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Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
Giornata di lavoro
Mathematical modeling and numerical analysis of quantum
systems with applications to nanosciences
Firenze, 16 dicembre 2005
MULTIBAND TRANSPORT MODELS
FOR SEMICONDUCTOR DEVICES
Giovanni Frosali
Dipartimento di Matematica Applicata “G.Sansone”
[email protected]
Multiband transport models for semiconductor devices
n. 1 di 28
Dipartimento di Matematica
Applicata Università di Firenze
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Research group on semicoductor modeling at University of Florence
Dipartimento di Matematica Applicata “G.Sansone”
 Giovanni Frosali
 Chiara Manzini (Munster)
 Michele Modugno (Lens-INFN)
Dipartimento di Matematica “U.Dini”
 Luigi Barletti
Dipartimento di Elettronica e Telecomunicazioni
 Stefano Biondini
 Giovanni Borgioli
 Omar Morandi
Università di Ancona
 Lucio Demeio
Others: G.Alì (Napoli), C.DeFalco (Milano), A.Majorana(Catania), C.Jacoboni,
P.Bordone et. al. (Modena)
Multiband transport models for semiconductor devices
n. 2 di 28
Dipartimento di Matematica
Applicata Università di Firenze
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
TWO-BAND APPROXIMATION
• The spectrum of the Hamiltonian of a quantum particle in a periodic
potential is continuous and characterized by (allowed) "energy bands“ separated
by (forbidden) “band gaps".
• In the presence of additional potentials, the projections of the wave function on
the energy eigenspaces (Floquet subspaces) are coupled by the Schrödinger
equation, which allows interband transitions to occur.
• Negibible coupling: single-band approximation
• In some nanometric semiconductor device
like Interband Resonant Tunneling Diode,
transport due to valence electrons becomes
important.
Multiband transport models for semiconductor devices
2
1
Energy (ev)
• This is no longer possible when the architecture of the device is such that other bands are
accessible to the carriers.
RITD Band Diagram
0
-1
-2
0
10
20
30
40
Position (nm)
50
60
n. 3 di 28
Dipartimento di Matematica
Applicata Università di Firenze
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
• Multiband models are needed: the charge carriers can be found in a
super-position of quantum states belonging to different bands.
• Different methods are currently employed for characterizing the band
structures and the optical properties of heterostructures, such as envelope
functions methods (effective mass theory), tight-binding, pseudopotential
methods,…
OUR APPROACH TO THE PROBLEM
 Schrödinger-like models (Barletti, Borgioli, Modugno, Morandi, etc.)
 Wigner function approach (Bertoni, Jacoboni, Borgioli, Frosali, Zweifel, Barletti,
Manzini, etc.)
 Hydrodynamics multiband formalisms (Alì, Barletti, Borgioli, Frosali,
Manzini, etc)
Multiband transport models for semiconductor devices
n. 4 di 28
Dipartimento di Matematica
Applicata Università di Firenze
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
MULTIBAND TRANSPORT
Isothermal
QDD
General Multiband
Models
WIGNER
APPROACH
HYDRODYNAMIC
MODELS
SCHRÖDINGER
APPROACH
QUANTUM
DRIFT-DIFFUSION
MODELS
KANE model
MeF model
Multiband transport models for semiconductor devices
ChapmanEnskog
expansion
n. 5 di 28
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
Envelope function models
Schrödinger equation
H  E
H 
2
2m0
We filter the solution
 ( x)
Hamiltonian
Multiband
“KP”
system
 2  Vper ( x )  U ext ( x )
1 ( x )
2 ( x )
n ( x )
The envelope functions c and v are the projections of  on the Wannier basis,
and therefore the corresponding multi-band densities represent the (cell-averaged)
probability amplitude of finding an electron on the conduction or valence bands,
respectively.
Multiband transport models for semiconductor devices
n. 6 di 28
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
MEF model: first order
2
 1
i t   2m*
c


