Radiative processes in poly(phenylene vinylene)

JOURNAL OF

LUMINESCENCE ELSEVIER

Journal of Luminescence 60&61 (1994) 849 852

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Radiative processes in poly(phenylene vinylene) F.X. Bronold*, A. Saxena, A.R. Bishop Theoretical Division and Center for Nonlinear Studies, Los A/amos National Laboratory, Los Alamos, NM 87545, USA

Abstract Within a single-electron Su-Schrieffer-Heeger-Iike model Hamiltonian for poly(phenylene vinylene) we investigate the effect of electron lattice coupling on the radiative decay of polaron excitons, using a lattice relaxation theory together with a real-space random-phase-approximation. In general, the phonon modes in the ground state and the polaron exciton configurations differ not only in their frequencies and equilibrium positions but also in their principal axes. However, in our calculation of the transition amplitude, we neglect the latter and consider only one decay channel for each (polaron—exciton) normal mode. Since our theoretically obtained photoluminescence spectra fit quite well with experiments, it seems unnecessary to include the difference in principal axes explicitly and, consequently, to allow for more decay channels.

Conjugated polymers with phenyl rings in their backbones bear great potential for electro-optic applications not only due to their unique optical properties but also due to their processibility (see e g. [I]). In particular, photoluminescence (PL) in poly(phenylene vinylene) (PPV) has attracted much interest from experimentalists and theorists. While there is persuasive evidence that singlet excitons are the source of PL, a conclusive exciton theory for these novel materials is still missing. Most importantly, the relative importance of electron—electron (e—e) versus electron lattice (e 1) interaction and consequently the character of excitons, namely polaron—excitons (PEs) with a significant lattice distortion versus excitons without lattice distortion, is an unresolved issue. A well developed Franck—’Condon progression in PL

*

Corresponding author. Also at: Physikalisches Institut, Universitãt Bayreuth, D-95440 Bayreuth, Germany.

spectra [2] indicates that e 1 interaction plays an essential role at least in radiative decay processes even though many theoretical investigations of excitons in conjugated polymers [3] neglect e I interaction completely. Here we adopt, as a complement to these studies, a Su-Schrieffer-Heeger (SSH)-like PPV model, considering PPV as a highly non-linear coupled e I system and calculate PL spectra for different e 1 coupling strengths at zero temperature. Since this model is an effective singleelectron model with emphasis on e 1 coupling, many-body effects are only included to the extent of renormalized effective (single-electron) parameters. To obtain PL spectra for varying e I coupling strengths, we [4] apply to a non-degenerate ground state SSH-like model [5] a lattice relaxation (LR) theory, developed [6] for conjugated polymers, that explicitly accounts for self-consistent, adiabatic multi-electron and multi-phonon states. Following Ref. [6], we split the lattice field ör1~into

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a static part ~ and a time-dependent part leading to the following 1-lamiltonian:

;‘~~

~

(1)

H,

=

=

H, +

~

{_(t

0

=

—

H,

ph’

~

+

h.c.)

~ K(~r~?5~}.

+

HPh

HPh +

~

( ir~-

+

(3)

3

ç H,~h = ~

~(c~c1~ + h.c4 + K~r~j°

(4)

Here ci,, is the annihilation operator for a ,t-electron at site j and spin t0, and K are the c--I coupling constant, the it-electron hybridization energy and the (a-bond) compressibility, respectively, it0 is the conjugate momentum to y~ and (for simplicity) M is taken to be the mass of a CH-unit for all sites in the unit cell, regardless of whether a H-atom is attached to the C-atom or not. The symbol denotes summation over bond vanables only. The lattice field describes deviation from the equilibrium bond length and phonons are described in terms of these bond variables. The inset in Fig. I shows a schematic PPV unit cell for reference. The numerical calculation of PL spectra consists of three steps: Firsi, we employ a Born—Oppenheimer approximation and calculate the self-consistent mean field configurations (MFCs) for the stoichiometric ground state and for the case where one electron is promoted from the top of the valence band to the bottom of the conduction band, creating an electron—hole pair which relaxes to a PE. The self-consistent treatment ensures that the (electronic) diagonal part of the c—I interaction. Eq. (4). is treated non-perturhative/v. Second, we consider small fluctuations around the MFCs to determine the normal modes for each MFC. This procedure is equivalent to treating the (electronic) nondiagonal part of (4) perturhaticely in a real-space random-phase-approximation (RPA) and leads to a significant renormalization of the original dispersionless Einstein oscillators of H~h, Eq. (3). Third, ~.

