*Corresponding author. Tel.: #49-3641-205621; fax: #49-3641-205601.

E-mail address: feller@mt.fh-jena.de (K.-H. Feller)1Permanent address. All-Russian Research Center Vavilov

State Optical Institute, Birzhevaya Liniya 12, 199034 Saint-Petersburg, Russia.

Journal of Luminescence 83}84 (1999) 291}295

Abrupt switching and hysteresis of transmittivity of anultrathin "lm consisting of J-aggregates

V.A. Malyshev!,1, H. Glaeske!, K.-H. Feller!,*

!Fachhochschule Jena, Fachbereich Medizintechnik/Physikalische Technik, Tatzendpromenade 1b, D-07745 Jena, Germany

Abstract

A theoretical study of transmittivity of an ultrathin "lm (of thickness smaller than an optical wavelength) comprized oflinear molecular aggregates is carried out. We report on an abrupt change of transmittivity of the "lm at a certainmagnitude of the incident "eld. The e!ect originates from the coupling of aggregates via the emitting "eld. Estimates ofdriving parameters show that such well-studied objects as J-aggregates of polymethine dyes provide a real basis to realizeit. ( 1999 Elsevier Science B.V. All rights reserved.

PACS: 42.65.Pc; 78.66.!w

Keywords: Assemblies of molecular aggregates; Optical bistability; Disorder

1. Introduction

Recently, the problem of bistable behavior in theoptical response of a single linear molecular ag-gregate [1}5] and even of a dimer [1,6}8] createdmuch interest. The e!ect consists of a suddenswitching of the population from a low level toa higher one as the pump intensity rises and orig-inates from the dynamical resonance frequencyshift depending on the population of the system.The shift mentioned re#ects, in fact, the fermioniccharacter of elementary excitations of an aggregatebeing Frenkel excitons [9]. An experimental real-

ization of the bistable behavior of an aggregatewould give us a real chance to create an all-opticalswitching unit, which being integrated in an arraycould serve as a macro-unit of a logical all-opticaldevice.

However, recent studies [2,5] showed that anaggregate of size smaller than the emissionwavelength did not display a bistable behavior.This "nding makes the above mechanism of theaggregate bistability to be hardly realizable, atleast, for such well-studied objects as J-aggregates.Despite the fact that they normally incorporatethousands of molecules, having subsequently a sizelarger than an optical wavelength, the presence ofdisorder in the surroundings leads to a reduction ofthe extension of the exciton eigenfunctions to a loc-alization segment of mean size NH determined bythe degree of disorder [10]. From the experimentaldata related to J-aggregates [11}15], it follows thateven at low temperatures NH does not exceed 100,

0022-2313/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 1 1 3 - 1

meaning that a typical size of a localization seg-ment is less than the emission wavelength.

The above mechanism of bistability is attributedto an individual homogeneous aggregate or a local-ization segment and originates from the dipolar(near-zone) coupling of molecules [2}4]. In thisconnection, one can raise the question whether anensemble of localization segments, being shorterthan the emission wavelength, may manifest a bis-table behavior, as it takes place for an ultrathin "lm(thickness smaller than an emission wavelength) ofhomogeneously broadened two-level molecules[16,17]. In the latter case, the bistable behavior ofthe optical response results from the coupling ofmolecules via the emission (far-zone) "eld.

In this paper, we argue that under the conditionsof low-temperature experiments, aggregated mo-lecular systems can be modeled by an ensemble ofinhomogeneously broadened two-level systems andmay manifest a bistable behavior of transmittivitydue to the coupling of localization segments via theemission "eld. Critical parameters for the occur-rence of such a behavior seems to be achievable forJ-aggregates of polymethine dyes.

