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”
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
Wigner approach to a new two-band envelope function model for quantum transport n. 2 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 3 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 4 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 5 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 6 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 7 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 8 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
MEF model: derivation MEF model: derivation
'( ) ( ) 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
Wigner approach to a new two-band envelope function model for quantum transport n. 9 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 10 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
MEF model: derivation MEF model: derivation
'( ) ( ) 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
Wigner approach to a new two-band envelope function model for quantum transport n. 11 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
,*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
MEF model: derivation MEF model: derivation
We retain only the first
order term
We retain only the first
order term
Second approximation:
Wigner approach to a new two-band envelope function model for quantum transport n. 12 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
MEF model: derivation MEF model: derivation
'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
Wigner approach to a new two-band envelope function model for quantum transport n. 13 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 14 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
• intraband dynamic
MEF model: first order MEF model: first order
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
Wigner approach to a new two-band envelope function model for quantum transport n. 15 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
• 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 16 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
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 approach to a new two-band envelope function model for quantum transport n. 17 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
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:
Wigner approach to a new two-band envelope function model for quantum transport n. 18 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 19 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
1x yH H
dfi f
dt -W W
Wigner picture:Wigner picture:
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).
Wigner approach to a new two-band envelope function model for quantum transport n. 20 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
1x yH H
dfi f
dt 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
Wigner approach to a new two-band envelope function model for quantum transport n. 21 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
*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
Wigner approach to a new two-band envelope function model for quantum transport n. 22 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
Wigner picture:Wigner picture: Two band Wigner model
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
f i Pi p f i f i f f
t m m E
Wigner approach to a new two-band envelope function model for quantum transport n. 23 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
*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
f i Pi p f i f i f f
t
f pf i f
t m
f pf i f
m m E
t m
Wigner picture:Wigner picture: Two band Wigner model
, iii if x v W
• intraband dynamic: zero coupling if the external potential is null
Wigner approach to a new two-band envelope function model for quantum transport n. 24 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
*
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
f i Pi p f i f i f f
t
Pf
m E
Pf
m E
m m E
Wigner picture:Wigner picture: Two band Wigner model
, 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
Wigner approach to a new two-band envelope function model for quantum transport n. 25 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 26 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
Mathematical settingMathematical setting
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
Wigner approach to a new two-band envelope function model for quantum transport n. 27 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
Mathematical settingMathematical setting
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
Hfthe two band Wigner system admits a unique solution
2, ( )ext xU W If the external potential
, ,cc vv cvf f ffT
Wigner approach to a new two-band envelope function model for quantum transport n. 28 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
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
Mathematical settingMathematical setting
Hfthe two band Wigner system admits a unique solution
2, ( )ext xU W If the external potential
, ,cc vv cvf f ffT
Wigner approach to a new two-band envelope function model for quantum transport n. 29 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
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
Mathematical settingMathematical setting
Hfthe two band Wigner system admits a unique solution
2, ( )ext xU W If the external potential
, ,cc vv cvf f ffT
0
0 0 2
0 0 2
0g
CP
m Ei i
Wigner approach to a new two-band envelope function model for quantum transport n. 30 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
0(0)
di A B C
dt
ff f
f f
The operator generate semigroup A B C
The unique solution of (1) is
Mathematical settingMathematical setting
0
i A B C te f f
Hfthe two band Wigner system admits a unique solution
2, ( )ext xU W If the external potential
, ,cc vv cvf f ffT
Wigner approach to a new two-band envelope function model for quantum transport n. 31 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 32 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 33 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
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
Wigner approach to a new two-band envelope function model for quantum transport n. 34 di 22
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Space coordinate Momentum co
ordinate
x p
,cc x pf Conduction band
Wigner approach to a new two-band envelope function model for quantum transport n. 35 di 22
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Space coordinate Momentum co
ordinate
x p
,cc x pf Conduction band
Space coordinate Momentum coordinate
x
p
Valence band ,vv x pf
Wigner approach to a new two-band envelope function model for quantum transport n. 36 di 22
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Wigner approach to a new two-band envelope function model for quantum transport n. 37 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
,cc x pf Conduction band
,cc x p f
Valence band ,vv x pf
,cv x p f
Stationary state: Thermal distribution
Wigner approach to a new two-band envelope function model for quantum transport n. 38 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
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