# Structure and Optical Properties of Transparent Cobalt-Doped ZnO Thin Layers
## 3 Results and Discussion
### Optical Measurements
#### 3.2.5 Quantitative Assessment of the Effects of Urbach Tail  
The calculated optical band gap for the layers is smaller than for the pure ZnO (3.37 eV). The reduction in the value of the optical band gap can be attributed to the location of the Co\({}_{\text{3p}}\) states in the ZnO layer above the top of the valence band and in combination with the O\({}_{\text{2p}}\) states, leading to a narrowing of the gap. To estimate the characteristics of ZnO:Co near the conduction band, the absorption behavior at lower photon energy is interpreted by [PERSON]'s rule: \(\alpha=K\exp\left(E/E_{U}\right)\), where \(K\) is constant and \(E_{U}\) is Urbach energy, which is interpreted as the formation of localized tails in the width of the gap. This region results from the transitions between states stretched in one band and states located in the exponential tail of the other, and as a consequence of all defects. It was observed that the increase in layer thickness leads to an increase in the value of dispersion energy.  
The presence of high concentrations of states located in thin layers causes the optical gap width to be reduced. Therefore, the addition of Co ions increases the concentration of states located in the layer, which reduces the band gap. The inclusion of Urbach endpoints in the linear selection method may reduce the resulting extrapolated gap width, and, therefore, also reduces the corresponding value of the slope. The plot showing the absorption of the Urbach tail, which can usually be described by an exponential function, is shown in Figure 7.  
To determine the size of Urbach tail in the graph, we introduced a value called the edge absorption coefficient (NEAR). In a perfect material that has absolutely no Urbach tail, the value (\(\alpha h\
u\))\({}^{2}\) will be zero with respect to the optical gap. Yet, there is always additional absorption in real materials. To estimate this, we compare the value (\(\alpha h\
u\))\({}^{2}\) at the extrapolated optical gap width with a value taken at slightly higher energy, where most of the signal will come from the band transition. The square root of this ratio is then taken into account to further extend the NEAR concept to potential applications in indirect absorb  
Figure 7: **(a,b)** Spectra for calculations of NEAR coefficients as grown (**a**) and annealed in 400 \({}^{\circ}\)C (**b**) thin layer of ZnO:Co.  
materials. If the Urbach tail is significant, the NEAR will approach one. If the material is without defects, the NEAR value will be small (ideally approaching 0). In the following, the NEAR value (Figure 7) for each of the plots was calculated. This way, we were able to examine how precise the Tauc slope values were and whether the precision decreased with more flawed material:  
\[NEAR=1.02\cdot\sqrt{\frac{\left(\alpha h\
u\right)^{2}\left| \right._{h\
u=E_{g}}}{\left(\alpha h\
u\right)^{2}\left|\right._{h\
u=1,02E_{ g}}}}=\frac{\alpha\left(E_{g}\right)}{\alpha\left(1.02\cdot E_{g}\right)} \tag{14}\]  
The following results were obtained from Equation (13) for ZnOCo layers annealed at different temperatures: NEAR = 0.98 (as growth) and NEAR = 0.91 (annealed in 400 \({}^{\circ}\)C).