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Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
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
[email protected]
Giovanni Frosali
Dipartimento di Matematica Applicata “G.Sansone”
[email protected]
Wigner approach to a new two-band envelope function model for quantum transport
n. 1 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
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
Wigner approach to a new two-band envelope function model for quantum transport
n. 2 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Problem setting: unperturbed system
• Homogeneous periodic crystal lattice:
d
i
 ( x)  H 0   x 
dt
H0  
2
2m0
2  Vper ( x)
• Time-dependent evolution semigroup:
( x, t0 )   n (k , x)
i
( x, t )  e
H0
t
( x, t0 )
i
( x, t )  e
En ( k )
t
 n (k , x)
No interband transition are possible if ( x, t0 ) is a Bloch function
H0 n (k , x)  Ek n (k , x)
Wigner approach to a new two-band envelope function model for quantum transport
n. 3 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Multiband models: derivation
Wannier envelope function
1

2
Bloch envelope function
 wn  x  Ri   n  Ri  
n
Ri
Wannier function
1
ik  x  Ri 
wn  x  Ri  
u
k
,
x
e
dk


n

2
• Non Homogeneous lattices
1
ikx

k
u
k
,
x
e
 dk
n  n

2
Bloch function
 n  Ri  is the (cell. averaged) probability to find the electron
2
in the site R and into n-th band
i
Wigner approach to a new two-band envelope function model for quantum transport
n. 4 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Multiband models: derivation
Wannier envelope function

In literature are proposed different approximations of un k , x
• We loose the simple interpretation of the envelope function
1

2

 w  x  R    R 
n
n
i
n
i
Ri
“kp” methods
Wannier function
1
ik  x  Ri 
Kane
Bloch
wn  x  Ri  
u
k
,
x
e
dk


u

u
n

n,k
n,0
Kane
2
un  k , x  High oscillating behaviour
Bloch

u
n,k
unL,k K  unBloch

L-K
,0
The direct use of Wannier basis is a difficult
task!!
k
k
k 0
Wigner approach to a new two-band envelope function model for quantum transport
n. 5 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
MEF model: derivation
Wannier function
1

2
Bloch function
 wn  x  Ri   n  Ri  
n
Ri
1
 n  Ri  
2

1
ikx

k
u
k
,
x
e
 dk
n  n

2
n  k eikR dk
i
FBZ
Our approach
To get our multiband model in Wannier basis:
We recover un approximate set of equation for n k in the Bloch basis
(momentum space)
• We Fourier transform the equations obtained (coordinate space)
 
Wigner approach to a new two-band envelope function model for quantum transport
n. 6 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
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
Wigner approach to a new two-band envelope function model for quantum transport
n. 7 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
MEF model: derivation
H0  Uext |  E | 
ikx
|
n
,
k

u
(
k
,
x
)
e
First approximation:
n
ˆ  k  k '  u | u 



n
,
k
|
U
|
n
,
k

U
ext n, k | U
n,k  n ',k '
( E  En )next
(k )    dk  
ext | n , k  n ' (k ')  0
 un ,k | un ',k ' 
n
 u (k , x)u
n
n'
(k ', x)dx
| un,k  un (k , x)
cell
By exploiting the periodicity of
potentials
un (k , x)
and for slow varying external
Wigner approach to a new two-band envelope function model for quantum transport
n. 8 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
MEF model: formal derivation
 un ,k | un ',k ' 
Evaluation of matrix elements
 u (k , x)u
n
n'
(k ', x)dx
cell
 un,k | un ',k '
k k'

Pn,n ' (k,k')
m0 En,n ' (k,k')
Pn,n  k , k '  2
 u ( k , x ) u
n
x n'
(k ', x)dx
Kane momentum matrix
cell
En,n ' (k , k )  En (k )  En ' (k ') 
2
2m0
k
2
 k2

