Wigner approach to a new two-band envelope function model for quantum transport

ICTT19 ICTT19

19th International Conference on Transport Theory 19th International Conference on Transport Theory

Budapest

Budapest, Ju , July ly 24- 24 - 30, 2005 30 , 2005

Giovanni Frosali

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

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

• Multiband-Wigner picture

• Description of the numerical algorithm

Problem setting: unperturbed system

• Homogeneous periodic crystal lattice:

• Time-dependent evolution semigroup:

0

 

( ) H

i d x x

dt   



H0

( , ) x t e

( , ) x t

 



2 2

0

0

H ( )

2 V

x

   m  

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

H



( , ) k x  E



( , ) k x

E ( )n

( , ) ( , )

i k t

x t e 

k x

 

( , ) x t



( , ) k x

 

Multiband models: derivation Multiband models: derivation

   

1

2

R w x R 

 ^

  

Wannier envelope function

  ¹   ^{ }

2 ,

n n

R

w x R

u k x e d k





 

Wannier function

   ^, 

1 2

i x n

k

u k x e

k dk

  

Bloch envelope function



Bloch function

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

 

n

R



• Non Homogeneous lattices

  ¹   ^{ }

2 ,

n n

R

w x R

u k x e d k





 

   

1

2

R w x R 

 ^

  

Wannier envelope function

Wannier function

High oscillating behaviour

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

 ^, 

u k x

L-K Kane

,

, ,0

0 Bloch L K Bloch n k

n k n

k

u u u k

k





  



, ,0

Kane Bloch

n k n

u  u

“k p” m eth od s

In literature are proposed different approximations of

• We loose the simple interpretation of the envelope function u k x

 ^, 

Multiband models: derivation

Multiband models: derivation

MEF model: derivation MEF model: derivation

  ¹  

2

ikRi

FB

n i

Z

n

k e dk

R 

   

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

2

R w x R 

 ^

  

Wannier function

   ^, 

1 2

i x n

k

u k x e

d

k 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

MEF model: derivation MEF model: derivation

(

) ( )

d , |

| ,

( ') 0

n

E E  k k n k U n k  k



  

     

| , n k  u k x e

( , )

H

 U

|   E |  

First approximation:

| u

 u k x

( , )

 

, |

| , ˆ

' u

|

n k U n k   U k k  u

    

' '

,

|

( , )

( ', )

n k n k

cell

u k x u k x

u dx

 u  

By exploiting the periodicity of and for slow varying external

potentials u k x

( , )

MEF model: formal derivation MEF model: formal derivation

0

, ,

,

, '

'

'

'

(k,k') ' P (k,k '

| )

n n

n k n n

n k

E

k k

u u m 

    

  2 ( , )

P , '

( ', )

c n,

e l n

l

u k x u k x dx

k k 



  

 

2

2 '

,

0 '

( , ) ( ) ( ')

n n

E k

E k

2 k k

E k k     m  

 

Evaluation of matrix elements

|

( , )

( ', )

cell

u k x u k x

u dx

 u  

Kane momentum matrix

MEF model: derivation MEF model: derivation

(

) ( )

d , |

| ,

( ') 0

n

E E  k k n k U n k  k



  

     

2

, '

0 , '

'

( ) ( ) ( ) ˆ ( ') ( ')d

' ˆ( ') P (k,k') ( ')d

n

n n

n n n

n

n n n n

E k E k k U k k k k

i k k U k k k

E k

m

  





   

 

 





  

   

,

2 2 *

, '

' '

0 0

'

, ' , '

ˆ M ˆ

(0,0) ' ' (

P (

d 0 ' d

0)

, ) ,

0

0

n n

n n n n n

n

n n

n

n

i k k k U k ik k k U k

E m E

m  

 

 

   

 

   

 

• Our aim: simplify the above equation.

“Interband term”:

“Interband term”:

 

2 2

, ' 0

' , '

0 , '

'

' ˆ P (k ' ˆ

, ' ,k')

d P (0 ,0)

d

n n n n

n n n n

n n n n

E k k

k k k k

i U k i U k

m  m E 

 

   

  

    

  

MEF model: derivation MEF model: derivation

We retain only the first order term

We retain only the first order term

Second approximation:

MEF model: derivation MEF model: derivation

ˆ

( ) ( ) ( ) ( ') ( ')d

n n n n

E  k  E k  k   U k k   k k ^

Approximate system Approximate system

, ,

2

' '

0 '

P (0,0

ˆ )

'

d

n n

n

n n

E

i k k U k

m 



 

 ^   

F

 

 

n

x

 = F 

(

) ( )

d , |

| ,

( ') 0

n

E E  k k n k U n k  k



  

     

We write it in coordinate space

We write it in coordinate space

MEF model: first order MEF model: first order

 

 

2

2 2

1 1

1 2

* 2

0 2

2 2

2 2

2 1

* 2

0

P 2

P 2

c

c g

v

v g

i E U U

t m x m E x

i E U U

t m x m E x

   

   

        

   

 

  

     

   



 



 



Physical meaning of the envelope function:

Physical meaning of the envelope function:

| ( )

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.

