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

$$ V(x) = \begin{cases} V_0 & -b < x < 0 \cr 0 & 0 < x < a \end{cases} $$

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

 %LatticeParameters 
   a + b
 %
 
 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

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
 
 %LatticeParameters 
   a + b
 %

 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 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.