2. Model

We start with an analysis of the spectroscopicdata related to J-aggregates from the viewpoint ofthe concept of hidden structure [18}20]. Accordingto this concept, any localization segment can beconsidered as a homogeneous aggregate with thecorresponding (to the size of segment) exciton en-ergy structure of few lowest states. The J-band isformed then by the transitions from the groundstate to the lowest states of the local (related toa certain segment of an aggregate) one-excitonband having dominating oscillator strengths[18}21]. The transition from a one- to two-excitonlocal band is blue-shifted with respect to that fromthe ground state to a one-exciton local band. Inparticular, for J-aggregates of pseudo-isocyanine-bromide (PIC-Br) this shift has an order of severaltens cm~1 (&30 cm~1) [13}16], while the radi-ative damping constant of the lowest one-excitonstates coupled to the light is at least one order ofmagnitude smaller [11,12], i.e., does not overlap

the blue shift. From this fact, we come to the con-clusion that the transition from the ground state tothe bottom of a local one-exciton band can beconsidered as an isolated two-level transition pro-vided the Rabi frequency of the actual "eld is small-er than the blue shift mentioned. Due to the#uctuations of sizes of localization segments, theedge energies of the `locala exciton bands alsoundergo #uctuations. This implies, under the con-ditions outlined, modeling an aggregate as a meso-ensemble of inhomogeneously broadened two-levelsystems and an aggregated sample as their macro-ensemble, only with characteristics attributed tolocalized excitons.

The mathematical basis of the above conceptimplies the use of the standard set of semiclassicalequations for the slowly varying (in time) ampli-tudes E and R, respectively, of the electric "eld ando!-diagonal density matrix element as well as forthe population di!erence between the one-excitonand ground states Z. For a sample in the form of anultrathin "lm of thickness ¸ (considered within thispaper) it has the form (for details see in Refs.[22,23])

RQN"!(iD

N#C

N)R

N#k

NXZ

N, (1a)

ZQN"!1

2kN

(XRHN#XHR

N)!c

N(Z

N#1), (1b)

X"X*#c

R+N

p(N)kNR

N. (1c)

Here the subscript N denotes the size of localiza-tion segment; D

N"u

N!u

*is the resonance detun-

ing between the incident (u*) and transition (u

N)

frequencies; cN

is the spontaneous emission con-stant of the optically active one-exciton state:cN+c

0N with c

0being the analogous constant for

an isolated molecule; CN"c

N/2#C

2is the dephas-

ing constant including that (C2) not connected with

the spontaneous emission; X"dNM E/+ and

X*"d

NM E

*/+ are the transmitted and incident elec-

tric "elds (in frequency units), respectively, withdNM being the transition dipole moment for a seg-

ment of mean size NM scaled as dNM +dJNM , where

d is the transition dipole moment of an isolatedmolecule; k

N"d

N/d

NM ; p(N) is the size distribu-

tion function for the localization segments;

292 V.A. Malyshev et al. / Journal of Luminescence 83}84 (1999) 291}295

Fig. 1. E!ect of the resonance detuning on the appearance ofbistability. The shadowed region represents the domain of para-meters DDD/C and c

R/C where the bistable behavior occurs.

Fig. 2. Above-threshold optical hysteresis loop of the transmis-sion ¹"X2X2

*(solid line) calculated by means of Eqs. (1a)}(1c)

at adiabatic up-and-down scan of the input intensityx"X2

*/cC(c

R"10C, the dispersion of sizes is neglected). The

dotted curve represents the stationary solution.

cR"2pd2

NMnk

*¸/+"2pd2n

0k¸/+, where k"u

N/c (c

is speed of light) and n0"NM n is the original

concentration of molecules in the "lm (beforeaggregation). The quantity c

R, expressed through

the spontaneous emission constant of an isolatedmolecule c

0"4d2k3/3+, reads c

R"(3/8p)c

0n0j2¸,

where j is the transition wavelength. Introducingthe surface density of molecules n

S"n

0¸, one can

interpret the factor n0j2¸"n

Sj2 as the number of

molecules within a j2-square.

3. Bistable behavior

First, we consider a simpli"ed model, assumingno dispersion of sizes (meaning no dependence onN of all the functions in Eqs. (1a)}(1c)) but witha non-zero detuning of resonance D, and look fora solution to Eqs. (1a)}(1c) under steady-state con-ditions (RQ

N"ZQ

N"0). Then, it is the matter of

simple algebra to arrive at the following cubic equa-tion for the dimensionless transmitted "eld inten-sity y"DXD2/cC

yCA1#a

1#b#yB2#A

ab

1#b#yB2

D"x, (2)

where the notations are introduced:x"X2

*/cC, a"c

R/C, b"(D/C)2. This equation

may have a three-valued solution, thus resulting inthe bistable behavior of the optical response weexpect.

