ISSN 1063-7834, Physics of the Solid State, 2007, Vol. 49, No. 4, pp. 708–714. © Pleiades Publishing, Ltd., 2007. Original Russian Text © O.A. Kavtreva, A.V. Ankudinov, A.G. Bazhenova, Yu.A. Kumzerov, M.F. Limonov, K.B. Samusev, A.V. Sel’kin, 2007, published in Fizika Tverdogo Tela, 2007, Vol. 49, No. 4, pp. 674–680.
OPTICAL PROPERTIES
Optical Characterization of Natural and Synthetic Opals by Bragg Reflection Spectroscopy O. A. Kavtreva, A. V. Ankudinov, A. G. Bazhenova, Yu. A. Kumzerov, M. F. Limonov, K. B. Samusev, and A. V. Sel’kin Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail:
[email protected] Received July 18, 2006; in final form, September 15, 2006
Abstract—The Bragg reflection of light from natural and synthetic opals was studied experimentally, and the samples were characterized by atomic-force microscopy. The reflection spectra were theoretically calculated within the model of a planar, periodically layered medium. A comparison of the experimental and calculated data made it possible to determine the parameters of the crystal structure of synthetic opals (lattice constants and sintering coefficients of a-SiO2 particles). It was concluded that the pores in the structure of natural opals are filled by a material with a refractive index close to that of a-SiO2. PACS numbers: 42.70.Qs, 42.25.Fx DOI: 10.1134/S106378340704018X
1. INTRODUCTION Synthetic opals are becoming classical objects for investigating three-dimensional photonic crystals whose photonic band gap lies in the visible range of electromagnetic radiation. Photonic crystals [1–4] are accepted to be structures with a spatially periodic modulation of the permittivity which have band gaps in the spectrum of normal electromagnetic states. A photonic band structure results from Bragg diffraction of electromagnetic waves in a crystal whose characteristic lattice constants are comparable to the light wavelength. The energy positions of the band gaps are determined by the periods of the spatial modulation of the permittivity of photonic crystals, and the band gap depends on the amplitude of this modulation, i.e., on the dielectric contrast [4]. The idea to investigate synthetic opals as photonic crystals was first proposed in [5]. In the decade elapsed since that study was published, many papers dealing with the structural and optical properties of opals and opal-type materials have appeared (see, for example, [6–19]). It should be pointed out that, in analyzing experimental data, those authors considered a model structure of opals to be made up of ideal undeformed aSiO2 spheres linked through point contacts to one another to form an ideal face-centered cubic (fcc) lattice. In such an fcc structure, spheres occupy 74% of the volume of the sample, with the remaining 26% occupied by pores [20]. An analysis of the process employed to fabricate synthetic opals, which includes annealing accompanied by sintering of the a-SiO2 spheres, shows that the commonly accepted model is
nothing more than an approximation which is at variance with reality. This point was stressed by the authors of [21–23], who analyzed the experimental spectra of “opal–semiconductor” photonic crystal composites. A similar situation (although with another mechanism of sintering of the structural elements) also occurs for polymer photonic crystals [19]. In this respect, the main objective of the present study was to demonstrate new approaches to the structural characterization of opals, which are based on spectroscopic measurements of the Bragg reflection of light. The studies were conducted with due regard for the surface profiles of photonic crystals measured with an atomic-force microscope (AFM). A comparison of the experimental data with the reflection spectra calculated in terms of the model of a planar, periodically layered medium [23, 24] made it possible to determine the structural parameters of opal samples. 2. OPAL STRUCTURE AND ITS MODEL REPRESENTATION Natural and synthetic opals consist of a-SiO2 particles. Natural opals are made up of individual disordered photonic crystal domains; in this case, a-SiO2 particles forming different domains can differ in size. It is these structural inhomogeneities that give rise to a remarkably beautiful fine play of spectral colors called opalescence. Unlike natural opals, synthetic opals used as photonic crystals are prized for the homogeneity of their structure, which accounts for the monochromic sample coloring. The particle diameter in different samples can vary in the range 200–800 nm.
