Somos uma comunidade de intercâmbio. Por favor, ajude-nos com a subida ** 1 ** um novo documento ou um que queremos baixar:

OU DOWNLOAD IMEDIATAMENTE

Physics Letters A 372 (2008) 4517–4520

Contents lists available at ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

A comparative study of colloidal silica spheres: Photonic crystals versus Bragg’s law K. Liu a,∗ , T.A. Schmedake b , R. Tsu a a b

Department of Electrical and Computer Engineering, and Department of Physics and Optical Science, University of North Carolina at Charlotte, Charlotte, NC 28223, USA Department of Chemistry, University of North Carolina at Charlotte, Charlotte, NC 28223, USA

a r t i c l e

i n f o

Article history: Received 25 March 2008 Accepted 4 April 2008 Available online 9 April 2008 Communicated by V.M. Agranovich PACS: 42.70.Qs 42.25.Bs 42.70.-a

a b s t r a c t A comparative study of colloidal crystals made of silica spheres using Bragg’s law and a model including a photonic band gap (PBG) was demonstrated. Optical properties of the crystals annealed at various temperatures were characterized by a procedure similar to X-ray diffraction technique. Experimentally, the PBG obtained from the transmission spectra agrees with the model of the photonic bands using two parameters, sphere size and the effective index. This procedure gives a better description than the traditional way using just the Bragg’s expression without a band gap, commonly referred to as Zonefolding. © 2008 Elsevier B.V. All rights reserved.

Keywords: Silica spheres Photonic crystals Photonic band gap Bragg’s law

Three-dimensional (3D) colloidal photonic crystals made of silica spheres have been extensively investigated due to their potential applications for photonic devices [1,2]. Such crystals are structures that exhibit 3D periodicity of their refractive indices resulting in a modiﬁcation of the dispersion characteristics from the normal ω(k) function in anisotropic materials, and this yields the formation of photonic band gaps (PBGs). Among the investigations related to photonic band structures [3–5], the width of the stop-band serves as an indicator for the strength of interaction between photons and the crystal [6]. No doubt the photonic band structures constitute the key to understanding of these phenomena reported [7–9]. In fact, the most fundamental observation is the direction of propagation inside a photonic crystal being normal to the equal-energy surface. For example, M. Notomi et al. experimentally determined the full photonic band structure of Si/SiO2 3D photonic crystals by monitoring only the beam propagation [10]. The usual method of treating the colloidal photonic crystals with a face-centered cubic (FCC) structure involves a modiﬁed Bragg’s law [11–13],

λmax = 2d111 n2eff − sin2 θi ,

*

Corresponding author. Tel.: +1 704 687 8435; fax: +1 704 687 8241. E-mail addresses: [email protected], [email protected] (K. Liu).

0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.04.008

(1)

where λmax is the wavelength of maximum reﬂected intensity, d111 is the inter-planar spacing between (111) planes, neff is the effective refractive index of the photonic crystals, and θi is the incident angle with respect to normal incidence. This equation takes into account Snell’s law at the surface of photonic crystal and Bragg diffraction which accounts for the array factor only. However, for colloidal crystals with an anisotropic structure, the propagation direction and wave vector are generally not collinear. In this work, a comparative study of the colloidal photonic crystals made of silica spheres using the Bragg’s law and a model based on photonic bands was undertaken. Optical properties of the crystals annealed at various temperatures were characterized by a procedure similar to the X-ray diffraction (XRD) technique–peak position in 2θ . A photonic band structure of the crystals is modeled by two parameters, the period of the close-pack spheres and the effective index. The forbidden gap measured from the transmission spectra agrees with our model using two parameters. Further details of the crystal structure after annealing will also be discussed. The details of syntheses of silica spheres were followed by a procedure similar to that of van Blaaderen and Vrij [14], and the surface properties of these spheres can be tuned by using siloxane chemistry. The colloidal crystal samples were prepared on Si/fused silica substrates using a vertical deposition method [15]. Silica powder was dispersed in ethanol solution, having a concentration of ∼ 1.5 wt%. The substrates were introduced and then immersed into the colloidal sphere solution vertically. The 3D photonic crys-

4518

K. Liu et al. / Physics Letters A 372 (2008) 4517–4520

(a)

(b)

Fig. 1. (a) Schematic 1D photonic band diagram with zone folding in the L direction, and the band gaps with a shaded region arises at the Brillouin zone boundary, and ω+ , ω0 , and ω− represent the upper band edge, band center, and lower band edge, respectively at the ﬁrst Brillouin zone boundary. (b) Dispersion relation without a photonic gap, generally referred to as simple zone folding.

