University of Groningen
Exciton superradiance in aggregates
Potma, E. O.; Wiersma, D. A.
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Journal of Chemical Physics
DOI:
10.1063/1.475898
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Potma, E. O., & Wiersma, D. A. (1998). Exciton superradiance in aggregates: The effect of disorder, higher
order exciton-phonon coupling and dimensionality.
Journal of Chemical Physics
,
108
(12), 4894-4903.
https://doi.org/10.1063/1.475898
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Exciton superradiance in aggregates: The effect of disorder, higher order
exciton-phonon coupling and dimensionality
Eric O. Potma and Douwe A. Wiersma
Ultrafast Laser and Spectroscopy Laboratory, Department of Chemical Physics, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands
~Received 19 September 1997; accepted 17 December 1997!
In this paper a detailed theoretical analysis is presented of the temperature dependent radiative decay
in aggregates of pseudoisocyanine ~PIC!. Our approach extends the original linear exciton-phonon
coupling model used by Spano, Kuklinsky, and Mukamel @Phys. Rev. Lett. 65, 212 ~1990!#
including static disorder and second order exciton-phonon interactions. It is shown that for a
one-dimensional exciton model neither of these additional effects alone or in combination with
linear electron–phonon coupling can explain the steep rise in radiative lifetime at 40 K observed in
the J-aggregate of PIC. However, when the aggregate assembles into a two-dimensional bricklike
structure its radiative dynamics can be simulated, with linear exciton-optical phonon coupling as the
only source for exciton scattering. Exciton-phonon scattering transfers oscillator strength from the
k5 0 state to other band states and also generates a nonequilibrium population among the exciton
states, which persists during the superradiant decay. These effects together explain the marked
temperature dependence of the radiative lifetime of the PIC J aggregate. When disorder limits the
coherence length at low temperatures to a few molecules, as seems the case in several light
harvesting complexes, the exciton population can equilibrate on the time scale of the superradiance.
This situation pertains to the strong collision limit of the master equation, where the radiative decay
is insensitive to details of the electron–phonon coupling, but only senses change in the thermal
population among the exciton states. © 1998 American Institute of Physics.
@S0021-9606~98!50912-9#
I. INTRODUCTION
One of the most fascinating optical properties of dipolar
coupled molecular assemblies is their enhanced rate of spon-
taneous emission compared to that of the monomer.
1,2
This
effect, which is due to the collective nature of the excited
states on the aggregate, is known as superradiance.
3,4
A fa-
mous example is the J aggregate of the dye pseudoisocya-
nine, where the low-temperature enhancement factor is about
100.
2,5
For a linear chain this number may be interpreted as
the number of molecules over which the optical excitation is
delocalized. Excitation delocalization leads also to a narrow-
ing and a shift of the aggregate’s absorption spectrum ~J
band! compared to the monomer.
6,7
Calculations show that
the linewidth of the excitonic J band is expected to be nar-
rower than that of the monomer by about the square root of
the number of coherently coupled molecules.
8
This line nar-
rowing effect is reminiscent of the well-known motional-
narrowing effect in nmr.
9
The spectral shift of the aggregate
absorption can be either to the blue or to the red, depending
on whether the dipolar coupling term has a positive or nega-
tive sign. The van der Waals shift also contributes to the
displacement of the aggregate absorption with respect to the
monomer.
In the past decade the theoretical description of the line
shape and radiative dynamics of J aggregates was based on
the one-dimensional Frenkel exciton model.
10–13
Diagonal
and off-diagonal disorder and exciton-phonon scattering
were shown to be particularly important in determining the
line shape and temperature dependence of the radiative dy-
namics in these systems.
14–17
Although many optical proper-
ties of J aggregates could be simulated well on basis of this
linear chain model, the recent interpretation of pump–probe
experiments, involving single- and two-exciton states, raised
doubt on one of the previous conclusions that diagonal
disorder in these self-assembled J aggregates is uncor-
related.
18,19
An alternative conclusion might be that the one-
dimensional exciton model is not applicable for these aggre-
gates.
