10 mm. Thus, information extracted from structuralanalysis could be directly correlated to the optical spectra. Ateach position on the sample 10 top-down images were cap-tured. From these images sphere size and lattice informationwere extracted.

In order to obtain a more quantitative description of dis-order in the crystal structure, we performed a two-dimensional radial distribution analysis of the SEM images.This involved the calculation of the dimensionless pair cor-relation functionsPCFd, gsrd f74g,

gsrd =1

krldnsr,r + drddasr,r + drd

.

The PCF is a two-dimensional statistical mapping of the dis-tances between the centers of pairs of colloids. It is evaluatedby counting the number of spheresdn that lie within aspherical shell,dr, of radiusr from an arbitrary origin. Thisis repeated for a range of radii and many arbitrary spheresselected as the origin. The statistical average of these num-bers, normalized with respect to the average particle numberdensitykrl and the sampling areada=2prdr for a particularradial distancer, gives the pair correlation function,gsrd.The value ofgsrd is large when the distance,r, happens tocoincide with a particular FCC lattice distance. Just beforeand beyond these distances the probability of encounteringspheres is much smallerssince spheres cannot overlapd.Hence we expect the PCF to consist of periodic peaks anddips. At distances larger than a characteristic correlationlength, the short-range order breaks down. Since the correla-tion is lost, the number of particles within a given samplingvolume is just the average particle density. Thusgsrd con-verges to unity for large values ofr. A highly ordered latticeresults in a PCF with many oscillation. We obtain a quanti-tative measure of the correlation length by extracting the fullwidth at half maximumsFWHMd, k, of the first peak of theFourier transform ofgsrd−1. A small value ofk indicatespersistent oscillations ingsrd, and therefore a well-orderedlattice.

The structural disorder from SEM images of colloidalcrystals could be quantified by obtaining the PCF using theprocedure described previouslyf75,76g. The first step to cal-culating the PCF was to find the center coordinates of all thespheres in a given image, using image analysis software suchas Digital MicrographsGatan Inc.d. Because the results aresensitive to the details of the data processing, we describehere our approach to developing a protocol that most accu-rately determines the sphere coordinates. First the image wasprocessed using a rank minimum filter. This took the pixelsin a defined neighborhood and filtered out all but the bright-est pixel resulting in a flattened image in which the sphereswere separated. The spheres were then isolated using an au-tomated thresholding process built into the software. Imagesused for PCF determination typically contained 2500 sepa-rated spheres. Several images were analyzed for each posi-tion on a particular sample. Figure 2 shows one such imagewith the coordinates of the spheres extracted. The dark filledcircles indicate the position of the spheres as extracted fromthe software. It is evident, from the overlap of the extractedcoordinates and the actual image, that the mapping of coor-dinates to spheres is very good.

Once the center coordinates of the spheres were found,the PCF was calculated, as described above. The outer circlein Fig. 2sad shows the maximum radius for which the PCFwas calculated for this image. This was determined as theshortest distance from the center to the edge of the image.The subset of colloids near the center of the image that arecircled were used as the origin points. The results from all ofthese origin points were averaged to obtain the final PCF.Figure 2sbd shows the calculated PCF from the image in Fig.

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2sad sthick solid lined and also from a perfectly ordered tri-angular crystal lattice. In the case of the perfect lattice, theprobability of finding a sphere at a particular location isknown exactly, for arbitrary distance from the origin. Hence,gsrd is a series of delta functions. The PCF for the actualcolloidal crystal shows a series of broader peaks. However,

the similarity between the idealized calculation and the mea-sured PCF is clear, particularly for smaller values ofr. Thepersistent oscillations in the experimental curve indicate thatthe structural correlation was preserved over a long range.The insetfFig. 2sbdg shows the Fourier transforms of thePCFs for the ideal and real crystals.

III. RESULTS AND DISCUSSION

We controlled the structural disorder in colloidal crystalsby varying the size distribution of the colloidal solutionsused to form the crystals. We characterize structural orderquite broadly here; it is defined as the full width at halfmaximumsFWHMd of the first peak of the Fourier transformof the PCF. Thus this general structural order is best de-scribed as an “in-plane” parameter as it is derived from im-ages of only the top layer of the films. We compute thevalues ofk normalized with respect to the result that wouldbe obtained from a perfect crystal of the same dimensions,imaged with the same field of view. Therefore a value ofk /k0=1 would indicate a colloidal crystal that is essentiallyperfect in its two-dimensional structural order on the lengthscale of the size of the imaged region. Figure 3 shows thisstructural order parameter as a function of the colloidal sizedistributiond.