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The technology for fabricating synthetic opals includes several stages [25]. First, monodisperse spherical a-SiO2 particles are deposited by gravity from solution. The structure thus formed consists of closely packed hexagonal layers. The sediment is subsequently calcined at a temperature of about 900°C. This procedure brings about sintering and deformation of the aSiO2 particles, which changes the particle shape and the ratio of the volumes occupied by a-SiO2 particles and pores in the opal crystal structure. Thus, the real structure of synthetic opals can differ substantially from the model assuming close packing of ideal spheres in an fcc lattice. In the present study, we took into account this difference in the analysis of the reflection spectra. Our consideration was based on the model of an opal-like photonic crystal, which allows for the sintering effects and possible modifications of the particle shape from spherical to spheroidal. The highly ordered hexagonal close-packed layers of a-SiO2 particles in synthetic opals are arranged perpendicular to the growth axis. Three-dimensional close packing allows three different stacking arrangements of such layers, denoted customarily by A, B, and C [20]. The periodic alternation of layers consisting of undeformed a-SiO2 spheres in the sequence ABCABC ... corresponds to the fcc lattice. The proposed model (Fig. 1) takes into account the sintering of spherical a-SiO2 particles in the course of preparation. During the sintering, the particles sweat and fuse with one another at points of contact. The deformation distorts the shape of particles from spherical to spheroidal. Thus, synthetic opals can be considered a structure made up of sintered spheroids with the rotation axis D|| directed along the [111] axis of the original fcc lattice. The average distance a00 between the centers of aSiO2 particles lying in the most ordered (111) growth planes of synthetic opals was determined with an NT MDT Smena P47H atomic-force microscope. The distance obtained for the sample under study was a00 = 280 ± 15 nm. 3. REFLECTION SPECTROSCOPY TECHNIQUE The spectra of Bragg reflection from samples of natural and synthetic opal were measured on a setup based on an MDR-23 spectrometer with a reciprocal linear dispersion of 13 Å/mm and an operating spectral range of 400–850 nm. The white light emitted by an incandescent lamp passed through a collimator consisting of an aperture and a lens, which provided a beam divergence angle of no greater than 5°. The beam was focused onto the sample, which was fixed to a goniometer. The fixation ensured high precision positioning of a crystal. Indeed, the desired angle of the incidence of light θ was set to within ±1°. The light reflected from the (111) growth surface of the sample was focused by lenses onto a spectrometer entrance slit 100 µm wide. PHYSICS OF THE SOLID STATE
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[111] Z a00
L
Fig. 1. Fragment of a photonic crystal consisting of sintered spheroids. The axes of rotation of spheroids are oriented along the [111] direction (Z) of the undeformed fcc lattice (L is the interplanar separation along the Z axis).
The spectra were measured in s polarization, which was isolated by a film polarizer and a film analyzer. 4. RESULTS OF THE SPECTROMETRIC EXPERIMENTS We studied the reflection spectra of samples of natural opal, synthetic opal without filling (i.e., with air in the pores), and synthetic opal with filling. Distilled water and isopropyl alcohol were used as fillers. The spectra of opal filled with water were measured under two different conditions, namely, with the waterimpregnated sample kept in air or in a flask with water. Figure 2 presents the spectra of light reflection from the (111) growth surface of unfilled synthetic opal and the spectra of the same sample but impregnated with water (and kept in air). The experimental spectra measured at different angles of incidence θ are presented by points, and the solid lines correspond to the theoretical calculations described below. As the angle θ increases, the reflection band shifts toward the short-wavelength range of the spectrum. As the Bragg wavelength λ, we accepted the spectral position of the maximum in the reflection profile. It can be seen from Fig. 2 that, as the angle θ increases from 20° to 40°, the maximum of the spectrum obtained with unfilled opal shifts from ~520 to ~470 nm. For the same sample impregnated with water, the maximum shifts from ~570 to ~540 nm as the angle θ increases from 10° to 30°. Figure 3 displays the reflection spectra obtained for different samples (natural opal and synthetic opal without a filler, as well as those filled with water and isopropyl alcohol) at a fixed angle of incidence θ = 30°. We readily see that the reflection bands of synthetic opal with fillers are substantially narrower than that obtained with unfilled opal. The full width at half-maximum (FWHM) of the band is ∆λ ≈ 45 nm for the unfilled opal and ∆λ ≈ 20 nm for the opal with filling. The narrowing of the reflection band should be assigned to the decrease in the permittivity contrast εa /εb, where εa and εb are the dielectric constants of aSiO2 and the filler, respectively. Indeed, we have the
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Theory Experiment
Air
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0.8 0.6 0.4 0.2
0.2 0 0 (b) 0.8
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500 550 Wavelength, nm
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Fig. 3. Spectra of light reflection from natural opal without a filler (solid line), synthetic opal without a filler (dashed line), and synthetic opal impregnated with distilled water (dotted line) and isopropyl alcohol (dot-dashed line). Angle of incidence θ = 30°.
