Transmission through a system of potential barriers

Nanny Fröman, Örjan Dammert

Örjan Dammert

The transmission coefficient for a one-dimensional system of N potential barriers of general shapes (unequal or equal) is expressed exactly, by means of the phase-integral method of N. Fröman and P. O. Fröman, in terms of quantities characterizing the separate barriers and wells of the system. The exact formula, being of interest for further rigorous studies of transmission properties, can for evaluation readily be converted into a useful approximate formula by insertion of available phase-integral expressions (of an arbitrary order) for the characteristic quantities mentioned above. Error estimates are given. The sub- and superbarrier cases are treated in a unified way. Some key facts about the phase-integral method are given in Appendix A.

Faqir Chand Auluck

P.N. Roy, P.N. Singh, D.K. Roy

Elias Wyss

Quantum tunnelling is a phenomenon in which a particle is able to traverse a potential barrier, even if said particle possesses less energy than the barrier. This process is highly important in many modern fields of physics, such as nuclear physics and semiconductor technology. In this thesis, this phenomenon was presented through analytical study, as well as the derivation of the transmission coefficient in one and three dimensions. Of even greater interest is the phenomenon of resonant tunnelling, where particles tunnel through two or more potential barriers, leading to the formation of quantum wells and quasi-bound energy levels that result in heightened transmission probabilities at certain energies. The transfer matrix method was used in order to find an expression for the resonant tunnelling transmission coefficient in double barrier systems. In addition, a programme was developed in Python in order to investigate resonant tunnelling in different barrier systems. The programme demonstrates how the probability of transmission fluctuates with the energy of the incoming particle, as well as how it is influenced by changes to the framework of the barrier system. Furthermore, effects caused by the inclusion of asymmetry into the system are studied. Finally, the process of spin-dependent tunnelling and the related Rashba spin-orbit coupling were discussed as an extension of the resonant tunnelling phenomenon.

Walter J. Rhein