tals were self-assembled onto the substrates via evaporation of the ethanol. Thermal treatment was used to provide the band gap variation, as well as the stabilization of the photonic crystals. In general, our process involves annealing for 3h in a quartz tube at a temperature of 300–950 ◦ C. The transmission spectra at normal incidence were measured by a Cary 300 Bio UV-Visible spectrophotometer. The reﬂection spectra at various incident angles were recorded using a LS-1 Tungsten Halogen light source and an OceanOptics USB2000 spectrometer, and with an angle-resolved goniometry setup, with incident angle varying from 10◦ to 50◦ and collecting the scattered light as in the XRD conﬁguration. The sphere size is determined by scanning electron microscopy (SEM) using 35 randomly chosen spheres by the Raith150 annotation tool. Fig. 1(a) shows a schematic photonic band diagram for 1D photonic crystal in the L direction with a period d, the lattice constant. Using the close-packing geometry, d1 and d2 represent the size of silica spheres after thermal treatment. The photon branch splits at the upper and lower band edges ω+ and ω− determined by either transmission or reﬂection measurements. Without losses, the shaded region is the band gap where the wave-vector k is imaginary, and the corresponding reﬂectivity R = 1. The slope of the bands as k → 0, deﬁne neff as shown in Fig. 1(a). Fig. 1(b) shows the dispersion relation without a photonic gap, generally referred to as simple zone folding. The PBGs determination of the crystals is related to three parameters including ω+ , ω− , and d at the Brillouin zone boundary. The lower branch may be closely approximated by a sine function [16],

ω = ω(k) = ω− sin(kd/2),

(2)

where ω is the angular frequency of the incident light. In Fig. 1(a), ω = ck/neff as k π /d, where c is the light velocity in free space, and Eq. (2) gives ω− = 2c /(neff d) for k π /d. Also, the upper branch can be approximated as a cosine function [16], and using the known parameter ω− and ω+ + ω− = 2ω0 , the upper branch expression is

√

ω=

2c

neff d

(π − 1)2 + 1 + (π − 1)4 + 1 + 2(π − 1)2 cos(kd). (3)

Unlike in homogeneous medium, the equal-energy surfaces in a photonic crystal are usually anisotropic, even for cubic symmetry. If we denote the wave vectors of the incident and refracted waves by k1 and k2 , the conservation of both transverse components leads to |k1 | sin θi = |k2 | sin θt , where θt is the angle of refraction. The dispersion relations in air and in the photonic crystals are ω/|k1 | = c /n1 (n1 = 1.0) and ω/|k2 | = c /neff , respectively.

At the surface of photonic crystal, the relation between θi and θt is given by, sin θi sin θt

=

c

ω/|k2 |

=

neff d · k2 2| sin(k2 d/2)|

.

(4)

The impinging external angle is thus converted into the internal angle by means of the above equation. Compared with the conventional Snell’s law, θt in a photonic crystal is always smaller than that in a uniform dielectric material, and the Snell’s law is only a fair approximation at low energies. It would be interesting to extend our measurements into the upper branch of the photonic dispersion represented by Eq. (3) compared with the case of Bragg’s law for m = 2, where m is the diffraction order. We now move to the refracted light inside the photonic crystals. By varying the angle θi , the refracted light would satisfy the XRD condition as long as a series of parallel planes are composed of uniform spheres and these dielectric spheres are regarded as point scatters. These planes scatter light, and for certain directions and wavelengths these scattered waves are added constructively. The wavelength of the reﬂection peak can thus be calculated using Bragg’s law, mλpeak = 2d111 cos θt , where λpeak is the peak wavelength of the Bragg diffraction, and d111 = (2/3)1/2 d for the FCC structure. Considering m = 1 and λpeak = λ0 /neff , where λ0 is the wavelength of the light wave in vacuum, our model based on photonic bands can be expressed as

4 sin2 (k2 d/2) λ0 = 2d111 neff 1 − sin2 θi . n2eff d2 · k22

(5)

For the case of the close-packed silica spheres, k2 = 2π neff /λ0 . Eq. (5) in term of two-parameter neff and d is given by

πn 2 2 2 eff 8d neff − 3λ0 . θi = arcsin √ 2 2λ0 | sin(π neff d/λ0 )|

(6)

Eq. (6) allows interaction to be modeled by a band-gap. For the case of the Bragg model represented by Fig. 1(b), it only allows zone folding without a photonic gap. Since our measurements clearly show the presence of gap as in Fig. 3 and in Fig. 5, the expression giving by Eq. (6) with the photonic bands allow a more general description, particularly with excitation close to the PBG. Since Eq. (6) has only two adjustable parameters, neff and d, the comparison between the gap ﬁxed by Eq. (2) with those measured constitutes a measure how good is our model with a twoparameter ﬁt. Fig. 2 shows a SEM picture of the top-viewed photonic crystals made of silica spheres. The inset (a) shows a 3D atomic force microscope (AFM) surface image of the crystal sample with a dimension of 2.0 μm × 2.0 μm, and gives a hexagonal close-packed

K. Liu et al. / Physics Letters A 372 (2008) 4517–4520

Fig. 2. AFM and SEM images of an as-grown colloidal photonic crystal made of ∼ 345 nm spheres, and inset (b) shows a thickness of the crystal ﬁlm with ∼ 30 layers.

