in asymmetric nanostructures

skillfulbuyerUrban and Civil

Nov 16, 2013 (3 years and 11 months ago)

82 views

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