Computational Solid State Physics 計算物性学特論 5 回

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 B crystal A

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

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

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. Effective mass approximation ・Suppose that a perturbationis 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

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

11. Effective mass approximation(2)

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

13. 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*.

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

15. Quantum well

16. Quantum corral

17. HEMT

18. 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.

19. 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.

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

21. 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

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

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

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

25. 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.