 

i

x



• intraband dynamic

MEF model: first order MEF model: first order

Effective mass dynamics:

Effective mass dynamics:

Zero external electric field: exact electron dynamic

 

 

2

2 2

1 1

1 2

* 2

0 2

2 2

2 2

2 1

* 2

0

P 2

P 2

c

c g

v

v g

i E U U

t m x m E x

i E U U

t m x m E x

   

   

        

   

 

  

     

   



 



 



• intraband dynamic

• interband dynamic

first order contribution of

transition rate of Fermi Golden rule

MEF model: first order MEF model: first order

Coupling terms:

Coupling terms:

( , )

T n  n k   k 

 

 

2

2 2

1 1

1 2

* 2

0 2

2 2

2 2

2 1

* 2

0

P 2

P 2

c

c g

v

v g

i E U U

t m x m E x

i E U 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:

2

2 2

1 1 2

2 1

0 0

2

2 2

2 2 1

2 2

0 0

2 2

Kane Kane Kane

Kane c

Kane Kane Kane

Kane v

i V P

t m x m x

i V P

t m x m x

   

   

   

   

   

 

  

    

   



 



 



Kane model Kane model

• Strong coupling between envelope function related

to different band index, even if the external field is null

• Poor physical interpretation ^{( )}

 

i

n x    x

Wigner function:

Phase plane representation: pseudo probability function ^{f x p}  ^, 

 ^,  ¹  ^{/ 2}   ^{/ 2} 

2

f x p x m x m e d



    

    ^  ^

Wigner equation Liouville equation Classical limit

 0



  ^x

^{n x}   ^{f x p dp}  ^, 

   

  ^{J x}   ^{p f x p dp}  ^, 

m ^       

Moments of Wigner function:

Wigner picture:

Wigner picture:



n-th band component

d H

i  dt  



 

,.., 



 

General Schrödinger-like model

matrix of operator

Wigner picture:

Wigner picture:

 ^H,   ^H

^H



i d

dt

     

 

   



1

,

n

n n n

x y

x y

   



   

 

 

  

 

 



  



Density matrix

 ^H

^H



i df f

dt  W  W

Wigner picture:

Wigner picture:

Evolution equation

Multiband Wigner function

 ^,    ¹  ^{/ 2 , / 2} 

2

ip

ij ij

f 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]

 ^H

^H



i df f

dt  W  W

Wigner picture:

Wigner picture:

Evolution equation

Multiband Wigner function

 

 

2 2 2

* 2

0

2 2 2

* 2

0

P H = 2

P

2

c

c g

v

g v

E U U

m x m E x

U E U

m E x m x

   

   

   

 

   

    

   

 

 

 

Two band MEF model

Two band Wigner model

 ^,    ¹  ^{/ 2 , / 2} 

2

ip

ij ij

f x p x m x m e d



 

  

 ^W     ^  ^

 

 

*

0

*

0

2

*

0

2 2

4

cc cc cc cc cv

g

vv

vv vv vv cv

g

cv cv cv cv cc vv

g

f p P

f i f f

t m m E

f p P

f i f f

t m m E

f i P

i p f i f i f f

t m m E

 

 

  





 

      

 

 

      

  

  

              

  







 



Wigner picture:

Wigner picture: Two band Wigner model

 ^,   

ii i

f 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.

 ^, 

f x p dp



Wigner picture:

Wigner picture: Two band Wigner model

  ^{ / 2   / 2 }

 

p



f

   V x

   m  V x

   m  

f

F F

  ^{ / 2 }

 

p



f

    V x    m  

f

F F

 

 

*

0

*

0

2

*

0

2 2

4

cc cc cc cv

g

vv

vv vv cv

g

cv cv c

cc

vv

c c vv

g

v v c

f p P

f i f f

t m m E

f p P

f i f f

t m m E

f i P

i p f i f i f f

t m m E

 

 

  





 

      

 

 

      

  

  

              

  







 



 

 

*

0

0

2

*

0

*

2 2

4

cv g

c

cc cc cc cc

vv

v g

cv cv

vv vv vv

cv cv cc vv

g

P f

m E

P f

m E

f i P

i p f i f i f f

t

f p

f i f

t m

f p

f i f

m m E

t m











 





 

  

 

 

 

 

   

              

  

  



    









 



Wigner picture:

Wigner picture: Two band Wigner model

  ^,  

ii i

f x v  W 

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

 

 

*

2

*

0 0

0

*

2 2

4

cv g

c

cc cc cc cc

vv

vv vv vv

cv cv cv c

v g

v cc vv

g

f p

f i f

t m

f p

f i f

t m

f i P

i p f i f i f f

t

P f

m E

P f

m E

m m E





  









 

    

 

 

    

  

  

