The study of energy harvesting in chain-like structures is important due to
its relevance to a variety of interesting physical systems. Harvesting is
understood as the combination of exciton transport through intra-band exciton
relaxation (via scattering on phonon modes) and subsequent quenching by a trap.
Previously, we have shown that in the low temperature limit different
harvesting scenarios as a function of the applied bias strength (slope of the
energy gradient towards the trap) are possible \cite{Vlaming07}. This paper
generalizes the results for both homogeneous and disordered chains to nonzero
temperatures. We show that thermal effects are appreciable only for low bias
strengths, particularly so in disordered systems, and lead to faster
harvesting.Comment: 8 pages, 2 fugures, to appear in Journal of Luminescenc

Abrahams

Bednarz

Crooker

Franzl

Heilemann

J. Knoester

Knoester

Malyshev

Mukai

S.M. Vlaming

Tilgner

V.A. Malyshev

Wagner

English

arXiv

Crossref

The study of energy harvesting in chain-like structures is important due to its relevance to a variety of interesting physical systems. Harvesting is understood as the combination of exciton transport through intra-band exciton relaxation (via scattering on phonon modes) and subsequent quenching by a trap. Recently, we have shown that in the low temperature limit different harvesting scenarios as a function of the applied bias strength (magnitude of the energy gradient towards the trap) are possible [S.M. Vlaming, V.A. Malyshev, J. Knoester, J. Chem. Phys. 127 (2007) 154719]. This paper generalizes the results for both homogeneous and disordered chains to nonzero temperatures. We show that thermal effects are appreciable only for low bias strengths, particularly so in disordered systems, and lead to faster harvesting. (C) 2007 Elsevier B.V. All rights reserved.</p

Vlaming, S. M.

Malyshev, V.A.

Knoester, J.

University of Groningen

   University of GroningenThermal effects in exciton harvesting in biased one-dimensional systemsVlaming, S. M.; Malyshev, V.A.; Knoester, J.Published in:Journal of LuminescenceDOI:10.1016/j.jlumin.2007.11.038IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2008Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vlaming, S. M., Malyshev, V. A., & Knoester, J. (2008). Thermal effects in exciton harvesting in biased one-dimensional systems. Journal of Luminescence, 128(5-6), 956-959.https://doi.org/10.1016/j.jlumin.2007.11.038CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019ARTICLE IN PRESS0022-2313/$ - sedoi:10.1016/j.jluCorrespondE-mail addrJournal of Luminescence 128 (2008) 956–959www.elsevier.com/locate/jluminThermal effects in exciton harvesting in biased one-dimensional systemsS.M. Vlaming, V.A. Malyshev, J. KnoesterCentre for Theoretical Physics and Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The NetherlandsAvailable online 20 February 2008AbstractThe study of energy harvesting in chain-like structures is important due to its relevance to a variety of interesting physical systems.Harvesting is understood as the combination of exciton transport through intra-band exciton relaxation (via scattering on phononmodes) and subsequent quenching by a trap. Recently, we have shown that in the low temperature limit different harvesting scenarios asa function of the applied bias strength (magnitude of the energy gradient towards the trap) are possible [S.M. Vlaming, V.A. Malyshev, J.Knoester, J. Chem. Phys. 127 (2007) 154719]. This paper generalizes the results for both homogeneous and disordered chains to nonzerotemperatures. We show that thermal effects are appreciable only for low bias strengths, particularly so in disordered systems, and lead tofaster harvesting.r 2007 Elsevier B.V. All rights reserved.PACS: 71.23.An; 71.35.CcKeywords: Frenkel excitons; Exciton transfer; Exciton trapping; Energy bias; Photonic wires1. IntroductionThe study of energy transport and trapping in molecularscale systems is an interesting topic, because of its relevanceto natural systems (photosynthetic antenna complexes [1])and applications in nanophysics and quantum computing,where energy should efﬁciently be transported oversome distance. Linear molecular aggregates [2,3], conju-gated polymers [4,5], photonic wires [6,7] and quantumdot arrays [8,9] may all be envisioned as realizations ofsuch energy transport complexes. A natural question thatarises is how this combined process of transport andeventual trapping, which we will from now on refer toas harvesting, can be optimized. In a previous paper [10],we performed a study on the effect of an energy gradient(bias) on the energy harvesting properties of a linearsystem. The scope there was limited to low temperatures.In this paper, we present numerical calculations of thermaleffects on the harvesting time and a qualitative analysis ofthe results.e front matter r 2007 Elsevier B.V. All rights reserved.min.2007.11.038ing author. Tel.: +3150 3634369; fax: +31 50 3634947.ess: j.knoester@rug.nl (J. Knoester).2. ModelWe use a Frenkel exciton model to describe the excitedstate dynamics in chain-like systems. The latter aremodeled as a regular one-dimensional array of two-levelmonomers that are coupled through their transition dipolemoments, which are assumed to be oriented parallel toeach other. We initially excite a monomer at one end of thechain ðn ¼ NÞ, and consider the other end ðn ¼ 1Þ as a trap.The harvesting time is calculated as the average time ittakes for an excitation to move across the chain and to bequenched by the trap. Including diagonal disorder and abias ðD40Þ towards the trap, the model Hamiltonian readsHex ¼XNn¼1½en þ ðn 1ÞDjnihnj þXNn;manJnmjnihmj, (1)where the state jni denotes that monomer n is excited whileall other monomers are not, the excitation energies en areuncorrelated Gaussian stochastic variables with zero meanand standard deviation s and the transfer integrals areJnm ¼ J=jnmj3, with the nearest-neighbor couplingJ40. Diagonalization of this Hamiltonian yields collectiveexcited states, jsi ¼Pn csnjni, where the site amplitudescan be chosen real. Both a bias and disorder result inARTICLE IN PRESS0.1 0.050|cs1|220 4010 30 5000.10.05state number, s|csN|2Δ=0Δ=0.01JΔ=0.1JΔ=0Δ=0.01JΔ=0.1JFig. 1. Bias dependence of the exciton probabilities c2sN and c2s1 thatdetermine the initial distribution of the exciton population Psð0Þ and thequenching rate Gs, respectively, for chains without disorder ðs ¼ 0Þ.The bias tends to shift the initial distribution of population to the top ofthe band (to higher s). Oppositely, the strongly quenched states occur atthe lower band edge (for small s).S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 957localization of the states, Bloch-like [11] and Anderson-like[12], respectively, so that the exciton states are spread outover ﬁnite segments of the chain only. In the case of a bias,there exists a correlation between the location of theeigenstate and its energy, as lower energy states are locatedin the vicinity of the trap.The dynamics of the populations Ps of the variousexciton states is analyzed within the framework of the Paulimaster equation,_Ps ¼ ðgs þ GsÞPs þXs0ðWss0Ps0 Ws0sPsÞ, (2)where gs ¼ g0ðPncsnÞ2 is the radiative rate of the state jsi(g0 being the radiative rate of a monomer) and Gs is thequenching rate, which is taken proportional to the excitonprobability at the trap ðn ¼ 1Þ: Gs ¼ Gc2s1 where G is thequenching amplitude [13], which cannot be arbitrarily largewithin the limitations of our model [10]. As at t ¼ 0 onlythe rightmost site n ¼ N is excited, the initial populationdistribution is given by Psð0Þ ¼ c2sN . The Wss0 describe thescattering of population between various exciton statesthrough interaction with phonons: Wss0 ¼ W 0 SðEs  Es0 ÞPNn¼1c2snc2s0nGðEs  Es0 Þ, where W 0 is the scattering ampli-tude, SðEÞ is the phonon spectral density function,PNn¼1c2snc2s0n is the probability overlap, and GðEÞ ¼ 1þnðEÞ if Eo0, while GðEÞ ¼ nðEÞ if E40 with nðEÞ ¼½expðjEj=TÞ  11 the phonon occupation factor. We use aDebye-like spectral density with cut-off oc,SðEÞ ¼ jE=Jj3 expðjEj=ocÞ. The harvesting time is de-ﬁned as follows: t1 ¼ t1trap  t10 , where ttrap and t0 are thedecay times with and without trap, respectively. They arecalculated asR10dthPsPsðtÞi, where h. . .i denotes thedisorder average.3. Disorder-free biased chainsWe begin with a discussion of the harvesting efﬁciency inhomogeneous chains ðs ¼ 0Þ. The bias-induced Blochlocalization dramatically changes the harvesting properties.Depending on the relative strengths of the quenching andrelaxation processes, both monotonic decreasing andnonmonotonic behavior of the harvesting time as afunction of the bias strength can be obtained. We refer toour earlier paper [10] for a full discussion of the possiblescenarios, while here we focus on the aspects relevant tounderstanding thermal effects in our model.In the limit of G5W 0 and for low temperature,relaxation to lower exciton states is the dominant processfor virtually all exciton states. As a result, most populationends up in the bottom states of the band. For zero bias, thequenching rate from these states is low because of the pooroverlap of the bottom states with the trap (see Fig. 1).However, turning on the bias increases this overlap,thereby accelerating the harvesting process. Temperaturewill enhance the harvesting efﬁciency at low bias by shiftingpopulation from the poorly quenched bottom state(s) tomore strongly quenched higher energy states. This en-hancement disappears for higher bias strengths, as thehigher energy states are no longer more strongly quenchedthan the bottom states, as is shown in Fig. 2(a).The interesting situation where GW 0 provides anonmonotonic bias dependency of t. A small bias isadvantageous for the harvesting process as it increases thequenching rate for the bottom states, but beyond someoptimal bias value the increase in relaxation time overtakesthe previous effect in importance and the net effect ofincreasing the bias becomes negative. The thermallyinduced changes in the harvesting efﬁciency are again onlypresent for small values of the bias, for the reasonsmentioned in the discussion of the limit of G5W 0. This iscorroborated in Fig. 2(b).In the limit GbW 0 our model does not adequatelydescribe the harvesting, in particular when the exciton statesare delocalized over the entire chain (i.e., small bias, smalldisorder, and relatively short chains). In this situation, theoff-diagonal density matrix elements (exciton coherences),which are also created by populating the site n ¼ Nðrss0 ð0Þ ¼ csNcs0NÞ, are sufﬁciently long-lived to be relevantto the transport process. In fact, neglecting coherences leadsto the unphysical artifact that there is instantaneous energytransfer over the chain. A more reﬁned approach isnecessary to remove the occurrence of unphysically fastenergy transfer; we refer to our earlier paper [10] for a moredetailed discussion of this limit and its complications.4. Disordered biased chainsThe inclusion of disorder slows down the energytransport process for all bias values; however, the effectARTICLE IN PRESS0.10.05010002000300040005000Δ (J)τ (J-1)0 0.05 0.1100101102103τ (J-1)increasing T increasing TΔ (J)Fig. 2. Harvesting time t versus the applied bias magnitude D calculated for disordered chains of 50 sites for the quenching amplitude G ¼ 101J (panel(a)) and G ¼ 10J (panel (b)) and the scattering amplitude W 0 ¼ 10J. The curves were calculated for different temperatures, T ¼ 0:02J, 0:2J and 0:4J,which are ordered as shown by the arrow. Solid, dotted and dashed curves correspond to disorder strength s ¼ 0, 0:2J and 0:4J, respectively.S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959958is especially strong for small biases. This will lead to a shiftof the optimal bias (when W 0G) to higher values withincreasing disorder strength. The overall shape of the tðDÞcurve does not change in most cases: only in the case of ahigh quenching amplitude Fig. 2(b), we see that an optimalbias appears as a result of the fact that the disorder doesmore damage for low biases than for higher ones.The explanation of these effects is straightforward.Disorder results in Anderson localization, even in theabsence of a bias. This leads to reduced relaxation rates,since the overlap of different states decreases. The effect isparticularly signiﬁcant for systems with a low bias, since ina disorder-free chain the excitons are then delocalized overthe whole chain. For higher biases, there is alreadyconsiderable localization because of the bias itself, andthe effect produced by the disorder is relatively small. Moreimportantly, for sufﬁciently strong disorder and small biasthe lowest energy states are not necessarily located near thetrap. At low temperatures, a large population is built up inthese states and the excitation may thus be unable to betrapped efﬁciently because of the blockage of diffusion[13,14]. A bias forces these low energy states to be locatednear the trap and thus accelerates the harvesting.Thermal effects are marginal for large bias strengths. Inthis case, the harvesting process consists largely of down-ward relaxation followed by quenching from the bottomstates. These states tend to be localized near the trap andthe quenching process is therefore hardly accelerated bythermally induced population shifts to higher energy states.In contrast, the thermal effects for small bias are quitestrong in disordered systems, even more so than inhomogeneous chains. As has been noted in the previousparagraphs, population tends to end up in the excitonstates at the bottom of the band. These, in general, are notlocated near the trap and thus are quenched poorly.Increasing the temperature has two effects. First, it rapidlyaccelerates the exciton diffusion, and second, the popula-tion is thermally excited to higher energy states, whichare quenched more strongly due to a better overlap withthe trap. In other words, the interaction of excitons withthe phonon bath can counteract some of the detrimentaleffects of Anderson localization on the harvesting process,as can be seen in Fig. 2.5. ConclusionsContrary to what one might naively expect, namely thatan energetic bias towards the trap will decrease theharvesting time, we have shown that various scenariosare possible, depending on the relative strengths of therelevant subprocesses. This is already apparent in lowtemperature systems (with and without disorder) [10], andthe same holds for systems at elevated temperature.Thermal effects are only signiﬁcant for small bias strengthsand tend to accelerate harvesting, especially in disorderedsystems.References[1] K. Mukai, S. Abe, H. Sumi, J. Phys. Chem. B 103 (1999) 6096.[2] T. Kobayashi (Ed.), J-aggregates, World Scientiﬁc, Singapore, 1996.[3] J. Knoester, in: V.M. Agranovich, G.C. La Rocca (Eds.), Proceedingsof the International School of Physics ‘‘Enrico Fermi’’; CourseCXLIX, Organic nanostructures: science and application, IOS Press,Amsterdam, 2002, pp. 149–186.[4] A. Tilgner, H.P. Trommsdorff, J.M. Zeigler, R.M. Hochstrasser,J. Chem. Phys. 96 (1992) 781.ARTICLE IN PRESSS.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 959[5] For a recent overview see, e.g., G. Hadziioannou, P. van Hutten(Eds.), ‘‘Semiconducting Polymers—Chemistry, Physics, and En-gineering’’; VCH, Weinheim, 1999.[6] R.W. Wagner, J.S. Lindsey, J. Amer. Chem. Soc. 116 (1994) 9759.[7] M. Heilemann, P. Tinnefeld, G. Sanchez Mosteiro, M. Garcia Parajo,N.F. van Hulst, M. Sauer, J. Amer. Chem. Soc. 126 (2004) 6514.[8] S.A. Crooker, J.A. Hollingsworth, S. Tretiak, L.V. Klimov, Phys.Rev. Lett. 89 (2002) 186802.[9] T. Franzl, T.A. Klar, S. Schietinger, A.L. Rogach, J. Feldmann,Nano Lett. 4 (2004) 1599.[10] S.M. Vlaming, V.A. Malyshev, J. Knoester, J. Chem. Phys. 127(2007) 154719.[11] F. Bloch, z. Phys. 52 (1927) 555; C. Zener, Proc. R. Soc. London, Ser.A 145 (1934) 523.[12] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrish-nan, Phys. Rev. Lett. 42 (1979) 673.[13] A.V. Malyshev, V.A. Malyshev, F. Domı´nguez-Adame, Chem. Phys.Lett. 371 (2003) 417;A.V. Malyshev, V.A. Malyshev, F. Domı´nguez-Adame, J. Phys.Chem. B 107 (2003) 4418.[14] M. Bednarz, V.A. Malyshev, J. Knoester, Phys. Rev. Lett. 91 (2003)217401;M. Bednarz, V.A. Malyshev, J. Knoester, J. Chem. Phys. 120 (2004)3827.

