Optical absorption spectra of Frenkel excitons in random one-dimensional
systems are presented. Two models of inhomogeneous broadening, arising from a
Gaussian distribution of on-site energies, are considered. In one case the
on-site energies are uncorrelated variables whereas in the second model the
on-site energies are pairwise correlated (dimers). We observe a red shift and a
broadening of the absorption line on increasing the width of the Gaussian
distribution. In the two cases we find that the shift is the same, within our
numerical accuracy, whereas the broadening is larger when dimers are
introduced. The increase of the width of the Gaussian distribution leads to
larger differences between uncorrelated and correlated disordered models. We
suggest that this higher broadening is due to stronger scattering effects from
dimers.Comment: 9 pages, REVTeX 3.0, 3 ps figures. To appear in Physical Review 

A. Sánchez

D. H. Dunlap

D. L. Huber

E. Diez

F. Domínguez Adame

F. Domínguez-Adame

H. L. Wu

J. C. Flores

P. K. Datta

P. Phillips

Th. Wagersreiter

English

arXiv

© 1995 The American Physical Society.
The author thanks A. Sanchez and E. Macia for illuminating conversations. This work was supported by UCM under Project No. PR161/93-4811.Optical-absorption spectra of Frenkel excitons in random one-dimensional  systems are presented. Two models of inhomogeneous broadening, arising from a Gaussian distribution of on-site energies, are considered. In one case the on-site energies are uncorrelated variables whereas in the second model the on-site energies are pairwise correlated (dimers). We observe a red shift and a broadening of the absorption line on increasing the width of the Gaussian distribution. In the two cases we And that the shift is the same, within our numerical accuracy, whereas the broadening is larger when dimers are introduced. The increase of the width of the Gaussian distribution leads to larger differences between uncorrelated and correlated disordered models. We suggest that this higher broadening is due to stronger scattering effects from dimers.UCMDepto. de Física de MaterialesFac. de Ciencias FísicasTRUEpu

