## 2 Probabilities of Spin-Dependent Scattering by Nanomagnets  
The Pauli equation of an electron moving in the magnetic field of the nanomagnet may be presented in the form  
\[\hat{H}=\frac{1}{2m}\big{[}\hat{p}+\frac{e}{c}\vec{A}\big{]}^{2}+\mu_{B}\hat{ \sigma}\cdot\vec{B} \tag{1}\]  
Here \(m\) and \(e\) are the electron mass and its absolute charge respectively, \(\hat{p}\) is its momentum operator, \(\vec{A}=\vec{\mu}\times\vec{r}/r^{3}\), is the vector potential of the magnetic moment \(\vec{\mu}\), \(\vec{r}\) is the radius vector, \(\vec{B}=\vec{\
abla}\times\vec{A}\), \(\mu_{B}=e\hbar/2 mc\) is the Bohr magneton, and \(\hat{\sigma}\) are the Pauli matrices. Below we consider the case when the magnetic moment of the nanomagnet \(\vec{\mu}\) is in the \(x-z\) plane. For this case the Equation (1) is reduced to the 2D problem and takes the form  
\[\hat{H}=\frac{1}{2m}\hat{p}_{2}^{2}+\hat{V}(\vec{\rho}) \tag{2}\]  
Here \(\hat{p}_{2}^{2}\) and \(\vec{\rho}\) are the two dimensional momentum operator and the radius vector of the electron, respectively. The second term in Equation (2) describes the interaction between the magnetic moments of the electron and nanomagnet in the dipole-dipole approximation, which can be written as  
\[\hat{V}(\vec{\rho})=\frac{\mu_{B}}{\rho^{3}}\bigg{(}3(\vec{\mu}\cdot\vec{n}) \vec{n}\cdot\hat{\sigma}-\vec{\mu}\cdot\hat{\sigma}\bigg{)} \tag{3}\]  
where \(\vec{n}\) is a unit vector along the radius vector \(\vec{\rho}\). Treating Equation (3) as a small perturbation, we use the Fermi golden rule and write down the probability of transition of the electron from the initial state \(i\) to the final state \(f\) as follows [11]  
\[dW_{fi}=\bigg{(}\frac{2\pi}{\hbar}\bigg{)}|V_{fi}|^{2}\delta(E_{f}-E_{i})d\
u_{f} \tag{4}\]  
where \(E_{i}\) and \(E_{f}\) are the energies of the electron in the initial and final state, respectively. \(V_{fi}\) is the matrix element of perturbation Equation (3), \(d\
u_{f}\) is the interval of the quantum numbers which corresponds to the final states. Now we apply this formula to the transition from the state of the incident particle with momentum \(\vec{p}_{i}\) to the state with momentum \(\vec{p}_{f}\). The interval of states \(d\
u_{f}\) can be written as \(d^{2}p_{f}/(2\pi\hbar)^{2}\). First, we express the difference between energies in terms of their momenta  
\[E_{f}-E_{i}=(p_{f}^{2}-p_{i}^{2})/2m \tag{5}\]  
Substituting Equation (5) into Equation (4) and using the property of delta function, we obtain  
\[dW_{fi}=\bigg{(}\frac{m}{\pi\hbar^{3}}\bigg{)}|\hat{V}_{fi}|^{2}\delta(p_{f}^{ 2}-p_{i}^{2})d^{2}p_{f} \tag{6}\]  
The wave functions of the incident and the scattered electron are the products of the plane waves and spin wave functions  
\[|i\rangle =\sqrt{\frac{m}{p}}\exp\big{(}\frac{i\vec{p}_{i}\cdot\vec{\rho}}{ \hbar}\big{)}\vec{\chi}(S_{i}) \tag{7}\] \[|f\rangle =\exp\big{(}\frac{i\vec{p}_{f}\cdot\vec{\rho}}{\hbar}\big{)}\vec {\chi}(S_{f}) \tag{8}\]  
The incident wave Equation (7) relates to the state with the momentum \(\vec{p}_{i}\) and \(S_{i}\), and the scattered wave Equation (8) relates to \(\vec{p}_{f}\) and \(S_{f}\), respectively. The incident wave function is normalized by the unit current density and the scattered wave function is normalized by the delta function [11]. With account of the above normalizations of the wave functions, the Equation (6) has dimension of length as it must be in 2D case and describes the differential scattering length.