Presentazione di PowerPoint

1

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 giovanni.frosali_at_unifi.it
2
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)

3
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
• This is no longer possible when the
  architect-ure 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.

4

• 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)

5
MULTIBAND TRANSPORT
QDD
General Multiband Models
WIGNER APPROACH
HYDRODYNAMIC MODELS
KANE model
QUANTUM DRIFT-DIFFUSION MODELS
SCHRÖDINGER APPROACH
MeF model
CE expansion
6
Envelope function models
We filter the solution
Multiband kp system
7
WIGNER APPROACH
Lucio Demeio - collaborazioni con Paolo Bordone,
Carlo Jacoboni Luigi Barletti Giovanni Frosali
collaborazione con Paul Zweifel Giovanni Borgioli
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).
8
Wigner picture for Schrödinger-like models
Density matrix
Multiband Wigner function
Evolution equation
9
Two band Wigner model
Wigner picture
10

• Two band Wigner model

Wigner picture

• intraband dynamic zero coupling if the external
  potential is null

11
Two band Wigner model
Wigner picture

• intraband dynamic zero coupling if the external
  potential is null

• interband dynamic coupling like G-R via

12
(No Transcript)
13
Kane model
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

14
MEF model first order
Physical meaning of the envelope function
The quantity represents the
mean probability density to find the electron
into n-th band, in a lattice cell.
15
MEF model first order
Effective mass dynamics

• intraband dynamic

Zero external electric field exact electron
dynamic
16
MEF model first order
Coupling terms

• intraband dynamic
• interband dynamic
• first order contribution of transition rate of
  Fermi Golden rule

17
Wigner picture
Wigner function
Phase plane representation pseudo
probability function
CLASSICAL LIMIT
Wigner equation
Liouville equation
Moments of Wigner function
18
MEF model characteristics
Hierarchy of kp multiband effective mass
models, where the asimptotic parmeter is the
quasi-momentum of the electron

• 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
  quntum-hydrodynamic formalism

19
First order two band MEF model
20
Well-posedness of the problem

• We get a bounded spectrum developing the
  diagonal term of
• the Hamiltonian to a higher
• order in k

21
Boundary condition

• The region of interest is bounded
• by two charge reservoirs

• Outside the domain we will
• assume that the electron is
• represented by a Bloch wave

22
Boundary condition
Envelope function approximation
Bloch wave
Sum of travelling waves
23
Boundary condition
Envelope function approximation
Bloch wave
Sum of travelling waves
Continuity of , on the
interface
24
Existence and uniqueness of the MeF problem
E(q) is the energy of the incident electron

• q is the liner momentum of the incident
• electron and it can vary from 0 to

25
Wigner picture
n-th band component
matrix of operator
General Schrödinger-like model
26
Existence and uniqueness
Non linear problem
Existence of solution of
the problem
27
Existence and uniqueness
Linear problem
Injective operator
28
Existence and uniqueness
Linear problem
The linear problem admit a solution for almost
every q
29
Existence and uniqueness
Non linear problem
Theorem the MFM-Poisson system admits a
unique solution with

Modified problem
Fixed point theorem
Asymptotic limit
30
Asymptotic limit
Non linear problem
Single band case
Energy estimate
31
Asymptotic limit
Non linear problem
A priori estimate Energy
32
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
Vai alla 8
33
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.
Vai alla 9
34
KANE MODEL
The Kane model consists into a couple of
Schrödinger-like equations for the conduction and
the valence band envelope functions.
35
Remarks on the Kane model

• The envelope functions are obtained
  expanding the wave function on the basis of the
  periodic part of the Bloch functions
  evaluated at k0,

where .
36
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 caused by the
applied electric field proportional to the
x-derivative of V. In the two-band case they
assume the form
37
Which are the steps to attain MEF model
formulation?

• Expansion of the wave function on the Bloch
  functions basis

• Introduction in the Schrödinger equation

38

• 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

MEF model can be obtained as follows
See Morandi, Modugno, Phys.Rev.B, 2005
Vai a lla 14
39
where the Wannier basis functions can be
expressed in terms of Bloch functions as
40
Comments on the MEF MODEL

• The envelope functions 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 increases, and vanishes in the
  absence of the external field V.

41
Physical meaning of the envelope functions
A more direct physical meaning can be ascribed to
the hydrodynamical variables derived from the MEF
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.
This simple picture does not apply to the Kane
model.
The Kane envelope functions and the MEF envelope
functions are linked by the relation
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.
42
(No Transcript)
43
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
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
Vai alla 21
44
with the dimensional
We introduce the rescaled Planck constant
where are typical dimensional
quantities
parameter
and the effective mass is assumed to be
equal in the two bands
MEF model reads in the rescaled form
45
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 can be expressed in terms of

plus the phase difference
46
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
47
Next, we derive a system of coupled equations for
phases , obtaining a system equivalent
to the coupled Schrödinger equations. Then we
obtain a system for the currents
and
The equations can be put in a more familiar form
with the quantum Bohm potentials
48
Recalling that and are given by
the hydrodynamic quantities and , we have
the HYDRODYNAMIC SYSTEM for the MEF model
49
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
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.
50
Hydrodynamic version of the MEF MODEL
51
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
We use the Madelung-type transform
We define the densities and the currents
corresponding to the two mixed states
Performing the analogous procedure and with an
appropriate closure, we get
52
Isothermal QUANTUM DRIFT-DIFFUSION for the MEF
MODEL
53
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 zero-temperature 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 and
similar to the temperature tensor of kinetic
theory.
NEXT STEPS

• Closure of the quantum hydrodynamic system
• Numerical treatment
• Heterogeneous materials
• Generalized MEF model

54
Thanks for your attention !!!!!
55
Non linear Schrödinger-like Poisson problem

• Description of the model

Multiband (MEF) model coupled with the Poisson
eqn.

• Mathematical problem

• Well-posedness and B.C.
• Existence and uniqueness of the solution for
  the MEF-P Ben Abdallah,Morandi

• Numerical applications

Application to IRTD