Domínguez-Adame Acosta, Francisco

Docta Complutense

PHYSICAL REVIEW 8 VOLUME 51, NUMBER 18 1 MAY 1995-IIBrief Reportsi f«po ts «ac««»fcompleted es«ch h~ch h le meeting the us al phys&&pf Reyjp~ standa ds ofsci ntifpc qual typublication schedule as for regular articles is folio~ed, and page proofs are sent to authors.Frenkel excitons in random systems with correlated Gaussian disorderF. Dominguez-AdameDepartamento de Fisica de Materiales, Facultad de Fisicas, Universidad Complutense, E 280/0-Madrid, Spain(Received 5 December 1994)Optical-absorption spectra of Frenkel excitons in random one-dimensional systems are presented.Two models of inhomogeneous broadening, arising from a Gaussian distribution of on-site energies,are considered. In one case the on-site energies are uncorrelated variables whereas in the secondmodel the on-site energies are pairwise correlated (dimers). We observe a red shift and a broadeningof the absorption line on increasing the width of the Gaussian distribution. In the two cases weAnd that the shift is the same, within our numerical accuracy, whereas the broadening is largerwhen dimers are introduced. The increase of the width of the Gaussian distribution leads to largerdifferences between uncorrelated and correlated disordered models. We suggest that this higherbroadening is due to stronger scattering eKects from dimers.I. INTRODUCTIONRecently, several researchers have shown that struc-tural correlated disorder has profound effects in randomsystems and produces a variety of unexpected phenom-ena. It is by now well known that a band of delocalizedelectrons appears in tight-binding Hamiltonians with cor-related diagonal and/or off-diagonal elements. A moredramatic occurrence of electron delocalization arises incontinuous models with dimer impurities as well as indisordered semiconductor superlattices with dimer quan-tum wells, where there exist infinitely many bands ofdelocalized states. What is most important, delocal-ization by correlation is not restricted to electrons, assuggested by the occurrence of delocalized vibrations inclassical random chains with paired disorder. ' Further-more, time-domain analysis of Frenkel excitons in sys-tems with traps randomly placed in an otherwise perfectlattice shows that the coherent quantum transport is alsoaffected by pairing of these traps, but trapping is less ef-fective as compared to systems with the saxne fraction ofunpaired traps.Roughly speaking, correlated disorder seems to retainsome of those special features characterizing periodic sys-tems; in other words, there exists a competition betweeenshort-range correlation and long-range disorder leading,for instance, to delocalization of electrons and phonons orto the occurrence of a major contribution of the slowlydecaying modes in exciton trapping processes, as men-tioned above. However, in this paper we point out thatthis conjecture is not always valid. In particular, weconsider optical absorption of Frenkel excitons in un-paired as well as in paired disordered models, focusingour attention on inhomogeneous broadening due to aGaussian distribution of on-site energies in a one-dimensional lattice. We show that inhomogeneous broad-ening is enhanced when structural correlations arise sinceexciton scattering from dimers is stronger as comparedto scattering from unpaired sites.II. MODKIWe consider % optically active centers in a regular lat-tice with spacing unity. For our present purposes, weneglect all thermal degrees of freedom (electron-phononcoupling and local lattice distortions). Therefore, theeffective Hamiltonian for the Frenkel-exciton problemcan be written in the tight-binding form with nearest-neighbor interactions as follows (we use units such that5 = 1):R ) ekakak + T) . (a'k k+1 + k+lak)k k1P(ek, e, o.) = — exp2' 0(ek —e)220 (2)where ak and ak are Bose operators creating or annihilat-ing an electronic excitation of on-site energy ek. Here Tis the nearest-neighbor coupling, which is assumed to beconstant in the whole lattice. On-site energies are subjectto diagonal disorder representing inhomogeneous broad-ening, for which a Gaussian distribution is the propertheoretical approximation. In what follows we considertwo different models, namely uncorrelated and correlateddisordered systems. This enables us to separate the ef-fects merely due to optical absorption in one-dixnensionfrom those which manifest the peculiarities of the corre-lation between random parameters. The first model canbe drawn from the distribution0163-1829/95/51(18)/12801(4)/$06. 00 12 801 1995 The American Physical SocietyBRIEF REPORTSwith average e and width o . It is important to stressthat neighbor on-site energies are uncorrelated randomvariables. On the other side, to build up our corre-lated disordered model, we choose e2k q according to theGaussian distribution given in (2) and then take e2k =~. This step is repeated at every odd site of the lat-tice, hence leading to a set of paired correlated on-siteenergies (dimers).Having presented our model we now describe themethod we have used to calculate the absorption spec-tra. The line shape I(E) of optical absorption in which asingle exciton is created after pulse excitation in a latticewith % sites can be obtained as follows:(dt e sin(Et) Im ) Gy(t))where the factor exp( —nt) takes into account the broad-ening due to the Lorentzian instrumental resolution func-tion of half-width o, . Hence the resulting spectra arethe convolution of the inhomogeneous broadening due toGaussian disorder and the instrumental resolution func-tion. The correlation functions Gk(t) obey the equationof motion. di —Gk(t) = ) Hk, G, (t),with Hk, . = eI, SI,~ + (1 —Sk, )T with j = k —1, k, k+ 1,and initial condition Gk(0) = 1. Therefore, the compu-tation of optical spectra reduces to solving the discreteSchrodinger-like equation (4) using standard numericaltechniques.III. NUMERICAL RESULTS AND DISCUSSIONSWe have solved the equation of motion (4) for chainsof % = 2000 sites using an implicit integration scheme.In order to minimize end efFects, spatial periodic bound-ary conditions are introduced. Energy will be measuredin units of T whereas time will be expressed in units ofT . To make contact with experiments, we note that ~T~is proportional to the exciton bandwidth in the perfectlattice, so that energy and time scales can be deducedfrom experimental data. Since we are mainly interestedin the comparison between uncorrelated and correlateddisordered models rather than in the efFects of the difFer-ent parameters on the optical absorption process, we willBx the values of e and T focusing our attention on thewidth o. In particular, we have set e = 4 and T = —1henceforth as representative values. The width of theinstrumental resolution was cr = I/O. The distributionwidth o ranged from 0 up to 0.5.In the absence of inhomogeneous broadening (a = 0),the absorption line shape is a Lorentzian function cen-tered at E = ~+ 2T, which with our choice of parametersis