We'll now begin examination of some of the techniques used to study surfaces. Most of these techniques utilize electron optics; so that's where we'll start.
The deflection of an electron entering an electric field is similar to the refraction of light as a light ray enters a medium with a differing refractive index than the medium it is leaving. The electron is subject to a force F = eE that is directed antiparallel to the electric field vector (because of the negative charge on the electron). Figure 12 shows an electron approaching the top plate of a capacitor grid with an angle of incidence a. The electric field is due to a voltage V that is applied across the grids. For the incident electron, the sin a = vx/v1; and for the transmitted electron, the sin b = vx/v2. Note that this is similar to Snell's law from the refraction of light. Using Snell's law, the ratio sin a/sin b = n2/n1. Let's use the analogy to identify the velocity ratio with the index ratio from Snell's law. First, write down the conservation of energy,
(17)
(1/2)mv12 + eV = (1/2)mv22
Figure 12. Electron trajectory through a parallel grid capacitor (indicated by dashed lines). The electric field within the capacitor changes the component of the electron's velocity in the vertical (y) direction, but doesn't affect the horizontal (x) component.
Let's also assume that the incident electron velocity was produced by an accelerating voltage V0. The "law of electron refraction" is then obtained:
(18)
This analysis applies as long as quantum interference effects are not significant. Equation (18) describes an electron moving normal to equipotential lines. The voltage V0 is used to adjust the deflection of the electron relative to the normal. If the electron were moving in an inhomogenous electric field, equation (18) could be used step wise to determine the electron's path.
Electron lenses are constructed using metallic aperatures as shown in Figure 13 below. The deflection of the electron beam is controlled again by adjusting the electric field via control of the relative potentials. See figure caption for description. The symmetric lens can be made into an electron mirror if the central negative potential is increased enough. Otherwise, the speed of the electron decreases as it approaches the center, or saddle, point when the magnitude of the central potential is less than that of the outer electrodes. If the potentials were switched, with the center positve and the outer ones negative, the lens would still focus, but the electrons would accelerate toward the center.
Figure 13. Electron lenses fromed using metallic aperatures. In part (a), the configuration is set to produce convergence; and in (b), divergence. Part (c) shows a single, symmetrical lens that is capable of focusing. In each case, equipotential lines are drawn.
The focal length of an electron lens can also be determined by thinking in terms of optical lenses again. For simple converging lenses with differing radii of curvature located in a homogeneous medium of refractive index n0, the relation for trajectories close to the optical axis is
(19)
Equation 19 could be generalized for a system with additional lenses by continuing the sum of the inverse radii of curvature. By analogy to the optical system then, we can write a similar relation for the electrostatic lens system.
(20)
where r(x) and n(x) are the radius of curvature and "electron refractive index" as described earlier in equation (18). The point x is on the central axis. One could write equation (20) in terms of the applied voltages; and the focal length may be found in terms of the voltages after a line integration.
Electrons may also be focused using magnetic lenses. One often sees magnetic lenses used in high energy particle applications. Consider Figure 14. The force on the electron has a magntitude F = ev x B; and its direction is given by the right hand rule. In the direction of B, the velocity of the electron is unchanged by the field. In the direction perpendicular to the field, the electron moves in a circle with an angular speed of w = eB/m. All charged particles entering the solonoid at point A in the drawing at different angles reach the point C after the same amount of time t. We can therefore say that they are focused at point C. The focus condition is only dependent on the charge/mass ratio. If one is using such a lens for electron imaging, it is easy to see why the image will almost always be tilted relative to the object.
Figure 14. A magnetic lens for electrons. Part (a) schematically shows the trajectory of an electron entering a "long" solenoid. The electron follows a helical path around the magnetic field lines with a period t. Part (b) shows an iron-shielded solenoid and representative magnetic field lines.
In surface physics, in addition to imaging applications such as the scanning electron microscope, on may also be interested in knowing the energy distribution of particles. An electron energy analyzer is shown in Figure 15.
Figure 15. A concentric cylindrical geometry is used in the construction of an electrostatic energy analyzer. The centripetal force, mv2/r, on the electron is determined by selecting the electrostatic force, qE0 where E0 is the electric field. Thus, only electrons with a chosen speed, or energy, will follow the central path shown.
This type of analyzer is found in many types of instruments. One example is shown in Figure 16 of a high-resolution electron energy loss spectrometer (HREELS). Here two analyzers are used in tandem, one to select the energy of electrons striking the surface (a monochromator) and one to measure the energy of the scattered electrons. Various focusing aperatures (electron lenses) are also used at the entrances and exits of the analyzers. The filament emits hot electrons that have a broad (Maxwell) distribution of energies. The monochromator selects electrons within a small energy window (on the order of an meV). The analyzer is used to measure to electron spectrum by varying the acceleration or deceleration voltage of the aperatures in front of the analyzer with an constant energy bandwidth window. The image is rectangular.
Figure 16. A schematic of an HREELS system which consists of a filament electron emitter and electron lens system, monochromator, and an energy analyzer within a UHV system. The monochromator is rotated around an axis relative to the surface of the sample.
A hemispherical energy alayzer is shown in Figure 17 where the electron trajectory is between two metallic hemispheres. The entrance and exit apertures are circular producing a circular image. Electrons that are deflected through an angle of 180o will converge. The energy resolution is given by equation 21
(21)
where x1 and x2 are the radii of the entrance and exit apertures; and a and b are the radii of the hemispheres. The angle a is the maximum deviation of the electron trajectory with respect to the normal to the entrance apertures.
Figure 17. A hemispherical electron energy analyzer consists of two entrance lenses that focus incoming electrons onto the entrance aperture, hemi spherically-shaped electrodes, and an electron detector.
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