Thermal effects in exciton harvesting in biased one-dimensional systems

University of Groningen Research Database

  
 University of Groningen
Thermal effects in exciton harvesting in biased one-dimensional systems
Vlaming, S. M.; Malyshev, V.A.; Knoester, Jasper
Published in:
Journal of Luminescence
DOI:
10.1016/j.jlumin.2007.11.038
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
2008
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Vlaming, S. M., Malyshev, V. A., & Knoester, J. (2008). Thermal effects in exciton harvesting in biased one-
dimensional systems. Journal of Luminescence, 128(5-6), 956-959. DOI: 10.1016/j.jlumin.2007.11.038
Copyright
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Download date: 10-02-2018
Journal of Luminescence 128 (2008) 956–959
Thermal effects in exciton harvesting in biased one-dimensional systems
S.M. Vlaming, V.A. Malyshev, J. Knoester
Centre for Theoretical Physics and Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Available online 20 February 2008
Abstract
The study of energy harvesting in chain-like structures is important due to its relevance to a variety of interesting physical systems.
Harvesting is understood as the combination of exciton transport through intra-band exciton relaxation (via scattering on phonon
modes) and subsequent quenching by a trap. Recently, we have shown that in the low temperature limit different harvesting scenarios as
a function of the applied bias strength (magnitude of the energy gradient towards the trap) are possible [S.M. Vlaming, V.A. Malyshev, J.
Knoester, J. Chem. Phys. 127 (2007) 154719]. This paper generalizes the results for both homogeneous and disordered chains to nonzero
temperatures. We show that thermal effects are appreciable only for low bias strengths, particularly so in disordered systems, and lead to
faster harvesting.
r 2007 Elsevier B.V. All rights reserved.
PACS: 71.23.An; 71.35.Cc
Keywords: Frenkel excitons; Exciton transfer; Exciton trapping; Energy bias; Photonic wires
1. Introduction
The study of energy transport and trapping in molecular
scale systems is an interesting topic, because of its relevance
to natural systems (photosynthetic antenna complexes [1])
and applications in nanophysics and quantum computing,
where energy should efﬁciently be transported over
some distance. Linear molecular aggregates [2,3], conju-
gated polymers [4,5], photonic wires [6,7] and quantum
dot arrays [8,9] may all be envisioned as realizations of
such energy transport complexes. A natural question that
arises is how this combined process of transport and
eventual trapping, which we will from now on refer to
as harvesting, can be optimized. In a previous paper [10],
we performed a study on the effect of an energy gradient
(bias) on the energy harvesting properties of a linear
system. The scope there was limited to low temperatures.
In this paper, we present numerical calculations of thermal
effects on the harvesting time and a qualitative analysis of
the results.
2. Model
We use a Frenkel exciton model to describe the excited
state dynamics in chain-like systems. The latter are
modeled as a regular one-dimensional array of two-level
monomers that are coupled through their transition dipole
moments, which are assumed to be oriented parallel to
each other. We initially excite a monomer at one end of the
chain ðn ¼ NÞ, and consider the other end ðn ¼ 1Þ as a trap.
The harvesting time is calculated as the average time it
takes for an excitation to move across the chain and to be
quenched by the trap. Including diagonal disorder and a
bias ðD40Þ towards the trap, the model Hamiltonian reads
Hex ¼
XN
n¼1
½en þ ðn 1ÞDjnihnj þ
XN
n;man
Jnmjnihmj, (1)
where the state jni denotes that monomer n is excited while
all other monomers are not, the excitation energies en are
uncorrelated Gaussian stochastic variables with zero mean
and standard deviation s and the transfer integrals are
Jnm ¼ J=jnmj3, with the nearest-neighbor coupling
J40. Diagonalization of this Hamiltonian yields collective
excited states, jsi ¼Pn csnjni, where the site amplitudes
can be chosen real. Both a bias and disorder result in
ARTICLE IN PRESS
www.elsevier.com/locate/jlumin
0022-2313/$ - see front matter r 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.jlumin.2007.11.038
Corresponding author. Tel.: +3150 3634369; fax: +31 50 3634947.
E-mail address: j.knoester@rug.nl (J. Knoester).
localization of the states, Bloch-like [11] and Anderson-like
[12], respectively, so that the exciton states are spread out
over ﬁnite segments of the chain only. In the case of a bias,
there exists a correlation between the location of the
eigenstate and its energy, as lower energy states are located
in the vicinity of the trap.
The dynamics of the populations Ps of the various
exciton states is analyzed within the framework of the Pauli
master equation,
_Ps ¼ ðgs þ GsÞPs þ
X
s0
ðWss0Ps0 Ws0sPsÞ, (2)
where gs ¼ g0ð
P
ncsnÞ2 is the radiative rate of the state jsi
(g0 being the radiative rate of a monomer) and Gs is the
quenching rate, which is taken proportional to the exciton
probability at the trap ðn ¼ 1Þ: Gs ¼ Gc2s1 where G is the
quenching amplitude [13], which cannot be arbitrarily large
within the limitations of our model [10]. As at t ¼ 0 only
the rightmost site n ¼ N is excited, the initial population
distribution is given by Psð0Þ ¼ c2sN . The Wss0 describe the
scattering of population between various exciton states
through interaction with phonons: Wss0 ¼ W 0 SðEs  Es0 ÞPN
n¼1c
2
snc
2
s0nGðEs  Es0 Þ, where W 0 is the scattering ampli-
tude, SðEÞ is the phonon spectral density function,PN
n¼1c
2
snc
2
s0n is the probability overlap, and GðEÞ ¼ 1þ
nðEÞ if Eo0, while GðEÞ ¼ nðEÞ if E40 with nðEÞ ¼
½expðjEj=TÞ  11 the phonon occupation factor. We use a
Debye-like spectral density with cut-off oc,
SðEÞ ¼ jE=Jj3 expðjEj=ocÞ. The harvesting time is de-
ﬁned as follows: t1 ¼ t1trap  t10 , where ttrap and t0 are the
decay times with and without trap, respectively. They are
calculated as
R1
0
dthPsPsðtÞi, where h. . .i denotes the
disorder average.
3. Disorder-free biased chains
We begin with a discussion of the harvesting efﬁciency in
homogeneous chains ðs ¼ 0Þ. The bias-induced Bloch
localization dramatically changes the harvesting properties.
Depending on the relative strengths of the quenching and
relaxation processes, both monotonic decreasing and
nonmonotonic behavior of the harvesting time as a
function of the bias strength can be obtained. We refer to
our earlier paper [10] for a full discussion of the possible
scenarios, while here we focus on the aspects relevant to
understanding thermal effects in our model.
In the limit of G5W 0 and for low temperature,
relaxation to lower exciton states is the dominant process
for virtually all exciton states. As a result, most population
ends up in the bottom states of the band. For zero bias, the
quenching rate from these states is low because of the poor
overlap of the bottom states with the trap (see Fig. 1).
However, turning on the bias increases this overlap,
thereby accelerating the harvesting process. Temperature
will enhance the harvesting efﬁciency at low bias by shifting
population from the poorly quenched bottom state(s) to
more strongly quenched higher energy states. This en-
hancement disappears for higher bias strengths, as the
higher energy states are no longer more strongly quenched
than the bottom states, as is shown in Fig. 2(a).
The interesting situation where GW 0 provides a
nonmonotonic bias dependency of t. A small bias is
advantageous for the harvesting process as it increases the
quenching rate for the bottom states, but beyond some
optimal bias value the increase in relaxation time overtakes
the previous effect in importance and the net effect of
increasing the bias becomes negative. The thermally
induced changes in the harvesting efﬁciency are again only
present for small values of the bias, for the reasons
mentioned in the discussion of the limit of G5W 0. This is
corroborated in Fig. 2(b).
In the limit GbW 0 our model does not adequately
describe the harvesting, in particular when the exciton states
are delocalized over the entire chain (i.e., small bias, small
disorder, and relatively short chains). In this situation, the
off-diagonal density matrix elements (exciton coherences),
which are also created by populating the site n ¼ N
ðrss0 ð0Þ ¼ csNcs0NÞ, are sufﬁciently long-lived to be relevant
to the transport process. In fact, neglecting coherences leads
to the unphysical artifact that there is instantaneous energy
transfer over the chain. A more reﬁned approach is
necessary to remove the occurrence of unphysically fast
energy transfer; we refer to our earlier paper [10] for a more
detailed discussion of this limit and its complications.
4. Disordered biased chains
The inclusion of disorder slows down the energy
transport process for all bias values; however, the effect
ARTICLE IN PRESS
0.1 
0.05
0
|c
s
1
|2
20 4010 30 50
0
0.1
0.05
state number, s
|c
s
N
|2
Δ=0
Δ=0.01J
Δ=0.1J
Δ=0
Δ=0.01J
Δ=0.1J
Fig. 1. Bias dependence of the exciton probabilities c2sN and c
2
s1 that
determine the initial distribution of the exciton population Psð0Þ and the
quenching rate Gs, respectively, for chains without disorder ðs ¼ 0Þ.
The bias tends to shift the initial distribution of population to the top of
the band (to higher s). Oppositely, the strongly quenched states occur at
the lower band edge (for small s).
S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 957
is especially strong for small biases. This will lead to a shift
of the optimal bias (when W 0G) to higher values with
increasing disorder strength. The overall shape of the tðDÞ
curve does not change in most cases: only in the case of a
high quenching amplitude Fig. 2(b), we see that an optimal
bias appears as a result of the fact that the disorder does
more damage for low biases than for higher ones.
The explanation of these effects is straightforward.
Disorder results in Anderson localization, even in the
absence of a bias. This leads to reduced relaxation rates,
since the overlap of different states decreases. The effect is
particularly signiﬁcant for systems with a low bias, since in
a disorder-free chain the excitons are then delocalized over
the whole chain. For higher biases, there is already
considerable localization because of the bias itself, and
the effect produced by the disorder is relatively small. More
importantly, for sufﬁciently strong disorder and small bias
the lowest energy states are not necessarily located near the
trap. At low temperatures, a large population is built up in
these states and the excitation may thus be unable to be
trapped efﬁciently because of the blockage of diffusion
[13,14]. A bias forces these low energy states to be located
near the trap and thus accelerates the harvesting.
Thermal effects are marginal for large bias strengths. In
this case, the harvesting process consists largely of down-
ward relaxation followed by quenching from the bottom
states. These states tend to be localized near the trap and
the quenching process is therefore hardly accelerated by
thermally induced population shifts to higher energy states.
In contrast, the thermal effects for small bias are quite
strong in disordered systems, even more so than in
homogeneous chains. As has been noted in the previous
paragraphs, population tends to end up in the exciton
states at the bottom of the band. These, in general, are not
located near the trap and thus are quenched poorly.
Increasing the temperature has two effects. First, it rapidly
accelerates the exciton diffusion, and second, the popula-
tion is thermally excited to higher energy states, which
are quenched more strongly due to a better overlap with
the trap. In other words, the interaction of excitons with
the phonon bath can counteract some of the detrimental
effects of Anderson localization on the harvesting process,
as can be seen in Fig. 2.
5. Conclusions
Contrary to what one might naively expect, namely that
an energetic bias towards the trap will decrease the
harvesting time, we have shown that various scenarios
are possible, depending on the relative strengths of the
relevant subprocesses. This is already apparent in low
temperature systems (with and without disorder) [10], and
the same holds for systems at elevated temperature.
Thermal effects are only signiﬁcant for small bias strengths
and tend to accelerate harvesting, especially in disordered
systems.
References
[1] K. Mukai, S. Abe, H. Sumi, J. Phys. Chem. B 103 (1999) 6096.
[2] T. Kobayashi (Ed.), J-aggregates, World Scientiﬁc, Singapore, 1996.
[3] J. Knoester, in: V.M. Agranovich, G.C. La Rocca (Eds.), Proceedings
of the International School of Physics ‘‘Enrico Fermi’’; Course
CXLIX, Organic nanostructures: science and application, IOS Press,
Amsterdam, 2002, pp. 149–186.
[4] A. Tilgner, H.P. Trommsdorff, J.M. Zeigler, R.M. Hochstrasser,
J. Chem. Phys. 96 (1992) 781.
ARTICLE IN PRESS
0.10.05
0
1000
2000
3000
4000
5000
Δ (J)
τ 
(J
-1
)
0 0.05 0.1
10
0
10
1
10
2
10
3
τ 
(J
-1
)
increasing T increasing T
Δ (J)
Fig. 2. Harvesting time t versus the applied bias magnitude D calculated for disordered chains of 50 sites for the quenching amplitude G ¼ 101J (panel
(a)) and G ¼ 10J (panel (b)) and the scattering amplitude W 0 ¼ 10J. The curves were calculated for different temperatures, T ¼ 0:02J, 0:2J and 0:4J,
which are ordered as shown by the arrow. Solid, dotted and dashed curves correspond to disorder strength s ¼ 0, 0:2J and 0:4J, respectively.
S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959958
[5] For a recent overview see, e.g., G. Hadziioannou, P. van Hutten
(Eds.), ‘‘Semiconducting Polymers—Chemistry, Physics, and En-
gineering’’; VCH, Weinheim, 1999.
[6] R.W. Wagner, J.S. Lindsey, J. Amer. Chem. Soc. 116 (1994) 9759.
[7] M. Heilemann, P. Tinnefeld, G. Sanchez Mosteiro, M. Garcia Parajo,
N.F. van Hulst, M. Sauer, J. Amer. Chem. Soc. 126 (2004) 6514.
[8] S.A. Crooker, J.A. Hollingsworth, S. Tretiak, L.V. Klimov, Phys.
Rev. Lett. 89 (2002) 186802.
[9] T. Franzl, T.A. Klar, S. Schietinger, A.L. Rogach, J. Feldmann,
Nano Lett. 4 (2004) 1599.
[10] S.M. Vlaming, V.A. Malyshev, J. Knoester, J. Chem. Phys. 127
(2007) 154719.
[11] F. Bloch, z. Phys. 52 (1927) 555; C. Zener, Proc. R. Soc. London, Ser.
A 145 (1934) 523.
[12] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrish-
nan, Phys. Rev. Lett. 42 (1979) 673.
[13] A.V. Malyshev, V.A. Malyshev, F. Domı´nguez-Adame, Chem. Phys.
Lett. 371 (2003) 417;
A.V. Malyshev, V.A. Malyshev, F. Domı´nguez-Adame, J. Phys.
Chem. B 107 (2003) 4418.
[14] M. Bednarz, V.A. Malyshev, J. Knoester, Phys. Rev. Lett. 91 (2003)
217401;
M. Bednarz, V.A. Malyshev, J. Knoester, J. Chem. Phys. 120 (2004)
3827.
ARTICLE IN PRESS
S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 959