These data show that colloidal size distribution has apowerful influence on the structural order of a colloidal crys-tal formed from controlled drying. Betweend=3% and 6%the colloidal crystals are near perfect, with ak /k0 of about1.5. Aboved=6%, the in-plane structural disorder increasesroughly linearly until at 9% the structural disorder is 5 timesworse than a perfect crystal. The fact thatk saturates at,1.5k0 for d,6%, and not atk=k0, indicates that this valuereflects the actual degree of order of the samples, and is notmerely an artifact resulting from the finite area sampled bythe SEM images. We conclude that a size distribution lessthan 6% ensures that the crystals have a reasonable degree oftwo-dimensional lattice order. Moreover, decreasing the sizedistribution beyond 6% does not lead to dramatic improve-

FIG. 2. An illustration of the method used to determine the paircorrelation function.sad Top-down SEM image of af111g crystalsurface with the center coordinates of the colloids extractedsdarkfilled circlesd and mapped back onto the image. The outer black ringshows the maximum area of the image used for the calculation. Thiscrystal was fabricated with spheres having a size distributiond=5.5%. The small number of spheres used to compute the paircorrelation function are indicated by the collection of circles at thecenter of the image.sbd The pair correlation function computedfrom the above imagessolid lined plotted against the scaled radiusr / r0, wherer0 is the mean sphere radius. Also shown is the series ofdelta functions which represent the PCF for an infinite perfect hex-agonal lattice. The inset shows the Fourier transforms of these twoPCFs. The thin line, the PCF for the ideal lattice, is computed for alattice of finite size in order to make accurate comparisons with thePCFs extracted from finite SEM images. We usek, the width of thefirst peak of this Fourier transform, as a measure of the degree ofstructural order shown in the SEM image.

FIG. 3. Variation of two-dimensional order with colloidal sizedistribution. k /k0 is the structural order parameter defined as thefull width at half maximum of the first peak of the Fourier trans-form of the pair correlation function, normalized as described in thetext. d is the colloidal size distribution.

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ments in the in-plane structural order. As we show below,decreasingd below 6% does lead to a slight decrease in themean defect density, an effect too small to be evident inFig. 3.

In addition to the in-plane structural disorder, we alsoevaluated whether these materials were disordered in planesperpendicular to thes111d plane. Stacking faults are a com-mon form of disorder in crystalline solids and their presencein gravity sedimented colloidal crystals has been studied byothersf17,48,52,56g. Figure 4 shows cross-sectional views ofa typical colloidal crystal produced through controlled dryingmethods. The size distribution for this particular sample is4%. The lattice was infiltrated with a polymerspolystyrenedto provide mechanical support for cross-sectional analysis.Though there are some spheres missing due to the cleavingprocess in sample preparation, the cross sections along thek100l andk110l lattice planes are visible. These images, typi-cal of many crystals that we have studied, clearly demon-strate that crystals formed using the controlled dryingmethod exhibit a negligible density of stacking faults, andconfirm that the lattice structure is face-centered cubicsfccd.This is to be distinguished from colloidal crystals formed viaother methods, which may produce a higher density of stack-ing faults f58g. Moreover, the crystal density in our films is

uniform throughout the entire depth of the crystal and miss-ing spheres are no more prevalent closer to the substrate thanon the surface. These results are consistent with earlier dis-cussions of crystals grown using a modification of this tech-niquef26g. For these reasons, we expect that the influence ofstacking faults on the optical spectra is small, compared tothat of other types of defects. So, the remainder of this articlefocuses on disorder arising from in-plane defects.

Several specific types of crystal defects contribute to thein-plane structural disorder as parametrized byk /k0. Lineand point defects are the most visually apparent features incolloidal crystal images. Fractional displacement of an entiresection of the crystal is a signature of a line defect, whereaspoint defects are single sphere vacancies in an otherwise per-fect crystal. Using the top-down SEM views, we have deter-mined the average density of both point and line defects.These results are shown in Fig. 5, as a function of sphere sizedistribution d, along with representative views of the twotypes of defect. For the most uniform colloidal crystalssd,4%d we found defect densities an order of magnitude lesss,106 cm−2d than those reported in earlier studies of planarcolloidsf53g. It is apparent from the data that line defects aremore abundant than point defects, typically by a factor of 2.We also found that the density of these defects weakly in-

FIG. 4. Cross-sectional SEM images of a colloidal crystal having a size distribution of 4%. The crystal showsf110g, f100g, andf111gfacets revealing it to be face-centered cubicsfccd. The insets corresponding to the images are cleaved images of a model fcc crystal. Thesecross-sectional views show no evidence of stacking faults.

