MSEG 667Nanophotonics: Materials and Devices

9: Absorption & Emission Processes

Prof. Juejun (JJ) Hu

hujuejun@udel.edu

Optical absorption in semiconductors

Band structure of silicon

0x y zk k k

2, 0x y zk k k

a

x y zk k ka

G:

L :

X :

a = 5.43 Å

Band structure of semiconductors near band edge

k

E

k

E

Direct gap semiconductors:compound semiconductors

Indirect gap semiconductors:group IV semiconductors

Eg Eg

Energy and crystal momentum conservation

k

E Energy conservation

Crystal momentum conservation

Thermalised electron momentum

Photon momentum

C C V VE k E k

C Vk k

* 8 11 3~ 2 ~ 2.2 10

2C e Bk m k T m

7 1~ 10 C

nm k

c

~C V e k k k

Energy and crystal momentum conservation

k

E Parabolic band approximation

Energy conservation

2 2 2 2

2 2

C e V e

e eC V g

r r

E k E k

k kE E E

m m

2 2

*2C

C C Ce

kE k E

m

2 2

*2V

V V Vh

kE k E

m

EC

EV

1* *1 1r e hm m m

Reduced mass:

Energy and crystal momentum conservation

k

E

Energy conservation2 2

2e

gr

kE

m

EC

EV

12e r gk m E

* *1g

C C Ce h

EE k E

m m

* *1g

V V Vh e

EE k E

m m

For a given photon energy, only a narrow band of states contribute to absorption!

Electronic density of states in energy bands

Number of states with energy between to E E dEE

V dE

In the k-space each electronic state occupies a volume of34

V

where V is the volume of the system

2 2

3 2

4

4C C C C C

C C C C

dk k dk k dkE k

VdE V VdE dE

2 2

*2e

C e Ce

kE k E

m

Dispersion relation

3 2*2 3

12C e CE m E E

Beware of spin degeneracy!

Absorption in direct gap semiconductors

Photon absorption rate is proportional to the joint density of states: the density of states (pairs) that can participate in the absorption process

2 2

2e

C e V e gr

kE k E k E

m

2

22e e e

j j e

dk k dkE k

VdE dE

3 2

2 32

12r

j g g

mE E

Optical transition typically does not flip the electron spin state

Direct band gap energy determination

21g gE E

V. Mudavakkat et al., Opt. Mater. 34, Issue 5, 893-900 (2012).

Conservation laws

Absorption in indirect gap semiconductors

k

E

q

C C V V q

C C V V

E k E k

E k E k

C V qk k k

Typical optical phonon energy in Si: 60 meV

Absorption in indirect gap semiconductors

2

1 21

exp 1

g q

g

q B

E E

k T

Phonon-assisted absorption:

Indirect band gap energy determination

Appl. Opt. 27, 3777-3779 (1988).

Si

Nanotechnology 23, 075601 (2012).

Amorphous solids

Short range order (SRO) Lack of long range order (LRO) Properties pertaining to SRO:

Energy band formation Density of states

Properties pertaining to lack of LRO: Crystal momentum (not a good quantum number anymore!) Localized states (Anderson localization)

Unlike gases, amorphous solids are NOT completely random

AmorphousRandom

vs.

Mobility gap in amorphous solids

Mobility edge: a well-defined energy that separates extended states with localized states

Tauc gap and Tauc plots

Tauc gap ET definition:

It is merely a fitting parameter and has little physical significance!

1 2

TE

ET = 3.3 eV

Quasi-Fermi levels in non-equilibrium semiconductors

Optical or electrical injection increases the density of both types of carriers

In semiconductors displaced from equilibrium, separate quasi-Fermi levels, EFn and EFp must be used for electrons and holes, respectively (EFn = EFp in equilibrium)

Quasi-thermal equilibrium within bands: electron relaxation time within a band is much lower than across the band gap

1

1 expV

Fp B

f EE E k T

1

1 expC

Fn B

f EE E k T

Occupation probability in the conduction band:

Occupation probability in the valence band:

Absorption saturation effect

At high injection levels, the available empty states for electrons in the conduction band are depleted

Equilibrium High injection level

k

E

k

E

Pumping

Absorption saturation effect

Absorption coefficient in the presence of saturation:

Equilibrium High injection level

1j V V C Cf E f E

k

E

k

E

Pumping

Here EC and EV are the energies of the initial and end states

Stimulated emission

The reverse process of optical absorption Electron-hole recombination to emit a

photon in the presence of an external electromagnetic field excitation

Energy conservation

Optical gain

k

E

C C V VE k E k

1j C C V Vs f E f E

where w is the angular frequency of the external field as well as the emitted photon

