in asymmetric nanostructures

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16 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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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 P
inter
.


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


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

Outline

Ultrafast experiments

k
1

(pump)

21

k
2

(probe)

FWM Signal

2
k
2
-

k
1

PP Signal

z

x

y

THz emission

SWM Signal, etc

2 1
3 2

k k
QW made asymmetric by
E
dc

E
dc

Energy or
frequency

w
c

Laser pulse

n=1

n=2

E
gap

+

-

G
(dipole mom.)

VB

CB

Biased SC Superlattice (BSSL)

energy or
frequency

w
2



1 0
'

intraband dipole:
'
;
(

)
e h
G e d G G
e

 

 


  
  
G r r

=2

d

E
dc

-

+

G
22

w
0


d


=2

d

E
dc


=0


=1


=
-
1

Bloch oscillations:
/
B dc
edE
w

frequency

0
B

w w w
 
w
B

(Stark ladder)

w
c

Laser pulse

<G>

-

+

-

-

Biased SC Superlattice (BSSL)

Bloch Oscillations

of dipole moment
(QM interference)

w
B

G
22

G
-
1
-
1

G
00

G
22

w
B

Exciton
: bound e and h in 2D
H
-
like

state, C of M wave vector
K

+

-

2a
0


=1

H
-
like binding
lowers



below
free e
-
h pair.


w
K

z

x,y

Basis {,} stands for {,H-like,,spin}.
 
K K
1s

w
c
=
w
0

Linear response (note H
-
like binding)

k
1
/k
2

interference: the polarization grating



13 by 2
intra
1 2
0
exp
m
m
im

 
  
 

P P k k R
2
p
/|
k
2
-
k
1
|

+ harmonics

z

x,y

k
2

2
k
2
-
k
1
=
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 K
K
Κ
K K
K K
P
P
P P
M
G
P
Inter and intraband polarization

PZW (multipolar) Hamiltonian
which we write as:



,
2


i
ex field
e
nt
in
x
t
field
V
V
H H H
H B B
H Kc a
H
H
a
  

 



w
 
   


  



 
K K K K
Κ Κ
Κ Κ
Κ
Κ Κ
Κ
D P P P
Dipole approximation

Hamiltonian

is
exact
,
P

is
approximate
,
includes self
-
energy
.









2
3
0
2
2
2
3
1
stationary dipole:
2

1
free dipole ~
self-energy negl
0
2
1
:
2 2
for N excitons if
igible
free.
d r ed
ed
ed
ed
V
N
d r
V




 





r r
EM field













,1

''''
2
2 2
2
dO
Heisenberg Picture: i
exp..
, (true bosons)
dynamics in
Using Heisenbergs twice:
,,, an
,
dt
cancels in

t
d
Kc
V
i a t i h c
a a
d
K K
dt
t t t
O H
 

  

 

 




  
 

 
 
 

K K
K
K K KK
Κ
Κ Κ
D e K R
D
D P
E R D R
D
P R
B
raband ,

for Kc>>
leaving

.

w

P
P
longitudinal/transverse P
intra

z

x

-
-
-
-
-
-
-

+
+
+
+
+
+
+

K

K
z

L

P
intra

L
~
.2

m

 >
1

m

K
z
>> K





intra
2 2
2
exp
sinc
K
K
z
L L
K L
P iKx z z
P
   
     
   
   

P z
z
K
z

p/
L

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

†';'†
'',',''''''''''''
'''''
,',
;''
';00 *''''''*
'''''
'
† †
,= - 2

B
B
B
B B
PSF
X
B X

   
 
    



  


 





 









KK
K
Κ K K Κ K
K
Κ Κ
k k
K k kK
K
k
k k
k
PSF

H
-
like excitons are

(approximate)

quasibosons.

+

-

k

-
k

eh
-
pair

+

-

H
-
like exciton

HP exciton dynamics


''';''
''

'';'''''''';''''



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


opt THz
S
S
dB
i
B PSF
PSF B PS
B
d
B
F
t

 
  

   

      
 

w
w




 

 
 
  
 
 
 
 
 
 

 
Κ Κ Κ
K
K
Κ Κ Κ Κ Κ
K
Κ
Κ
Κ
K K
K K
E
E
G
M
M
G
To solve numerically, must take
expectation value
.

inter
intra
Note that .

  
K
K K
K
D P
E P
PSF

~ n/n
0

n= exciton areal density =10
9

to 10
10
cm
-
2

n
0
= 1/
p
a
0
2

= 2x10
11
cm
-
2

n/n
0
< 0.1

Will omit PSF in numerical calculations here
.

(1)

(1)

'
2inter
1
1st order interband dynamics:

1
ext
opt
d B
i
i B
T
dt

  
w
 
    
 
 
Κ
Κ
E M
Can solve to any definite order in
E
opt



(2)

(2)

2intra
(1)
(1)
* †
(2) (2)
* † †
''''''
''
Can then get intraband dynamics:


2nd orde


r

ext
opt
ext
THz
d B B
i
i B B
dt T
B B
B B B B
 
   
   
     

w w
 
   
 
 
  
 
  
 
 

Κ P
Κ P
P
Κ
Κ P Κ Κ
K
E M M
E G G

(1)

'''
''
+
ext
THz
B
 



Κ
K
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 B
i B B
dt

    
  
w

 
   
 
 
 
Κ
Κ K Κ
K K K
E M G
Need infinite order, factored, like SBEs







* † †
''''''''
''''''
1
+ terms
d B B
i B B
dt
dB
B B B B
dt
 
   

     
 
w w

  
 
  
 
 
 
Κ P
Κ P
P
K P
Κ P Κ
K K
E M G


Retains exciton
-
exciton correlations, no biexcitons.

intr
inter
a
where .

  
K K
K K
P
D P
E



''
inte
''''''
'''',''''
r
1


+ higher order
i
d B
B
T
i B
dt

 
  
  
w

 
  
 
 
 
  
 
 
 
Κ
Κ
K
Κ
K K K
E M G
with phenomenological decay





* † †
''''''''
'''
2
'''
intra
1
+ terms
d B B
i B B
dt
dB
B B B B
dt
i
T
 
   

     
 
w w

 
   
 
 
 
  
 
 
 
Κ P
Κ P
P
K P
Κ P Κ
K K
E M G


Convergence: n
0
=3 (dash),
5(dot) and 13 (solid)

FWM

EWM

SWM











-2
-1
0
1
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3

=+1

=0

=-1
Spectrally-Resolved FWM Intensity
for Different time delays,

21


n=6.36 x 10
9
cm
-2
w
c
=
w
0
-2.27
w
B

=-3

=-2
FWM Spectrum (arb. units)
(
w
-
w
0
)/
w
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
w

k k
2
, 2


1
k k
2
',

k
2
'', 2


1
k k
2 1
THz
w

k k
0
THz
w
'
.
opt
w
.
opt
w
2
, 2


1
k 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