wigner approach to a two-band electron-hole semi-classical model n. 1 di 22 graz june 2006 wigner...

Wigner approach to a two-band electron-hole semi-classical model n. 1 di 22

Graz June 2006Graz June 2006

Wigner approach to a two-band electron-hole semi-classical model

Omar Morandi Dipartimento di Elettronica e Telecomunicazioni

[email protected]



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:L-Z effect:



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: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



• intraband dynamic

MEF model: first order MEF model: first order

Effective mass dynamics:

Zero external electric field: exact electron dynamic

22 21 1

1 2* 20

22 22 2

2 1* 20

P

2

P

2

cc g

vv g

Ui E U

t m x m E x

Ui E U

t m x m E x



• intraband dynamic

• interband dynamic

MEF model: first order MEF model: first order

Coupling terms:

22 21 1

1 2* 20

22 22 2

2 1* 20

P

2

P

2

cc g

vv g

Ui E U

t m x m E x

Ui E U

t m x m E x



n n-th band component

Hd

idt

1,..,t

n

General Schrödinger-like model

matrix of operator

Wigner picture:Wigner picture:

x yH, H Hd

idt

1 1 1

1

,n

n n n

x y

x y

Density matrix



1x yH H

dfi fdt W W -

Wigner picture:Wigner picture:

Evolution equation

Multiband Wigner function

2 2 2

* 20

2 2 2

* 20

P

2H =

P

2

cc g

vg v

UE U

m x m E x

UE U

m E x m x

Two band

MEF model

Two band Wigner model

1, / 2 , / 2

2ip

ij ijf x p x m x m e d

W



*0

*0

2*

0

2

2

4

cccc cc cc cv

g

vvvv vv vv cv

g

cvcv cv cv cc vv

g

f p Pf i f f

t m m E

f p Pf i f f

t m m E

f i Pi p f i f i f f

t m m E

Wigner picture:Wigner picture: Two band Wigner model

2,ii if x p dp x

Moments of the multiband Wigner function:

represents the mean probability density to find the electron into n-th band, in a lattice cell.

,iif x p dp




1p/ 2 / 2p ij i j ijij f V x m V x m f F F

-1/ 2p ij p ijf V x m f F F

*0

*0

2*

0

2

2

4

cccc cc cv

g

vvvv vv cv

g

cvcv c

cc

vv

c c vvg

v v c

f p Pf i f f

t m m E

f p Pf i f f

t m m E


t m m E



*0

0

2*

0

*

2

2

4

cvg

c

cccc cc cc

vvv

g

cvcv

vv vv vv

cv cv cc vvg

Pf

m E

Pf

m E


t

f pf i f

t m

f pf i f

m m E

t m


, iii if x v W

• intraband dynamic: zero coupling if the external potential is null



*

2*

0

0

0*

2

2

4

cvg

c

cccc cc cc

vvvv vv vv

cvcv cv c

vg

v cc vvg

f pf i f

t m

f pf i f

t m


t

Pf

m E

Pf

m E

m m E


, iii if x v W

• intraband dynamic: zero coupling if the external potential is null

• interband dynamic: coupling via ,cvf x p



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



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



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



We study the W-MEF system in the particular cases

(no approximation of the equations)• Uniform electric field U(x)= E x



• Constant in space initial data for the distributions function



•Charge conservation

time dependence



• Constant in space initial data for the distributions function



•Charge conservation

time depend comp.



1-1

Time evolution of f(p)



1-1

el. in Conduction band

Band transition:

We consider an initial momentum p



1-1el. in Valence band

Band transition



1-1




1-1


Strong transition



1-1


We want to Evaluate this term



We expect the following temporal evolution of



Approximate solution




Baker-Hausdorf formula We neglect terms of order

The “interesting dynamic”

is for




Step Function



Gain

Loss

wigner approach to a two-band electron-hole semi-classical model n. 1 di 22 graz june 2006 wigner...

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