Wigner approach to a new two-band envelope function model for quantum transport
n. 9 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
MEF model: derivation
( E  En )n (k )    dk   n, k | Uext | n, k   n ' (k ')  0
n
En (k )  En (k )n (k )   Uˆ (k  k ')n (k ')dk 
k k' ˆ
i
U (k  k ')Pn,n ' (k,k')n ' (k ')dk 


m0 n n En,n '
2
Wigner approach to a new two-band envelope function model for quantum transport
n. 10 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
MEF model: derivation
• Our aim: simplify the above equation.
“Interband term”:
2
k k'
ˆ P (k,k') dk  i
i
U

n,n '
n
m0 nn  En,n '  k , k '
m0
2
k k' ˆ
U Pn,n ' (0,0)n 'dk 


n n En , n '
*
2
Pn,n ' (0,0) Second approximation:
M
n,n '
ˆ
ˆ  dk 
We
retain
only
the
first

i
k
'
k

k
'
U

d
k

ik
k

k
'
U






n
'
n'


m0 n n En,n ' (0,0)
m0 n n En,n ' (0,0)
order term
2
Wigner approach to a new two-band envelope function model for quantum transport
n. 11 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
MEF model: derivation
( E  En )n (k )    dk   n, k | Uext | n, k   n ' (k ')  0
n
Approximate system
En (k )  En (k )n (k )   Uˆ (k  k ')n ' (k ')dk 
i
Fk
k k' ˆ
U Pn,n ' (0,0)n 'dk 


m0 n n En,n '
2
We write it in coordinate space
n  x  = Fk n 
Wigner approach to a new two-band envelope function model for quantum transport
n. 12 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
MEF model: first order
2
 1
 *
i
2mc
 t

2


i
2
 *
 t
2mv

2
 2 1
P U

E

U


2
 c  1
2
x
m0 Eg x
2
2 2
P U

E

U


1
 v  2
2
x
m0 Eg x
Physical meaning of the envelope function:

 | n  dx   n ( Ri )
2
|n
Ri cell
The quantity  i  x  represents the mean probability density to
find the electron into n-th band, in a lattice cell.
2
Wigner approach to a new two-band envelope function model for quantum transport
n. 13 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
MEF model: first order
2
 1
 *
i
2mc
 t

2


i
2
 *
 t
2mv

2
 2 1
P U

E

U


2
 c  1
2
x
m0 Eg x
2
2 2
P U

E

U


1
 v  2
2
x
m0 Eg x
Effective mass dynamics:
• intraband dynamic
Zero external electric field: exact
electron dynamic
Wigner approach to a new two-band envelope function model for quantum transport
n. 14 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
MEF model: first order
2
 1
 *
i
2mc
 t

2


i
2
 *
 t
2mv

2
 2 1
P U

E

U


2
 c  1
2
x
m0 Eg x
2
2 2
P U

E

U


1
 v  2
2
x
m0 Eg x
Coupling terms:
• intraband dynamic
• interband dynamic
first order contribution of
T (n  n, k  k )
transition rate of Fermi Golden rule
Wigner approach to a new two-band envelope function model for quantum transport
n. 15 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Kane model
2
2
 1Kane
 2 1Kane
 2Kane
Kane

 Vc 1 
P
i
2
t
2m0 x
m0
x


Kane
2 Kane
Kane
2
2






Kane
i
2
2
1


V


P
v 2

t
2m0 x 2
m0
x
Problems in the practical use of the Kane model:
• Strong coupling between envelope function related
to different band index, even if the external field is null
• Poor physical interpretation n( x )   
Kane
i
 x
2
i
• Critical choice in the cut off for the band index
Wigner approach to a new two-band envelope function model for quantum transport
n. 16 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Wigner picture:
1
ip
f  x, p  

x


/
2
m

x


/
2
m
e
d





2 
Wigner function:
Phase plane representation: f  x, p  pseudo probability function
Classical limit
0
Wigner equation
Liouville equation
Moments of Wigner function:
  x   n  x    f  x, p dp
2
m
   J  x    p f  x, p dp
Wigner approach to a new two-band envelope function model for quantum transport
n. 17 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Wigner picture:
   1 ,..,  n 
t
n
n-th band component
General Schrödinger-like model
matrix of operator
d
i
 H
dt
Density matrix
 1  x  1  y 

  x, y   

 n 1

1  n 


 n  n 
d
i
  H,     H x  H y  
dt
Wigner approach to a new two-band envelope function model for quantum transport
n. 18 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Wigner picture:
Multiband Wigner function
fij  x, p   W    
1
ip

x


/
2
m
,
x


/
2
m
e
d


ij

2 
Evolution equation
df
i
 W  H x  H y  W -1 f
dt
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).
Wigner approach to a new two-band envelope function model for quantum transport
n. 19 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Wigner picture:
Multiband Wigner function
fij  x, p   W    
1
ip

