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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
KINETIC EQUATIONS:
Direct and Inverse Problems
Università degli Studi di Pavia (sede di Mantova)
Mantova
May 15-17, 2005
TWO-BAND MODELS FOR ELECTRON TRANSPORT
IN SEMICONDUCTOR DEVICES
Giovanni Frosali
Dipartimento di Matematica Applicata “G.Sansone”
[email protected]
Two-band models for electron transport in semiconductor devices
n. 1 di 42
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
University of Florence research group on semicoductor modeling
Dipartimento di Matematica Applicata “G.Sansone”
 Giovanni Frosali
Dipartimento di Matematica “U.Dini”
 Luigi Barletti
Dipartimento di Elettronica e Telecomunicazioni
 Stefano Biondini
 Giovanni Borgioli
 Omar Morandi
Università di Ancona
 Lucio Demeio
Scuola Normale Superiore di Pisa (Munster)
 Chiara Manzini
Others: G.Alì (Napoli), C.DeFalco (Milano), M.Modugno (LENS-INFM Firenze),
A.Majorana(Catania), C.Jacoboni, P.Bordone et. al. (Modena)
Two-band models for electron transport in semiconductor devices
n. 2 di 42
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
SINGLE-BAND APPROXIMATION
In the standard semiconductor devices, like the Resonant Tunneling
Diode, the single-band approximation, valid if most of the current is carried
by the charged particles of a single band, is usually satisfactory. Together with
the single-band approximation, the parabolic-band approximation is also
usually made. This approximation is satisfactory as long as the carriers
populate the region near the minimum of the band.
Also in the most bipolar electrons-holes models, there is no coupling
mechanism between energy bands which are always decoupled in the
effective-mass approximation for each band and the coupling is heuristically
inserted by a "generation-recombination" term.
Most of the literature is devoted to single-band problems, both from the
modeling and physical point of view and from the numerical point of view.
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
It is well known that the spectrum of the Hamiltonian of a quantum
particle moving in a periodic potential is a continuous spectrum which can be
decomposed into intervals called "energy bands". In the presence of external
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.
RITD Band Diagram
2
Energy (ev)
1
0
-1
-2
0
10
20
30
40
Position (nm)
50
60
Two-band models for electron transport in semiconductor devices
The single-band approximation
is no longer valid when the
architecture of the device is
such that other bands are
accessible to the carriers. In
some nanometric semiconductor
device like Interband Resonant
Tunneling Diode, transport due
to valence electrons becomes
important.
n. 4 di 42
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
It is necessary to use more sophisticated models, in which 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 based on the effective mass theory (Liu, Ting,
McGill, Chao, Chuang, etc.)
• tight-binding (Boykin, van der Wagt, Harris, Bowen, Frensley, etc.)
• pseudopotential methods (Bachelet, Hamann, Schluter, etc.)
Various mathematical tools are employed to exploit the multiband
quantum dynamics underlying the previous models:
• the Schrödinger-like models (Sweeney,Xu, etc.)
• the nonequilibrium Green’s function (Luke, Bowen, Jovanovic, Datta, etc.)
• the Wigner function approach (Bertoni, Jacoboni, Borgioli, Frosali, Zweifel, Barletti, etc.)
• the hydrodynamics multiband formalisms (Alì, Barletti, Borgioli, Frosali, Manzini, etc)
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
MULTI-BAND, NON-PARABOLIC ELECTRON TRANSPORT
• Wigner-function approach
• Formulation of general models for multi-band non-parabolic electron
transport
• Use of Bloch-state decomposition (Demeio, Bordone, Jacoboni)
• Envelope functions approach (Barletti)
• Wigner formulation of the two-band Kane model (Borgioli, Frosali, Zweifel)
•
Numerical applications (Demeio, Morandi)
•
The Wigner function for thermal equilibrium of a two-band (Barletti)
•
Multiband envelope function models (MEF models) (Modugno, Morandi)
•
Two-band hydrodynamic models (Two-band QDD equations) (Alì, Biondini,
Frosali, Manzini)
Two-band models for electron transport in semiconductor devices
n. 6 di 42
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
QUANTUM MECHANICS LEVEL
In this talk we present different Schrödinger-like models.
The first one is well-known in literature as the Kane model.
The second, based on the Luttinger-Kohn approach, disregards the
interband tunneling effect.
The third, recently derived within the usual Bloch-Wannier formalism, is
formulated in terms of a set of coupled equations for the electron envelope
functions by an expansion in terms of the crystal wave vector k (MEF model).
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 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 k is 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
Two-band models for electron transport in semiconductor devices
an external potential.
n. 8 di 42
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
KANE MODEL
The Kane model consists in 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 )
c
