Excitonic enhancement of nonradiative energy transfer
from a quantum well in the optical near field of energy gradient
quantum dots
Sedat Nizamoglu1,2, Pedro Ludwig Hernández-Martínez1,3, Evren Mutlugun1,3, Durmus
Ugur Karatay1, and Hilmi Volkan Demir1,3*
1
Department of Electrical and Electronics Engineering, Department of Physics,
UNAM — National Nanotechnology Research Center, and Institute of Materials Science
and Nanotechnology, Bilkent University, Ankara, Turkey TR-06800
2
Wellman Center, Harvard Medical School, 65 Landsdowne Street, Cambridge,
Massachusetts 02139, United States
3
Luminous! Center of Excellence for Semiconductor Lighting and Displays, School of
Electrical and Electronic Engineering, School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 639798, Singapore
Theoretical Discussion
The excitonic energy transfer in the hybrid nanostructure was model considering the properties
of individual NQDs. The important parameters include the exciton recombination rate,    1  ;
the sublevel relaxation rate,  ; and the interaction rate, U ij . Here,   is the intrinsic exciton
lifetime.  denotes the nanocrystal’s type, which in our case can be quantum well, (QW); greenemitting nanocrystal quantum dot, (NQDG); or red-emitting nanocrystal quantum dot, (NQDR).
Here we start with our theoretical analysis and numerical results for the excitation transfer in the
case of a QW to a bilayer of red-emitting NQDs (Case I: QW  NQDR  NQDR ) and then
continue with the case of a QW to a layer of green-emitting NQD and a second layer of redemitting NQDs (Case II: QW  NQDG  NQDR ).
Case I: QW  NQDR  NQDR
In this case, we consider the excitation energy transfer between a QW to a bilayer of red-emitting
NQDs. Letting  AB be the density matrix corresponding to the energy level E1 (Fig. 1) then, the
master equation is given by
* Email: volkan@stanfordalumni.org
 AB
i
  VNF , AB ,  AB    N AB  AB   AB N AB 
t
(1)
where N AB is a diagonal matrix whose diagonal elements are given by   and  . VNF , AB is the
interaction between levels with energy E1 given by U AB . The master equation, corresponding to
the density matrix  BC with energy E2, can be written as
 BC
i
  VNF ,BC ,  BC    N BC  BC   BC N BC   P   AB 
t
(2)
where P   AB  represents the relaxation from the energy level E1 to E2; VNF ,BC and N BC are
defined similar in Eq. (1).
VNF,12
E1
NQD1
E2
QW
Vvib
VNF,23
NQD1
NQD2
Fig. 1: Energy diagram for the excitation transfer from a QW to a bilayer of red-emitting NQDs (Case I). Horizontal
arrows represent the interaction energy between the energy levels. Vertical arrow shows relaxation process between
the excited levels. Dash vertical arrows depict the relaxation between the lowest excited state and ground state.
Case II: QW  NQDG  NQDR
In the following, we consider the excitation transfer from a QW to a layer of green-emitting
NQDs and a second layer of red-emitting NQDs. Similar to the previous case,  AB is the density
matrix corresponding to the energy level E1 (Fig. 2). Here, the master equation is given by
 AB
i
  VNF , AB ,  AB    N AB  AB   AB N AB 
t
(3)
where N AB is a diagonal matrix whose diagonal elements are given by   and  . VNF , AB is the
interaction between levels with energy E1 given by U AB . The master equation, corresponding to
the density matrix  BC with energy E2, can be written as
 BC
i
  VNF ,BC ,  BC    N BC  BC   BC N BC   P   AB 
t
(4)
where P   AB  represents the relaxation form the energy level E1 to E2; VNF ,BC and N BC are
defined similar to Eq. (3). Finally, the master equation for the density matrix  C with energy E3
is written as
 C
   N C  C   C N C   P   BC 
t
(5)
where P   BC  represents the relaxation from the energy level E2 to E3; and N C is a diagonal
matrix with matrix elements   .
VNF,12
E1
NQD1
Vvib,1
E2
VNF,23
QW
E3
NQD1
NQD2
Vvib,2
NQD2
Fig. 2: Energy diagram for the excitation transfer from a QW to a layer of green-emitting NQDs and a layer of redemitting NQDs (Case II). Horizontal arrows represent the interaction energy between the energy levels. Vertical
arrows show relaxation process between the excited levels. Dash vertical arrows depict the relaxation between the
lowest excited state and ground state.
Numerical Results
Here, we calculate the exciton population for the complexes (Case I and II) described in the previous
section. For the NQDs we assume:  NQD1   NQD 2  5 1ns and  NQD1   NQD 2  101ps . For the QW we use:
 QW  1.321 ns . For the interaction between the nanocrystal we take U AB  U BC  1001 ps .1 Next, we
calculate the exciton population for each complex. In case I, we use the following matrices:
 11AB
 AB  
  21
AB
12AB 

