UNIT - III Light-Semiconductor Interaction · UNIT - III Light-Semiconductor Interaction. Table of Contents •Optical transitions in bulk semiconductors: absorption, spontaneous

UNIT - III Light-Semiconductor Interaction

Table of Contents

• Optical transitions in bulk semiconductors: absorption, spontaneous emission, and stimulated emission

• Joint density of states, Density of states for photons

• Transition rates (Fermi's golden rule)

• Optical loss and gain

• Photovoltaic effect, Exciton

• Drude model

Optical transitions in bulk semiconductors

Absorption • Spontaneous event in which an atom or molecule absorbs a photon from an incident optical field

• The asborption of the photon causes the atom or molecule to transition to an excited state

Spontaneous Emission

• Statistical process (random phase) – emission by an isolated atom or molecule

• Emission into 4π steradians

Stimulated Emission

• Same phase as “stimulating” optical field

• Same polarization

• Same direction of propagation

E2

hn

E1

2hn

Putting it all together…

• Assume that we have a two state system in equilibrium with a blackbody radiation field.

E2

E1

Stimulated emission

AbsorptionSpontaneous

emission

Joint Density of States

The probability of absorption or emission will depend on the overlap

and energy difference of the initial and final state, and the density of

these states.

In order to determine the probability or amplitude of the absorption we

must find the overlap of the initial and final wavefunctions.

Instead of single initial and final states in single-particle picture, we

have in principle a large density of final states -(k)

In quantum structures case

Eg

Probe

EgHHx

Pump

E2

E1

HH1

LH1

K||

Eg

Choice of the wavefunctions for the initial and final states

Two different kinds of possibilities in quantum structure

Transitions between the valence and conduction bands

Transitions between the quantum-confined states within a given band,

so-called "intersubband“ transitions

E

V.B

C.B

Probe

HH1

LH1

E2

E1

Eg

Barrier QW Barrier

Emission

LHx HHx

e1-e2 ISBT

Resonant optical transition

Photovoltaic effectThe photovoltaic effect is the creation of voltage and electric currentin a material upon exposure to light and is a physical and chemicalphenomenon.

The photovoltaic effect is closely related to the photoelectric effect.In either case, light is absorbed, causing excitation of an electron orother charge carrier to a higher-energy state. The main distinction isthat the term photoelectric effect is now usually used when theelectron is ejected out of the material (usually into a vacuum) andphotovoltaic effect used when the excited charge carrier is stillcontained within the material. In either case, an electric potential (orvoltage) is produced by the separation of charges, and the light has tohave a sufficient energy to overcome the potential barrier forexcitation. The physical essence of the difference is usually thatphotoelectric emission separates the charges by ballistic conductionand photovoltaic emission separates them by diffusion, but some"hot carrier" photovoltaic device concepts blur this distinction.

55Excitonic Effect : Two particle (e-h) interaction

Absorption via Excitons

Electron-Hole interaction: Excitons

Let the electric field of optical wave in an atom be

E=E0e-iwt

the electron obeys the following equation of motion

EXXX emdtdm

dtdm

2

02

2

w

X is the position of the electron relative to the atom

m is the mass of the electron

w0 is the resonant frequency of the electron motion

is the damping coefficient

Classical Electron Model ( Lorentz Model)

+ -

w0

X

E

The solution is tieim

e w

www

)( 22

0

0EX

The induced dipole moment is

EEXp www

)( 22

0

2

im

ee

)( 22

0

2

www

im

e

is atomic polarizability

The dielectric constant of a medium depends on the manner in which the atoms are assembled. Let N be the number of atoms per unit volume. Then the polarization can be written approximately as

P = N p = N a E = e0 c E

Classical Electron Model ( Lorentz Model)

)(1

22

00

2

0

2

www

im

Nen

)(21

22

00

2

www im

Nen

If the second term is small enough then

The dielectric constant of the medium is given by

= 0 (1+c) = 0 (1+N/ 0)

If the medium is nonmagnetic, the index of refraction is

n= ( /0)1/2 = (1+N/ 0 )1/2

Classical Electron Model ( Lorentz Model)

])[(2])[(2

)(1

22222

00

2

22222

00

22

0

2

www

w

www

ww

m

Nei

m

Neinnn ir

The complex refractive index is

-6 -4 -2 0 2 4 6

(w-w0)/

nr

ni

Normalized plot of n-1 and k versus ww0

])2/()[(8])2/()[(4

)(1

22

000

2

22

000

0

2

www

www

ww

m

Nei

m

Neinn ir

at w ~w0 ,

Classical Electron Model ( Lorentz Model)

j jj

j

i

f

mNen

)(1

220

22

www

For more than one resonance frequencies for each atom,

Zfj

j

Classical Electron Model ( Drude model)

If we set w0=0, the Lorentz model become Drude model. This model can be used in free electron metals

)(1

2

0

22

ww im

Nen

0

2

n

ir innn

21 i

0

2/1

1

2/12

2

2

1 /]})[(21{ rn

0

2/1

1

2/12

2

2

1 /]})[(21{ in

By definition,

We can easily get:

Relation Between Dielectric Constant and Refractive Index

Real and imaginary part of the index of refraction of GaN vs. energy;

An Example to Calculate Optical Constants

The real part and imaginary part of the complex dielectric function w) are not independent. they can connected by Kramers-Kronigrelations:

P indicates that the integral is a principal value integral.

K-K relation can also be written in other form, like

www

ww

w

dP0 22

201

)(2)(

www

w

ww

dP0 22

012

)(2)(

Kramers-Kronig Relation

dPn0 2)/(1

)(1)(

Typical experimental setup

( 1) halogen lamp;

(2) mono-chromator; (3) chopper; (4) filter;

(5) polarizer (get p-polarized light); (6) hole diaphragm;

(7) sample on rotating support (); (8) PbS detector(2)

A Method Based on Reflection

xirx

xirx

xx

xxp

kinnk

kinnk

knkn

knknr

1

2

2

1

2

2

1

2

22

2

1

1

2

22

2

1

)(

)(

Snell Law become:

Reflection of p-polarized light

sin2

21 zz kk

2/122

1 ])2[( xk

2/1222

2 ])()2[( irx innk

Reflection coefficient:

In this case, n1=1, and n2=nr+i n i

Reflectance:R(1, , nr, n i)=|r p|2

From this measurement, they got R, for each wavelength ,Fitting the experimental curve, they can get nr and n i .

Calculation

E1

E1'

n1=1

z

x

n2=nr+i n i

Results Based on Reflection Measurement

22

0

021

EE

EEn d

r

)ln(122

24

3

0

2

0

2

EE

EEE

E

EE

E

En

fddr

FIG. 2. Measured refractive indices at 300 K vs.

photon energy of AlSb and AlxGa1-xAsySb1-y

layers lattice matched to GaSb (y~0.085 x).

Dashed lines: calculated curves from Eq. ( 1);

Dotted lines: calculated curves from Eq. (2)

E0: oscillator energyEd: dispersion energyE: lowest direct band gap energy

22

0

22 EEE f

)(222

0

3

0

EEE

Ed

Single effective oscillator model

(Eq. 1)

(Eq. 2)