# Tutorial:Kronig-Penney Model

The Kronig-Penney model is a 1D system that demonstrates band gaps, which relate to the allowed energies for electrons in a material. In this tutorial we calculate the bandstructure for Kronig-Penney Model. The Kronig-Penney Model has a periodic potential of

Where b is the width of each barrier, and a is the spacing between them.

## Input

The following input file will be used for the ground state calculation:

`CalculationMode`

= gs`ExtraStates`

= 4`PeriodicDimensions`

= 1`Dimensions`

= 1`TheoryLevel`

= independent_particles a = 5 b = 1 V = 3`Lsize`

= (a + b)/2`Spacing`

= .0075 %`Species`

"A" | species_user_defined | potential_formula | "(x>-b)*V*(x<0)" | valence | 1 % %`Coordinates`

"A" | 0 | % %`KPointsGrid`

11 | % %`KPointsPath`

11 | 0.0 | 0.5 | %`ConvEigenError`

= true

The first two wavefunctions plotted alongside the potential.

## Bandstructure

To calculate the bandstructure simply change the `CalculationMode`

to unocc.

`CalculationMode`

= unocc`ExtraStates`

= 4`PeriodicDimensions`

= 1`Dimensions`

= 1`TheoryLevel`

= independent_particles a = 5 b = 1 V = 3`Lsize`

= (a + b)/2`Spacing`

= .0075 %`Species`

"A" | species_user_defined | potential_formula | "(x>-b)*V*(x<0)" | valence | 1 % %`Coordinates`

"A" | 0 | % %`KPointsGrid`

11 | % %`KPointsPath`

11 | 0.0 | 0.5 | %`ConvEigenError`

= true

To plot the bandstructure, we will use the same command from the Getting started with periodic systems (assuming you are using gnuplot).

plot for [col=5:5+9] 'static/bandstructure' u 1:(column(col)) w l notitle ls 1

Reference: Sidebottom DL. Fundamentals of condensed matter and crystalline physics: an introduction for students of physics and materials science. New York: Cambridge University Press; 2012.