Fig. 3. Wavelength of the reﬂection peak versus incident angles of light for a crystal sample annealed at various temperatures, and the inset shows an example of the reﬂection spectra of the crystal sample for a constant incidence angle of 10◦ .

alignment, and the inset (b) shows a thickness of the crystal ﬁlm with ∼ 30 layers for our sample. Fig. 3 shows the relationship between incident angles and peaks of reﬂection spectra for a crystal sample annealed at various temperatures T , and the inset shows an example of the reﬂection spectra of the sample annealed at various T for a constant incidence angle of 10◦ . Eq. (6) indicates that the increasing incident angle of light shifts the stop-band toward the short wavelengths due to 0.5 < (neff · d)/λ0 < 1 for our case. The inset in Fig. 4 shows sphere sizes measured by SEM for a crystal sample annealed at different T , and the sphere size continues to decrease as an increase of T . The unconsolidated spheres are constructed using organo-silicates sol–gel and hydrolysis polymerization processes that include water, solvents, and micropores, resulting in a continuous shrinkage of the silica spheres [17]. Replacing d as the size measured by SEM and using the data of λ0 versus θi (Fig. 3), Fig. 4 shows a comparison of the determination of neff by a two-parameter ﬁt of Eqs. (1) and (6) with a non-linear least squares method, and the increased effective index of the colloidal crystals is observed as T increases. The index ﬁtted using the Bragg model is larger than that using the photonic band model. On the other hand, we know the dielectric constant ε is related to the refractive index by ε ∝ (n2 − 1), and actually the refractive index of medium is the sum of the average interaction between light ﬁeld and every particle in the medium according to a lind-

4519

Fig. 4. Comparison of the ﬁtted effective index as a function of annealing temperature using the Bragg model (Eq. (1)) and the photonic band model (Eq. (6)), respectively, and the inset shows sphere sizes measured by SEM for a crystal sample as a function of annealing temperature.

Fig. 5. Transmission spectra at normal incidence of the crystals on fused silica substrates annealed at various T , and FWHM of the dip is considered as ω . The inset shows the percentage of the relative PBGs width under various annealing conditions.

hard formula [18]. n2 − 1 is proportional to its volume change V , namely (n2 − 1) ∝ V , and V ∝ (d1 /d2 )3 . The V between the as-grown sample and the sample annealed at 950 ◦ C is 1.28 with its corresponding ratio ε of 1.44 for the case of the lower curve. The reason for a larger ε compared to its V is because these silica spheres consist of a partially condensed siloxane structure [14], leading to a tissue region surrounding an inner core, allowing signiﬁcant shrinkage of the core during annealing. The seemingly bigger sphere size measured by SEM may result in a larger ratio of ε compared to its V and a smaller value of the ﬁtting effective index. Fig. 5 shows the transmission spectra at normal incidence of the crystal samples annealed at various T , and a blue shift of the stop-band is observed as T increases. There is a total change of ∼ 70 nm in the Bragg reﬂected wavelength between as-grown samples and those annealed at 950 ◦ C. The band gap relative to the center frequency is deﬁned by W PBG = ω/ω0 [19], where ω = ω+ − ω− . For the case of our model, the width of the PBG is 2(π − 2)/π (7.3%). Fig. 5 experimentally shows the full width at half maximum (FWHM) of the transmission dip as ω in these spectra, and the inset shows the corresponding percentage of the

