PHY3707
Assignment 6
Solid State Physics
Due 16 September 2025
, Student Name: PHY3707 Assignment 6
PHY3707 Assignment 6
Due date: 16 September 2025
Question 7.3
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 is
2πx 2πy
V (x, y) = −4V0 cos cos .
a a
Solution:
The corner of the Brillouin zone is at
π π
k= ,
a a
.
At this point, degenerate plane wave states are mixed by the periodic potential.
The free-electron energy is
ℏ2 k 2 ℏ2 π 2 2
π 2
E0 = = , k =2 .
2m ma2 a
Expanding the potential, only reciprocal lattice vectors
2π 2π 2π
, − 2π
G=± a
, a , ± a a
contribute, with Fourier coefficients
VG = −V0 .
The secular matrix reduces to !
E0 −V0
.
−V0 E0
The eigenvalues are
λ = E0 ± V0 .
Thus, the band gap is
Eg = (E0 + V0 ) − (E0 − V0 ) = 2V0 .
Final Answer:
Eg ≈ 2V0
Page 1
Assignment 6
Solid State Physics
Due 16 September 2025
, Student Name: PHY3707 Assignment 6
PHY3707 Assignment 6
Due date: 16 September 2025
Question 7.3
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 is
2πx 2πy
V (x, y) = −4V0 cos cos .
a a
Solution:
The corner of the Brillouin zone is at
π π
k= ,
a a
.
At this point, degenerate plane wave states are mixed by the periodic potential.
The free-electron energy is
ℏ2 k 2 ℏ2 π 2 2
π 2
E0 = = , k =2 .
2m ma2 a
Expanding the potential, only reciprocal lattice vectors
2π 2π 2π
, − 2π
G=± a
, a , ± a a
contribute, with Fourier coefficients
VG = −V0 .
The secular matrix reduces to !
E0 −V0
.
−V0 E0
The eigenvalues are
λ = E0 ± V0 .
Thus, the band gap is
Eg = (E0 + V0 ) − (E0 − V0 ) = 2V0 .
Final Answer:
Eg ≈ 2V0
Page 1