   University of GroningenThermal effects in exciton harvesting in biased one-dimensional systemsVlaming, S. M.; Malyshev, V.A.; Knoester, J.Published in:Journal of LuminescenceDOI:10.1016/j.jlumin.2007.11.038IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2008Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vlaming, S. M., Malyshev, V. A., & Knoester, J. (2008). Thermal effects in exciton harvesting in biased one-dimensional systems. Journal of Luminescence, 128(5-6), 956-959.https://doi.org/10.1016/j.jlumin.2007.11.038CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 26-04-2023ARTICLE IN PRESS0022-2313/$ - sedoi:10.1016/j.jluCorrespondE-mail addrJournal of Luminescence 128 (2008) 956–959www.elsevier.com/locate/jluminThermal effects in exciton harvesting in biased one-dimensional systemsS.M. Vlaming, V.A. Malyshev, J. KnoesterCentre for Theoretical Physics and Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The NetherlandsAvailable online 20 February 2008AbstractThe study of energy harvesting in chain-like structures is important due to its relevance to a variety of interesting physical systems.Harvesting is understood as the combination of exciton transport through intra-band exciton relaxation (via scattering on phononmodes) and subsequent quenching by a trap. Recently, we have shown that in the low temperature limit different harvesting scenarios asa function of the applied bias strength (magnitude of the energy gradient towards the trap) are possible [S.M. Vlaming, V.A. Malyshev, J.Knoester, J. Chem. Phys. 127 (2007) 154719]. This paper generalizes the results for both homogeneous and disordered chains to nonzerotemperatures. We show that thermal effects are appreciable only for low bias strengths, particularly so in disordered systems, and lead tofaster harvesting.r 2007 Elsevier B.V. All rights reserved.PACS: 71.23.An; 71.35.CcKeywords: Frenkel excitons; Exciton transfer; Exciton trapping; Energy bias; Photonic wires1. IntroductionThe study of energy transport and trapping in molecularscale systems is an interesting topic, because of its relevanceto natural systems (photosynthetic antenna complexes [1])and applications in nanophysics and quantum computing,where energy should efficiently be transported oversome distance. Linear molecular aggregates [2,3], conju-gated polymers [4,5], photonic wires [6,7] and quantumdot arrays [8,9] may all be envisioned as realizations ofsuch energy transport complexes. A natural question thatarises is how this combined process of transport andeventual trapping, which we will from now on refer toas harvesting, can be optimized. In a previous paper [10],we performed a study on the effect of an energy gradient(bias) on the energy harvesting properties of a linearsystem. The scope there was limited to low temperatures.In this paper, we present numerical calculations of thermaleffects on the harvesting time and a qualitative analysis ofthe results.e front matter r 2007 Elsevier B.V. All rights reserved.min.2007.11.038ing author. Tel.: +3150 3634369; fax: +31 50 3634947.ess: j.knoester@rug.nl (J. Knoester).2. ModelWe use a Frenkel exciton model to describe the excitedstate dynamics in chain-like systems. The latter aremodeled as a regular one-dimensional array of two-levelmonomers that are coupled through their transition dipolemoments, which are assumed to be oriented parallel toeach other. We initially excite a monomer at one end of thechain ðn ¼ NÞ, and consider the other end ðn ¼ 1Þ as a trap.The harvesting time is calculated as the average time ittakes for an excitation to move across the chain and to bequenched by the trap. Including diagonal disorder and abias ðD40Þ towards the trap, the model Hamiltonian readsHex ¼XNn¼1½en þ ðn 1ÞDjnihnj þXNn;manJnmjnihmj, (1)where the state jni denotes that monomer n is excited whileall other monomers are not, the excitation energies en areuncorrelated Gaussian stochastic variables with zero meanand standard deviation s and the transfer integrals areJnm ¼ J=jnmj3, with the nearest-neighbor couplingJ40. Diagonalization of this Hamiltonian yields collectiveexcited states, jsi ¼Pn csnjni, where the site amplitudescan be chosen real. Both a bias and disorder result inARTICLE IN PRESS0.1 0.050|cs1|220 4010 30 5000.10.05state number, s|csN|2Δ=0Δ=0.01JΔ=0.1JΔ=0Δ=0.01JΔ=0.1JFig. 1. Bias dependence of the exciton probabilities c2sN and c2s1 thatdetermine the initial distribution of the exciton population Psð0Þ and thequenching rate Gs, respectively, for chains without disorder ðs ¼ 0Þ.The bias tends to shift the initial distribution of population to the top ofthe band (to higher s). Oppositely, the strongly quenched states occur atthe lower band edge (for small s).S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 957localization of the states, Bloch-like [11] and Anderson-like[12], respectively, so that the exciton states are spread outover finite segments of the chain only. In the case of a bias,there exists a correlation between the location of theeigenstate and its energy, as lower energy states are locatedin the vicinity of the trap.The dynamics of the populations Ps of the variousexciton states is analyzed within the framework of the Paulimaster equation,_Ps ¼ ðgs þ GsÞPs þXs0ðW ss0Ps0 W s0sPsÞ, (2)where gs ¼ g0ðPncsnÞ2 is the radiative rate of the state jsi(g0 being the radiative rate of a monomer) and Gs is thequenching rate, which is taken proportional to the excitonprobability at the trap ðn ¼ 1Þ: Gs ¼ Gc2s1 where G is thequenching amplitude [13], which cannot be arbitrarily largewithin the limitations of our model [10]. As at t ¼ 0 onlythe rightmost site n ¼ N is excited, the initial populationdistribution is given by Psð0Þ ¼ c2sN . The W ss0 describe thescattering of population between various exciton statesthrough interaction with phonons: W ss0 ¼W 0 SðEs  Es0 ÞPNn¼1c2snc2s0nGðEs  Es0 Þ, where W 0 is the scattering ampli-tude, SðEÞ is the phonon spectral density function,PNn¼1c2snc2s0n is the probability overlap, and GðEÞ ¼ 1þnðEÞ if Eo0, while GðEÞ ¼ nðEÞ if E40 with nðEÞ ¼½expðjEj=TÞ  11 the phonon occupation factor. We use aDebye-like spectral density with cut-off oc,SðEÞ ¼ jE=Jj3 expðjEj=ocÞ. The harvesting time is de-fined as follows: t1 ¼ t1trap  t10 , where ttrap and t0 are thedecay times with and without trap, respectively. They arecalculated asR10dthPsPsðtÞi, where h. . .i denotes thedisorder average.3. Disorder-free biased chainsWe begin with a discussion of the harvesting efficiency inhomogeneous chains ðs ¼ 0Þ. The bias-induced Blochlocalization dramatically changes the harvesting properties.Depending on the relative strengths of the quenching andrelaxation processes, both monotonic decreasing andnonmonotonic behavior of the harvesting time as afunction of the bias strength can be obtained. We refer toour earlier paper [10] for a full discussion of the possiblescenarios, while here we focus on the aspects relevant tounderstanding thermal effects in our model.In the limit of G5W 0 and for low temperature,relaxation to lower exciton states is the dominant processfor virtually all exciton states. As a result, most populationends up in the bottom states of the band. For zero bias, thequenching rate from these states is low because of the pooroverlap of the bottom states with the trap (see Fig. 1).However, turning on the bias increases this overlap,thereby accelerating the harvesting process. Temperaturewill enhance the harvesting efficiency at low bias by shiftingpopulation from the poorly quenched bottom state(s) tomore strongly quenched higher energy states. This en-hancement disappears for higher bias strengths, as thehigher energy states are no longer more strongly quenchedthan the bottom states, as is shown in Fig. 2(a).The interesting situation where GW 0 provides anonmonotonic bias dependency of t. A small bias isadvantageous for the harvesting process as it increases thequenching rate for the bottom states, but beyond someoptimal bias value the increase in relaxation time overtakesthe previous effect in importance and the net effect ofincreasing the bias becomes negative. The thermallyinduced changes in the harvesting efficiency are again onlypresent for small values of the bias, for the reasonsmentioned in the discussion of the limit of G5W 0. This iscorroborated in Fig. 2(b).In the limit GbW 0 our model does not adequatelydescribe the harvesting, in particular when the exciton statesare delocalized over the entire chain (i.e., small bias, smalldisorder, and relatively short chains). In this situation, theoff-diagonal density matrix elements (exciton coherences),which are also created by populating the site n ¼ Nðrss0 ð0Þ ¼ csNcs0NÞ, are sufficiently long-lived to be relevantto the transport process. In fact, neglecting coherences leadsto the unphysical artifact that there is instantaneous energytransfer over the chain. A more refined approach isnecessary to remove the occurrence of unphysically fastenergy transfer; we refer to our earlier paper [10] for a moredetailed discussion of this limit and its complications.4. Disordered biased chainsThe inclusion of disorder slows down the energytransport process for all bias values; however, the effectARTICLE IN PRESS0.10.05010002000300040005000Δ (J)τ (J-1)0 0.05 0.1100101102103τ (J-1)increasing T increasing TΔ (J)Fig. 2. Harvesting time t versus the applied bias magnitude D calculated for disordered chains of 50 sites for the quenching amplitude G ¼ 101J (panel(a)) and G ¼ 10J (panel (b)) and the scattering amplitude W 0 ¼ 10J. The curves were calculated for different temperatures, T ¼ 0:02J, 0:2J and 0:4J,which are ordered as shown by the arrow. Solid, dotted and dashed curves correspond to disorder strength s ¼ 0, 0:2J and 0:4J, respectively.S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959958is especially strong for small biases. This will lead to a shiftof the optimal bias (when W 0G) to higher values withincreasing disorder strength. The overall shape of the tðDÞcurve does not change in most cases: only in the case of ahigh quenching amplitude Fig. 2(b), we see that an optimalbias appears as a result of the fact that the disorder doesmore damage for low biases than for higher ones.The explanation of these effects is straightforward.Disorder results in Anderson localization, even in theabsence of a bias. This leads to reduced relaxation rates,since the overlap of different states decreases. The effect isparticularly significant for systems with a low bias, since ina disorder-free chain the excitons are then delocalized overthe whole chain. For higher biases, there is alreadyconsiderable localization because of the bias itself, andthe effect produced by the disorder is relatively small. Moreimportantly, for sufficiently strong disorder and small biasthe lowest energy states are not necessarily located near thetrap. At low temperatures, a large population is built up inthese states and the excitation may thus be unable to betrapped efficiently because of the blockage of diffusion[13,14]. A bias forces these low energy states to be locatednear the trap and thus accelerates the harvesting.Thermal effects are marginal for large bias strengths. Inthis case, the harvesting process consists largely of down-ward relaxation followed by quenching from the bottomstates. These states tend to be localized near the trap andthe quenching process is therefore hardly accelerated bythermally induced population shifts to higher energy states.In contrast, the thermal effects for small bias are quitestrong in disordered systems, even more so than inhomogeneous chains. As has been noted in the previousparagraphs, population tends to end up in the excitonstates at the bottom of the band. These, in general, are notlocated near the trap and thus are quenched poorly.Increasing the temperature has two effects. First, it rapidlyaccelerates the exciton diffusion, and second, the popula-tion is thermally excited to higher energy states, whichare quenched more strongly due to a better overlap withthe trap. In other words, the interaction of excitons withthe phonon bath can counteract some of the detrimentaleffects of Anderson localization on the harvesting process,as can be seen in Fig. 2.5. ConclusionsContrary to what one might naively expect, namely thatan energetic bias towards the trap will decrease theharvesting time, we have shown that various scenariosare possible, depending on the relative strengths of therelevant subprocesses. This is already apparent in lowtemperature systems (with and without disorder) [10], andthe same holds for systems at elevated temperature.Thermal effects are only significant for small bias strengthsand tend to accelerate harvesting, especially in disorderedsystems.References[1] K. Mukai, S. Abe, H. Sumi, J. Phys. Chem. B 103 (1999) 6096.[2] T. Kobayashi (Ed.), J-aggregates, World Scientific, Singapore, 1996.[3] J. Knoester, in: V.M. Agranovich, G.C. La Rocca (Eds.), Proceedingsof the International School of Physics ‘‘Enrico Fermi’’; CourseCXLIX, Organic nanostructures: science and application, IOS Press,Amsterdam, 2002, pp. 149–186.[4] A. Tilgner, H.P. Trommsdorff, J.M. Zeigler, R.M. Hochstrasser,J. Chem. Phys. 96 (1992) 781.ARTICLE IN PRESSS.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 959[5] For a recent overview see, e.g., G. Hadziioannou, P. van Hutten(Eds.), ‘‘Semiconducting Polymers—Chemistry, Physics, and En-gineering’’; VCH, Weinheim, 1999.[6] R.W. Wagner, J.S. Lindsey, J. Amer. Chem. Soc. 116 (1994) 9759.[7] M. Heilemann, P. Tinnefeld, G. Sanchez Mosteiro, M. Garcia Parajo,N.F. van Hulst, M. Sauer, J. Amer. Chem. Soc. 126 (2004) 6514.[8] S.A. Crooker, J.A. Hollingsworth, S. Tretiak, L.V. Klimov, Phys.Rev. Lett. 89 (2002) 186802.[9] T. Franzl, T.A. Klar, S. Schietinger, A.L. Rogach, J. Feldmann,Nano Lett. 4 (2004) 1599.[10] S.M. Vlaming, V.A. Malyshev, J. Knoester, J. Chem. Phys. 127(2007) 154719.[11] F. Bloch, z. Phys. 52 (1927) 555; C. Zener, Proc. R. Soc. London, Ser.A 145 (1934) 523.[12] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrish-nan, Phys. Rev. Lett. 42 (1979) 673.[13] A.V. Malyshev, V.A. Malyshev, F. Domı́nguez-Adame, Chem. Phys.Lett. 371 (2003) 417;A.V. Malyshev, V.A. Malyshev, F. Domı́nguez-Adame, J. Phys.Chem. B 107 (2003) 4418.[14] M. Bednarz, V.A. Malyshev, J. Knoester, Phys. Rev. Lett. 91 (2003)217401;M. Bednarz, V.A. Malyshev, J. Knoester, J. Chem. Phys. 120 (2004)3827.

