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ADIABATIC NONLINEAR DYNAMICS IN '10DEL, OF QUASIONEDIMEN:JSIOrNAL CO[1JUGATED POLYMERS BY SIMON ROBERT PHILLPOT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985 To my Father and to the Memory of my Mother. Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/adiabaticnonline00phil AC'riOLLEDGMENTS It is with great pleasure that I thank my mentor and friend, Pradeep Kumar, who has provided both inspiration and guidance thrn :it my radiate career. I also owe a great debt to Alan Bishop, who flJeted and then skillfully and patiently led me through my thesis problem. If I can call myself a professional physicist it is largely due to the influences of the above two gentlemen. I have also tenefited from collaborating with Dionys Baeriswyl, David Campbell, Baruch Horovitz, and Peter Lomdahl, each of whom has left his intellectual mark. I should like to thank my committee, Drs. Beatty, Dufty, Tanner, and Trickey, for their stimulating sigestions and incisive questions. Sheri Hill's preparation of this manuscript has been both quick and accurate as has Chris Fombarlet's preparation of the figures. Both John Aylmer and Nick Jelley were strong influences on my career. I am grateful to Lord Trend and President Marston for making it possible for me to come to Florida. I have also enjoyed the hospitality of CNLS and T11 at Los Alamos over the last two years. There are many others, whose friendship has made my last five years productive and pleasurable. They know who they are and they have my gratitude. Althoijh my family has been physically far away, they have provided a constant source of support and encouragement. Finally I should like to thank Melanie for making the last year and a half a very happy' time for me. TUBLE OF COi.JT7iT ACKNOWLEDGMENTS.................................... ........ ..........iii ABSTRACT..............................................................vi CHAPTER I [L TRODUCTION ................... .................................1 II BASIC T.EORFi ....................................................5 TransPolyacetylene........................................5 The H Model..............................................7 The TLM Model.............................................10 Limitations of the SSH and TLM models....................15 III 0i0iAMICS OF A SINGLE KINK IN THE SSH AND TLM MODELS............ 26 The Numerical Technique..................................26 Boundary Conditions......................................29 Statics of the SSH Model.................................31 Optical Absorption within the SSH Model..................33 Kink Dynamics in the TLM Model...........................37 Kink Dynamics in the SSH Model...........................41 IV PHOTIriE:. STATION IN TRANSPOLYACETYLENE. ........................59 The Analytic Breather....................................60 Breather Dynamics and Optical Absorption.................62 Photoexcitation in the Presence of Intrinic Gap State....66 Above Band Edge Photoexcitation..........................67 Neutron Scattering Cross Section of the Breather.........68 Quantization of the Breather.............................71 Comparison Between Theory and Experiment.................73 V DYNAMICS IN CISPOLYACETYLENE AND RELATED MATERIALS............94 Statics in CisPolyacetylene.............................94 Dynamics in CisPolyacetylene............................97 Breather Dynamics in CisPolyacetylene...................99 Dynamics in Finite Polyenes..............................99 Photoexcitation in the Presence of Damping..............101 VI DYNAMICS IN DEFECTED SYSTEMS..................................118 Static Model for the Single Site Impurity............... 119 Dynamics in the Presence of a Single Site Impurity...... 122 KinkSite Impurity Interactions ........................1 Photoexcitation in the Presence of a Single Site Impurity............................128 Photoexcitation with Many Site Impurities...............131 The Sin'l e Bond Impurity................................ 132 KinkBond Impurity Interactions.........................134 rnore1taitation in the Presence of a Single Bond Impurity............................1>_ VII STATICS AND D':YAMICS IN POL11.:F............................... 153 Formalism............................................... 163 The Continuum Model......................................167 The Grs..leve ? Model ................................... 169 Nonlinear Excitations in Polyyne........................171 Adiabatic Nonlinear D,;'rinlics in Polyyne.................. 173 VIII CONCLUSIONS............................ .......................185 APPENDIX A PAR.A' E1T79 OF THE SSH AND TLM MODELS..........................189 B OPTICAL ABSOCRPTION IN THE SSH MODEL...........................191 C THE ANALYTIC BREATHER.........................................199 D CLASSICAL DYNAMIC STRUCTURE FACTOR OF THE BREATHER.............204 E CONTINUED FRACTION SCHE'IE FOR IMPURITY LEVELS...................2 BIBLIOC ;APHY.........................................................214 BIOGRAPHICAL SKETCH..................................................219 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ADIABATIC NONLINEAR DYNAMICS IN MODELS OF QUASIONEDIMENJSI.;.NAL CONJUGAL D POLYMERS By Simon Robert Phillpot May 1985 Chairman: Pradeep Kumar Major Department: Physics We undertake a systematic study of the adiabatic dynamics of nonlinear excitations within the Su, Schrieffer and Heger (SSH) model for polyacetylene. The SSH model is a tightbinding electronphonon coupled system and admits both kink soliton and polaron solutions. Analytic and numerical studies show that, due to the finite response time of the lattice to changes in its dimerization, the kink has a maximum free propagation velocity. In a numerical simulation of a simple photoexcitation experiment in transpolyacetylene we find that a kinkantikink pair and a coherent optical phonon package, a breather, are produced. We calculate the optical absorption spectrum of the photoexcited system and suggest that the breather may account for the anomalous subband edge absorption in transpolyacetylene. We compare two suggested Hamiltonians for cispolyacetylene and compare their photoexcitation dynamics with those of transpolyacetylene. We further investigate the role of coherent anharmonicity in simple models of a finite polyene and of pol.':,"i. As the observed transport properties of polyacetylene j.pend critically on the presence of external impurities we study the dynamics in the presence of model defects and we find that a single kink may be trapped at a single site or confined to a short segment of the system. In photoexcitation experiments on the defected system we find that kinks, breathers, polarons, excitons and traoped kinks may be produced. CHAPTER