- Email: [email protected]
Hot luminescence in polyacetylene
Solid State Communications, Vol. 44, No. 6, pp. 827-831, 1982. Printed in Great Britain.
0038-1098/82/420827-05503.00/0 Pergamon Press Ltd.
HOT LUMINESCENCE IN POLYACETYLENE E.J. Mele Department of Physics and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, PA 19104-3859, U.S.A.
(Received 19 March 1982 by G. Burns) We propose that the unusual frequency dependent lineshapes observed in Raman scattering experiments on polyacetylene are due to hot luminescence in very long polyene chains. Employing a generalized equation of motion technique we find that a photoexcited polyene is unstable with respect to a variety of symmetry lowering distortions which vary as a function of the frequency of the exciting radiation. Radiative decay during this relaxation is observably strong in the inelastically scattered photon spectrum and would explain the dominant observed lineshape variations.
RECENT INELASTIC LIGHT scattering studies on polyacetylene, (CH)x, have led to some puzzling conclusions about the microscopic structure of this polymer. The first order Raman sidebands obtained from these films exhibit dramatic lineshape changes as the primary excitation frequency is varied between 2 and 2.7 eV [1-3]. This behaviour has long been interpreted as an inhomogeneous effect arising from the presence of finite chains of various lengths in the material. Recent quantitative analyses within this model, to be described in more detail below, yield a mean conjugation length no longer than 40 double bonds in the polymer [2] and suggest a second peak in the distribution of chain lengths near 5 double bonds [3]. These are surprisingly short estimates and are difFwult to reconcile with estimated conjugation lengths of 200-3000 double bonds from NMR [4], ESR [5] and infrared absorption studies [6]. In this paper we propose an alternative interpretation of the inelastic light scattering results which suggests that short chains are not the primary cause of the unusual lineshapes. We propose that the Raman lineshape results from the dynamic response of the lattice of an infinite (or at least very long) polymer to photoexcitation. We believe that the observed Stokes shifted light is radiated in a hot or fast luminescence process in the material and not a true coherent Raman process [7]. By theoretically examining the initial stages of the relaxation of the polymer backbone following photoexcitation above the band edge, we find that the lattice undergoes a remarkably fast structural relaxation which is highly specific to the frequency of the absorbed photon. The results provide a natural explanation for the observed lineshape variations solely on the basis of inelastic scattering from long chains. While we clearly
cannot exclude the presence of a certain number of short chains in a quasi-crystalline sample like (CH)x , our results do show that the short chains are not required to explain the inelastic light scattering spectra. Consequently we believe that conjugation lengths in (CH)x are underestimated in present analyses of Raman data. Lineshape analyses of Raman spectra in (CH)x have focused primarily on the C=C stretching vibration about which a good deal is known from work on finite polyenes. It is known, for example, that the frequency of the longest wavelength C=C stretching mode in a finite polyene increases as the chain length decreases [8]. It is even better known that the optical absorption edge in a finite polyene increases as the chain length decreases. Further, Raman experiments on polyacetylene films indicate that when the primary excitation frequency exceeds the optical absorption threshold for the film, the Stokes shifted Raman line is skewed, showing a well defined band edge, but tailing towards lower scattered photon energy, or higher scattered phonon frequency [1-3]. It is inferred therefore that the band edge is due to Raman excitation in the longest chains in the film and that the skewing is due to excitations in the shorter chains which are resonantly enhanced at the higher exciting frequencies. This model has been studied in some detail, fits to observed lineshapes assuming a unimodal distribution of chain lengths yields an estimated mean chain length of 15 double bonds [2], whereas a refined model, hypothesizing a bimodal distribution suggests a distribution which peaks at both 40 and 5 double bonds [3]. A large density of short chains is an unavoidable conclusion in such an analysis because the skewing is very large, even taking the form of a well defined second peak displaced from the band edge by as much as 100 cm -~ for exciting radiation at 2.7 eV. 827
HOT LUMINESCENCE IN POLYACETYLENE
828
A number of recent studies have drawn attention to the distinction between the properties of (CH)x and those of a prototypical inorganic semiconductor [9-12]. Of particular relevance are a series of numerical simulations undertaken by Su and Schrieffer [11] (SS) studying the structural response of a model polyene as a function of time following a variety of perturbations. For the case of photoexcitation at the band edges SS observe a very rapid gemination of bond alternation defects which trap the photoexcited band edge carriers. In the present work we have generalized the SS equation of motion technique to study the response of the polyene to photoexcitation above the band edge. This generalized procedure allows for transitions between excited states of the system in parallel with the response of the lattice, and thus allows us to study the various channels of relaxation for a photoexcited carrier. As previously [13], we start from the model Hamiltonian: H = ~
t(+)c . .t+ l ,
oC.,o
+
+tlte ) ~Lt- . - l , OL'n) ~ 0
+ h.c.
