1Review Problem

22 0.98

2

d

2

d

d

60cmf

1

2 0.90 23 0.98

1 2 0.98

Scattering Loss at

lens =3% per surface

T T

0

gl

1T

2T

0

0

0 ( )

/ 2F c d

1 ... M

allowed modes

- Loss per unit of length

- Gain per unit of length

B

Region of possible oscillations

1/ 2

a) Find the photon lifetime of the passive cavity; that is, with 0() = 0.

b) What is the passive cavity Q?

c) What is the free spectral range of the cavity?

d) What is the minimum gain coefficient 0(0) necessary to sustain oscillations in this cavity?

e) If the gain coefficient 0(0) were 2 x 10-2 cm-1 (at line center) and the line shape of the transition was approximated by a Lorentzian with h = 1.5 GHz, then how many TEM0,0,q modes are above the threshold?

f) What is the stimulated emission cross section (at line center)?

g) What is the absorption cross section?

h) The characteristic beam parameter z0 is related to the dimensions of the cavity and the focal length of the lens by

1) Where is z = 0 in the cavity?

2) Find a formula for the resonant frequency of the TEMm,p,q mode.

3) What is the difference in resonant frequencies (in MHz) of the TEM 0,0,q and the TEM1,0,q modes?

5 121

2

1

50 cm

30cm

0.8μm

1.5GHz

10 sec

1

2

g

h

d

l

A

J

J

1/ 22

1/ 200

33 1

4

w dz df

f

2Introduction to Optical Electronics

Quantum (Photon) Optics (Ch 12)

Resonators (Ch 10)

Electromagnetic Optics (Ch 5)

Wave Optics (Ch 2 & 3)

Ray Optics (Ch 1)

Photons & Atoms (Ch 13)

Laser Amplifiers (Ch 14)

Lasers (Ch 15) Photons in Semiconductors (Ch 16)

Semiconductor Photon Detectors (Ch 18)

Semiconductor Photon Sources (Ch 17)

Optics Physics Optoelectronics

3Semiconductors

201

212

1

2

1R

2R 1iW

E1

E2

h21

k

E

Ec

Ev

h

Photons & Atoms

• Distinct Energy Levels

• Probability: Boltzmann

• Gain: Population Inversion

Photons in Semiconductors

• Energy Bands

• Probability: Fermi-Dirac

• Gain: Quasi-Fermi Energies Efc – Efv > Eg

21

21

E1

E2

4Review of Quantum Mechanics

• Free Electron Theory of Solids

• Free Electrons are Waves, (r,t)

• Obey Schrodingers’ Equation

– Time-Independent Schrodinger’s Equation

22( , ) ( , )

2j r t r t

m

22 ( ) ( )

2r E r

m

5Free-Electron’s Energy Spectrum

E

E dE Number of states: ?EN dE

Number of states: ?

Volume EdE

6Fermi Gas

0E

0T K

fE E fE

10 f E

7Fermi Gas

0E

0T K

fE E fE

10 f E

( ) /

1

1f BE E k Tf Ee

8Band Theory of Solids

1n

2n

3n

infn

antibondbond

1 Hydrogen Atom 2 Hydrogen Atoms 3 Hydrogen Atoms N Hydrogen Atoms

N levels

N levels

N levels

9Energy BandsConduction & Valence

Ele

ctro

n E

ner

gy E

ConductionBand

ValenceBand

Bandgap energy Eg

Electron

Hole

En

erg

y Bandgap Energy • Eg=1.11 eV (Si)• Eg=1.42 eV (GaAs)

Conduction Band

Valence Band

10

Direct & Indirect Semiconductor Bandgaps

Ec

Ev

Ec

Ev

Eg=1.11 eV

Silicon (Si)

Eg=1.42 eV

[111] [100]k

E

[111] [100]k

EGallium Arsenide (GaAs)

k

E

k

E

cE

cE

11Semiconducting Materials

II III IV V VIAluminum (Al) Silicon (Si) Phosphorus (P) Sulfur (S)

