Bands interception

1. Oct 3, 2012

hokhani

Is it possible in the first brillouin zoon that two energy bands crossed?

2. Oct 4, 2012

PhysTech

No, if there were band crossing, then at the point of intersection we would have two states with the same $(E,\textbf{k})$. But by Pauli's exclusion principle two states cannot have same quantum numbers. Consequently, this degeneracy will cause a gap to open at the intersection point.

PS: Throughout this argument it was implicitly assumed that all bands are spin degenerate. Therefore we do not consider the spin quantum number.

3. Oct 4, 2012

DrDu

Yes, two energy bands can cross. The crossing can be due to symmetry, e.g. two bands transforming as a two or three-dimensional representation of the little group at some special point.
But also without symmetry intersections are possible. Specifically, assume that the effective one-particle hamiltonian h(k) has been brought already to block diagonal form leaving only the sub-space of the two states which may potentially intersect, called |1> and |2> in the following. The exact states are then obtained by diagonalizing the matrix with elements <1|h|1>, <1|h|2>, <2|h|1> and < 2|h|2>. Now a hermitian 2x2 matrix has at most 3 independent entries, which means that varying k, which has also 3 independent components, we can achieve to reach a point of degeneracy. Taking time reversal into account, the matrix has effectively only two free parameters, so that lines of degeneracy in k-space become possible.

In contrast to what physchem has stated, Pauli principle plays no role as the two bands which become degenerate still have different quantum numbers, so that energetic degeneracy is not forbidden.

4. Oct 4, 2012

PhysTech

In a crystal you label the Bloch states using $\textbf{k}$. Therefore $\textbf{k}$ is a quantum number (or numbers if you count the three components) (pg. 141 of Ashcroft and Mermin). So yes, at the intersection point the quantum numbers are in fact the same.

Also, can you please provide specific examples DrNo? I want to know which crystals permit band crossing.

5. Oct 4, 2012

DrDu

The point I wanted to make is that k is not the only quantum number but the label of the bands is also a quantum number.

The crossing of bands which has created most furor in the last years are maybe the "diabolic points" in graphene.

Almost any material like e.g. Si will show plenty of intersections:
https://wiki.fysik.dtu.dk/gpaw/exercises/band_structure/bands.html

6. Oct 4, 2012

7. Oct 4, 2012

DrDu

Yes, Herring has worked out the proof I only sketched.