# Investigation of Ytterbium Incorporation in Lithium Niobate for Active Waveguide Devices
## 3 Results and Discussion
### Diffusion Theory
The dotted lines show the measured profiles and the solid lines indicate complementary error function fits. \[C(y)=C_{\max}\operatorname{erfc}(y/d_{eff}), \tag{2}\]1 Confocal microscope images of LiNbO\({}_{3}\) surfaces coated with 20 nm thick Yb films after annealing for 30 h. Results for x-cut surfaces annealed at (**a**) 1000, (**b**) 1060, and (**c**) 1130 \({}^{\circ}\)C; results for z-cut surfaces annealed at (**d**) 1000, (**e**) 1060, and (**f**) 1130 \({}^{\circ}\)C.  
The profiles obtained using the highest temperature were fitted by Gaussian functions; see \[C(y)=C_{\max}\operatorname{erfc}(y/d_{eff}), \tag{2}\]2 The results obtained for the surface concentration and for the depths \(d_{\textit{eff}}\) or \(d_{\textit{exp}}\) are summarized in \[C(y)=C_{\max}\operatorname{erfc}(y/d_{eff}), \tag{2}\]3  
With the values for the diffusion depth, the diffusion coefficients \(D\) are calculated. The temperature dependence of the diffusion coefficient is given by Equation (7). Figure 5 shows the corresponding Arrhenius plots for the two crystal-cut directions used. By correlating the fitted lines with Equation (7), the values for the diffusion coefficients and the activation energy for the x- and z-direction were obtained. The results are summarized in \[C(y)=C_{\max}\operatorname{erfc}(y/d_{eff}), \tag{2}\]4 \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Temperature** & **Time (h)** & \(\begin{array}{c}C_{0}\times 10^{19}\\ \textbf{(\lx@math@degree{}c)}\end{array}\) & \(\begin{array}{c}C_{\textbf{max}}\times 10^{19}\\ \textbf{(ions/cm^{3})}\end{array}\) & \(\begin{array}{c}d_{\textbf{eff}}\ (\textbf{\mu}\textbf{)}\end{array}\) & \(d_{\textbf{exp}}\ (\textbf{\mu}\textbf{)}\) \\ \hline z-cut & & & & \\
930 & 30 & \(6.96\pm 0.10\) & & \(1.99\pm 0.13\) & \\
1000 & 30 & \(11.53\pm 0.09\) & & \(2.32\pm 0.14\) & \\
1060 & 30 & \(16.89\pm 0.10\) & & \(3.47\pm 0.19\) & \\
1130 & 30 & & \(10.94\pm 0.05\) & & \(4.68\pm 0.26\) \\ x-cut & & & & \\
930 & 30 & \(6.89\pm 0.10\) & & \(1.24\pm 0.08\) & \\
1000 & 30 & \(13.21\pm 0.12\) & & \(1.66\pm 0.10\) & \\
1060 & 30 & \(20.06\pm 0.12\) & & \(2.70\pm 0.15\) & \\
1130 & 30 & & \(13.62\pm 0.06\) & & \(3.84\pm 0.21\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters obtained for the Yb diffusion in LiNbO\({}_{3}\). Figure 4: SNMS depth profiles of Yb in LiNbO\({}_{3}\) for in-diffused layers with an initial thickness of 20 nm annealed for 30 h at 1130 \({}^{\circ}\)C for different cut directions. Figure 5: Arrhenius plot of the diffusion coefficients of Yb in LiNbO\({}_{3}\) for diffusion parallel to the x- and z-axis. The fitted lines correspond to Equation (7). The temperature dependence of the surface concentration is presented in Figure 6. As can be seen, the measured values for the surface concentration of Yb for the three lowest temperatures indicate that, in contrast to the diffusivity, the solubility of Yb in LiNbO\({}_{3}\) shows no anisotropic behavior. The solid line is a fit to the data with Equation (8). This way, the values of \(\Delta H=(1.04\pm 0.23)\) eV and \(\hat{C}=(1.6\pm 0.35)\times 10^{24}\) cm\({}^{-3}\) were found. With these characteristic values for the diffusion of Yb in LiNbO\({}_{3}\), a first test was performed for diffusion doping of a ridge waveguide prepared in an x-cut LiNbO\({}_{3}\) substrate oriented parallel to the y-axis by diamond blade dicing. Figure 7a shows the simulated Yb concentration profile in the ridge for the case of two 22 nm thick Yb layers being coated under \(\pm 60^{\circ}\) with respect to the x-axis obtained by numerically solving [PERSON]'s second law. For in-diffusion, the sample is annealed for 216 h at 1125 \({}^{\circ}\)C. The concentration profile also shows that at the upper right and left corners of the ridge, the expected concentration is below the solubility limit. Thus, no additional surface defects are to be expected. To achieve waveguiding, the upper part of the ridge is additionally doped by in-diffusion of Ti.