Zinc (Zn) Gallium (Ga) Germanium (Ge) Arsenic (As) Selenium (Se)

Cadmium (Cd) Indium (In) Antimony (Sb) Tellurium (Te)

Mercury (Hg)

Al

Ga

In

P

As

Sb

Al

Ga As

Ga

In

P

As

x

1-x

x

1-x

y

1-y

Binary Ternary QuaternaryElemental

Si

Ga

12Lattice Constants

13Density of Statesnear the band edges

3/ 2

1/ 2

2 3

2( ) ,

2c

c c c

mE E E E E

3/ 2

1/ 2

2 3

2( ) ,

2v

v v v

mE E E E E

E

k

E

Ec

Ev

Eg

d

Ec

Ev

E

Allowed energy levels(at all k)

Ec

Ev

Density of states

( )v E

( )c E

14Semiconductor’s Density of States

Light HoleHeavy Hole

Electrons

3/ 2

1/ 2

2 3

2( ) ,

2c

c c c

mE E E E E

3/ 2

1/ 2

2 3

2( ) ,

2v

v v v

mE E E E E

15

Fermi-Dirac Distribution f(E)

Ec

Ev

EfEg Ef

Ec

Ev

Ef

Ec

Ev

0.50 1 0.50 1

f(E)

T > 0 KT = 0 KE E

1-f(E)

f(E)

1( )

exp 1f

B

f EE E

k T

16

Carrier Concentration (n & p)

Ec

Ev

EfEg

( )p E

( )n E

( ) ( ) ( )c c

c

E E

n n E dE E f E dE

( ) ( ) 1 ( )v vE E

vp p E dE E f E dE

( )f E 1 ( )f E

17

n- and p-type Semiconductors

Ef

0 1 f(E)Carrier

concentration

Ec

Ev

Ef

Carrierconcentration

Ec

Ev

( )p E

( )p E

( )n E

( )n E

0 1 f(E)

ED

EA

E

E

Donor level

Acceptor level

18Exercise 16.1-2Exponential Approximation of the Fermi Function

When ,f BE E k T the Fermi function f(E) may be approximated by an exponential

function. Similarly, when , 1 ( )f BE E k T f E may be approximated by an

exponential function. These conditions apply when the Fermi level lies within the bandgap, but away from its edges by an energy of at least several times Bk T (at room

temperature 0.026 eVBk T whereas 1.11eVgE in Si and 1.42 eV in GaAs). Using

these approximations, which apply for both intrinsic and doped semiconductors, show that (15.1-8) gives

exp

exp

exp

c fc

B

f vv

B

gc v

B

E En N

k T

E Ep N

k T

Enp N N

k T

where 3/ 222 2 /c c BN m k T h and 3/ 222 2 /v v BN m k T h . Verify that if fE is closer

to the conduction band and , then v cm m n p whereas if it is closer to the valence band,

then p n

19

Semiconductors

Density of StatesProbability of Occupation

Concentration of Carriers

Concentration of Carriers

(Approximation)

3/ 2

1/ 2

2 3

2( )

2c

c c

mE E E

cE E

3/ 2

1/ 2

2 3

2( )

2v

v v

mE E E

vE E

( ) /

1( )

e 1f BE E k Tf E

1 ( )f E

( ) ( )c

c

E

n E f E dE

( ) 1 ( )vE

vp E f E dE

exp c fc

B

E En N

k T

3/ 222 2 /c c BN m k T h

if f BE E k T

exp f vv

B

E Ep N

k T

3/ 222 2 /v v BN m k T h

if f BE E k T

Law of Mass Action:2in p n

/1/ 2where the intrinsic carrier conce (ntration )is g BE k T

i c vn N N e

20

Quasi-Equilibrium Carrier Concentrations

Ec

Ev

Efc

Eg

( )p E

( )n E

Efv Efv

Ec

Ev

22/32 2/3

/

22 /32 2/3

/

1for : where 3

21

1for : 1 1 where 3

21

fc B

fv B

fc cE E k Tc

fv vE E k Tv

n E f E E E nme

p E f E E E pme

21Exercise 16.1-3Determination of the Quasi-Fermi Levels Given

the Electron and Hole Concentrations

(a) Given the concentrations of electrons n and holes p in a semiconductor at T = 0 K, use (15.1-7) and (15.1-8) to show that the quasi-Fermi levels are