• 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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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 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 valence band, with the wave
number becoming imaginary.
Then the electron continues
propagating into the valence
band.
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
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 Eg increases (the opposite of physical evidence).
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
LUTTINGER-KOHN model
This model is a k P model, i.e. the crystal momentum k is used as a perturbation
parameter of the Hamiltonian. The wave function is expanded on a different basis
with respect to Kane model:
where n, n' are the band index and
m0
the bare electron mass.
As a result, if we limit ourselves to the two-band case, we have:
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
where  c ,v are envelope functions in the conduction and valence bands,
*
*
respectively and mc and mv are, respectively, the isotropic effective masses
in the conduction and valence bands.
As it is manifest, disregarding the off-diagonal terms implies the achievement of
two uncoupled equations for the envelope functions in the two bands. This
means that the model, at this stage of approximation, is not able to describe an
interband tunneling dynamics.
Two-band models for electron transport in semiconductor devices
n. 13 di 42
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
MEF MODEL (Morandi, Modugno, Phys.Rev.B, 2005)
The MEF model consists in a couple of Schrödinger-like equations as
follows.
A different procedure of approximation leads to equations describing the
intraband dynamics in the effective mass approximation as in the LuttingerKohn model, which also contain an interband coupling, proportional to the
momentum matrix element P. This is responsible for tunneling between
different bands induced by the applied electric field proportional to the xderivative of V. In the two-band case they assume the form:
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
*
• mc* (and mv ) is the isotropic effective mass
• c and  v are the conduction and valence envelope functions
• E is the energy gap
g
• P is the coupling coefficient between the two bands
Which are the steps to attain MEF model formulation?
• Expansion of the wave function on the Bloch functions basis
• Insert in the Schrödinger equation
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
• Approximation
• Simplify the interband term in
k 0
• Introduce the effective mass approximation
• Develope the periodic part of the Bloch functions
order
un (k , x)
to the first
• The equation for envelope functions in x-space is obtained by inverse
Fourier transform
For more rigorous details:
See: Morandi, Modugno, Phys.Rev.B, 2005
Two-band models for electron transport in semiconductor devices
Vai a lla 19
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
More rigorously MEF model can be obtained as follows:
•
W
projection of the wave function on the Wannier basis
n
on x  Ri where Ri are the atomic sites positions, i.e.
which depends
where the Wannier basis functions can be expressed in terms of Bloch
functions as
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
The use of the Wannier basis has some advantages.
As a matter of fact the amplitudes  n ( Ri ) that play the role of envelope
functions on the new basis, can be obtained from the Bloch coefficients by a
simple Fourier transform
Moreover they can be interpreted as the actual wave function of an electron
in the n-band. In fact, ''macroscopic'' properties of the system, like charge
density and current, can be expressed in term of  n ( Ri ) averaging on a
scale of the order of the lattice cell.
Performing the limit to the continuum to the whole space and by
using standard properties of the Fourier transform, equations for the
coefficients n ( x) are achieved.
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
Comments on the MEF MODEL
• The envelope functions  c ,v can be interpreted as the effective wave
functions of the electron in the conduction (valence) band
• The coupling between the two bands appears only in presence of an external
(not constant) potential
• The presence of the effective masses (generally different in the two bands)
implies a different mobility in the two bands.
• The interband coupling term reduces as the energy gap
vanishes in the absence of the external field V.
Two-band models for electron transport in semiconductor devices
Eg
increases, and
n. 19 di 42
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
QUANTUM HYDRODYNAMICS LEVEL
From the point of view of practical applications, the approaches
based on microscopic models are not completely satisfactory.
Hence, it is useful to formulate semi-classical models in terms of
macroscopic variable. Using the WKB method, we obtain a system for
densities and currents in the two bands.
In this context zero and nonzero temperatures quantum drift
diffusion models are derived, for the Kane and MEF systems.
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
Hydrodynamic version of the KANE MODEL
We can derive the hydrodynamic version of the Kane model using the WKB
method (quantum system at zero temperature).
Look for solutions in the form
 c ( x, t )  nc ( x, t ) exp 
iSc ( x, t ) 