 22AB 
 0
VNF , AB  
 U AB
N
AB

1   QW
 
2 0
 11BC
 BC  
  21
U AB 
0 
0 
 NQD1 
  NQD1 22AB
AB


P    
0

 0
VNF ,BC  
 U BC
N
0

0
BC
12BC 

 22BC 
BC

1   NQD1
 
2 0
U BC 
0 
0 
 NQD 2 
(6)
where 11AB represents the exciton population of the first excited state in the QW,  22AB is the exciton
population of the second excited state of the first NQD layer, 11BC represents the exciton population of
BC
the first excited state in the first NQD layer, and  22
is the exciton population of the first excited state of
the second NQD layer. Numerical results for the exciton population are shown in Fig. 3. From this plot, it
is observed that it takes about 1 ns for an exciton in the QW to be transfer to the second NQD layer.
Moreover, the nutation process between the first and second NQD layer is observed; i.e., the exciton
oscillates between the first excited state of both NQDs because their energy levels are in resonance. This
is expected due to the fact that both layers are made of the same NQDs (in this case red-emitting NQDs).
The transfer efficiency from the QW to the last layer of NQDs is about 45%, as depicted in Fig. 3 (Inset).
This result agrees with the experimental value of 50.7%. In case II, the used matrices are:
 11AB
 AB  

AB
21
12AB 

 22AB 
 0
VNF , AB  
 U AB
N
AB

1   QW
 
2 0
 11BC
 BC  

U AB 
0 
0 
 NQD1 

 AB
P   AB    NQD1 22
0

 0
VNF ,BC  
 U BC
N
0

0
BC
21
12BC 

 22BC 
BC

1   NQD1
 
2 0
 C
C  
 0
U BC 
0 
0 
 NQD 2 

 BC
P   BC    NQD 2 22
0

NC 
0

0
1   NQD 2
2  0
0
0 
(7)
0

0
where  C is the exciton population of the first excited state of the second NQDs layer and the other
parameter are defined similar to case I. Numerical results are shown in Fig. 4. From this figure, it is
observed that it takes about 1 ns for an exciton in the QW to be transferred to the red-emitting NQD layer,
similar to the previous case. Also, this plot shows the exciton population of the intermediate states. The
transfer efficiency from the QW to the second layer is about 70%, as depicted in Fig. 4 (Inset). This result
also agrees with the experimental value of 83.3%. The advantage of the energy gradient NQDs system is
that the exciton is funneling to the lowest energy state via near field interaction. This process is
unidirectional because of the fast interband relaxation between the energy levels avoiding the nutation
process between the dots. Therefore, the probability of exciton transfer to the first excited state of the
second NQD from the other dot could significantly be improved. In summary, the results suggested that
in order to have a better exciton transfer from the QW to the last NQD layer, we need to have an energy
difference between the NQD layers.
1.0
AB11 QW. 1st Excited state
AB22 1st Red NQD. 2nd Excited state
BC11 1st Red NQD. 1st Excited state
BC22 2nd Red NQD. 1st Excited state
0.5
0.6
0.4
Exciton population
Exciton population
0.8
0.4
0.2
0.3
0.2
0.1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Time (ns)
0.0
0
2
4
6
8
10
12
14
16
Time (ns)
Fig. 3: Calculated exciton population for Case I of energy transfer from a QW to a bilayer of red-emitting NQDs.
Red line shows the exciton population of the first excited state in the QW. Cyan line depicts the exciton population
of the second excited state for the first red-emitting NQD layer; green line shows the exciton population of the first
excited state of the first red-emitting NQD layer; and the blue line depicts the exciton population of the first excited
state of the second red-emitting NQD layer. Inset: Zoom-in for this plot.
1.0
AB11 QW. 1st Excited state
AB22 Green NQD. 2nd Excited state
BC11 Green NQD. 1st Excited state
BC22 Red NQD. 2nd Excited state
C Red NQD. 1st Excited state
0.8
0.6
Exciton population
Exciton population
0.8
0.4
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Time (ns)
0.0
0
2
4
6
8
10
12
14
16
Time (ns)
Fig. 4: Calculated exciton population for Case II of energy transfer from a QW to a layer of green-emitting NQD
and a second layer of red-emitting NQDs. Red line shows the exciton population of the first excited state in the QW.
Cyan line depicts the exciton population of the second excited state for the green-emitting NQD layer; green line
shows the exciton population of the first excited state of the green-emitting NQD layer; dark yellow line depicts the
exciton population of the second excited state of the red-emitting NQD layer; and the blue line shows the exciton
population of the first excited state of the red-emitting NQD layer. Inset: Zoom-in for this plot.
1
M. Naruse, E. Runge, K. Kobayashi, and M. Ohtsu, Phys. Rev. B 82, 125417 (2010).