Download presentation

Presentation is loading. Please wait.

Published byKimberly Hurst Modified over 4 years ago

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

Similar presentations

Presentation is loading. Please wait....

OK

1 Chapter 13 Nuclear Magnetic Resonance Spectroscopy.

1 Chapter 13 Nuclear Magnetic Resonance Spectroscopy.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on any one mathematician rene Ppt on job evaluation answers Ppt on steps of farming Ppt on project report writing Ppt on vehicle chassis Ppt on clutch system in automobile Ppt on hydrogen fuel cell vehicles price Ppt on ram and rom diagram Ppt on describing words for class 1 Ppt on team management skills