PHY3707
Assignment 6
Solid State Physics
Due 16 September 2025
, PHY3707 Assignment 6 — Due date: 16 September 2025
PHY3707 Assignment 6
Due date: 16 September 2025
Q7.3 – Band Gap Estimate
Problem Statement:
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
2πx 2πy
V (x, y) = −4V0 cos cos .
a a
Solution:
The corner point of the Brillouin zone is at
π π
k= , .
a a
The free electron energy at this point is
ℏ2 k 2 ℏ2 ℏ2 π 2
π 2
E0 = = 2 = .
2m 2m a ma2
Expanding the potential,
2π(x + y) 2π(x − y)
V (x, y) = −2V0 cos + cos ,
a a
the Fourier coefficients are
2π 2π 2π 2π
VG = −V0 , G=± , , ± ,− .
a a a a
The secular matrix becomes
E0 −V0
.
−V0 E0
Page 1
Assignment 6
Solid State Physics
Due 16 September 2025
, PHY3707 Assignment 6 — Due date: 16 September 2025
PHY3707 Assignment 6
Due date: 16 September 2025
Q7.3 – Band Gap Estimate
Problem Statement:
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
2πx 2πy
V (x, y) = −4V0 cos cos .
a a
Solution:
The corner point of the Brillouin zone is at
π π
k= , .
a a
The free electron energy at this point is
ℏ2 k 2 ℏ2 ℏ2 π 2
π 2
E0 = = 2 = .
2m 2m a ma2
Expanding the potential,
2π(x + y) 2π(x − y)
V (x, y) = −2V0 cos + cos ,
a a
the Fourier coefficients are
2π 2π 2π 2π
VG = −V0 , G=± , , ± ,− .
a a a a
The secular matrix becomes
E0 −V0
.
−V0 E0
Page 1
