# Valley's quantum number

1. Sep 27, 2015

### hokhani

According to "Nat. Nanotechnol., vol. 7, no. 8, pp. 488–489, 2012 (http://www.nature.com/nnano/journal/v7/n8/full/nnano.2012.117.html?WT.ec_id=NNANO-201208)":
Valley quantum number is associated with different crystal axes that differ only in their orientations. Such axes can support electron waves that are also identical apart from their direction (or, more precisely, their momentum),so an additional discrete index known as the valley quantum number is needed to fully describe these waves.

This argument raises a question:
Si and Ge have the same crystal structure. Therefore according to this statement they have to have the same valleys. However Si has 6 valleys in the $\Gamma X$ direction while Ge has 4 valleys in the $\Gamma L$ direction. I don'n know what goes wrong with my thought.

Last edited by a moderator: May 7, 2017
2. Oct 2, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Oct 3, 2015

### hokhani

I think the post clearly shows the ambiguity and rewording it may not lead to any response.

4. Oct 3, 2015

### Daz

Hi, I don’t really understand the question, but I wonder whether this might be any help.

Germanium and silicon both have valleys in their conduction bands along <100> direction and the <111> direction.

They both have valleys along both directions.

So the symmetry of their band structure is, in fact, the same – as it must be because it’s determined by the crystal lattice symmetry which as you point out is the same. In that sense, they do have the same valleys, at least the same symmetry, but in silicon the Γ-X valleys are a bit deeper than the Γ-L valleys whereas in germanium it’s the other way around.

So, when we talk about the silicon band gap, we’re referring to the indirect gap between Γ and the six X points. When we talk about the germanium band gap the shortest distance (in energy) between valence and conduction band just happens to be between Γ and the four L points.

Does that help at all?