# Investigation of Ytterbium Incorporation in Lithium Niobate for Active Waveguide Devices
## 3 Results and Discussion
### Diffusion Theory  
The Yb concentration profiles obtained in this work can be well approximated utilizing the one-dimensional diffusion theory described by [PERSON]'s second law of diffusion. \[\frac{\partial C}{\partial t}=D\frac{\partial^{2}C}{\partial y^{2}}, \tag{1}\]  
where \(D\) is the diffusion coefficient and \(y\) quantifies the direction normal to the surface. Assuming an inexhaustible source (i.e., a thick film) of Yb on the surface, the solution of [PERSON]'s law (1) can be approximated by \[C(y)=C_{\max}\operatorname{erfc}(y/d_{eff}), \tag{2}\]  
with \(C_{\max}\) being the concentration of Yb at the crystal surface and the diffusion depth \(d_{eff}=\sqrt{4 Dt}\). In the thick-film diffusion regime. \(C_{\max}\) describes the solubility of Yb in LiNbO\({}_{3}\). This value is given by  
\[C_{\max}=\sqrt{\frac{\pi}{4 Dt}}\int\limits_{0}^{\infty}C(y)dy. \tag{3}\]  
Using the law of mass conservation, the time needed for the Yb source to be depleted can be calculated with the metallic Yb density \(\rho\), the atomic mass \(m_{\Yb}\), and the thickness \(\tau\) of the initial Yb layer  
\[t_{d}=\left(\frac{\rho\tau}{2m_{\Yb}\,C_{\max}}\right)^{2}\frac{\pi}{D}. \tag{4}\]  
For sufficiently thin films and/or sufficiently long in-diffusion times (\(t>t_{d}\)), the Yb source is depleted and the diffusion profile can be approximated by a Gaussian function  
\[C(y)=C_{0}\exp(-\frac{y^{2}}{d_{\exp}^{2}}). \tag{5}\]  
In this case, \(d_{\exp}=\sqrt{4 Dt}\) describes the 1/e-depth of the profile. The surface concentration \(C_{0}\) is below the solubility and given by  
\[C_{0}=\frac{\tau\,C_{\max}}{\sqrt{\pi Dt}}. \tag{6}\]  
The temperature dependence of the diffusion coefficient \(D\) is given by the Arrhenius relation  
\[D(T)=D_{0}\exp(-\frac{E_{A}}{k_{B}T}) \tag{7}\]  
Here, \(D_{0}\) is the diffusion constant, \(E_{A}\) the activation energy, \(k_{B}\) the Boltzmann constant, and \(T\) is the temperature used for in-diffusion. As the solubility describes the equilibrium between Yb in LiNbO\({}_{3}\) and the pure metallic Yb on the surface, the concentration at the surface can also be written in terms of an Arrhenius-type equation as  
\[C_{\max}(T)=\hat{C}\exp(-\frac{\Delta H}{k_{B}T}), \tag{8}\]  
where \(\Delta H\) is the mixing enthalpy and \(\hat{C}\) is a pre-factor. Figure 1: Secondary neutral mass spectrometry (SNMS) spectra of x-cut Yb:LiNbO\({}_{3}\). The two samples were coated with 20 nm Yb, and the diffusion parameters are (**a**) \(T=1060\)\({}^{\circ}\)C, \(t=30\) h and (**b**) \(T=1130\)\({}^{\circ}\)C, \(t=30\) h, respectively. Figure 2 shows the high-resolution images of the crystal surfaces obtained using a confocal microscope and applying the differential interference contrast method for the two crystal orientations. When the temperature used for annealing is less than 1130 \({}^{\circ}\)C, an increased surface roughness caused by insufficient diffusion is visible. For the x-cut surfaces, the arithmetical mean height (_Sa_) slightly decreases from 12 to 10 nm when the temperature increases from (a) 1000 to (b) 1060 \({}^{\circ}\)C.  
For the z-cut surface shown in (d) and (e), the _Sa_ is reduced from 11 to 7 nm. Annealing for 30 h at 1130 \({}^{\circ}\)C results in a smooth surface with _Sa_ = 2 nm for both (c) x- and (f) z-cut surfaces, indicating that the Yb ion source at the surface is completely exhausted, and the Yb profile can be described by the thin-film diffusion regime of Equation (5). For calibration of the SNMS data, the samples annealed at 1130 \({}^{\circ}\)C were used assuming a linear relation between ion yield and concentration as well as mass conservation in the diffusion process. For lower temperatures, the experimentally obtained profiles can be best fitted by complementary error functions, as can be seen in Figure 3.  
\[C(y)=C_{\max}\operatorname{erfc}(y/d_{eff}), \tag{2}\]0 SNMS depth profiles of Yb in LiNbO\({}_{3}\) for in-diffused layers with an initial thickness of 20 nm annealed for 30h at different temperatures for (**a**) x-cut and (**b**) z-cut substrates.