# Andreev reflection graphene - metallic superconductor

We have Bogoliubov-De Gennes (BdG) equation, $$\left(\begin{array}{cc} \mathbf{p}\cdot\boldsymbol{\sigma} - V & \Delta_{0}e^{i\phi} \\ \Delta_{0}e^{-i\phi} & V - \mathbf{p}\cdot\boldsymbol{\sigma} \end{array} \right) \left( \begin{array}{c}u\\v\end{array}\right) = \mathcal{E}\left( \begin{array}{c}u\\v\end{array}\right)$$ with definition, $$\mathbf{p}\cdot\boldsymbol{\sigma} = -i\hbar v_{F}(\partial_{x}\sigma_x + \partial_y\sigma_y)$$ We solve the equation through ansatz, $$\Phi = {\rm Const}\cdot \exp(iqy + ik_{0}x + \kappa x)$$ for $$k_{0} > 0$$ and $$\kappa > 0$$. The question is: How to use the assumption that $$V >> \Delta_{0}, \mathcal{E}$$ in the calculations to get nice looking formulas like in Beenakker (A13)?

• Did you at least tried to find $k_{0}$, $q$ and $\kappa$ ? The assumption come from their expressions ... Sep 2 '14 at 9:14
• Yes I did, but it involves double square root (nested). You can find Yourself expressions for $k_{0}$ and $\kappa$ in Mathematica which depend on $q$. Sep 2 '14 at 20:04

• solve the expressions for the eigenvalues for $$\kappa$$
• neglect terms proportional to $$\kappa^2$$, $$\kappa \Delta$$ in the expressions for the eigenvectors, expand them to first order in $$\kappa$$, insert the expression for $$\kappa$$ you got from the eigenvalues
• simplify everything (using the expression for $$\epsilon$$ and approximate $$\hbar v k \approx E_F+U$$) and arrive at Beenakker's simple-looking expressions (tedious)