I I :TF:DjCU'i riON Polymers have long been valued as plastics for their low cost and lightweight, durable, flexible structure. The observation in 19771 that by adding small quantities of metallic and nonmetallic dopants the conductivity of polyacetylene can be varied over twelve orders of magnitude, from that of a :rJ insulator to that of a fair metal, has not surprisingly, produced a great deal of interest. Most of the eniir. ; activity has been in four broad areas: technological applications, synthesis, physical testing and characterization, and theoretical modeling. Arri.m n2,t the suggested technological applications2 for polyacetylene and related materials are Schottky diodes, solar cells, li:;ht;wei.iht metals, and high power storage batteries. Ito et al.3 first synthesized selfsupporting films of polyacetylene by the direct polrimerization of acetylene gas in the presence of catalysts. This method is essentially uncontrolled and produces samples of widely varying structure and physical properties. Since then much effort2 has been made to produce samples with predictable and reproducible physical characteristics, as well as the synthesis of forms of polyacetylene that are air stable and soluble in organic solvents. Another area of major interest has been in the synthesis of other conducting polymers. Some coni'ugated polyaromatic heterocyclic polymers and some conjugated block copolymers can be 1 synthesized u:ing conventional ioni.3n1c techniques. T'dir conductivities are only, however, at most 110% that of polyacetylene. The full barrage of standard solid state spectroscopies has been used to characterize polyacetylene : electron and Xray diffraction have been used to study its structure; Raman scattering to determine its conjugation lenlith; infrared and optical studies of its excitation spectrum; conductivity measurements of its charge transport properties; and ESR and susceptibility measurements of its magnetic properties. The extreme sensitivity of the morphology, structure and transport properties of polyacetylene as currently synthesized has, to some extent, been re!pri'3ible for the widely differing experimental data and interpretation. However one particularly significant conclusion has emerged: ESR experiments have shown that the charce carriers in polyacetylene are spinless and therefore are not electrons. The theoretical work has been in three general areas. Ab initio calculations,5 usually at the HartreeFock level, give a good understanding of the structure and properties of the groundstate. They cannot, however, even qualitatively describe the properties of the excited system. Phenomenological Hamiltonians8 that include strong electronelectron interactions can well describe both the groundstate and the excited state system. Analytic calculations, however, are extremely difficult and computational difficulties limit studies to the static properties of short chain systems. In 1979 Su, Schrieffer and Heeger (SSH)911 proposed a simple tightbinding electronphonon coupled model, which naturally explained the reverse spincharge relation as arising from kink solitons. This static discrete model and its continuum limit 1213 proved to be analytically tractable and their properties have been extensively investigated. The success of this tightbinding model is particularly remarkable as neither electron electron interactions nor quantum lattice effects are included. Each of these is separately difficult to include and studies continue.2 None of the above theoretical viewpoints has included dynamics, which are clearly essential if the unusual transport properties of the conjugated polymers are to be understood. Here we shall undertake a systematic study of the dynamics of the nonlinear excitations in the SSH model. The focus will be in two areas: first, to obtain a basic understanding of the dynamics of nonlinear excitations; and second, to simulate real experiments and make qualitative and quantitative comparisons with data. The major task is to establish whether the simple electronphonon model can even qualitatively describe the dynamics of the real system. In particular it is necessary to identify physically plausible production mechanisms for nonlinear excitations; to establish their robustness in the presence of deviations from ideality e.g. damping and disorder; and to investigate their transport and thermodynamic properties. In Chapter II the relevant parts of the theory of the electron phonon model are outlined. The idea of nonlinear excitations is introduced and the limitations of the model discussed. In Chapter III the numerical algorithm for the solution of the discrete SSH model is discussed. It is shown that the static excitations found in the continuum analysis are good approximate solutions to the discrete equations and that their optical properties can be understood from the continuum calculations. Analytic and numerical studies of the dynamics of a free kink show that the kink propagation velocity is limited by the finite response time of the lattice. In Chapter IV a simple photoexcitation experiment in tr:an.c lyacetylene is modeled. It is found that both kinks and a coherent optical phonon package, a "breather", are produced in such an experiment and that this photoexcitation scenario is consistent with the experimental data. In Chapter V cispolyacetylene and the finite polyenes are discussed. In Chapter VI it is shown that a wide range of nonlinear excitations can be produced during the photoexcitation of disordered systems. Their robustness is established and their tr an rt properties discussed. Chapter VII introduces the polyynes and suggests that nonlinear excitations may be iinportant here, also. Chapter VIII presents our conclusions. CHAPTER II BASIC THEORY TransPolyacetylene Polyacetylene is a linear chain polymer consisting of a spine of carbon atoms with a single hydrogen atom bonded to each. It exists in two isomeric forms: cispolyacetylene and the energetically more stable transpolyacetylene. The present discussion will concentrate on transpol;.:etyene. (A full discussion of cispolyacetylene will be undertaken in CVa.iitr V.) Of the four carbon valence electrons, three are in a bonds to a h:,d'r ge: atom and to the two nei;h :.boring carbon atoms. The remaining electron is in the i band, which is half filled. One thus expects transpolyacetylene to be a conductor. However, undoped transpolyacetylene is a semiconductor with a bandgap 2A 1.41.6 eV. This gap is, at least in part, due to the Peierls effectl7: for a one dimensional system any nonzero electronphonon coupling lowers the energy by inducing a gap at the Fermi level. For a half filled band the band gap at wavevector k = i/2a increases the unit cell size from one (CH) unit to two, thus dimerizing the lattice: i.e. there is an alternation of long ("single") and short ("double") bonds. This symmetry breaking can occur with a single bond either to the right (A phase) or to the left (B phase) of an even site carbon (fig. 2.1). Thus there are two structurally distinct though energetically degenerate groundstates in the long chain limit. A kink soliton can 5 interpolate between them. Physically this kink comes from the joining of a section of A phase to a section of B phase producing a pair of (say) single bonds next to each other (fig. 2.2) and thus a localized electron at this defect. This picture of a kink localized over a si.le lattice spacing is, however, oversimplified; rather it is extended over a few lattice spacings due the competition of two effects: on the one hand the interpolating region between the two groundstates has a higher :ne?' than the dimerized state and thus favors a narrow kink; on the other hand the stiffness of the electron gas favors a wide kink. For "realistic" polyacetylene parameters the width of the kink is 25 10 20a (a is the lattice constant). As we shall see this large width justifies the use of a continuum approximation. That the kink has an associated localized electronic level can be understood from the following argument. Consider a neutral chain of 2n+1 atoms of a single phase. Tnere are 2n bonds joining the 2n+1 (CH) units. On joining the two ends of the chain to form a ring an additional bond is added, giving 2n+1 bonds. Thus, at some point on the chain there must be two single (say) bonds adjacent. A ring with an odd number of sites, therefore, has a kink in it. With 2n+1 sites there are 2n+1 associated energy levels. So for any Hamiltonian that has charge conjugation symmetry (i.e. for every electronic level at energy e there is a level at energy E) there is a level exactly at miidgap associated with the kink. There are 2n electrons in the valence band that pair to give zero spin. For a neutral kink the midgap state is singly occupied and thus the kink has spin 1/2. Adding an electron (hole) gives a kink charge of 1 (+1) and spin 0. These novel spincharge relations provide a natural explanation for the simultaneous observation of high electrical conductivity and low Curie susceptibility : the charge carriers in doped polyacetylene are charged, zero spin, kinks. The SSH Model The SSH Hamiltonian Su, Schrieffer and Heeger9 (SSH) have proposed a simple Hamiltonian for transpolyacetylene based on a number of reasonable assumptions: 1. All manybody effects can be incorporated in a si"ile particle Hamiltonian. 2. The effects of a electrons can be accounted for in the chain cohesion and lattice dynamics. Xr 18 and NMR19 data show that a 1.22A and u0 0.03 0.01 A (a is the lattice constant and u. is the dimerization). Thus excursions of a (CH) unit from its Bravais lattice site are small and the harmonic approximation is a useful approach to the lattice dynamics. The nr electrons are highly mobile and delocalized. They must, therefore, be treated explicitly. 3. The atomic displacements can be treated as classical variables and electron states can be evaluated with respect to fixed atomic displacements. The adiabatic approximation can be expected to be valid if the ratio of the sound velocity to the Fermi velocity is small. For "realistic" polyacetylene parameters this ratio is 0.05. Brazovskii and Dzyaloshinskii20 have shown rigorously that the adiabatic approximation requires u << 1, where y = Hio /A For transpolyacetylene p 0.1. o o 4. A linear combination of atomic orbitals can be used as a basis set. 5. Only matrix elements between states with bonded sites need be considered (tight binding approximation). 6. The system is onedimensional and interchain effects can be neglected. Experiments suggest that the interchain overlap is 100 times less than the intrachain overlap in well oriented samples. The effects of interchain coupling and crosslin1:4 may, however, be important in many materials. The SSH Hamiltonian is H E2 K H(u u ) [t +a(u u )][IcC +C C] (2.1) SSH 2 n 2+ 1 0 n n1 n n n1 n n n where M is the mass of a (CH) unit, K is the force constant, t0 is the intrinsic electron looping matrix element, a is the electronphonon coupling constant. C (C ) creates (annihilates) an electron on site n; n n un is the displacement of the nth (CH) unit from its lattice site. The first term is the lattice kinetic energy; the second is the lattice elastic energy (due to a electrons); the third is the in electronic energy, consisting of intrinsic and phononassisted contributions. Dimerized Groundstate9 The SSH Hamiltonian has two jezenerate uniform solutions un = (1)n uo. (2.2) Now defining a "staggered order parameter", n = (1) /uo (2.3) then S= 1 (2.4) It is interestirn to note that with two d.'e:er'ate groundstates and with a local n'!';,' maximum at u=0O the SSH model is topologically identical to the Q model. This similarity will be exploited at various times, but it must be remembered that D4 is a lattice theory whilst in the SSH model the electrons play an important role. Certain analytic results have been obtained for the SSH model. In particular it can be shown in the weak :0iling limit that the band gap, 2A is given by 2AO = 8au (2.5) The major problem of the SSH model is that very few exact results can be obtained. In particular no analytic form for the kink soliton has been found. Numerical studies, however, show that for the variational kink profile, A(x) = A tanh(x/E) (the exact 4 kink solution), the kink half width, ,, is 710a. This suggests that a continuum approximation to the SSH Hamiltonian may provide a useful basis for further analytic studies. The TLM Model The TLM Hamiltonian The continuum apppr'.: imation is valid if the electronphonon 2 coupling is weak: <1 where A = 2a /wt k. For .:'! parameters 0 X 0.2. The fermion operators C and C can be considered as the sim n n of left and rightgoing waves.12,13 Cns (n) e iv n) e. (2.6) /a For the halffilled band the system dimerizes to give a lattice period 2a. Thus we define 1 a+( 2i _na 2ik na u [A(n) e + A(n) e F2 (2.7) n 4(a To order a, and dropping the rapidly varying terms of the type (1)n the SSH Hamiltonian becomes the classical TLM (Tak3yama, LinLiu, Maki) Hamiltonian. 2 HTLM dxA2(x) + dx Y+(x) a[iv + A(x)o,]V(x) (2.8) 2g where Y(x) = ) ,2 4K (a 1/2 (x) v(x) ,Q = 4a g = 4(F =2t0a (2.9) and a. is the ith Pauli matrix. 