handout 29 optical transitions in solids, optical gain ...k q k cv k q k c k v k k q k c k v k r c k...

1

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Handout 29

Optical Transitions in Solids, Optical Gain, and Semiconductor Lasers

In this lecture you will learn:

• Electron-photon Hamiltonian in solids• Optical transition matrix elements• Optical absorption coefficients • Stimulated absorption and stimulated emission• Optical gain in semiconductors• Semiconductor heterostructure lasers


Interactions Between Light and Solids

The basic interactions between light and solids cover a wide variety of topics that can include:

• Interband electronic transitions in solids

• Intraband electronic transitions and intersubband electronic transitions

• Plasmons and plasmon-polaritons

• Surface plasmons

• Excitons and exciton-polaritons

• Phonon and phonon-polaritons

• Nonlinear optics

• Quantum optics

• Optical spintronics

2


Fermi’s Golden Rule: A Review

Now suppose a time dependent externally applied potential is added to the Hamiltonian:

titio eVeVHH

ˆˆˆˆ

Consider a Hamiltonian with the following eigenstates and eigenenergies:

integerˆ mEH mmmo

Suppose at time t = 0 an electron was in some initial state k: pt 0

Fermi’s golden rule tells that the rate at which the electron absorbs energy from the time-dependent potential and makes a transition to some higher energy level is given by:

pmpm

mEEVpW

2ˆ2

The rate at which the electron gives away energy to the time-dependent potential and makes a transition to some lower energy level is given by:

pmpm

mEEVpW

2ˆ2

E

E


Optical Transitions in Solids: Energy and Momentum Conservation

k

For an electron to absorb energy from a photon energy conservation implies:

infm kEkE

Final energy

Initialenergy

Photonenergy

Momentum conservation implies:

qkk if

Initial momentum

Final momentum

Photonmomentum

Intraband photonic transitions are not possible:

For parabolic bands, it can be shown that intraband optical transitions cannot satisfy both energy and momentum conservation and are therefore not possible

Intraband

Intraband

Interband

Note that the momentum conservation principle is stated in terms of the crystal momentum of the electrons. This principle will be derived later.

E

3


Electromagnetic Wave BasicsConsider an electromagnetic wave passing through a solid with electric field given by:

trqEntrE o

.sinˆ,

The vector potential associated with the field is:

t

trAtrE

,

,

trqAn

trqE

ntrA

o

o

.cosˆ

.cosˆ,

The power per unit area or the Intensity of the field is given by the Poynting vector:

2

ˆ2

ˆ,,,222oo A

qE

qtrHtrEtrSI

noo

The photon flux per unit area is:

2

2oAI

F

E

H

q

The divergence of the field is zero:

0,.,. trAtrE


Consider electrons in a solid. The eigenstates (Bloch functions) and eigenenergies satisfy:

knnkno kEH

,,ˆ

rVmPP

H latticeoˆ

2

ˆ.ˆˆ

Electron-Photon Hamiltonian in Solids

where:

In the presence of E&M fields the Hamiltonian is:

Pnm

eeAeH

PtrAme

H

PtrAme

trAPme

H

trAtrAm

ePtrA

me

trAPme

rVmPP

rVm

trAePH

tirqitirqio

o

o

o

lattice

lattice

ˆˆ2

ˆ

ˆ.,ˆˆ

ˆ.,ˆ2

,ˆ.ˆ2

ˆ

,ˆ.,ˆ2

ˆ.,ˆ2

,ˆ.ˆ2

ˆ2

ˆ.ˆ

ˆ2

,ˆˆˆ

ˆ.ˆ.

2

2

Assume small

Provided: 0,. trA

trqAntrA o

.cosˆ,

4


k

Interband

E

Optical Interband Transitions in Solids

knnkno kEH

,,ˆ

Suppose at time t = 0 the electron was sitting in the valence band with crystal momentum :

ikvt

,0

nPm

eeAeHH

tirqitirqio

o ˆ.ˆ2

ˆˆ

ˆ.ˆ.

ik

The transition rate to states in the conduction band is given by the Fermi’s golden rule:

ivfckvkck

i kEkEVkWif

f

2

,,ˆ2

The summation is over all possible final states in the conduction band that have the same spin as the initial state. Energy conservation is enforced by the delta function.

ik

Comparison with:

nPmeAe

Vrqi

o ˆ.ˆ2

ˆˆ.

nPm

eAeV

rqio ˆ.ˆ2

ˆˆ.

titio eVeVHH

ˆˆˆˆgives:


Optical Matrix Element

ivfckvrqi

kck

o

ivfckvkck

i

kEkEnPem

eA

kEkEVkW

iff

iff

2

,ˆ.

,

2

2

,,

ˆ.ˆ2

2

ˆ2

Now consider the matrix element:

rnPerrd

nPe

if

if

kvrqi

kc

kvrqi

kc

,.

,*3

,ˆ.

,

ˆ.ˆ

ˆ.ˆ

k

Interband

E

k

Interband

E

ruenPeruerdi

if

fkv

rkirqikc

rki

,..