Aggregates also play an important role as light-
harvesting complexes in photobiology. Here their function is
to transfer the captured sunlight to the photosynthetic reac-
tion center as efficiently as possible. In the past the question
has often been raised to what extent this energy transfer pro-
cess proceeds via excitonic motion ~coherent energy transfer!
or via a site-to-site hopping process ~incoherent energy
transfer!.
20–22
Now that the spatial structure of some of these
light-harvesting antennas has been resolved,
23–25
showing
that these systems are truly one-dimensional circular chains,
a direct confrontation between Frenkel exciton theory, with
disorder and exciton-phonon scattering incorporated, and ex-
periment is possible. Key observables in these systems are
spectral line shape, homogeneous linewidth, and radiative
lifetime. Recently it has been shown that many of the con-
cepts developed for J aggregates also apply well for these
light-harvesting complexes.
26,27
When the antennea systems
are compared to PIC a major difference is the magnitude of
JOURNAL OF CHEMICAL PHYSICS VOLUME 108, NUMBER 12 22 MARCH 1998
48940021-9606/98/108(12)/4894/10/$15.00 © 1998 American Institute of Physics
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disorder, which is much larger in the antenna systems. This
limits the exciton delocalization length and thus the impor-
tance of coherent energy transfer in these biological aggre-
gates.
Recently Higgins et al.
28
performed near-field imaging
experiments on aggregates of PIC which confirmed the elon-
gated structure of the J aggregate, but at the same time
showed that on a microsopic scale the aggregates are not
one-dimensional chains but rather fibrous strands woven into
a three-dimensional network. Of course, from an optical
point of view the aggregates may still behave like a one-
dimensional chain, but such a simple description can no
longer be taken for granted. The interpretation of the pump–
probe spectra and the modeling of the temperature dependent
superradiant lifetime ~vide infra!, hereto based on a one-
dimensional exciton model, thus should be questioned.
In this paper we return to the theoretical description of
superradiance in aggregates. De Boer and Wiersma showed
that the fluorescence lifetime of PIC aggregates in a water/
ethylene glycol glass is about 30 ps at low temperature, but
rapidly increases at temperatures above 50 K to a few hun-
dred ps at room temperature. A similar effect was observed
in other aggregates,
29
in light harvesting antennas
30
and in
excitonic systems like multiple quantum wells ~MQWs!.
31
This dramatic shortening of the low-temperature fluores-
cence lifetime in these excitonic systems is attributed to co-
operative emission known as superradiance. The lengthening
of the superradiant lifetime at higher temperatures is gener-
ally attributed to the fact that the exciton scatters during its
lifetime to other ~dark! states in the exciton band. When the
excitons are assumed to be in thermal equilibrium with the
phonon bath, a one-dimensional model does not yield a
proper description of the temperature dependent lifetime of
PIC.
2
A much better description of superradiance quenching
was given by Spano et al., who included explicitly the
exciton-phonon interaction in their calculations.
16,17
Re-
cently, however, Fidder et al. reported their final analysis of
the temperature dependent radiative lifetime, taking into ac-
count a temperature dependent fluorescence quantum yield.
5
This correction of the data leads to a much steeper activation
of the superradiant lifetime, and this raises the question
whether the Spano et al. model can still capture these results.
In this paper we discuss the temperature dependence of
the coherence length starting with the one-dimensional
model discussed by Spano et al. The model Hamiltonian for
this case is presented in Sec. II. In Sec. III we show that the
existing theory fails to describe the radiative lifetime of the J
aggregate at higher temperatures. Several additional quench-
ing mechanisms are considered, including the contribution of
static inhomogeneities and anharmonic exciton-phonon scat-
tering processes. Although their effects on the radiative life-
time are pronounced, the fit to the experimental lifetimes is
not improved. In Sec. IV we show that a much improved fit
of the data can be achieved when the J aggregate is assumed
to be a two-dimensional system. In Sec. V we contrast su-
perradiance in J aggregates and light-harvesting complexes.
In the last section we summarize our findings.
II. MODEL CONSIDERATIONS
In describing the radiative properties of the aggregate, an
effective Hamiltonian is used, which includes radiative
damping of the lowest exciton state.