2


2
i
 *
 t
2mv

2
 2 1
P U
  Ec  U  1 
2
2
x
m0 E g x
2
2 2
P U
  Ev  U   2 
1
2
x
m0 E g x
Physical meaning of the envelope function:

 | n  dx   n ( Ri )
Ri cell
The quantity
i  x 
2
2
represents the mean probability density to find the
electron into the n-th band, in a lattice cell.
Multiband transport models for semiconductor devices
n. 7 di 28
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
MEF model: first order
2
 1
i t   2m*
c


2


2
i
 *
 t
2mv

2
 2 1
P U
  Ec  U  1 
2
2
x
m0 E g x
2
2 2
P U
  Ev  U   2 
1
2
x
m0 E g x
Effective mass dynamics:
• intraband dynamic
Zero external electric field: exact
electron dynamic
Multiband transport models for semiconductor devices
n. 8 di 28
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
MEF model: first order
2
 1
i t   2m*
c


2


2
i
 *
 t
2mv

2
 2 1
P U
  Ec  U  1 
2
2
x
m0 E g x
2
2 2
P U
  Ev  U   2 
1
2
x
m0 E g x
Coupling terms:
• intraband dynamic
• interband dynamic
T (n  n, k  k )
first order contribution of transition rate of Fermi Golden rule
Multiband transport models for semiconductor devices
n. 9 di 28
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
Wigner picture:
Wigner function:
f  x, p  
1
ip

x


/
2
m

x


/
2
m
e
d





2 
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
Multiband transport models for semiconductor devices
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Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
WIGNER APPROACH
Wigner picture for Schrödinger-like models
Density matrix
 1  x  1  y 

  x, y   

 n 1

1  n 


 n  n 
d
i
  H,     H x  H y  
dt
Multiband Wigner function
f ij  x, p   W    
1
ip

x


/
2
m
,
x


/
2
m
e
d


ij

2 
Evolution equation
Multiband transport models for semiconductor devices
df
i
 W  H x  H y  W -1 f
dt
n. 11 di 28
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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
i
 cv   i *   p 2  f cv  icv f cv
 4m

 t
ij
i
2
P
  f cv 
m0 E g
2
P
  f cv 
m0 E g
P
  f cc    f vv 
m0 E g
pseudo-differential operators:
Fp ij f ij   Vi  x   / 2m   V j  x   / 2m   Fp1  f ij 
Fp   f ij   V  x   / 2m   Fp -1  f ij 
Multiband transport models for semiconductor devices
n. 12 di 28
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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
i
 cv   i *   p 2  f cv  icv f cv
 4m

 t
i
2
P
  f cv 
m0 E g
2
P
  f cv 
m0 E g
P
  f cc    f vv 
m0 E g
fii  x, v   W  ii 
 intraband dynamic: zero coupling if the external potential is null
Multiband transport models for semiconductor devices
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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
i
 cv   i *   p 2  f cv  icv f cv
 4m

 t
i
2
P
  f cv 
m0 E g
2
P
  f cv 
m0 E g
P
  f cc    f vv 
m0 E g
fii  x, v   W  ii 
• intraband dynamic: zero coupling if the external potential is null
• interband dynamic: coupling like G-R via f cv  x, p 
Multiband transport models for semiconductor devices
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Mathematical setting
1 D problem:
Hilbert space:
H

Weighted spaces: X 1
If the external potential U ext  W 2, (
x
X1  X1  X1
f : f  L2
Multiband transport models for semiconductor devices
; 1  p 2  dx dp

)
the two-band Wigner system admits a unique solution
 df
 Af   B  C  f
i
 dt
f (0)  f0