              

  



 

 



 





Wigner picture:

Wigner picture: Two band Wigner model

  ^,  

ii i

f x v  W 

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

• interband dynamic: coupling like G-R via f

 x p ^, 

Mathematical setting Mathematical setting

 

 





1

: ; 1

f f L p dx dp

X    

1 1 1

X X

H    X

   

, ,

H



f g

f g 

Hilbert space:

Weighted spaces:

  ^{x p ,} _

1 D problem:

,  H

f g

Mathematical setting Mathematical setting

 

(0)

i d B C

dt A

   

 

 



f f f

f f

 H f

the two band Wigner system admits a unique solution

2,

( )

ext x

U  W

 If the external potential

0

 D A    H f

 f

^, f

^, f





f

Mathematical setting Mathematical setting

2

2

* * * 2

, , 1

4

p p

diag i i p

m x m x x

A m

    

         





e iAt

Unbounded operator unitary semigroup on H

Stone theorem

 

(0)

i d B C

dt A

   

 

 



f f f

f f

2

3 1

1 2

f : ,

( ) f f , f

H

A x

D    x  x X 

 

    

 



 

(0)

i d B C

dt A

   

 

 



f f f

f f

 H f

the two band Wigner system admits a unique solution

2,

( )

ext x

U  W

 If the external potential

 f

^, f

^, f





f

 

(0)

d B C

i A

dt

  



 

 



f f f

f f

  ^{   }

 

p



f  ^  V x

   / 2 m  V x

   / 2 m ^ 

f

F F

  ^{ }

 

p



f

    U x    / 2 m  

f

F F



^,

^,



d

B  iag   

   

   

   

0

0 0 2

0 0 2

g

0

C P

m E i i





 





 

  

 

 

  

 

  

 



 

 

Mathematical setting Mathematical setting

 H f

the two band Wigner system admits a unique solution

2,

( )

ext x

U  W

 If the external potential

 f

^, f

^, f





f

Symmetric bounded operators

 

(0)

d B C

i A

dt

  



 

 



f f f

f f

  ^H

,

B C  B

1, ( _x)

ij ij

f

c U

f

 



2, ( _x)

W X

ij X ext ij

f c U f











^,

^,



d

B  iag   

Mathematical setting Mathematical setting

 H f

the two band Wigner system admits a unique solution

2,

( )

ext x

U  W

 If the external potential

 f

^, f

^, f





f

   

   

   

0

0 0 2

0 0 2

g

0

C P

m E i i





 





 

  

 

 

  

 

  

 



 

 

 

(0)

i d A B C

dt

   

 

 



f f f

f f

The operator generate semigroup A B C  

The unique solution of (1) is

Mathematical setting Mathematical setting

 

0 i A B C t

e



f f

 H f

the two band Wigner system admits a unique solution

2,

( )

ext x

U  W

 If the external potential

 f

^, f

^, f





f

Numerical implementation: splitting scheme Numerical implementation: splitting scheme

Linear evolution semigroup

 

0

i t

 e

f f

,_j

( ,

)

i

x p

f = f is a three element vector





e

e e e e

   

 A B A C

A+ B C



  



f

( , )

i t i

i j t

e

f x p    e

  t n  

Discrete operator

,

f

Uniform mesh x

  i

p

  j

Numerical implementation: splitting scheme Numerical implementation: splitting scheme

2 2

ij x

ˆ

f f

t t

i i

e e







  

 

A

F

ij x ij

ˆf  F     f ^f.f.t.

2

2

* * * 2

, , 1 4

p p

diag i i p

m x m x m x

    

          

A 



2

* * *

2 , 2 , 2 1

4

j j

p p

diag i i p

m  m  m 

 

     

 

A 







e

e e e e

   

 A B A C

A+ B



Approximate solution of

(0) i d

dt

  

 

 



f A

f

f f

Numerical implementation: splitting scheme Numerical implementation: splitting scheme

ij p ij

ˆf  F     f f.f.t.

(0)

i d dt

  

 

 



f B

f

f f



^,

^,



diag   

 B

   

'

/ 2 / 2

1 '

2

i j i v v

ij ij

v

V x m V x m

f e

dv d



 

 



  

  ^ _ ^

   

 

ij

1 / 2 / 2

2 V x

m V x

m

  

      





^,

^,



diag   

 B

Approximate solution of

2 2

ij p

ˆ

f f

t t

i i

e e







  

 

B

F





e

e e e e

   

 A B A C

A+ B



Space coo

rdinate _Mo^mentu

m coordinate

x p

 ^, 

cc

x p

f Conduction band

Space coo

rdinate _Mo^mentu

m coordinate

x p

 ^, 

cc

x p

f Conduction band

Space coordinate Momentum coordinate

x

p Valence band

 ^, 

vv

x p

f

 ^, 

cc

x p

f Conduction band

 



^{x p ,} 

 f

Valence band

 ^, 

vv

x p

f

 



^{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