THEORY OF INELASTIC LIGHT SCATTERING PROCESSES IN (CH)x

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THEORY OF INELASTIC LIGHT SCATTERING PROCESSES IN (CH)x

E. Melé

To cite this version:

E. Melé. THEORY OF INELASTIC LIGHT SCATTERING PROCESSES IN (CH)x. Journal de

Physique Colloques, 1983, 44 (C3), pp.C3-267-C3-272. �10.1051/jphyscol:1983351�. �jpa-00222703�

JOURNAL DE PHYSIQUE

Colloque C3, supplément au n°6, Tome 44, juin 1983 page C3-267

THEORY OF INELASTIC LIGHT SCATTERING PROCESSES IN ( C H )X

E.J. Mele

Department of Physios, University of Pennsylvania, Philadelphia, PA 19104, U.S.A.

Résumé - Nous décrivons une analyse de la forme de raie de dispersion de Stokes dans (CH)X gui attribue l'effet à la dépen- dance du processus de relaxation avec l'énergie d'excitation, pour le polymère après photoexcitation, plutôt qu'à une diffusion Raman sur des segments trans courts.

abstract - We outline an analysis of the Stokes lineshape dispersion in trans(CH)„ which attributes the effect to the excitation energy dependence of the relaxation pathway for the polymer following photoexcitation rather than to Raman scattering in short trans segments.

I - INTRODUCTION

Since the synthesis of free standing films of (CH)x by Shirikawa, Raman scattering has been routinely used to probe the structural quality of these s a m p l e s / 1 - 5 / . There is c o n s i d e r a b l e experimental motivation for such studies: it is well known that the energy of the lowest lying electronic excitation in finite polyenes decreases with increasing chain length / 6 / and also that the frequency of the prominent Raman active A double bond stretching mode increases with decreasing chaiTi length /!/.

Experimentally the Stokes shifted C=C lineshapes in (CH) films exhibit dramatic changes as the primary exciting frequency is varied above the (CH) fundamental absorption edge /1-5/. These variations in the Stokes shifted lineshapes have been interpreted as resulting from a distribution of conjugation lengths in the samples; the evolution of the high frequency shoulder in the observed lineshapes is then attributed to Raman events in the shorter chains in the film which are resonantly selectively enhanced at the higher exciting frequencies. Supportive evidence for this model is drawn from studies of the dependence of this lineshape on degree of isomerization and sample degradation / 8 / . Considerable effort has been devoted to quantify this explanation and thus to use the Raman spectra as a quantitative monitor of sample quality. Initial analyses employing a very simple model for the resonance excitation profile for a single polyene yield estimates of a mean conjugation length of 30 double bonds, modelled by a Gaussian distribution with a half width of 20 double bonds / 3 / . The distribution must heavily weight the short chains because the Stokes lineshape is dominated by the high frequency shoulder at the higher exciting frequencies, suggesting a very high density of short segments in the material. Refined models, hypothesizing a bimodal distribution of chain lengths yield only slightly different estimates (a distribution peaked at 40 and 5 double bonds)/3,9/. £ recent independent analysis, again assuming a bimodal distribution, but with a 1/N dependence of the shift of the Raman active frequency with chain length, suggest a mean conjugation length between 100 and 200 double

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983351

C3-268 JOURNAL DE PHYSIQUE

bonds /lo/. In all cases an unexpectedly large number of very short chains is obtained which have generally not been identified by other spectroscopic probes, leading to the suggestion that the Raman data m a y be sensitive to subtle disruptions in the c o n j u g a t i o n path ( e m b e d d e d c i s s e g m e n t s , d i h e d r a l a n g l e variations, etc.) /8-10/ which have to date eluded other studies.

Recently we have identified an additonal phenomena which should contribute to the observed dispersion of the Stokes lineshape in (CHI and consequently affect the chain length distributions whicff are extracted from these data /11,12/. We emphasize that since the primary exciting frequency exceeds the threshold for pair creation, fast luminescence from photogenerated carriers will also contribute to the radiated Stokes shifted light.

Unlike the Raman scattered light, this recombination radiation probes the structural dynamics of the photoexc'ited polyene on a s i n g l e e x c i t e d s t a t e surface. T h e e x c i t a t i o n f r e q u e n c y dependence of this relaxation is found to provide a natural and simple explanation for the evolution of the Stokes lineshape with primary exciting frequency.

I1 Coherent

and

Incoherent Scattering Mechanisms

A useful formal distinction between Raman scattering and hot luminescence has been deduced by Shen /13/. Making use of his notation for the third order response of a three level system to the time dependent radiation field, one obtains for the third order density matrix /3/:

where F(w) oscillates as e i W t l

w

is the primary exciting frequency,

Us

the Stokes frequency and

der

is the coupling between the electrons and the radiation field. The first term on the right hand side describes the usual Raman scattering process while the second term describes the radiative recombination of electron hole pairs created by the primary exciting light. The two channels are distinguished by different transient behavior following primary excitation, potentially different selection ru!es relating incoming and outgoing polarizations and different resonance excitation profiles as the incident frequency is tuned though the resonance at V e = Wn

-

IAJ i: A very important physical distinction is possible for the experimental situation in which a continuum of intermediate states is present. The Raman amplitude is complex with real (imaginary) contributions from electronic excitations off (on) resonance; the luminescence channel requires a real excitation to a single intermediate state on the energy conserving "shell". As a consequence, while the Rarnan process probes the structural response to a weighted average of all possible excitations of the electron density, the luminescence channel can probe a structural relaxation o n a single excited state surface.