The study of energy harvesting in chain-like structures is important due to its relevance to a variety of interesting physical systems. Harvesting is understood as the combination of exciton transport through intra-band exciton relaxation (via scattering on phonon modes) and subsequent quenching by a trap. Recently, we have shown that in the low temperature limit different harvesting scenarios as a function of the applied bias strength (magnitude of the energy gradient towards the trap) are possible. This paper generalizes the results for both homogeneous and disordered chains to nonzero temperatures. We show that thermal effects are appreciable only for low bias strengths, particularly so in disordered systems, and lead to faster harvesting.

Vlaming, S.M.,

Malyshev, V.A.,

Knoester, J.,

University of Groningen Digital Archive

ARTS repository - University of Groningen

   University of GroningenThermal effects in exciton harvesting in biased one-dimensional systemsVlaming, S. M.; Malyshev, V.A.; Knoester, J.Published in:Journal of LuminescenceDOI:10.1016/j.jlumin.2007.11.038IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2008Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vlaming, S. M., Malyshev, V. A., & Knoester, J. (2008). Thermal effects in exciton harvesting in biased one-dimensional systems. Journal of Luminescence, 128(5-6), 956-959.https://doi.org/10.1016/j.jlumin.2007.11.038CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 01-02-2024ARTICLE IN PRESS0022-2313/$ - sedoi:10.1016/j.jluCorrespondE-mail addrJournal of Luminescence 128 (2008) 956–959www.elsevier.com/locate/jluminThermal effects in exciton harvesting in biased one-dimensional systemsS.M. Vlaming, V.A. Malyshev, J. KnoesterCentre for Theoretical Physics and Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The NetherlandsAvailable online 20 February 2008AbstractThe study of energy harvesting in chain-like structures is important due to its relevance to a variety of interesting physical systems.Harvesting is understood as the combination of exciton transport through intra-band exciton relaxation (via scattering on phononmodes) and subsequent quenching by a trap. Recently, we have shown that in the low temperature limit different harvesting scenarios asa function of the applied bias strength (magnitude of the energy gradient towards the trap) are possible [S.M. Vlaming, V.A. Malyshev, J.Knoester, J. Chem. Phys. 127 (2007) 154719]. This paper generalizes the results for both homogeneous and disordered chains to nonzerotemperatures. We show that thermal effects are appreciable only for low bias strengths, particularly so in disordered systems, and lead tofaster harvesting.r 2007 Elsevier B.V. All rights reserved.PACS: 71.23.An; 71.35.CcKeywords: Frenkel excitons; Exciton transfer; Exciton trapping; Energy bias; Photonic wires1. IntroductionThe study of energy transport and trapping in molecularscale systems is an interesting topic, because of its relevanceto natural systems (photosynthetic antenna complexes [1])and applications in nanophysics and quantum computing,where energy should efficiently be transported oversome distance. Linear molecular aggregates [2,3], conju-gated polymers [4,5], photonic wires [6,7] and quantumdot arrays [8,9] may all be envisioned as realizations ofsuch energy transport complexes. A natural question thatarises is how this combined process of transport andeventual trapping, which we will from now on refer toas harvesting, can be optimized. In a previous paper [10],we performed a study on the effect of an energy gradient(bias) on the energy harvesting properties of a linearsystem. The scope there was limited to low temperatures.In this paper, we present numerical calculations of thermaleffects on the harvesting time and a qualitative analysis ofthe results.e front matter r 2007 Elsevier B.V. All rights reserved.min.2007.11.038ing author. Tel.: +3150 3634369; fax: +31 50 3634947.ess: j.knoester@rug.nl (J. Knoester).2. ModelWe use a Frenkel exciton model to describe the excitedstate dynamics in chain-like systems. The latter aremodeled as a regular one-dimensional array of two-levelmonomers that are coupled through their transition dipolemoments, which are assumed to be oriented parallel toeach other. We initially excite a monomer at one end of thechain ðn ¼ NÞ, and consider the other end ðn ¼ 1Þ as a trap.The harvesting time is calculated as the average time ittakes for an excitation to move across the chain and to bequenched by the trap. Including diagonal disorder and abias ðD40Þ towards the trap, the model Hamiltonian readsHex ¼XNn¼1½en þ ðn 1ÞDjnihnj þXNn;manJnmjnihmj, (1)where the state jni denotes that monomer n is excited whileall other monomers are not, the excitation energies en areuncorrelated Gaussian stochastic variables with zero meanand standard deviation s and the transfer integrals areJnm ¼ J=jnmj3, with the nearest-neighbor couplingJ40. Diagonalization of this Hamiltonian yields collectiveexcited states, jsi ¼Pn csnjni, where the site amplitudescan be chosen real. Both a bias and disorder result inARTICLE IN PRESS0.1 0.050|cs1|220 4010 30 5000.10.05state number, s|csN|2Δ=0Δ=0.01JΔ=0.1JΔ=0Δ=0.01JΔ=0.1JFig. 1. Bias dependence of the exciton probabilities c2sN and c2s1 thatdetermine the initial distribution of the exciton population Psð0Þ and thequenching rate Gs, respectively, for chains without disorder ðs ¼ 0Þ.The bias tends to shift the initial distribution of population to the top ofthe band (to higher s). Oppositely, the strongly quenched states occur atthe lower band edge (for small s).S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 957localization of the states, Bloch-like [11] and Anderson-like[12], respectively, so that the exciton states are spread outover finite segments of the chain only. In the case of a bias,there exists a correlation between the location of theeigenstate and its energy, as lower energy states are locatedin the vicinity of the trap.The dynamics of the populations Ps of the variousexciton states is analyzed within the framework of the Paulimaster equation,_Ps ¼ ðgs þ GsÞPs þXs0ðW ss0Ps0 W s0sPsÞ, (2)where gs ¼ g0ðPncsnÞ2 is the radiative rate of the state jsi(g0 being the radiative rate of a monomer) and Gs is thequenching rate, which is taken proportional to the excitonprobability at the trap ðn ¼ 1Þ: Gs ¼ Gc2s1 where G is thequenching amplitude [13], which cannot be arbitrarily largewithin the limitations of our model [10]. As at t ¼ 0 onlythe rightmost site n ¼ N is excited, the initial populationdistribution is given by Psð0Þ ¼ c2sN . The W ss0 describe thescattering of population between various exciton statesthrough interaction with phonons: W ss0 ¼W 0 SðEs  Es0 ÞPNn¼1c2snc2s0nGðEs  Es0 Þ, where W 0 is the scattering ampli-tude, SðEÞ is the phonon spectral density function,PNn¼1c2snc2s0n is the probability overlap, and GðEÞ ¼ 1þnðEÞ if Eo0, while GðEÞ ¼ nðEÞ if E40 with nðEÞ ¼½expðjEj=TÞ  11 the phonon occupation factor. We use aDebye-like spectral density with cut-off oc,SðEÞ ¼ jE=Jj3 expðjEj=ocÞ. The harvesting time is de-fined as follows: t1 ¼ t1trap  t10 , where ttrap and t0 are thedecay times with and without trap, respectively. They arecalculated asR10dthPsPsðtÞi, where h. . .i denotes thedisorder average.3. Disorder-free biased chainsWe begin with a discussion of the harvesting efficiency inhomogeneous chains ðs ¼ 0Þ. The bias-induced Blochlocalization dramatically changes the harvesting properties.Depending on the relative strengths of the quenching andrelaxation processes, both monotonic decreasing andnonmonotonic behavior of the harvesting time as afunction of the bias strength can be obtained. We refer toour earlier paper [10] for a full discussion of the possiblescenarios, while here we focus on the aspects relevant tounderstanding thermal effects in our model.In the limit of G5W 0 and for low temperature,relaxation to lower exciton states is the dominant processfor virtually all exciton states. As a result, most populationends up in the bottom states of the band. For zero bias, thequenching rate from these states is low because of the pooroverlap of the bottom states with the trap (see Fig. 1).However, turning on the bias increases this overlap,thereby accelerating the harvesting process. Temperaturewill enhance the harvesting efficiency at low bias by shiftingpopulation from the poorly quenched bottom state(s) tomore strongly quenched higher energy states. This en-hancement disappears for higher bias strengths, as thehigher energy states are no longer more strongly quenchedthan the bottom states, as is shown in Fig. 2(a).The interesting situation where GW 0 provides anonmonotonic bias dependency of t. A small bias isadvantageous for the harvesting process as it increases thequenching rate for the bottom states, but beyond someoptimal bias value the increase in relaxation time overtakesthe previous effect in importance and the net effect ofincreasing the bias becomes negative. The thermallyinduced changes in the harvesting efficiency are again onlypresent for small values of the bias, for the reasonsmentioned in the discussion of the limit of G5W 0. This iscorroborated in Fig. 2(b).In the limit GbW 0 our model does not adequatelydescribe the harvesting, in particular when the exciton statesare delocalized over the entire chain (i.e., small bias, smalldisorder, and relatively short chains). In this situation, theoff-diagonal density matrix elements (exciton coherences),which are also created by populating the site n ¼ Nðrss0 ð0Þ ¼ csNcs0NÞ, are sufficiently long-lived to be relevantto the transport process. In fact, neglecting coherences leadsto the unphysical artifact that there is instantaneous energytransfer over the chain. A more refined approach isnecessary to remove the occurrence of unphysically fastenergy transfer; we refer to our earlier paper [10] for a moredetailed discussion of this limit and its complications.4. Disordered biased chainsThe inclusion of disorder slows down the energytransport process for all bias values; however, the effectARTICLE IN PRESS0.10.05010002000300040005000Δ (J)τ (J-1)0 0.05 0.1100101102103τ (J-1)increasing T increasing TΔ (J)Fig. 2. Harvesting time t versus the applied bias magnitude D calculated for disordered chains of 50 sites for the quenching amplitude G ¼ 101J (panel(a)) and G ¼ 10J (panel (b)) and the scattering amplitude W 0 ¼ 10J. The curves were calculated for different temperatures, T ¼ 0:02J, 0:2J and 0:4J,which are ordered as shown by the arrow. Solid, dotted and dashed curves correspond to disorder strength s ¼ 0, 0:2J and 0:4J, respectively.S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959958is especially strong for small biases. This will lead to a shiftof the optimal bias (when W 0G) to higher values withincreasing disorder strength. The overall shape of the tðDÞcurve does not change in most cases: only in the case of ahigh quenching amplitude Fig. 2(b), we see that an optimalbias appears as a result of the fact that the disorder doesmore damage for low biases than for higher ones.The explanation of these effects is straightforward.Disorder results in Anderson localization, even in theabsence of a bias. This leads to reduced relaxation rates,since the overlap of different states decreases. The effect isparticularly significant for systems with a low bias, since ina disorder-free chain the excitons are then delocalized overthe whole chain. For higher biases, there is alreadyconsiderable localization because of the bias itself, andthe effect produced by the disorder is relatively small. Moreimportantly, for sufficiently strong disorder and small biasthe lowest energy states are not necessarily located near thetrap. At low temperatures, a large population is built up inthese states and the excitation may thus be unable to betrapped efficiently because of the blockage of diffusion[13,14]. A bias forces these low energy states to be locatednear the trap and thus accelerates the harvesting.Thermal effects are marginal for large bias strengths. Inthis case, the harvesting process consists largely of down-ward relaxation followed by quenching from the bottomstates. These states tend to be localized near the trap andthe quenching process is therefore hardly accelerated bythermally induced population shifts to higher energy states.In contrast, the thermal effects for small bias are quitestrong in disordered systems, even more so than inhomogeneous chains. As has been noted in the previousparagraphs, population tends to end up in the excitonstates at the bottom of the band. These, in general, are notlocated near the trap and thus are quenched poorly.Increasing the temperature has two effects. First, it rapidlyaccelerates the exciton diffusion, and second, the popula-tion is thermally excited to higher energy states, whichare quenched more strongly due to a better overlap withthe trap. In other words, the interaction of excitons withthe phonon bath can counteract some of the detrimentaleffects of Anderson localization on the harvesting process,as can be seen in Fig. 2.5. ConclusionsContrary to what one might naively expect, namely thatan energetic bias towards the trap will decrease theharvesting time, we have shown that various scenariosare possible, depending on the relative strengths of therelevant subprocesses. This is already apparent in lowtemperature systems (with and without disorder) [10], andthe same holds for systems at elevated temperature.Thermal effects are only significant for small bias strengthsand tend to accelerate harvesting, especially in disorderedsystems.References[1] K. Mukai, S. Abe, H. Sumi, J. Phys. Chem. B 103 (1999) 6096.[2] T. Kobayashi (Ed.), J-aggregates, World Scientific, Singapore, 1996.[3] J. Knoester, in: V.M. Agranovich, G.C. La Rocca (Eds.), Proceedingsof the International School of Physics ‘‘Enrico Fermi’’; CourseCXLIX, Organic nanostructures: science and application, IOS Press,Amsterdam, 2002, pp. 149–186.[4] A. Tilgner, H.P. Trommsdorff, J.M. Zeigler, R.M. Hochstrasser,J. Chem. Phys. 96 (1992) 781.ARTICLE IN PRESSS.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 959[5] For a recent overview see, e.g., G. Hadziioannou, P. van Hutten(Eds.), ‘‘Semiconducting Polymers—Chemistry, Physics, and En-gineering’’; VCH, Weinheim, 1999.[6] R.W. Wagner, J.S. Lindsey, J. Amer. Chem. Soc. 116 (1994) 9759.[7] M. Heilemann, P. Tinnefeld, G. Sanchez Mosteiro, M. Garcia Parajo,N.F. van Hulst, M. Sauer, J. Amer. Chem. Soc. 126 (2004) 6514.[8] S.A. Crooker, J.A. Hollingsworth, S. Tretiak, L.V. Klimov, Phys.Rev. Lett. 89 (2002) 186802.[9] T. Franzl, T.A. Klar, S. Schietinger, A.L. Rogach, J. Feldmann,Nano Lett. 4 (2004) 1599.[10] S.M. Vlaming, V.A. Malyshev, J. Knoester, J. Chem. Phys. 127(2007) 154719.[11] F. Bloch, z. Phys. 52 (1927) 555; C. Zener, Proc. R. Soc. London, Ser.A 145 (1934) 523.[12] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrish-nan, Phys. Rev. Lett. 42 (1979) 673.[13] A.V. Malyshev, V.A. Malyshev, F. Domı́nguez-Adame, Chem. Phys.Lett. 371 (2003) 417;A.V. Malyshev, V.A. Malyshev, F. Domı́nguez-Adame, J. Phys.Chem. B 107 (2003) 4418.[14] M. Bednarz, V.A. Malyshev, J. Knoester, Phys. Rev. Lett. 91 (2003)217401;M. Bednarz, V.A. Malyshev, J. Knoester, J. Chem. Phys. 120 (2004)3827.

