Electron Trajectories in a Field Emission Microscope

Abstract

Trajectories of electrons in a field emission microscope have been calculated for the case of a tip and screen taken as confocal hyperboloids of revolution. The calculations were done in prolate spheroidal coordinates by an iterative method utilizing Hamilton's equations. An IBM digital computer has been programmed to accept values for the initial position and momentum of the electron, the tip and screen radii (each measured at the apex), the tip-to-screen distance and the applied voltage. A calculation yields the final values of position and momentum components at the screen as well as the transit time for the electron. The results of the calculation have been used to predict a field emission pattern which is compared with an experimental pattern. Good agreement is outained without the use of any adjustable parameters.

Electrostatic potential for a hyperbolic probe tip near a semiconductor

“…1(a). For the case of a metallic sample this choice of coordinates immediately provides an exact solution of the electrostatic problem [11]. For the present problem of a semiconducting sample the coordinate system does not present a trivial solution to the problem, but it nevertheless allows a convenient (and exact) means of specifying the boundary condition in the vacuum and it also enables the use of a moderate size finite-element grid in the vacuum.…”

“…To handle the electrostatic problem of a metallic probe tip in proximity to a semiconducting surface, we use in the vacuum region the well-known prolate spheroidal coordinate system [10][11][12]. These coordinates form families of confocal hyperbolas and ellipses, as illustrated in Fig.…”

“…This type of coordinate system is well-suited to this problem; it has been previously used for the problem of field emission [11] as well as other problems involving a sharp tip in proximity to a metallic surface [12]. Here, we apply such coordinates to the case of the tip-semiconductor problem as encountered in scanning probe microscopy.…”

Observation of Metal-Vacuum-Metal Tunneling, Field Emission, and the Transition Region

“…For a fixed emitter field, the emitter-to-surface voltage has been determined by solving Laplace's equation in prolate spheroidal coordinates. 4 By comparing this calculation with experimental curves for emitter-to-surface voltage versus distance it is possible to determine the emitter radi-us of curvature. 2 When the emitter-to-surface spacing is less than one tenth of the emitter radius of curvature, the voltage approaches a linear function of emitter-to-surface spacing (parallelplate approximation), and the field at the surface of the emitter can be determined directly from emitter voltage versus distance.…”

Atomic scale images of acceptors in III-V semiconductors

“…However, modeling and self-consistent solving of the realistic three dimensional situation is time consuming, in particular when many different tip potentials need to be evaluated. Recently Feenstra proposed a numerical model based on the prolate spheroidal coordinate system, previously applied for field emission microscopy [62]. The model approximates the tip with a hyperboloid characterized by an apex radius and a shank opening angle (see Fig.…”