~,

we use Fermi’s Golden Rule together with Condon~sapproximation to express the zero temperature transition probability between the PE and the GS by emitting one photon of the appropriate energy and the necessary number of phonons to conserve the energy in the usual form

(2)

K

/994, 549 F52

a(w) =


J~e~>~&(w). .

-

(~) ,

where J is the Lurrent operator and the line shape function G(w) is basically the Fourier transform of the multi-phonon overlap integral (characteristic function) summed over all final (phonon) states. The RPA [7] consists of augmenting the “bare” phonon Hamiltonian by the polarizability of the free electron gas for the GS and for the PE configuration, respectively. This procedure leads to phonon modes which differ not only in frequency and equilibrium position but also in principal axes. making the calculation of the characteristic function fairly complicated. To obtain quantitative resuits we have simplified the characteristic function by ignoring the different principal axes. In other words, we neglect off-diagonal terms in the augmented GS phonon Hamiltonian expressed in PE normal coordinates. Then the characteristic function factorizes and the line shape function G(o~i)can be written as a convolution of individual line shape functions obtained with the help of Manneback’s recursion formulas. We can furthermore limit the convolution only to modes with significant Huang—Rhys factors and/or change in force constants indicating a strong coupling to the transition process. Before discussing PL spectra in PPV we emphasize two features in the phonon spectrum which are special for the chosen SSH-like Hamiltonian: First. it is well known that SSH-like models can exhibit localized phonon modes (“shape” modes) [8] if the lattice is inhomogeneously distorted as in the PE configuration. These shape modes occur not due to external forces or defects but due to c—-I interaction and therefore represent an intrinsic dynamical effect. Second, in the PE phonon spectrum the lowest frequency always corresponds to the “center-ofmass” motion which eventually becomes the Goldstone translation mode in the limit of large coherence length (PE width). Although not much of

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Fig. 1. Photoluminescence spectrum for i. = 0.15. The dashed line corresponds to neglecting differences in the force constants for initial and final states. Note that we underestimate the PE binding energy by neglecting e—e interactions. Consequently, the spectrum is shifted to lower energies compared to experimental photoluminescence. The inset shows a schematic unit cell of poly(phenylene vinylene).

Fig. 2. Photoluminescence spectrum for )~ 0.20. The dashed line corresponds to neglecting differences in the force constants for initial and final states. The PE binding energy is underestimated by neglecting e—e interactions as in Fig. 1.

a problem in real materials where this mode is probably pinned by imperfections, it marks (con-

varied the dimensionless electron—phonon coupling constant ). = 2~2/ict0K, namely A = 0.15 (Fig. 1)

ceptual) limitations of the present LR theory applied to SSH-like models. Namely, the phonon modes should be calculated in a way that the translational symmetry broken by the inhomogeneous PE configuration is restored by promoting the PE center to a (collective) dynamical variable. The phonon modes then must be chosen orthogonal to the (translational) motion connected with this collective variable leading to a much more complicated Hamiltonian. However, for the present purpose we ignore the contribution of the center-ofmass motion completely in all PL spectra. As already mentioned, as far as real materials are concerned this approach is reasonable, In Figs. 1 and 2 we present two numerically obtained PL spectra for a chain of 10 unit cells with periodic boundary conditions. Although to date there exists no unique parametrization of a SSHlike (single-particle) PPV Hamiltonian primarily due to the unresolved issue of excitonic bandedge effects we have chosen t0 = 2.5eV and K = 99.0eV/A to reproduce the experimentally observed Peierls distortion and bandwidth. In order to demonstrate trends in the PL line shape we

