3. Colloidal films of SiO2 spheres: effect of thickness

This section is devoted to the effect of thickness of a colloidal film on its optical response. We consider a film built by N layers made of SiO2 spheres with a diameter of 200 nm, ordered in HCP structure (AB sequence). A scheme of the thin film is shown in Figure 2; there, the film is on the yz plane and its thickness goes along the x axis. Also, the incident electromagnetic (EM) plane wave has a wave vector K in direction of the positive x axis.

The TUC for each N layer thin film, with N from 1 to 12, is conveniently built. The TUC for the monolayer contains two spheres, for the bilayer four spheres, for

Figure 2.

(a) The wave vector K of the incident EM plane wave, perpendicular to the thin film. (b) Lateral view of a thin film composed of 6 layers of SiO2 spheres in an ABAB sequence.

#### Metallo-Dielectric Colloidal Films as SERS Substrate DOI: http://dx.doi.org/10.5772/intechopen.90313

between the dipole located at r<sup>j</sup><sup>00</sup> and the field induced by the dipoles of the TUC

If there are Ndip dipoles in the TUC with a similar equation to Eq. (3) existing for each dipole, then, a system of 3Ndip complex coupled equations needs to be solved for p<sup>j</sup><sup>00</sup> values. Once the equation system is solved, then far and near field optical

Usually, Ndip dipoles are of the order of 10<sup>5</sup> � <sup>10</sup>6; hence, numerical tools are necessary to solve the system of 3Ndip equations. A robust numerical implementation of DDA is the DDSCAT code [26] that assumes an incident plane wave along the positive direction of x axis and a bidimensional target resting on the yz plane. Among other interesting physical quantities, the 2 � 2 scattering amplitude matrix elements Si can be calculated, and consequently, the 4 � 4 scattering intensity

For the specific case of unpolarized incident light, R and T are related to the Sαβ

where ksx and k0<sup>x</sup> are the wave vector components of the scattered light and of the incident light, respectively. Both of them are along the direction of the incident

<sup>2</sup> <sup>þ</sup> j j <sup>S</sup><sup>3</sup>

<sup>2</sup>

This section is devoted to the effect of thickness of a colloidal film on its optical response. We consider a film built by N layers made of SiO2 spheres with a diameter of 200 nm, ordered in HCP structure (AB sequence). A scheme of the thin film is shown in Figure 2; there, the film is on the yz plane and its thickness goes along the x axis. Also, the incident electromagnetic (EM) plane wave has a wave vector K in

The TUC for each N layer thin film, with N from 1 to 12, is conveniently built. The TUC for the monolayer contains two spheres, for the bilayer four spheres, for

(a) The wave vector K of the incident EM plane wave, perpendicular to the thin film. (b) Lateral view of a

thin film composed of 6 layers of SiO2 spheres in an ABAB sequence.

<sup>2</sup> <sup>þ</sup> j j <sup>S</sup><sup>2</sup>

R ¼ S<sup>11</sup> for ð Þ ksxk0<sup>x</sup> < 0 , (4) T ¼ S<sup>11</sup> for ð Þ ksxk0<sup>x</sup>>0 , (5)

<sup>2</sup> <sup>þ</sup> j j <sup>S</sup><sup>4</sup>

: (6)

and its replicas at the rkmn positions; for more details, see [24].

response of the periodic target can be calculated.

matrix elements Sαβ [27].

Nanorods and Nanocomposites

elements through the next expressions:

light. The S<sup>11</sup> value is related to Si elements by:

direction of the positive x axis.

Figure 2.

96

<sup>S</sup><sup>11</sup> <sup>¼</sup> <sup>1</sup>

<sup>2</sup> j j <sup>S</sup><sup>1</sup>

3. Colloidal films of SiO2 spheres: effect of thickness

the three-layer six spheres, and so on. A 10 nm separation distance between two adjacent dipoles is used, implying about 4600 dipoles per sphere.

In the visible range, the refractive index of the silica, nSiO2 , is almost constant, and its variations go from 1.48 to 1.45 for wavelengths from 400 to 700 nm [28, 29]. For simplicity and without loss of information, we choose nSiO2 ð Þλ = 1.46. It is noteworthy to mention that in the visible the imaginary part of nSiO2 is negligible compared to the real part. However, in the ultraviolet interval, special care needs to be taken as the imaginary part comes to be significant and it is associated to the absorption coefficient.