4,17
With periodic bound-
ary conditions, the Hamiltonian of an aggregate consisting of
N molecules is given by (\5 1)
H
ex
5
(
k50
N21
H
v
~
k
!
1i
N
2
g
d
k,0
J
B
k
†
B
k
. ~1!
Here B
k
†
and B
k
are the creation and annihilation operators of
a Frenkel exciton with wave vector k, while the exciton dis-
persion is given by
v
~k!. The second term in this Hamil-
tonian accounts for the superradiant decay of the k5 0 exci-
ton with an emission rate N
g
, with
g
denoting the decay rate
of the monomer. When exciton-phonon and exciton-disorder
scattering terms are incorporated into the Hamiltonian the
coherence length of the optical excitation is generally re-
duced. To account for this effect, the superradiant decay is
defined in terms of an effective rate N
coh
g
, where N
coh
is
the number of coherently coupled molecules. In principle,
the coherence length of an exciton can be affected by both
diagonal and off-diagonal disorder. Diagonal disorder is
characterized by a molecular offset frequency from the aver-
age transition frequency
v
0
. Off-diagonal disorder comes
from a variation in the dipolar coupling strength J
nm
between
molecules n and m. Both types of disorder tend to localize
the excitation. Disorder in the aggregate system origins from
static as well as from dynamical contributions. The latter can
be modeled by coupling the exciton to a phonon bath.
32
When both static diagonal disorder and exciton-phonon in-
teraction are taken into account, the total Hamiltonian be-
comes
H5 H
ex
1H
ph
1H
ex-ph
1H
ex-dis
, ~2!
where H
ex-dis
and H
ex-ph
represent the change in energy due
to scattering of excitons on static disorder impurities and
phonons, respectively. The phonon energies are given by
H
ph
. In the momentum representation, the explicit forms can
be written as
H
ph
5
(
s,q
V
s
~
q
!
$
b
q,s
†
b
q,s
11/2
%
,
H
ex-ph
5
1
A
N
(
s,q,k
F
s
~
k,q
!
B
k1q
†
B
k
~
b
q,s
1b
2q,s
†
!
, ~3!
H
ex-dis
5
1
A
N
(
k,k
8
F
dis
~
k
8
2k
!
B
k
†
B
k
8
.
The operators b
q,s
†
(b
q,s
) create ~annihilate! a molecular vi-
bration with wave vector q. The energy of the phonon modes
is given by V
s
(q), where s specifies the type of phonon
branch. The function F
s
(k,q) represents the scattering
strength of the kth exciton on phonon q, F
dis
(k) stands for
the corresponding process involving disorder. The Hamil-
tonian displayed in Eq. ~3! is similar to the one used by
Spano et al.
16,17
in their microscopic description of superra-
4895J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 E. O. Potma and D. A. Wiersma
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diance quenching by lattice vibrations. The only difference is
the last term H
ex-dis
, which describes the contribution of
static diagonal disorder to exciton scattering.
To calculate the spontaneous emission lifetime of an ag-
gregate described by the Hamiltonian in Eq. ~3! we use the
same method as Spano et al.
17
Their approach consists in
calculating the time envelope of the averaged exciton popu-
lation of the lowest state (k5 0). The expectation value of
the exciton population operator B
k
†
B
k
is evaluated by insert-
ing it into the Heisenberg equation of motion. The infinite
hierarchy of equations is truncated at second order by impos-
ing a factorization approximation for the expectation values.
We arrive at the following ~non-Markovian! master equation:
]
^
B
k
†
B
k
~
t
!
&
]
t
52N
g
d
k,0
^
B
k
†
B
k
~
t
!
&
2
(
q
E
0
t
K
phon,dis
~
k,q;t2t
8
!
~
^
B
k
†
B
k
~
t
8
!
&
2
^
B
k1q
†
B
k1q
~
t
8
!
&
!
dt
8
. ~4!