x


/
2
m
,
x


/
2
m
e
d


ij

2 
Evolution equation
Two band
MEF model
df
i
 W  H x  H y  W -1 f
dt
2

2
  * 2   Ec  U 
2mc x

H=
2
P U



m0 Eg x




2

2
  Ev  U  
*
2

2mv x


P U
m0 Eg x
2
Two band Wigner model
Wigner approach to a new two-band envelope function model for quantum transport
n. 20 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Wigner picture:
Two band Wigner model
 f cc
p


f cc  icc f cc

*
m
 t
 f
p
 vv


f vv  ivv f vv

*
m
 t
 f
 cv   i *   i p 2  f cv  icv f cv
 t
 4m

i
2
P
  f cv 
m0 Eg
2
P
  f cv 
m0 Eg
P
  f cc    f vv 
m0 Eg
Moments of the multiband Wigner function:

 f  x, p dp
ii
fii  x, p dp   i  x 
2
represents the mean probability density to find the
electron into n-th band, in a lattice cell.
Wigner approach to a new two-band envelope function model for quantum transport
n. 21 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Wigner picture:
Two band Wigner model
 f cc
p


f cc  icc f cc

*
m
 t
 f
p
 vv


f vv  ivv f vv

*
m
 t
 f
 cv   i *   i p 2  f cv  icv f cv
 t
 4m

i
2
P
  f cv 
m0 Eg
2
P
  f cv 
m0 Eg
P
  f cc    f vv 
m0 Eg
F p ij f ij   Vi  x   / 2m   V j  x   / 2m   F p 1  f ij 
F p   f ij   V  x   / 2m   F p -1  f ij 
Wigner approach to a new two-band envelope function model for quantum transport
n. 22 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Wigner picture:
Two band Wigner model
 f cc
p


f cc  icc f cc

*
m
 t
 f
p
 vv


f vv  ivv f vv

*
m
 t
 f
 cv   i *   i p 2  f cv  icv f cv
 t
 4m

i
2
P
  f cv 
m0 Eg
2
P
  f cv 
m0 Eg
P
  f cc    f vv 
m0 Eg
fii  x, v   W  ii 
• intraband dynamic: zero coupling if the external potential is null
Wigner approach to a new two-band envelope function model for quantum transport
n. 23 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Wigner picture:
Two band Wigner model
 f cc
p


f cc  icc f cc

*
m
 t
 f
p
 vv


f vv  ivv f vv

*
m
 t
 f
 cv   i *   i p 2  f cv  icv f cv
 t
 4m

i
2
P
  f cv 
m0 Eg
2
P
  f cv 
m0 Eg
P
  f cc    f vv 
m0 Eg
fii  x, v   W  ii 
• intraband dynamic: zero coupling if the external potential is null
• interband dynamic: coupling like G-R via fcv  x, p 
Wigner approach to a new two-band envelope function model for quantum transport
n. 24 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Mathematical setting
 x, p 
1 D problem:
2
Hilbert space: H
X1  X1  X1
Weighted spaces:
X1

f : f  L2
 
2
f , g H  fi , gi  X
i

; 1  p 2 dx dp

f,g H
1
Wigner approach to a new two-band envelope function model for quantum transport
n. 25 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Mathematical setting
2,
If the external potential Uext W ( x )
the two band Wigner system admits a unique solution f
 df
i  Af   B  C  f
 dt

f (0)  f0
H
f   f cc , f vv , f cv 
T
f0  D  A  H
Wigner approach to a new two-band envelope function model for quantum transport
n. 26 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Mathematical setting
2,
If the external potential Uext W ( x )
the two band Wigner system admits a unique solution f
 df
i  Af   B  C  f
 dt

f (0)  f0
H
f   f cc , f vv , f cv 
T
Stone theorem

p 
p 
2 1 2 
A  diag  i * , i * , * 2  p 
 m x m x 4m x

Unbounded operator
e
iAt
unitary semigroup on
H


f1 f 2  2 f3
D( A)  f  H : ,
, 2  X1 
x x x


Wigner approach to a new two-band envelope function model for quantum transport
n. 27 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Mathematical setting
2,
If the external potential Uext W ( x )
the two band Wigner system admits a unique solution f
 df
i  Af   B  C  f
 dt