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creases as the size distribution parameter increases. Thisweak trend is too small to have a significant impact on theresults of the PCF analysisssee Fig. 3d, and is therefore alsonot likely to have a strong impact on the optical properties ofthe films near the first stop band. This is confirmed by ouroptical studies, discussed further below.

Another source of disorder affecting the optical propertiesof colloidal crystals are cracks that develop during drying ofthe films. After the colloids are deposited on the glass sub-strate during the growth phase, drying cracks develop as thesolvent evaporates. We expect that the size and number ofthese cracks should be independent of the colloid size distri-bution, since they are formed after the deposition process.The cracks, unlike grain boundaries, do not destroy the over-all order of the crystal; they only introduce small gaps in anotherwise uniform single crystal. In order to estimate thesignificance of cracks, we digitally processed the SEM im-ages to extract the surface area coverage of the cracks in agiven image. It should be noted that the very process ofimaging these crystals in an ultrahigh vacuum causes thecracks to expand. Hence the data presented here representupper limits of crack coveragesi.e., the worst cased. Theinset in Fig. 5 shows the percentage of crack coverage perunit area as a function of colloidal size distribution. The plotshows no discernable trend with size distribution, as ex-pected. On average only about 2.7% of the image area werecracks. These were generally spaced by 10 to 50 microns.

Disorder in colloidal crystals leads to many distinctivechanges in the optical properties. This is apparent in Fig. 6,

which displays the reflection spectra as a function of illumi-nation spot size for two crystals with very different degreesof order. The figure shows superimposed reflection spectrafor two crystals, formed with spheres havingd=5.7% sup-perd and d=9.0% slowerd, respectively. The spectra weremeasured at the same central position on each sample, andspot sizes were varied from 50mm to 1 mm. The moremonodisperse samples5.7%d shows spectra that are identicaland independent of spot size. The more polydisperse film, onthe other hand, exhibits more evidence of structural disorder.As can be seen from the variability of the spectra, the opticalspot size has a drastic effect on the appearance of the spec-trum in this case. In contrast to the case of thed=5.7%sample, the peak reflectivity changes significantly over therange probed in the experiments,13%d. Also, the peak re-flectivity for smaller spot sizess100 and 150mmd is largerthan when measured with larger spot sizes, suggesting thatthe domain size is less than 150mm. It is clear that a pho-tonic crystal formed with polydisperse colloids has poor op-tical uniformity even in regions as small as 150mm.

Having obtained optical spectra from specific regions ofthe crystal that have also been characterized for structuraldisorder, it is now possible to correlate the optical propertiesand the structural uniformity of the colloidal crystals as afunction of the size distribution of the colloids. Figure 7shows reflection spectra, SEM images, and PCFs for fourtypical samples with different size distributions. The hori-zontal axes of the reflection spectra have been shifted so asto facilitate comparison of samples with varying radii. Onthis normalized frequency axis, the lowest stop band lies at

FIG. 5. Estimation of in-plane defect densities as a function ofcolloidal size distribution. The SEM images show examples of lineand point defects. The defect densities for the linesopen squaredand point ssolid squared defects are fitted to a linear regression.These densities are about an order of magnitude less than thosepreviously reported for colloidal crystalsf53g. The inset analyzescrack densities from drying processes and plots the percentage ofthe surface area of the crystal occupied by cracks. As expected,there is no discernable trend with size distribution. The averagecrack coverage is approximately 2.7±0.6% per unit area.

FIG. 6. Normal incidence reflectivity spectra as a function ofspot size.sad Spectra from a crystal withd=5.7%. The differentspectra overlap with minimal fluctuation in the peak reflectivitysavariation in the peak amplitude of less than 3%d. sbd Spectrum froma crystal with d=9.0%. These spectra show large variations inshape, with significant fluctuations in the peak reflectivitysvariationgreater than 13%d, as well as substantial variations in crystal thick-ness. The legend applies to bothsad and sbd.

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2r0/l=0.48. There is a dramatic change in the reflectionspectrum as the size distribution parameterd increases. How-ever, size distribution has a smaller effect on the appearanceof the materials in the SEM images; the top and bottomimages, for example, do not seem so drastically different oncasual inspection, as compared to the reflection spectra.However, when the PCF is extracted from these images, thesubtle but important differences in structural order are moreobvious. In particular, the number of oscillations in the PCFdecreases asd increases, clearly indicating a deterioration ofthe in-plane structural order.