Water
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ral opal consisting of a-SiO2 particles of different sizes; so, the light beam is reflected from two closely lying, differently “colored” domains of the sample.
0.4 0.2 0
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Fig. 2. Experimental and calculated spectra of light reflection from the (111) growth surface of a sample of synthetic opal at different angles of incidence θ: (a) opal without a filler and (b) opal impregnated with distilled water and kept in air.
dielectric contrast εa /εb = 1.97 for unfilled opal, εa /εb = 1.11 for the opal filled with distilled water, and εa /εb = 1.04 for the opal with isopropyl alcohol. The reflection profile obtained with natural opal is even narrower, i.e., ∆λ ≈ 7 nm, which can be attributed to the very weak dielectric contrast. It is known [25] that pores between a-SiO2 particles in the structure of natural opal are usually filled with amorphous SiOx (x < 2) and that the contrast in this system is close to unity. Thus, the analysis of the spectra presented in Fig. 3 confirmed the conclusion that the decrease in the dielectric contrast leads to a decrease in the photonic band gap associated with the width of the Bragg reflection profile [4, 23]. Therefore, the natural opal characterized by the smallest dielectric contrast exhibits the narrowest profile. We can also see that the reflection profile of natural opal is inhomogeneous. Indeed, an additional maximum is present in the short-wavelength part of the spectrum. This maximum is due to the natu-
5. STRUCTURAL AND OPTICAL PARAMETERS OF OPALS The parameters of the synthetic opals were calculated within a model that considers a spatially periodic structure made up of closely packed ellipsoids of revolution (spheroids) with the axes D|| and D⊥. The D|| axis, which is parallel to the [111] axis of the original fcc lattice, is the axis of revolution and is oriented perpendicular to the growth surface of the structure. If the adjacent spheroids are in point contact, the interplanar distance L (Fig. 1) in the structure along the surface normal is L = d111 = 2/3 D||. The sintering process will be considered as the interpenetration of spheroids, as a result of which the distance a00 between the spheroid centers in the (111) plane decreases by ∆a and the value of L, by ∆L. If the relative change in the parameters ∆a and ∆L is characterized by the sintering coefficient χ, the coefficient f0 accounting for the filling of the structure by spheroids, which enters into the expression for the average permittivity of the opal with filler, ε 0 = ε a f 0 + ε b ( 1 – f 0 ),
(1)
can be written for the case of isotropic sintering in the form [24] 2
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540
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Fig. 4. Dependences λB(θ) for synthetic opal with different fillers: (1) unfilled opal in air, (2) water-impregnated opal in air, and (3) opal in water. Symbols are experimental data. Solid lines correspond to the results of fitting.
2 8 --- A ε 0 – ε v sin θ , 3
(3)
where λB is the Bragg wavelength, A ≡ a00η, η is the ellipticity of the a-SiO2 particles, εv is the permittivity of the medium into which the sample was submerged, and θ is the angle of incidence of light on the [111] growth surface of the sample. The experimental data obtained were used to construct the dependences of the Bragg wavelength λB on the angle of incidence of light θ for a sample of synthetic opal with different fillers (Fig. 4). The data prePHYSICS OF THE SOLID STATE
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Fig. 5. Dependence of the permittivity εa of a-SiO2 particles on the sintering coefficient χ for an opal sample without a filler (solid line) and an opal sample impregnated with water (dashed line). The vertical arrow indicates the intersection point of the curves at which the values of εa and χ are determined.