University of GroningenThermal effects in exciton harvesting in biased one-dimensional systemsVlaming, S. M.; Malyshev, V.A.; Knoester, J.Published in:Journal of LuminescenceDOI:10.1016/j.jlumin.2007.11.038IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2008Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vlaming, S. M., Malyshev, V. A., & Knoester, J. (2008). Thermal effects in exciton harvesting in biased one-dimensional systems. Journal of Luminescence, 128(5-6), 956-959.https://doi.org/10.1016/j.jlumin.2007.11.038CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 24-12-2025Journal of Luminescence 128 (2008) 956–959Thermal effects in exciton harvesting in biased one-dimensional systemsS.M. Vlaming, V.A. Malyshev, J. KnoesterCentre for Theoretical Physics and Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The NetherlandsAvailable online 20 February 2008AbstractThe study of energy harvesting in chain-like structures is important due to its relevance to a variety of interesting physical systems.Harvesting is understood as the combination of exciton transport through intra-band exciton relaxation (via scattering on phononmodes) and subsequent quenching by a trap. Recently, we have shown that in the low temperature limit different harvesting scenarios asa function of the applied bias strength (magnitude of the energy gradient towards the trap) are possible [S.M. Vlaming, V.A. Malyshev, J.Knoester, J. Chem. Phys. 127 (2007) 154719]. This paper generalizes the results for both homogeneous and disordered chains to nonzerotemperatures. We show that thermal effects are appreciable only for low bias strengths, particularly so in disordered systems, and lead tofaster harvesting.r 2007 Elsevier B.V. All rights reserved.PACS: 71.23.An; 71.35.CcKeywords: Frenkel excitons; Exciton transfer; Exciton trapping; Energy bias; Photonic wires1. IntroductionThe study of energy transport and trapping in molecularscale systems is an interesting topic, because of its relevanceto natural systems (photosynthetic antenna complexes [1])and applications in nanophysics and quantum computing,where energy should efficiently be transported oversome distance. Linear molecular aggregates [2,3], conju-gated polymers [4,5], photonic wires [6,7] and quantumdot arrays [8,9] may all be envisioned as realizations ofsuch energy transport complexes. A natural question thatarises is how this combined process of transport andeventual trapping, which we will from now on refer toas harvesting, can be optimized. In a previous paper [10],we performed a study on the effect of an energy gradient(bias) on the energy harvesting properties of a linearsystem. The scope there was limited to low temperatures.In this paper, we present numerical calculations of thermaleffects on the harvesting time and a qualitative analysis ofthe results.2. ModelWe use a Frenkel exciton model to describe the excitedstate dynamics in chain-like systems. The latter aremodeled as a regular one-dimensional array of two-levelmonomers that are coupled through their transition dipolemoments, which are assumed to be oriented parallel toeach other. We initially excite a monomer at one end of thechain ðn ¼ NÞ, and consider the other end ðn ¼ 1Þ as a trap.The harvesting time is calculated as the average time ittakes for an excitation to move across the chain and to bequenched by the trap. Including diagonal disorder and abias ðD40Þ towards the trap, the model Hamiltonian readsHex ¼XNn¼1½en þ ðn 1ÞDjnihnj þXNn;manJnmjnihmj, (1)where the state jni denotes that monomer n is excited whileall other monomers are not, the excitation energies en areuncorrelated Gaussian stochastic variables with zero meanand standard deviation s and the transfer integrals areJnm ¼ J=jnmj3, with the nearest-neighbor couplingJ40. Diagonalization of this Hamiltonian yields collectiveexcited states, jsi ¼Pn csnjni, where the site amplitudescan be chosen real. Both a bias and disorder result inARTICLE IN PRESSwww.elsevier.com/locate/jlumin0022-2313/$ - see front matter r 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jlumin.2007.11.038Corresponding author. Tel.: +3150 3634369; fax: +31 50 3634947.E-mail address: j.knoester@rug.nl (J. Knoester).localization of the states, Bloch-like [11] and Anderson-like[12], respectively, so that the exciton states are spread outover finite segments of the chain only. In the case of a bias,there exists a correlation between the location of theeigenstate and its energy, as lower energy states are locatedin the vicinity of the trap.The dynamics of the populations Ps of the variousexciton states is analyzed within the framework of the Paulimaster equation,_Ps ¼ ðgs þ GsÞPs þXs0ðW ss0Ps0 W s0sPsÞ, (2)where gs ¼ g0ðPncsnÞ2 is the radiative rate of the state jsi(g0 being the radiative rate of a monomer) and Gs is thequenching rate, which is taken proportional to the excitonprobability at the trap ðn ¼ 1Þ: Gs ¼ Gc2s1 where G is thequenching amplitude [13], which cannot be arbitrarily largewithin the limitations of our model [10]. As at t ¼ 0 onlythe rightmost site n ¼ N is excited, the initial populationdistribution is given by Psð0Þ ¼ c2sN . The W ss0 describe thescattering of population between various exciton statesthrough interaction with phonons: W ss0 ¼W 0 SðEs  Es0 ÞPNn¼1c2snc2s0nGðEs  Es0 Þ, where W 0 is the scattering ampli-tude, SðEÞ is the phonon spectral density function,PNn¼1c2snc2s0n is the probability overlap, and GðEÞ ¼ 1þnðEÞ if Eo0, while GðEÞ ¼ nðEÞ if E40 with nðEÞ ¼½expðjEj=TÞ  11 the phonon occupation factor. We use aDebye-like spectral density with cut-off oc,SðEÞ ¼ jE=Jj3 expðjEj=ocÞ. The harvesting time is de-fined as follows: t1 ¼ t1trap  t10 , where ttrap and t0 are thedecay times with and without trap, respectively. They arecalculated asR10dthPsPsðtÞi, where h. . .i denotes thedisorder average.3. Disorder-free biased chainsWe begin with a discussion of the harvesting efficiency inhomogeneous chains ðs ¼ 0Þ. The bias-induced Blochlocalization dramatically changes the harvesting properties.Depending on the relative strengths of the quenching andrelaxation processes, both monotonic decreasing andnonmonotonic behavior of the harvesting time as afunction of the bias strength can be obtained. We refer toour earlier paper [10] for a full discussion of the possiblescenarios, while here we focus on the aspects relevant tounderstanding thermal effects in our model.In the limit of G5W 0 and for low temperature,relaxation to lower exciton states is the dominant processfor virtually all exciton states. As a result, most populationends up in the bottom states of the band. For zero bias, thequenching rate from these states is low because of the pooroverlap of the bottom states with the trap (see Fig. 1).However, turning on the bias increases this overlap,thereby accelerating the harvesting process. Temperaturewill enhance the harvesting efficiency at low bias by shiftingpopulation from the poorly quenched bottom state(s) tomore strongly quenched higher energy states. This en-hancement disappears for higher bias strengths, as thehigher energy states are no longer more strongly quenchedthan the bottom states, as is shown in Fig. 2(a).The interesting situation where GW 0 provides anonmonotonic bias dependency of t. A small bias isadvantageous for the harvesting process as it increases thequenching rate for the bottom states, but beyond someoptimal bias value the increase in relaxation time overtakesthe previous effect in importance and the net effect ofincreasing the bias becomes negative. The thermallyinduced changes in the harvesting efficiency are again onlypresent for small values of the bias, for the reasonsmentioned in the discussion of the limit of G5W 0. This iscorroborated in Fig. 2(b).In the limit GbW 0 our model does not adequatelydescribe the harvesting, in particular when the exciton statesare delocalized over the entire chain (i.e., small bias, smalldisorder, and relatively short chains). In this situation, theoff-diagonal density matrix elements (exciton coherences),which are also created by populating the site n ¼ Nðrss0 ð0Þ ¼ csNcs0NÞ, are sufficiently long-lived to be relevantto the transport process. In fact, neglecting coherences leadsto the unphysical artifact that there is instantaneous energytransfer over the chain. A more refined approach isnecessary to remove the occurrence of unphysically fastenergy transfer; we refer to our earlier paper [10] for a moredetailed discussion of this limit and its complications.4. Disordered biased chainsThe inclusion of disorder slows down the energytransport process for all bias values; however, the effectARTICLE IN PRESS0.1 0.050|cs1|220 4010 30 5000.10.05state number, s|csN|2Δ=0Δ=0.01JΔ=0.1JΔ=0Δ=0.01JΔ=0.1JFig. 1. Bias dependence of the exciton probabilities c2sN and c2s1 thatdetermine the initial distribution of the exciton population Psð0Þ and thequenching rate Gs, respectively, for chains without disorder ðs ¼ 0Þ.The bias tends to shift the initial distribution of population to the top ofthe band (to higher s). Oppositely, the strongly quenched states occur atthe lower band edge (for small s).S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 957is especially strong for small biases. This will lead to a shiftof the optimal bias (when W 0G) to higher values withincreasing disorder strength. The overall shape of the tðDÞcurve does not change in most cases: only in the case of ahigh quenching amplitude Fig. 2(b), we see that an optimalbias appears as a result of the fact that the disorder doesmore damage for low biases than for higher ones.The explanation of these effects is straightforward.Disorder results in Anderson localization, even in theabsence of a bias. This leads to reduced relaxation rates,since the overlap of different states decreases. The effect isparticularly significant for systems with a low bias, since ina disorder-free chain the excitons are then delocalized overthe whole chain. For higher biases, there is alreadyconsiderable localization because of the bias itself, andthe effect produced by the disorder is relatively small. Moreimportantly, for sufficiently strong disorder and small biasthe lowest energy states are not necessarily located near thetrap. At low temperatures, a large population is built up inthese states and the excitation may thus be unable to betrapped efficiently because of the blockage of diffusion[13,14]. A bias forces these low energy states to be locatednear the trap and thus accelerates the harvesting.Thermal effects are marginal for large bias strengths. Inthis case, the harvesting process consists largely of down-ward relaxation followed by quenching from the bottomstates. These states tend to be localized near the trap andthe quenching process is therefore hardly accelerated bythermally induced population shifts to higher energy states.In contrast, the thermal effects for small bias are quitestrong in disordered systems, even more so than inhomogeneous chains. As has been noted in the previousparagraphs, population tends to end up in the excitonstates at the bottom of the band. These, in general, are notlocated near the trap and thus are quenched poorly.Increasing the temperature has two effects. First, it rapidlyaccelerates the exciton diffusion, and second, the popula-tion is thermally excited to higher energy states, whichare quenched more strongly due to a better overlap withthe trap. In other words, the interaction of excitons withthe phonon bath can counteract some of the detrimentaleffects of Anderson localization on the harvesting process,as can be seen in Fig. 2.5. ConclusionsContrary to what one might naively expect, namely thatan energetic bias towards the trap will decrease theharvesting time, we have shown that various scenariosare possible, depending on the relative strengths of therelevant subprocesses. This is already apparent in lowtemperature systems (with and without disorder) [10], andthe same holds for systems at elevated temperature.Thermal effects are only significant for small bias strengthsand tend to accelerate harvesting, especially in disorderedsystems.References[1] K. Mukai, S. Abe, H. Sumi, J. Phys. Chem. B 103 (1999) 6096.[2] T. Kobayashi (Ed.), J-aggregates, World Scientific, Singapore, 1996.[3] J. Knoester, in: V.M. Agranovich, G.C. La Rocca (Eds.), Proceedingsof the International School of Physics ‘‘Enrico Fermi’’; CourseCXLIX, Organic nanostructures: science and application, IOS Press,Amsterdam, 2002, pp. 149–186.[4] A. Tilgner, H.P. Trommsdorff, J.M. Zeigler, R.M. Hochstrasser,J. Chem. Phys. 96 (1992) 781.ARTICLE IN PRESS0.10.05010002000300040005000Δ (J)τ (J-1)0 0.05 0.1100101102103τ (J-1)increasing T increasing TΔ (J)Fig. 2. Harvesting time t versus the applied bias magnitude D calculated for disordered chains of 50 sites for the quenching amplitude G ¼ 101J (panel(a)) and G ¼ 10J (panel (b)) and the scattering amplitude W 0 ¼ 10J. The curves were calculated for different temperatures, T ¼ 0:02J, 0:2J and 0:4J,which are ordered as shown by the arrow. Solid, dotted and dashed curves correspond to disorder strength s ¼ 0, 0:2J and 0:4J, respectively.S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959958[5] For a recent overview see, e.g., G. Hadziioannou, P. van Hutten(Eds.), ‘‘Semiconducting Polymers—Chemistry, Physics, and En-gineering’’; VCH, Weinheim, 1999.[6] R.W. Wagner, J.S. Lindsey, J. Amer. Chem. Soc. 116 (1994) 9759.[7] M. Heilemann, P. Tinnefeld, G. Sanchez Mosteiro, M. Garcia Parajo,N.F. van Hulst, M. Sauer, J. Amer. Chem. Soc. 126 (2004) 6514.[8] S.A. Crooker, J.A. Hollingsworth, S. Tretiak, L.V. Klimov, Phys.Rev. Lett. 89 (2002) 186802.[9] T. Franzl, T.A. Klar, S. Schietinger, A.L. Rogach, J. Feldmann,Nano Lett. 4 (2004) 1599.[10] S.M. Vlaming, V.A. Malyshev, J. Knoester, J. Chem. Phys. 127(2007) 154719.[11] F. Bloch, z. Phys. 52 (1927) 555; C. Zener, Proc. R. Soc. London, Ser.A 145 (1934) 523.[12] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrish-nan, Phys. Rev. Lett. 42 (1979) 673.[13] A.V. Malyshev, V.A. Malyshev, F. Domı́nguez-Adame, Chem. Phys.Lett. 371 (2003) 417;A.V. Malyshev, V.A. Malyshev, F. Domı́nguez-Adame, J. Phys.Chem. B 107 (2003) 4418.[14] M. Bednarz, V.A. Malyshev, J. Knoester, Phys. Rev. Lett. 91 (2003)217401;M. Bednarz, V.A. Malyshev, J. Knoester, J. Chem. Phys. 120 (2004)3827.ARTICLE IN PRESSS.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 959

