Semiconductors and Light - PowerPoint Presentation

Slide1

Semiconductors and Light

A non-standard way to deal with

it

Assumes

confidence

with

Basic

S

olid

S

tate

Physics

:

Crystals

Electrons

,

Holes

,

direct-indirect

Band Gap

Their

foundation

on the

Bloch’s

Theorem

p

n

junction

DC

characteristics

Basic light-

matter

interaction

theory

Quantum-

mechanic

perturbation

theory

Spontaneous

and

stimulated

emission

,

absorptionSlide2

Semiconductors and Light

Does

NOT

aim

to be complete

Aims

to

build

up a self-

consistent

view

of:

Electron,

Hole

and

Photon

densities

Their

link with

measurable

quantities

: I, V, P

Their

call for

technological

solutionsSlide3

Semiconductors and Light

It

gives

a new

insight

on:

Threshold

current

Gain and

loss

coefficients

The

theoretical

foundation

of

known

empirical

formulas

It

should

be

compared

with

literature

.

Coldren

LA, Corzine SW,

Mašanović

ML.

Diode lasers and photonic integrated circuits

, Wiley series in microwave and optical engineering.

Jhon

Wiley & Sons, Inc., Hoboken, New Jersey; Second Edition 2012

.

J.T.

Verdeyen

.

Laser Electronics

. Third Edition, Prentice Hall, 1995. Slide4

Semiconductors and Light

Absorption

Emission

Transparency

Refraction

Partial

reflection

Easily

explained

by

Eg:

Absorbption

Transparency

E

g

n

Refraction

Partial

ReflectionSlide5
Slide6

Diamond

has

the

closest

lattice

spacing

It

is

Mechanically

Hard

Electrically

InsulatingOptically Transparent

Optically RefractiveSlide7

…

but

Light

Emission

is

another

tale

Fermi Golden

Rule

(holds for any system

)

Momentum

Selection Rule

(specific for crystals)

High

probability

Low

probability

Si excluded from light emitting devices

Recombination

rate

is no more

It

must

obey

the

selection

rulesSlide8

Something

may

be

anticipated

about

light

emission in direct gap semiconductors

Many

electrons

Many

holes

Many

photons

No

electrons

No

holesNo photons

Few

electrons

Few holes

Few photons

h

𝜈

E

g

𝜙

𝜈

=

photon

density

Typical

40kT

Typical

≈kTSlide9

How

t

o

get

many

electrons

and many holes

together?

By

injecting

a forward current into a pn

junction

But

in an

ideal diode

, they meet inside the depletion layer without recombining

.Then recombine, separately, entering the neutral regionsSlide10

The

practical

solution

is

to force e-h

recombination

inside the depletion layer introducing

a thin layer at

smaller bandap between the p and n

regions.

In a

well designed

device, a forward bias V will cause a current I to flow and an

optical power POUT

to leave the structure.

This is a Light Emitting Diode

= LEDSlide11

But

something

more

may

be

anticipated about light emission

in direct gap semiconductors

𝜙

n

𝜙

p

More

empty

than

filled

states

More

filled

than

empty

states

More

filled

than

empty

states

More

empty

than

filled

states

Possibility

for

population

inversion

More top-down (

emitting

)

than

bottom-up (

absorbing

)

transitions

stimulated

by light.

GAIN = Light

Amplification

The LED

achieves

L

ight

A

mplification

by

S

timulated

E

mission

of

R

adiation

This

is

a LASER DIODESlide12

Form qualitative to quantitative:

we

should

:

List

all

mechanisms

involving photons and electrons and

holesBalance them

Bring optical properties properties inside the world of

diodes:Spectrum

𝜙𝜈, gain g, loss

α, optical power

POUT

Include concurring phenomena non involving photons

Find lumped equations for V, I and POUT

Bring

diode properties

properties inside the world of lasers:Current I, Voltage V

The starting point will be a Rate Equation for photonsSlide13

Our

program

:

Consider

a Double

Heterostructure

ideal

diode

made of direct gap semiconductor

where all

recombinations are radiativeand happens only

inside the central active

layer

qV

p

n

depletion layer

active layer

Inside

that

layer

all

electron,

holes

and

photon

densities

are

uniform

We

will

look first for the

electro-optical

characteristics

of

such

ideal

diode

a

nd

later

on,

we

will

include

Non-radiative

recombination

inside the

layer

Other

recombinations

and

currents

outside

the

layerSlide14

A

quick

recall

of the

Einstein’s

treatment of Black Body

radiation

(1905)

Search for the spectral

power density

u𝜈 at

equilibrium Fermi’s Golden rule not

yet discovered

Classical results from Thermodynamics

Stefan’s

Law

Wien’s

(

Displacement

) Law

Rayleigh

and Jeans’

Ultraviolet

Catastrophe . But

good for low 𝜈.

No Fermi-

Dirac

or Bose-Einstein

statistics

available

:

only

Boltzmann

. No

exclusion

principle

No quanta.