where f00 = π/3 2 ≈ 0.74. In the absence of sintering, i.e., for χ = 0, we have f0 = f00. This result does not depend on the ratio of the ellipsoid axes and, in particular, is valid for a cubic lattice of closely packed spheres when D|| = D⊥. The spheroidal pattern of the structural elements of the lattice leads to a lowering of the symmetry of the structure from Oh to D3d. This lowering of the symmetry corresponds to a uniaxial strain of the cubic lattice along the [111] direction, which, in our case, coincides with the normal to the opal growth surface. This strain can be described qualitatively by the coefficient of uniaxial compression η = L/Lc, where Lc ≡ a 00 2/3 is the interplanar distance along the [111] direction in the cubic lattice. In the case of isotropic sintering, the uniaxial compression coefficient η within our model coincides with the ellipticity ηell = D|| /D⊥ of the spheroid, η = ηell. The characteristic pattern of the reflection spectra discussed here (Fig. 2) originates from the Bragg diffraction of light from the (111) opal growth planes. The Bragg reflection condition for our case can be presented in the following form: λB =
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sented relate to the opal in air and water-impregnated opal kept in air or in water. The experimental spectra were fitted with relationship (3). The parameters of a sample of synthetic opal (namely, the lattice parameter a00, the average permittivity of the opal ε0, the permittivity εa of a-SiO2 particles, the sintering coefficient χ, and the filling factor f0) were determined in the following order. The approximation of the data for unfilled opal (εv = εb = 1) led to the coefficient A = 252 nm, which was used subsequently to construct the approximating curves for the water-impregnated opal sample. We also determined the average permittivity ε0, which turned air
water
out to be ε 0 ≈ 1.74 for the opal in air and ε 0 ≈ 1.94 for the water-impregnated opal. In order to determine the lattice parameter a00 entering into the coefficient A in relationship (3), we used the available value for synthetic opal, η = 0.93 [23], and obtained a00 ≈ 270 nm, which fits well the value a00 ≈ 280 ± 15 nm derived from the AFM data. Then, relationships (1) and (2) were employed to find the dielectric constant εa of a-SiO2 particles, the sintering coefficient χ, and the filling factor f0. This was done by constructing the theoretical dependences εa(χ) air
water
(Fig. 5) with the use of the values of ε 0 and ε 0 derived previously from the data plotted in Fig. 4. For air εb, we used ε b = 1 for the filler-free synthetic opal and water
εb
= 1.77 for the opal impregnated with water.
Different points on each of the curves in Fig. 5 correspond to reflection profiles of the same Bragg wavelength but for different values of the parameters εa and
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χ. The true values of these parameters for the sample under study are given by the point of crossing of the curves, which is indicated by a vertical arrow. In this way, we established the values of the sintering coefficient χ = 0.011 and the permittivity of a-SiO2 particles, εa = 1.97. For the filling factor for an fcc lattice of ideal undeformed spheres, we obtained f00 = 0.74. Knowing the sintering coefficient χ, we used relationship (2) to calculate the filling coefficient f0 = 0.77. This means that, upon sintering, the volume occupied by a-SiO2 particles in the opal lattice increases by 3%. 6. ANALYSIS OF THE BRAGG REFLECTION SPECTRA In our theoretical analysis of the reflection spectra, we used a model in which the opal lattice is described as a spatially periodic structure with the sites occupied by mutually overlapping (sintered) spheroids of a dielectric material. The spectra were calculated by the transfer matrix technique in the approximation of a planar, periodically layered medium. As was shown in [24], this approximation leads to practically the same result as the more general theory of dynamic diffraction taking into account the essentially three-dimensional pattern of photonic crystals, provided we do not consider special conditions allowing manifestation of the multiwave Bragg diffraction of light [12, 19, 26]. In the approximation of a planar, spatially periodic structure [23, 24], the permittivity of a three-dimensional periodic structure is averaged in the (111) plane perpendicular to the Z axis (along the [111] direction). After the averaging, we obtain the effective permittivity of the medium: ε s ( z ) = ε a f s ( z ) + ε b ( 1 – f s ( z ) ),
(4)
which depends only on one coordinate, z, and can be expressed through the effective filling function fs(z). Within one period of variation, 0 ≤ z ≤ L, this function can be written in the form f s ( z ) = u ( z ) + u ( z – L ),
(5)
where 2
α0 -2 u ( z ) = ------------------------(6) 3(1 – χ) × [ 2π + 6 ( sin ( 2β 0 ) – 2β 0 ) + 3 ( sin ( 2β 1 ) – 2β 1 ) ]. 2
In this expression, α0 = Re 1/4 – ( z/D || ) , βi = 1 2 arcsin ( Re 1 – ( ρ i /α 0 ) ) , (i = 0, 1), ρ0 = --- (1 – χ), ρ1 = 2 a 3ρ 0 – 2 z /D || , D|| = ηD⊥, D⊥ = ------------ , and L = 1–χ D|| 2/3 (1 – χ) = a 00 η 2/3 .
