Wigner approach to a new two-band envelope function model for quantum transport n. 1 di 22 Facoltà...

Wigner approach to a new two-band envelope function model for quantum transport n. 1 di 22

Facoltà di Ingegneria ICTT19 – 19ICTT19 – 19ththInternational Conference on Transport Theory International Conference on Transport Theory Budapest, July 24-30, 2005Budapest, July 24-30, 2005

ICTT19ICTT1919th International Conference on Transport Theory19th International Conference on Transport Theory

BudapestBudapest, Ju, Julyly 2424--3030, 2005, 2005

Giovanni Frosali Dipartimento di Matematica Applicata “G.Sansone”

[email protected]

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

[email protected]



Plain of the talk

• Description of the model• Description of the model

• Numerical applications• Numerical applications

• Mathematical problem• 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



Problem setting: unperturbed system

• Homogeneous periodic crystal lattice:

• Time-dependent evolution semigroup:

0( ) Hd

i x xdt

0H

0( , ) ( , )i t

x t e x t

22

00

H ( )2 perV xm

No interband transition are possible if is a Bloch function0 ( , )x t

0H ( , ) ( , )n k nk x E k x

nE ( )

( , ) ( , )k

i t

nx t e k x

0( , ) ( , )nx t k x



Multiband models: derivation Multiband models: derivation

1

2i

n i n in R

Rw x R

Wannier envelope function

1

2, iik x

n nR

iw x R u k dx e k

Wannier function

,1

2i x

nk

nu k x ek dk

Bloch envelope function

Bloch function

is the (cell. averaged) probability to find the electron in the site Ri and into n-th band

2

n iR

• Non Homogeneous lattices



1

2, iik x

n nR

iw x R u k dx e k

1

2i

n i n in R

Rw x R

Wannier envelope function

Wannier function

High oscillating behaviour

The direct use of Wannier basis is a difficult task!!

,nu k x

L-K

Kane

,, ,0

0

Blochn kL K Bloch

n k n

k

uu u k

k

, ,0Kane Blochn k nu u

“kp” methods

In literature are proposed different approximations of • We loose the simple interpretation of the envelope function

,nu k x

Multiband models: derivation Multiband models: derivation



MEF model: derivation MEF model: derivation

1

2iikR

FB

n i

Z

n k e dkR

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)

n k

Our approach

1

2i

n i n in R

Rw x R

Wannier function

,1

2i x

nk

nu k x e dk k

Bloch function



MEF model characteristics: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




'( ) ( ) d , | | , ( ') 0n n ext nn

E E k k n k U n k k

| , ( , ) ikxnn k u k x e

0H | |extU E

First approximation:

,| ( , )n k nu u k x

, ', 'ˆ, | | , ' |ext e t n k n kx un k U n k U uk k

' ', , ' ( , ) )| ( ',n nn k n k

cell

u k x u k xu dxu

By exploiting the periodicity of and for slow varying external potentials

( , )nu k x



MEF model: formal derivation MEF model: formal derivation

0, ,,

,'

''' (k,k')

P (k,k'

'| )n n

n k n nn k E

k k

mu u

'2 ( , )P , (' ', )n x n

c

n,

e l

n

l

u k x u k x dxk k

2

2',

0'

2( , ) ( ) ( ')2n n nn E k E k k km

E k k

Evaluation of matrix elements ' ', , ' ( , ) )| ( ',n nn k n k

cell

u k x u k xu dxu

Kane momentum matrix




'( ) ( ) d , | | , ( ') 0n n ext nn

E E k k n k U n k k

2

, '0 , '

'

ˆ( ) ( ) ( ) ( ') ( ')d

' ˆ ( ') ( ')dP (k,k')

n

n n

n n n

nn n n n

E k E k k U k k k k

k ki U k k k

Ek

m



,*2 2

, '' '

0 0

'

, ' , '

Mˆ ˆ(0,0) (

' 'P (

0d ' d

0)

, )

,

0

0 n nn n

n n nn n

n

n n

n

n

i k k k U k ik k k U kE m Em

• Our aim: simplify the above equation.

“Interband term”: “Interband term”:

,

2 2

, '0' , '

0 , ''

' 'ˆ ˆP (k, '

,k') P (0 d,0)dn nn n n n

n n n nn n n nE k k

k k k ki U k i U km m E


We retain only the first

order term

We retain only the first

order term

Second approximation:




'ˆ( ) ( ) ( ) ( ') ( ')dn n n nE k E k k U k k k k

Approximate systemApproximate system

,,

2

''0 '

P (0,0ˆ )'

dn nn n

nn n E

k ki U km

kF

kn nx = F

'( ) ( ) d , | | , ( ') 0n n ext nn

E E k k n k U n k k

We write it in coordinate spaceWe write it in coordinate space



MEF model: first order MEF model: first order

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

Physical meaning of the envelope function: Physical meaning of the envelope function:

2| ( )

i

n i

R cell

n dx R | n

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

2

i x



• intraband dynamic


Effective mass dynamics: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

first order contribution of

transition rate of Fermi Golden rule


Coupling terms:Coupling terms:

( , )T n n k k

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



Problems in the practical use of the Kane model:Problems in the practical use of the Kane model:

22 21 1 2

120 0

22 22 2 1

220 0

2

2

Kane Kane KaneKane

c

Kane Kane KaneKane

v

i V Pt m x m x

i V Pt m x m x

Kane modelKane 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 2( ) Kane

ii

n x x



Wigner function:

Phase plane representation: pseudo probability function ,f x p

1, / 2 / 2

2ipf x p x m x m e d

Wigner equation Liouville equation

Classical limit

0

2,x n x f x p dp

,J x p f x p dpm

Moments of Wigner function:

Wigner picture:Wigner picture:



n n-th band component

Hd

idt

1,..,t

n

General Schrödinger-like model

matrix of operator


x yH, H Hd

idt

1 1 1

1

,n

n n n

x y

x y

Density matrix



1x yH H

dfi f

dt -W W


Evolution equation

Multiband Wigner function

1, / 2 , / 2

2ip

ij ijf x p x m x m e d

W

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).



1x yH H

dfi f

dt W W -


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 like G-R via ,cvf x p



Mathematical settingMathematical setting

2 21

2: ; 1f f L p dx dpX

1 1 1X XH X

1

, , ii

H i Xf gf g

Hilbert space:

Weighted spaces:

2,x p 1 D problem:

, Hf g




0(0)

di B C

dtA

ff f

f f

Hfthe two band Wigner system admits a unique solution

2, ( )ext xU W If the external potential

0 HD A f

, ,cc vv cvf f ffT




22

* * * 2

1, ,

4

p pdiag i i p

m x m x xA

m

iAte

Unbounded operator unitary semigroup on H

Stone theorem

0(0)

di B C

dtA

ff f

f f

23

11 2

2f : ,( ,)

ff fHA

xD

x xX

0(0)

di B C

dtA

ff f

f f



, ,cc vv cvf f ffT



0(0)

B Cd

i Adt

ff f

f f

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

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

, ,cc vv cvdB iag

0

0 0 2

0 0 2

0g

CP

m Ei i




, ,cc vv cvf f ffT



Symmetric bounded operators

0(0)

B Cd

i Adt

ff f

f f

H,B CB

1, ( )xij ij ext ijWX X

f c U f

2, ( )xW XXij ext ijf c U f

, ,cc vv cvdB iag




, ,cc vv cvf f ffT

0

0 0 2

0 0 2

0g

CP

m Ei i



0(0)

di A B C

dt

ff f

f f

The operator generate semigroup A B C

The unique solution of (1) is


0

i A B C te f f



, ,cc vv cvf f ffT



Numerical implementation: splitting schemeNumerical implementation: splitting schemeLinear evolution semigroup

0i te A+ B+Cf f

, ( , )ij ji x pf = f is a three element vector

+ 2 2ti i

t ti t i tie e e e e

B CA AA+ B C

+ijf( , ) t

ni t i

i j te x p e t n AA+ B+C + B Cf

Discrete operator

,i jf

Uniform mesh i x j px i p j



Numerical implementation: splitting schemeNumerical implementation: splitting scheme

2 2ij x ij

ˆf ft t

i ie e

A A-1F

ij x ijf̂ f F f.f.t.

22

* * * 2

1, ,

4

p pdiag i i p

m x m x m x

A

2* * *

12 , 2 , 2

4j jp p

diag i i pm m m

A

2 2+C t

t ti t i ti iie e e e e

B CA AA+ B

Approximate solution of

0(0)

di

dt

Af

f

f f



Numerical implementation: splitting schemeNumerical implementation: splitting scheme

ij p ijf̂ f F f.f.t.

0(0)

di

dt

Bf

f

f f

, ,cc vv cvdiag B

'

'

/ 2 / 21'

2i j i v v

ij ij

v

V x m V x mf e dv d

ij

1/ 2 / 2

2 i jV x m V x m

, ,cc vv cvdiag B

Approximate solution of

2 2ij p ij

ˆf ft t

i ie e

B B-1F

2 2+C t

t ti t i ti iie e e e e

B CA AA+ B



Space coordinate Momentum co

ordinate

x p

,cc x pf Conduction band



Space coordinate Momentum co

ordinate

x p


Space coordinate Momentum coordinate

x

p

Valence band ,vv x pf






,cc x p f

Valence band ,vv x pf

,cv x p f

Stationary state: Thermal distribution



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

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Wigner approach to a new two-band envelope function model for quantum transport n. 1 di 22 Facoltà...

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