. ) O"
+ LT~K(u.+,-u.)' + Zm~r(u.. 1 -u.), 2. . (1) where cn, o annihilates arr electron of spin o on site n, u , labels the longitudinal displacement of each carbon atom from its location in the ideal metallic (equal bond length) structure and M is the CH mass. The t~-+)are linearized in the u , , i.e. t(n+-)(u) = to + e I u . +_l -- un]
ffk)
and F = 4aflr. We choose the parameters to = - - 3 eV, K = 68.6 e V A -2 and a = 8 e V A - ] which yields a 12 eV
rr electron bandwidth, a 1.4 eV band gap in the ground state and an equilibrium dimerization amplitude lu, I = 0.0228 A, results which are representative of transpolyacetylene. It is convenient to reformulate the equation of motion approach [11 ] as follows. For the polymer described by equation (1) with some configuration {u,} we first find the eigenstates {~,} which diagonalize the electronic part of equation (1): He,~. = ~k.~..
(3)
We then construct a density operator f r o m :
/5 = Z/"-OntO.
(4)
11
where f , is the occupation probability in the nth eigenstate. This charge density exerts a force on the nth ion coordinate in the polymer calculable from:
Fn -
0 aUn
Vol. 44, No. 6
tr b n a,
(5)
which can be combined with the lattice part of equation (1) to yield an equation for the evolution of the u,(t): mff,(t) = K l u , . ~(t) + u , _ ~(t) -- 2u,,(t)l + F , .
(6)
In earlier studies [11 ], equations (3)-(6) are numerically integrated to study the evolution of un(t ) from some fixed starting configuration. In the present work, since transitions between excited states cannot be excluded a priori as a mode of relaxation we include in this procedure an equation of motion for the density operator, /~ [7, 14]:
O^
1
/)
]
-~ p = -~ [H(u),p] + ~-/~]damping"
(7)
Thus we replace the construction of the density matrix [equation (4)] by a calculation of the proper time evolution of/5 as given by equation (7). Qualitatively, this generalization may be thought of as allowing the occupation probabilities in equation (4) to evolve in time as well, i.e. f , = f,(t). The damping of the off diagonal terms in equation (7) ultimately limits the transition rates between electronic states of the system [14]. In the present application this damping is dominated by random fields generated by the zero point motion of the lattice. This non-adiabatic relaxation channel, included above, is obviously crucial for describing the long time relaxation of a hot carrier in the polymer. However, we have found that the characteristic time for transitions between excited states is considerably longer (typically 10 -12 sec) than the characteristic time for an adiabatic relaxation in a single excited state ( ~ 10 -14 sec). Thus in the initial stages of the relaxation of the photoexcited carrier, with which we are concerned presently, the adiabatic approach works very well. For times up to 3 x 10 -]4 sec following a photoexcitation the adiabatic and fully non-adiabatic procedures yield qualitatively similar results; a quantitative difference is that the lattice relaxation in the non-adiabatic model proceeds slightly more slowly reflecting the non-zero inertia of the electronic charge density. We are currently extending these simulations to study the long time relaxation of the excited carrier which should be strongly affected by the non-adiabatic channel. The most striking result to emerge from this study is that the short time lattice is highly specific to the frequency of the absorbed photon. This is demonstrated in Fig. 1 in which we examine the modulation of the bond alternation amplitude at a fixed time (1 x 10 -14 sec) following photoexcitation for the four lowest allowed
Vol. 44, No. 6
HOT LUMINESCENCE IN POLYACETYLENE
OAC
OOE 006 0.04 n
o02
4 ~.~ ~ -'~-,
,.
.-.~ .." \.
3
Z
"~'" / / "~/"
' , " - ..... -~, "'~ ./ ".
