# Revisiting the Meandering Instability During Step-Flow Epitaxy
## 2 Stability Analysis
### The \(\Theta\) Dependence of the Stability Function \(\omega(k)\)
From the straight forward calculation, we can get  
\[f(k)=\frac{NM\Gamma\Lambda_{k}[\Lambda_{k}(d_{-}+d_{+})\tanh(\Lambda_{k}l)+2( 1-\text{sech}(\Lambda_{k}l))]}{\Lambda_{k}(d_{-}-d_{+})(\text{sech}(\Lambda_{k }l)-1)\Theta+(\Lambda_{k}^{2}d_{+}d_{-}+1)\tanh(\Lambda_{k}l)+\Lambda_{k}(d_{ +}+d_{-})} \tag{21}\]  
and  
\[g(k)=\frac{NM\Phi}{x_{s}^{2}}\cdot\mathfrak{g}, \tag{22}\]  
and  
\[g(k)\rightarrow\frac{NM\Phi}{x_{s}^{2}}\cdot\frac{\Lambda_{k}\Theta[(1+\text{ sech}(l/x_{s}))(\cosh(\Lambda_{k}l)-1)-\Lambda_{k}x_{s}\sinh(\Lambda_{k}l)\tanh(l/x_{s})]}{ \begin{array}{c}+\Lambda_{k}[\text{sech}(l/x_{s})+\Lambda_{k}x_{s}\sinh( \Lambda_{k}l)\tanh(l/x_{s})-\cosh(\Lambda_{k}l)]\\ \begin{array}{c}[(\text{sech}(l/x_{s})-1)\Theta+1+d_{+}/x_{s}\tanh(l/x_{s})] \\ \begin{array}{c}[\Lambda_{k}(1-\cosh(\Lambda_{k}l))\Theta+\Lambda_{k}^{2}d_{+} \sinh(\Lambda_{k}l)+\Lambda_{k}\cosh(\Lambda_{k}l)]\end{array}\end{array} }. \tag{25}\]  
Therefore, \(\omega(k)=0\), that is, \(k^{2}f(k)=g(k)\) gives that  
\[k^{2}\Gamma\Lambda_{k}^{2}\tanh(\Lambda_{k}l) = \frac{\Theta\Lambda_{k}}{x_{s}^{2}}\cdot\frac{\begin{array}{c} \Theta[(1+\text{sech}(l/x_{s}))(1-\text{sech}(\Lambda_{k}l))-\Lambda_{k}x_{s} \tanh(\Lambda_{k}l)\tanh(l/x_{s})]\\ \begin{array}{c}+[\text{sech}(l/x_{s})\operatorname{sech}(\Lambda_{k}l)+ \Lambda_{k}x_{s}\tanh(\Lambda_{k}l)\tanh(l/x_{s})-1)]\\ \begin{array}{c}(\text{sech}(l/x_{s})-1)\Theta+1+d_{+}/x_{s}\tanh(l/x_{s}) -1)]\\ \begin{array}{c}(\text{sech}(l/x_{s})-1)\Theta+1+d_{+}/x_{s}\tanh(l/x_{s})-1 )]\\ \end{array}}.\end{array}  \tag{26}\]  
The analytic expressions for \(k_{c}\) can be obtained from (26) in the limiting cases of very large and very small terrace widths. When \(l>>x_{s}\), we have  
\[k_{c}x_{s}= \begin{cases}\sqrt{\frac{4}{3}\left(1-\frac{2\zeta}{x_{s}}\cdot\frac {(1-\Theta)+d_{+}/x_{s}}{1-\Theta}\right)}&\text{if $k_{c}x_{s}<<1$}\\ \sqrt{\frac{x_{s}}{\zeta}\cdot\frac{1-\Theta}{(1-\Theta)+d_{+}/x_{s}}}&\text{ if $k_{c}x_{s}>>1$},\end{cases}  \tag{27}\]  
where \(\zeta=\frac{\Gamma}{\Phi}\) is the surface capillary length. Since \(k_{c}\) is real, the prerequisite for morphological instability is \(\zeta<\frac{x_{s}(1-\Theta)}{2(1-\Theta)+2d_{+}/x_{s}}\). Figure 3 illustrates the dependence of \(\Theta\) on the cutoff \(\zeta\) for different values of \(d_{+}/x_{s}\). When \(l<<x_{s}\) (and \(l^{2}<<\lambda_{c}^{2}\), the critical wavelength \(\lambda_{c}=2\pi/k_{c}\)),  
\[k_{c}x_{s}=\sqrt{\frac{1}{2\zeta}\left(\frac{l}{1+\frac{d_{+}l}{x_{s}^{2}}} \right)-1}. \tag{28}\]  
The cutoff here is \(\frac{x_{l}^{2}l}{2(x_{s}^{2}+d_{+}l)}\). **Remark 1**. : _By using a similar method, we can compute for the limiting situation where the step attachment is infinitely fast from the lower terrace and infinitely slow from the upper terrace (\(d_{-}\rightarrow\infty,d_{+}=0\)). We can show that_  
\[V_{0}=MN\left(\frac{A}{x_{s}}-\frac{A}{x_{s}}\cosh(l/x_{s})-\frac{B}{x_{s}}\sinh (l/x_{s})\right) \tag{29}\]  
_and_  
\[\omega(k)=MN\left(\frac{B}{x_{s}^{2}}+A_{k}\Lambda_{k}-\frac{A\sinh(l/x_{s})} {x_{s}^{2}}-\frac{B\cosh(l/x_{s})}{x_{s}^{2}}-A_{k}\Lambda_{k}\cosh(\Lambda_{ k}l)-B_{k}\Lambda_{k}\sinh(\Lambda_{k}l)\right) \tag{30}\]  
_which is in the form of \(g(k)-k^{2}f(k)\). From the straight forward calculation, we obtain_  
\[f(k)=-\frac{NM\Gamma\Lambda_{k}\sinh(\Lambda_{k}l)}{\Theta(\cosh(\Lambda_{k}l )-1)-\cosh(\Lambda_{k}l)} \tag{31}\]  
_and_  
\[g(k) = -\frac{NM\Phi}{x_{s}^{2}}\times \tag{32}\] \[\bigg{\{}\frac{\Theta[1+\sinh(\Lambda_{k}l)\sinh(l/x_{s})\Lambda _{k}x_{s}-\cosh(\Lambda_{k}l)\cosh(l/x_{s})-\cosh(\Lambda_{k}l)+\cosh(l/x_{s} )]}{[\Theta(\cosh(\Lambda_{k}l)-1)-\cosh(\Lambda_{k}l)][\Theta(\cosh(l/x_{s} )-1)-\cosh(l/x_{s})]}\] \[+\frac{\cosh(\Lambda_{k}l)\cosh(l/x_{s})-1-\sinh(\Lambda_{k}l) \sinh(l/x_{s})\Lambda_{k}x_{s}}{[\Theta(\cosh(\Lambda_{k}l)-1)-\cosh(\Lambda_{ k}l)][\Theta(\cosh(l/x_{s})-1)-\cosh(l/x_{s})]}\bigg{\}}.\]  
_Notice that \(0<\Theta<1\) implies that \(\Theta(\cosh(\Lambda_{k}l)-1)-\cosh(\Lambda_{k}l)<0\) and \(\Theta(\cosh(l/x_{s})-1)-\cosh(l/x_{s})<0\). Hence \(f(k)\) is positive.