1. Wigner approach to a two-band electron-hole semi-classical model n. 1 di 22 Graz June 2006 Wigner approach to a two-band electron-hole semi-classical model Omar Morandi Dipartimento di Elettronica e Telecomunicazioni omar.morandi@unifi.it

2. Wigner approach to a two-band electron-hole semi-classical model n. 2 di 22 Graz June 2006 Quantum correction to a Semi-classical electron-hole system to take into account inteband transition in presence of strong electric field (Landau-Zener effect). Strong electric field can give rise transition between conduction and valence electrons. L-Z effect:

3. Wigner approach to a two-band electron-hole semi-classical model n. 3 di 22 Graz June 2006 Quantum correction to a Semi-classical electron-hole system to take into account inteband transition in presence of strong electric field (Landau-Zener effect). L-Z effect: Semiclassical Boltzmann equations describe the inteband dynamics in the low field regions. In collisionless limit Semiclassical equation provide no transition between conduction and valence band. Wigner Formalism for a multiband system to derive a semi-classical correction

4. Wigner approach to a two-band electron-hole semi-classical model n. 4 di 22 Graz June 2006 intraband dynamic MEF model: first order Effective mass dynamics: Zero external electric field: exact electron dynamic

5. Wigner approach to a two-band electron-hole semi-classical model n. 5 di 22 Graz June 2006 intraband dynamic interband dynamic MEF model: first order Coupling terms:

6. Wigner approach to a two-band electron-hole semi-classical model n. 6 di 22 Graz June 2006 n-th band component General Schrödinger-like model matrix of operator Wigner picture: Density matrix

7. Wigner approach to a two-band electron-hole semi-classical model n. 7 di 22 Graz June 2006 Wigner picture: Evolution equation Multiband Wigner function Two band MEF model Two band Wigner model

8. Wigner approach to a two-band electron-hole semi-classical model n. 8 di 22 Graz June 2006 Wigner picture: Two band Wigner model Moments of the multiband Wigner function: represents the mean probability density to find the electron into n-th band, in a lattice cell.

9. Wigner approach to a two-band electron-hole semi-classical model n. 9 di 22 Graz June 2006 Wigner picture: Two band Wigner model

10. Wigner approach to a two-band electron-hole semi-classical model n. 10 di 22 Graz June 2006 Wigner picture: Two band Wigner model intraband dynamic: zero coupling if the external potential is null

11. Wigner approach to a two-band electron-hole semi-classical model n. 11 di 22 Graz June 2006 Wigner picture: Two band Wigner model intraband dynamic: zero coupling if the external potential is null interband dynamic: coupling via

12. Wigner approach to a two-band electron-hole semi-classical model n. 12 di 22 Graz June 2006 Solution of W-MEF system: fast oscillating behaviour of the solution fast oscillating in time: Simple interpretation Given the eigenfunctions We have the following temporal evolution of the Wigner functions

13. Wigner approach to a two-band electron-hole semi-classical model n. 13 di 22 Graz June 2006 Some analogies: electron-phonon coupling Optical Phonon bath: fast oscillating field Number of phonon Electron gas Number of electron in the i-th band slow dynamics Band transition

14. Wigner approach to a two-band electron-hole semi-classical model n. 14 di 22 Graz June 2006 L-Z effect: Band transition without phonon coupling Optical Phonon bath: fast oscillating field Number of phonon Electron gas Number of electron in the i-th band slow dynamics Band transition Strong El. field

15. Wigner approach to a two-band electron-hole semi-classical model n. 15 di 22 Graz June 2006 We study the W-MEF system in the particular cases (no approximation of the equations) Uniform electric field U(x)= E x

16. Wigner approach to a two-band electron-hole semi-classical model n. 16 di 22 Graz June 2006 Constant in space initial data for the distributions function We study the W-MEF system in the particular cases (no approximation of the equations) Uniform electric field U(x)= E x Charge conservation time dependence

17. Wigner approach to a two-band electron-hole semi-classical model n. 17 di 22 Graz June 2006 Constant in space initial data for the distributions function We study the W-MEF system in the particular cases (no approximation of the equations) Uniform electric field U(x)= E x Charge conservation time depend comp.

18. Wigner approach to a two-band electron-hole semi-classical model n. 18 di 22 Graz June 2006 1 Time evolution of f(p)

19. Wigner approach to a two-band electron-hole semi-classical model n. 19 di 22 Graz June 2006 1 el. in Conduction band Band transition: We consider an initial momentum p

20. Wigner approach to a two-band electron-hole semi-classical model n. 20 di 22 Graz June 2006 1 el. in Valence band Band transition

21. Wigner approach to a two-band electron-hole semi-classical model n. 21 di 22 Graz June 2006 1 Time evolution of f(p)

22. Wigner approach to a two-band electron-hole semi-classical model n. 22 di 22 Graz June 2006 1 Time evolution of f(p)

23. Wigner approach to a two-band electron-hole semi-classical model n. 23 di 22 Graz June 2006 1 Time evolution of f(p)

24. Wigner approach to a two-band electron-hole semi-classical model n. 24 di 22 Graz June 2006 1 Time evolution of f(p) Strong transition

25. Wigner approach to a two-band electron-hole semi-classical model n. 25 di 22 Graz June 2006 1 Time evolution of f(p) We want to Evaluate this term

26. Wigner approach to a two-band electron-hole semi-classical model n. 26 di 22 Graz June 2006 We expect the following temporal evolution of

27. Wigner approach to a two-band electron-hole semi-classical model n. 27 di 22 Graz June 2006

28. Wigner approach to a two-band electron-hole semi-classical model n. 28 di 22 Graz June 2006 Approximate solution

29. Wigner approach to a two-band electron-hole semi-classical model n. 29 di 22 Graz June 2006 Approximate solution Baker-Hausdorf formula We neglect terms of order The interesting dynamic is for

30. Wigner approach to a two-band electron-hole semi-classical model n. 30 di 22 Graz June 2006 Approximate solution

31. Wigner approach to a two-band electron-hole semi-classical model n. 31 di 22 Graz June 2006 Approximate solution

32. Wigner approach to a two-band electron-hole semi-classical model n. 32 di 22 Graz June 2006 Approximate solution Step Function

33. Wigner approach to a two-band electron-hole semi-classical model n. 33 di 22 Graz June 2006 Gain Loss