1. Computational Solid State Physics 計算物性学特論 5回
5.Band offset at hetero-interfaces 　　and effective mass approximation

2. Energy gaps vs. lattice constants

3. Band alignment at hetero-interfaces
: conduction band edge
: valence band edge
crystal A
crystal B

4. Anderson’s rule for the band alignment (1)
χ:electron affinity
None of the interface effects are considered.

5. Anderson’s rule for the band alignment (2)
: conduction band offset
: valence band offset

6. Types of band alignment
type I
type II
type III

7. Band bending in a doped hetero-junction (1)

8. Band bending in a doped hetero-junction (2)

9. ■ interface state ■ charge transfer
Evaluation of valence band offset from averaged local potential and band edge energies
valence band edge
ΔV: difference between average potential energies at the interface
All the interface effects are included in ΔV.
■ interface state ■ charge transfer
T.Nakayama et al.,J. Phys. Soc. Jpn. 61 (1992) 2434

10. Effective mass approximation
・Suppose that a perturbation 　 is added to a perfect crystal.
・How is the electronic state?
Examples of perturbations
an impurity, a quantum well, barrier, superlattice,
potential from a patterned gate, space charge potential

11. Effective mass approximation (1)
: Bloch function
V : external potential
assume: conduction band 　　　　　　n is minimum at k=0

12. Effective mass approximation(2)

13. Effective mass approximation(3)If
Schroedinger equation for envelope function χ(r)

14. Effective mass approximation (4)
・Schroedinger equation for an envelope 　function χ(r)
All the effects of crystal potential are included in εc and effective mass m*.

15. Impurity
:potential from a donor ion
Ry=13.6 eV: Rydberg constant

16. Quantum well

17. Quantum corral

18. HEMT

19. 2D-electron confinement in HEMT
The sub-band structure at the interface of the GaAs active channel in a HEMT structure. E1 and E2 are the confined levels. The approximate positions of E1 and E2 as well as the shape of the wave functions are indicated in the lower part of the diagram. In the uper part, an approximate form of the potential profile is shown, including contributions of the conduction band offset and of the space charge potential.

20. Superlattice Crystal A Crystal B
The Kronig-Penney model, a simple superlattice, showing wells of width w alternating with barriers of thickness b and height V0. The (super)lattice constant is a=b+w.

21. Kronig-Penny model (1)
Schroedinger equation in the effective mass approximation
Bloch condition for superlattice
k: wave vector of Bloch 　 function in the superlattice

22. Kronog-Penney model (2)
Boundary condition at x=0
(1)continuity of wavefunction
(2)connection condition for the 1st derivative of wavefunction
(2’)
Solution of Schroedinger equation
for

23. Kronig-Penney model (3)
(1)
(2’)
Simultaneous equation for E(k)

24. Kronig-Penney model (4)
allowed range of cos(ka)

25. Kronig-Penney model (5)
Conduction band of crystal A is split into mini-bands with mini-gaps by the Bragg reflection of the superlattice.

26. Problems 5
Calculate the lowest energy level for electrons and light and heavy holes in a GaAs well 6 nm wide sandwiched between layers of Al0.35Ga0.65As. Calculate the photoluminescence energy of the optical transition.
Calculate the two-dimensional Schroedinger equation for free electrons confined in a cylindrical well with infinitely high walls for r>a.