1 Varying eq. 2.8 with respect to u (x) and v (x) gives iv u + A(x)v(x) = e u(x) F 3x v (2.10) +iv +) A(x)u(x) = e v(x). F Ox v These are a pair of coupled Dirac equations for massless particles with eigenvalues that are unbounded below. As the Fermi sea of electrons is not appropriate for modeling a bounded Tr band an artificial energy cutoff W is introduced to simulate the band edges. This is expected only to alter the e'i; scale. Varying eq. 2.8 with respect to A gives the consistency equation 2 2 A(x) = g /UQ E' [u*(x) v(x) + v (x) u(x)] V,3 (2.1 1) where the sum is over spin components and occupied electron levels. Defining the new functions f,(x) = u(x) + iv(x) f_(x) = u(x) iv 2.10 become 3f E f (x) = iv F (x) iA(x) f_(x) v2 o 2 A(x) A (x)] f (x) = 0. F 32 + E F Ox + Ox (x) (2.12) (2.13) (2.14) equations and then Thus the problem has reduced to the solution of a "Schrtdingerlike" equation, with the potential belriu a function of the order parameter A(x). Of course the consistency conditions (2.11) must also be obe';.j. Solutions are particularly easy to obtain for the class of potentials with 2 5Ao 2 A (x) vF = A F 3x o (2.15) Two particular solutions that fall into this class are the purely dimerized lattice and the kink solution. The Dimerized Lattice1 For the purely dimerized lattice: A(x) = A one gets 2 [ F2 + E 2 2 f (x) = 0. F 2 x + ax_ (2.16) This has solutions Snkx vFk + k w + A ikx r F oi u (x)= N e e vk n k vFk ikx VFk o v (x) = Nk e [ vK n k L v k (2.17) N = 1i [ o 1/2, k 2 v2 w 1Tr These plane wave eigenfunctions have the energy dispersion relation with 2 2 2 2 i = A + vk 0 F (2.18) where the consistency equation is satisfied by 1/2X A =We O (2.19) Here one sees that W merely sets the scale of energies. The Kink Soliton16,21 There is also a kink solution (fig. 2.3a) with order parameter A(x) = A tanh(x/) (2.20) where = vF/A (The minus sign gives the antikink solution.) This has both planewave solutions and a single localized level at E = 0 with wavefunctions uo(x) = N sech(x/) v (x) = iN sech(x/E) (2.21) and No = [A /4 v ]112. This electronic level is localized on the kink with the same coherence length as the lattice distortion. The kink creation en.r L,'V16,22 can also be calculated as Ek = 2A /T. (2.22) Although the TLM modil is neither Galilean nor Lorentzian invariant, the kink mass can be calculated in the ansatz x(t)=xVt by equating the mass to the coefficient of the (l/2)V2 term. This livess the mass of the kink, '1. (where M is the mass of a (CH) unit) M = (a1 ) 2 >1 (2.23) k a for SSH parameters Mk 36m (2.24) where me is the electron mass. Tnis the kink is very light and kink dynamics can be expected to be important. The Polaron21,23 It is natural to look for a static solution to the TLM equations that does not involve a change of phase. The polaron is formed by the selftrapping of an added electron or hole into the dimerized lattice (fig. 2.3b). The polaron is topologically equivalent to a KK pair, as is clear from its order parameter A(x) = A0 k vF [tanh k (x+x ) tanh k (xx )] o o F o o o o  2 2 2 2 where o = ko v + 0 (2.25) and tanh k x = (A w )/k v_. This selfconsistency condition is, however, only satisfied in trans polyacetylene if k v = A //2. (2.26) This lattice deformation is supported by a pair of intragap levels symmetrically about the Fermi level, which in transpolyacetylene are at = A //2. For an electron polaron the lower level is doubly occupied and the upper is sinrly occupied. For the hole polaron the lower level is singly occupied and the upper is empty. The polaron creation ener,7; is Ek = 2/2 A o/ (2.27) i.e. the polaron rest energy is greater than that of a single kink, but less than that of two kinks or the energy (A ) of the added electron (hole). Limitations of the SSH and the TLM Models Although the SSH and TLM models may be good zeroth approximations to the real behavior of polyacetylene, a number of potentially important effects are not explicitly included. ElectronElectron Interactions Experiments show the first excited state of the finite polyenes4 ((CH)n, n=2,3,...) has 1A symmetry. Band structure calculations predict it to be 1Bu. This qualitative disagreement, and its explanation as a breakdown in the single particle model, has been attributed to the effects of electronelectron interactions. By anal"y',', in polyacetylene the single particle model cannot be expected to be valid. On the other hand certain other properties of polyacetylene are well explained by a single particle model, e.g. the energies of the infrared absorption peaks are correctly predicted.25 A number of studies on electronelectron effects in a Peierls distorted phase have been undertaken with conflicting conclu sions.68,2628 Calculations at the HartreeFock level generally show that the dimerization is decreased by electronelectron effects, whilst calculations that go beyond HartreeFock conclude that the dimerization is increased. Al'horuh electronelectron effects are not considered explicitly, they are included in an average way. The fermion operators, as in Landau fermi liquid theory, can be considered as fermionn quasi particle" rather than simple electron operators. Also the values of the various parameters of the SSH model are deduced from experimental data and thus can be considered as being renormalized to include the electronelectron effects. However, without fourbody operators in the Hamiltonian some electronelectron effects are evidently omitted. Given the renormalization already considered and with Horovitz's demonstration that the optical properties of the SSH model are unaffected by the addition of electronelectron effects (see Chapter III) the assumption that the nonlinear dynamics and statistical thermodynamics of polyacetylene can be considered as mainly arising from the electronphonon coupling seems to be justified. All manybody effects can, therefore, be considered as perturbations to the single particle Hamiltonian. Largely because of the controversy over how to include electron electron effects no single parameterization of the SSH model is accepted by all authors. 15 Here two parameterizations are chosen. "SSH parameters"9 are chosen to reproduce three experimental observables of transpolyacetylene: the '*.n.id.p (2 Ao 1.4eV), the dimerization (u0 0.04A) and the full bandwidth (W 10eV). The spring constant (K 21eV/A ) is the value for ethane. This uniquely fixes the electron phonon coupling constant (a 4.1eV/A) and gives the coherence length S~ 7a. For many of the simulations performed the parameterization W = 10eV, 2A. = 3.8eV, u. = 0.1A, E = 2.7a will be used. These rescaledd parameters"29 are numerically more convenient as they enable the consideration of systems with fewer sites. Also, where data for both sets of parameters are compared, the effects of discreteness and acoustic phonons can be studied. It will be shown that there are no qualitatively important differences in the numerical results between the two parameter sets. Of the quantitative differences none is important on experimentally relevant timescales. Quantum effects The lattice is treated classically. The two phases are separated by a potential barrier and are classically stable. However, quantum mechanically their wavefunctions overlap and there is a finite transition rate between them. Explicit calculations show that the dimerization is reduced by 1525%.30,31 