Modelling & Simulation of Semiconductor Devices - Weebly

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Nov 1, 2013 (3 years and 11 months ago)

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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


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



.

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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