The study of energy harvesting in chain-like structures is important due to its relevance to a variety of interesting physical systems. Harvesting is understood as the combination of exciton transport through intra-band exciton relaxation (via scattering on phonon modes) and subsequent quenching by a trap. Recently, we have shown that in the low temperature limit different harvesting scenarios as a function of the applied bias strength (magnitude of the energy gradient towards the trap) are possible [S.M. Vlaming, V.A. Malyshev, J. Knoester, J. Chem. Phys. 127 (2007) 154719]. This paper generalizes the results for both homogeneous and disordered chains to nonzero temperatures. We show that thermal effects are appreciable only for low bias strengths, particularly so in disordered systems, and lead to faster harvesting. (C) 2007 Elsevier B.V. All rights reserved

NARCIS 

University of GroningenThermal effects in exciton harvesting in biased one-dimensional systemsVlaming, S. M.; Malyshev, V.A.; Knoester, J.Published in:Journal of LuminescenceDOI:10.1016/j.jlumin.2007.11.038IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2008Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vlaming, S. M., Malyshev, V. A., & Knoester, J. (2008). Thermal effects in exciton harvesting in biased one-dimensional systems. Journal of Luminescence, 128(5-6), 956-959.https://doi.org/10.1016/j.jlumin.2007.11.038CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 30-12-2025Journal of Luminescence 128 (2008) 956–959Thermal effects in exciton harvesting in biased one-dimensional systemsS.M. Vlaming, V.A. Malyshev, J. KnoesterCentre for Theoretical Physics and Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The NetherlandsAvailable online 20 February 2008AbstractThe study of energy harvesting in chain-like structures is important due to its relevance to a variety of interesting physical systems.Harvesting is understood as the combination of exciton transport through intra-band exciton relaxation (via scattering on phononmodes) and subsequent quenching by a trap. Recently, we have shown that in the low temperature limit different harvesting scenarios asa function of the applied bias strength (magnitude of the energy gradient towards the trap) are possible [S.M. Vlaming, V.A. Malyshev, J.Knoester, J. Chem. Phys. 127 (2007) 154719]. This paper generalizes the results for both homogeneous and disordered chains to nonzerotemperatures. We show that thermal effects are appreciable only for low bias strengths, particularly so in disordered systems, and lead tofaster harvesting.r 2007 Elsevier B.V. All rights reserved.PACS: 71.23.An; 71.35.CcKeywords: Frenkel excitons; Exciton transfer; Exciton trapping; Energy bias; Photonic wires1. IntroductionThe study of energy transport and trapping in molecularscale systems is an interesting topic, because of its relevanceto natural systems (photosynthetic antenna complexes [1])and applications in nanophysics and quantum computing,where energy should efficiently be transported oversome distance. Linear molecular aggregates [2,3], conju-gated polymers [4,5], photonic wires [6,7] and quantumdot arrays [8,9] may all be envisioned as realizations ofsuch energy transport complexes. A natural question thatarises is how this combined process of transport andeventual trapping, which we will from now on refer toas harvesting, can be optimized. In a previous paper [10],we performed a study on the effect of an energy gradient(bias) on the energy harvesting properties of a linearsystem. The scope there was limited to low temperatures.In this paper, we present numerical calculations of thermaleffects on the harvesting time and a qualitative analysis ofthe results.2. ModelWe use a Frenkel exciton model to describe the excitedstate dynamics in chain-like systems. The latter aremodeled as a regular one-dimensional array of two-levelmonomers that are coupled through their transition dipolemoments, which are assumed to be oriented parallel toeach other. We initially excite a monomer at one end of thechain ðn ¼ NÞ, and consider the other end ðn ¼ 1Þ as a trap.The harvesting time is calculated as the average time ittakes for an excitation to move across the chain and to bequenched by the trap. Including diagonal disorder and abias ðD40Þ towards the trap, the model Hamiltonian readsHex ¼XNn¼1½en þ ðn 1ÞDjnihnj þXNn;manJnmjnihmj, (1)where the state jni denotes that monomer n is excited whileall other monomers are not, the excitation energies en areuncorrelated Gaussian stochastic variables with zero meanand standard deviation s and the transfer integrals areJnm ¼ J=jnmj3, with the nearest-neighbor couplingJ40. Diagonalization of this Hamiltonian yields collectiveexcited states, jsi ¼Pn csnjni, where the site amplitudescan be chosen real. Both a bias and disorder result inARTICLE IN PRESSwww.elsevier.com/locate/jlumin0022-2313/$ - see front matter r 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jlumin.2007.11.038Corresponding author. Tel.: +3150 3634369; fax: +31 50 3634947.E-mail address: j.knoester@rug.nl (J. Knoester).localization of the states, Bloch-like [11] and Anderson-like[12], respectively, so that the exciton states are spread outover finite segments of the chain only. In the case of a bias,there exists a correlation between the location of theeigenstate and its energy, as lower energy states are locatedin the vicinity of the trap.The dynamics of the populations Ps of the variousexciton states is analyzed within the framework of the Paulimaster equation,_Ps ¼ ðgs þ GsÞPs þXs0ðW ss0Ps0 W s0sPsÞ, (2)where gs ¼ g0ðPncsnÞ2 is the radiative rate of the state jsi(g0 being the radiative rate of a monomer) and Gs is thequenching rate, which is taken proportional to the excitonprobability at the trap ðn ¼ 1Þ: Gs ¼ Gc2s1 where G is thequenching amplitude [13], which cannot be arbitrarily largewithin the limitations of our model [10]. As at t ¼ 0 onlythe rightmost site n ¼ N is excited, the initial populationdistribution is given by Psð0Þ ¼ c2sN . The W ss0 describe thescattering of population between various exciton statesthrough interaction with phonons: W ss0 ¼W 0 SðEs  Es0 ÞPNn¼1c2snc2s0nGðEs  Es0 Þ, where W 0 is the scattering ampli-tude, SðEÞ is the phonon spectral density function,PNn¼1c2snc2s0n is the probability overlap, and GðEÞ ¼ 1þnðEÞ if Eo0, while GðEÞ ¼ nðEÞ if E40 with nðEÞ ¼½expðjEj=TÞ  11 the phonon occupation factor. We use aDebye-like spectral density with cut-off oc,SðEÞ ¼ jE=Jj3 expðjEj=ocÞ. The harvesting time is de-fined as follows: t1 ¼ t1trap  t10 , where ttrap and t0 are thedecay times with and without trap, respectively. They arecalculated asR10dthPsPsðtÞi, where h. . .i denotes thedisorder average.3. Disorder-free biased chainsWe begin with a discussion of the harvesting efficiency inhomogeneous chains ðs ¼ 0Þ. The bias-induced Blochlocalization dramatically changes the harvesting properties.Depending on the relative strengths of the quenching andrelaxation processes, both monotonic decreasing andnonmonotonic behavior of the harvesting time as afunction of the bias strength can be obtained. We refer toour earlier paper [10] for a full discussion of the possiblescenarios, while here we focus on the aspects relevant tounderstanding thermal effects in our model.In the limit of G5W 0 and for low temperature,relaxation to lower exciton states is the dominant processfor virtually all exciton states. As a result, most populationends up in the bottom states of the band. For zero bias, thequenching rate from these states is low because of the pooroverlap of the bottom states with the trap (see Fig. 1).However, turning on the bias increases this overlap,thereby accelerating the