,*.3 ˆ.ˆ

runPeeruerd

runkPeeruerd

ii

ff

ii

ff

kvrkirqi

kcrki

kvirkirqi

kcrki

,..

,*.3

,..

,*.3

ˆ.ˆ

ˆ.ˆ

0

Rapidly varying in spaceSlowly

varying in space

5


Optical Matrix Element

if kvrqi

kc nPe

,ˆ.

,ˆ.ˆ

k

Interband

E

k

Interband

E

jif

jfi

ii

ff

Rkvkc

j

Rkqki

kvrkirqi

kcrki

runPrurde

runPeeruerd

,,*

cell primitive th-

3.

,..

,*.3

ˆ.ˆ

ˆ.ˆ

nP

runPrurd

runPrurdN

runPrurde

cvkqk

kvkckqk

kvkckqk

Rkvkc

Rkqki

fi

iffi

iffi

jif

jfi

ˆ.

ˆ.ˆ

ˆ.ˆ

ˆ.ˆ

,

,,*

crystal entire

3,

,,*

cell primitive one any

3,

,,*

cell primitive one any

3.

N = number of primitive cells in the crystal

Interband momentum matrix element


Crystal Momentum Selection Rule

iviccvo

ivfckqkcvk

o

ivfckvrqi

kck

oi

kEqkEnPm

eA

kEkEnPm

eA

kEkEnPem

eAkW

fif

iff

22

,

22

2

,ˆ.

,

2

ˆ.2

2

ˆ.2

2

ˆ.ˆ2

2

k

Interband

E

The wave vector of the photons is very small since the speed of light is very large

Therefore, one may assume that:

Crystal Momentum Selection Rule:

We have the crystal momentum selection rule:

qkk if

if kk

Optical transitions are vertical in k-space

6


iviccv

oi kEkEnP

meA

kW2

2ˆ.

22

Transition Rates per Unit Volume

Generally one is not interested in the transition rate for any one particular initial electron state but in the number of transitions happening per unit volume of the material per second

The upward transition rate per unit volume is obtained by summing over all the possible initial states per unit volume weighed by the occupation probability of the initial state and by the probability that the final state is empty:

R

ik

icivi kfkfkWV

R

1

2

If we assume, as in an intrinsic semiconductor, that the valence band is full and the conduction band is empty of electrons, then:

ik

ikWV

R

2

ikiviccv

o kEkEnPm

eAV

2

2ˆ.

222

k

Interband

E

k

Interband

E

ikW

ik

R



FBZ3

322

FBZ3

322

FBZ

22

3

3

22

22ˆ.

22

22ˆ.

22

ˆ.2

2

22

ˆ.2

22

kEkEkd

nPm

eA

kEkEkd

nPm

eA

kEkEnPm

eAkd

kEkEnPm

eAV

R

vccvo

ivici

cvo

iviccvoi

kiviccv

o

i

The integral in the expression above is similar to the density of states integral:

ce

cD EEm

EkEkd

Eg

23

22FBZ

3

3

32

2

1

22

e

cc mk

EkE2

22Suppose:

Then:

k

Interband

E

R

Joint density of states

7


Transition Rates per Unit Volume and Joint Density of States

FBZ3

322

22ˆ.

22

kEkE

kdnP

meA

R vccvo

e

cc mk

EkE2

22Suppose:

Then:

h

vv mk

EkE2

22

r

ghe

vcvc mk

Emm

kEEkEkE

211

2

2222

gr

cvo E

mnP

meA

R

23

22

22 2

2

1ˆ.

22

R

gE

Joint density of states

Reduced effective mass

k

Interband

E

R



gr

cvo E

mnP

meA

R

23

22

22 2

2

1ˆ.

22

2

22oA

I

The power per unit area or the Intensity of the E&M wave is:

gr

cv Em

nPI

me

R

23

22

2

2

2 2

2

1ˆ.

22

2

k

Interband

E

R

Interband Momentum Matrix Elements:

g

cv

her E

nP

mmmm

2

2

ˆ.4111

Recall the result from homework 7:

k

E

Eg

The above result assumed diagonal isotropic conduction and valence bands and also that no other bands are present, and is therefore oversimplified

8


Because of transitions photons will be lost from the E&M wave traveling inside the solid. This loss will result in decay of the wave Intensity with distance travelled:

xIxxIxxI

xIdxxI

xIexI x

0

Loss coefficient or absorption coefficient

xI xxI

x xx

R

Transition Rates per Unit Volume and the Absorption or Loss Coefficient

Momentum Matrix ElementsIn bulk III-V semiconductors with cubic symmetry, the average value of the momentum matrix element is usually expressed in terms of the energy Ep as,

6ˆ.

2 pcv

mEnP

Parameters at 300K GaAs AlAs InAs InP GaP

Ep (eV) 25.7 21.1 22.2 20.7 22.2

Note that the momentum matrix elements are independent of the direction of light polarization!


