Polarization driven exciton dynamics in asymmetric nanostructures
description
Transcript of Polarization driven exciton dynamics in asymmetric nanostructures
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