I11 Excited State Dynamics Following Photoexcitation

The structural response of a polyene to photoexcitation has been explored in t w o numerical studies to date. Su and Schrieffer /14/ monitored the lattice response to sudden creation ofan electron hole pair at the band edges, obtaining a spontaneous, fast (10-l3 sec) symmetry lowering relaxation which traps the photoexcited carriers in charged solitons. In extension of these calculations to photoexcitation above the band edge /11,12/ we have noticed that the relaxation pathway depends strongly on the

primary exciting frequency. This is demonstrated in Figure 1 in which we plot the bond alternation profile in a 48 atom chain 10-l4 seconds following photoexcitation of the third,f ifth and seventh allowed optical excitations of the chain. The ensuing

Fig. 1: Dimerization profile on a t o m chain 10-14 sec after excitation of the third, fifth and seventh allowed optical excitations of the chain.

spontaneous lattice response follows the third, fifth and seventh allowed standing wave patterns in the bond alternation profile.

T h e s u b s e q u e n t l a t t i c e r e s p o n s e is i n t e r e s t i n g , t h o u g h complicated /12/, ultimately also leading to the formation of charged photogenerated solitons. However, we have generally found that the nonadiabatic relaxation pathway is substantially slower than the adiabatic "band edge" relaxation so that the incubation time for these defects following photoexcitation deep in the band should extend into the picosecond regime.

A useful diagramatic description of this relaxation is given in Figure 2, illustrating the dependence of the total energy of a molecule on a configurational coordinate Qi. On the ground

Fig. 2 : Structural energy as a functlon of the configurational coordinate Qi for the ground state

(i), excited state in a short finite chain (n) and excited state in an infinite defect free chain (n' )

.

state surface (i) we assume that Qi is harmonically bound to Qi =

0. On the excited surface (n) the minimum is displaced in response to the photoexcited density. For the molecule the excited state surface has at nonzero gradient at Q = 0, so that the relaxation indeed involves first order coupling of the electron density to the lattice. For an infinite periodic structure the situation is more complicated; there the excited state relaxation has the form of a Peierls distortion and the potential follows the form given by n' in the figure. This is even in Qi so that

aEhr

0 and the lattice response occurs only in even order in the

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electron lattice coupling. As a consequence first order coupling of the excited electron to phonon field is expected for a finite molecule but is s y m m e t r y forbidden in the perfectly periodic structure. The characteristic length which separates these regimes is the chain length Nc for which the lifetime of an electronic state (e.g. for decay via phonon emission) is less than the time required to resolve the eigenstate in energy; or

where W is band width. Taking W = 10 eV and 'i;= 10 -I3 sec as representative values Nc

>

1500 s o that w e expect first order excitation of the lattice following photoexcitation for N

<

lu3 carbons.

IV Lineshape

and

Quantum Efficiency

Hot luminescence at the first order Stokes shifted frequency results from radiative decay from the excited state surface after excitation of a single vibrational quantum. Two quantities are then relevant: (1) the distribution of this Stokes shifted radiation in frequency and ( 2 ) the yield for this emission compared to the usual Raman channel. The structural behavior illustrated in Figure 1 determines the frequency of the Stokes shifted light. For the nth excitation above threshold in a chain of N atoms the lattice response takes the form of the standing wave

s o that a s the excitation frequency increases the dominant wavelength of the given lattice distortion decreases. This is precisely the dependence provided by the models which invoke an inhomogeneous distribution of chain lengths. More detailed analysis of the distribution of phonons driven in the relaxation of the hot carrier yield a qualitatively correct description of the lineshape dispersion /12/. In the luminescence model, the emission at the larger Stokes shifted frequencies is attributed not to a static distribution of chain lengths but to the strong excitation energy dependence of the relaxation pathway. As Figure 1 demonstrates, the dominant wavelength of the lattice response depends strongly on the exciting frequency in a single chain, and thus is not essentially related to the chain length.

W e have also undertaken a calculation of excitation profiles comparing the scattering cross sections in the R a m a n and luminescence channels. The results are shown in Figure 3 /15/.

As expected the Raman profile is resonantly enhanced at the interband threshold (2 eV in the model calculation) and is flat and structureless thereafter. The frequency independence of the scattering cross section above threshold is primarily a result of the real part of the Raman amplitude, resulting from virtual electronic excitations, which d o m i n a t e the scattering. By contrast, the luminescence profile is peaked in the interband continuum and is somewhat stronger than the Raman cross section in the frequency regime of interest. Experimental measurements /5,16/ (integrated over all first order Stokes shifted light) are given by the dashed curves in the figure. The size and shape are consistent with the luminescence model; however unresolved quantitative discrepancies in peak position and width remain.

RAMAN

1 2 3 4

THEORY

Fig. 3: Top: Theoretical (solid) and experimental (dashed)

resonance excitation profiles for

'\ first order Stokes processes in (Ca) x. sottorn: Absorption coefficient in (C9) x.

O 0 1 2 3 4

Measurements directed at time resolving the Stokes shifted radiation or measuring the depolariztion of the outgoing radiation m a y be of further use for distinguishing these t w o processes.

V Acknowledgements

,This work was supported in part by NSF Grant D M R 82-03484. I would also like to thank the Organizing C o m m i t t e e and the Department of Physics at the University of Pennsylvania for additional support which made travel to this meeting possible.

References

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