Polarization driven exciton dynamics in asymmetric nanostructures

Margaret Hawton, Lakehead University

Marc Dignam, Queens University

Ontario, Canada

• Excitons with a dipole moment are created by a laser pulse, giving polarization Pinter.

• This results in a diffraction grating and an internal electric field, E (Pintra).

• Simulation retains inter and intraband coherence, results shown are for a BSSL.

Outline

Ultrafast experiments

k1 (pump)21

k2 (probe)

FWM Signal2k2- k1

PP Signal

z

x

y

THz emission

SWM Signal, etc2 13 2k k

QW made asymmetric by Edc

Edc

Energy or frequency

c Laser pulsen=1

n=2

Egap

+

-G (dipole mom.)

VB

CB

Biased SC Superlattice (BSSL)

energy or frequency

1 0

'

intraband dipole:

'

;( )e h

G e d G G

e

G r r

=2

d

Edc

-

+

G22

d

=2

d

Edc

=0=1

=-1

Bloch oscillations:/B dcedE

frequency

0 B

B

(Stark ladder)

c Laser pulse

<G>

-

+

--

Biased SC Superlattice (BSSL)

Bloch Oscillationsof dipole moment (QM interference)

B

G22

G-1 -1

G00

G22

B

Exciton: bound e and h in 2D H-like state, C of M wave vector K

+-

2a0

=1

H-like binding lowers

below free e-h pair.

Kz

x,y

Basis { , } stands for { ,H-like, ,spin}. K K

1s

c=0

Linear response (note H-like binding)

k1/k2 interference: the polarization grating

13 by 2

intra1 2

0

expmm

im

P P k k R

2/|k2-k1|+ harmonics

z

x,y

k2

2k2-k1= K-3

FWM Signal

thus Ks are discrete

1 2

2 1 2

0 0

0

2

1

2

: for

:

to by steps of 2 for grati

intraband even

interband

ng

odd

c onverged at 1 )n 3(

m

m

m

m

m

n

m

m n

K k k

K k k k

intra

intr

†

inter

int

a †' ' '

e

', '

r *

',

1

1

creates an exciton

Polarization density:

. .V

V

B

B c

B

h

B

K

K Κ K KK

Κ

K K

K K

P

P

P P

M

G

P

Inter and intraband polarization

PZW (multipolar) Hamiltonian which we write as:

†

†

,

2

iex field

e

nt

in

x

t

field

VV

H H H

H B B

H Kc a

H

H

a

K K K KΚ Κ

Κ ΚΚ

Κ ΚΚ

D P P P

Dipole approximation

Hamiltonian is exact, P is approximate, includes self-energy.

230

2

223

1stationary dipole: 2

1free dipole ~

self-energy negl

02

1:

2 2

for N excitons if igible free.

d r ed

ed

eded

V Nd r

V

r r

EM field

, 1

†' ' ' '

22 2

2

dOHeisenberg Picture: i

exp . .

, (true bosons)

dynamics in

Using Heisenbergs twice:

, , , an

,dt

cancels in td

KcV

i a t i h c

a a

dK K

dt

t t t

O H

K KK

K K KK

ΚΚ Κ

D e K R

DD P

E R D R DP R

B

raband , for Kc>> leaving .

PP

longitudinal/transverse Pintra

z

x

-------

+++++++

K Kz

L

Pintra

L .2m

1m

Kz >> K

intra

2 2

2

exp

sinc

K

Kz

L L

K L

P iKx z z

P

P z

z

Kz

L

For GaAs/Ga.7Al0.3As (67A/17A) 30 period superlattice

† '; ' †' ' , ' , ' '' '' ' ''' '' '

'' '' '

, ' , ; ' '

';00 * ' '' '''*'' '''

'

† †

, = - 2

B

B

B

B B

PSF

X

B X

KKK Κ K K Κ K

K Κ Κ

k k

K k kK

K

k

k kk

PSF

H-like excitons are (approximate) quasibosons.

