# Non-Interacting Electrons in a Two-Dimensional Square Lattice

At Charles Kittel's "Introduction to Solid State Physics", 7th ed., Chapter 7 "Energy Bands",
[Problem 1 Square lattice, free electron energies] and [Problem 6 Square lattice].
## Free Electron Gas in a Two-Dimensional Empty Square Lattice

## Non-Interacting Electrons in a Simple Two-Dimensional Square-Lattice Potential

Consider the potential U(x,y) = -4U cos(2πx/a) cos(2πy/a) .
Band structures for U=0.05 and U=0.10 are shown in Fig.3 and Fig.4.
- TwoDim.F Fortran90 source code
- Writen in Fortran90 free format. You may need a proper compile option,
such as -free, -FR, -ffree-form, etc. It depends on your compiler.
- With a `-DDEBUG' compile option,
the Hamiltonian and its eigenvalues will be printed out.

- TwoDim.F.pdf Colored source code
- TwoDim.gp GNUPLOT source code
- TwoDim.pdf PDF file (Fig.3)
- TwoDim.ps PS file
- TwoDimLargeU.pdf PDF file (Fig.4)
- TwoDimLargeU.ps PS file
- Makefile for an example

## Problem

(a) Visualize the potential U(x,y).
Use GNUPLOT and its `set nosurface' and `set contour'.

(b) Discuss the splitting of levels at M-point for small and large U's.

(c) Consider another potential U'(x,y) = -2U' [ cos(2πx/a) + cos(2πy/a) ].
Visualize it.

(d) Make a little change in TwoDim.F
to adopt H=p^2/(2m)+U+U'.
What happens with the U'(x,y)?
Discuss the splitting of levels at M-point.

## Figures

[A figure of the indexed 1st BZ should be put here.] Figure 1

Figure 2

Figure 3

Figure 4