A New, Classical-Field Theory of Polarized Electron Reflection: Amplitude Reflection Coefficient for an Electron Pure State, at a Single Step in Electric Potential and Magnetic Flux Density

Figure 2.2: Surface of a Bulk Magnetic Sample

Figure 2.3: 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 (figures 2.2, 2.3.) The existence of sudden steps in magnetic flux density follows [35] from Maxwell's equations, for an in-plane magnetization. Shape anisotropy encourages [36,37] the magnetizations of film structures to be in-plane, and experiments confirm [38,39,40,41,42] that the permanent magnetization in $Co/Cu(001)$ is in-plane. The existence of sudden steps in electrostatic potential is supported by a comment in Calculated Electronic Properties of Metals [43], although this book is concerned primarily with bulk materials, and there is the possibility that steps, which are sudden on the scale of bulk materials, may be gradual on the scale of electron de Broglie wavelengths. I suspect that such a possibility would be easily detectable in the experiments, because there would be no reflection without a sudden potential step.

In reality, I expect the ``magnetic flux density'' part of the potential not to be a genuine magnetic flux density, but a Weiss field; in this paragraph, I will explain what the Weiss field is, and why I expect it, rather than a genuine magnetic flux density, to determine the behaviour of the electron beam. The Weiss field is a quantum-mechanical effect related to symmetry requirements on the joint wave-function of a pair of electrons. To illustrate the effect, following loosely arguments in Solid State Physics [44] and Quantum Physics [45], consider two electrons, labelled $i = 1$ and $i = 2$, occupying two spatial states $\vert\psi_i>$ and $\vert\phi_i>$, which are similar, atomic-orbital-like states, centred on two neighbouring atomic sites, in a solid, and which also have possible spin states $\vert\uparrow_i>$ and $\vert\downarrow_i>$. It is a requirement of quantum mechanics that the two electrons are indistinguishable from one another; therefore, their joint spatial state cannot be a simple state such as $\vert\psi_1>\vert\phi_2>$, which would identify a particular electron as being on each atomic site. To fulfil the indistinguishableness requirement, the state must be such that the only effect of exchanging the particle labels is to introduce a uniform phase shift in the wave-function, i.e. the joint spatial state must be $\frac{1}{\sqrt{2}}(\vert\psi_1>\vert\phi_2>+\exp{}(i\theta_A)\vert\psi_2>\vert\phi_1>)$, where $\theta_A$ is the uniform phase shift in question. Similarly, indistinguishableness limits the joint spin states to $\vert\uparrow_1>\vert\uparrow_2>$, $\vert\downarrow_1>\vert\downarrow_2>$, or $\frac{1}{\sqrt{2}}(\vert\uparrow_1>\vert\downarrow_2>+\exp{}(i\theta_B)\vert\uparrow_2>\vert\downarrow_1>)$, where $\theta_B$ is the uniform phase shift involved in particle exchange due to the spin part of the wave-function alone. Given that electrons are fermions, quantum mechanics, further, demands that the total phase shift on particle exchange, $\theta_A+\theta_B$, or simply $\theta_A$, where no $\theta_B$ is involved in the spin wave-function, is $\pi$. One can identify the concept of the two electrons' spins being parallel with the spin states where no $\theta_B$ is involved, or where $\theta_B = 0$; therefore, when the electrons' spins are parallel, $\theta_A = \pi$, and the joint spatial state is $\frac{1}{\sqrt{2}}(\vert\psi_1>\vert\phi_2>-\vert\psi_2>\vert\phi_1>)$; this is an odd function of the spatial displacement between the two electrons, and must, therefore, vanish when this displacement is zero, prohibiting the two electrons from being spatially coincident and, providing that the wave-function varies smoothly with this displacement, rendering the probability of them being very close together low. Given that there is an electrostatic repulsion between the electrons, this is a low-energy state. Similarly, one can identify the concept of the two electrons' spins being anti-parallel with the $\theta_B = \pi$ spin state, giving a joint spatial state $\frac{1}{\sqrt{2}}(\vert\psi_1>\vert\phi_2>+\vert\psi_2>\vert\phi_1>)$; this even function of the displacement between the two electrons does not prevent the electrons from being very close together, or even spatially coincident; their electrostatic repulsion, therefore, renders it a high-energy state. To summarize, the state in which the electrons' spins are anti-parallel has a higher energy, by some amount $\Delta{}V$, than the state in which their spins are parallel; it is as if each electron's spin generates a magnetic flux density $B = \frac{m_e\Delta{}V}{e\hbar}$, acting on the spin magnetic moment of the other electron. The Weiss field is this effective magnetic flux density, produced by quantum-mechanical symmetry considerations, scaled up to a situation where an electron interacts not just with one other electron, but with the whole population of other electrons in a solid. I expect the Weiss field, not the genuine magnetic flux density, to be the effect detected with the reflected electron beam, because the Weiss field in transition metals is [44] roughly a factor of $100$ larger than the genuine magnetic flux density in the remnant state; however, some caution about this expectation is in order, because, as is clear from the argument above, the spatial states of the electrons are crucial in producing the Weiss field, and the spatial states of the unbound electrons being reflected are very similar neither to the single-site atomic orbitals described above, nor to the states of the transition metals' $3d$ electrons, for which the factor of $100$ is known to hold good.