The first term on the right-hand side of Eq. ~4! describes the
superradiant decay of the k5 0 exciton population, the sec-
ond term accounts for the propagation of the k-state popula-
tion, which is affected by scattering processes, while the last
term (
^
B
k1 q
†
B
k1 q
&
) describes the propagation of all k
8
Þk
populations. The evolution of these populations is dictated
by the kernel K
phon,dis
which is given in Appendix A. Ac-
cording to equation Eq. ~4!, the initial exciton population,
which is assumed to reside in the lowest exciton level, is
subject to upward and downward scattering by phonons and
static disorder.
For a homogeneous circular exciton all oscillator
strength is accumulated in the optical transition from the
ground to the k5 0 state. When static disorder and exciton-
phonon coupling are taken into account, the k5 0 state no
longer is an eigenstate of the ‘‘dressed’’ exciton system. Pro-
jected onto the exciton space this means that oscillator
strength is transferred from the superradiant k50 state to
other, originally dark, kÞ0, exciton states. Since the radia-
tive decay time is directly related to the oscillator strength of
the k5 0 transition, disorder- and phonon-induced exciton-
scattering processes thus quench the excitonic superradiant
decay. Physically this can also be interpreted as a reduction
of the coherence length (N), the length over which the op-
tical excitation is delocalized.
The temperature dependence of the superradiant lifetime
enters the equations in the form of the averaged phonon oc-
cupation number, which equals a Bose–Einstein distribution
~see Appendix A!. With increasing temperature, the average
number of excited phonons increases and consequently the
probability that excitonic population is being scattered by
phonons rises. In the description given above, however, only
linear interactions between the exciton and the phonon bath
are included. Higher order exciton-phonon interactions are
expected to give rise to a different temperature dependence.
In order to study the effects of higher order electron-phonon
couplings we also considered the second-order interaction of
excitons with phonons. This additional higher order exciton-
phonon coupling contribution to the Hamiltonian can be ex-
pressed as
H
ex-ph
~
2
!
5
1
A
N
(
k,q,q
8
F
~
2
!
~
k,q,q
8
!
B
k1q1q
8
†
B
k
~
b
q
1b
2q
†
!
3
~
b
q
8
1b
2q
8
†
!
. ~5!
The function F
(2)
(k,q,q
8
) describes the second order
exciton-phonon interaction and is derived in Appendix B. In
the second order scattering process the wave vector of the
exciton is changed due to phonon scattering involving two
modes at the same time. The resulting distribution of popu-
lation among band states is thus expected to be different
from that arising from linear electron-phonon coupling. Dis-
regarding the coupling between linear and higher order
exciton-phonon scattering processes, the equation of motion
for exciton phonon interactions can be completed by adding
the following term to the r.h.s. of Eq. ~4!
2
(
q,q
8
E
0
t
K
~
2
!
~
k,q,q
8
;t2 t
8
!
~
^
B
k
†
B
k
~
t
8
!
&
2
^
B
k1 q1 q
8
†
B
k1 q1 q
8
~
t
8
!
&
!
dt
8
. ~6!
The kernel K
(2)
describes the evolution of the exciton popu-
lation under the effect of second order phonon scattering
events; its explicit form can also be found in Appendix B.
While an analytical solution of the integro-differential
equation ~4! is not possible, a numerical solution can be
found. The calculations were performed using the so-called
coarse grained approximation ~CGA!; for details of this
method we refer to Ref. 17.
III. ONE-DIMENSIONAL AGGREGATES
Let us start by considering a one-dimensional circular
chain, in which the transition dipoles of all molecules are
aligned at an angle
f
with the aggregate axis. The equilib-
rium molecular positions are assumed to be fixed.
When only nearest neighbour coupling is taken into ac-
count, the excitonic energies are given by
v
~
k
!
5
v
0
2 2J cos
S
2
p
k
N
D
,
where J denotes the nearest neighbour dipole–dipole cou-
pling energy.
A. Phonon-induced dephasing
In the absence of diagonal disorder and higher order
phonon-scattering, the radiative decay of the exciton is solely
determined by linear exciton-phonon scattering. In a one-
dimensional chain, there is only a single acoustical- and
optical-phonon branch for the exciton to interact with and
which can induce optical dephasing processes. If coupling to
the low-energy acoustic phonon branch were important, the
excitons fluorescence lifetime should already be affected at
very low temperatures. Experimentally the J-aggregate’s ra-
diative lifetime is constant up to ;30 K, so exciton-acoustic
phonon coupling can be ruled out. We therefore restrict our
4896 J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 E. O. Potma and D. A. Wiersma
Downloaded 06 Feb 2006 to 129.125.25.39. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
discussion to the effect of exciton-optical phonon scattering
on superradiance. The relevant lattice vibrations involve col-
lective angular changes in the orientation
f
of the transition
dipoles.