2
f H
f   f cc , f vv , f cv 
T
f0  D  A  H
n. 15 di 28
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Mathematical setting
Stone theorem

p 
p 
2 1 2 
A  diag  i * , i * , * 2  p 
 m x m x 4m x

unbounded operator
e
unitary group on


f1 f 2  2 f 3
D( A)  f  H :
,
, 2  X1 
x x x


B  diag cc ,vv ,cv 
 0
P 
C m E  0
0
g
 i  

iAt
2  
 


2  
0
0

i

H

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 
Multiband transport models for semiconductor devices
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Mathematical setting
Symmetric bounded
operators
B, C B  H 
If the external potential U ext W 2, (
a unique solution
f H
ij fij
X
 c Uext
W 1, (
  fij
X
 c Uext
W 2, (
x
A B  C
fij
fij
x)
X
X
) the two band Wigner system admits
 df
 Af   B  C  f
i
 dt

f (0)  f0
The operator
x)
f   f cc , f vv , f cv 
T
generates semigroup
The unique solution is given by
Multiband transport models for semiconductor devices
i A BC t
f e
f0
Simulation
n. 17 di 28
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Multiband transport models for semiconductor devices
Giornata di lavoro sulle Nanoscienze
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Hydrodynamic version of the MEF MODEL
We can derive the hydrodynamic version of the MEF model using the WKB
method (quantum system at zero temperature).
Look for solutions in the form
 iS ( x, t ) 
 c ( x, t )  nc ( x, t ) exp  c

  
 v ( x, t )  nv ( x, t ) exp 
iSv ( x, t ) 




we introduce the particle densities
Then n
nij ( x, t )   i ( x, t ) j ( x, t ).
  c c   v v is the electron density in conduction and valence bands.
We write the coupling terms in a more manageable way, introducing the complex
quantity
ncv :  c v  nc nv e
Multiband transport models for semiconductor devices
i
with
 :
Sc  Sc

Vai alla 21
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We introduce the rescaled Planck constant
mlR2
parameter  
tR
and the effective mass
where
m
lR , t R


with the dimensional
are typical dimensional quantities
is assumed to be equal in the two bands
MEF model reads in the rescaled form:
m P V
with K 
mEg
Multiband transport models for semiconductor devices
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Quantum hydrodynamic quantities
• Quantum electron current densities
J ij   Im( i  j )
when i=j , we recover the classical current densities
J c  ncSc J v  nvSv
• Osmotic and current velocities
uc  uos ,c  iuel ,c
uos ,i
 ni

,
ni
uel ,i
uv  uos ,v  iuel ,v
Ji
 Si 
, i  c, v
ni
• Complex velocities given by osmotic and current velocities can be
expressed in terms of
nc , nv , J c , J v
Multiband transport models for semiconductor devices
plus the phase difference

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The quantum counterpart of the classical continuity equation
Taking account of the wave form, the MEF system gives rise to
Summing the previous equations, we obtain the balance law
where, compared to the Kane model, the “interband density”
Is missing.
 c v
The previous balance law is just the quantum counterpart of the classical
continuity equation.
Multiband transport models for semiconductor devices
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Next, we derive a system of coupled equations for phases S c , S v , obtaining a
system equivalent to the coupled Schrödinger equations. Then we obtain a
system for the currents J c and J v
The equations can be put in a more familiar form with the quantum Bohm
potentials
It is important to notice that, differently from the uncoupled model, equations
for densities and currents are not equivalent to the original equations, due to
the presence of  .
Multiband transport models for semiconductor devices
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Recalling that
and

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ncv , uc , and uv are given by the hydrodynamic quantities nc , nv , J c , J v
, we have the HYDRODYNAMIC SYSTEM for the MEF model
Multiband transport models for semiconductor devices
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The DRIFT-DIFFUSION scaling

We rewrite the current equations, introducing a relaxation time , in order
to simulate all the mechanisms which force the system towards the
statistical mechanical equilibrium.
In analogy with the classical diffusive limit for a one-band system, we introduce
the scaling
t
t  , J c   J c , J v   J v ,    ,

Finally, after having expressed the osmotic and current velocities, in terms of
the other hydrodynamic quantities, as
tends to zero, we formally obtain the
ZER0-TEMPERATURE QUANTUM DRIFT-DIFFUSION MODEL for the MEF
system.

