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 is the (cell. averaged) probability to find the electron in the site R i and into n-th band Non Homogeneous lattices

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 Wannier envelope function Wannier function High oscillating behaviour The direct use of Wannier basis is a difficult task!! L-K Kane kp methods In literature are proposed different approximations of We loose the simple interpretation of the envelope function Multiband models: derivation

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 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

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 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

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 First approximation: By exploiting the periodicity of and for slow varying external potentials

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: formal derivation Evaluation of matrix elements Kane momentum matrix

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 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:

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: derivation Approximate system We write it in coordinate space

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: 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.

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 intraband dynamic MEF model: first order Effective mass dynamics: Zero external electric field: exact electron dynamic

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 intraband dynamic interband dynamic first order contribution of transition rate of Fermi Golden rule MEF model: first order Coupling terms:

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 Problems in the practical use of the Kane model: Kane model Strong coupling between envelope function related to different band index, even if the external field is null Critical choice in the cut off for the band index Poor physical interpretation

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 Wigner function: Phase plane representation: pseudo probability function Wigner equationLiouville equation Classical limit Moments of Wigner function: Wigner picture:

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 n-th band component General Schrödinger-like model matrix of operator Wigner picture: Density matrix

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 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).

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 picture: Evolution equation Multiband Wigner function Two band MEF model Two band Wigner model

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 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.

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: Two band Wigner model

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: Two band Wigner model intraband dynamic: zero coupling if the external potential is null

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 intraband dynamic: zero coupling if the external potential is null interband dynamic: coupling like G-R via

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 Mathematical setting Hilbert space: Weighted spaces: 1 D problem:

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 Mathematical setting the two band Wigner system admits a unique solution If the external potential

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 Mathematical setting Unbounded operator unitary semigroup on Stone theorem the two band Wigner system admits a unique solution If the external potential

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 the two band Wigner system admits a unique solution If the external potential

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 Symmetric bounded operators 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 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

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 Numerical implementation: splitting scheme Linear evolution semigroup is a three element vector Discrete operator Uniform mesh

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 Numerical implementation: splitting scheme f.f.t. Approximate solution of

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 Numerical implementation: splitting scheme f.f.t. Approximate solution of

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 Space coordinate Momentum coordinate xp Conduction band

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 Space coordinate Momentum coordinate xp Conduction band Space coordinate Momentum coordinate x p Valence band

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

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 Conduction bandValence band Stationary state: Thermal distribution

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 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