A more detailed analysis of the optical properties showsthat the two features of the optical reflectivity that are mostsensitive to disorder are the peak reflectivity and the fullwidth at half maximumsFWHMd of the stop band. Figure8sad shows the peak reflectivity of the stop band as a functionof the sphere size distribution, for constant optical spot size.It is known that the peak of the reflection spectrum increasesexponentially with increasing film thicknessf53g, as long as

the film is thicker than a critical thickness ofNc<13 layersof colloids f62g. Because of this strong thickness depen-dence, comparisons between samples of different thicknessare difficult unless the results are first normalized in someway. Since all of our samples are thicker than the criticalthickness, we can correct our measured peak reflectivities tofacilitate such comparisons. In Fig. 8sad, we have normalizedall of our measured data using this exponential scaling, toshow what each of their reflectivities would be if the films allhad a thickness of 30 layers. For reference, an ideal crystalwith 30 layers should reflect about 80% of the incident lightat the center of the stop band. We observe that the bestsamplessthose with the smallest values ofdd indeed ap-proach this ideal value. Further, it is evident that there is acritical size distribution,d=6%, beyond which the reflectiv-ity at the center of the stop band drops rapidly. For the mostpolydisperse crystals the peak reflectivity falls to less than10%. For smaller size distributions the peak reflectivity isindependent ofd, within our experimental uncertainty.

FIG. 7. Qualitative correlation of the optical spectrum and structural disorder with increased polydispersity of the colloids. Each rowrepresents data from one particular spot on a crystal. The first column shows the reflectivity spectrum. The horizontal axis is normalized withrespect to the sphere diameter, 2r0. In these frequency units, the stop band occurs at 0.48. The second column is the corresponding top-downSEM image. The images are magnified for clear viewing. The last column represents the pair correlation function of the corresponding SEMimage plotted against a normalized radial distance. From top to bottom, the size distributions ared=4.9%, 5.7%, 6.5%, and 7.4%.

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Figure 8sbd shows the fractional bandwidthsDl /ld as afunction of the sphere size distribution. Because the frac-tional bandgap is only dependent on the crystal structuresitis invariant for a given dielectric contrast, for fcc colloidalcrystalsd, it is an ideal parameter for comparing the unifor-mity of different samples. Here, a normalization procedure isagain required for meaningful comparisons of samples withdifferent film thickness. In this case, the dependence of theFWHM on the film thickness is not so simple as an expo-nential relationship. Instead, we have used the theoreticallypredicted relation derived from the scalar wave approxima-tion sSWAd, which has been shown to provide a reasonablygood description of the variation of bandwidth with thick-nessf62,77g. Despite the approximate nature of the SWA, weexpect that this normalization procedure should be valid, par-ticularly since the variation of FWHM with thickness is notvery strong for samples thicker thanNc f62g. As above, wenormalize the measured FWHM to the value predicted for acrystal with a thickness of 30 layers, and then divide by thecentral wavelength in order to obtain the normalizedDl /l.The SWA predicts that the spectrum of a perfect 30 layerscrystal should have a fractional bandwidth of 5.8%. Thisvalue is the same as the result obtained from a numericalsimulation based on the transfer matrix methodf78g. Asshown in Fig. 8sbd, the fractional bandwidth is constant atapproximately 5.5% ford,6%, slightly lower than the valueof 5.8% for a perfect crystal. Ford.6% the fractional band-width increases approximately linearly with further increasein disorder. One way to understand this result is to considerthat in a photonic crystal, increasing the level of disorder can

lead to significant changes in the photon density of states.The local density of states, which shows a pseudogap mainlyalong thef111g direction, is a key property in these systems.As a perfect crystal experiences a small increase in disorder,propagating modes may be introduced near the edges of thepartial band gap. This may cause the pseudogap to closeslightly resulting in the observed reduction of the bandwidthfor low levels of disorder, i.e.,d,6%. However, at a certainpoint the number of states inside the partial gap may besufficient to fill the gap. Since the total number of states isconstant, one may expect that the density of states nearsbutnot ind the gap could decrease while the number inside thegap increases. As a result, the dip in the density of states maybecome shallower and broader. This could explain why theobserved bandwidth increases as the degree of disorder in-creases beyondd=6%. A similar effect has been described inrecent theoretical studies of templated photonic crystalsf49g.