It will be noted that, in the model, both the electrostatic potential and the Weiss field are taken to be laterally homogeneous, i.e. not to vary with position in the plane of the film; this gives the appearance of being a rather strong assumption about the configuration of the magnetic film structure being studied. In particular, it gives the appearance of an implicit assumption of single-domain magnetization; this is an assumption that has been made in other workers' [46,47,41,42] explanations of the behaviour of $Co/Cu(001)$ film structures, and which has [39] some justification from Brillouin light scattering experiments; it is (section 2.9) also supported by magneto-optical Kerr effect measurements. In reality, the use of laterally homogeneous models does not imply any assumption about the configuration of the film structure or its domain structure; this study is concerned with the specular reflection, which is the scattered beam, the in-plane component of whose wave-vector is the same as the in-plane component of wave-vector of the incident beam. In the general theory of scattering of matter waves by a potential, a scattered beam whose in-plane wave-vector component differs from that of the incident beam by $\mathbf{k}$ results from the Fourier component of the scattering potential with in-plane wave-vector $\mathbf{k}$: specifically, the specularly reflected beam results from the average value of the potential, over the area of the sample covered by the incident beam, and is unaffected by variations in the scattering potential, such as domain structure, which are encoded in the Fourier components of the scattering potential with non-zero in-plane wave vectors, the effects of which appear only in the diffraction orders. Of course, if there are domains of varying magnetization directions, within the region illuminated by the incident beam, then the average Weiss field detected by polarized electron reflection will be smaller than the saturation Weiss field within a single domain, but the situation is still within the capability of this theory to handle.

Next, I need to discover the amplitude reflection coefficient, for a pure, coherent, electron wave, at a single step (figure 2.3.) 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 2.5.) The amplitude reflection coefficient is [30] this:

$$r_{01} = \frac{q_0-q_1}{q_0+q_1}\textrm{,}$$ (2.1)

or, for a general interface, this one:

$$r_{ij} = \frac{q_i-q_j}{q_i+q_j}\textrm{.}$$ (2.2)

Next, I 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 [30]. 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 that includes a magnetic field; more about this later (section 2.5.) This leads to this expression

$$q_i = \left(\frac{2m_eE\cos^2I}{\hbar^2}\right)^{1/2}(1+x_i)^{1/2}\textrm{,}$$ (2.3)

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

$$x_i = y_i+z_i\cos{}S_i\textrm{,}$$ (2.4)

$$y_i = \frac{eV_i}{E\cos^2I}\textrm{,}$$ (2.5)

$$z_i = -\frac{e\hbar{}B_i}{2m_eE\cos^2I}\textrm{;}$$ (2.6)

$S_i$ is the angle between the electron spin direction and the magnetic flux density in region $i$, and $E_b$ is the total energy of the incident electrons, and therefore, by conservation of energy, of all the electrons.

I now use a binomial expansion [48] for the case where the potential energy terms are much smaller than the total electron energy, where the dimensionless numbers I've just devised are small. The magnetic term associated with the Weiss field in a ferromagnet is [43] a few tenths of an electron-volt, and the electrostatic contact potentials in the metals which I study will [49,50] not be more than a few volts, whereas, in my 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:

$$r_{ij} = \frac{1}{4}x_i-\frac{1}{4}x_j-\frac{1}{8}x_i^2-\frac{1}{8}x_ix_j+\frac{1}{8}x_j^2+O(\{x_i,x_j\}^3)\textrm{.}$$ (2.7)