22/32 2/3

22/32 2/3

32

32

fc cc

fv vv

E E nm

E E pm

(b) Show that these equations are approximately applicable at an arbitrary

temperature T if n and p are sufficiently large so that fc c BE E k T and

v fv BE E k T , i.e., if the quasi-Fermi levels lie deeply within the conduction

and valence bands.

22

Electron-Hole Generation & Recombination

Ec

Ev

Generation Recombination

Ec

Ev

Trap

0

1where

( )o

nR

r n p n

23Exercise 16.1-4Electron-Hole Pair Injection in GaAs

Assume that electron-hole pairs are injected into n-type GaAs

0 0( 1.42eV, 0.07 , 0.5 )g c vE m m m m at a rate 23 310 per cm per second.R The

thermal equilibrium concentration of electrons is 16 30 10 .n cm If the recombination

parameter 11 310 / and 300 , determine:r cm s T K (a) The equilibrium concentration of holes 0p .

(b) The recombination lifetime . (c) The steady-state excess concentration n (d) The separation between the quasi-Fermi levels fc fvE E , assuming that 0T K

24How to Handle an Inverted SemiconductorVerdeyen’s Approach

1/ 23/ 2

2 /

21( ) ( ) ( )

2 1fc B

c c c

ccc E E k T

E E E

E Emn n E dE E f E dE dE

e

1/ 23/ 2

2 /

21( ) ( ) 1 ( )

2 1

v v v

fv B

E E Evv

v E E k T

E Emp p E dE E f E dE dE

e

/

/

or

/

fv v B

c fc B

c v

B

E E k T

E E k T

u E E u E E

x u k T

a e

b e

Setting:

25Inverted Semiconductor Example: GaAs

3/ 2 1/ 2

/

2 20

21

2 1/c fc BE E k Te B

x

m k T xn e dx

e b

3/ 2 1/ 2

/

2 20

21

2 1/fv v BE E k Te B

x

m k T xp e dx

e a

1/ 2

2

0

2let

1/ ,x

xI dx

e a b

/

/

fv v B

c fc B

E E k T

E E k T

a e

b e

26Semiconductors

Density of States

Probability of Occupancy

Carrier Concentrations

Law of Mass Action

3/21/2

2 3

3/21/2

2 3

2,

2

2,

2

cc c c

vv v v

mE E E E E

mE E E E E

p

p

/

1

1f BE E k Tf E

e

1v

c

E

c v

E

n E f E dE p E f E dE

p p

2 where exp2

gi i c v

B

En p n n N N

k T

27

Generation, Recombination & Injection

Rate of Recombination

Recombination Lifetime,

Internal Quantum Number

(low concentrations)

0 0 0 0

0

0

where G

and

and

r n p R G r n p

n n n

p p n

0 0

1where

( )

nR

r n p n

0 0

1 where

( )

r r

r nri

nr

r r nr

r r

r r r

r n p

28

Semiconductor Fermi Energy Levels

p n

Ca

rrie

rC

on

cen

tra

tion

Ele

ctro

n E

ne

rgy

Before Contact

p

n

n

p

Position

E0Neutral p Neutral n

After Contact

DepletionLayer

eV0

Ca

rrie

rC

on

cen

tra

tion

Ele

ctro

n E

ne

rgy

p(x)

n(x)

n(x)

p(x)

x

29

Forward-Biased p-n JunctionForward Bias

E0 - ENeutral p Neutral n

eV

Ca

rrie

rC

on

cen

tra

tion

Ele

ctro

n E

ne

rgy

p(x)

