Modelling & Simulation of
Semiconductor Devices
Lecture 7 & 8
Hierarchy of Semiconductor Models
Introduction
•
Nowadays,
semiconductor
materials
are
contained
in
almost
all
electronic
de

vices
.
•
Some
examples
of
semiconductor
devices
and
their
use
are
described
in
the
following
.
•
Some
examples
of
semiconductor
devices
and
their
use
are
described
in
the
following
.
–
Photonic
devices
capture
light
(photons)
and
convert
it
into
an
electronic
signal
.
They
are
used
in
camcorders,
solar
cells,
and
light

wave
communication
systems
as
optical
fibers
.
Introduction
–
Optoelectronic
emitters
convert
an
electronic
signal
into
light
.
Examples
are
light

emitting
diodes
(LED)
used
in
displays
and
indication
lambs
and
semiconductor
lasers
used
in
compact
disk
systems,
laser
printers,
and
eye
surgery
.
–
Flat

panel
displays
create
an
image
by
controlling
light
that
either
passes
through
the
device
or
is
reflected
off
of
it
.
They
are
made,
for
instance,
of
liquid
crystals
(liquid

crystal
displays,
LCD)
or
of
thin
semiconductor
films
(
electroluminescent
displays
)
.
–
In
field

effect
devices
the
conductivity
is
modulated
by
applying
an
electric
field
to
a
gate
contact
on
the
surface
of
the
device
.
The
most
important
field

effect
device
is
the
MOSFET
(metal

oxide
semiconductor
field

effect
transistor
),
used
as
a
switch
or
an
amplifier
.
Integrated
circuits
are
mainly
made
of
MOSFETs
.
Introduction
–
Quantum
devices
are
based
on
quantum
mechanical
phenomena,
like
tunneling
of
electrons
through
potential
barriers
which
are
impenetrable
classically
.
Examples
are
resonant
tunneling
diodes,
super
lattices
(
multi

quantum

well
structures),
quantum
wires
in
which
the
motion
of
carriers
is
restricted
to
one
space
dimension
and
confined
quantum
mechanically
in
the
other
two
directions,
and
quantum
dots
.
•
Clearly,
there
are
many
other
semiconductor
devices
which
are
not
mentioned
(
for
instance,
bipolar
transistors,
Schottky
barrier
diodes
,
thyristors)
.
•
Other
new
developments
are,
for
instance,
nanostructure
devices
(
hetero

structures
)
and
solar
cells
made
of
amorphous
silicon
or
organic
semiconductor
materials
.
Introduction
•
Usually,
a
semiconductor
device
can
be
considered
as
a
device
which
needs
an
input
(an
electronic
signal
or
light)
and
produces
an
output
(light
or
an
electronic
signal
)
.
•
The
device
is
connected
to
the
outside
world
by
contacts
at
which
a
voltage
(potential
difference
)
is
applied
.
•
We
are
mainly
interested
in
devices
which
produce
an
electronic
signal,
for
instance
the
macroscopically
measurable
electric
current
(electron
flow
),
generated
by
the
applied
bias
.
•
In
this
situation
,
the
input
parameter
is
the
applied
voltage
and
the
output
parameter
is
the
electric
current
through
one
contact
.
Introduction
•
The
relation
between
these
two
physical
quantities
is
called
current

voltage
characteristic
.
It
is
a
curve
in
the
two

dimensional
current

voltage
space
.
•
The
current

voltage
characteristic
does
not
need
to
be
a
monotone
function
and
it
does
not
need
to
be
a
function
(but
a
relation
)
.
•
The
main
objective
of
this
subject
is
to
derive
mathematical
models
which
describe
the
electron
flow
through
a
semiconductor
device
due
to
the
application
of
a
voltage
.
Introduction
•
Depending
on
the
device
structure,
the
main
transport
phenomena
of
the
electrons
may
be
very
different
,
for
instance,
due
to
drift,
diffusion
,
convection
,
or
quantum
mechanical
effects
.
•
For
this
reason,
we
have
to
devise
different
mathematical
models
which
are
able
to
describe
the
main
physical
phenomena
for
a
particular
situation
or
for
a
particular
device
.
•
This
leads
to
a
hierarchy
of
semiconductor
models
.
Hierarchy
of
Semiconductor Models
•
Roughly
speaking,
we
can
divide
semiconductor
models
in
three
classes
:
–
Quantum
models
–
Kinetic
models
–
Fluid
dynamical
(macroscopic)
models
•
In
order
to
give
some
flavor
of
these
models
and
the
methods
used
to
derive
them,
we
explain
these
three
view

points
:
quantum
,
kinetic
and
fluid
dynamic
in
a
simplified
situation
.
Quantum Models
•
Consider
a
single
electron
of
mass
m
and
elementary
charge
q
moving
in
a
vacuum
under
the
action
of
an
electric
field
E
=
E(x
;
t
)
.
•
The
motion
of
the
electron
in
space
𝑥
∈
ℝ
𝑑
and
time
t
>
0
is
governed
by
the
single

particle
Schrodinger
equation
•
With
some
initial
condition
Quantum Models
Quantum Models
Fluid Dynamic Model
•
In
order
to
derive
fluid
dynamical
models,
for
instance
,
for
the
evolution
of
the
particle
density
n
and
the
current
density
J
;
we
assume
that
the
wave
function
can
be
decomposed
in
its
amplitude
𝑛
𝑥
,
𝑡
>
0
and
phase
𝑆
𝑥
,
𝑡
∈
ℝ
.
Fluid Dynamic Model
•
The
current
density
is
now
calculated
as
Semiconductor Crystal
•
A
solid
is
made
of
an
infinite
three

dimensional
array
of
atoms
arranged
according
to
a
lattice
Semiconductor Crystal
•
The
state
of
an
electron
moving
in
this
periodic
potential
is
described
Schrodinger
equation
:
Home Work
•
Apply
Madelung
Transform
on
above
equation
to
obtain
Fluid
dynamical
model
of
electron
moving
in
periodic
potential
in
semiconductor
crystal
.
END OF LECTURES 7

8
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