Net gain and optical amplification

Net gain = gain – loss

In practical applications, additional loss sources (scattering, FCA, etc.) need to be considered in the loss term as well

1j C C V Vg s f E f E

0 C C V V Fn Fpg f E f E E E Population inversion

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits

Stimulated and spontaneous emission

Stimulated emission

Spontaneous emission

Single mode rate equation

Consider a semiconductor with uniform carrier density

n : carrier density np : photon number

abs stim sp

nR R R

t

1abs abs p V V C CR k n f E f E

1stim stim p C C V VR k n f E f E

1sp sp C C V VR k f E f E

kabs , kstim , ksp : absorption, stimulated and spontaneous emission coefficients, which are environment-independent

Absorption rate

Stimulated emission rate

Spontaneous emission rate

No carrier injection

Single mode blackbody radiator

In thermal equilibrium

Energy conservation:

Hot blackbody ( ):

Cool blackbody ( ):

1

1 expC C

C F B

f EE E k T

1

1 expV V

V F B

f EE E k T

1

exp 1pB

nk T

Fermi-Dirac distribution

Planck distribution

Bk T 1p Bn k T

Bk T 0pn

C VE E

Single mode “hot” blackbody radiator

In thermal equilibrium Bk T

,1Bp abs stim sp

k Tn R R R

0abs stim abs stim

nR R R R

t

exp11

1 expV F BC C V Vabs

stim C C V V C F B

E E k Tf E f Ek

k f E f E E E k T

Thermal equilibrium

abs stim k k

Absorption and stimulated emission dominate over spontaneous emission in a “hot” blackbody

At high temperatures, occupation probability of all states are equal

Single mode “cool” blackbody radiator

In thermal equilibrium

0abs stim sp

nR R R

t

1

1V V C C

sp abs p stim pV V C C

f E f Ek k n k n

f E f E

exp1 exp 1 1

expC F Bsp

p p Babs V F B

E E k Tk n n k T

k E E k T

a e abs stim sp k k k k

All the rate constants are the same specified by detailed balance

Note: the rate constant ka-e is NOT a material constant as it depends on the optical mode under investigation

Rereading Einstein on Radiation

Optical gain in semiconductors

The absorption curve at zero injection level and the gain curve at complete inversion are symmetric with respect to the horizontal axis

Their shape reflects the joint DOS in the material

Optical amplification

Under steady-state carrier injection:

Optical amplification:

2 0abs stim sp nr n

nR R R R G

t

r

J

Gain medium

Net radiative recombination

Non-radiative recombination

Current injection

0I 0gdI e

d

stim abs a eC C V V

g p g

R R kg f E f E

v n v

1j C C V Vg f E f E

p pp g p stim abs g p

n ngn v gn R R v gn

z t

a e jk

Optical injection

Carrier density in semiconductor devices

Solving the current continuity and the Poisson equations

2 0abs stim sp nr n

nR R R R G

t

r

J Current continuity

f D D E E

2 fA D

en p N N

Poisson equation

0 V bias 0.5 V forward bias 1-D semiconductor

device simulator:

SimWindows download

Stimulated emission and spontaneous emission in technical applications

Stimulated emission Spontaneous emission

Optical amplifiers

Lasers

LEDs

Fluorescence imaging

Photoluminescence spectroscopy

Laser threshold

Ligh

t in

tens

ity (

L)

Current (I)

ThresholdBelow threshold Spontaneous

emission into multiple cavity modes dominates

Gain < loss Incoherent output

Above threshold Stimulated

emission into usually a single mode dominates

Gain ~ loss Coherent output

log(

L)

Current (I)

b

Ligh

t in

tens

ity (

L)

Wavelength (l)Li

ght

inte

nsity

(L)

Wavelength (l)

× 100

Engineering spontaneous emission rate: Purcell effect

Fermi’s golden rule:

No photon states: suppressed spontaneous emission (SE) Photonic band gap

Large photon density of states: enhanced (i.e. faster) SE rate Optical resonant cavity

Enhancement factor (Purcell factor) of SE in a cavity:

2

all finalstates

2( )f i f i

f

V E E

Initial state yi

electron in CB + 0 photonFinal state yf

electron in VB + 1 photon

3

2

3

4P

QF

n V

n : refractive index Q: cavity Q-factor*l : wavelength V: cavity mode volume

Engineering spontaneous emission rate: Purcell effect

SE suppression SE enhancement

"Investigation of spontaneous emission from quantum dots embedded in two-dimensional photonic-crystal slab," Electron. Lett. 41, 1402-1403 (2005).