R and T of a thin film composed of N layers, when light comes parallel to the normal of the surface, are shown in Figure 3. As the number of layers increases from 1 to 12, that is, as the film becomes thicker, a maximum of R emerges defining a photonic band, getting sharper around 450 nm. The optical spectrum to the left and right of the PBG is not symmetric because we have chosen the wavelength as the independent variable and not the wave number. The thickness, wavelength of the BG center (λc), width of the BG (ΔBG), effective refractive index of the thin film (neff), and optical path length (L) of each thin film are given in Table 1. Each quantity is estimated as follows:

The thickness is determined by considering the diameter of the sphere, D, and geometrical aspects of the AB sequence, as is explained next. The base of a tetrahedron is formed by the center of three spheres on a layer A, its height goes from its base to the center of a sphere of layer B, and the last is resting on the void left by the three spheres on layer A. The height of the tetrahedron coincides with the separation distance, d(1 1 1), between two adjacent planes with Miller indices (1 1 1). Then, the thickness of a film with <sup>N</sup> layers is (<sup>N</sup> � 1)d(1 1 1) + <sup>D</sup>, with d(1 1 1) = ffiffiffiffiffiffiffiffiffi <sup>ð</sup>2=<sup>3</sup> <sup>p</sup> <sup>D</sup>.

λ<sup>c</sup> is extracted from the calculated R(λ) spectrum, noting that it moves to the blue as the thickness increases, stopping around 444 nm. Latter, we compare this stop value to the λ<sup>c</sup> of a 3D opal, and with experimental results. ΔBG is associated to the full width at the half maximum (FWHM) of the spectrum, and to estimate it, a Gaussian centered at the Bragg peak (the maximum of the R spectrum) was

#### Figure 3.

Reflectance and transmittance of a thin film composed of N layers of 200 nm SiO2 spheres. The incident light comes normal to the surface and is unpolarized.


#### Table 1.

Thickness of the thin film composed of N layers, BG center position (λc), width of the BG (ΔBG), effective refractive index (neff ), and optical path length (L).

adjusted. The trend is that the PB becomes narrow as the film is getting thicker. In Table 1, the ΔBG values are given in nm and eV.

Because the film is a heterogeneous material composed of SiO2 spheres with air at the interspace, a refractive index can be assigned to it in order to treat the film as a homogeneous material with an effective property, neff. This can be done relating the refractive index of each involved material and its filling fraction (ff)<sup>1</sup> , that is, the fraction of volume occupied by the spheres and air. neff is deduced using [30]:

$$n\_{\rm eff}^2(\lambda) = n\_{\rm air}^2(1 - \rm ff) + n\_{\rm SiO\_2}^2(\lambda) \rm ff. \tag{7}$$

neff ¼ 1:35 for more than seven layers (see Table 1), the same value of a 3D opal with HCP structure.

In the last column of Table 1, the optical path length (L) associated to the inhomogeneous thin film is determined simply using the generalized Bragg equation at normal incidence [31]:

$$\mathcal{L} = \frac{\lambda\_{\rm c}}{2n\_{\rm eff}},\tag{8}$$

the wavelength of the center of the PBG, λc,opal, can be estimated. For the specific case of normal incidence and constructive interference due to planes 111 ð Þ, λc,opal = 442.7 nm, a value very close to 444 nm. Then, the PBG of films with more than eight layers resembles that of a 3D artificial opal (see Figure 4). Excluding thin films with less than three layers, the best fitting for λ<sup>c</sup> as a function of the number of layers is a polynomial of third order: <sup>λ</sup><sup>c</sup> (N) = 600.57 nm � 48.04 <sup>N</sup> + 5.11 <sup>N</sup><sup>2</sup> � 0.18 <sup>N</sup><sup>3</sup>

Metallo-Dielectric Colloidal Films as SERS Substrate DOI: http://dx.doi.org/10.5772/intechopen.90313

In Figure 5, the reflectance of an opal of 200 nm SiO2 spheres has a PBG with a

Wavelength position of the center of the band gap, λc, as the number of layers increases. λ<sup>c</sup> of a 3D opal is about

Reflectance of an opal of 200 nm silica spheres; its PBG has a maximum around 474 nm. In the AFM image,

some defects are noticed in the orderliness. Image modified with permission from Ref. [17].

maximum around 474 nm [17], agreeing to our results shown in Figure 3. The

Figure 4.

442.7 nm.

Figure 5.

99

.

L is interpreted as the path that follows the light when going through the heterogeneous film in order to produce constructive interference.

Now, Let's keep in mind the trend observed by the position of λ<sup>c</sup> as the thickness of the film increases: a shift to the blue stopping at a value around 444 nm. On the other hand, consider a 3D opal with an HCP structure and a ff = 0.74. Then, using (Eq. 7) and generalized Bragg's law

$$
\lambda\_{\text{c,equal}} = 2d\_{(111)} \sqrt{n\_{\text{eff}}^2 - \sin^2 \theta},
\tag{9}
$$

<sup>1</sup> The ff is calculated considering a cell of a hexagonal base with a length side D and a high defined by the thickness of the N layers. Then, ff¼ mVsphere=Vcell, with Vsphere and Vcell being the volume of a sphere and the cell, respectively. m is the number of spheres in the cell.

