optical transitions in semiconductors - cornell universitycv c v o c i v i i cv o cv c i v i i o k...

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Optical Transitions in Semiconductors

In this lecture you will learn:

• Electron-photon Hamiltonian in semiconductors• Optical transition matrix elements• Optical absorption coefficients • Stimulated absorption and stimulated emission• Optical gain in semiconductors• Spontaneous emission


InP

850 nm

980 nm

1300 nm1550 nm

Semiconductors

2


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


Fermi’s Golden Rule: A Review

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

titio eHeHHH

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

mEEHW

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

mEEHW

2ˆ2

E

E

3


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


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 carried by the field is given by the Poynting vector:

c

nAq

cnE

qtrHtrEtrSo

o

o

o

2ˆ

2ˆ,,,

222

The photon flux per unit area is:

cnAS

Fo

o

2

ˆ2

E

H

q

The divergence of the field is zero:

0,.,. trAtrE

cnA

Io

o

2

22

4


Photons and Photon Density

Photons and Photon Flux:Energy in radiation fields can be added or subtracted in discrete units only, called photons. Each photon has energy .

So far a plane wave, whose intensity is =

the photon flow per unit area per second =

2

2||

2 oo

Ac

n

20 ||

2A

cn

o

Photon Density:The photon density (number of photons per unit volume) for a plane wave equals:

222

222

2

||21

||21

=||21

=

||2

=photonsofvelocity

secondperareaunitperflowPhoton=

oor

ooroo

oo

p

EAAn

nc

Ac

n

n


Photon Density in Dispersive Media

Phase Velocity and Group VelocityIn dispersive media, where the refractive index is frequency dependent, a distinction needs to be made between the phase velocity of a wave and the group velocity of a wave:

pv gv

ggp n

cddnn

cv

nc

v

index groupgn

The group velocity is the velocity at which the energy, and therefore the photons, travel.

Photon Density:The photon density for a plane wave in a dispersive mdeium equals:

222

2

||21

=||21

=

||2

=photonsofvelocity

secondperareaunitperflowPhoton=

oor

oog

g

oo

p

EAnn

nc

Ac

n

n

5


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

eeAqH

PtrAmq

H

PtrAmq

trAPmq

H

trAtrAm

qPtrA

mq

trAPmq

rVmPP

rVm

trAqPH

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,


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 kEkEHkWif

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

Where:

nPmeAq

Hrqi

o ˆ.ˆ2

ˆˆ.

nPmeAq

Hrqi

o ˆ.ˆ2

ˆˆ.

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Optical Matrix Element

ivfckvrqi

kck

o

ivfckvkck

i

kEkEnPem

qA

kEkEHkW

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


Optical Matrix Element

if kvrqi

kc nPe

,ˆ.

,ˆ.ˆ

k

Interband

E

k

Interband

E

jif

fi

ii

ff

Rkvkc

rkqki

kvrkirqi

kcrki

runPruerd

runPeeruerd

,,*.

cell primitive th-j

3

,..

,*.3

ˆ.ˆ

ˆ.ˆ

nP

runPrurdN

runPrurde

runPrurde

cvkqk

kvkckqk

Rkvkc

Rkqki

Rkvkc

Rkqki

fi

iffi

jif

jfi

jif

jfi

ˆ.

ˆ.ˆ

ˆ.ˆ

ˆ.ˆ

,

,,*

cell primitive one any

3,

,,*

cell primitive one any

3.

,,*

cell primitive th-j

3.

7


Crystal Momentum Selection Rule

iviccvo

ivfckqkcvk

o

ivfckvrqi

kck

oi

kEqkEnPm

qA

kEkEnPm

qA

kEkEnPem

qAkw

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


iviccv

oi kEkEnP

mqA

kW2

2

ˆ.2

2

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

qAV

2

2

ˆ.2

22

k

Interband

E

k

Interband

E

ikW

ik

R

8



FBZ3

322

FBZ3

322

FBZ

22

3

3

22

22ˆ.

22

22ˆ.

22

ˆ.2

2

22

ˆ.2

22

kEkEkd

nPm

qA

kEkEkd

nPm

qA

kEkEnPm

qAkd

kEkEnPm

qAV

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



FBZ3

322

22ˆ.

22

kEkE

kdnP

mqA

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

m

qAR

23

22

22 2

2

1ˆ.

22

R

gE

Joint density of states

Reduced effective mass

k

Interband

E

R

Joint density of states

9



gr

cvo E

mnP

m

qAR

23

22

22 2

2

1ˆ.

