Optical nonlinearities in polyenes: role of intrinsic localized modes

Within a simple theoretical model we demonstrate analytically the effect of intrinsic localized modes such as breathers on nonlinear optical properties. We have earlier observed enhancement of optical nonlinearities in ab initio numerical simulations of ®nite polyenes, speci®cally C40H42. We also discuss experimental implications of our results. # 2001 Elsevier Science B.V. All rights reserved. PACS: 42.65.-k; 63.20.Kr; 63.20.Ry Keywords: Optical nonlinearities; Intrinsic localized modes; C40H42 Energy localization due to nonlinearity in a material can lead to novel types of physical excitations. In discrete anharmonic lattices, highly localized nonlinear waves (intrinsic localized modes or discrete breathers) can exist with dimensions comparable to the lattice constant [1±3]. Such localized modes have been identi®ed in numerical molecular dynamics simulations of many different systems [4], but only very recently the ®rst experimental observation of localized states have been reported in mixed-valence transition metal complexes [5], Josephson junction arrays and ladders [6,7], and quasi-one-dimensional antiferromagnetic chains [8]. Our goal is to understand the effect of such spatially localized nonlinear modes, if excited, on the electronic, vibrational, transport and optical properties of conjugated polymers. Intrinsic localized modes (or breathers) are charge-neutral, spatially localized, time-periodic modes with the symmetry of the ground state. Thus, a direct experimental study of these modes is usually dif®cult, and it requires some special, generally indirect methods. On the other hand, nonlinear effects are well studied in optics where there exist a variety of methods to measure different types of nonlinear properties, such as a nonlinearity-induced change of the effective refractive index characterizing optical properties of nonlinear materials. Therefore, it is very important to understand the effect of nonlinear localized modes on the nonlinear optical response. Polyenes can be described by coupled ®elds (e.g. electron±phonon) and can be studied in terms of a single lattice distortion ®eld with nonlinearity when the electronic ®eld is integrated out in a prescribed manner. Often the electronic, vibrational, transport and thermodynamic properties of conjugated polymers crucially depend on nonlinear excitations such as solitons, polarons and excitons. Breathers on the other hand are intrinsically dynamic excitations and can be viewed as bound states of linear excitations (e.g. phonons). They are expected to be important for energy and charge localization as well as for transport. In particular, they have been proposed for understanding the transient photodynamics of conjugated polymers [9]. Indeed, the emerging ®eld of femtosecond chemistry [10,11] is well-suited to probe the photophysics of breather-like excitations. Breathers are also closely related to intrinsic localized modes in anharmonic lattices [12], with the anharmonicity arising from the coupled ®elds. In polyenes strong lattice distortions introduce localized states within electronic gaps between occupied and unoccupied levels due to broken ground state symmetries such as Peierls distortions. The deformability of the p-conjugated backbone implies large polarizability and thus large nonlinear optical (NLO) response. Here we ®rst study the role of breathers on static polarizability (a) and hyperpolarizability (g) for a representative polyene C40H42. Our main conclusion is that there is a signi®cant enhancement of a and g in the presence of breathers, which are found to be persistent nonlinear excitations even in the presence of realistic polyene geometries and dynamics. We also distinguish between Synthetic Metals 116 (2001) 45±48 * Corresponding author. Tel.: ‡1-505-667-5227; fax: ‡1-505-665-4063. E-mail address: [email protected] (A. Saxena). 0379-6779/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 9 6 7 7 9 ( 0 0 ) 0 0 5 1 2 9 the contribution of phonons and breathers to NLO enhancement. These results are obtained for a model of the polyene with its true geometry and all its constituent atoms described at a quantum-chemical level. However, it is plausible that the qualitative physics of such a nonlinearity enhancement can be found in simple, nonlinear models describing the interaction of coupled ®elds (e.g. charge and lattice deformation in the polyenes). Later in this paper we introduce such a simple model, and use it to explain the basic mechanism of the breather-induced enhancement of the NLO response. First we summarize the results of the quantum chemical molecular dynamics (MD) simulations [13]. Breather dynamics have been studied previously in discrete model systems, in nonlinear lattices using tight-binding models [14,15], or as a solution of corresponding continuum equations [9]. We represent the polyene with its true geometry and all its constituent atoms at a quantum chemical level. In addition, to study dynamical evolution of electronic properties and, to identify their signatures in experiments, we investigate the nonlinear dynamics of breather formation. We study the dynamics of NLO response using MOPAC93 [16]. We numerically computed the static polarizability (a) and third-order hyperpolarizability (g) in the time-dependent Hartree±Fock (TDHF) approximation [16±18]. a and g are plotted as a function of time (over one period of the breather in Fig. 1) in Fig. 2. The minimum (a ˆ 740 au at 0 fs) and maximum (a ˆ 940 au at 9 fs) represent a variation of 27%. Note that for the undistorted ground state a ˆ 810 au. The evolution in time of the atomic displacement is shown in Fig. 1. First consider the short-time behavior. Between t ˆ 0 and 9 fs, the displacement of the 19th carbon atom relative to the dimerized chain (r19) oscillates with a period of 18 fs between the values of 0 and 2. (rn is normalized to its average value at t ˆ 0, rave ˆ 6:1 10ÿ2.) In order to compare this frequency with that of an extended (phonon-like) mode, we also examined a trajectory in which a small displacement (0.001 A ) was added to each un of the optimized ground state in the shorter chain C24H26. With this initial condition, the period of the long-wavelength oscillation for interior values of rn was determined to be 18 fs, which corresponds to a vibrational energy …hn† ˆ 5:3 kcal=mol ˆ 1860 cmÿ1 ˆ 0:23 eV. Since a breather state periodically reduces the electronic gap energy, we expect the breather period to be larger than the optical phonon period. Next consider the long-time behavior of the MD trajectory. The initial pulse in rn bifurcates (Fig. 1) until it reaches the two free ends of the chain at about 1.6±1.7 ps. The bifurcation refocuses at r19 at about 3.3 ps (which is 183 breather periods, 33 000 MD timesteps). Continuing the trajectory, the peak bifurcates again and reaches the free ends at about 5.4 ps. A second refocussing occurs at about 6.8 ps, although the height of the peak is only 2=3 of the peak at 3.3 ps (due to energy dissipation). The simulation was not continued beyond 7.3 ps, but at longer times the localized pulse will presumably begin to couple with the extended phonon-like states and other polyene distortions, and slowly dissipate. Variations in the semi-empirical g are larger; for the breather in Fig. 2, the minimum value of g ˆ 6:2 10 au at 0 fs and the maximum value of g ˆ 1:7 10 au at 9 fs represent a change of about a factor of three. For comparison, in the undistorted ground state g ˆ 9:5 10. The values for a and g calculated during one cycle at 110, 201, and 274 fs all yield similar trends. The oscillating behavior is thus not con®ned to the ®rst 10 fs. The contribution of Fig. 1. Atomic displacement (rn) for C40H42. Each trace is r1 through r37 running from bottom to top. The initial (t ˆ 0) pulse is centered about r19 (from [13]). Fig. 2. Static polarizability (a, open circles, in units of 10 au) and thirdorder hyperpolarizability (g, closed circles, in units of 10 au) for C40H42 vs. time at zero temperature (from [13]). 46 A. Saxena et al. / Synthetic Metals 116 (2001) 45±48