The radiation decay of excitons in a slightly disordered molecule chain is considered. The degree of disordering dictates the excitonic state length that determines the enhancement of the decay rate Г (superradiance). Г was calculated versus disordering degree and temperature and compared with the case of a regular chain where the length of excitonic state is determined by the chain length, N. For every degree of disordering, the value of N was found that provides practically the same decay rate as a function of temperature

Ratner, M.A.

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Fizika Nizkikh Temperatur, 2003, v. 29, No. 7, p. 804–808Superradiance of J-aggregates: correspondencebetween infinite disordered chain and regular chainof finite lengthM.A. RatnerInstitute for Single Crystals of the National Academy of Sciences of Ukraine60 Lenin Ave., Kharkov 61001, UkraineÅ-mail: ratner@isc.kharkov.comReceived November 19, 2002The radiation decay of excitons in a slightly disordered molecule chain is considered. The de-gree of disordering dictates the excitonic state length that determines the enhancement of the de-cay rate  (superradiance).  was calculated versus disordering degree and temperature and com-pared with the case of a regular chain where the length of excitonic state is determined by thechain length, N. For every degree of disordering, the value of N was found that provides practi-cally the same decay rate as a function of temperature.PACS: 73.20.Dx, 78.55.–m, 78.66.–wA term «J-aggregate» denotes an almost periodicchain of organic impurity molecules in an organic ma-trix (J-aggregates are called after the name of one oftheir discoverers E. Jelley (1937) [1] ). Since seven-ties, J-aggregates attracted attention of researchers inseveral aspects. Of great physical interest is a phenom-enon of superradiance that consists in a shortening ofthe luminescence decay time of J-aggregates as com-pared with that of the corresponding monomer. Thedegree of shortening (considered below) depends onthe chain regularity degree and can be used for the di-agnostics of the chain structure and for studying themolecule—matrix interaction responsible for a slightdisordering of the chain [211].The study of J-aggregates is perspective also for bi-ological applications; in particular, electronic relaxa-tion processes in a molecule chain reveal some si-milarity with the mechanisms of light energytransformation in biological photosynthesis systems[12,13]. In 1991 , it was proposed to use the excitonicluminescence of some J-aggregates as a probe for theliving state of biological cells [14]. J-aggregates areformed on the cell membrane if they are attracted to itby the membrane potential that exists only in an alivecell and disappears if the cell is dead (the formation ofJ-aggregates is revealed through their luminescence).For all aspects of the study and applications ofJ-aggregates, of importance is the correspondence be-tween the degree of their disordering and the shorten-ing of the luminescence decay time  that can be mea-sured experimentally and used for the diagnostics ofthe structure. Up to now, this correspondence was es-tablished within the model of a regular (strictly peri-odic) chain of a finite length N (length is expressed inunits of the chain period a) [29,15,16].The essence of this model is physically clear andconsists in what follows. Consider an one-dimensionalexcitonic band corresponding to the lowest excitedelectronic state  of a separate molecule. Thisexcitonic state can be presented as a linear combina-tion of the orthonormalized molecular states centeredat the chain sites n: 11Ncn nnN(1)(subscript  numbers excitonic states which for a re-gular chain correspond to the wave vectors k). At alow temperature, when only the lowest excitonicstate is occupied, the rate  of the radiative transitionto the ground state is proportional to the squared ma-trix element of dipole moment between the excitedand ground states. For the lowest excitonic state,© M.A. Ratner, 2003since all the coefficients cn have the same sign and thecorresponding matrix element for a separate moleculeis independent of n, the matrix element between thestate (1) and the ground state is proportional to N1/2;thus,  is proportional to the chain length N. The co-efficient of proportionality can be found at N = 1 asthe radiative decay rate, 0, of a separate excited mo-lecule. Thus, the radiative decay rate of the lowestexcitonic state  exceeds that of a separate moleculeabout N times: 	 N 0 . (2)Relation (2) is valid only for the lowest excitonicstate. For a higher excitonic state, the coefficients cnin (1) have alternate signs and  is much less than thequantity (2). It follows that  has a decreasing tem-perature dependence. For a regular chain, the depend-ence of  on N and temperature is well known; belowit will be inferred to compare with the case of a disor-dered chains of an infinite length.The model of a regular (periodic) chain of a finitelength N qualitatively describes the behavior of exci-tons in an infinite slightly disordered chain where thelowest excitonic state is always localized; the localiza-tion radius determines an effective value of N. Thismodel is broadly used in view of its simplicity andillustrativity. However, up to now the model hadqualitative