Quantum fluctuations also reduce the kink creation energy by 25%.32 One can consider these effects as being included in the renormalized ..H parameters. However there are certain effects that can only be understood from a full quantum treatment. For example, classically a kink may be pinned to a discrete lattice. This is not, however, experimentally observable, even in principle, because of the quantum uncertainty in the kink location. Similarly, nonadiabatic effects may have important consequences for electronic transitions, absorption coefficients, phonon absorption and transport properties. Interchain coupling and intrinsic disorder The model treats each polyacetylene chain as independent. However, Xray data show that there is three dimensional ordering in trans polyacetylene with the antiparallel alignment of dimerization patterns on neighboring chains. A kink introduces a defect33 in this alignment with an energy that increases linearly with the separation of kinks. Thus one expects a KK pair to be confined thoughh not necessarily on the same chain) to reduce the misalignment energy. Experiments suggest that this confinement energy is 310K and that a KK pair is confined over 100 lattice sites. (As will be seen in Chapter V this confinement is similar to the intrinsic confinement in cis polyacetylene.) Interchain hopping34 of charge between kinks has also been suggested as the dominant conduction mechanism for electrical conductivity at low dopant concentrations. Other morphological effects that have been omitted are partial crystalinity, incomplete isomerization, variations in the chain length, cross linking and the proximity of chain ends to a neighboring carbon chain. Finite Temperature Effects Both the SSH and the TLM models are zero temperature models. As the band gap of transpolyacetylene is 1.4eV one expects thermal production of electrons into the conduction band to be negligble. However, the selffocussing of thermal :ritr' into nonlinear Lhvn.rin wavepackets, breathers, may be important. Extrinsic Doping The most striking experimental feature of transpolacetylene is that its conductivity can be varied over thirteen orders of .Tirnitude by extrinsic doping.35 It is therefore essential that the system be studied in the presence of defects. This will be the subject of Chapter VI. Dynamics If the mechanism for electrical conduction in polyacetylene is to be understood, it is clearly necessary to understand the dynamics of the nonlinear excitations of the model. Attempts to find time dependent nonlinear solutions to TLMlike equations have proven unsuccessful (the TLM Hamiltonian is neither Lorentzian nor Gaililean invariantthere are two velocity scales, the Fermi velocity and the sound velocity). 20 All of the above deficiencies of the basic model need to be addressed. A first logical step is to study the dynamics of the basic SSH model. This is the subject of Chapter III. Figure 2.1 The two degenerate phases of transpolyacetylene. c. b. EF ..........F .' ^ : ^ :^ v.b. v' ^y .*.*.*>>^ '" 1:^ *>^ X^X< Al> H H H 1 I C C C I I ' H H H H I C C ( I H * *C 1 A0 +A0 C.** I H A C*** . * UU * 0I0 Un U oI 01 ^ I z 0  1 0 I 0 (/) (n < z OI < 0 fl "T" ' , E0 ^ Iu / / I0 10 0 (, 0 * CO a. 4 1 0  o 0,0 S * r Y c c CO CC () ) ) 30.1.0 CC 0) 3.X tC I4 {HI 10 ~ I0 a, U> 03 0m .,4 4 C'4 Oa, ca (U) a CL OC *O * 0 Ca, 0 0 Q. ca, U () C ) CO O *O  *i ( ^1 ' o 0 3 3 CN I I I o (A)v 0 n 0 " I ; 0 c Y L0 . 0 N ON Nr CHAPTER III DYNAMICS OF A SINGLE :I;iK IN THE SSH AND TLM 1'LELS In this chapter a systematic study of the dynamics of a single kink in both the TLM and SSH models will be pursued. Before this is done it is useful to outline the numerical method employed and discuss boundary conditions. A brief study of the optical absorption of some simple static lattice configurations shows the similarity in nonlinear behavior of the SSH and TLM models. The Numerical Technique29 The energy of the SSH Hamiltonian can be formally written as SSH 2 n n+ J (31) n where the first term is the lattice kinetic energy and the second is the potential energy (V), which is a functional of the atomic displacements un. The potential energy is K 2 V = (u u ,) + E m e (3.2) 2 n n1 v,s vs n v,s where the first term is the lattice strain energy and the second is the lattice electronic energy. The sum is over occupied states of energy E and occupation m. The electronic energy can be found by the direct diagonalization of the hopping matrix T, which has components T.. = {t + [u. iu]}{6 + 6 (3.3) 0 o 1 1j ,j1 i,j+1 This has eigenvalues e given by (T E I) = 0 (3.4) where E = (e ,E E ) Newton's equations of motion are MU6V(u }) M1 = 6.({ n (3.5) The time derivative can be evaluated by finite differences as u (m) =  [un(m) (m1)] n dt n n (3.6) where dt is the time between the m1 and the m iterations. This gives the equation of motion as u (m) = 2u (m1) u (m2) t2 (.(m1)}). n n n M u n n (3.7) The functional derivative of the potential can also be calculated by finite differences as 6V({u ] 6un [V(1 ...un+6u...uN) V(1u2.. ....u ...u)]/u n where 6u << . 0 Energy conservation is used as a check on the stability of the algorithm. In general znerof is conserved to better than 99.99%. At some points in this study spatially homogeneous velocity dependent dumping is applied by the addition of the term n dt n to the equations of motion. This gives evolution equations 1 (dt)2 n(u }) u (1+) (2+p)u (m1) u (m2) M 6u n (3.10) (3.11) Also in some studies instantaneous damping is applied by setting the velocity to zero at every timestep, i.e., un(m1) = u (m2). This modifies the equations of motion, eq. 3.7, to (dt)2 n u (m) = u (m1) (dt2 6Vu n n M 6u (3.12) (3.13) (3.3) (3.9) Boundary Conditions Two types of boundary conditions are used in this numerical study.36 Ring Boundary Conditions A ring is a system in which there is a finite coupling between the 1st and the Nth sites of a system of N lattice sites. Ring boundary conditions are imposed by setting u = UNn (3.14) and C =C (3.15) n N+n As discussed in Chapter II a ring with an odd number of sites is topologically restricted to contain a single kink (or, more generally, an odd number of kinks). If the kink (antikink) is assigned a topological charge of +1 (1) then an odd length ring must have topological charge of 1. On the other hand a ring with an even number of sites must contain an even number of solitons (including zero). It thus has topological charge 0. This difference between even and odd length rings persists for all values of N and leads to the boundary condition on the staggered parameter H n+N  u = (1) N n n+N There is also a more subtle distinction between systems of lern'th 4n and 4n+2. For an undimerized system (uo = 0) the eigenstates of the SSH Hamiltonian are Bloch states with energy E = 2t cos k a (3.17) n o n where the wavevector,kn, satisfies the