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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 giovanni.frosali_at_unifi.it
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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)

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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.
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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.
The single-band approximation is no longer valid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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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 Greens 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)

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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)

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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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The physical environment
Electromagnetic and spin effects are disregarded,
just like the field generated by the charge
carriers themselves. Dissipative phenomena like
electron-phonon 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 (non-degenerate) valence band, around the
point is the "crystal"
wave vector. The point is assumed to
be a minimum for the conduction band and a
maximum for the valence band.
where
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KANE MODEL
The Kane model consists in a couple of
Schrödinger-like equations for the conduction and
the valence band envelope functions.
c
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Remarks on the Kane model
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LUTTINGER-KOHN model
This model is a model, i.e. the crystal
momentum 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 the
bare electron mass. As a result, if we limit
ourselves to the two-band case, we have
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where are envelope functions in the
conduction and valence bands, respectively and
and 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.
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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
Luttinger-Kohn 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
x-derivative of V. In the two-band case they
assume the form
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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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• Approximation

• Simplify the interband term in
• Introduce the effective mass approximation
• Develope the periodic part of the Bloch
  functions to the first order
• 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
Vai a lla 19
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• More rigorously MEF model can be obtained as
  follows
• projection of the wave function on the Wannier
  basis which depends on
  where are the atomic sites positions, i.e.

where the Wannier basis functions can be
expressed in terms of Bloch functions as
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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 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 are achieved.
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Comments on the MEF MODEL
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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.
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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
we introduce the particle densities
Then is the
electron density in conduction and valence bands.
We write the coupling terms in a more manageable
way, introducing the complex quantity
with
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Quantum hydrodynamic quantities

• Quantum electron current densities

when ij , we recover the classical current
densities

• Osmotic and current velocities

• Complex velocities given by osmotic and
  current velocities, which can be expressed in
  terms of

plus the phase difference
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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
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The previous balance law is just the quantum
counterpart of the classical continuity equation.
Next, we derive a system of coupled equations for
phases , obtaining an equivalent
system to the coupled Schrödinger equations. Then
we obtain a system for the currents
(similar equation for ).
The left-hand sides can be put in a more familiar
form with the quantum Bohm potentials
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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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Recalling that and are given by
the hydrodynamic quantities and , we have
the HYDRODYNAMIC SYSTEM
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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

• We express the osmotic and current
  velocities, in terms of the other hydrodynamic
  quantities.

Vai alla 31
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ZER0-TEMPERATURE QUANTUM DRIFT-DIFFUSION MODEL
for the Kane system.
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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 the densities and the currents
corresponding to the two mixed states
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HYDRODINAMIC SYSTEM for the KANE MODEL
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QUANTUM DRIFT-DIFFUSION for the KANE MODEL
with
Vai alla 33
Vai alla 32
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HYDRODINAMIC SYSTEM for the MEF MODEL
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QUANTUM DRIFT-DIFFUSION for the MEF MODEL
with
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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
and 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. The Wannier basis element arises
from applying the Fourier transform to the Bloch
functions related to the same band index .
This simple picture does not apply to the Kane
model.
The Kane envelope functions and the MEF envelope
functions are linked by the relation
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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.
See Alì, F.,Morandi, SCEE2005 Proceedings
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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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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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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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QUANTUM KINETICS LEVEL
Vai alla 42
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Wigner picture
Wigner function
MEF-Wigner Model
Vai alla 42
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Numerical results Kane-Wigner
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Numerical results MEF-Wigner
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Numerical results MEF-WIGNER model
Valence band
Conduction band
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Thanks for your attention !!!!!
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Wigner picture
Wigner function
Kane-Wigner Model
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REMARKS

• We have derived a set of quantum hydrodynamic
  equations from the two- band 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
and ,
similar to the temperature tensor of kinetic
theory.

• 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.