(2) Understanding various scattering process:
Let us consider the scattering of an electron with the spin up direction injected into the SC from the FM as shown in Figure 19.1. Here, we have four possible scattering processes (see Figure 19.2) in which the incident electron is indicated as io. The scattering processes S1 , S2 , S3, and S4 correspond to AR, normal reflection, transmission as a hole-like quasiparticle, and transmission as an electron-like quasiparticle, respectively.
Therefore, the wavefunction in the FM is expressed as the sum of the incident wave, AR wave, and normal reflected wave:
 |
(19) |
where aσ and bσ,nl are the coefficients for AR and normal reflections, respectively.
In the SC region, we have |
(20) |
where cσ and dσ are the coefficients for transmission waves as a electron-like and hole-like quasiparticle, respectively. All these coefficients can be determined by matching the wavefunctions at the interface:
 |
(21) |
Substituting the eqn.(21) into eqns.(19) and (20) results wavefunction, which subsequently helps to evaluate the probabilities for each scattering process.
(3) Finding the probability current density and probability of each scattering:
The probability density P(z) for finding either an electron or a hole at a particular time and place is expressed as
 |
(22) |
where fσ(z) and gσ(z) are the electron and hole components of the wavefunction in eqn.(13). The probability density satisfies the following conservation relation
 |
(23) |