Download Lecture 8 - KFUPM Faculty List

Charge Carrier Related Nonlinearities
Bandgap Renormalization (Band Filling)
Before Absorption
After Absorption
Conduction Band

Egap
E
Egap
kx
Valence Band
Absorption induced transition
of an electron from valence to
conduction band conserves
kx,y!

ky
Egap> Egap
Recombination
time
Kramers-Kronig  n( ) 
c   ()

0
(   )
2
2
d
- frequency at which  occurs
 - frequency at which n measured
 
n
0.01
Egap


Egap
Exciton Bleaching
Charge Carrier Nonlinearities Near Resonance
- Most interesting case is GaAs, carrier lifetimes are nsec  effective e (linewidths) meV
classical dispersion (Haug & Koch) is of form [(  Ee  Eh ) 2  e2 ]1 .
near resonance, as discussed before
Ee – electron energy level to which electron excited in conduction band
Eh – electron energy level in valence band from which electron excited by absorption
x   / Egap
-Simplest case of a 2 band model:
n
NL

 R Ne
k vac
meh  reduced electron - hole mass
 2e 2
1
1

Ne
2
2
2
2 0 n0 meh E gap x x  1


N e  conduction electron density
 R - absorption cross - section per electron
N e,ss R 1 R
d
I (t ) N e state
 
N e  1

 n 

I  n2,eff  1 R
dt
 
k vac
k vac
k vac
steady
Active Nonlinearities (with Gain)
Optical or
electrical
pumping
-

-
Stimulated
emission
Get BOTH an index
change AND gain!
Kramers-Krönig used to calculate
index change n() from ().
Ultrafast Nonlinearities Near Transparency Point
At the transparency point, the losses are balanced by
gain so that carrier generation by absorption is no
longer the dominant nonlinear mechanism for
index change. Of course one gets the Kerr effect +
other ps and sub-ps phenomena which now dominate.
“Transparency point”
Gain
0
Loss
Ne
Evolution of carrier density in time
“Spectral Hole Burning”
“hole” in conduction band due to
to stimulated emission at maximum
gain determined by maximum
product of the density of occupied
states in conduction band and
density of unoccupied states in
valence band
“Carrier Heating”
(Temperature Relaxation)
electron collisions return carrier
distribution to a Fermi distribution
at a lower electron temperature
SHB – Spectral Hole Burning
Experiments have confirmed these calculations!
Semiconductor Response for Photon Energies Below the Bandgap
As the photon frequency decreases away from the bandgap, the contribution to the electron
population in the conduction band due to absorption decreases rapidly. Thus other mechanisms
become important. For photon energies less than the band gap energy, a number of passive
ultrafast nonlinear mechanisms contribute to n2 and 2. The theory for the Kerr effect is based
on single valence and conduction bands with the electromagnetic field altering the energies of
both the electrons and “holes”.
There are four processes which contribute, namely the Kerr Effect, the Raman
effect (RAM), the Linear Stark Effect (LSE) and the Quadratic (QSE) Stark Effect. Shown
schematically below are the three most important ones.
c   ()
NL
The theoretical approach is to calculate first the nonlinear
n ( ) 
d
2
2
0
absorption  NL () and then to use the Kramers-Kronig
 (   )
NL
Relation to calculate the nonlinear index change n ( ) .  - frequency at which  occurs
 - frequency at which n calculated

NL
Kerr
 (1, 2 )  K
i
xi 
Egap
Ep
n01n02 Eg3
25 1
K
5  0 2
F2 ( x1, x2 )
e4
m0 c
2
,
( x1  x2  1)3 / 2  1 1 
  
x1  x2  1 : F ( x1 , x2 ) 
7
2
2 x1 x2
 x1 x2 
Here Ep (“Kane energy”) and the constant K are
given in terms of the semiconductor’s properties.
K=3100 cm GW-1 eV5/2
Ep
cK
K.K.  n2 (1 , 2 ) 
G2 ( x1 , x2 )
4
2 n01n02 E g
2
G2 ( x1 , x2 )  H ( x1 , x2 )  H ( x1 , x2 )
5 3 2 9 2 2 9

2 3 3 1 3 2
3 / 2 1
x
x

x
x

x
x

x

x
x
(
1

x
)
 ( x2  x1 ) 2 [(1  x2  x1 )3 / 2  (1  x1 )3 / 2 ]
1
16 2 1 8 2 1 4 2 1 4 2 32 2 1
2