Multiband transport models for semiconductor devices
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Hydrodynamic version of the MEF MODEL
Multiband transport models for semiconductor devices
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NON ZERO TEMPERATURE hydrodynamic model
We consider an electron ensemble which is represented by a mixed quantum
mechanical state, to obtain a nonzero temperature model for a Kane system.
We rewrite the MEF system for the k-th state, with occupation probability k
We use the Madelung-type transform
We define
 ik  nik exp  iSik /   , i  c, v
J ck , J vk ,  k , ncvk , uck , uvk .
We define the densities and the currents corresponding to the two mixed states
Performing the analogous procedure and with an appropriate closure, we get
Multiband transport models for semiconductor devices
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Isothermal QUANTUM DRIFT-DIFFUSION for the MEF MODEL
Multiband transport models for semiconductor devices
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Thanks for your attention !!!!!
Multiband transport models for semiconductor devices
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REMARKS
We derived a set of quantum hydrodynamic equations from the two-band
MEF model. This system, which is closed, can be considered as a zerotemperature quantum fluid model.
Starting from a mixed-states condition, we derived the corresponding non
zero-temperature quantum fluid model, which is not closed.
In addition to other quantities, we have the tensors  vc and  c ,  v , cv
similar to the temperature tensor of kinetic theory.
NEXT STEPS
• Closure of the quantum hydrodynamic system
• Numerical treatment
• Heterogeneous materials
• Generalized MEF model
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Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
Kane model
2
2
 1Kane
 2 1Kane
 2Kane
Kane

 Vc 1 
P
i
2
t
2m0 x
m0
x


Kane
2
2 Kane
2
Kane






Kane
2
2
1
i


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
Multiband transport models for semiconductor devices
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Dipartimento di Matematica
Applicata Università di Firenze
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
The physical environment
Electromagnetic and spin effects are disregarded, just like the field generated
by the charge carriers themselves. Dissipative phenomena like electronphonon collisions are not taken into account.
The dynamics of charge carriers is considered as confined in the two highest
energy bands of the semiconductor, i.e. the conduction and the (nondegenerate) valence band, around the point k  0 where kis the "crystal"
wave vector. The point k  0 is assumed to be a minimum for the conduction
band and a maximum for the valence band.
The Hamiltonian introduced in the Schrödinger equation is
H  Ho  V ,
where
h2
H o     Vper
2m
V per is the periodic potential of the crystal and V an external potential.
Multiband transport models for semiconductor devices
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Dipartimento di Matematica
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Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Interband Tunneling: PHYSICAL PICTURE
Interband transition in the 3-d
dispersion diagram.
The transition is from the bottom of
the conduction band to the top of
the val-ence band, with the wave
number becoming imaginary.
Then the electron continues
propagating into the valence band.
Kane model
Multiband transport models for semiconductor devices
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Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
KANE MODEL
The Kane model consists into a couple of Schrödinger-like equations for
the conduction and the valence band envelope functions.
Let  c ( x , t ) be the conduction band electron envelope function and
be the valence band envelope function.
 v ( x, t )
• m is the bare mass of the carriers, Vi  Ei  V ,
i  c, v
• Ec ( Ev ) is the minimum (maximum) of the conduction (valence) band energy
• P is the coupling coefficient between the two bands (the matrix element of the
gradient operator between the Bloch functions)
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Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
Remarks on the Kane model
• The envelope functions  c ,v are obtained expanding the wave function on the
ikx
basis of the periodic part of the Bloch functions bn ( x, t )  e un (k , x), evaluated
at k=0,
0
0
 ( x)   c ( x)uc  v ( x)uv
where
uc0,v ( x)  uc,v (0, x) .
• The external potential V affects the band energy terms Vc (Vv ), but it does not
appear in the coupling coefficient P .
• There is an interband coupling even in absence of an external potential.
• The interband coefficient P increases when the energy gap between the two
bands E g increases (the opposite of physical evidence).
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