In Fig. 1, the results of numerical solution ofEq. (2) are depicted showing the domain of para-meters a and b for which the bistable behavior canbe realized. As is seen from this "gure, the thresholdof bistability is a

#"(c

R/C)

#"8 [15]. Introducing

a detuning of resonance D acts in such a mannerthat higher values of c

R/C are required to get the

bistability e!ect.Fig. 2 represents the numerical solution of Eqs.

(1a)}(1c) at adiabatic up-and-down scanning of theinput intensity X2

*displaying the possibility of get-

ting the hysteresis and high-contrast switching oftransmittivity of the "lm. At the zero detuningof resonance, the critical values of the input inten-sity x, at which the "lm transmittivity changesabruptly, can be calculated analytically and read

x1,2

"(z1,2

!1)(1#a/z1,2

)2, where z1,2

"(a/2)[1$(1!8/a)1@2]. Slightly above the threshold ofbistability (a'8) one has z

1,2+4 and thus

X2*+27cC. It provides an estimate of the input

"eld magnitude (in terms of relaxation constants)needed for switching the transmittivity. Note thatthe corresponding estimate for the "eld inside the"lm reads X2+cC.

Let us now turn to a discusion of the e!ect of sizedispersion. It should be noted, that J-bands at lowtemperatures are inhomogeneously broadened[11], which means that the main e!ect of size dis-persion results in #uctuations of the transition en-ergies. Since we have no information relative to thecorresponding distribution function, it seems to be

V.A. Malyshev et al. / Journal of Luminescence 83}84 (1999) 291}295 293

Fig. 3. E!ect of the inhomogeneous broadening (or size disper-sion) on the appearence of bistability in optical transmittivity.The shadowed region represents the domain of parameters G/Cand c

R/C where the bistable behavior can appear.

reasonable to replace the distribution over sizes bythat over detunings taking a suitable distributionfunction, which allows us to carry out the averageobviously. A Lorentz-shaped distribution just pro-vides such a possibility. Thus, making inEq. (1a)}(1c) the substitution p(N)Pp(D)"(G/p)[(D!D

0)2#G2]~1, where G plays the role of

the J-band width and D0"u

0!u

iis the deviation

of the external frequency ui

on the central fre-quency of the Lorentzian, and replacing all theother quantities by their means, the problem can besolved analytically. Fig. 3 shows a map of para-meters c

R/C and G/C where the shadowed part

represents a bistable domain.

4. Conclusions

Here, we provide an estimate of the driving para-meter c

R, taking the low-temperature data for J-

aggregates of PIC-Br [12], one of the most studiedreal systems in the last few years. Usingc0"(1/3.7 ns) and assuming ¸+j+5.73]

10~5 cm (red J-band) and n0"2]1018 cm~3

(an accessible value), one arrives at cR+13 ps~1.

This value is remarkably larger than the width ofthe red J-band G+1 ps~1 (+30 cm~1). Manipu-lating then the temperature, the required ratios ofcR/C and G/C can be achieved to be within the

bistable region (shown in Fig. 3 as shadowed).

Recall that switching of the transmittivity occursat a saturating magnitude of the "eld inside the"lm, X2&cC (see previous Section), meaning thatthis "eld will never exceed the blue shift (+G)provided G'C and C<c.

In conclusion, the conditions for the model pro-posed in this paper seems to be indeed realizable foran ensemble of J-aggregates of PIC-Br at low tem-peratures. A certain support for our "ndings isgiven by the recent publications reporting on thecooperative emission in p-conjugated polymer thin"lm [24,25] as well as the superradiant lasing fromthe J-aggregated BIC molecules adsorbed onto col-loidal silica or silver [26,27], which means a collec-tivization of polymers and J-aggregates via theemission "eld we need for the e!ect under consid-eration.

Acknowledgements

H.G. and K.-H. F. greatly acknowledge the sup-port from the Bundesministerium fuK r Bildung, Wis-senschaft, Forschung und Technologie within theHochschulsonderprogramm III. V.A.M. thanksa partial support from Deutsche Bundesminis-terium fuK r Bildung, Forschung und Technologiewithin the TRANSFORM-WissenschaftsunterstuK t-zungsprogramm (project No. 01 BP 820/7) as wellas acknowledges a partial support from the RussianFoundation for Basic Research (project 97-03-09221).

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