4 8--..,~.~_.~--~
\ / ,',~ "\ / J "
4~
-o.o2F 004~
_ZL Fig. 1. The bond alternation amplitude, Uo, plotted as a function of location in a 48 atom chain for a time 1 x 10-14 sec following photoexcitation of the four lowest allowed transitions of the chain. optical transitions of a 48 atom chain. For the lowest optical excitation (between band edge states) we observe a strong depression of the bond alternation amplitude at the center of the chain. The zero crossings near sites 12 and 36 signal the emergence of photogenerated solitons as the band edge electronic states relax into the band gap [11 ]. However, for excitations above the band gap the situation is qualitatively different. For the next higher lying excitation we obtain a uniform suppression of the bond alternation amplitude and a shorter wavelength modulation of the bond alternation amplitude, with wavelength given by the length of the finite segment. For the higher lying excitations the shorter wavelength modulation takes the form of a standing wave in the chain of length L, with wavelength ~n = L / ( n -- 1) for the nth excitation above the optical threshold. The origin of this behaviour is easily traced directly back to the effect of photoexcitation on the density matrix in equation (4). The off diagonal elements ( n l ~ l n + 1) where In) are the Ir electron basis states undergo a long wavelength modulation when the electron is excited. This produces an instantaneous force [equation (5)] which is strong along a single normal coordinate of the structure. In a quantum model the system would respond by the emission of a quantum of the vibration corresponding to the driven coordinate. In this finite system the normal coordinates are approximately given by the standing waves allowed in the finite chain. In an infinite or very large system the excited state charge density will similarly lower the translational symmetry of the chain; in this case the spontaneous relaxation would correspond to emission of a zone center phonon in the appropriate folded Brillouin zone. A more descriptive explanation of this phenomenon is that following excitation the polymer is unstable with respect to a relaxation which will lower (raise) the one electron eigenvalue of the final (initial) state connected
829
in the absorption. The relaxation then corresponds to a non-degenerate Jahn-Teller distortion on the excited state potential surface. This interpretation has a particularly elegant realization for the case of a large ring. For the dimerized ring all of the eigenstates except for the band edge states are two-fold degenerate. For photoexcitation above the gap we find that the excited state is unstable with respect to a symmetry lowering distortion which removes this degeneracy for the levels connected in the absorption process. This relaxation has the form of a dynamic Peierls effect. Importantly, because of the relative stability of the system with respect to the non-adiabatic relaxation described above, the initial stages of the lattice response are dominated by dynamics on a single excited state potential surface. We now seek to relate these observations to the measured Raman band-shapes. The equation of motion analysis proceeds from an initial configuration in which there is a real excitation to an electronic excited state. Thus phonon emission in this process would not correspond to a true Raman process (or even a resonant Raman process) which proceed via a virtual excitation to the intermediate state. Instead, the short time radiative decay of the excess population out of the excited intermediate state corresponds in this picture to inelastic light scattering via the hot luminescence channel. This distinction between hot luminescence and Raman scattering has been discussed by Shen [7]. Since hot luminescence is an incoherent process which depends on the population of the intermediate state, the crosssection for this scattering can be calculated from the frequency dependent absorption cross-section oa(co) and the radiative and non-radiative relaxation times rr and ~'nr- We obtain
on,.(~)
=
oa(~)~,,(~)/rr(co).
(8)
The optical absorption cross-section is taken directly from experimental studies and varies from 10 x 10 -la cm 2 to 5 x 10-18cm 2 within 1 e V o f t h e absorption threshold [ 15 ]. The non-radiative relaxation time is evidently of order 10-14 sec (see Fig. 1) and the radiative decay time calculated from the eigenstates of the dimerized chain is approximately 2 x 10 -7 sec. Thus we estimate that OHL(O~)-- 1.2 X 10 -24 cm 2 and weakly dependent on photon frequency. This behaviour contrasts with a calculation of the Raman excitation profile for a linear chain described by the Hamiltonian of equation (1). This analysis yields a peak scattering cross-section near 1 x 10 -2s cm 2 which decreases precipitously with increasing excitation energy. Experimentally the inelastic cross-section is "" 1 x 10 -24 cm 2 and weakly dependent on photon energy [16]. Thus the large magnitude of the cross-section and its weak frequency dependence are both consistent with the hot luminescence model.
HOT LUMINESCENCE IN POLYACETYLENE
830 ~oo-~-
~
~
~
60
l
~
(CDxl :
8 4 0 c m -t
CO ( e V )
Fig. 2. The shift of the satellite peak from the primary line, A(co), plotted as a function of exciting frequency, u~, for the five prominent lines observed in Raman scattering experiments on (CH)~ and (CD)~. (The inset frequencies are the experimental band edge frequencies.) The photoinduced relaxations illustrated in Fig. 1 provide an alternative explanation for the observed Raman lineshape variations. For a dimerized onedimensional chain with a 12 eV bandwidth and a gap Eg = 2 eV, [15] absorption of a photon of energy ~ will occur by excitation between states at wave-vector +- k where: +
k
--
(9)
where v is the Fermi velocity in the metallic state (by = 7.2 eVA). The static lattice relaxation which is matched to the photoexcited population at this level is a standing wave constructed from phonons of wavevector +-q where q = 2k = h~ x/(hw)= - e f t .