There are also a few anomalous but experimentally con-sistent data points withDl /l values in the range of 8–9 %.In assessing the origin of these data points, we have consid-ered whether stacking faults could be increasing the band-width of the partial gap in monodisperse samples. It has beenshown in theoretical investigations that stacking faults in anotherwise perfectly ordered crystal can broaden the full bandgap f48g; however, our samples have a low incidence ofstacking faultsssee Fig. 4d. We also note that this theoreticalwork assumed stacking faults on the order of 20%, which isclearly significantly larger than what we observe, or indeedwhat is seen in crystals grown using any controlled dryingmethodf26g. In addition, even for this large fault density theeffect on the partial gap was much smaller than the observedeffect. It seems unlikely that stacking faults could producethe broadening observed in the bandwidth of these crystals.At present we do not have an explanation as to why theobserved bandwidth in some monodisperse samples is nearlytwice the predicted value for the very uniform crystals.

Although we are not able to confirm that subtle changes inthe midgap density of states are responsible for the variationof the fractional bandwidth in near-perfect crystals, this datadoes suggest that sphere size distribution is an importantparameter to control when generating colloidal crystal tem-plates for photonic applications. For our particular growthconditions, there is a critical transition atd=6% for the fab-rication of high-quality colloidal crystals. This result is re-flected in several independent measurements of both thestructural and optical quality of the materialsfsee, for ex-ample, Fig. 3 and 8sadg. It demonstrates for the first time thatto achieve optimal optical properties, the size distribution ofthe starting colloidal materials must be below a certain criti-cal value. Theoretical studies have indicated that the full gapsbetween the 8th and 9th bandsd is more sensitive to disorderthan is the lower gap investigated heref48,49g, so it is likelythat even smaller values ofd are required for studies whichrely on this higher gap. We note that starting with monodis-perse colloids is a necessary condition for growth of high-quality films, but of course it is also necessary to carefullycontrol the growth conditions, as described earlier. Whilemuch of the recent work on colloidal crystals has focusedspecifically on the optical properties near the stop band, anequally important parameter to characterize is the diffuse

FIG. 8. Quantitative trends in stop band parameters as a functionof colloidal size distribution.sad Peak reflectivity, normalized to 30layers as described in the text. A perfect lattice has a peak reflec-tivity of 0.8 for a thickness of 30 layers.sbd FWHM of the reflec-tion peak, normalized to 30 layers. A perfect lattice has a fractionalbandwidth of 5.8%, as predicted by both scalar wave and transfermatrix theories.

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broadband scatteringf79g. In particular, the majority of pub-lished spectra have exhibited a diffuse background, increas-ing with decreasing wavelength as roughly a power law. Of-ten this spectral feature is simply fitted to a power law, andsubtracted from the experimental data prior to further pro-cessing. This broadband feature has generally been inter-preted as scattering from random defects within the other-wise ordered crystalf80g, but there has been no systematicstudy of its variation with crystalline order.

We have evaluated the scattering spectrum as a functionof the degree of disorder, in order to better understand itsorigins. The spectrum of the scattered radiation is obtainedfrom Ssld=1−Rsld−Tsld, whereRsld andTsld are the re-flection and transmission coefficients, respectively, measuredat the same location on the sample. Figure 9 shows a repre-sentative scattering spectrum of a colloidal crystal samplewith d=8.2%, plotted on a log scale. The horizontal axis isthe normalized frequency 2r0/l, as in Fig. 7, also plotted ona log scale.

We expect that the nature of the scattering profile canprovide information about the size of the defects responsiblefor diffuse scattering. From Fig. 9 it is apparent that forwavelengths longer than the Bragg wavelength, the scatter-ing approximately follows Rayleigh’s law. The Rayleighscattering regime is generally assumed to apply only to verysmall scatterers, whenkd,0.1, whered is the diameter ofthe scattererf81g. This implies that in colloidal crystals, thescattering site is less than 20 nm in size. The imperfectionsin this size range include the variations in the sphere diam-eterssand the sizes of the interstitial regionsd, surface rough-ness and the random variations in the sphere positions. Onthe short wavelength side of the stop band, the scattering is

evidently not well described by a simple power law. It ispossible that the contributions to diffuse scattering in thisregime include effects from larger scatterers in the sample,e.g., drying cracks or line defects.

We have attempted to describe the long-wavelength scat-tering for all of our samples using a power law of the forml−n, wheren is the scattering powersn=4 for Rayleigh scat-teringd. The inset to Fig. 9 shows the extracted value of thisexponent as a function of size distribution. Although there isconsiderable variation in the results, the general trend is thatn,4 for smaller size distributions, andn.4 for larger sizedistributions. This is consistent with a simple Mie theorysimulation, which predicts that point vacancies result in ascattering power ofn,4. These results suggest that, forcrystals with a greater degree of disordersd.5–6 %d theprimary source of scattering is not from sphere vacancies orstacking faults, but from disorder resulting from the finitesphere size distributions. This would incorporate both sizeand site randomness. However for smalld valuess,5%d,the disorder due to site and size randomness of the spheres isminimized due to the monodispersity of the colloids. Here,the primary source of scattering is point defects, i.e., struc-tures similar in size to the spheres such as vacancies.