4520

K. Liu et al. / Physics Letters A 372 (2008) 4517–4520

relative PBGs width under various annealing conditions. Compared to the theoretical value of 7.3%, there is a very good agreement with the as-grown sample, showing that our model using Eq. (6) is perfect. However, the percentages of W PBG are usually bigger for the annealed samples. This is because the uniform dispersion of the spheres, and the disordered zones or dislocations becomes worse for those annealed samples [20], resulting in a deterioration of the crystal quality with manifestation of the shallow transmission dip, as shown in Fig. 5. Two possible structures of the colloidal crystals may result from a thermal treatment. One structure is that the spheres with shrinkage are located in the original lattice position without changing lattice constant, leading to a smaller ω given by ω = 2c (π − 2)/(neff · d). However, this is not what was observed since the product of neff · d is further decreased after each annealing process for our case. The other structure is that the spheres are contacting resulting in a smaller lattice constant, leading to a blue shift of PBG according to ω0 = c π /(neff · d). Such thermal treatment may produce a matter viscous ﬂow causing the spheres to be bound together [21]. In conclusion, optical properties of the colloidal crystals made of silica spheres annealed at various temperatures were characterized by measuring their reﬂection/transmission spectra. The photonic band structure, such as the Brillouin zone, is determined by the same technique as in the XRD conﬁguration with the sphere size from SEM analysis, and an additional ﬁtting parameter, the effective refractive index. The forbidden gap experimentally measured from the transmission stop bands is close to what is predicted using our model with the PBGs from Eq. (2). A comparative study of the colloidal crystals using a model with a gap not only gives better ﬁt but also provides better physical insight than the traditional way using just the Bragg’s expression. This simple approach is not designed to replace photonic band structures, rather, giving descriptions as modiﬁcations of the Snell’s law, and Bragg’s law. Further reﬁnement may include a three-parameter ﬁt capable for separately ﬁtting the position of ω0 and the gap ω+ − ω− . We would like to take this opportunity to make an important point. Band structure computation is always important in solid-state physics, however, to most experimentalists, the effective masses and energy bandgaps constitute the relevant parameters. What we point out is the similar fact that in dealing with the photonic crystals, parameters such as neff and the size of the Brillouin zone k B form the principle parameters. Just as in solids, we can

further introduce non-isotropic parameters if needed. The bottom line is the fact that simple zone folding, representing the use of Bragg’s law is not suﬃcient, which unfortunately is widely used by researchers. Acknowledgements We acknowledge the support of this research by the Center for Optoelectronics and Optical Communications under DARPA/ARL grant to UNC-Charlotte (W911NF-05-2-0053), and we are also grateful for partial support of this work by the Charlotte Research Institute. References [1] Y.N. Xia, B. Gates, Y.D. Yin, Y. Lu, Adv. Mater. 12 (2000) 693. [2] K. Busch, S. Lölkes, R.B. Wehrspohn, H. Föll, Photonic Crystals: Advances in Design, Fabrication, and Characterization, Wiley–VCH Verlag GmbH & Co., Weinheim, 2004. [3] I.I. Tarhan, G.H. Watson, Phys. Rev. Lett. 76 (1996) 315. [4] H. Takeda, K. Yoshino, J. Appl. Phys. 93 (2003) 3188. [5] E. Pavarini, L.C. Andreani, C. Soci, M. Galli, F. Marabelli, D. Comoretto, Phys. Rev. B 72 (2005) 045102. [6] W.L. Vos, H.M. van Driel, Phys. Lett. A 272 (2000) 101. [7] H.M. van Driel, W.L. Vos, Phys. Rev. B 62 (2000) 9872. [8] V.N. Astratov, A.M. Adawi, S. Fricker, M.S. Skolnick, D.M. Whittaker, P.N. Pusey, Phys. Rev. B 66 (2002) 165215. [9] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, Phys. Rev. B 58 (1998) 10096. [10] M. Notomi, T. Tamamura, Y. Ohtera, O. Hanaizumi, S. Kawakami, Phys. Rev. B 61 (2000) 7165. [11] H. Miguez, C. Lopez, F. Meseguer, A. Blanco, L. Vazquez, R. Mayoral, M. Ocana, V. Fornes, A. Mifsud, Appl. Phys. Lett. 71 (1997) 1148. [12] Z.Z. Gu, Q.B. Meng, S. Hayami, A. Fujishima, O. Sato, J. Appl. Phys. 90 (2001) 2042. [13] L.M. Goldenberg, J. Wagner, J. Stumpe, B.R. Paulke, E. Gornitz, Langmuir 18 (2002) 3319. [14] A. van Blaaderen, A. Vrij, J. Colloid Interface Sci. 156 (1993) 1. [15] P. Jiang, J.F. Bertone, K.S. Hwang, V.L. Colvin, Chem. Mater. 11 (1999) 2132. [16] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York, 1976. [17] A.A. Chabanov, Y. Jun, D.J. Norris, Appl. Phys. Lett. 84 (2004) 3573. [18] R. Tsu, Superlattice to Nanoelectronics, Elsevier, Boston, 2005. [19] I.I. Tarhan, G.H. Watson, Phys. Rev. B 54 (1996) 7593. [20] H. Miguez, A. Blanco, C. Lopez, F. Meseguer, H.M. Yates, M.E. Pemble, F. LopezTejeira, F.J. Garcia-Vidal, J. Sanchez-Dehesa, J. Lightwave Technol. 17 (1999) 1975. [21] C. Lopez, Adv. Mater. 15 (2003) 1679.