iS ( x, t ) 
 v ( x, t )  nv ( x, t ) exp  v




we introduce the particle densities
Then
nij ( x, t )   i ( x, t ) j ( x, t ).
n   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
i
Two-band models for electron transport in semiconductor devices
with
 :
Sc  Sv

n. 21 di 42
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
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, which can
be expressed in terms of
nc , nv , J c , J v
Two-band models for electron transport in semiconductor devices
plus the phase difference

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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
The quantum counterpart of the classical continuity equation
Taking account of the wave form, the Kane system gives rise to
Summing the previous equations, we obtain the balance law
where we have used the “interband density” and the complex velocities
 c v  ncvuv  nc nv (cos  i sin  )(uos,v  iuel ,v )
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
The previous balance law is just the quantum counterpart of the classical
continuity equation.
Next, we derive a system of coupled equations for phases Sc , Sv , obtaining an
equivalent system to the coupled Schrödinger equations. Then we obtain a
system for the currents
(similar equation for J v ).
The left-hand sides can be put in a more familiar form with the quantum
Bohm potentials
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
We express the right-hand sides of the previous equations in terms of the
hydrodynamic quantities
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  .
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Dipartimento di Matematica Applicata
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Recalling that
and
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
ncv , uv ,and uvare given by the hydrodynamic quantities nc , nv , J c , J v
 , we have the HYDRODYNAMIC SYSTEM
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
The DRIFT-DIFFUSION scaling
• We rewrite the current equations, introducing a relaxation time term in
order to simulate all the mechanisms which force the system towards the
statistical mechanical equilibrium characterized by the relaxation time

• 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 ,    .

