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)?
 A: When you write down the formulas A15-A18, apply this assumption to these definitions and put it into wave functions, which are first calculated by solving BdG equation. You can reach the Beenakker results.
A: This is actually exactly the same question I was asking myself a while ago, and it took me quite some time to figure it out.
What I ended up doing was:

*

*diagonalise the BdG Hamiltonian in Mathematica

*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)

I hope that helps some, in case you haven't figured it out yourself in the meantime. If you need more info/details don't hesitate to ask, I can elaborate some more if needed!