Transition Rates per Unit Volume and Loss Coefficient

xIxxIxxI

The wave power loss (per unit area) in small distance x is x I(x)

The wave power loss in small distance x must also equal:

xI xxI

x xx

R

I

R

IxxR

Therefore:

xR

gr

cv Em

nPme

23

22

22 2

2

1ˆ.

22

2

gr

cvo

Em

nPcnm

e

23

22

22 2

2

1ˆ.

Values of () for most semiconductors can range from a few hundred cm-1 to hundred thousand cm-1

k

Interband

E

R

9


Loss Coefficient of Semiconductors


Direct Bandgap and Indirect Bandgap Semiconductors

Direct bandgap (Direct optical transitions)

Direct bandgap(Indirect phonon-assisted transitions)

ehr mmm111

If me approaches -mh , then mr becomes very large

23

rmThen, also becomes very large

10


Stimulated Absorption and Stimulated Emission

FBZ3

322

12

2ˆ.2

2

kEkEkfkf

kdnP

meA

R vccvcvo

Throwing back in the occupation factors one can write more generally:

Stimulated Absorption:The process of photon absorption is called stimulated absorption (because, quite obviously, the process is initiated by the incoming radiation or photons some of which eventually end up getting absorbed)

Stimulated absorption

k

EAn incoming photon can also cause the reverse process in which an electron makes a downward transition

This reverse process is initiated by the term in Hamiltonian that has the time dependence:tie

titio eVeVHH

ˆˆˆˆ

nPmeAe

Vrqi

o ˆ.ˆ2

ˆˆ.

nPm

eAeV

rqio ˆ.ˆ2

ˆˆ.


Following the same procedure as for stimulated absorption, one can write the rate per unit volume for the downward transitions as:

FBZ3

322

12

2ˆ.2

2

kEkEkfkf

kdnP

meA

R vcvccvo

Stimulatedemission

k

E

Stimulated Emission:In the downward transition, the electron gives off its energy to the electromagnetic field, i.e. it emits a photon! The process of photon emission caused by incoming radiation (or by other photons) is called stimulated emission.

Spontaneous Emission: Electrons can also make downward transitions even in the absence of any incoming radiation (or photons). This process is called spontaneous emission.

Stimulated Emission and Spontaneous Emission

Spontaneousemission

k

E

11


Stimulated Absorption and Stimulated Emission

The net stimulated electronic transition rate is the difference between the stimulated emission and stmulated absorption rates:

FBZ3

322

22ˆ.

22

kEkEkfkf

kdnP

meA

RR vccvcvo

And the more accurate expression for the loss coefficient is then:

I

RR

Stimulated absorption

FBZ3

322

22ˆ.

kEkEkfkfkd

nPcnm

evccvcv

o

Stimulatedemission

k

E

k

E


Optical Gain in Semiconductors

FBZ3

322

22ˆ.

kEkEkfkf

kdnP

cnme

vccvcvo

Note that the Intensity decays as: 0 xIexI x

What if were to become negative? Optical gain !!

Optical gain is possible if:

k

E

RR0

kEkEkfkf

kEkEkfkf

vcvc

vccv

for0

for0

A negative value of implies optical gain (as opposed to optical loss) and means that stimulated emission rate exceeds stimulated absorption rate

12


Optical Gain in SemiconductorsOptical gain is only possible in non-equilibrium situations when the electron and hole Fermi levels are not the same

Suppose:

KTEkEfhvv

KTEkEfecc

fhv

fec

eEkEfkf

eEkEfkf

1

11

1

fhfe

vcfhfe

fhvfec

EE

kEkEEE

EkEfEkEf 0

Then the condition for optical gain at frequency is:k

E

Efe

Efh

The above is the condition for population inversion (lots of electrons in the conduction band and lots of holes in the valence band)

kEkE vc


Optical Gain in Semiconductors

The loss coefficient is a function of frequency:

FBZ3

322

22ˆ.

kEkEkfkf

kdnP

cnme

vccvcvo

gE

Optical gain for frequencies for which:

fhfeg EEE

Increasing electron-hole density (n = p)

Net Loss

Net Gain

0k

E

Efe

Efh

13


Semiconductor PN Heterojunctions2cE

2vE

fE2cE

2vEfE

1cE

1vE

P-dopedN-doped

2vE

fE fE

P-doped

N-doped

fE

fLEP-doped

N-dopedfRE

PN heterojunction in equilibrium

PN heterojunction in forward biaseVEE fRfL

PN heterojunction

Electron and hole Fermi level splitting


photons

stimulated andspontaneousemission

holes

electrons

A Heterostructure Laser (Band Diagram)

N-dopedP-doped

metal

A Ridge Waveguide Laser Structure

All lasers used in fiber optical communications are semiconductor lasers

Semiconductor Heterostructure Lasers

Efe

Efh

Non-equilibrium electron and hole populations can be sustained in a forward bias pn junction

eV

EE fhfe

14


A commercial packagedsemiconductor laser

Semiconductor laser chip

actual laser(the long strip)

Wire bond

fiber

Semiconductor Heterostructure Lasers

handout 29 optical transitions in solids, optical gain ...k q k cv k q k c k v k k q k c k v k r c k...

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