(10)
Thus in addition to the normal Raman excitation of the q = 0 mode, the photo-induced relaxation should manifest itself by the emission of vibrational quanta at frequency v(q). This appears as a "satellite" line in the Raman spectrum whose deviation from the primary line depends on the incident photon energy. Taking phonon dispersions from our previous force field calculations on (CH)x and (CD)x [17] the expected frequency shift of this luminescence emission from the primary line, A(6o), is plotted as a function of exciting frequency, fie, in Fig. 2. The curves are calculated for the five prominent lines observed in Raman scattering experiments on (CH)x and (CD)x. The results satisfactorily describe the principle trends observed experimentally. In particular, the ordering of the broadening of the five Raman bands
Vol. 44, No. 6
is given correctly (experimentally the 1450 and 1055 cm -1 lines in (CH)x show nearly equivalent lineshape dispersion in agreement with the similar behaviour calculated for these modes in Fig. 2). In fact, the calculated shifts fall within approximately 10% of the shifts of the centroids of the skewed experimental lineshapes, although there is a systematic overestimate in this model. The conclusion is that the dramatic frequency dependence of the Raman lineshape can be understood as a dynamic effect on a long chain, obviating the need for a large density of short chains. The non-adiabatic channel for the relaxation of the hot carrier may be responsible for part of the width of the secondary emission. Our calculations indicate a mean residence time of the electron in the photoexcited state on the order of 10 -12 sec. This time provides a characteristic time over which correlations in the ion motions can persist, implying an intrinsic lifetime broadening on the order of 10 cm -1 for the satellite line. The linewidth of the secondary line observed for exciting frequency near 2.7 eV exceeds this estimate; for lower primary frequencies the shift A(u;) is on the order of this estimate linewidth, yielding a high frequency tail on the observed line. Inhomogeneous broadening may also contribute to this broadening. This effect is particularly suspect since the details of the lineshapes appear to be somewhat sample dependent [16, 18]. Disorder would also perturb the one electron eigenstates and hence the photoexcited charge density. Consequently, in the presence of weak disorder a distribution of normal modes of the polymer should be excited in the relaxation, inducing some sample dependence in the measured spectra. Effects such as chain twists, bends or local variations in the packing density rather than disruption of the conjugation path could be of relevance here. Despite such disorder related variations in spectral details the frequency dependent skewing of the Raman lines is apparently a general feature of all samples studied. Finally, while we expect that this phenomenon should be intrinsic to a quasi-one-dimensional solid, a fortunate confluence of several factors makes this a very dramatic effect in (CH)x. First, the interesting electronic excitation in this system are in a convenient optically accessible frequency range unlike many of the other one-dimensional solids previously studied. Secondly, as a consequence of the covalently bonded chain structure and light ion mass the phonon bands are extremely broad, making the dispersion of the lineshape a very noticeable effect. In principle it should be possible to find other quasi-one-dimensional systems with properties similar to (CH)x which also exhibit this form of luminescence in their inelastic light scattering spectra. In summary we suggest a reinterpretation of the lineshape variations observed in Raman experiments on
Vol. 44, No. 6
HOT LUMINESCENCE OF POLYACETYLENE
(CH)x. We find that the polyene lattice is intrinsically unstable following photoexcitation, undergoing an adiabatic relaxation which is highly specific to the frequency of the exciting radiation. We estimate that radiative recombination during the relaxation is observably strong in the inelastically scattered photon spectrum, and would take the form of a satellite line to the usual Raman peak in the radiated spectrum. Since such secondary scattering should be expected for long chain polyene a large density of short chains is apparently not required for interpretation of the excitation frequency dependence of the observed Raman lineshapes.
Acknowledgements - It is a pleasure to acknowledge helpful discussions with D.P. DiVincenzo, S. Etemad, A. Heeger and L. Lauchlan, who has performed the absolute cross-section measurements quoted here. This work is supported by the National Science Foundation MRL program under Grant No. DMR-79-23647 and NSF Grant DMR-82-03484.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
REFERENCES 1. 2. 3.
I. Harada, M. Tasumi, H. Shirakawa & S. Ikeda, Chem. Lett. 1411 (1978). L.S. Lichtmann, D.B. Fitchen & H. Temkin, Synthetic Metals 1,139 (1980); L.S. Lichtmann, Unpublished Ph.D. thesis, Cornell University. H. Kuzmany, Phys. Status Solidi 97, 521 (1980).
17. 18.