and A = 0.2 (Fig. 2). Depending on ). the characteristic spatial extent of a PE is four ~ = 0.15) and three ~. = 0.2) unit cells, respectively. In both cases we find a “linkage PE” (PE centered around a vinylene linkage) is favored over a “ring PE” (PE centered around a benzene ring), although the energy difference is quite small due to the relatively large (compared to a lattice constant) typical width of a PE. Fig. 1 = 0.15) shows a clear Franck— Condon progression with a dominant zero-phonon spectral line and weaker phonon satellites. By ignoring the frequency change when the system relaxes from the PE to the GS configuration we obtain the dashed line, clearly indicating that the frequency change affects the distribution of oscillator strength among various phonon lines. For A = 0.2 (Fig. 2) the PL spectrum is significantly Stokes-shifted with the 0—2 line as the most dominant phonon satellite. For this stronger e—l coupling the difference between the PL spectrum calculated with and without the change in force constants is smaller than in the former case. Note that the absolute position of the PL spectrum is related to the PE binding energy which we underestimate

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Journal of LUinIIICSCS’OCC 60&6 / ( /994 M9 552

by neglecting all c—c interactions. Consequently, a comparison with experiments can only be quailtative. However, at this level the FL spectrum of Fig. I fits well with the experimentally observed PL spectrum [2]. We find that all accepting modes lie within the same GS phonon branch with the optical phonon at the center of the Brillouin zone (q = 0) and the phonon with wave number approximately the inverse of the width of the FE as the modes which couple most strongly. However, the dispersion of this branch is small making an effective singlemode description of PL in PPV feasible. Further, at least at zero temperature, it seems that the structure of FL spectra in PPV can be well described by assuming a “one-to-one-correspondence” (neglecting the difference in principal axes) between FE and GS normal modes. We tested this approximation by expanding FE modes in terms of GS normal modes and verified that the expansion for each individual mode clusters around one of the two most dominant accepting modes. Consequently, until experiments show evidence for novel signatures in the FL spectrum it does not appear necessary to elaborate on the difference in principal axes. To summarize, we combined a LR approach and a real-space RPA to study within a SSH-like PPV model Hamiltonian the effect of c—I interaction on PL spectra at zero temperature. We assumed a “one-to-one-correspondence” between FE and GS normal modes but explicitly allowed for frequency changes during the relaxation process. The structure of the theoretically obtained PL spectrum for moderate c—I coupling (A = 0.15) coincides with the experimentally observed PL spectrum except for the absolute position of the spectrum which we

underestimate by neglecting all Coulomb contributions to the PE binding energy. We also pointed out conceptual problems associated with the broken translational symmetry in the PE configuration. giving rise to a (translational) Goldstone mode.

Acknowledgements We like to thank D. Cai and D. Smith for illuminating discussions about LR theory, and one of the authors (F.X.B.) acknowledges discussions with R. McKenzie.

References [I] D.D.C, Bradley et a!.. Synth. Met. 29 (1989) E12t. R. Lazzaronl ci at., in: Conjugated PolymerIc Materials: Opportunities in Electronics. Optoelectronics and Molecular Electrorlics. eds. iL Brédas and R.R. Chance, NATO Advanced Research Workshop Series. Vol. El82 (Kiuwer Academic Publishers. Dordrecht, 1990) p. 149. 12] T.W. Hagler et at.. Phys. Rev. B 44)1991)8652: H.S. Woo et al., Phys. Rev. B 46 (1992) 7379; S. Heun et at.. J. Phys.: C’ondens Matter 5)1993)247 and references therein These are only some of the most recent experimenial papers concerning photoluminescence in poly)phenylene vinylene) and derivatives. [3] 5. Abe,J. Yu and W.P. Su, Phys. Res. B 45)1992)8264. H. Hayashi and K. Nasu, Phys. Rev B 32 (1985) 5295 [4] F.X. Bronold, A. Saxena and A.R. Bishop, B 48 (1993! 13162. [5] H.-Y. Choi and Mi. Rice, Phys. Rev. B 44)1991) 10521. [6] Su Zhao-bin et a!.. Commun. Theor. Phys. 2 (1983) 1203; Commun. Theor. Phys. 2 (1983) 323; Commun. Theor. Phys.Hicks 2)1983)1341. [7] iC. and J.T. (jammel. Phys. Rev. B 37 (1988) 6315, [8] H. Ito ci al.. J. Phys. Soc. Japan. 53 (1984) 352)).