IV. CONCLUSIONS

In conclusion, we have carried out an in-depth study onthe role of sphere size disorder on the optical properties ofcolloidal crystals. We have estimated the density of both lineand point defects for samples grown using the controlleddrying method, and obtained values substantially lower thanthose previously reported for other types of colloidal crys-tals. We have shown quantitatively how the the sphere sizedistribution influences both the optical and structural proper-ties of colloidal films; in particular, to generate ordered crys-tals requires a critical value in size distributionsd,6%dbeyond which the lowest frequency stop band deterioratesrapidly. Also for very monodisperse colloids the main sourceof scattering is from point defects, which suggests that inwell ordered crystals stacking faults and sphere size random-ness play secondary roles insofar as their effect on the opticalspectrum. We also report an anomalous behavior in the varia-tion of the width of the optical stop band withd, which mayresult from defect-induced variations in the photon density ofstates near the band edges. These data improve our under-standing of the connection between structural and opticalproperties in colloidal photonic crystals, and suggest meth-ods to improve their fabrication.

ACKNOWLEDGMENTS

The authors would like to acknowledge Silvia Rubio forher contributions during the early phase of this work. Theauthors would also like to thank William Bosworth for hisassistance with the measurements of the colloid size distri-butions. This work was supported in part by the NationalScience FoundationsECS-0103174d and by the R. A. WelchFoundationsC-1538d.

FIG. 9. A plot of the diffuse scattering spectrum of a filmformed with colloids havingd=8.2%. The scattering spectrum,plotted on a log10 scale, is defined asSsld=1−R−T, whereR andTare the reflectivity and the transmissivity measured at the samelocation on the crystal. The horizontal axis is the normalized fre-quency 2r0/l, as in Fig. 7, also plotted on a log10 scale. The longwavelength region of the spectrum approximately follows Ray-leigh’s law, superimposed on the usual Fabry-Pérot oscillations. Alinear fit to this region illustrates the power law behavior. In con-trast, the short wavelength region is not well described by a powerlaw. The inset shows the power law exponent, extracted from thelong-wavelength regions, as a function of colloidal size distribution.

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f1g E. Yablonovitch, Phys. Rev. Lett.58, 2059s1987d.f2g S. John, Phys. Rev. Lett.58, 2486s1987d.f3g J. D. Joannopoulos, R. D. Meade, and J. N. Winn,Photonic

Crystals: Molding the Flow of LightsPrinceton UniversityPress, Princeton, NJ, 1995d.

f4g T. F. Krauss, R. M. DeLaRue, and S. Brand, NaturesLondond383, 699 s1996d.

f5g O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, Science284, 1819s1999d.

f6g E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D.Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H.Hou, and A. Alleman, NaturesLondond 407, 983 s2000d.

f7g M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg,and M. C. Netti, NaturesLondond 404, 740 s2000d.

f8g C. Weisbuch, H. Benisty, M. Rattier, C. J. M. Smith, and T. F.Krauss, Synth. Met.116, 449 s2001d.

f9g M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, Phys.Rev. B 65, 195306s2002d.

f10g Y. Akahane, T. Asano, B. S. Song, and S. Noda, NaturesLondond 425, 944 s2003d.

f11g D. N. Christodoulides, F. Lederer, and Y. Silberberg, NaturesLondond 424, 817 s2003d.

f12g A. Chutinan, S. John, and O. Toader, Phys. Rev. Lett.90,123901s2003d.

f13g R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F.Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y.Cho, and F. Capasso, Science302, 1374s2003d.

f14g P. Russell, Science299, 358 s2003d.f15g Y. G. Roh, S. Yoon, S. Kim, H. Jeon, S. H. Han, Q. H. Park,

and I. Park, Appl. Phys. Lett.83, 231 s2003d.f16g S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R.

Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz,and J. Bur, NaturesLondond 394, 251 s1998d.

f17g H. Miguez, F. Meseguer, C. Lopez, A. Mifsud, J. S. Moya, andL. Vazquez, Langmuir13, 6009s1997d.

f18g A. van Blaaderen, R. Ruel, and P. Wiltzius, NaturesLondond385, 321 s1997d.

f19g J. Wijnhoven and W. L. Vos, Science281, 802 s1998d.f20g Z. D. Cheng, W. B. Russell, and P. M. Chaikin, Nature

sLondond 401, 893 s1999d.f21g S. A. Johnson, P. J. Ollivier, and T. E. Mallouk, Science283,

963 s1999d.f22g O. D. Velev, A. M. Lenhoff, and E. W. Kaler, Science287,

2240 s2000d.f23g P. Jiang, J. F. Bertone, K. S. Hwang, and V. L. Colvin, Chaos

11, 2132s1999d.f24g S. H. Park, B. Gates, and Y. N. Xia, Adv. Mater.sWeinheim,

Ger.d 11, 462 s1999d.f25g R. M. Amos, J. G. Rarity, P. R. Tapster, T. J. Shepherd, and S.

C. Kitson, Phys. Rev. E61, 2929s2000d.f26g Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, Nature

sLondond 414, 289 s2001d.f27g S. G. Romanov, A. V. Fokin, H. M. Yates, M. E. Pemble, N. P.

Johnson, and R. M. De La Rue, IEE Proc.: Optoelectron.147,138 s2000d.

f28g M. E. Turner, T. J. Trentler, and V. L. Colvin, Adv. Mater.sWeinheim, Ger.d 13, 180 s2001d.

f29g P. Jiang, J. F. Bertone, and V. L. Colvin, Science291, 453s2001d.

f30g Q. B. Meng, C. H. Fu, Y. Einaga, Z. Z. Gu, A. Fujishima, and

O. Sato, Chem. Mater.14, 83 s2002d.f31g F. Meseguer, A. Blanco, H. Miguez, F. Garcia-Santamaria, M.

Ibisate, and C. Lopez, Colloids Surf., A202, 281 s2002d.f32g L. Y. Hao, M. You, X. Mo, W. Q. Jiang, Y. R. Zhu, Y. Zhou, Y.

A. Hu, X. M. Liu, and Z. Y. Chen, Mater. Res. Bull.38, 723s2003d.

f33g B. Gates, Y. D. Yin, and Y. N. Xia, Chem. Mater.11, 2827s1999d.

f34g P. Jiang, K. S. Hwang, D. M. Mittleman, J. F. Bertone, and V.L. Colvin, J. Am. Chem. Soc.121, 11630s1999d.

f35g A. A. Zakhidov, R. H. Baughman, Z. Iqbal, C. X. Cui, I.Khayrullin, S. O. Dantas, I. Marti, and V. G. Ralchenko,Science282, 897 s1998d.

f36g P. Jiang, J. Cizeron, J. F. Bertone, and V. L. Colvin, J. Am.Chem. Soc.121, 7957s1999d.

f37g O. D. Velev, P. M. Tessier, A. M. Lenhoff, and E. W. Kaler,NaturesLondond 401, 548 s1999d.

f38g H. W. Yan, C. F. Blanford, B. T. Holland, M. Parent, W. H.Smyrl, and A. Stein, Adv. Mater.sWeinheim, Ger.d 11, 1003s1999d.

f39g K. M. Kulinowski, P. Jiang, H. Vaswani, and V. L. Colvin,Adv. Mater. sWeinheim, Ger.d 12, 833 s2000d.

f40g J. Aizenberg, P. V. Braun, and P. Wiltzius, Phys. Rev. Lett.84,2997 s2000d.

f41g K. M. Chen, X. P. Jiang, L. C. Kimerling, and P. T. Hammond,Langmuir 16, 7825s2000d.

f42g Y. H. Ye, S. Badilescu, V. V. Truong, P. Rochon, and A. Na-tansohn, Appl. Phys. Lett.79, 872 s2001d.

f43g E. Kumacheva, R. K. Golding, M. Allard, and E. H. Sargent,Adv. Mater. sWeinheim, Ger.d 14, 221 s2002d.

f44g H. Miguez, S. M. Yang, and G. A. Ozin, Appl. Phys. Lett.81,2493 s2002d.

f45g Y. Yin, Z. Y. Li, and Y. Xia, Langmuir 19, 622 s2003d.f46g S. Wong, V. Kitaev, and G. A. Ozin, J. Am. Chem. Soc.125,

15589s2003d.f47g A. S. Dimitrov and K. Nagayama, Langmuir12, 1303s1996d.f48g V. Yannopapas, N. Stefanou, and A. Modinos, Phys. Rev. Lett.

86, 4811s2001d.f49g Z. Y. Li and Z. Q. Zhang, Phys. Rev. B62, 1516s2000d.f50g P. N. Pusey, W. Vanmegen, P. Bartlett, B. J. Ackerson, J. G.