Lihat lebih banyak...
Contents lists available at ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

A comparative study of colloidal silica spheres: Photonic crystals versus Bragg’s law K. Liu a,∗ , T.A. Schmedake b , R. Tsu a a b

Department of Electrical and Computer Engineering, and Department of Physics and Optical Science, University of North Carolina at Charlotte, Charlotte, NC 28223, USA Department of Chemistry, University of North Carolina at Charlotte, Charlotte, NC 28223, USA

a r t i c l e

i n f o

Article history: Received 25 March 2008 Accepted 4 April 2008 Available online 9 April 2008 Communicated by V.M. Agranovich PACS: 42.70.Qs 42.25.Bs 42.70.-a

a b s t r a c t A comparative study of colloidal crystals made of silica spheres using Bragg’s law and a model including a photonic band gap (PBG) was demonstrated. Optical properties of the crystals annealed at various temperatures were characterized by a procedure similar to X-ray diffraction technique. Experimentally, the PBG obtained from the transmission spectra agrees with the model of the photonic bands using two parameters, sphere size and the effective index. This procedure gives a better description than the traditional way using just the Bragg’s expression without a band gap, commonly referred to as Zonefolding. © 2008 Elsevier B.V. All rights reserved.

Keywords: Silica spheres Photonic crystals Photonic band gap Bragg’s law

Three-dimensional (3D) colloidal photonic crystals made of silica spheres have been extensively investigated due to their potential applications for photonic devices [1,2]. Such crystals are structures that exhibit 3D periodicity of their refractive indices resulting in a modiﬁcation of the dispersion characteristics from the normal ω(k) function in anisotropic materials, and this yields the formation of photonic band gaps (PBGs). Among the investigations related to photonic band structures [3–5], the width of the stop-band serves as an indicator for the strength of interaction between photons and the crystal [6]. No doubt the photonic band structures constitute the key to understanding of these phenomena reported [7–9]. In fact, the most fundamental observation is the direction of propagation inside a photonic crystal being normal to the equal-energy surface. For example, M. Notomi et al. experimentally determined the full photonic band structure of Si/SiO2 3D photonic crystals by monitoring only the beam propagation [10]. The usual method of treating the colloidal photonic crystals with a face-centered cubic (FCC) structure involves a modiﬁed Bragg’s law [11–13],

λmax = 2d111 n2eff − sin2 θi ,

*

Corresponding author. Tel.: +1 704 687 8435; fax: +1 704 687 8241. E-mail addresses: [email protected], [email protected] (K. Liu).

0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.04.008

(1)

where λmax is the wavelength of maximum reﬂected intensity, d111 is the inter-planar spacing between (111) planes, neff is the effective refractive index of the photonic crystals, and θi is the incident angle with respect to normal incidence. This equation takes into account Snell’s law at the surface of photonic crystal and Bragg diffraction which accounts for the array factor only. However, for colloidal crystals with an anisotropic structure, the propagation direction and wave vector are generally not collinear. In this work, a comparative study of the colloidal photonic crystals made of silica spheres using the Bragg’s law and a model based on photonic bands was undertaken. Optical properties of the crystals annealed at various temperatures were characterized by a procedure similar to the X-ray diffraction (XRD) technique–peak position in 2θ . A photonic band structure of the crystals is modeled by two parameters, the period of the close-pack spheres and the effective index. The forbidden gap measured from the transmission spectra agrees with our model using two parameters. Further details of the crystal structure after annealing will also be discussed. The details of syntheses of silica spheres were followed by a procedure similar to that of van Blaaderen and Vrij [14], and the surface properties of these spheres can be tuned by using siloxane chemistry. The colloidal crystal samples were prepared on Si/fused silica substrates using a vertical deposition method [15]. Silica powder was dispersed in ethanol solution, having a concentration of ∼ 1.5 wt%. The substrates were introduced and then immersed into the colloidal sphere solution vertically. The 3D photonic crys-

4518

K. Liu et al. / Physics Letters A 372 (2008) 4517–4520

(a)

(b)

Fig. 1. (a) Schematic 1D photonic band diagram with zone folding in the L direction, and the band gaps with a shaded region arises at the Brillouin zone boundary, and ω+ , ω0 , and ω− represent the upper band edge, band center, and lower band edge, respectively at the ﬁrst Brillouin zone boundary. (b) Dispersion relation without a photonic gap, generally referred to as simple zone folding.