meaning for two reasons. First, no quanti-tative dependence was established between N and thedisordering degree of an infinite chain (this was con-nected with difficulties of modeling an infinite disor-dered chain). Second, it was not known with what ac-curacy the model (even with an optimally chosen N)describes the temperature dependence of the super-radiance rate of excitons in a disordered chain.The purpose of the present work is to impart aquantitative character to the model of finite chain. Tothat end, first, the quantitative correspondence willbe established between the model chain length N andthe degree of disordering of the true infinite chain.Second, the temperature dependence of the super-radiance rate of excitons in a disordered chain will becalculated and compared with the case of the regularchain of the best fitting length. It is assumed thatsuperradiance occurs under condition of the completethermodynamic equilibrium.This consideration will be carried on the base ofgeneral relations for the excitonic states and super-radiance rate of a molecule chain with an arbitrarylength M and an arbitrary degree of disordering. Thechain is placed into an inorganic crystalline matrix;the discrepancy between the periods of the chain andmatrix lattice is the origin of a slight aperiodicity ofthe chain potential. The excitonic band is assumed tocorrespond to the lowest excited electronic state of aseparate molecule of the chain (the higher states of themolecule are not taken into account). In the absenceof disordering at M = N, this general case turns to themodel of the regular chain of a finite length N; in thepresence of disordering at a large enough M, the caseof a true infinite disordered chain is realized.The Hamiltonian of the electronic system of thechain can be written down asH = H0n  V . (3)Here H0n is the Hamiltonian of a separate molecule in-cluding the Hamiltonian of the free molecule and thecrystalline field acting on the nth molecule in thechain; term V describes interaction between elec-tronic subsystems of different molecules. Accordingto (1), excitonic states , i.e., eigenfunctions of theHamiltonian (3), are expanded in the orthonorma-lized states, n , of the chain containing excited mole-cule in the nth site (n = 1, ..., M). By definition, nis an eigenfunction of the Hamiltonian H0n that,hence, has no off-diagonal matrix elements, whereasV has no diagonal elements (the diagonal part of V iscarried over to the first term in (3) ):H0n n = n n, H0nm = nnm ,  Vnn = 0. (4)The interaction potential V is assumed to be of ex-change character which permits one to neglect all thematrix elements Vnm except those between adjacentsites, Vn,n+1 and Vn,n1. The latter take random va-lues around their average V, and site levels n takerandom values around their average position that isput equal to zero. In the absence of disordering, theexcitonic band has the width W = 4V extending from2V to 2V.The Shröedinger equation with Hamiltonian (3)can be written down in the site representation in ausual way. Making allowance for (1) and (4), one ob-tains finally a system of linear homogeneous equationsrelative to the expansion coefficients cn :  n n n n n n n n nc V c V c E c     , , , ,1 1 1 1(, n = 1,..., M)(5)(subscript  numbers excitonic states with energyE). Equation (5) should be complemented by theboundary condition that runs for a linear chain of thelength M:c0 = 0, cM +1 = 0 . (6)The chain can be also closed into a ring of the lengthM with periodic boundary conditionscn+M cn, Vn+M,n+M+1= Vn,n+1. (7)Superradiance of J-aggregates: correspondence between infinite disordered chainFizika Nizkikh Temperatur, 2003, v. 29, No. 7 805The exciton energy levels Ecan be found viaequating to zero the determinant of the equation sys-tem (5). Then the orthonormalized set of solutions{cn} is obtained from equation (5) after substitutingfor every  the corresponding value of E.The rate, , of the radiation decay from the thexcited state to the ground state is expressed in anusual way through the solution of Eq. (5):  012cnnM. (8)The total decay rate  under thermodynamic equilib-rium is:( , )exp ( )exp ( )M TE /TE /TMM11(9)(T stands for temperature in energy units). It is thelowest excited state with  = 1 that makes the pre-dominant contribution to the decay rate (9) due to aconstant sign of cn1; hence a decreasing temperatureof  follows .In a regular chain, n and Vn,n+1 = V are independ-ent of n, and Eq. (5) has a simple solutioncikanMnk exp ( ),E V kak  2 cos , k = 2/Ma,  = 1,..., M .