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Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap Recombination time dgap Renormalization (Band Filling) E k x k y Absorption induced transition of an electron from valence to conduction band conserves k x,y ! W - frequency at which occurs - frequency at which n measured Kramers-Kronig Conduction Band Valence Band E gap E gap 0.01 W W W d c n 0 2 2 ) ( ) ( ) ( n

Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap Recombination time Bandgap Renormalization (Band Filling)

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Page 1: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

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!

d

cn

0 22 )(

)()(

W- frequency at which occurs - frequency at which n measured

Kramers-Kronig

Conduction Band

Valence Band

Egap Egap

n

0.01

Page 2: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

Exciton Bleaching

Page 3: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

- 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

11

2 222ap00

22

vace

geh

eRNL NxxEmn

e

k

Nn

electronper section -cross absorption -

densityelectron conduction

mass hole-electron reduced

/

R

gap

e

eh

N

m

Ex -Simplest case of a 2 band model:

vac

1,2

vac

1

vac

,state steady

1 )(

knI

kk

Nn

NtIN

dt

d Reff

RRsseee

Page 4: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

- Get BOTH an index change AND gain!

- Stimulatedemission

Active Nonlinearities (with Gain)

Optical orelectricalpumping

Kramers-Krönig used to calculate

index 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”

Page 5: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

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!

Page 6: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

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.

d

cn

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

Page 7: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

),(),(),( ),(2

),( K.K. 212121221240201

212 xxHxxHxxGxxGEnn

EcKn

g

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

,1

5

22

0

4

20

5

cm

eK

gapEx i

i

])1(1)[(2

1)1(

8

3)1)((

4

3)1(

2

3

)1(2

3])1()1[(

16

3

])1()1[()(2

1)1(

32

1

4

3

4

9

8

9

16

5

2

1),(

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

xxxxxxxxxxxxx

xxxxxxx

xxxxxxxxxxxxxxx

xxxxH

11

2

)1(),( :1

2

21221

7

2/321

2121

xxxx

xxxxFxx

Kerr

),(),(),( 11

2

)1(),(:1 RAM 2121212

2

21221

7

2/321

2121 xxHxxHxxGxxxx

xxxxFxx

Page 8: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

])1()1[()(

)3(2])1()1[(

)(

)3(2

])1()1[(44

2

1

2

1),(

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

21

22

21

921

xxxxx

xxxxx

xxx

xxx

xx

xx

x

x

xx

xxxxG

)(0 211 xxx

22

21

222

21

22

211

22

21

12/1

1221

9211)1(812

)1(2

1),( :1

x

x

xx

xxx

xx

x

xxxxxFx

2

1

8

)1()1()1()1(

4

3

2

1),(

2/31

2/31

1

2/11

2/11

41

921xx

x

xx

xxxG )(0 211 xxx

QSE

Kerr

Page 9: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

Quantum Confined SemiconductorsWhen 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 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

Page 10: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

Example of Multi-Quantum Well (MQW) Nonlinearities

-Nonlinear 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

ele

ctro

n

Page 11: Charge Carrier Related Nonlinearities E gap Before Absorption After Absorption E gap E gap > E gap  Recombination time Bandgap Renormalization (Band Filling)

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!