2 Amplitude Reflection Coefficient for an Electron Pure State, at a Single Step in Electric Potential and Magnetic Flux Density

Figure 2: Reflection of an Electron Wave by a Single Step in Electric Potential and Magnetic Flux Density

The first step in the analysis of reflection is to build a potential-theory model of the sample, as a series of steps in electric potential and magnetic flux density (figure 1, figure 2.) Next, we need to discover the amplitude reflection coefficient, for a pure, coherent, electron wave, at a single step (figure 2.) The incident and transmitted electron waves are modelled as plane waves, with well-defined wave-vector components $p$ in the plane of the interface, and $q_i$ perpendicular to the interface. $p$ must be the same for all the waves, in order to satisfy the boundary condition of continuity of the wave-function at the interface. Strictly, the eigen-states of a Hamiltonian which includes a magnetic field are not plane waves; more about this later (section 5.) The amplitude reflection coefficient is [16] this:

$$r_{01} = \frac{q_0-q_1}{q_0+q_1}\textrm{,}$$ (3)

or, for a general interface, this one:

$$r_{ij} = \frac{q_i-q_j}{q_i+q_j}\textrm{.}$$ (4)

Next, we need to build an expression for the energy of the electrons. There will be kinetic energy terms, along with an electrostatic potential energy, and a term due to the torque, on the electron magnetic moment, in a magnetic field [16]. The form used for this last term assumes a well-defined energy for all directions of the electron spin. Strictly, only certain spin directions are eigen-states of a Hamiltonian which includes a magnetic field; more about this later (section 5.) This leads to this expression

$$E = E\sin^2I+\frac{\hbar^2q_i^2}{2m_e}-eV_i+\frac{e\hbar{}B_i\cos{}S_i}{2m_e}$$ (5)

for the perpendicular wave-vector component, where $I$ represents an angle of incidence, and the potential energy terms are represented by these dimensionless numbers:

$$x_i = y_i+z_i\cos{}S_i\textrm{,}$$ (6)

$$y_i = \frac{eV_i}{E\cos^2I}\textrm{,}$$ (7)

$$z_i = -\frac{e\hbar{}B_i}{2m_eE\cos^2I}\textrm{;}$$ (8)

I have a big enough ego to call them the Hatton numbers, but I suspect I wouldn't get away with it. $S_i$ is the angle between the electron spin direction and the magnetic flux density in region $i$, and $E$ is the total energy of the incident electrons, and therefore, by conservation of energy, of all the electrons.

We now use a binomial expansion [6] for the case where the potential energy terms are much smaller than the total electron energy, where the dimensionless numbers we've just devised are small. The magnetic term associated with the Weiss field in a ferromagnet is [15] a few tenths of an electron-volt, and the electrostatic contact potentials in the metals which we study will not be more than a few volts, whereas, in our experimental set-up, the incident electron energies range from a few hundred to a few thousand electron volts, so this approximation seems reasonable. With this expansion, the amplitude reflection coefficient is this:

$$r_{ij} = \frac{1}{4}x_i-\frac{1}{4}x_j-\frac{1}{8}x_i^2-\frac{1}{8}x_ix_j+\frac{1}{8}x_j^2+O(\{x_i,x_j\}^3)\textrm{.}$$ (9)