(10)The expression (8) , with allowance for (10), takesthe formkM ka/ Mka/ ka/M ka4 2 2 22 2sin [( ) ] sin ( ) )sin ( ) co1 ctans [( ) ] sin ( ).M ka Mka 1(11)Let us consider now an infinite slightly disorderedchain. Two possible types of disordering will be con-sidered: (1) diagonal disordering, i.e., the random,straggling of site levels n around their average   0 and (2) off-diagonal disordering, i.e., thestraggling of the matrix elements of interaction,Vn,n+1, near their average   V Vn n, 1  0.25W (Wis the band width of a long regular chain). The strag-gling is described by the Gaussian distribution withthe dispersion dia or off in the former or latter case,respectively.The cases of diagonal and off-diagonal disorderingwere considered separately. For a given dia or off,1000 realizations of the chain with random sets of n orVn n, 1 were constructed. For every realization, theequation system (5) was solved numerically. Theboundary condition (7), related to the chain closed toa ring, was used in order to avoid edge effects; suchring of the length M >>N practically exactly modelsan infinite chain (N is the length of the equivalentregular chain). It was numerically verified that a ringwith the length M = 100 practically exactly models aninfinite chain with the equivalent length N  35.The range of dispersion  was taken from 0.05W to0.3W for diagonal disorder, and from 0.01W to 0.1Wfor off-diagonal disorder. For every dia or off, onecan indicate the equivalent regular chain length N ()which provides the same decay rate at zero tempera-ture. Figure 1 shows the dependence of N on dia oroff . It can be seen from Fig. 1 that the off-diagonaldisordering with dispersion off is approximately equi-valent to the diagonal disordering with dispersion3off . For estimation, considering diagonal and off-di-agonal disordering as independent random processes,one can reduce the simultaneous diagonal disorderingwith dispersion dia and off-diagonal disordering withdispersion off to the diagonal disordering with disper-sion (dia2 + 9off2)1/2.It was found that the temperature dependence ofthe decay rate ( )T for an infinite disordered chain iswell approximated by that of the regular chain of thecorresponding equivalent length without restriction ofthe temperature range. Figures 2 and 3 show the ex-amples of such approximation.806 Fizika Nizkikh Temperatur, 2003, v. 29, No. 7M.A. Ratner1020300 0.1 0.2 0.3NFig. 1. The dependence of the equivalent length N of theregular chain on the disordering degree, , of the infinitechain. Solid curve — N(dia ) at off = 0, dotted curve —N(off) at dia = 0. Dashed line shows the dependenceN(off/3) at dia = 0.It should be taken into account that Figs. 2 and 3show the decay rate averaged over a large number ofrandom realizations of the disordered chain. Such av-eraging takes place at a single excitation pulse, if it isstrong enough to excite a large number of chains. Ifevery pulse excites only one molecule (single moleculespectroscopy), observed  has a significant straggling,whose dispersion is shown in Figs. 2 and 3 by dottedlines.In the indicated meaning, an infinite irregularchain can be modeled by a regular chain of a finitelength. However, this correspondence is restricted tothe superradiance rate and cannot be extended to ab-sorption spectrum (consisting of discrete lines for aregular chain and smeared to a band for a disorderedchain).The author is grateful to Yu.V. Malyukin and A.M.Ratner for helpful discussions.Superradiance of J-aggregates: correspondence between infinite disordered chainFizika Nizkikh Temperatur, 2003, v. 29, No. 7 807a24680 0.5 1.0Temperature / bandwidthDecay  rateb1020300 0.05 0.1 0.15 0.2Temperature / bandwidthDecay  rateFig. 2. Temperature dependence of radiative decay rate for a chain with diagonal disorder in units of that of themonomer. Thick solid line relates to the infinite disorderedchain with dia = 0.3 (a) and 0.05 (b). Thin solid line re-lates to the regular chain with the length N = 8 (a) andN = 31 (b) that best models the disordered chain. Dottedline shows the dispersion of  over random realizations ofthe disordered chain.a24680 0.5 1Temperature/bandwidthDecay  rateb102030400 0.1 0.2Temperature/bandwidthDecay  rateFig. 3. Temperature dependence of radiative decay rate for a chain with off-diagonal disorder in units of that ofthe monomer. Thick solid line relates to the infinite disor-dered chain with off = 0.1 (a) and 0.01 (b). Thin solidline relates to the regular chain with the length N = 8 (a)and N = 35 (b) that best models the disordered chain.Dotted line shows the dispersion of  over random realiza-tions of the disordered chain.This work was carried out within the projectINTAS 2001-240.1. E.E. Jelley, Nature (London) 139, 631 (1937).2. E.W. Knapp, Chem. Phys. 85, 73 (1984).3. H. Fidder and D.A. Wiersma, Phys. Rev. Lett. 66,1501 (1991).4. H. Fidder, J. Knoester, and D.A. Wiersma, J. Chem.Phys. 95, 7880 (1991).5. J. Knoester, J. Chem. Phys. 99, 7846 (1993).6. M. Wubs and J. Knoester, Chem. Phys. Lett. 284, 63(1998).7. H. Fidder, J. Knoester, and D.A. Wiersma, J. Chem.Phys. 98, 6564 (1993).8. J.R. Durrant, J. Knoester, and D.A. Wiersma, Chem.Phys. Lett. 222, 450 (1994).9. J. Knoester, J. Lumin. 58, 107 (1994).10. J. Klafter and J. Jortner, J. Chem. Phys. 68, 1513(1978).11. M. Schreiber and Y. Toyozawa, J. Phys. Soc. Jpn. 51,1528 (1981).12. Th. Renger and V. May, Phys. Rev. Lett. 78, 3406(1997).13. Th. Renger and V. May, J. Phys. Chem. B101, 7211(1997).14. M. Reers, T.W. Smith, and L.B. Chen, Biochemistry30, 4480 (1991).15. S. Boer and D.A. Wiersma, J. Lumin. 45, 241 (1990).16. V. Sundstrom, T. Gillbro, R.A. Gadonas, and A. Pis-karskas, J. Chem. Phys. 89, 2754 (1988).808 Fizika Nizkikh Temperatur, 2003, v. 29, No. 7M.A. Ratner

Superradiance of J-aggregates: correspondence between infinite disordered chain and regular chain of finite length

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Superradiance of J-aggregates: correspondence between infinite disordered chain and regular chain of finite length

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