tals were self-assembled onto the substrates via evaporation of the ethanol. Thermal treatment was used to provide the band gap variation, as well as the stabilization of the photonic crystals. In general, our process involves annealing for 3h in a quartz tube at a temperature of 300–950 ◦ C. The transmission spectra at normal incidence were measured by a Cary 300 Bio UV-Visible spectrophotometer. The reﬂection spectra at various incident angles were recorded using a LS-1 Tungsten Halogen light source and an OceanOptics USB2000 spectrometer, and with an angle-resolved goniometry setup, with incident angle varying from 10◦ to 50◦ and collecting the scattered light as in the XRD conﬁguration. The sphere size is determined by scanning electron microscopy (SEM) using 35 randomly chosen spheres by the Raith150 annotation tool. Fig. 1(a) shows a schematic photonic band diagram for 1D photonic crystal in the L direction with a period d, the lattice constant. Using the close-packing geometry, d1 and d2 represent the size of silica spheres after thermal treatment. The photon branch splits at the upper and lower band edges ω+ and ω− determined by either transmission or reﬂection measurements. Without losses, the shaded region is the band gap where the wave-vector k is imaginary, and the corresponding reﬂectivity R = 1. The slope of the bands as k → 0, deﬁne neff as shown in Fig. 1(a). Fig. 1(b) shows the dispersion relation without a photonic gap, generally referred to as simple zone folding. The PBGs determination of the crystals is related to three parameters including ω+ , ω− , and d at the Brillouin zone boundary. The lower branch may be closely approximated by a sine function [16],

ω = ω(k) = ω− sin(kd/2),

(2)

where ω is the angular frequency of the incident light. In Fig. 1(a), ω = ck/neff as k π /d, where c is the light velocity in free space, and Eq. (2) gives ω− = 2c /(neff d) for k π /d. Also, the upper branch can be approximated as a cosine function [16], and using the known parameter ω− and ω+ + ω− = 2ω0 , the upper branch expression is

√

ω=

2c

neff d

(π − 1)2 + 1 + (π − 1)4 + 1 + 2(π − 1)2 cos(kd). (3)

Unlike in homogeneous medium, the equal-energy surfaces in a photonic crystal are usually anisotropic, even for cubic symmetry. If we denote the wave vectors of the incident and refracted waves by k1 and k2 , the conservation of both transverse components leads to |k1 | sin θi = |k2 | sin θt , where θt is the angle of refraction. The dispersion relations in air and in the photonic crystals are ω/|k1 | = c /n1 (n1 = 1.0) and ω/|k2 | = c /neff , respectively.

At the surface of photonic crystal, the relation between θi and θt is given by, sin θi sin θt

=

c

ω/|k2 |

=

neff d · k2 2| sin(k2 d/2)|

.

(4)

The impinging external angle is thus converted into the internal angle by means of the above equation. Compared with the conventional Snell’s law, θt in a photonic crystal is always smaller than that in a uniform dielectric material, and the Snell’s law is only a fair approximation at low energies. It would be interesting to extend our measurements into the upper branch of the photonic dispersion represented by Eq. (3) compared with the case of Bragg’s law for m = 2, where m is the diffraction order. We now move to the refracted light inside the photonic crystals. By varying the angle θi , the refracted light would satisfy the XRD condition as long as a series of parallel planes are composed of uniform spheres and these dielectric spheres are regarded as point scatters. These planes scatter light, and for certain directions and wavelengths these scattered waves are added constructively. The wavelength of the reﬂection peak can thus be calculated using Bragg’s law, mλpeak = 2d111 cos θt , where λpeak is the peak wavelength of the Bragg diffraction, and d111 = (2/3)1/2 d for the FCC structure. Considering m = 1 and λpeak = λ0 /neff , where λ0 is the wavelength of the light wave in vacuum, our model based on photonic bands can be expressed as

4 sin2 (k2 d/2) λ0 = 2d111 neff 1 − sin2 θi . n2eff d2 · k22

(5)

For the case of the close-packed silica spheres, k2 = 2π neff /λ0 . Eq. (5) in term of two-parameter neff and d is given by

πn 2 2 2 eff 8d neff − 3λ0 . θi = arcsin √ 2 2λ0 | sin(π neff d/λ0 )|

(6)

Eq. (6) allows interaction to be modeled by a band-gap. For the case of the Bragg model represented by Fig. 1(b), it only allows zone folding without a photonic gap. Since our measurements clearly show the presence of gap as in Fig. 3 and in Fig. 5, the expression giving by Eq. (6) with the photonic bands allow a more general description, particularly with excitation close to the PBG. Since Eq. (6) has only two adjustable parameters, neff and d, the comparison between the gap ﬁxed by Eq. (2) with those measured constitutes a measure how good is our model with a twoparameter ﬁt. Fig. 2 shows a SEM picture of the top-viewed photonic crystals made of silica spheres. The inset (a) shows a 3D atomic force microscope (AFM) surface image of the crystal sample with a dimension of 2.0 μm × 2.0 μm, and gives a hexagonal close-packed

K. Liu et al. / Physics Letters A 372 (2008) 4517–4520

Fig. 2. AFM and SEM images of an as-grown colloidal photonic crystal made of ∼ 345 nm spheres, and inset (b) shows a thickness of the crystal ﬁlm with ∼ 30 layers.