831
W.G. Clark, K. Glover, S. Etemad & M. Maxfield (to be published). M. Nechtschein, F. Devreaux, R.L. Greene, T.C. Clarke & G.B. Street, Phy~ Rev. Lett. 44,356 (1980). H.W. Gibson et al., Proc. Int. Conf. on Low Ih'mensional Conductors, biol. Crystals and Liquid Crystals (in press). Y.R. Shen, Phys. Rev. B9,622 (1974). R.M. Gavin & S.A. Rice, J. Chem. Phys. 55, 1675 (1971). W.P. Su, J.R. Schrieffer & A.J. Heeger, Phy~ Rev. Lett. 42, 1698 (1979). M.J. Rice, Phys. Lett. 71A, 1521 (1979). W.P. Su & J.R. Schrieffer, Proc. Nat. Acad. Sc~, New York 77, 5626 (1980). E.J. Mele & M.J. Rice, Phys. Rev. B23, 5397 (1981). D. Vanderbilt & E.J. Mele, Phys. Rev. B22, 3939 (1980). N. Bloembergen, Nonlinear Optics. Benjamin, New York (1965). N. Suzuki, M. Ozaki, S. Etemad, A.J. Heeger & A.G. MacDiarmid,Phys. Rev. Lett. 45, 1209 (1980). L. Lauchlan, Private communication and to be published. E.J. Mele, Proc. Int. Conf. on Low Dimensional Conductors, Mol. Crystals and Liquid Crystals 77, 25 (1981). S. Lefrant, E. Faulques, L. Lauchlan, M.J. Kletter & S. Etemad, Proc. 6th Int. Conf. on Low Dimensional Conductors, Mol. Crystals and Liquid Crystals (in press).
0038-1098/82/420827-05503.00/0 Pergamon Press Ltd.
HOT LUMINESCENCE IN POLYACETYLENE E.J. Mele Department of Physics and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, PA 19104-3859, U.S.A.
(Received 19 March 1982 by G. Burns) We propose that the unusual frequency dependent lineshapes observed in Raman scattering experiments on polyacetylene are due to hot luminescence in very long polyene chains. Employing a generalized equation of motion technique we find that a photoexcited polyene is unstable with respect to a variety of symmetry lowering distortions which vary as a function of the frequency of the exciting radiation. Radiative decay during this relaxation is observably strong in the inelastically scattered photon spectrum and would explain the dominant observed lineshape variations.
RECENT INELASTIC LIGHT scattering studies on polyacetylene, (CH)x, have led to some puzzling conclusions about the microscopic structure of this polymer. The first order Raman sidebands obtained from these films exhibit dramatic lineshape changes as the primary excitation frequency is varied between 2 and 2.7 eV [1-3]. This behaviour has long been interpreted as an inhomogeneous effect arising from the presence of finite chains of various lengths in the material. Recent quantitative analyses within this model, to be described in more detail below, yield a mean conjugation length no longer than 40 double bonds in the polymer [2] and suggest a second peak in the distribution of chain lengths near 5 double bonds [3]. These are surprisingly short estimates and are difFwult to reconcile with estimated conjugation lengths of 200-3000 double bonds from NMR [4], ESR [5] and infrared absorption studies [6]. In this paper we propose an alternative interpretation of the inelastic light scattering results which suggests that short chains are not the primary cause of the unusual lineshapes. We propose that the Raman lineshape results from the dynamic response of the lattice of an infinite (or at least very long) polymer to photoexcitation. We believe that the observed Stokes shifted light is radiated in a hot or fast luminescence process in the material and not a true coherent Raman process [7]. By theoretically examining the initial stages of the relaxation of the polymer backbone following photoexcitation above the band edge, we find that the lattice undergoes a remarkably fast structural relaxation which is highly specific to the frequency of the absorbed photon. The results provide a natural explanation for the observed lineshape variations solely on the basis of inelastic scattering from long chains. While we clearly
cannot exclude the presence of a certain number of short chains in a quasi-crystalline sample like (CH)x , our results do show that the short chains are not required to explain the inelastic light scattering spectra. Consequently we believe that conjugation lengths in (CH)x are underestimated in present analyses of Raman data. Lineshape analyses of Raman spectra in (CH)x have focused primarily on the C=C stretching vibration about which a good deal is known from work on finite polyenes. It is known, for example, that the frequency of the longest wavelength C=C stretching mode in a finite polyene increases as the chain length decreases [8]. It is even better known that the optical absorption edge in a finite polyene increases as the chain length decreases. Further, Raman experiments on polyacetylene films indicate that when the primary excitation frequency exceeds the optical absorption threshold for the film, the Stokes shifted Raman line is skewed, showing a well defined band edge, but tailing towards lower scattered photon energy, or higher scattered phonon frequency [1-3]. It is inferred therefore that the band edge is due to Raman excitation in the longest chains in the film and that the skewing is due to excitations in the shorter chains which are resonantly enhanced at the higher exciting frequencies. This model has been studied in some detail, fits to observed lineshapes assuming a unimodal distribution of chain lengths yields an estimated mean chain length of 15 double bonds [2], whereas a refined model, hypothesizing a bimodal distribution suggests a distribution which peaks at both 40 and 5 double bonds [3]. A large density of short chains is an unavoidable conclusion in such an analysis because the skewing is very large, even taking the form of a well defined second peak displaced from the band edge by as much as 100 cm -~ for exciting radiation at 2.7 eV. 827
HOT LUMINESCENCE IN POLYACETYLENE
828
A number of recent studies have drawn attention to the distinction between the properties of (CH)x and those of a prototypical inorganic semiconductor [9-12]. Of particular relevance are a series of numerical simulations undertaken by Su and Schrieffer [11] (SS) studying the structural response of a model polyene as a function of time following a variety of perturbations. For the case of photoexcitation at the band edges SS observe a very rapid gemination of bond alternation defects which trap the photoexcited band edge carriers. In the present work we have generalized the SS equation of motion technique to study the response of the polyene to photoexcitation above the band edge. This generalized procedure allows for transitions between excited states of the system in parallel with the response of the lattice, and thus allows us to study the various channels of relaxation for a photoexcited carrier. As previously [13], we start from the model Hamiltonian: H = ~
t(+)c . .t+ l ,
oC.,o
+
+tlte ) ~Lt- . - l , OL'n) ~ 0
+ h.c.