Excesselectrons

n(x)

Excess holes

x

+_

e(V0-V)

V

Efc

Efv

n

p

E0Neutral p Neutral n

Neutral

eV0

Ca

rrie

rC

on

cen

tra

tion

Ele

ctro

n E

ne

rgy

p(x)

n(x)

n(x)

p(x)

x

Ef

30Current-Voltage Characteristics of an ideal p-n Junction Diode

is

i

V

i

_

+

V

i p

n_

+

V

exp 1sB

eVi i

k T

31PIN Diodein Thermal Equilibrium

p i n

Depletion layer

Electric Field

Ec

Ec

Electronenergy

-

+x

Fixed-chargedensity

x

Electric-field magnitude

32

Photon Absorption & Emission Mechanics

Eg=1.42 eV

Ec

Ev

Band-to-Band Transitions

EA = 0.088 eV Eg=0.66 eV

Acceptor-LevelTransition

Free-CarrierTransition

33Absorption

c E

v E

Intensity

34Stimulated Emission

c E

v E

35Absorption Phenomenon

36Band-To-Band Photon Interactions

E

Ec

Ev

E1

E2

k

h

k

h

k

h h

h

ih

i

h

ih

Absorption SpontaneousEmission

StimulatedEmission

37Optical Joint Density of States

3/ 2

1/ 2

2 3

2( ) ,

2c

c c c

mE E E E E

3/ 2

1/ 2

2 3

2( ) ,

2v

v v v

mE E E E E

2 2( ) ( )d E dE

22

3/ 21/ 22

2

( ) ( )

2,g g

dEE

d

mh E h E

2 ( )rc g

c

mE E h E

m

1 2( )rv g

v

mE E h E E h

m

Eg

h

38Band-To-Band Photon Interactions

Photon AbsorptionIndirect-gap Semiconductor

Photon EmissionIndirect-gap Semiconductor

k

Photon

Phonon

k

Thermalization

PhotonAbsorption

h

39Exercise 16.2-1

Requirement for the Photon Emission Rate to Exceed the Absorption Rate

(a) For a semiconductor in thermal equilibrium, show that ( )ef is always smaller

than ( )af so that the rate of photon emission cannot exceed the rate of photon

absorption.

(b) For a semiconductor in quasi-equilibrium fc fvE E , with Radiative transitions

occurring between a conduction-band state of energy 2E and a valence-band state

of energy 1E with the same k, show that emission is more likely than absorption if

the separations between the quasi-Fermi levels is larger than the photon energy, i.e., if

fc fE E h

What does this condition imply about the locations of fcE relative cE and fE ,

relative to E ?

40Spontaneous Emission Spectral Densityin Thermal Equilibrium

Eg . . . . .h

Eg

rsp()

kBT

1/ 2

0( ) exp ,gg

sp gB

h Eh Er D h E

k T

3/ 2

0 2

2where exp gr

r B

EmD

k T

41

0 1 2

0.5x104

104

Absorption Coefficientin Thermal Equilibrium

h- Eg

()

(cm

-1)

1/ 2

1 1 2( ) ( ) ( )gD h E f E f E

3/ 2 2

1 2

2where r

r

mD

h

1

( ) ( ) ( )exp / 1

c v

f B

f E f E f EE E k T

42Exercise 16.2-2Wavelength of Maximum Band-to-Band Absorption

Use

2 3/ 2

1/ 2

2

2 1rg

r

c mh E

h

(15.2-28)

to determine the (free-space) wavelength p at which the absorption coefficient of a

semiconductor in thermal equilibrium is maximum. Calculate the values of p for GaAs.

Note that this result applies only to absorption by direct band-to-band transitions.