Atilla Ozgur Cakmak, PhD Nanophotonics Atilla Ozgur Cakmak, PhD

Lecture 15: Tight Binding Model-Part2 Unit 2 Lecture 15: Tight Binding Model-Part2

Outline The Transfer Matrix Impurities Tying up to Kronig-Penney Single Impurity N Identical Impurity 2N Alternating Impurity Tying up to Kronig-Penney

A couple of words… There is another model effectively used for periodic wave propagation, which will be very useful for our understanding. It is called the tight binding approach and it is based on the wave functions which are very much localized at their sites. The wave propagation is in a way akin to hopping from site to site. There are studies which treat periodic configurations within the perspective of tight binding approach as we will see in the upcoming lectures. Suggested reading: Peter Markos, Costas M. Soukoulis Wave Propagation From Electrons to Photonic Crystals and Left-Handed Materials, 5th Chapter.

The Transfer Matrix

The Transfer Matrix (problem) Find the explicit formula for K, the transfer matrix for tight binding problem given in the previous slide.

The Transfer Matrix (solution) Find the explicit formula for T, the transfer matrix for tight binding problem given in the previous slide.

The Transfer Matrix

The Transfer Matrix

Impurities: Single Impurity

Impurities: N-identical Impurity

Impurities (problem) How many perfect transmission (T=1) peaks do appear for N=26 impurities? εA=0.5 a) 20 b) 25 c) 19 d) 23

Impurities (solution) How many perfect transmission (T=1) peaks do appear for N=26 impurities? εA=0.5

Impurities (solution) How many perfect transmission (T=1) peaks do appear for N=26 impurities? εA=0.5 We have to decrease E spacing to see T=1 bands clearly. The first band becomes T=1 when the spacings are small enough.

Impurities: 2N Alternating Impurity

Impurities: 2N Alternating Impurity

Impurities: 2N Alternating Impurity

Impurities (problem) How many perfect transmission (T=1) peaks do appear for 2N=26 impurities? εA=0.5 a) 20 b) 25 c) 19 d) 23

Impurities (solution) How many perfect transmission (T=1) peaks do appear for 2N=26 impurities? εA=0.5

Impurities (solution) How many perfect transmission (T=1) peaks do appear for 2N=26 impurities? εA=0.5 2N=26

Tying up to Kronig-Penney