(1) Determining the wavefunction:
The distribution functions 
and 
are obtained by solving the scattering problem described by following Bogoliubov–de Gennes (BdG) equation [5]:
 |
(9) |
The single-particle Hamiltonian H0 is defined as
 |
(10) |
where 
, m is the effective mass of the electrons, and 
is the Fermi wavevector. The third term in eqn.(10) represents a δ function type scattering potential at the interface characterized by the dimensionless parameter Z. The exchange field hex(z) is given by
 |
......(11) |
and the SC gap is
 |
........(12) |
Therefore, the solution for BdG equation takes the form
 |
........(13) |
Where fσ(z) and gσ(z) are the electron and hole components of the wavefunction, respectively. Substituting eqn.(13) in eqn.(9) results;
  |
.......(14) |
Where uo and vo are the coherence factors defined as
  |
(15) |