E = 2. The full width at half maximum (FWHM)is 2a, For o. g 0 a broadening of this main line is ob-served accompanied by a shift of its position to lower en-ergies on increasing o. , as shown in Fig. 1 for correlatedas well as uncorrelated disorder. Unlike optical spectraIIII I I I I I1 2 3 4FIG. 1. Absorption spectra for one-dimensional randomlattices with Gaussian distribution of correlated (solid lines)and uncorrelated (dashed lines) on-site energies with cr = 0.15and o = 0.50. Vertical scale is the same in all cases.O.B—0.6—10.0II i I I I0.1 0.2 0.3 0.4 0.5FIG. 2. Full width at half maximum (FWHM) as a functionof o for correlated (solid lines) and uncorrelated (dashed lines)disordered systems.in random binary systems, where ek takes only two dif-ferent unpaired or paired values, there are no signa-tures of satellites appearing in the high-energy region ofthe spectra, at least in the range of parameters we haveconsidered. Only a long high-energy tail is observed onincreasing o.From Fig. 1 several conclusions can be drawn. First,the shift to lower energies increases with increasing o inboth models. A similar shift is also found using the co-herent potential approximation (CPA) but the CPA failsto predict its value correctly, bringing up a smaller shiftthan that observed in numerical simulations. Second,this red shift is the same in both models for a given valueof o, within our numerical accuracy. Hence, we are ledto the conclusion that the efFects of correlation cannotbe deduced from this shift. Third, and more importantfrom an experimental viewpoint, there are substantialdifFerences regarding the width of the optical spectra.In all cases we have studied we found that the FWHMis larger for correlated inhomogeneous broadening, andthat the difference increases on increasing o. This resultis shown in Fig. 2, where it is seen that the FWHM is2o. = 0.5 close to o. = 0, as expected, but increases moreBRIEF REPORTS 12 803pronouncedly when correlation is present. This result issomewhat unexpected since it indicates that correlatedGaussian disorder disturbs exciton dynamics much morethan uncorrelated disorder. This is to be compared withprevious results on random systems with traps appear-ing as pairs in an otherwise perfect lattice, where pair-ing causes less disruption of Frenkel-exciton dynamics ascompared to those systems without this constraint, asmentioned in the Introduction. In particular, the de-pletion of the q = 0 exciton mode is slower when trapsare paired. However, our present results suggest that inthe case of inhomogeneous broadening the behavior isjust the opposite, i.e. , correlations create centers causinglarger scattering effects on excitons. To validate or dis-card this suggestion we have evaluated the probability offinding an exciton in the q = 0 mode at time t after pulseexcitation, obtained as follows:0.10 8 4 10FIG. 3. Probability of finding an exciton in the q = 0 modeat time t for correlated (solid lines) and uncorrelated (dashedlines) disordered systems with rr = 0.50.Figure 3 shows the results for 0 = 0.50, although wehave found similar behavior in all cases. From this fig-ure we can see that P(t) decays faster when correlationsare present. Since in our system there is no trapping,the q = 0 mode is only depleted by disorder, thus indi-cating that scattering by dimers is more important thanscattering by uncorrelated single sites. In this sense cor-related systems are more disordered than uncorrelatedones, thus explaining the larger inhomogeneous broaden-ing in the former systems.Finally, let us comment that the CPA predicts a muchsmaller FWHM than that found by numerical integra-tion in the case of inhomogeneous broadening in uncor-related disordered lattices. Therefore, the CPA failureis even more dramatic in the case of correlated disorder,strongly suggesting that future theoretical works shouldrely on different grounds. Probably, a renormalization ofthe lattice considering the dimer defect as a single entity,as much as in the line we have recently proposed in thecase of trapping of classical excitons in correlated dis-ordered systems, could be a good starting theoreticalscenario to fully account for optical spectra.dom systems. Two different models have been consid-ered; in both cases broadening arises from a Gaussiandistribution of on-site energies. In uncorrelated disor-dered systems on-site energies are chosen according to theGaussian distribution at every site, whereas in correlateddisordered systems this selection is made only at oddsites and even sites take the value of the preceding site.Therefore, the correlation length is of the order of thelattice spacing. Our results show that the presence ofstructural correlations in random systems can be thenreadily determined from the analysis of the absorptionspectra of these samples. By comparing the obtainedspectra in both models we found an identical red shiftof the absorption line while inhomogeneous broadeningis more pronounced whenever correlations are present inthe lattice. We have realized that the q = 0 mode isdepleted faster in systems with correlated disorder, thusindicating stronger scattering effects from dimers. Froma theoretical viewpoint, we have pointed out that CPAfailures are even more noteworthy in the case of corre-lated disordered systems. Hence, the necessity of a newtheoretical framework to deal with random systems withstructural correlations becomes a very appealing task.IV. CONCLUSIONS ACKNOWLEDGMENTSIn summary, we have studied the effects of inhomo-geneous broadening on the absorption spectrum corre-sponding to the Frenkel-exciton Hamiltonian for ran-The author thanks A. Sanchez and E. Macia for illumi-nating conversations. This work was supported by UCMunder Project No. PR161/93-4811.J. C. Flores, J. Phys. Condens. Matter 1, 8471 (1989).D. H. Dunlap, H.-L. Wu, and P. Phillips, Phys. Rev. Lett.65, 88 (1990).H.-L. Wu and P. Phillips, Phys. Rev. Lett. 66, 1366 (1991).P. Phillips and H.-L. Wu, Science 252, 1805 (1991).J. C. Flores and M. Hilke, J. Phys. A 26, L1255 (1993).A. Sanchez, E. Macia, and F. Dominguez-Adame, Phys.Rev. B 49, 147 (1994).E. Diez, A. Sanchez, and F. Dominguez-Adame, Phys. Rev.B 50, 14 359 (1994).F. Dominguez-Adarne, A. Sanchez, and E. Diez, Phys. Rev.B 50, 17736 (1994).F. Dominguez-Adarne, E. Macia, and A. Sanchez, Phys.Rev. B 48, 6054 (1993).P. K. Datta and K. Kundu, J. Phys. Condens. Matter 6,4465 (1994).12 804 BRIEF REPORTS 51F. Dominguez-Adame, B. Mendez, A. Sanchez, and E.Macia, Phys. Rev. B 49, 3839 (1994).D. L. Huber and W. Y. Ching, Phys. Rev. B 39, 8652(1989).Th. Wagersreiter and H. F. Kau8'mann, Phys. Rev. B 49,8655 (1994).F. Dominguez-Adame, E. Macia, and A. Sanchez, Phys.Rev. B 50, 6453 (1994).D. L. Huber and W. Y. Ching, Chem. Phys. 146, 409(1990).A. Sanchez, F. Dominguez-Adame, and E. Macia, Phys.Rev. B 51, 173 (1995).

Frenkel Excitons in Random Systems With Correlated Gaussian Disorder

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