Solutions of Poisson and the time-independent Schroedinger equations in prolate spheriodal coordinate system with hyperboloid boundary and quantum filtering

Atomically sharp metal and semiconductor micro field emitting tips are widely used in the field of science and engineering. They form the central part of a class of electron-tunneling devices and instruments such as geometrically asymmetric nanotunnel structures (GANTS) and scanning tunneling microscopes (STM). They are also used as sources of coherent electrons in electronic holography and electronic lithography. Efforts in quantifying the operational characteristics of these devices have been limited to the modification of a one-dimensional theory to cope with the three-dimensional features of these devices. In this study, we have solved the Poisson and Schrödinger equations in the prolate spheroidal coordinate system, a coordinate system that gives the realistic geometric representation of a tip-base junction. We have studied the behavior of the 3-D wavefunctions that represents electrons existing in a very sharp metal tip, and concluded that the electron exhibit directional coherency due to “quantum filtering” when the cone angle of the tip is extremely small. We have also concluded that the cone angle of sharp metal tip plays a vital role in the operation of these devices. We have also demonstrated the strong 3-D feature of the potential barrier in the tip-base junction of these devices.