. ) O"
+ LT~K(u.+,-u.)' + Zm~r(u.. 1 -u.), 2. . (1) where cn, o annihilates arr electron of spin o on site n, u , labels the longitudinal displacement of each carbon atom from its location in the ideal metallic (equal bond length) structure and M is the CH mass. The t~-+)are linearized in the u , , i.e. t(n+-)(u) = to + e I u . +_l -- un]
ffk)
and F = 4aflr. We choose the parameters to = - - 3 eV, K = 68.6 e V A -2 and a = 8 e V A - ] which yields a 12 eV
rr electron bandwidth, a 1.4 eV band gap in the ground state and an equilibrium dimerization amplitude lu, I = 0.0228 A, results which are representative of transpolyacetylene. It is convenient to reformulate the equation of motion approach [11 ] as follows. For the polymer described by equation (1) with some configuration {u,} we first find the eigenstates {~,} which diagonalize the electronic part of equation (1): He,~. = ~k.~..
(3)
We then construct a density operator f r o m :
/5 = Z/"-OntO.
(4)
11
where f , is the occupation probability in the nth eigenstate. This charge density exerts a force on the nth ion coordinate in the polymer calculable from:
Fn -
0 aUn
Vol. 44, No. 6
tr b n a,
(5)
which can be combined with the lattice part of equation (1) to yield an equation for the evolution of the u,(t): mff,(t) = K l u , . ~(t) + u , _ ~(t) -- 2u,,(t)l + F , .
(6)
In earlier studies [11 ], equations (3)-(6) are numerically integrated to study the evolution of un(t ) from some fixed starting configuration. In the present work, since transitions between excited states cannot be excluded a priori as a mode of relaxation we include in this procedure an equation of motion for the density operator, /~ [7, 14]:
O^
1
/)
]
-~ p = -~ [H(u),p] + ~-/~]damping"
(7)
Thus we replace the construction of the density matrix [equation (4)] by a calculation of the proper time evolution of/5 as given by equation (7). Qualitatively, this generalization may be thought of as allowing the occupation probabilities in equation (4) to evolve in time as well, i.e. f , = f,(t). The damping of the off diagonal terms in equation (7) ultimately limits the transition rates between electronic states of the system [14]. In the present application this damping is dominated by random fields generated by the zero point motion of the lattice. This non-adiabatic relaxation channel, included above, is obviously crucial for describing the long time relaxation of a hot carrier in the polymer. However, we have found that the characteristic time for transitions between excited states is considerably longer (typically 10 -12 sec) than the characteristic time for an adiabatic relaxation in a single excited state ( ~ 10 -14 sec). Thus in the initial stages of the relaxation of the photoexcited carrier, with which we are concerned presently, the adiabatic approach works very well. For times up to 3 x 10 -]4 sec following a photoexcitation the adiabatic and fully non-adiabatic procedures yield qualitatively similar results; a quantitative difference is that the lattice relaxation in the non-adiabatic model proceeds slightly more slowly reflecting the non-zero inertia of the electronic charge density. We are currently extending these simulations to study the long time relaxation of the excited carrier which should be strongly affected by the non-adiabatic channel. The most striking result to emerge from this study is that the short time lattice is highly specific to the frequency of the absorbed photon. This is demonstrated in Fig. 1 in which we examine the modulation of the bond alternation amplitude at a fixed time (1 x 10 -14 sec) following photoexcitation for the four lowest allowed
Vol. 44, No. 6
HOT LUMINESCENCE IN POLYACETYLENE
OAC
OOE 006 0.04 n
o02
4 ~.~ ~ -'~-,
,.
