# Structure and Optical Properties of Transparent Cobalt-Doped ZnO Thin Layers
## 3 Results and Discussion
### Optical Measurements
#### 3.2.2 Refractive Index and Thickness  
Assuming that the layer absorbs poorly and the substrate is completely transparent, the refractive index \(n\) and the light extinction coefficient \(k\) of the layers can be estimated from the transmission spectra using the envelope method [42; 43]. A typical transmission spectrum \(T\%(\lambda)\) of the ZnO:Co layer (annealed at 400 \({}^{\circ}\)C) with interference patterns is shown in Figure 2b. Three approximations are also shown: \(T_{M}\), \(T_{m}\), and \(T_{a}\) for layers of ZnO:Co. The theoretical transmittance \(T_{th}\) is also illustrated. For comparison, the line \(T_{a}\) is an experimental approximation. The refractive index \(n\) is an important parameter for optical materials for optoelectronics applications [42; 44]. From the curves \(T_{M}(\lambda)\) and \(T_{m}(\lambda)\) of the layers in the spectral range of weak and medium absorption, the index \(n\) can be calculated using Equation (4):  
\[n=\sqrt{N+\left(N^{2}-n_{s}^{2}\right)^{1/2}} \tag{4}\]  
\[N=2n_{s}.\frac{T_{M}-T_{m}}{T_{M}\cdot T_{m}}+\frac{n_{s}^{2}+1}{2} \tag{5}\]  
\[n_{s}=\frac{1}{T_{S}}+\sqrt{\frac{1}{T_{S}^{2}}-1} \tag{6}\]  
where \(T_{M}\) and \(T_{m}\)--the maximum and minimum transmission at a given wavelength, respectively; \(T_{s}\)--the transmission of the substrate; \(n_{s}\)--the refractive index of the substrate (\(n_{s}=1.62\)). The layer thickness \(d\) can be calculated from the equation:  
\[d=\frac{\lambda_{1}\cdot\lambda_{2}}{2(\lambda_{1}n_{2}-\lambda_{2}n_{1})} \tag{7}\]  
where \(\lambda_{1}\) and \(\lambda_{2}\)--wavelengths corresponding to successive interference maxima; \(n_{1}\) and \(n_{2}\)--the corresponding refractive indexes. The thickness values are given in Table 1. \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **\(\lambda\), nm** & \(d\), nm** \\ \hline
1 & 1063 & - \\
2 & 893 & 1446 \\
3 & 777 & 1541 \\
4 & 701 & 1847 \\
5 & 624 & 1464 \\
6 & 571 & 1735 \\ \hline \end{tabular}
\end{table} 
Table 1: The maximum interference pattern \(\lambda\) and the layer thickness \(d\). Practical modelled interferences for thin layers on a transparent substrate are shown in Figure 3a. The transparent substrate with the refractive index \(n_{s}\) and the transmittance \(T_{S}\) has a thickness of several orders of magnitude greater than \(d\). If the layer thickness is homogeneous, the interference effects give rise to a spectrum, shown by the full curves. The thickness, refractive index, absorption index, and transmittance of the ZnO:Co layer were estimated. The transmittance spectrum has the description of the Swanepoel method to match the optical parameters [43; 45; 46]. The transmission \(T\) for normal incidence resulting from the interference of the wave transmitted from the interfaces can be written as:  
\[T=T(n,x)=\frac{Ax}{B-Cxcos(\varphi)+Dx^{2}} \tag{8}\]  
where the matching parameters are: \(A\) = 68, \(B\) = 106, \(C\) = 0.11, \(D\) = \(-\)0.24, \(\varphi\) = 4\(\pi\)\(nd\)/\(\lambda\), \(x\) = _exp_(-\(\alpha\)), and \(\alpha\) = 4 \(\pi\)\(x\)/\(\lambda\). The curves for \(d\) = 1500 nm were adjusted for the two parameter values \(\alpha\) = 10\({}^{4}\) cm\({}^{-1}\) and \(\alpha\) = 2 \(\times\) 10\({}^{4}\) cm\({}^{-1}\). For a thin layer on a non-absorbent substrate, the optical transmission of \(T\) for normal incidence is determined by the Swanepoel equations. Parameters \(n\) and \(k\) are the real and imaginary parts of the refractive index of the thickness of the thin layer, \(d\) is the thickness of the layer, and \(n_{s}\) is the actual refractive index of the substrate (Equation (3)) [47; 48]. The computational work was carried out on a simple theoretical model built on the basis of this algorithm. The substrate refractive index and the \(n\) and \(k\) values are calculated from the equations. Using the estimation, we can obtain the theoretical value of the transmittance, and, then, by appropriately evaluating the various predicted values of thickness d, we can calculate the exact thickness of the layer.