harvesting process. Temperaturewill enhance the harvesting efficiency at low bias by shiftingpopulation from the poorly quenched bottom state(s) tomore strongly quenched higher energy states. This en-hancement disappears for higher bias strengths, as thehigher energy states are no longer more strongly quenchedthan the bottom states, as is shown in Fig. 2(a).The interesting situation where GW 0 provides anonmonotonic bias dependency of t. A small bias isadvantageous for the harvesting process as it increases thequenching rate for the bottom states, but beyond someoptimal bias value the increase in relaxation time overtakesthe previous effect in importance and the net effect ofincreasing the bias becomes negative. The thermallyinduced changes in the harvesting efficiency are again onlypresent for small values of the bias, for the reasonsmentioned in the discussion of the limit of G5W 0. This iscorroborated in Fig. 2(b).In the limit GbW 0 our model does not adequatelydescribe the harvesting, in particular when the exciton statesare delocalized over the entire chain (i.e., small bias, smalldisorder, and relatively short chains). In this situation, theoff-diagonal density matrix elements (exciton coherences),which are also created by populating the site n ¼ Nðrss0 ð0Þ ¼ csNcs0NÞ, are sufficiently long-lived to be relevantto the transport process. In fact, neglecting coherences leadsto the unphysical artifact that there is instantaneous energytransfer over the chain. A more refined approach isnecessary to remove the occurrence of unphysically fastenergy transfer; we refer to our earlier paper [10] for a moredetailed discussion of this limit and its complications.4. Disordered biased chainsThe inclusion of disorder slows down the energytransport process for all bias values; however, the effectARTICLE IN PRESS0.1 0.050|cs1|220 4010 30 5000.10.05state number, s|csN|2Δ=0Δ=0.01JΔ=0.1JΔ=0Δ=0.01JΔ=0.1JFig. 1. Bias dependence of the exciton probabilities c2sN and c2s1 thatdetermine the initial distribution of the exciton population Psð0Þ and thequenching rate Gs, respectively, for chains without disorder ðs ¼ 0Þ.The bias tends to shift the initial distribution of population to the top ofthe band (to higher s). Oppositely, the strongly quenched states occur atthe lower band edge (for small s).S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 957is especially strong for small biases. This will lead to a shiftof the optimal bias (when W 0G) to higher values withincreasing disorder strength. The overall shape of the tðDÞcurve does not change in most cases: only in the case of ahigh quenching amplitude Fig. 2(b), we see that an optimalbias appears as a result of the fact that the disorder doesmore damage for low biases than for higher ones.The explanation of these effects is straightforward.Disorder results in Anderson localization, even in theabsence of a bias. This leads to reduced relaxation rates,since the overlap of different states decreases. The effect isparticularly significant for systems with a low bias, since ina disorder-free chain the excitons are then delocalized overthe whole chain. For higher biases, there is alreadyconsiderable localization because of the bias itself, andthe effect produced by the disorder is relatively small. Moreimportantly, for sufficiently strong disorder and small biasthe lowest energy states are not necessarily located near thetrap. At low temperatures, a large population is built up inthese states and the excitation may thus be unable to betrapped efficiently because of the blockage of diffusion[13,14]. A bias forces these low energy states to be locatednear the trap and thus accelerates the harvesting.Thermal effects are marginal for large bias strengths. Inthis case, the harvesting process consists largely of down-ward relaxation followed by quenching from the bottomstates. These states tend to be localized near the trap andthe quenching process is therefore hardly accelerated bythermally induced population shifts to higher energy states.In contrast, the thermal effects for small bias are quitestrong in disordered systems, even more so than inhomogeneous chains. As has been noted in the previousparagraphs, population tends to end up in the excitonstates at the bottom of the band. These, in general, are notlocated near the trap and thus are quenched poorly.Increasing the temperature has two effects. First, it rapidlyaccelerates the exciton diffusion, and second, the popula-tion is thermally excited to higher energy states, whichare quenched more strongly due to a better overlap withthe trap. In other words, the interaction of excitons withthe phonon bath can counteract some of the detrimentaleffects of Anderson localization on the harvesting process,as can be seen in Fig. 2.5. ConclusionsContrary to what one might naively expect, namely thatan energetic bias towards the trap will decrease theharvesting time, we have shown that various scenariosare possible, depending on the relative strengths of therelevant subprocesses. This is already apparent in lowtemperature systems (with and without disorder) [10], andthe same holds for systems at elevated temperature.Thermal effects are only significant for small bias strengthsand tend to accelerate harvesting, especially in disorderedsystems.References[1] K. Mukai, S. Abe, H. Sumi, J. Phys. Chem. B 103 (1999) 6096.[2] T. Kobayashi (Ed.), J-aggregates, World Scientific, Singapore, 1996.[3] J. Knoester, in: V.M. Agranovich, G.C. La Rocca (Eds.), Proceedingsof the International School of Physics ‘‘Enrico Fermi’’; CourseCXLIX, Organic nanostructures: science and application, IOS Press,Amsterdam, 2002, pp. 149–186.[4] A. Tilgner, H.P. Trommsdorff, J.M. Zeigler, R.M. Hochstrasser,J. Chem. Phys. 96 (1992) 781.ARTICLE IN PRESS0.10.05010002000300040005000Δ (J)τ (J-1)0 0.05 0.1100101102103τ (J-1)increasing T increasing TΔ (J)Fig. 2. Harvesting time t versus the applied bias magnitude D calculated for disordered chains of 50 sites for the quenching amplitude G ¼ 101J (panel(a)) and G ¼ 10J (panel (b)) and the scattering amplitude W 0 ¼ 10J. The curves were calculated for different temperatures, T ¼ 0:02J, 0:2J and 0:4J,which are ordered as shown by the arrow. Solid, dotted and dashed curves correspond to disorder strength s ¼ 0, 0:2J and 0:4J, respectively.S.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959958[5] For a recent overview see, e.g., G. Hadziioannou, P. van Hutten(Eds.), ‘‘Semiconducting Polymers—Chemistry, Physics, and En-gineering’’; VCH, Weinheim, 1999.[6] R.W. Wagner, J.S. Lindsey, J. Amer. Chem. Soc. 116 (1994) 9759.[7] M. Heilemann, P. Tinnefeld, G. Sanchez Mosteiro, M. Garcia Parajo,N.F. van Hulst, M. Sauer, J. Amer. Chem. Soc. 126 (2004) 6514.[8] S.A. Crooker, J.A. Hollingsworth, S. Tretiak, L.V. Klimov, Phys.Rev. Lett. 89 (2002) 186802.[9] T. Franzl, T.A. Klar, S. Schietinger, A.L. Rogach, J. Feldmann,Nano Lett. 4 (2004) 1599.[10] S.M. Vlaming, V.A. Malyshev, J. Knoester, J. Chem. Phys. 127(2007) 154719.[11] F. Bloch, z. Phys. 52 (1927) 555; C. Zener, Proc. R. Soc. London, Ser.A 145 (1934) 523.[12] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrish-nan, Phys. Rev. Lett. 42 (1979) 673.[13] A.V. Malyshev, V.A. Malyshev, F. Domı́nguez-Adame, Chem. Phys.Lett. 371 (2003) 417;A.V. Malyshev, V.A. Malyshev, F. Domı́nguez-Adame, J. Phys.Chem. B 107 (2003) 4418.[14] M. Bednarz, V.A. Malyshev, J. Knoester, Phys. Rev. Lett. 91 (2003)217401;M. Bednarz, V.A. Malyshev, J. Knoester, J. Chem. Phys. 120 (2004)3827.ARTICLE IN PRESSS.M. Vlaming et al. / Journal of Luminescence 128 (2008) 956–959 959

Thermal effects in exciton harvesting in biased one-dimensional systems

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