quantization condition N N k = 2rn/N ( < n S . (3.18) n 2 2 Thus for N=4n and a half filled band the Fermi level lies at E=O. By the Peierls effect the system then spontaneously dimerizes. On the other hand for N=4n+2 the Fermi level lies at the center of a gap of width '4t sin (n/N). Thus the 4n+2 length ring only dimerizes for sufficiently large values of N. (The critical value of N depends on the coupling constant, A.) This difference between 4n and 4n+2 length rings decreases as 1/N. Most of the numerical studies performed here are for N=98 where the difference is small. (Finite N dependence is, however, interesting and will be studied briefly in Chapter V.) Chain Boundary Conditions On a chain there is no direct coupling between the 1st and Nth sites of a system of N sites. This results in a breaking of the electronhole symmetry to order 1/N, which for all reasonable chain lengths is small. Here two different t,r; of chain t'i.ind, conditions will be used. 1. In "free boundary conditions" no constraint is applied to the end sites. To conserve the chain length, however, it is necessary to add to the Hamiltonian the "pressure term" (ul LI1). 2. In "fixed boundary conditions" each of the end sites is bound to its nearest neighbor by the constraint u, = '2' UN = (3.19) At no point in this study has any qualitative, or significant quantitative, difference been found between the dynamics of chains with free and fixed boundaries. Statics of the SSH Model As a preliminary to the study of the dynamics of the SSH model it is important to verify that the static analytic solutions to the TLM Hamiltonian well approximate the solutions to the SSH equations. As an initial condition we use the t=O lattice configuration and minimize the energy of the system by varying the dimerization uo from its continuum value. The shape of the initial profile remains fixed however. The solution to the SSH Hamiltonian is then found by numerically relaxing around this initial configuration (see eq. 3.13). This allows the shape of the nonlinear excitation to change by the emission of phonons. It has been shown analytically that the dimerized lattice is an exact solution to the SSH equations: the energetically minimized dimerized groundstate produces few phonons and appears to be dynamically stable for all times. Although no analytic form for the kink in the SSH model exists it is expected that the TLM kink profile should be a good approximation. On relaxing the kink profile there is a strong emission of acoustic phonons, which appear in the staBgerei lattice distortion as a "saw tooth" (fig. 3.1a). The effect of the acoustic phonons can most easily be seen by plotting the short and long range components of the staggered order parameter separately. The actual ion displacement at site n is u = (1) u n + n.6a (3.20) where 6a is the change in the lattice constant and un is the staggered order parameter with respect to the local lattice constant. Now define37 r = (1)n 2u u u (3.21) n 4 n n+1 n1 s = 1 [2un + + u1]. (3.22) n 4 n n+1 n1 Then rn un and sn n.6a. In these variables the kink profile is more easily seen (fig 3.1b). Note that the dip in sn corresponds to a contraction of the lattice constant around the kink (for proof see Appendix C of ref. 15). Optical Absorption within the SSH Model Before discussing the optical absorption within the SSH model it is useful to review the optical absorption in the TLM model.21,36 The optical absorption is defined within the linear response dipole approximation as A a(w) = T A E <4 203 I ,>2 6(E2 E W) 1,2 (3.23) where a is the current operator, the 6function ensures the conservation of energy and A is an unimportant constant. Figure 3.2 shows the available electronic transitions for the dimerized lattice, the kink and the polaron. Using the analytic form of the wavefunctions (eq. 2.17) the optical absorption of the purely dimerized lattice can easily be calculated as (3.24) a(a) = A(2A /wa)[ 2 42)1/2 This is divergent at the band edge and decreases as the inverse square of the frequency. Great care is needed in calculating the optical absorption of the single kink.21,36 Although the kink is a local excitation it results in a global change in the dimerization of the system. (To one side of the kink the system is in the A phase, on the other side it is in the B has.?.) This alters the phase relations in the valence and conduction bands, causing diagonal transitions (Ak=O) to be explicitly forbidden. For the midgap transition acc (w) = A 1 se (w (w A2)12 (3.25) k 4 L s 2A0 o whilst for the interband absorption a (W) = a (w) [1 2y/L]. (3.26) It is important to note that the kink removes weight essentially uniformly in frequencies above the band edge because the alteration in the dimerization over a semiinfinite region requires a large number of Fourier modes for its description. The polaron absorption21 has similarly been calculated and consists of three contributions plus the interband. Transitions between the two intragap polaron levels give a (w). There is a second contribution, p a (W), arising from transitions between the valence band and the lower polaron level (for the hole polaron) and between the upper polaron level and the conduction band (for the electron polaron). Contributions between the valence band and the upper polaron level and between the lower polaron level and the conduction band give the contribution a (w). The relative intensities of each of these is shown p in fig. 3.3 as a function of the location of the intragap polaron level. (For the polaron in transpolyacetylene wo = 1//2. For cis polyacetylene see Chapter V.) For most gap locations a (a) is much stronger than a (w) even though it involves only one transition, p whilst a (3) involves many transitions. This uprising intensity ratio is again a result of a fortuitous phase relation in the continuum equations. This continuum theory suffers from a number of limitations; the most important of which, is that only a few simple lattice configurations can be studied. Further it is not clear whether the phase relations discussed above will hold in the more realistic discrete system. A theory for the optical absorption in the SSH model has been constructed by Horovitz.3 A review is given here (full details are given in Appendix C). Starting from the definition of the charge density, p(n) = eCnC and relating it to the current density through the continuity equation, he found the current operator j(n) to be given by j(n) = ie[t0 + a(u u )][CC C C c. (3.27) Then in the linear response dipole approximation a(w) = ffe t2 1 26( e2 W) (3.28) 1,2 where M 1,2 [1 + (u u )][f (n)f (n+1) f (n+1)f (n)] (3.29) n o and f is the wavefunction of the a level. a He further showed that ado) obeys a sum rule fJ a()du = e 2N ez 0 where H is the electronic eneri:' of the configuration. This algorithm for the