Metallo-Dielectric Colloidal Films as SERS Substrate DOI: http://dx.doi.org/10.5772/intechopen.90313

the wavelength of the center of the PBG, λc,opal, can be estimated. For the specific case of normal incidence and constructive interference due to planes 111 ð Þ, λc,opal = 442.7 nm, a value very close to 444 nm. Then, the PBG of films with more than eight layers resembles that of a 3D artificial opal (see Figure 4). Excluding thin films with less than three layers, the best fitting for λ<sup>c</sup> as a function of the number of layers is a polynomial of third order: <sup>λ</sup><sup>c</sup> (N) = 600.57 nm � 48.04 <sup>N</sup> + 5.11 <sup>N</sup><sup>2</sup> � 0.18 <sup>N</sup><sup>3</sup> .

In Figure 5, the reflectance of an opal of 200 nm SiO2 spheres has a PBG with a maximum around 474 nm [17], agreeing to our results shown in Figure 3. The

Figure 4.

adjusted. The trend is that the PB becomes narrow as the film is getting thicker. In

Thickness of the thin film composed of N layers, BG center position (λc), width of the BG (ΔBG), effective

Number of layers Thickness (nm) λ<sup>c</sup> (nm) ΔBG (nm)/(eV) neff L (nm) 200 650 — 1.29 251.9 363.3 549.7 — 1.3 211.4 526.4 499 — 1.33 187.6 689.9 476 — 1.34 177.6 853.2 465 73.64/16.7 1.34 173.5 1016.5 457 62.96/19.54 1.34 170.5 1179.8 453 55.39/22.21 1.35 169.0 1343.1 450 50.04/24.58 1.35 166.7 1806.4 449 46.05/26.71 1.35 166.3 1669.9. 447 43.28/28.42 1.35 165.5 1833.0 444.7 42.47/28.96 1.35 164.7 1996.3. 444.3 37.57/32.74 1.35 164.5

the fraction of volume occupied by the spheres and air. neff is deduced using [30]:

In the last column of Table 1, the optical path length (L) associated to the inhomogeneous thin film is determined simply using the generalized Bragg

> <sup>L</sup> <sup>¼</sup> <sup>λ</sup><sup>c</sup> 2neff

L is interpreted as the path that follows the light when going through the

Now, Let's keep in mind the trend observed by the position of λ<sup>c</sup> as the thickness of the film increases: a shift to the blue stopping at a value around 444 nm. On the other hand, consider a 3D opal with an HCP structure and a ff = 0.74. Then, using

n2

q

The ff is calculated considering a cell of a hexagonal base with a length side D and a high defined by the thickness of the N layers. Then, ff¼ mVsphere=Vcell, with Vsphere and Vcell being the volume of a sphere

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

eff � sin <sup>2</sup>

θ

heterogeneous film in order to produce constructive interference.

λc,opal ¼ 2dð Þ <sup>111</sup>

and the cell, respectively. m is the number of spheres in the cell.

airð Þþ <sup>1</sup> � ff <sup>n</sup><sup>2</sup>

neff ¼ 1:35 for more than seven layers (see Table 1), the same value of a 3D opal

SiO2

the refractive index of each involved material and its filling fraction (ff)<sup>1</sup>

Because the film is a heterogeneous material composed of SiO2 spheres with air at the interspace, a refractive index can be assigned to it in order to treat the film as a homogeneous material with an effective property, neff. This can be done relating

, that is,

ð Þλ ff: (7)

, (8)

, (9)

Table 1, the ΔBG values are given in nm and eV.

refractive index (neff ), and optical path length (L).

Nanorods and Nanocomposites

n2

with HCP structure.

1

98

Table 1.

equation at normal incidence [31]:

(Eq. 7) and generalized Bragg's law

effð Þ¼ <sup>λ</sup> <sup>n</sup><sup>2</sup>

Wavelength position of the center of the band gap, λc, as the number of layers increases. λ<sup>c</sup> of a 3D opal is about 442.7 nm.

#### Figure 5.

Reflectance of an opal of 200 nm silica spheres; its PBG has a maximum around 474 nm. In the AFM image, some defects are noticed in the orderliness. Image modified with permission from Ref. [17].

asymmetry of the spectrum and difference in position of the gap center is presumably due to the presence of defects and polydispersity of the synthesized thin film.