• We express the osmotic and current velocities, in terms of the other
hydrodynamic quantities.
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
ZER0-TEMPERATURE QUANTUM DRIFT-DIFFUSION MODEL
for the Kane system.
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
NON ZERO TEMPERATURE hydrodynamic model
We consider an electron ensemble which is represented by a mixed quantum
mechanical state, with a view to obtaining a nonzero temperature model for a
Kane system.
We rewrite the Kane system for the k-th state
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
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
HYDRODINAMIC SYSTEM for the KANE MODEL
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
QUANTUM DRIFT-DIFFUSION for the KANE MODEL
with
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
HYDRODINAMIC SYSTEM for the MEF MODEL
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
QUANTUM DRIFT-DIFFUSION for the MEF MODEL
with
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
Physical meaning of the envelope functions
A more direct physical meaning can be ascribed to the hydrodynamical variables
derived from the second approach.
The envelope functions  c and  v are the projections of  on the Wannier
basis, and therefore the corresponding multi-band densities represent the (cellaveraged) probability amplitude of finding an electron on the conduction or
valence bands, respectively.
The Wannier basis element arises from applying the Fourier transform to
the Bloch functions related to the same band index n .
M
M
This simple picture does not apply to the Kane model.
The Kane envelope functions and the MEF envelope functions
are linked by the relation
   i
K
j
M
j
2
P
 hM
m0 ( E j  Eh )
Two-band models for electron transport in semiconductor devices
, j, h  c, v.
n. 34 di 42
Dipartimento di Matematica Applicata
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Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
This fact confirms that even in absence of external potential , when no interband
transition can occur, the Kane model shows a coupling of all the envelope
functions.
NUMERICAL SIMULATION
We consider a heterostructure
which consists of two
homogeneous regions separated
by a potential barrier and which
realizes a single quantum well in
valence band.
See: Alì, F.,Morandi,
SCEE2005 Proceedings
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kane
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
MEF
The incident (from the left) conduction electron beam is mainly
reflected by the barrier and the valence states are almost unexcited.
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kane
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
MEF
The incident (from the left) conduction electron beam is partially
reflected by the barrier and partially captured by the well
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kane
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
MEF
When the electron energy approaches the resonant level, the electron can
travel from the left to the right, using the bounded valence resonant state as a
bridge state.
Two-band models for electron transport in semiconductor devices
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
QUANTUM KINETICS LEVEL
Two-band models for electron transport in semiconductor devices
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n. 39 di 42
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
Wigner picture
Wigner function: fij  x, v  
1
*
iv

x


/
2
m

x


/
2
m
e
dv ' d




i
j

2 
MEF-Wigner Model
 f cM
p
P
M
M
 M



f

i

f

2
Im

f cv 


c
cc c
*
m
m0 E g
 t
 f M
p
P
M
M
 M
v



f

i

f

2
Im

f cv 


v
vv v
*
m
m0 E g
 t
 f M
i
P
 cv  i * f cvM  p 2 f cvM  icv f cvM  i
  f cM    f vM 
4m
m0 E g
 t
ij fij 
1
2

Vi  x   / 2m   V j  x   / 2m 
ei v v ' dv ' d
 v'
Two-band models for electron transport in semiconductor devices
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n. 40 di 42
Numerical results: Kane-Wigner
Numerical results: MEF-Wigner
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
Numerical results: MEF-WIGNER model
Conduction band
Two-band models for electron transport in semiconductor devices
Valence band
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
Thanks for your attention !!!!!
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Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
Wigner picture
Wigner function: fij  x, v  
1
*
iv

x


/
2
m

x


/
2
m
e
dv ' d




i
j

2 
Kane-Wigner Model
 f ccKane
P
Kane
Kane
Kane
Kane


v

f

i

f


Im
f

2
Pv
Re
f
 cv 
 cv 
cc
cc cc
 t
m
0

 f vvKane
P
Kane
Kane
Kane
Kane


v

f

i

f


Im
f

2
Pv
Re
f





vv
vv vv
cv
cv
m0
 t
 f Kane
P
Kane
Kane
cv
 vf cv  icv f cv  i
  f ccKane  f vvKane   Pv  f ccKane  f vvKane 

2m0
 t
Vi  x   / 2m   V j  x   / 2m  i v v '
1
iv
ij fij 
e
dv
'
d

e
dv ' d


2  v '
Two-band models for electron transport in semiconductor devices
n. 45 di 42
Dipartimento di Matematica Applicata
Università di Firenze
Kinetic Equations: direct and inverse problems
Mantova, May 15-17, 2005
REMARKS
We have derived a set of quantum hydrodynamic equations from the twoband Kane model, and from the MEF model. These systems, which can be
considered as a nonzero-temperature quantum fluid models, are not closed.
In addition to other quantities, we have the tensors
similar to the temperature tensor of kinetic theory.
c ,v ,cv
and
vc ,
• Closure of the quantum hydrodynamic system
• Numerical treatment
• Heterogeneous materials
Numerical validation for the quantum drift-diffusion equations (Kane and
MEF models) are work in progress.
Two-band models for electron transport in semiconductor devices
n. 46 di 42