Fig. 3. Wavelength of the reﬂection peak versus incident angles of light for a crystal sample annealed at various temperatures, and the inset shows an example of the reﬂection spectra of the crystal sample for a constant incidence angle of 10◦ .

alignment, and the inset (b) shows a thickness of the crystal ﬁlm with ∼ 30 layers for our sample. Fig. 3 shows the relationship between incident angles and peaks of reﬂection spectra for a crystal sample annealed at various temperatures T , and the inset shows an example of the reﬂection spectra of the sample annealed at various T for a constant incidence angle of 10◦ . Eq. (6) indicates that the increasing incident angle of light shifts the stop-band toward the short wavelengths due to 0.5 < (neff · d)/λ0 < 1 for our case. The inset in Fig. 4 shows sphere sizes measured by SEM for a crystal sample annealed at different T , and the sphere size continues to decrease as an increase of T . The unconsolidated spheres are constructed using organo-silicates sol–gel and hydrolysis polymerization processes that include water, solvents, and micropores, resulting in a continuous shrinkage of the silica spheres [17]. Replacing d as the size measured by SEM and using the data of λ0 versus θi (Fig. 3), Fig. 4 shows a comparison of the determination of neff by a two-parameter ﬁt of Eqs. (1) and (6) with a non-linear least squares method, and the increased effective index of the colloidal crystals is observed as T increases. The index ﬁtted using the Bragg model is larger than that using the photonic band model. On the other hand, we know the dielectric constant ε is related to the refractive index by ε ∝ (n2 − 1), and actually the refractive index of medium is the sum of the average interaction between light ﬁeld and every particle in the medium according to a lind-

4519

Fig. 4. Comparison of the ﬁtted effective index as a function of annealing temperature using the Bragg model (Eq. (1)) and the photonic band model (Eq. (6)), respectively, and the inset shows sphere sizes measured by SEM for a crystal sample as a function of annealing temperature.

Fig. 5. Transmission spectra at normal incidence of the crystals on fused silica substrates annealed at various T , and FWHM of the dip is considered as ω . The inset shows the percentage of the relative PBGs width under various annealing conditions.

hard formula [18]. n2 − 1 is proportional to its volume change V , namely (n2 − 1) ∝ V , and V ∝ (d1 /d2 )3 . The V between the as-grown sample and the sample annealed at 950 ◦ C is 1.28 with its corresponding ratio ε of 1.44 for the case of the lower curve. The reason for a larger ε compared to its V is because these silica spheres consist of a partially condensed siloxane structure [14], leading to a tissue region surrounding an inner core, allowing signiﬁcant shrinkage of the core during annealing. The seemingly bigger sphere size measured by SEM may result in a larger ratio of ε compared to its V and a smaller value of the ﬁtting effective index. Fig. 5 shows the transmission spectra at normal incidence of the crystal samples annealed at various T , and a blue shift of the stop-band is observed as T increases. There is a total change of ∼ 70 nm in the Bragg reﬂected wavelength between as-grown samples and those annealed at 950 ◦ C. The band gap relative to the center frequency is deﬁned by W PBG = ω/ω0 [19], where ω = ω+ − ω− . For the case of our model, the width of the PBG is 2(π − 2)/π (7.3%). Fig. 5 experimentally shows the full width at half maximum (FWHM) of the transmission dip as ω in these spectra, and the inset shows the corresponding percentage of the