+ -k-k

eh-pair

+ -H-like exciton

HP exciton dynamics

†' ' ' ; ' '

' '

†' ' ; ' ' ' '' ' '' ; '' ''

††

†' ' '' ' ''

', ' , '''

''

' ' '' ''

opt THz

S

S

dBi B PSF

PSF B PS

Bd

B

F

t

Κ Κ ΚK

K Κ Κ Κ Κ Κ

K

ΚΚ

ΚK K

K K

E

E

G

M

M

G

To solve numerically, must take expectation value.

inter intraNote that . KK KKD PE P

PSF ~ n/n0

n= exciton areal density =109 to 1010 cm-2

n0 = 1/a02 = 2x1011 cm-2

n/n0 < 0.1

Will omit PSF in numerical calculations here.

(1)†(1)†

'2inter

1

1st order interband dynamics:

1 ext

opt

d B ii BTdt

Κ

Κ E M

Can solve to any definite order in Eopt

(2)†(2)†

2intra

(1)(1)* †

(2) (2)* † †' ' ' ' ' '

' '

Can then get intraband dynamics:

2nd orde

r

extopt

extTHz

d B B ii B B

dt T

B B

B B B B

Κ P

Κ P

P Κ

Κ P Κ ΚK

E M M

E G G

(1)†' ' '

' '

+ extTHz B

ΚK

E G

etc, etc

Lyssenko et al PRL 79, 301 (1997)

but solving to any finite order isn’t good enough - experiments show peaks oscillate

†† †

' ' ' '' ' ''' ' ' ', '' ''

1 -

d Bi B Bdt

Κ

Κ K ΚK K K

E M G

Need infinite order, factored, like SBEs

†

†

* † †' ' ' '' ' ''

' ' '' ''

1 + terms

d B Bi B B

dt

dBB B B B

dt

Κ P

Κ P

PK P Κ P Κ

K K

E M G

Retains exciton-exciton correlations, no biexcitons.

intrinter awhere . K KK KPD PE

††

†' '

inte

' '' ' ''' ' ' ', '' ''

r

1

+ higher order

id B

B

Ti Bdt

ΚΚ

K ΚK K K

E M G

with phenomenological decay

†

†

* † †' ' ' '' ' ''

' ' '

2

' ''

intra

1 + terms

d B Bi B B

dt

dBB B B B

dt

i

T

Κ P

Κ P

PK P Κ P Κ

K K

E M G

Convergence: n0=3 (dash), 5(dot) and 13 (solid)

FWM

EWMSWM

-2 -1 0 1

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

=+1

=0

=-1

Spectrally-Resolved FWM Intensityfor Different time delays,

21

n=6.36 x 109 cm-2

c=

0-2.27

B

=-3

=-2

FW

M S

pe

ctru

m (

arb

. un

its)

(-0)/

B

21

=0.235 ps

21=0.340 ps

21

=0.445 ps

21=0.550 ps

21

=0.655 ps

Origin of peak oscillations is quantum interference

2 1

THz k k 2, 2 1k k

2', k2'', 2 1k k

2 1

THz k k0THz

'.opt

.opt

2, 2 1k k

2', k

+ higher order processes

back to PSF † † †

'

, '

' '''' ' ''' ''' ' '' '''

† †' ''' '' '',1

'',1 ' '',1 1

If 0 , ' 0 , etc.

| '

' | '' '''

1 1 1 1 1 | ' '''

s

s s s

B B B

X

s s B B B s s s

Work on PSF in the exciton basis is in progress.

Summary

• Our model is a system of excitons described by and K, driven and scattered by E=D-P.

• Infinite order calculations retain exciton-exciton correlations and show observed oscillations due to internal field, P/.

• The chief merit of our approach is sufficient simplicity for numerical work and a direct connection to the physics.

Acknowledgements

• Collaborator: Marc Dignam, Queens University

• Financial support: NSERC Canada