f (0)  f0
B  diag cc ,vv ,cv 
H
f   f cc , f vv , f cv 
T
 0

P 
C
0
m0 Eg 
 i  

2  
 

2    
0
0

i


0



F p ij f ij   Vi  x   / 2m   V j  x   / 2m   F p 1  f ij 
F p   f ij   U  x   / 2m   F p -1  f ij 
Wigner approach to a new two-band envelope function model for quantum transport
n. 28 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Mathematical setting
2,
If the external potential Uext W ( x )
the two band Wigner system admits a unique solution f
 df
i  Af   B  C  f
 dt

f (0)  f0
B  diag cc ,vv ,cv 
H
f   f cc , f vv , f cv 
T
 0

P 
C
0
m0 Eg 
 i  

2  
 

2    
0
0

Symmetric bounded
operators
ij fij
X
 c U ext
B, C B  H
  fij
X
 c U ext
i
W 1, (


fij
x)
W 2, (
x)
Wigner approach to a new two-band envelope function model for quantum transport
0



X
fij
X
n. 29 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Mathematical setting
2,
If the external potential Uext W ( x )
the two band Wigner system admits a unique solution f
 df
i  Af   B  C  f
 dt

f (0)  f0
The operator
A B C
The unique solution of (1) is
H
f   f cc , f vv , f cv 
T
generate semigroup
f  ei A BC t f0
Wigner approach to a new two-band envelope function model for quantum transport
n. 30 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Numerical implementation: splitting scheme
f e
i A+B+C t
f0
Uniform mesh
xi  i x
Linear evolution semigroup
p j  j p
fi , j
fi, j = f ( xi , p j )
i A+B+C t
e
is a three element vector
i  A+B +C t
e

f ( xi , p j )
n
 fij t  n t

Discrete operator
i  A+ B +C  t
e
iA
e
t
2
e
iBt iA t iC t
2
e
e
Wigner approach to a new two-band envelope function model for quantum transport
n. 31 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Numerical implementation: splitting scheme
i  A+ B +C t
e
e
iA
t
2
iA
e
t iBt
2
e
e
iA
t
iA


-1
2 ˆ
fij  F x e fij 


t iC t
2
e
fˆij  F x  f ij 
f.f.t.
pj
 pj
1 
A  diag  i * 2 , i * 2 ,  * 2  p 2 
m
4m
 m

Approximate solution of
 df
  Af
i
 dt

 f (0)  f0

p 
p 
2 1 2 
A  diag  i * , i * ,  * 2  p 
 m x m x 4m x

Wigner approach to a new two-band envelope function model for quantum transport
n. 32 di 22
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Facoltà di Ingegneria
Numerical implementation: splitting scheme
i  A+ B +C t
e
e
iB
t
2
iA
e
t iBt
2
e
e
iA
t iC t
2
e
f.f.t.
B  diag cc , vv , cv 
t
iB


-1
2 ˆ
fij  F p e fij 


 ij 
Approximate solution of
 df
  Bf
i
 dt
 f (0)  f 0
fˆij  F p  f ij 
1
2
V  x  
i
/ 2m   V j  x   / 2m  
B  diag cc ,vv ,cv 
ij fij 
1
2

Vi  x   / 2m   V j  x   / 2m 
ei v v ' dv ' d
 v'
Wigner approach to a new two-band envelope function model for quantum transport
n. 33 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
fcc  x, p  Conduction band
x
p
Wigner approach to a new two-band envelope function model for quantum transport
n. 34 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
fvv  x, p  Valence band
fcc  x, p  Conduction band
x
x
p
Wigner approach to a new two-band envelope function model for quantum transport
p
n. 35 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Wigner approach to a new two-band envelope function model for quantum transport
n. 36 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
Stationary state: Thermal distribution
fcc  x, p  Conduction band
 fcv  x, p 
fvv  x, p  Valence band
 fcc  x, p 
Wigner approach to a new two-band envelope function model for quantum transport
n. 37 di 22
Facoltà di Ingegneria
ICTT19 – 19thInternational Conference on Transport Theory
Budapest, July 24-30, 2005
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
Wigner approach to a new two-band envelope function model for quantum transport
n. 38 di 22