1
3 2 2
3
1 / 2
1 / 2
2
1/ 2


H ( x1, x2 ) 
 x x [(1  x1 )
 (1  x2 )
]  x2 x1 (1  x2 )
6 4 4  16 2 1

2
2 x1 x2


3
3
3
1
2
1
/
2
2
2
1
/
2
3

1
/
2
2
2
3
/
2
 x2 x (1  x )  x ( x2  x1 )(1  x1 )  x2 x (1  x1 )

 ( x2  x1 )[1  (1  x2 ) ]
1
1
1
 2

4 2
8
2
( x1  x2  1)3 / 2  1 1 
  
RAM  x1  x2  1 : F ( x1, x2 ) 
7
2
2 x1x2
 x1 x2 
2
G2 ( x1, x2 )  H ( x1, x2 )  H ( x1, x2 )
QSE


 x
2
2
2


2
x

1
x

x
8
(
x

1
)


1
1
1
2
1
x1  1 : F ( x1 , x2 )   9 2


2

2
2 2
2 x1 x2 ( x1  1)1 / 2  x12  x22
x
2
x

x
1
2


x1  0( x1  x2 )  1

1




 


2
2
2
2
 2 x1 x2 x1

x1  x2
1
G ( x1, x2 ) 


2
2
2
29 x12 x22  2 x 2 (3x 2  x 2 )
1
2
1 [(1  x )1 / 2  (1  x )1 / 2 ]  2 x2 (3 x1  x2 ) [(1  x )1 / 2  (1  x )1 / 2 ]



2
2
1
1
2 2
2 2
2 2
2 2
x
(
x

x
)
x
(
x

x
)
2 1
2
1 1
2


4
4
1 / 2
 (1  x1 ) 1 / 2 ]
x22 [(1  x1 )
 3 (1  x ) 1 / 2  (1  x ) 1 / 2 (1  x ) 3 / 2  (1  x ) 3 / 2 1 
1
1
1
1
x1  0( x1  x2 ) G ( x1, x2 ) 


 
9 4 4
8
2
2 x1 
x1

1
Kerr
Quantum Confined Semiconductors
When the translational degrees of freedom of electrons in both the valence and conduction bands
are confined to distances of the order of the exciton Bohr radius aB, the oscillator strength is
redistributed, the bandgap increases, the density of states e(E) changes and new bound states
appear. As a result the nonlinear optical
properties can be enhanced or reduced)
in some spectral regions.
Quantum Wells
-Absorption edge moves
to higher energies.
-Multiple well-defined
absorption peaks due to
transitions between
confined states
-Enhanced absorption
spectrum near band edge
-Nonlinear absorption change (room temp.)
measured versus intensity and converted
to index change via Kramers-Kronig
A factor of 3-4 enhancement!!
Index change per
excited electron
Example of Multi-Quantum Well (MQW) Nonlinearities
Quantum Dots
Quantum dot effects become important when the
crystallite size r0  aB (exciton Bohr radius). For example, the exciton Bohr radius for
CdS aB = 3.2nm, CdSe aB = 5.6nm, CdTe aB = 7.4nm and GaAs aB = 12.5nm.
Definitive measurements were performed
on very well-characterized samples by
Banfi. De Giorgio et al. in range aB  r0  3 aB
Measurements at1.2m (), 1.4m () and
1.58m () for CdTe
Measurements at 0.79m (+) for CdS0.9Se0.1
Note the trend that Im{(3)} seems to fall
when aB  r0 !
Nonlinear Refraction and Absorption in Quantum Dots for aB  r0  3 aB:
II-VI Semiconductors
Experimental QD test of the previously discussed off-resonance universal F2(x,x) and G2(x,x)
functions for bulk semiconductors (discussed previously) by M. Sheik-Bahae, et. al., IEEE J.
Quant. Electron. 30, 249 (1994).
2
10-20
Nanocrystals
+ 0.79m
 2.2 m
 1.4 m
 1.58m
10-21
Bulk
CdS 0.69m
▼ CdTe 12, 1.4, 1.58m
10-19
1.0
Egap /  1.5
2.0
Real{(3)} in units of 10-19m2V-2
(/0)4 Imag{(3)} in units of m2V-2
10-18
0
 / Egap
0.5
0.6
0.7
-2
-4
To within the experimental uncertainty (factor of 2), no enhancements were
found in II-VI semiconductors for the far off-resonance nonlinearities!
0.8