MSEG 667
Nanophotonics: Materials and Devices
9: Absorption & Emission Processes
Prof. Juejun (JJ) Hu
hujuejun@udel.edu
Optical absorption in semiconductors
Band structure of silicon
G : kx  k y  kz  0
L : kx 
2
, k y  kz  0
a
X : kx  k y  kz 
a = 5.43 Å

a
Band structure of semiconductors near band edge
E
E
Eg
Eg
k
Direct gap semiconductors:
compound semiconductors
k
Indirect gap semiconductors:
group IV semiconductors
Energy and crystal momentum conservation

E
Energy conservation
  EC  kC   EV  kV 

Crystal momentum conservation
  kC  kV  kC  kV ~ ke


k
Thermalised electron momentum
kC ~

1
3
2m  kBT ~ 2.2 108 m1
2
*
e
Photon momentum
n 

~ 107 m 1 kC
c
Energy and crystal momentum conservation

E
2
kC 2
EC  kC   EC 
2me*
EC
2
kV 2
EV  kV   EV 
2mh*

k
EV
Parabolic band approximation

Energy conservation
  EC  ke   EV  ke 
2
2 2
ke 2
ke
 EC  EV 
 Eg 
2mr
2mr

Reduced mass: mr  1 m  1 m
*
e
*
h

1
Energy and crystal momentum conservation

E
2
ke 2
  Eg 

2mr
1
ke 
2mr    Eg 
EC

k
EV
Energy conservation
EC  kC   EC 
EV  kV   EV 
  Eg
1  me* mh*
  Eg
1  mh* me*
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 E to E  dE
V  dE
4 3
In the k-space each electronic state occupies a volume of
V
 E 
where V is the volume of the system
2
ke 2
EC  ke   EC 
2me*
Dispersion relation
dkC
4 kC 2 dkC
kC 2 dkC
C  EC   C  kC  


 2 
3
VdE 4 V VdE 
dE
C  E  
1

2
3
 
 m
*
e
32
 2  E  EC 



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
ke 2
  EC  ke   EV  ke   Eg 
2mr
 j E   j 
j 
dke
ke 2 dke
     ke  


2
VdE 2 dE





mr 3 2
1
   2 3  2    E g       
2

  Eg
Optical transition typically does not flip the electron spin state
Direct band gap energy determination
   
1


  Eg        Eg
2
V. Mudavakkat et al., Opt. Mater. 34, Issue 5, 893-900 (2012).
Absorption in indirect gap semiconductors

Conservation laws
  EC  kC   EV  kV   q
 EC  kC   EV  kV 
  kC  kV  kq
E

q
k
Typical optical phonon
energy in Si: 60 meV
Absorption in indirect gap semiconductors
Phonon-assisted absorption:
   
1


  E g  q 
2
 exp  q k BT   1

12
      Eg
Indirect band gap energy determination
Si
Appl. Opt. 27, 3777-3779 (1988).
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)
Amorphous
Random
Unlike gases,
amorphous solids
are NOT completely
random
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:       ET
It is merely a fitting parameter and has little physical
significance!
12
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
Occupation probability in
the conduction band:
fC  E  
1
1  exp  E  EFn  k BT 
Occupation probability in
the valence band:
fV  E  
1
1  exp  E  EFp  k BT 
Absorption saturation effect

At high injection levels, the available empty states for
electrons in the conduction band are depleted
E
E

Equilibrium
Pumping
k

High injection level
k
Absorption saturation effect
E
E

Pumping
k
Equilibrium


High injection level
Absorption coefficient in the presence of saturation:
   j     fV  EV   1  fC  EC  
Here EC and EV are the energies of the initial and end states
k
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
  EC  kC   EV  kV 
where  is the angular frequency
of the external field as well as the
emitted photon

Optical gain
s   j     f C  EC   1  fV  EV  
E

k
Net gain and optical amplification

Net gain = gain – loss
g     s       
1
  j      f C  EC   fV  EV  

g    0  fC  EC   fV  EV   EFn  EFp  

In practical applications,
additional loss sources
(scattering, FCA, etc.)
need to be considered
in the loss term as well
L. A. Coldren and S. W.
Corzine, Diode Lasers and
Photonic Integrated Circuits
Population
inversion
Stimulated and spontaneous emission
Stimulated
emission
Spontaneous
emission
Single mode rate equation

