1. Wigner approach to a new two-band envelope function model for quantum transport n. 1 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 ICTT19 19th International Conference on Transport Theory Budapest, July 24-30, 2005 Giovanni Frosali Dipartimento di Matematica Applicata G.Sansone giovanni.frosali@unifi.it Wigner approach to a new two-band envelope function model for quantum transport Wigner approach to a new two-band envelope function model for quantum transport Omar Morandi Dipartimento di Elettronica e Telecomunicazioni omar.morandi@unifi.it

2. Wigner approach to a new two-band envelope function model for quantum transport n. 2 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Plain of the talk Description of the model Numerical applications Mathematical problem Multiband (MEF) model Mathematical setting Well posedness of the Multiband-Wigner system Application to IRTD Multiband-Wigner picture Description of the numerical algorithm

3. Wigner approach to a new two-band envelope function model for quantum transport n. 3 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Problem setting: unperturbed system Homogeneous periodic crystal lattice: Time-dependent evolution semigroup: No interband transition are possible if is a Bloch function

4. Wigner approach to a new two-band envelope function model for quantum transport n. 4 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Multiband models: derivation Wannier envelope function Wannier function Bloch envelope function Bloch function

5. Wigner approach to a new two-band envelope function model for quantum transport n. 5 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Multiband models: derivation Wannier envelope function Wannier function is the (cell. averaged) probability to find the electron in the site R i and into n-th band Non Homogeneous lattices

6. Wigner approach to a new two-band envelope function model for quantum transport n. 6 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Wannier envelope function Wannier function High oscillating behaviour The direct use of Wannier basis is a difficult task!! Multiband models: derivation

7. Wigner approach to a new two-band envelope function model for quantum transport n. 7 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 L-K Kane kp methods In literature are proposed different approximations of We loose the simple interpretation of the envelope function Multiband models: derivation

8. Wigner approach to a new two-band envelope function model for quantum transport n. 8 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 MEF model: derivation To get our multiband model in Wannier basis: We recover un approximate set of equation for in the Bloch basis (momentum space) We Fourier transform the equations obtained (coordinate space) Our approach Wannier functionBloch function

9. Wigner approach to a new two-band envelope function model for quantum transport n. 9 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 MEF model characteristics: Direct physical meaning of the envelope function Easy approximation (cut off on the index band) Highlight the action of the electric field in the interband transition phenomena Easy implementation: Wigner and quantum-hydrodynamic formalism Hierarchy of kp multiband effective mass models, where the asimptotic parmeter is the quasi-momentum of the electron

10. Wigner approach to a new two-band envelope function model for quantum transport n. 10 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 MEF model: derivation

11. Wigner approach to a new two-band envelope function model for quantum transport n. 11 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 MEF model: derivation First approximation: By exploiting the periodicity of and for slow varying external potentials

12. Wigner approach to a new two-band envelope function model for quantum transport n. 12 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 MEF model: formal derivation Evaluation of matrix elements Kane momentum matrix

13. Wigner approach to a new two-band envelope function model for quantum transport n. 13 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 MEF model: derivation

14. Wigner approach to a new two-band envelope function model for quantum transport n. 14 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Our aim: simplify the above equation. Interband term: MEF model: derivation

15. Wigner approach to a new two-band envelope function model for quantum transport n. 15 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Our aim: simplify the above equation. Interband term: MEF model: derivation We retain only the first order term We retain only the first order term Second approximation:

16. Wigner approach to a new two-band envelope function model for quantum transport n. 16 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 MEF model: derivation Approximate system We write it in coordinate space

17. Wigner approach to a new two-band envelope function model for quantum transport n. 17 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 MEF model: first order Physical meaning of the envelope function: The quantity represents the mean probability density to find the electron into n-th band, in a lattice cell.