Rarity, and S. M. Underwood, Phys. Rev. Lett.63, 2753s1989d.

f51g M. M. Sigalas, C. M. Soukoulis, C. T. Chan, R. Biswas, and K.M. Ho, Phys. Rev. B59, 12767s1999d.

f52g E. A. Monroe, D. B. Sass, and S. H. Cole, Acta Crystallogr.,Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr.25, 578s1969d.

f53g Y. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyan-skii, O. Z. Karimov, and M. F. Limonov, Phys. Rev. E61,5784 s2000d.

f54g J. Huang, N. Eradat, M. E. Raikh, Z. V. Vardeny, A. A. Zakhi-dov, and R. H. Baughman, Phys. Rev. Lett.86, 4815s2001d.

f55g V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M.Whittaker, and P. N. Pusey, Phys. Rev. B66, 165215s2002d.

f56g J. P. Hoogenboom, D. Derks, P. Vergeer, and A. van Blaaderen,J. Chem. Phys.117, 11320s2002d.

f57g J. F. Lopez and W. L. Vos, Phys. Rev. E66, 036616s2002d.f58g S. A. Asher, J. M. Weissman, A. Tikhonov, R. D. Coalson, and

R. Kesavamoorthy, Phys. Rev. E69, 066619s2004d.f59g P. A. Rundquist, P. Photinos, S. Jagannathan, and S. A. Asher,

RENGARAJAN et al. PHYSICAL REVIEW E 71, 016615s2005d

016615-10

J. Chem. Phys.91, 4932s1989d.f60g P. A. Rundquist, R. Kesavamoorthy, S. Jagannathan, and S. A.

Asher, J. Chem. Phys.95, 1249s1991d.f61g S. Johnson and J. Joannopoulous, http://ab-initio.mit.edu/mpb,

1999.f62g J. F. Bertone, P. Jiang, K. S. Hwang, D. M. Mittleman, and V.

L. Colvin, Phys. Rev. Lett.83, 300 s1999d.f63g H. S. Sozuer, J. W. Haus, and R. Inguva, Phys. Rev. B45,

13962s1992d.f64g K. Busch and S. John, Phys. Rev. E58, 3896s1998d.f65g A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S.

W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia,G. A. Ozin, O. Toader, and H. M. van Driel, NaturesLondond405, 437 s2000d.

f66g S. H. Park and Y. N. Xia, Chem. Mater.10, 1745s1998d.f67g O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, Chem.

Mater. 10, 3597s1998d.f68g P. D. Yang, T. Deng, D. Y. Zhao, P. Y. Feng, D. Pine, B. F.

Chmelka, G. M. Whitesides, and G. D. Stucky, Science282,2244 s1998d.

f69g B. T. Holland, C. F. Blanford, T. Do, and A. Stein, Chem.Mater. 11, 795 s1999d.

f70g G. Subramanian, V. N. Manoharan, J. D. Thorne, and D. J.Pine, Adv. Mater.sWeinheim, Ger.d 11, 1261s1999d.

f71g O. D. Velev and A. M. Lenhoff, Curr. Opin. Colloid InterfaceSci. 5, 56 s2000d.

f72g W. Stober, A. Fink, and E. Bohn, J. Colloid Interface Sci.26,62 s1968d.

f73g Y. A. Vlasov, M. Deutsch, and D. J. Norris, Appl. Phys. Lett.76, 1627s2000d.

f74g S. M. Allen and E. L. Thomas,The Structure of MaterialssWiley, New York, 1999d.

f75g J. Ballato, J. DiMaio, E. Gulliver, D. G. Han, R. E. Riman, andD. L. Carroll, Opt. Mater.sAmsterdam, Neth.d 20, 51 s2002d.

f76g J. Ballato, J. DiMaio, T. D. Taylor, E. Gulliver, D. G. Han, R.E. Riman, and D. L. Carroll, J. Am. Ceram. Soc.85, 1366s2002d.

f77g D. M. Mittleman, J. F. Bertone, P. Jiang, K. S. Hwang, and V.L. Colvin, J. Chem. Phys.111, 345 s1999d.

f78g D. Cassagne, A. Reynolds, and C. Jouanin, Opt. QuantumElectron. 32, 923 s2000d.

f79g A. Femius Koenderink and W. L. Vos, Phys. Rev. Lett.91,213902s2003d.

f80g A. Femius Koenderink, M. Megens, G. van Soest, W. L. Vos,and A. Lagendijk, Phys. Lett. A268, 104 s2000d.

f81g A. Ishimaru,Wave Propagation and Scattering in Random Me-dia sAcademic, New York, 1978d.

EFFECT OF DISORDER ON THE OPTICAL… PHYSICAL REVIEW E 71, 016615s2005d

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