Consider a semiconductor with uniform carrier density
n : carrier density
n
 Rabs  Rstim  Rsp
t
np : photon number
No carrier injection
Rabs  kabs n p fV  EV   1  f C  EC  
Absorption rate
Rstim  k stim n p f C  EC   1  fV  EV  
Stimulated emission rate
Rsp  ksp fC  EC   1  fV  EV  
Spontaneous emission rate
kabs , kstim , ksp : absorption, stimulated and spontaneous
emission coefficients, which are environment-independent
Single mode blackbody radiator

In thermal equilibrium
fC  EC  
1
1  exp  EC  EF  k BT 
1
fV  EV  
1  exp  EV  EF  k BT 
np 
exp 
1
 kBT   1
Fermi-Dirac distribution
Planck distribution

Energy conservation:   EC  EV

Hot blackbody ( 

Cool blackbody ( 
k BT
): n p  k BT 
k BT
): n p  0
1
Single mode “hot” blackbody radiator

In thermal equilibrium
np 
k BT

1  Rabs , Rstim

k BT
Rsp
Absorption and stimulated emission dominate over spontaneous
emission in a “hot” blackbody
n
 Rabs  Rstim  0  Rabs  Rstim
t
Thermal equilibrium
fC  EC  1  fV  EV  exp  EV  EF  k BT 
kabs




1
kstim 1  fC  EC 
fV  EV 
exp  EC  EF  k BT 
At high temperatures, occupation probability of all states are equal
 kabs  kstim
Single mode “cool” blackbody radiator

In thermal equilibrium
n
 Rabs  Rstim  Rsp  0
t
ksp  kabs n p 

ksp
kabs
fV  EV 
1  fV  EV 

1  f C  EC 
f C  EC 
Rereading
Einstein on
Radiation
 k stim n p

 exp  EC  EF  k BT  

 np  
 1  n p  exp   k BT   1  1

 exp  EV  EF  k BT  

 ka e  kabs  k stim  k sp
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
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
d
Optical amplification

Under steady-state carrier injection:
n  r 
t
Optical injection
Gain
medium
I 0 e G gd
 Rabs  Rstim  Rsp  Rnr  G   2 J n  0
Net radiative
recombination

I0
Non-radiative
recombination
Current
injection
Optical amplification:
n p
z
g
 G gn p 
n p
t
 vg G gn p  Rstim  Rabs  vg G gn p
Rstim  Rabs ka e

  fC  EC   fV  EV 
G vg n p
G vg
g   
1
  j      fC  EC   fV  EV  


  ka  e  G   j   

Carrier density in semiconductor devices

Solving the current continuity and the Poisson equations
n  r 
t
 Rabs  Rstim  Rsp  Rnr  G   2 J n  0
 D  f
D E
E   
f e
 
   n  p  N A  ND 


2
0 V bias
Current continuity
Poisson equation
0.5 V forward bias
1-D
semiconductor
device
simulator:
SimWindows
download
Stimulated emission and spontaneous emission in
technical applications
Stimulated emission
Optical amplifiers
Lasers
Spontaneous emission
LEDs
Fluorescence
imaging
Photoluminescence
spectroscopy
Laser threshold
Threshold
Below threshold
 Spontaneous
emission into
multiple cavity
modes dominates
 Gain < loss
 Incoherent output
Current (I)
Above threshold
 Stimulated
emission into
usually a single
mode dominates
 Gain ~ loss
 Coherent output

× 100
Wavelength (l)
Current (I)
Wavelength (l)
Engineering spontaneous emission rate: Purcell effect

Fermi’s golden rule: G 
2

2
 f V  i  ( E f  Ei )
all final
states f
Initial state i
electron in CB + 0 photon

No photon states: suppressed spontaneous emission (SE)


Photonic band gap
Large photon density of states: enhanced (i.e. faster) SE rate


Final state f
electron in VB + 1 photon
Optical resonant cavity
Enhancement factor (Purcell factor) of SE in a cavity:
l Q
FP 

  
4 2  n  V
3
3
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 twodimensional photonic-crystal slab," Electron. Lett. 41, 1402-1403 (2005).