Understanding Valley Quantum Number in Si and Ge Band Structures

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SUMMARY

The discussion centers on the valley quantum number in silicon (Si) and germanium (Ge) band structures, highlighting their differing valley configurations despite having the same crystal structure. Si possesses six valleys in the Γ-X direction, while Ge has four valleys in the Γ-L direction. The symmetry of their band structures is identical due to their shared crystal lattice symmetry, but the energy depths of the valleys differ, with Si's Γ-X valleys being deeper than Ge's Γ-L valleys. This distinction is crucial for understanding the band gaps of both materials.

PREREQUISITES
  • Understanding of valley quantum numbers in solid-state physics
  • Familiarity with band structure concepts in semiconductors
  • Knowledge of crystal lattice symmetry and its implications
  • Basic principles of energy band gaps in materials
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  • Research the role of valley quantum numbers in semiconductor physics
  • Explore the differences in band structures between Si and Ge
  • Study the implications of crystal lattice symmetry on electronic properties
  • Investigate the effects of valley depth on semiconductor performance
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Researchers in semiconductor physics, materials scientists, and engineers working with silicon and germanium in electronic applications will benefit from this discussion.

hokhani
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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.
 
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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?
 
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