Figure 1 displays a plot of the relative radiative lifetime
as reported by Fidder et al.
5
The dashed line presents the
result of a calculation which assumes PIC to be a one-
dimensional chain and for Boltzmann equilibrium among the
exciton states. This model clearly fails to capture the steep
rise in radiative lifetime above 40 K. The solid line corre-
sponds to the best fit obtained by a calculation based on Eq.
~4! and by employing the CGA approximation. The tempera-
ture T
*
at which a break occurs in the radiative dynamics is
mainly determined by the frequency of the optical mode
V
op
; varying the coupling strength F
op
affects only the scat-
tering efficiency. In the calculations the aggregate is assumed
to comprise about 100 molecules, yielding the lifetime mea-
sured below 30 K. For an optical phonon frequency (V
op
)of
150 cm
21
and a scattering strength (F
op
) of 250 cm
21
, the
temperature dependence of the radiative decay time can be
satisfactorily described for temperatures up to 80 K. How-
ever, at higher temperatures the calculated lifetimes deviate
strongly from the experimental ones. Instead of a steep in-
crease in radiative lifetime with rising temperature, the slope
of the calculated curve decreases for temperatures above 80
K. In this temperature range the population of the lowest
exciton level is not much changed when the temperature in-
creases. This implies a balancing between the upward and
downward scattering processes of the excitonic population in
the lowest band states. Increase of the coupling parameter
F
op
or increasing the number of active phonons does not
improve the quality of the fits. Clearly, the model cannot
capture the temperature dependence of the radiative lifetime
of the J aggregate above 80 K.
B. Diagonal static disorder
Several studies have shown that disorder can have a dra-
matic effect on the excitonic states on the aggregate.
15,33
One
of the important effects of disorder is that it leads to local-
ization of the optically active states near the band edges. In
the current model disorder can be viewed as yet another
source for exciton scattering, causing the initial exciton
population to be redistributed over the lowest few states. As
a result the lifetime of the ‘‘k5 0’’ exciton is lengthened
indicating a reduction of the coherence length on the aggre-
gate. The disorder induced exciton scattering probability is,
however, temperature independent. When the combined ef-
fect of diagonal disorder (H
ex-dis
) and phonon scattering
(H
ex-ph
) is considered, additional contributions appear in the
scattering kernel ~see Appendix A!. These extra terms do
have temperature dependent components that go with the
square root of the phonon occupation number.
Figure 2 displays the combined effect of static disorder
and exciton-phonon coupling on the radiative lifetime of an
one-dimensional aggregate. The first thing to note is that
with increasing disorder the superradiant lifetime lengthens.
This effect results from disorder spreading the oscillator
FIG. 1. Relative radiative lifetimes of PIC aggregates in a water/ethylene
glycol glass. Triangles indicate experimental data, taken from Ref. 5; the
solid line is calculated for an aggregate using N5 100, F
op
5250 cm
21
and
V
op
5150 cm
21
. The dotted curve is the predicted temperature dependence
for a one-dimensional chain where the exciton population is in Boltzmann
equilibrium. The dipole–dipole coupling for the PIC aggregate is taken to be
J5 600 cm
21
and the monomer lifetime is taken to be 3.7 ns.
FIG. 2. Effect of static disorder impurities on exciton-phonon radiative dy-
namics. From top to bottom; D5 30,20,10,0 cm
21
, the aggregate length N
5 75 and is coupled to a phonon V
op
50.15J with strength F
op
50.1J. The
curves are averaged over 100 realizations of the disorder parameter. Param-
eters of PIC are used in the calculation. Triangles indicate experimental
relative radiative lifetimes of PIC.
4897J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 E. O. Potma and D. A. Wiersma
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