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
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