This document extends the Kronig-Penney model for modeling electron interactions with atomic lattices in semiconductors to arbitrary periodic potentials using numerical matrix mechanics. It introduces the Kronig-Penney model and represents periodic potentials using matrix representations in different basis sets. Numerical solutions are obtained and band structures are compared for different periodic potentials, including the Kronig-Penney, harmonic oscillator, and inverted harmonic oscillator potentials. Effective mass is also calculated from the band structure curvature.

1. KRONIG-PENNEY MODEL EXTENDED TO
ARBITRARY POTENTIALS VIA NUMERICAL
MATRIX MECHANICS
NAME-Ashish Ahlawat
ROLL NO-18510015
NAME-Neeraj Kumar Meena
ROLL N0-18510045
Instructor- Dr. Anand sengupta
T.A. -Amit Reza

2. Outline
• Introduction
• FORMALISM
– Infinite square well
– Square well with periodic boundary conditions
• HARMONIC OSCILLATOR
– Infinite square well basis
– Periodic boundary conditions
– Numerical comparison periodic and square well basis sets
• PERIODIC POTENTIALS
– Matrix method for the Kronig-Penney model
– Numerical solution
– Comparing band structures

3. Introduction
KRONIG-PENNEY MODEL
• An effective way to understand the energy gap in semiconductors is to model
the interaction between the electrons and the lattice of atoms.
• An effective way to understand the energy gap in semiconductors is to model
the interaction between the electrons and the lattice of atoms.
• Kronig and Penney assumed that an electron experiences an infinite one-
dimensional array of finite potential wells.
• The free electron model implies that, a conduction electron in a metal
experiences constant (or zero) potential and free to move inside the crystal
but will not come out of the metal because an infinite potential exists at the
surface.

4. Introduction (Contd.)
• the periodic potentials due to the positive ions in a metal have been
considered. shown in Fig.a
• if an electron moves through these ions, it experiences varying potentials.
The potential of an electron at the positive ion site is zero and is maximum
in between two ions. The potential experienced by an electron, when it
passes along a line through the positive ions is as shown in Fig.b and Fig.c
Fig.a
Fig. b and Fig. c

5. Formalism
1) Infinite square well
a potential in the infinite square well potential and study the bound states in that
basis(square well basis).
the wave function in the infinite well basis
Basis states of infinite square well are
and the eigenvalues are
a is the width of the well

6. 2) Square well with periodic boundary conditions
The general periodic boundary condition is that
our wave function satisfies is
energy eigenvalues are for periodic boundary conditions
here we have taken n …-2,-1,0,1,2,3…

7. Harmonic Oscillator
1) Infinite Square Well Basis
Potential of Infinite Square Well Basis
dimensionless form of infinite square well basis is

8. Harmonic Oscillator(contd.)

9. Harmonic Oscillator(contd.)
2) For periodic conditions in harmonic oscillator:
dimensionless hamiltonian matrix is
here we have used order of the quantum number n = {0,1,-1,2,-2,...}

10. Harmonic Oscillator(contd.)
3) Numerical comparison of Periodic and Square well Potential
Fig.3: E vs n for Infinite Square Well and
Periodic Square Well Potential
Numerically calculated

11. Harmonic Oscillator(contd.)
FOURIER COEFFICIENTS
Original From paper

12. PERIODIC POTENTIALS
1) Matrix method for the Kronig-Penney model
Hamiltonian matrix elements is given by the equation

13. 2) E-K Diagram For Different Shapes of Potentials
• Kronig-Penney potential with no well/barrier: for v0 = 0
Original E-K diagram
numerically calculated E-K diagram

14. E-K Diagram For Different Shapes of Potentials(contd.)
• Kronig-Penney potential with no well/barrier: for v0 = 10
numerically calculated E-K diagram
Original E-K diagram

15. E-K Diagram For Different Shapes of Potentials(contd.)
● For Harmonic Oscillator Potential
Hamiltonian matrix is given by
We use ɣ =4.84105
numerically calculated E-K diagram Original E-K diagram

16. E-K Diagram For Different Shapes of Potentials(contd.)
● Inverted Harmonic Oscillator Potential
Hamiltonian matrix is given by
We use ɣ =7.84105
numerically calculated E-K diagram Original E-K diagram

17. From the Kronig-Penney Model (1 dimensional periodic potential function)

18. Effective Mass:
From this curvature plot we can find the
effective mass of electrons and holes and
their mass ratio which is given by the
following formulas.

19. References
1. https://aapt.scitation.org/doi/10.1119/1.4923026
2. https://sites.google.com/site/puenggphysics/home/unit-5/kronig-penny-model
3. https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice
4. http://cmt.dur.ac.uk/sjc/thesis_prt/node26.html

20. THANK YOU