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Equations for filling factor estimation in opal matrix

We consider two equations for the filling factor estimation of infiltrated zinc oxide (ZnO) in silica (SiO_2) opal and gallium nitride in ZnO opal. The first equation is based on the effective medium approximation, while the second one - on Maxwell-Garnett approximation. The comparison between two filling factors shows that both equations can be equally used for the estimation of the quantity of infiltrated nanocrystals inside opal matrix.


I. Introduction
Photonic crystals (PhCs) with forbidden band-gaps, proposed by Eli Yablonovitch 1  One of the kinds of PhCs is an opal matrix consisting of spherical sub-micron balls packed into face centered cubic (FCC) structure by means of self-sedimentation in a fluid suspension 6, 7 . Silicon dioxide (SiO 2 ) or silica is frequently used as a host material in artificial opals. Silica balls are synthesized by Stöber-Fink-Bohn process through the hydrolysis of tetraethylorthosilicate in the ethanol solution mixed with ammonium hydroxide and water 8 .
The applications of opal PhCs have number of significant advantages over others. For instance, the opal matrix can be grown over a large practically unlimited plane area. Their fabrication is very technological without requirement for expensive equipment. It has been recently shown that by means of electro-deposition the high quality two-and three-dimensional porous films, patterned in inverted opal, can be successfully realized 9 . Thus, the nanocrystals grown in artificial opal can be regarded as inexpensive and efficient alternative for electro-and photolithography.
The fabrication of high quality artificial opal by natural self-sedimentation in monodispersed fluid suspension may continue for a long period, up to ten months 6 . However this drawback is resolved in electrophoretically assisted sedimentation involving an external electric filed. Such an original technology enables one to accelerate a sedimentation velocity up to 0.2-0.7 mm per hour for the balls ranging in diameter between 300 -550 nm 10 .
Consider Figure 2 showing the SEM image of the opal, which FCC structure has dislocations, formed during the self-sedimentation process in a fluid suspension. The sample contains micro-size domains with facing up (111) and (100) planes. These domains are chaotically distributed within the sample and differently tilted with respect to its surface. As a result, the blue-shift in reflectance (or in transmittance) becomes insignificant and irregular with increasing angle 17, 18 . It signifies that in highly imperfect or in amorphous opal the color remains practically stable at any θ ( Figure 1).
Despite of the fact that such sample does not exhibit the blue-shift with increasing angle, the influence of photonic band-gap (PBG) in opal with disordered FCC structure is possible to observe conclusively either in evolution of photoluminescence arising due to gradual increase of the filling factor or in temperature-dependent photoluminescence 16 .
It is worth remarking that imperfect opal structures exhibit spectrum with greater FWHM in reflection (or transmission) 19 . While perfectly ordered opal matrix may find its applications in various light emitting devices 3, 4 , imperfectly assembled and/or amorphous opal with embedded luminescent nanoparticles might be useful for applications in full-color displays 16 . Nanoparticles infiltrated in interglobular spaces can considerably alter the optical properties of opal matrix. Therefore the estimation of the quantity of infiltrated nanocrystals plays significant role in practice. Particularly, the amount of infiltrated material has to be properly controlled during its deposition in the voids.
This paper reports two filling factor equations based on effective medium and Maxwell-Garnett approximations. The comparison between them shows that both equations can be equally used for quantity estimation of infiltrated nanoparticles in interglobular spaces between FCC packed spheres.

Refractive index based on effective medium approximation
The quantitative analysis of the optical characteristics of opal matrix can be significantly simplified introducing the effective refractive index according to effective medium approximation 20 .
Effective refractive index can be defined as a weighted sum of indices of refraction n 1 , n 2 , n 3 , for spherical balls, infiltrated nanoparticles and air, respectively. For the bare and infiltrated opals, the effective refractive indices accordingly are The red-shift in reflectance or transmittance spectra arising due to infiltrated nanocrystals inside opal matrix can be found as ( ) (3) Alternatively, the filling factor can be derived through ratio between wavelengths 1 λ , 2 λ for the bare and infiltrated opals 1 , , Substitution of effective refractive indices (2a, b) into (5) gives Clearly that (4) and (6) (6) leads to (4).

Effective refractive index based on Maxwell-Garnett approximation
Another definition for effective refractive indices, also widely used in practice, is based on Maxwell-Garnett approximation 21  (7b) Assume again a low angle of incidence. Substitution of definitions (7a, b) into (1) provides two squared wavelengths and corresponding to the bare and infiltrated opals, respectively.
The difference between them is From (7a, b) and (8) the filling factor can be found as   Figure 3 shows the refractive indices for silicon dioxide and zinc oxide vs. wavelength.

Refractive indices of silicon dioxide and zinc oxide
Analytical form of ( ) Below this point, the refractive index of ZnO has strong wavelength dependence and its curve rapidly rises due to resonance occurring between valence and conduction bands.
Contrarily, the curve for the silicon dioxide is nearly flat and consequently its refractive index can be considered a constant over the wide optical range covering near infrared (IR) to near ultraviolet (UV) spectra. Taking this into account and considering the fact that refractive index of air is very close to unity, (4) and (6) can be simplified and approximated as Equation (15) has been used for ZnO filling factor estimation in our previous work 27 .

Discrepancies between filling factors
The relative error defined as % 100 (16) is used in the present work to evaluate discrepancies between filling factors. Figure 4 shows the filling factors and the their relative errors vs. red-shift. The filling factors and , calculated according to (13) and (14), are shown by solid and dashed curves, respectively.    At the high infiltrations the relative errors are negligible for UV and purple-blue opals. For all other opals they do not exceed 5%. Discrepancies between filling factors show that each of two simplified equations (13) and (14) can be used for the quantity estimation of infiltrated material inside silica opal. However, it should be taken into account that for the small amount of infiltration in UV opal the discrepancy may be relatively high, nearly 10%.

IV. Filling factors for opal matrix comprising ZnO balls
In fact, both equations (2a) and (7a) contain term , which is itself, generally, may depend on  The right part of Table 1 converges faster to the desired values due to squared form of (7a). Figure 6 shows filling factors and their relative error for ZnO opal infiltrated with GaN. The filling factors do not intercept. The curve for grows faster than that for , consequently the relative error monotonically increase. At the origin the relative error is small, less than 6%.
However, at complete infiltration the discrepancy between filling factors becomes relatively high, reaching almost 16%.

Maxwell-Garnett approximation
Step Trial λ 1 , nm Calculated λ 1 , nm n 1 (λ 1 ) Step Trial λ 1 , nm Calculated λ 1 , nm n 1 (λ 1 )   (Figure 7). As a result, the relative error for infiltrated ZnO opal is higher. It should be noted, however, that for the visible spectral range, the relative error between filling factors does not exceed 17.5 % even at complete infiltration.

V. Conclusion
Two equations for the filling factor estimation of infiltrated ZnO in silica opal and GaN in ZnO opal have been considered. The first equation is based on effective medium approximation, while the second one -on Maxwell-Garnett approximation.
Filling factor equations can be simplified for silica opal because of its week dependence of refractive index on wavelength. However, the filling factor equations for opal comprising ZnO balls requires a numerical solution.
The comparison between filling factors shows that both of them can be equally used for quantity estimation of infiltrated material inside opal matrix. However, at complete infiltration the relative error between filling factors for ZnO opal infiltrated with GaN may exceed 15% due to small refractive index contrast between zinc oxide balls and infiltrated gallium nitride nanoparticles.