1. Wigner approach to a new two-band
ICTT19
19th International Conference on Transport Theory
Budapest, July 24-30, 2005
Wigner approach to a new two-band
envelope function model for quantum transport
Omar Morandi
Dipartimento di Elettronica e Telecomunicazioni
Giovanni Frosali
Dipartimento di Matematica Applicata “G.Sansone”

2. Description of the model
Plain of the talk
Description of the model
Multiband (MEF) model
Multiband-Wigner picture
Mathematical problem
Mathematical setting
Well posedness of the Multiband-Wigner system
Numerical applications
Description of the numerical algorithm
Application to IRTD

3. Problem setting: unperturbed system
Homogeneous periodic crystal lattice:
Time-dependent evolution semigroup:
No interband transition are possible if is a Bloch function

4. Multiband models: derivation
Wannier envelope function
Bloch envelope function
Bloch function
Wannier function

5. Multiband models: derivation
Wannier envelope function
Wannier function
is the (cell. averaged) probability to find the electron in the site Ri and into n-th band
Non Homogeneous lattices

6. Multiband models: derivation
Wannier envelope function
Wannier function
High oscillating behaviour
The direct use of Wannier basis is a difficult task!!

7. Multiband models: derivation
In literature are proposed different approximations of
We loose the simple interpretation of the envelope function
L-K
Kane
“kp” methods

8. MEF model: derivation Wannier function Bloch function Our approach
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)

9. MEF model characteristics:
Hierarchy of “kp” multiband effective mass models, where the asimptotic parmeter is the “quasi-momentum” of the electron
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

10. MEF model: derivation

11. MEF model: derivation First approximation:
By exploiting the periodicity of and for slow varying external potentials

12. MEF model: formal derivation
Evaluation of matrix elements
Kane momentum matrix

13. MEF model: derivation

14. MEF model: derivation Our aim: simplify the above equation.
“Interband term”:

15. We retain only the first
MEF model: derivation
Our aim: simplify the above equation.
“Interband term”:
We retain only the first
order term
Second approximation:

16. MEF model: derivation Approximate system
We write it in coordinate space

17. Physical meaning of the envelope function:
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. Effective mass dynamics:
MEF model: first order
Effective mass dynamics:
Zero external electric field: exact electron dynamic
intraband dynamic

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

20. Wigner picture: Wigner equation Liouville equation Classical limit
Wigner function:
Phase plane representation: pseudo probability function
Wigner equation
Liouville equation
Classical limit
Moments of Wigner function:

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

22. Wigner picture: Multiband Wigner function Evolution equation
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 , (2003).

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

24. Two band Wigner model Wigner picture:
Moments of the multiband Wigner function:
represents the mean probability density to find the electron into n-th band, in a lattice cell.

25. Two band Wigner model
Wigner picture:

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

27. Two band Wigner model Wigner picture:
intraband dynamic: zero coupling if the external potential is null
interband dynamic: coupling like G-R via

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

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

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

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

32. Mathematical setting If the external potential
the two band Wigner system admits a unique solution
Symmetric bounded
operators

33. Mathematical setting If the external potential
the two band Wigner system admits a unique solution
The operator generate semigroup
The unique solution of (1) is

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

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

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

37. Space coordinate
Momentum coordinate x p
Conduction band

38. x p Valence band Conduction band x p Space coordinate Space coordinate
Momentum coordinate x p
Valence band
Conduction band x p
Momentum coordinate
Space coordinate

40. Stationary state: Thermal distribution
Conduction band
Valence band

41. Conclusion Next steps 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