Schrodinger equation in periodic

1. EEE 3103 Electrical Properties of Materials
Semester: December 2018
S. M. Jahadun-Nobi
Lecturer
Department of EEE, NSTU
Email: smjnobi.eee@nstu.edu.bd
Office: EEE 520, Adminstrative Building

2. References

3. Schrodinger equation in periodic
U(x)
E
U(x)
a b
p=a+ b
 But N atoms have two 2N unknown constants to find ….
 For large N, isn’t there a better way ?

4. Steps of Finding Energy Levels in
Crystals

5. Periodic U(x) and Bloch’s
Theorem
E U(x)=U(x+a+b)=U(x+p)
U(x)
a b

6. Bloch’s Theorem: Phase‐factor for
N‐cells
E U(x)
N
1 3
2
Ψ(x) Ψ(x+p)
Ψ(x+2p) Ψ(x+Np)

7. Bloch’s Theorem: Periodic
Boundary Condition

8. KP model: Boundary
Condition

9. KP model: Boundary
Condition

10. KP model: Boundary
Condition

11. Graphical solution to Energy
Levels
k = 0
k=2π/Np
ζ= E/U

12. Energy Band Diagram
k = 0
k=2π/Np
ζ= E/U
0 k

13. Brillouin Zone and Number of States
E
-π/p π/p
2π/Np

14. Wave Packet and Group Velocity
E
k

15. Effective Mass
• Wavepacket is a packet of waves, consists of a linear
combination of constant E wave function solutions
closely grouped about a peak or center energy.
• For a packet of travelling waves with center frequency ω
and center wave number k, group velocity will be
•
1
g
d dE E
v
dk dk


  

16. Effective Mass for a Given Band
/m*
1
v
E
k
k