absorption is easily implemented. It is important to note that all these results remain unchanrii when electronelectron effects are added at the Hubbard level, i.e. H = E V p(n) p(m) is added to the ee rm nm SSH Hamiltonian. This is because [H p(n)] = 0. Figure 3.4 shows the numerically calculated optical absorption of the purely dimerized lattice. The 6function in eq. 3.28 has been approximated by a Lorentzian of width G (for all the present studies G A /10). The sum rule is obeyed to better than 99.92%. Fijgre 3.5 shows the optical absorption of two widely spaced kinks on a ring of 98 sites. As expected there is a strong midgap absorption in addition to the interband continuum. Here the sum rule is obeyed to > 99.97% In fig. 3.6 the difference, Aa(w), between the absorption of the ring with the two kinks and the absorption of the purely dimerized ring is plotted. Here the expected interband bleaching is evident over all frequencies above the band edge. Figure 3.7 shows the optical absorption of a single polaron on a 98 1 2 site ring. The contributions a (w) and a (2) are evident at 3A /2 and A /4 respectively. The weak contribution, a (w), is present but o p only 2% of that of a (w). Figure 3.8 shows Aa(w) for the polaron. Note that the polaron removes intensity over a wide energy range above the band edge. This is because, for these parameters (C 2.7), the polaron is is narrow and requires a large number of Foirier modes for its description. Thi:e simple calculations show that the optical absorption of these static configurations is in good agreement with the continuum calculations. The validity of the sum rule is also confirmed. In subsequent chapters this formalism will be used to calculate the optical absorption spectrum of various d;n3mical excitations of the system. Kink Dynamics in the TLM Model It was noted in Chapter II that the TLM equations are neither Galilean nor Lorentzian invariant. There is thus no obvious velocity scale in this problem. However due to the finite response time of the lattice to cbnir'; in its dimerization, the kink has a maximum free 2pr jo'ition velocity as can be seen from the following simple argument.37,39'40 For a kink of velocity v the lattice changes from the A to B phase (say) over a distance (v) and in time (v)/v. The minimum response time of the lattice is the inverse of the renormalized 1/2 phonon frequency, wR, where wR = (2A) 1 wo 0( being the bare phonon frequency). This gives the maximum kink propagation velocity (vm) as vm WR (v ). Assuming that the kink width is not strongly velocity dependent, then 1/2 v /v (2X) (E/a). (3.31) r S For SSH parameters (7 7a, A 0.2) this gives (vy/vs)2 19the maximum velocity is greater than the sound velocity. The conclusion of the above arvjment can be confirmed from an approximate analytic calculation. Using the ansatz for the kink profile A(x,t) = A tanh (x vt)/E(v) it is straightforward to show that the lattice kinetic eneri'i of the kink (T) is M(v/v M r *2 s T M j A (x)dx = AY ) (3.33) where (3.34) 2  Y = A /24AnXt'. o 0 (3.35) It has been shown numerically9 that the soliton width is velocity dependent (contrary to the assumption of the simple argument above); the excess energy due to the lattice deformation being ED = A0o(1 r)2 (3.36) where numerically we find B(E = 7) 0.20 8(E = 2.7) 0.10. (3.37) Then the Langragian of a kink of finite velocity is (3.32) L = A (1 r)2 Yq_/r}. Thence the momentum (Pq) conjugate to the displacement q (q = v/vs) is 2Yq P = q r and the momentum, pp, conjugate to the width r is p, = 0. Now extremizing the Lagrangian with respect to r and q gives Y2 + Lir (1 r) 0 I = 0. r r (3.41) (3.42) Then for consistency r = 0, q = 0. (3.43) Thus from eq. 3.41 the velocity has an extremum at r=2/3 independent of the parameters Y and 3. Substituting back for r the kink has a maximum propagation velocity (vi) given by (v / = () B/8 (3.44) M. s 27 (3.38) (3.39) (3.40) 40 which gives the momentum at the maximum kink velocity as 2 8 P 2 Y m 3 Thus the enrr.g, at maximum kink velocity is (3. 46) 5 Ek 9 o The kink energy can also be written Ek(p) = A [p2/4Y p 24/64Y2]. (3.47) This gives a maximum in the energy at pM = 8Y8. (3.48) This apparent contradiction between eqs. 3.45 and 3.48 is resolved by noting that the previous calculation was for the momentum at the maximum kink velocity, whilst this calculation gives the momentum at which the energy is a maximum. Now defining the effective kink mass by S= d2E m* dp2 (3.49) gives 1 1 3 2 m 2a 16 p /YB (3.45) (3.50) 41 Thus in the low momentum limit the kink behaves like a Newtonian particle of mass 2 a. However, the effective mass divr':cs at the critical value of the momentum 2 8 P Y. (3.51) This is the same as the kink momentum for free propagation at the maximum velocity (eq. 3.45). Using appropriate value for 6 and Y one gets for SSH parameters (v /v ) 11, m 5.6 m E 0.1 'A (3.52) n s e m o and for rescaled parameters (v, /v ) 1.2 m 80 m E 0.06 A (3.53) The maximum velocity calculated here for SSH parameters is in excellent agreement with that calculated above. The mass of the kink is also in good agreement with that calculated in Chapter II. Kinks Dynamics in the SSH Model37 To test the validity of the above calculations, a detailed numerical study of kink dynamics in the SSH model was undertaken. A single kink, given by its continuum analytic form, was given a Galilean boost towards the center of a 79 site chain with fixed boundary conditions. (All results were found to be similar for free boundary conditions and for chains of differing 1r;ths.) In fig 3.9 the short time kink velocity squared is plotted as a function of the ene.:i input. Both axes are in dimensionless units, where unity corr3sp;njL to 2 the theoretical value of vm and the theoretically calculated energy, L, at this maximum velocity. The solid line comes from the continuum calculation of the previous section. It is clear that the data for the two parameter sets lie approximately on a si'lile curve. Indeed by treating 6 and Y as free parameters and force fitting, better agreement may be obtained. At low energy inputs the data are in ,oud agreement with the continuum theory. The maximum kink velocity is also well predicted by the continuum calculation. However the continuum theory st.ongly underestimates the erier;y input required to reach this velocity. This is a result of the simplicity of the ansatz, which allows for kink translation and uniform contraction only. In the numerical studies the kink is also coupled to the optic and acoustic phonon fields, which provide efficient routes for energy dissipation. Indeed, numerically we find that phonon production is strongly favored over kink contraction; the width of the kink only decreasing by 10% even at the highest velocities. Short time dynamics have only been considered up to this point; as after