The theoretical model includes the possibility of partially truncating the outer spheroids by the “front” v and “rear” u surfaces. In the calculations, this truncation corresponded to a displacement of the boundary planes by distances ∆lv and ∆lu into the bulk of the photonic crystal. For ∆lv, ∆lu = 0, the boundary planes are tangents to the untruncated outer spheroids. The truncation effects were quantitatively included by introducing the truncation coefficients ζv = 2∆lv /D|| and ζu = 2∆lu /D|| for the front and the rear crystal surface, respectively. By properly varying the values of these coefficients from 0 to 1, we can change the symmetry of the reflection profiles from “dispersion-type” (the reflection R λ > λ B at the long-wavelength edge of the spectrum is larger than that at the short-wavelength edge R λ < λ B ) to “antidispersion-type” ( R λ > λ B is larger than R λ < λ B ). The reflection spectra were calculated with the parameters A, χ, and εa obtained using expressions (1)– (3) and the approximation described in Section 5. The program compiled for the calculations of the spectra involved variable parameters, such as the truncation coefficient for the “front” surface v and the imaginary part ε 0'' of the permittivity ε0. After fitting the theoretical profiles to the experimental spectra, we obtained ζv = 0.5 and ε 0'' = 0.1 for the unfilled opal in air and ζv = 0.3 and ε 0'' = 0.065 for the water-impregnated opal in air. The value of the truncation coefficient ζu for the rear surface of the sample does not affect the reflection profile in our case of thick (~1 mm) photonic crystals, because light virtually does not reach the rear surface of the sufficiently thick photonic crystal plate with due regard for real attenuation. The imaginary part ε 0'' of the permittivity εs(z), which was introduced into the computer program, effectively took into account the absorption and scattering of light in the real crystal structure of the opal. By properly choosing this parameter, one could reach the best fit between the theoretical and experimental spectral profiles. The reflection spectra obtained in the calculations are displayed in Fig. 2. The calculated dependences are seen to fit the experimental curves fairly well. It should be stressed that the theoretical Bragg reflection spectra were constructed within the model of a planar, periodically layered medium and were based on the parameters derived from an analysis of the curves depicted in Fig. 4 for the Bragg law (3). A certain difference in shape between the experimental and theoretical reflection profiles in Fig. 2 should apparently be assigned to some finer effects of inhomogeneous profile broadening traceable to structural imperfection of the samples, which are disregarded in the theoretical model. PHYSICS OF THE SOLID STATE
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A theoretical analysis of the experimental spectrum of Bragg reflection from natural opal (Fig. 3) would also be of interest. Unfortunately, the small sizes and large scatter in the structural parameters of photonic crystal domains in natural opal preclude unambiguous characterization of its structure in terms of the schematic representation chosen for synthetic opal. In particular, during spectral measurements, problems arise regarding the identification of the surface region of the natural sample studied with the use of atomic-force microscopy. Therefore, spectral measurements performed under these conditions can offer only a qualitative conclusion that the degree of spatial modulation of the permittivity in natural opal is very low as compared to that for synthetic opal. 7. CONCLUSIONS We carried out experimental and theoretical investigations of the Bragg reflection of light from synthetic opals with different filling for the purpose of determining their real crystal structure. The spectroscopic results were substantiated by the AFM measurements. The theoretical calculations of the spectra of Bragg reflection from synthetic opals were performed in terms of the model of a periodically layered medium. The crystal lattice of the opal was considered a spatially periodic structure with its sites occupied by mutually overlapping (sintered) spheroids of dielectric material. The calculations of the spectra were conducted by the transfer matrix method. The parameters of the sample of synthetic opal studied were found to be as follows: the lattice parameters a00 ≈ 270 nm (obtained by processing the optical spectra) and a00 = 280 ± 15 nm (from the AFM data); the average permittivity of unfilled opal air
