Nanophotonics Atilla Ozgur Cakmak Ph D Unit 2

Nanophotonics Atilla Ozgur Cakmak, Ph. D

Unit 2 Lecture 10: Multiple Potentials and Tunneling-Part 1

Outline • Tunneling • Bound States

A couple of words… The waves do not always propagate nicely through the potentials. In this lecture we are going to investigate two scenarios: tunneling and bound states, for which the waves will become evanescent (decaying) in certain regions rather than propagating. Accordingly, the propagation constants will become imaginary. Suggested reading: Peter Markos, Costas M. Soukoulis Wave Propagation From Electrons to Photonic Crystals and Left-Handed Materials, 2 nd Chapter.

Tunneling 0<E<V 0

Tunneling 0<E<V 0

Tunneling 0<E<V 0 0. 95 V 0 =E β=10

Tunneling 0<E<V 0 (problem) 1) Use your previous MATLAB codes and wave functions to figure out the transmission for E=0. 5 V 0 when β=2 and 0. 5, respectively. a) T=0. 014, T=0. 63 b) T=0. 14, T=0. 73 c) T=0. 012, T=0. 8 d) T=0. 12, T=0. 8

Tunneling 0<E<V 0 (solution) 1) Use your previous MATLAB codes and wave functions to figure out the transmission for E=0. 5 V 0 when β=2 and 0. 5, respectively. β=2 β=0. 5 V 0 =E

Bound States V 0<E<0

Bound States V 0<E<0

Bound States V 0<E<0 β=1

Bound States V 0<E<0

Bound States V 0<E<0

Bound States V 0<E<0 β=10

Bound States V 0<E<0 n=1 n=2 n=4 n=7 β=10

Some Useful Links I would like to share two very useful links that address the same problems that we have covered here. 1) https: //phet. colorado. edu/en/simulation/quantum-tunneling 2) https: //phet. colorado. edu/en/simulation/bound-states These Java based programs are extremely easy to operate and are developed by University of Colorado. They are very interactive and can solve the wave propagation and bound state problems very easily. The only drawback is that they do not accept decimal values smaller than 0. 01. For example, a potential with 10. 02 e. V is rounded to 10 e. V or 10. 1 e. V. Yet, the interactive programs might be helpful for the students at the beginning.