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Charge Carrier Related Nonlinearities
Egap
Before Absorption
After Absorption
Egap Egap> Egap Recombination
time
Bandgap Renormalization (Band Filling)
E
kx
ky
Absorption induced transitionof an electron from valence toconduction band conserves kx,y!
dcn 0 22 )()()(
- frequency at which occurs - frequency at which n measured
Kramers-Kronig
Conduction Band
Valence Band
Egap Egap
n
0.01
Exciton Bleaching
Most interesting case is GaAs, carrier lifetimes are nsec effective e (linewidths) meV classical dispersion (Haug & Koch) is of form . 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
122 ])[( ehe EE
Charge Carrier Nonlinearities Near Resonance
1
112 222
ap00
22
vace
geh
eRNL NxxEmn
ek
Nn
electronper section -cross absorption - densityelectron conduction
mass hole-electron reduced
/
R
gap
e
ehNm
Ex Simplest case of a 2 band model:
vac
1,2
vac
1
vac
,state steady
1 )(k
nIkk
NnNtIN
dtd R
effRRssee
e
- Get BOTH an index change AND gain!
- Stimulatedemission
Active Nonlinearities (with Gain)
Optical orelectricalpumping
Kramers-Krönig used to calculateindex change n() from ().
Ultrafast Nonlinearities Near Transparency Point
At the transparency point, the losses are balanced bygain so that carrier generation by absorption is no longer the dominant nonlinear mechanism forindex change. Of course one gets the Kerr effect + other ps and sub-ps phenomena which now dominate.
0
Gain
Loss
eN
“Transparency point”
Evolution of carrier density in time “Spectral Hole Burning”“hole” in conduction band due toto stimulated emission at maximumgain determined by maximumproduct of the density of occupiedstates in conduction band and density of unoccupied states invalence band
“Carrier Heating” (Temperature Relaxation)electron collisions return carrierdistribution to a Fermi distributionat 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 Ramaneffect (RAM), the Linear Stark Effect (LSE) and the Quadratic (QSE) Stark Effect. Shownschematically below are the three most important ones.
dcn
NLNL
0 22 )()()(
- frequency at which occurs - frequency at which n calculated
The theoretical approach is to calculate first the nonlinearabsorption and then to use the Kramers-Kronig Relation to calculate the nonlinear index change .)(NLn
)( NL
),(),(),( ),(2
),( K.K. 212121221240201
212 xxHxxHxxGxxGEnn
EcKng
p
Here Ep (“Kane energy”) and the constant K are given in terms of the semiconductor’s properties. K=3100 cm GW-1 eV5/2
),(),( 21230201
21 xxFEnn
EK
g
p
,152
20
4
20
5
cmeK
gapEx i
i
])1(1)[(21)1(
83)1)((
43)1(
23
)1(23])1()1[(
163
])1()1[()(21)1(
321
43
49
89
165
21),(
2/32
21
22
2/111
32
2/11
21
222
2/111
22
2/12
212
2/12
2/11
21
22
2/31
2/312
212
2/31
21
32
32
212
21
22
21
32
42
41
621
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xxxxH
112
)1(),( :12
21221
7
2/321
2121
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Kerr
),(),(),( 112
)1(),(:1 RAM 2121212
2
21221
7
2/321
2121 xxHxxHxxGxxxx
xxxxFxx
])1()1[()(
)3(2])1()1[()(
)3(2
])1()1[(4421
21),(
2/11
2/1122
221
21
22
21
222/1
22/1
2222
21
22
21
22
21
22
21
2/11
2/11
1
22
22
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921
xxxxx
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xxxxG
)(0 211 xxx
2
2
21
222
21
22
211
22
21
12/1
1221
9211)1(812
)1(21),( :1
xx
xx
xxxxx
xxxx
xxFx
21
8)1()1()1()1(
43
21),(
2/31
2/31
1
2/11
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921xx
x
xx
xxxG )(0 211 xxx
QSE
Kerr
Quantum Confined SemiconductorsWhen the translational degrees of freedom of electrons in both the valence and conduction bandsare 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 statesappear. As a result the nonlinear opticalproperties can be enhanced or reduced) in some spectral regions.
-Absorption edge movesto higher energies.-Multiple well-definedabsorption peaks due totransitions betweenconfined states-Enhanced absorptionspectrum near band edge
Quantum Wells
Example of Multi-Quantum Well (MQW) NonlinearitiesNonlinear absorption change (room temp.) measured versus intensity and convertedto index change via Kramers-Kronig
A factor of 3-4 enhancement!!
Quantum Dots
Quantum dot effects become important when thecrystallite size r0 aB (exciton Bohr radius). For example, the exciton Bohr radius forCdS aB = 3.2nm, CdSe aB = 5.6nm, CdTe aB = 7.4nm and GaAs aB = 12.5nm.
Definitive measurements were performedon very well-characterized samples byBanfi. De Giorgio et al. in range aB r0 3 aB
Measurements at1.2m (), 1.4m () and1.58m () for CdTe
Measurements at 0.79m (+) for CdS0.9Se0.1
Note the trend that Im{(3)} seems to fallwhen aB r0 !
Inde
x ch
ange
per
exc
ited
elec
tron
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).
gapE/0.80.6 0.70.5
2
0
-2
-4Re
al{
(3) } i
n un
its o
f 10-1
9 m2 V-2
10-18
10-19
10-21
10-20
1.0 2.01.5(/
0)4 Imag
{(3
) } in
units
of m
2 V-2
/gapE
Nanocrystals+ 0.79m 2.2 m 1.4 m 1.58m
Bulk CdS 0.69m
▼ CdTe 12, 1.4, 1.58m
To within the experimental uncertainty (factor of 2), no enhancements werefound in II-VI semiconductors for the far off-resonance nonlinearities!