.-.~ .." \.
3
Z
"~'" / / "~/"
' , " - ..... -~, "'~ ./ ".
4 8--..,~.~_.~--~
\ / ,',~ "\ / J "
4~
-o.o2F 004~
_ZL Fig. 1. The bond alternation amplitude, Uo, plotted as a function of location in a 48 atom chain for a time 1 x 10-14 sec following photoexcitation of the four lowest allowed transitions of the chain. optical transitions of a 48 atom chain. For the lowest optical excitation (between band edge states) we observe a strong depression of the bond alternation amplitude at the center of the chain. The zero crossings near sites 12 and 36 signal the emergence of photogenerated solitons as the band edge electronic states relax into the band gap [11 ]. However, for excitations above the band gap the situation is qualitatively different. For the next higher lying excitation we obtain a uniform suppression of the bond alternation amplitude and a shorter wavelength modulation of the bond alternation amplitude, with wavelength given by the length of the finite segment. For the higher lying excitations the shorter wavelength modulation takes the form of a standing wave in the chain of length L, with wavelength ~n = L / ( n -- 1) for the nth excitation above the optical threshold. The origin of this behaviour is easily traced directly back to the effect of photoexcitation on the density matrix in equation (4). The off diagonal elements ( n l ~ l n + 1) where In) are the Ir electron basis states undergo a long wavelength modulation when the electron is excited. This produces an instantaneous force [equation (5)] which is strong along a single normal coordinate of the structure. In a quantum model the system would respond by the emission of a quantum of the vibration corresponding to the driven coordinate. In this finite system the normal coordinates are approximately given by the standing waves allowed in the finite chain. In an infinite or very large system the excited state charge density will similarly lower the translational symmetry of the chain; in this case the spontaneous relaxation would correspond to emission of a zone center phonon in the appropriate folded Brillouin zone. A more descriptive explanation of this phenomenon is that following excitation the polymer is unstable with respect to a relaxation which will lower (raise) the one electron eigenvalue of the final (initial) state connected
829
in the absorption. The relaxation then corresponds to a non-degenerate Jahn-Teller distortion on the excited state potential surface. This interpretation has a particularly elegant realization for the case of a large ring. For the dimerized ring all of the eigenstates except for the band edge states are two-fold degenerate. For photoexcitation above the gap we find that the excited state is unstable with respect to a symmetry lowering distortion which removes this degeneracy for the levels connected in the absorption process. This relaxation has the form of a dynamic Peierls effect. Importantly, because of the relative stability of the system with respect to the non-adiabatic relaxation described above, the initial stages of the lattice response are dominated by dynamics on a single excited state potential surface. We now seek to relate these observations to the measured Raman band-shapes. The equation of motion analysis proceeds from an initial configuration in which there is a real excitation to an electronic excited state. Thus phonon emission in this process would not correspond to a true Raman process (or even a resonant Raman process) which proceed via a virtual excitation to the intermediate state. Instead, the short time radiative decay of the excess population out of the excited intermediate state corresponds in this picture to inelastic light scattering via the hot luminescence channel. This distinction between hot luminescence and Raman scattering has been discussed by Shen [7]. Since hot luminescence is an incoherent process which depends on the population of the intermediate state, the crosssection for this scattering can be calculated from the frequency dependent absorption cross-section oa(co) and the radiative and non-radiative relaxation times rr and ~'nr- We obtain
on,.(~)
=
oa(~)~,,(~)/rr(co).
(8)
The optical absorption cross-section is taken directly from experimental studies and varies from 10 x 10 -la cm 2 to 5 x 10-18cm 2 within 1 e V o f t h e absorption threshold [ 15 ]. The non-radiative relaxation time is evidently of order 10-14 sec (see Fig. 1) and the radiative decay time calculated from the eigenstates of the dimerized chain is approximately 2 x 10 -7 sec. Thus we estimate that OHL(O~)-- 1.2 X 10 -24 cm 2 and weakly dependent on photon frequency. This behaviour contrasts with a calculation of the Raman excitation profile for a linear chain described by the Hamiltonian of equation (1). This analysis yields a peak scattering cross-section near 1 x 10 -2s cm 2 which decreases precipitously with increasing excitation energy. Experimentally the inelastic cross-section is "" 1 x 10 -24 cm 2 and weakly dependent on photon energy [16]. Thus the large magnitude of the cross-section and its weak frequency dependence are both consistent with the hot luminescence model.