ε 0 = 1.74, the average permittivity of water-impregwater
nated opal ε 0 = 1.94, and the permittivity of a-SiO2 particles εa = 1.97; the sintering coefficient χ = 0.011; and the filling factor f0 = 0.77. Thus, the total volume occupied by a-SiO2 particles in the samples amounted to 77%. Upon sintering during opal fabrication, the volume of pores in the structure decreased by 3% as compared to the theoretical estimate of 26% corresponding to the model of undeformed spheres in point contact in a closely packed fcc lattice. We also studied the spectra of Bragg reflection from natural opals. It was shown that natural opals consist of individual domains formed by a-SiO2 particles. The pores in the crystal structure of these opals are filled by a material with a refractive index close to that of aSiO2. Therefore, the dielectric contrast in natural opals is extremely small, which accounts for the appearance of Bragg reflections from such samples in the form of very narrow reflection bands. PHYSICS OF THE SOLID STATE
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ACKNOWLEDGMENTS We are grateful to M.I. Samoœlovich for providing the samples used in the measurements and to A.A. Kaplyanskiœ for helpful discussions of the results. This study was supported by the Russian Foundation for Basic Research (project nos. 05-02-17809 and 05-02-17776), the Federal Goal-Oriented Research Program (project no. 02.434.11.2009), and the PHOREMOST Foundation (project no. FP6/2003/IST/2-511616). REFERENCES 1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987). 2. S. John, Phys. Rev. Lett. 58, 2486 (1987). 3. J. D. Joannopoulos, R. D. Mead, and J. D. Winn, Photonic Crystals: Molding of Flow of Light (Princeton University Press, Princeton, 1995). 4. K. Busch and S. John, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 58, 3896 (1998). 5. V. N. Astratov, V. N. Bogomolov, A. A. Kaplyanskii, A. V. Prokofiev, L. A. Samoilovich, S. M. Samoilovich, and Yu. A. Vlasov, Nuovo Cimento Soc. Ital. Fis., D 17, 1349 (1995). 6. Yu. A. Vlasov, V. N. Astratov, O. Z. Karimov, A. A. Kaplyanskii, V. N. Bogomolov, and A. V. Prokofiev, Phys. Rev. B: Condens. Matter 55, R13357 (1997). 7. V. N. Bogomolov, S. V. Gaponenko, I. N. Germanenko, A. M. Kapitonov, E. P. Petrov, N. V. Gaponenko, A. V. Prokofiev, A. N. Ponyavina, N. I. Silvanovich, and S. M. Samoilovich, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 55, 7619 (1997). 8. S. G. Romanov, N. P. Johnson, A. V. Fokin, V. Y. Butko, and C. M. Sotomayor Torres, Appl. Phys. Lett. 70, 2091 (1997). 9. H. M’iguez, C. L’opez, F. Meseguer, A. Blanco, L. Vazguez, R. Mayoral, M. Osana, V. Forn’es, and A. Mifsud, Appl. Phys. Lett. 71, 1148 (1997). 10. A. A. Zakhidov, R. H. Baughman, Z. Iqbal, C. Cui, I. Khairulin, S. O. Dantas, J. Marti, and V. G. Ralchenko, Science (Washington) 282, 897 (1998). 11. Yu. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 61, 5784 (2000). 12. H. M. van Driel and W. L. Vos, Phys. Rev. B: Condens. Matter 62, 9872 (2000). 13. Yu. A. Vlasov, X.-Z. Bo, J. C. Sturm, and D. J. Norris, Nature (London) 414, 289 (2001). 14. V. G. Golubev, J. L. Hutchison, V. A. Kosobukin, D. A. Kurdyukov, A. V. Medvedev, A. B. Pevtsov, J. Sloan, and L. M. Sorokin, J. Non-Cryst. Solids 299– 302, 1062 (2002). 15. V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, Phys. Rev. B: Condens. Matter 66, 165215 (2002). 16. A. V. Baryshev, A. A. Kaplyanskii, V. A. Kosobukin, M. F. Limonov, K. B. Samusev, and D. E. Usvyat, Physica E (Amsterdam) 17, 426 (2003).
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Translated by G. Skrebtsov
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