Facoltà di Ingegneria
WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
WASCOM05
XIII Int. Conf. on Waves and Stability in Continuous Media
Acireale - Santa Tecla, June 19-25, 2005
TWO-BAND DYNAMICS
IN SEMICONDUCTOR DEVICES:
PHYSICAL AND NUMERICAL VALIDATION
Giovanni Borgioli
Dipartimento di Elettronica e Telecomunicazioni
[email protected]
Giovanni Frosali
Dipartimento di Matematica Applicata “G.Sansone”
[email protected]
Two-band dynamics in semiconductor devices: physical and numerical validation
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Facoltà di Ingegneria
WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 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 assumed. This approximation is satisfactory as long as the carriers
populate the region near the minimum of the band.
Also in the most of 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 dynamics in semiconductor devices: physical and numerical validation
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
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Facoltà di Ingegneria
TWO-BAND APPROXIMATION
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
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is no longer valid when the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.
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Facoltà di Ingegneria
WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 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.)
We quote here only our different approaches to the problem:
• Schrödinger-like models (Barletti, Borgioli, Modugno, Morandi, etc.)
• Wigner function approach (Bertoni, Jacoboni, Borgioli, Frosali, Zweifel, Barletti, etc.)
• hydrodynamics multiband formalisms (Alì, Barletti, Borgioli, Frosali, Manzini, etc)
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 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.
Two-band dynamics in semiconductor devices: physical and numerical validation
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Facoltà di Ingegneria
WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 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
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
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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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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 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 E g increases (the opposite of physical evidence).
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Facoltà di Ingegneria
WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 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 caused by the applied electric field proportional to the xderivative of V. In the two-band case they assume the form:
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
*
• mc (and mv* ) is the isotropic effective mass
•  c and  v are the conduction and valence envelope functions
• Eg is the energy gap
• 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
• Introduction in the Schrödinger equation
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 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
MEF model can be obtained as follows:
See: Morandi, Modugno, Phys.Rev.B, 2005
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Facoltà di Ingegneria
WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
• projection of the wave function on the Wannier basis
x  Ri
where
Ri
nW
which depends on
are the atomic sites positions, i.e.
where the Wannier basis functions can be expressed in terms of Bloch
functions as
• 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
Two-band dynamics in semiconductor devices: physical and numerical validation
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Facoltà di Ingegneria
WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
Performing the limit to the continuum to the whole space and by using
standard properties of the Fourier transform, equations for the coefficients
 n ( Ri ) are achieved.
Comments on the MEF MODEL
• The envelope functions  c ,v can be interpreted as the effective wave
functions of the 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 dynamics in semiconductor devices: physical and numerical validation
Eg
increases, and
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
Facoltà di Ingegneria
Physical meaning of the envelope functions
A more direct physical meaning can be ascribed to the hydrodynamical variables
derived from the MEF approach.
 cM
M
 on the Wannier

and
v are the projections of
The envelope functions
basis, and therefore the corresponding multi-band densities represent the (cellaveraged) probability amplitude of finding an electron on the conduction or
valence bands, respectively.
This simple picture does not apply to the Kane model.
The Kane envelope functions and the MEF envelope functions
are linked by the relation
2
 
K
j
M
j
P
i
 hM ,
m0 ( E j  Eh )
j , h  c, v.
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.
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Facoltà di Ingegneria
WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
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: G.Alì, G.F., O.Morandi,
SCEE2005 Proceedings
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
Kane
MEF
The incident (from the left) conduction electron beam is mainly
reflected by the barrier and the valence states are almost unexcited.
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
Kane
MEF
The incident (from the left) conduction electron beam is partially
reflected by the barrier and partially captured by the well
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
Kane
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.
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Facoltà di Ingegneria
WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
Hydrodynamic version of the MEF 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
 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
i
with
 :
Two-band dynamics in semiconductor devices: physical and numerical validation
Sc  Sc

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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 
mE g
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
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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
plus the phase difference
Two-band dynamics in semiconductor devices: physical and numerical validation

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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
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, on the contrary of the Kane model, the “interband density”
Is missing.
 c v
The previous balance law is just the quantum counterpart of the classical
continuity equation.
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
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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  .
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Recalling that
and

WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
ncv , uv , and uv are given by the hydrodynamic quantities nc , nv , J c , J v
, we have the HYDRODYNAMIC SYSTEM for the MEF model
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
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.

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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
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Hydrodynamic version of the MEF MODEL
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 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 MEF 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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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
obtaining
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WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
QUANTUM DRIFT-DIFFUSION for the MEF MODEL
with
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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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Facoltà di Ingegneria
WASCOM05 – XIII Int.Conf. on Waves and Stability in Continuous Media
Santa Tecla (Catania), June 19-25, 2005
Thanks for your attention !!!!!
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