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.


(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.


[A figure of the indexed 1st BZ should be put here.] Figure 1
Figure 2 Figure 2
Figure 3 Figure 3
Figure 4 Figure 4