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Make an approximate estimate of the band gap at the corner point of the Brillouin zone for a square lattice in two dimensions, assuming the crystal potential to be given by 2πχ V(x, y) = -4V₁ cos (²x) COS (2πy)

Make an approximate estimate of the band gap at the corner point of the Brillouin zone for a square lattice in two dimensions, assuming the crystal potential to be given by 2πχ V(x, y) = -4V₁ cos (²x) COS (2πy)

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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6. Make an approximate estimate of the band gap at the corner point of the Brillouin zone for a square lattice in two dimensions, assuming the crystal potential to be given by V(x, y) = -4V cos πχ (ZTZx), a COS (2пу a
Transcribed Image Text:6. Make an approximate estimate of the band gap at the corner point of the Brillouin zone for a square lattice in two dimensions, assuming the crystal potential to be given by V(x, y) = -4V cos πχ (ZTZx), a COS (2пу a
Expert Solution
Step 1

The Crystal Potential is given as:

V(x,y)=-4V0 cos2πxa cos2πya

Using the central equation to find the Energy gap Eg, at corner point:

The Brillouin zone for a square lattice is, (x,y)=πa,πa

The Fourier transform with direction of period in box and the potential with, G=2πa,2πa

 

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