4520

K. Liu et al. / Physics Letters A 372 (2008) 4517–4520

relative PBGs width under various annealing conditions. Compared to the theoretical value of 7.3%, there is a very good agreement with the as-grown sample, showing that our model using Eq. (6) is perfect. However, the percentages of W PBG are usually bigger for the annealed samples. This is because the uniform dispersion of the spheres, and the disordered zones or dislocations becomes worse for those annealed samples [20], resulting in a deterioration of the crystal quality with manifestation of the shallow transmission dip, as shown in Fig. 5. Two possible structures of the colloidal crystals may result from a thermal treatment. One structure is that the spheres with shrinkage are located in the original lattice position without changing lattice constant, leading to a smaller ω given by ω = 2c (π − 2)/(neff · d). However, this is not what was observed since the product of neff · d is further decreased after each annealing process for our case. The other structure is that the spheres are contacting resulting in a smaller lattice constant, leading to a blue shift of PBG according to ω0 = c π /(neff · d). Such thermal treatment may produce a matter viscous ﬂow causing the spheres to be bound together [21]. In conclusion, optical properties of the colloidal crystals made of silica spheres annealed at various temperatures were characterized by measuring their reﬂection/transmission spectra. The photonic band structure, such as the Brillouin zone, is determined by the same technique as in the XRD conﬁguration with the sphere size from SEM analysis, and an additional ﬁtting parameter, the effective refractive index. The forbidden gap experimentally measured from the transmission stop bands is close to what is predicted using our model with the PBGs from Eq. (2). A comparative study of the colloidal crystals using a model with a gap not only gives better ﬁt but also provides better physical insight than the traditional way using just the Bragg’s expression. This simple approach is not designed to replace photonic band structures, rather, giving descriptions as modiﬁcations of the Snell’s law, and Bragg’s law. Further reﬁnement may include a three-parameter ﬁt capable for separately ﬁtting the position of ω0 and the gap ω+ − ω− . We would like to take this opportunity to make an important point. Band structure computation is always important in solid-state physics, however, to most experimentalists, the effective masses and energy bandgaps constitute the relevant parameters. What we point out is the similar fact that in dealing with the photonic crystals, parameters such as neff and the size of the Brillouin zone k B form the principle parameters. Just as in solids, we can

further introduce non-isotropic parameters if needed. The bottom line is the fact that simple zone folding, representing the use of Bragg’s law is not suﬃcient, which unfortunately is widely used by researchers. Acknowledgements We acknowledge the support of this research by the Center for Optoelectronics and Optical Communications under DARPA/ARL grant to UNC-Charlotte (W911NF-05-2-0053), and we are also grateful for partial support of this work by the Charlotte Research Institute. References [1] Y.N. Xia, B. Gates, Y.D. Yin, Y. Lu, Adv. Mater. 12 (2000) 693. [2] K. Busch, S. Lölkes, R.B. Wehrspohn, H. Föll, Photonic Crystals: Advances in Design, Fabrication, and Characterization, Wiley–VCH Verlag GmbH & Co., Weinheim, 2004. [3] I.I. Tarhan, G.H. Watson, Phys. Rev. Lett. 76 (1996) 315. [4] H. Takeda, K. Yoshino, J. Appl. Phys. 93 (2003) 3188. [5] E. Pavarini, L.C. Andreani, C. Soci, M. Galli, F. Marabelli, D. Comoretto, Phys. Rev. B 72 (2005) 045102. [6] W.L. Vos, H.M. van Driel, Phys. Lett. A 272 (2000) 101. [7] H.M. van Driel, W.L. Vos, Phys. Rev. B 62 (2000) 9872. [8] V.N. Astratov, A.M. Adawi, S. Fricker, M.S. Skolnick, D.M. Whittaker, P.N. Pusey, Phys. Rev. B 66 (2002) 165215. [9] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, Phys. Rev. B 58 (1998) 10096. [10] M. Notomi, T. Tamamura, Y. Ohtera, O. Hanaizumi, S. Kawakami, Phys. Rev. B 61 (2000) 7165. [11] H. Miguez, C. Lopez, F. Meseguer, A. Blanco, L. Vazquez, R. Mayoral, M. Ocana, V. Fornes, A. Mifsud, Appl. Phys. Lett. 71 (1997) 1148. [12] Z.Z. Gu, Q.B. Meng, S. Hayami, A. Fujishima, O. Sato, J. Appl. Phys. 90 (2001) 2042. [13] L.M. Goldenberg, J. Wagner, J. Stumpe, B.R. Paulke, E. Gornitz, Langmuir 18 (2002) 3319. [14] A. van Blaaderen, A. Vrij, J. Colloid Interface Sci. 156 (1993) 1. [15] P. Jiang, J.F. Bertone, K.S. Hwang, V.L. Colvin, Chem. Mater. 11 (1999) 2132. [16] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York, 1976. [17] A.A. Chabanov, Y. Jun, D.J. Norris, Appl. Phys. Lett. 84 (2004) 3573. [18] R. Tsu, Superlattice to Nanoelectronics, Elsevier, Boston, 2005. [19] I.I. Tarhan, G.H. Watson, Phys. Rev. B 54 (1996) 7593. [20] H. Miguez, A. Blanco, C. Lopez, F. Meseguer, H.M. Yates, M.E. Pemble, F. LopezTejeira, F.J. Garcia-Vidal, J. Sanchez-Dehesa, J. Lightwave Technol. 17 (1999) 1975. [21] C. Lopez, Adv. Mater. 15 (2003) 1679.

Somos uma comunidade de intercâmbio. Por favor, ajude-nos com a subida ** 1 ** um novo documento ou um que queremos baixar:

OU DOWNLOAD IMEDIATAMENTE