22

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

k

Interband

E

R

22

||2

= oo

Ac

nI

2

2

||21

= oogp Annn

gr

cvog

p Em

nPnn

n

mq

R

23

22

22 2

2

1ˆ.

gr

cvo E

mnP

n

Ic

mq

R

23

22

2

2

2 2

2

1ˆ.

The transition rate can be written as:

The transition rate depends on the photon density or the wave intensity


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!

10


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

cvo

Em

nPcnm

q

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


Loss Coefficient of Semiconductors

11


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


Stimulated Absorption

FBZ3

322

12

2ˆ.2

2

kEkEkfkf

kdnP

mqA

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

Stimulated absorption

k

E

12


Stimulated Emission An 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 eHeHHH

ˆˆˆˆnP

m

eAqH

rqio ˆ.ˆ2

ˆˆ.

nPm

eAqH

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

m

qAR vcvccv

o

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! This process of photon emission caused by incoming radiation (or by other photons) is called stimulated emission.


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

mqA

RR vccvcvo

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

gp vn

RR

I

RR

Stimulated absorption

FBZ3

322

22ˆ.

kEkEkfkfkd

nPcnm

qvccvcv

o

Stimulatedemission

k

E

k

E

pg nvRR

13


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

Spontaneous Emission

Spontaneousemission

k

E Single radiation modeMode volume = Vp

Semiconductor

Temperature = T

Spontaneous Emission Rate: Consider an optical cavity with a single mode of radiation at temperature T

The number of photons in the radiation mode is given by the Bose-Einstein factor:

1

1

KTpe

N


Spontaneous Emission Rate

kEkEkfkf

kEkEkfkfkd

nPV

N

nnmq

RR

vcvc

vcvccvp

p

og

for0

02

2ˆ.

0

FBZ3

322

In thermal equilibrium, the emission and absorption rates must be in balance:

The above condition does not hold in thermal equilibrium since: RR

Assume another emission mechanism such that: 0 RRR sp

Also assume:

FBZ3

322

FBZ3

322

12

2ˆ.

12

2ˆ.

kEkEkfkfkd

nPnnm

qCR

kEkEkfkfkd

nPnnm

qR

vcvccvog

sp

vcvccvog

sp

14


Spontaneous Emission Rate

p

vcKT

pKTkEkE

vcp

p

vc

cv

vcvcvcp

p

sp

VC

kEkEeCVe

kEkEN

CV

kfkf

kfkf

kEkEkfkfCkfkfV

N

RRR

vc

1

for11

for1

for01

0

In thermal equilibrium, the emission and absorption rates must be in balance:

FBZ3

322

12

2ˆ.1

kEkEkfkf

kdnP

Vnnmq

R vcvccvpog

sp

The spontaneous emission rate is therefore:

The rate of spontaneous emission of a photon into a radiation mode is the same as the rate of stimulated emission into the same radiation mode provided one assumes that there is only one photon inside the radiation mode!


Optical Gain in Semiconductors

FBZ3

322

22ˆ.

kEkEkfkf

kdnP

cnmq

vccvcvo

Note that power 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

RR

15


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 and electron and hole densities:

FBZ3

322

22ˆ.

kEkEkfkf

kdnP

cnmq

vccvcvo

gE

Optical gain for frequencies for which:

fhfeg EEE

Increasing electron-hole density (n = p)

Net Loss

Net Gain

0k

E

Efe

Efh

16


Photon Density of StatesConsider a medium of volume Vp

The radiation modes are plane waves:

ti

p

rqioti

p

rqio e

V

eAe

V

eAntrA

22ˆ=),(

n

cqThe wave dispersion is:

In q-space, in a volume of size there are different plane wave modes

qd3 33 22 qdVp

The dispersion relation gives: dqvdcdqdddn

n g

dgV

vd

cn

Vdqq

Vqd

V ppg

ppp

2

2

2

3

3

22

Since:

there are different radiation modes per unit volume per unit frequency interval pg

g

p vcn

g1

2

Photon density of states


Total Spontaneous Emission Rates in Semiconductors

Spontaneous emission into different radiation modes

1

2

3

FBZ3

322

12

2ˆ.1

kEkEkfkf

kdnP

Vnnmq

R vcvccvpog

sp

The spontaneous emission rate into a single radiation mode is:

The spontaneous emission rate (per unit volume per second) into all radiation modes is:

0 dgRR pspTsp

0 1

de

gvR

KTqVp

gTsp

It can also be written as:

k

E

Efe

Efh

qV

optical transitions in semiconductors - cornell universitycv c v o c i v i i cv o cv c i v i i o k...

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