This paper presents a simple method for accurately calculating quantum mechanical transmission probability and current across arbitrary potential barriers by using the multistep potential approximation. This method is applicable to various potential barriers and wells, including continuous variations of potential energy and electron effective mass. Various potential barrier structures and a hot‐electron transistor are analyzed to show the feasibility of this method.

Calculation of transmission tunneling current across arbitrary potential barriers

A simple, accurate, and fast algorithm for solving the one-dimensional time-independent Schrodinger equation is presented. The algorithm is based on the transfer matrix method. This makes it possible to calculate all bound and quasi-bound energy levels and the corresponding wave functions for an arbitrarily shaped potential profile. The results of calculations are compared with those obtained by other authors for various types of problems. A central part of this study deals with solving the Schrodinger equation in quantum-well structures. The results show that the transfer matrix method is as accurate as other methods, but it is easier to implement and, hence, is superior for calculations on small computer, such as a PC. >

Solving the Schrodinger equation in arbitrary quantum-well potential profiles using the transfer matrix method

This paper presents a simple yet accurate method for solving the Schrodinger equation to calculate the quantum mechanical transmission probability across arbitrary potential barriers. It is based on the concept of wave impedance analogous to transmission line theory. The quantum mechanical transmission probability in a resonant tunneling device can be easily calculated using this method.

Transmission line analogy of resonance tunneling phenomena: the generalized impedance concept

Scattering of X-rays and neutrons at interfaces

High‐temperature charge transport across an oxide‐nitride‐oxide sandwich of erasable programmable read only memories is mainly governed by the oxide conductivity as experimentally determined. It was verified in the examined devices that charge loss is not due to mobile ions. Since hole injection from the control gate into the nitride can be blocked by a 70‐A‐thick top oxide we conjecture that charge loss is due to leakage of electrons; however, the observed leakage current is too large to be explained by pure electrode‐limited charge transport (Richardson emission and direct tunneling). It was also verified that field gain on asperities and along edges cannot increase the charge loss current to the required range. Numerical evaluation of trap tunneling and resonant tunneling indicated that both mechanisms are weakly temperature dependent while charge loss has a typical activation energy of 1.2 eV in the range of 250–350 °C. Consequently, a multiphonon‐assisted tunneling mechanism is proposed where electrons stored on the floating gate tunnel to oxide traps, then are emitted into the nitride. The coupling of the trap level to oxide phonons results in virtual energy levels in the oxide which allow for more effective transition paths. As a consequence of the electron‐phonon coupling, the emission occurs close to the oxide conduction‐band edge at temperatures between 250 and 350 °C, producing a strong temperature dependence for the mechanism.

Field and high‐temperature dependence of the long term charge loss in erasable programmable read only memories: Measurements and modeling

A simple yet exact method of solving the Schrodinger equation across an arbitrary one‐dimensional piecewise‐linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer‐matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.

Masao Fukuma

Papers

Exact solution of the Schrodinger equation across an arbitrary one‐dimensional piecewise‐linear potential barrier