18. Wigner approach to a new two-band envelope function model for quantum transport n. 18 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 intraband dynamic MEF model: first order Effective mass dynamics: Zero external electric field: exact electron dynamic

19. Wigner approach to a new two-band envelope function model for quantum transport n. 19 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 intraband dynamic interband dynamic first order contribution of transition rate of Fermi Golden rule MEF model: first order Coupling terms:

20. Wigner approach to a new two-band envelope function model for quantum transport n. 20 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Wigner function: Phase plane representation: pseudo probability function Wigner equationLiouville equation Classical limit Moments of Wigner function: Wigner picture:

21. Wigner approach to a new two-band envelope function model for quantum transport n. 21 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 n-th band component General Schrödinger-like model matrix of operator Wigner picture: Density matrix

22. Wigner approach to a new two-band envelope function model for quantum transport n. 22 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Wigner picture: Evolution equation Multiband Wigner function Introduced by Borgioli, Frosali, Zweifel [1] Well-posedness of the two band Kane-Wigner System [1] G. Borgioli, G. Frosali and P. Zweifel, Wigner approach to the two-band Kane model for a tunneling diode, Transp. Teor.Stat. Phys. 32 3, 347-366 (2003).

23. Wigner approach to a new two-band envelope function model for quantum transport n. 23 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Wigner picture: Evolution equation Multiband Wigner function Two band MEF model Two band Wigner model

24. Wigner approach to a new two-band envelope function model for quantum transport n. 24 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 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.

25. Wigner approach to a new two-band envelope function model for quantum transport n. 25 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Wigner picture: Two band Wigner model

26. Wigner approach to a new two-band envelope function model for quantum transport n. 26 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Wigner picture: Two band Wigner model intraband dynamic: zero coupling if the external potential is null

27. Wigner approach to a new two-band envelope function model for quantum transport n. 27 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Wigner picture: Two band Wigner model intraband dynamic: zero coupling if the external potential is null interband dynamic: coupling like G-R via

28. Wigner approach to a new two-band envelope function model for quantum transport n. 28 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Mathematical setting Hilbert space: Weighted spaces: 1 D problem:

29. Wigner approach to a new two-band envelope function model for quantum transport n. 29 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Mathematical setting the two band Wigner system admits a unique solution If the external potential

30. Wigner approach to a new two-band envelope function model for quantum transport n. 30 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Mathematical setting Unbounded operator unitary semigroup on Stone theorem the two band Wigner system admits a unique solution If the external potential

31. Wigner approach to a new two-band envelope function model for quantum transport n. 31 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Mathematical setting the two band Wigner system admits a unique solution If the external potential

32. Wigner approach to a new two-band envelope function model for quantum transport n. 32 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Symmetric bounded operators Mathematical setting the two band Wigner system admits a unique solution If the external potential

33. Wigner approach to a new two-band envelope function model for quantum transport n. 33 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 The operator generate semigroup The unique solution of (1) is Mathematical setting the two band Wigner system admits a unique solution If the external potential

34. Wigner approach to a new two-band envelope function model for quantum transport n. 34 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Numerical implementation: splitting scheme Linear evolution semigroup is a three element vector Discrete operator Uniform mesh

35. Wigner approach to a new two-band envelope function model for quantum transport n. 35 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Numerical implementation: splitting scheme f.f.t. Approximate solution of

36. Wigner approach to a new two-band envelope function model for quantum transport n. 36 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Numerical implementation: splitting scheme f.f.t. Approximate solution of

37. Wigner approach to a new two-band envelope function model for quantum transport n. 37 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Space coordinate Momentum coordinate xp Conduction band

38. Wigner approach to a new two-band envelope function model for quantum transport n. 38 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Space coordinate Momentum coordinate xp Conduction band Space coordinate Momentum coordinate x p Valence band

39. Wigner approach to a new two-band envelope function model for quantum transport n. 39 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005

40. Wigner approach to a new two-band envelope function model for quantum transport n. 40 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Conduction bandValence band Stationary state: Thermal distribution

41. Wigner approach to a new two-band envelope function model for quantum transport n. 41 di 22 Facoltà di Ingegneria ICTT19 – 19 th International Conference on Transport Theory Budapest, July 24-30, 2005 Conclusion Multiband-Wigner model Well posedness of the Multiband-Wigner system Application to IRTD Next steps Extention of MEF model to more general semiconductor Well posedness of Multiband-Wigner model coupled with Poisson eq. Calculation of I-V IRDT characteristic for self-consistent model