Planck

(1901)

not

sure

of

their

existence

. Einstein

going

to

explain

the

phototelectric

effect

on the

same

year

(1905)Slide15

E

2

g

2

Level 2

Level 1

2-level

system

with N

0

particles

E

nergy

Density

of

states

Population

E

1

g

1

Rate of

spontaneous

2→1 transitions

Rate of

stimulated

2→1

transitions

Rate of

stimulated

1→

2

transitions

Coefficients

A and B can

depend

on

𝜈

but

not

on T

At

equilibrium

2→1 = 1 →

2Slide16

At high temperature:

Do

not

change

with T

Go to

unity

Increase

with T

4

Do

not

include T

Wien’s

(

Displacement

) LawSlide17

Must be (

Rayleigh

and Jeans):

Planck’s

LawSlide18
Slide19
Slide20

Top-down

optical

transitions

proportional

to

Bottom-up

optical

transitions

proportional

toSlide21
Slide22
Slide23

The

diode

at

equilibrium

must

give

back the Black Body formula

Conduction

-to-

valence

spontaneous

transitions (e-h recombination,

photon emission)

Conduction-to-

valence stimulated transitions(e-h recombination, photon emission)

Valence-to-conduction stimulated transitions (e-h generation,

photon absorption)

At

equilibrium

V=0

As

for

Einstein’s

treatment:Slide24

Out of

equilibrium

,

rates

can be

not

constant

, V≠0 and an escape term must be introduced

, that must vanish at

equilibrium

In the steady state:

w

e

can solve

for

Using the

previous

forms

Slide25

where

h

𝜈

E

g

𝜙

𝜈

=

photon

density

𝜙

0𝜈

B

may

be

and

𝜙

0

𝜈

is

a slow

function

of h

𝜈.

We

can

safely

assume for

both

:

B and

𝜙

0

𝜈

are

constant

Is

the joint

density

of

states

for

electrons

and

holes

.

It

is

null

for

a

nd

is

linear

in for

thick

layers

and

constant

for

thin

layers

(the

normal

case)

𝜏

C

is the average permanence time of radiation inside the active layer. It is a function of

𝜈

because of refraction and resonances. But we start keeping it constant.Slide26

Let

us

suppose

E

g

= 1

eV

and qV=0.5 eV

h

𝜈

E

g

𝜙

𝜈

=

photon

density

calculated

Expected

(qualitative)

Linear scale

Thick

layer case. Linear

vertical scale. Pay attention to the

values

in the

abscissaSlide27

Let

us

now

change

qV:

Logarithmic

scale

Thick

layer case. Log vertical scale.

Pay attention

to the values of qVSlide28

Thin

layer

case. Log

vertical

scale

.

Pay

attention to the values of

qV

Logarithmic

scale

At

qV

=1.02

eV

Something happens when

qV approaches EgSlide29

For

possible

emission

,

we

have

For

e

xponentials

at

the denominator are

extremely

smallSlide30

The

total

optical

power

i

ncreases

unlimited

Infinite

energy

is

required

to

further increase qV.

But the denominator vanishes

as

Voltage

clamps

at a threshold value

t

hat

is

the minimum

among

the

ones

allowed

For a

thin

layer

this

isSlide31

What

is

it

the happening?

R

sp

R

st

R

abs

R

esc

qV

th

E

g

Spontaneous

emission

and

absorption

dominate

o

ver stimulated emission. It is

the LED regime.

Stimulated

emission

balances

absorption

.

It

the

the

transparency

condition

.

Stimulated

emission

overcomes

absorption

.

Light

starts

to be

amplified

(super-

radiance

)

Spontaneous

emission

is

blocked

. Voltage

is

calmped

.

Stimulated

emission

dominates

.

It

is

the

LASER regime

.Slide32

P

TOT

qV

The more the

bathtube

is

filled

, the

higher

is

the output

flux

But

when

the

edge

is

reached

, the

flux

can

increase

without

increasing

the water

levelSlide33

Current in the ideal

laser

diode

The

current

is

q

times the net recombination rate.

For the

ideal LED/Laser diode, where recombination

is always radiative

h

𝜈

E

g

𝜙

𝜈

=

photon

density

Integration

factor

.Slide34

Shockley

region

(LED)

Laser

region

V<

V

th

: LED

range

V=

V

th

: Laser

range

Equation of an

ideal

diode

with

saturation

current

=

This

is

the DC transfer

function

I(V) of an

ideal

laser

diode

.Slide35

Optical

Power

in the

ideal

laser

diode

Optical

Power

=

Photon Density x

Photon Energy x Volume / Lifetime

But

:

Collection

efficiency

(

coupling+conversion

+…)

I

ph

P

OUTSlide36

measured

Optical

Power

in the

real

laser

diode

I

ph

is

not

the

only

current

For V<

Vth non-radiative recombination dominates

A non-radiative current Inr exists

The

total

current

is made of I= Inr

+ IphAs

qV clamps at qV

th

,

I

nr

stops

, and

I

ph

grows

alone

A

threshold

current

I

th

defines

the

transiton

In

practise

:

qV

th

I

th

Quantum

Efficiency

:Slide37

Threshold

currentSlide38

Collected

escaping

power

Total

escaping

power

Conversion

efficiency

(

often

omitted

)

My opinion: a

big

mistake.

It must go to unity for I>

Ith

f

or

any

V,I

f

or V>

V

th

(

that

is

I>

I

th

)

What

do

textbooks

tell

?Slide39

Lumped

equations

for the DC regime

I(V), P

OUT

(V) →

direct

substitution

P

OUT

(I) → eliminate

It

comes

out a

huge

formula,

but it

is

analytic under the form I(

POUT)calculates and embeds the threshold condition

deals with the sub-thresholdregimeSlide40