HOT LUMINESCENCE IN POLYACETYLENE
830 ~oo-~-
~
~
~
60
l
~
(CDxl :
8 4 0 c m -t
CO ( e V )
Fig. 2. The shift of the satellite peak from the primary line, A(co), plotted as a function of exciting frequency, u~, for the five prominent lines observed in Raman scattering experiments on (CH)~ and (CD)~. (The inset frequencies are the experimental band edge frequencies.) The photoinduced relaxations illustrated in Fig. 1 provide an alternative explanation for the observed Raman lineshape variations. For a dimerized onedimensional chain with a 12 eV bandwidth and a gap Eg = 2 eV, [15] absorption of a photon of energy ~ will occur by excitation between states at wave-vector +- k where: +
k
--
(9)
where v is the Fermi velocity in the metallic state (by = 7.2 eVA). The static lattice relaxation which is matched to the photoexcited population at this level is a standing wave constructed from phonons of wavevector +-q where q = 2k = h~ x/(hw)= - e f t .
(10)
Thus in addition to the normal Raman excitation of the q = 0 mode, the photo-induced relaxation should manifest itself by the emission of vibrational quanta at frequency v(q). This appears as a "satellite" line in the Raman spectrum whose deviation from the primary line depends on the incident photon energy. Taking phonon dispersions from our previous force field calculations on (CH)x and (CD)x [17] the expected frequency shift of this luminescence emission from the primary line, A(6o), is plotted as a function of exciting frequency, fie, in Fig. 2. The curves are calculated for the five prominent lines observed in Raman scattering experiments on (CH)x and (CD)x. The results satisfactorily describe the principle trends observed experimentally. In particular, the ordering of the broadening of the five Raman bands
Vol. 44, No. 6
is given correctly (experimentally the 1450 and 1055 cm -1 lines in (CH)x show nearly equivalent lineshape dispersion in agreement with the similar behaviour calculated for these modes in Fig. 2). In fact, the calculated shifts fall within approximately 10% of the shifts of the centroids of the skewed experimental lineshapes, although there is a systematic overestimate in this model. The conclusion is that the dramatic frequency dependence of the Raman lineshape can be understood as a dynamic effect on a long chain, obviating the need for a large density of short chains. The non-adiabatic channel for the relaxation of the hot carrier may be responsible for part of the width of the secondary emission. Our calculations indicate a mean residence time of the electron in the photoexcited state on the order of 10 -12 sec. This time provides a characteristic time over which correlations in the ion motions can persist, implying an intrinsic lifetime broadening on the order of 10 cm -1 for the satellite line. The linewidth of the secondary line observed for exciting frequency near 2.7 eV exceeds this estimate; for lower primary frequencies the shift A(u;) is on the order of this estimate linewidth, yielding a high frequency tail on the observed line. Inhomogeneous broadening may also contribute to this broadening. This effect is particularly suspect since the details of the lineshapes appear to be somewhat sample dependent [16, 18]. Disorder would also perturb the one electron eigenstates and hence the photoexcited charge density. Consequently, in the presence of weak disorder a distribution of normal modes of the polymer should be excited in the relaxation, inducing some sample dependence in the measured spectra. Effects such as chain twists, bends or local variations in the packing density rather than disruption of the conjugation path could be of relevance here. Despite such disorder related variations in spectral details the frequency dependent skewing of the Raman lines is apparently a general feature of all samples studied. Finally, while we expect that this phenomenon should be intrinsic to a quasi-one-dimensional solid, a fortunate confluence of several factors makes this a very dramatic effect in (CH)x. First, the interesting electronic excitation in this system are in a convenient optically accessible frequency range unlike many of the other one-dimensional solids previously studied. Secondly, as a consequence of the covalently bonded chain structure and light ion mass the phonon bands are extremely broad, making the dispersion of the lineshape a very noticeable effect. In principle it should be possible to find other quasi-one-dimensional systems with properties similar to (CH)x which also exhibit this form of luminescence in their inelastic light scattering spectra. In summary we suggest a reinterpretation of the lineshape variations observed in Raman experiments on
Vol. 44, No. 6
HOT LUMINESCENCE OF POLYACETYLENE
(CH)x. We find that the polyene lattice is intrinsically unstable following photoexcitation, undergoing an adiabatic relaxation which is highly specific to the frequency of the exciting radiation. We estimate that radiative recombination during the relaxation is observably strong in the inelastically scattered photon spectrum, and would take the form of a satellite line to the usual Raman peak in the radiated spectrum. Since such secondary scattering should be expected for long chain polyene a large density of short chains is apparently not required for interpretation of the excitation frequency dependence of the observed Raman lineshapes.
Acknowledgements - It is a pleasure to acknowledge helpful discussions with D.P. DiVincenzo, S. Etemad, A. Heeger and L. Lauchlan, who has performed the absolute cross-section measurements quoted here. This work is supported by the National Science Foundation MRL program under Grant No. DMR-79-23647 and NSF Grant DMR-82-03484.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
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