approximately one phonon period, phonon emission becomes important. Optic phonons couple to the nonlinear excitations to 2 order (a/i) whilst acoustic phonons couple to order (a/i) Thus for rescaled parameters phonon effects should be much stronger than for the SSH parameters. Figure 3.9 also shows the energy input as a function of the kink velocity squared over the time period 0.150.30 psecs, during which the kink propagates at an approximately constant velocity. Here the phonon effects reduce the maximum kink velocity by a factor of 2, for ~ 7, but by a factor of 10 for the more discrete kink with 2.7. Figures 3.10 (a) and (b) compare the lattice dynamics of a single kink, with rescaled parameters, with energy inputs E0.3 E and E2.0 E respectively. At low energy inputs the kink propagates essentially uniformly, whilst at higher energy inputs kinetic nrs, is lost by the emission of an optical phonon 40ac'r.3 0 This may be viewed as the classical analog to the emission of Cerenkov radiation by a particle traveling faster than its maximum r';ir'3tion velocity in the medium. Although, as we shall see in Chapter VI charge conduction by the ballistic transport of kinks is unlikely, kink propagation is essential to the underst rridi q of the short time jrinrinics of the photoexcited system. 2.0 Un 2.0 III 30 n 60 2.0 rn, n sn (b) 2.   I  I  I_____ 2.0 I I I 30 n 60 Figure 3.1 The Single fink (a) unsmoothed order parameter, u for single kink. (b) Smoothed order parameter, r for the same single kink and sn (dashed line). I9 11) Pl > 9 9 ) rJ C3 Y 4 0 r.  aP P Co 4' Ca C .s0 Is S4 C 0 a, .0 Hc a, ^13 .r: E .rs.0 H 0 0 Q 0 I  a 0 1Q j * 0.7 0.6 0.5 0.4r 0.2 OI ( 3) ap 0 0.2 0.4 0.6 0.8 1.0 Figure 3.3 Relative strength of transitions in the polaron as a function of the location, /A0 of the intragap levels. For transpolyacetylene w /A 0.7. 300 200 100 2.0 4.0 6.0 E/A, Figure 3.14 Numerically calculated optical absorption, a(w), of a 98 length dimerized ring with rescaled parameters. The full band gap 2A = 3.9 eV and the full bond with B = 4t = 10 eV. The 6functions in energy have been approximated by Lorentzian lineshapes of width A /10. In this and subsequent figures the vertical axis is in arbitrary (but consistent) units. 300 200 3 a 100 0 2.0 4.0 E /A0 Figure 3.5 Optical absorption, a(w), of two widely spaced kinks on a 98 site ring. 6.0 2.0 4.0 E/AO Figure 3.6 Difference between optical absorption of two kinks and the dimerized lattice of a 98 site ring. (Figure 3.6 = Figure 3.5 Figure 3.4). 80 40 0 40 80 120 d K 6.0 300 200 100 0 2.0 4.0 E/A, Fi 'n're 3.7 Optical absorption, a(w), of a polaron on a 98 site ring. 6.0 2.0 4.0 E/Ao Figure 3.8 Difference in absorption of a single polaron and the purely dimerized lattice of a 98 site ring. (Figjre 3.8 = Figure 3.7 Figure 3.4) < ,3 L, <1 40 20 0 20 40 6.0 coa rt CO e 4O X C 4 c C 0 II U) L LPC o CC) O'2 O L Y CC rl 4 C 0 UOC 0 0 .0 Cja c .0 CLU tO L) 4a a) , C Q. 0 3 0 U)OU) 0 ' E 0 * 0 0 Q . 0, 0 < 0 .4 0 0 CE E r1 O) 0 () ., 0 '4 to a c  2 C b0 oo r: 0 4) 2 ~ E' a, 0 cd 2 0^ 1) 42 >,^ W0 C o  C*^ Cd 42 a. b0 CC C CC C iil a0 Co a. ci, Cci) ci .C O 0 N3X CHAPTER IV PHOTOEXCITATION IN TRANSPOLYACETYLENE We begin this study of photoexcitation in transpolyacetylene with a simple experiment:29,37 the excitation of a single electron from the top of the valence band to the bottom of the conduction band. To simulate an infinite system 98 sites are used and periodic boundary conditions applied. In the rest of this work the rescaledd parameters" will be used as they allow smaller systems to be studied, thus red .n' the computation time. (The dynamics are expected to be similar for the more realistic SSH parameters.) At t=O a single electron is manually removed from the top of the valence band and placed in the bottom of the conduction band. All electrons then remain in the same levels throughout the experiment and the system simply evolves adiabatically. As electronic transitions should only be important on the nanosecond timescale this should be a good approximation for the subpicosecond dynamics considered here. The single electron photoexcitation adds energy 2A to the system. In less than 0.1 psecs a kinkantikink (KK) pair is formed. The kink and antikink separate with their maximum free propagation velocity, vy, (fig. 4.1). Each kink has creation energy 2A /i and kinetic energy <0.1 A Thus the two kinks have a total energy of <1.5 A Figure 4.1 shows that a substantial fraction of the remaining energy from the photoexcitation is localized as a nonlinear lattice excitationan amplitude breather. The breather dynamics were 59 studied by removing the central 42 atoms from the photoexcitation experiment at t=0.301 psecs and allowing them to evolve separately (fig. 4.2a). It was also shown numerically that the breather is stable with no appreciable change in frequency or amplitude after 5 psecs. The Analytic Breather As the breather is formed by small oscillations about the purely dimerized lattice, one expects that an approximate analytic form ia.' be obtained from the known effective Lagrangian41 for the halffilled band '2 *2 L = 2 {1/2 A 2[n(2E /A) + 1/2] A2/4x v 2 A A (4.1) F F 24A 2w where E. is the electronic cutoff energy and the dot and prime denote time and spatial derivatives respectively. (This Lagrangian cannot be used to find kinklike solutions as it is illdefined as A0.) Firom eq. 4.1 the equation of motion is '2 " 2 (A AA 2 A/2A = A in 2E /A vAA A/ (4.2) 12 A To find spatially localized time periodic solutions for small deviations from the dimerized lattice define37'42 A(x,t) = A[1 + 6(x,t)] (4.3) OL1 (4xt) where (4.4) and t and T are the two time scales. Using multiple timescale asymptotic perturbation theory one finds the approximate breather solution (see Appendix C for details) 6(x,t)=2E sech(x/d)cos w t+ esech2 (x/d)[cos 2w t3] (4.5) where d = c/2/2 (4.6a) (4.6b) = 1 1 ' ) : . It is easy to show37,42 that the effective energy density is 2 1 2 2 2 3 4 E = { + 6, + + 6 Tv F 2R2 24 2 6 24 F 2wR and, thus, that the classical breather energy is 2V2 10 2 4 E = 1 7E + 0( )]. (4.7) (4.8) (xt)c[(XT)exp iwt0T)expiw +251 (xt) 6(x,t)=e[A(X,T)exp int+A (X,T)expiuit]+E 5 ,(x,t )+E"6 (x,t). Breather Dynamics and Optical Absorption The approximate analytic form for the breather (q. 4.5) was used as an initial condition and allowed to evolve under the adiabatic 3': equations of motion. Figure 4.2(b) shows that for E=0.75 the dynamics of the analytic and numerical breathers (fig. 4.2(a)) are very similar. The simpler lattice distortion of the analytic breather may be attributed to the absence of higher order terms in e in the analytic form, eq. 4.5. Equation 4.9 gives the classical breather energy as 0.55 A for e=0.75 and thus essentially accounts for the energy "missing" from the photoexcitation. Spatial localization of nonlinear excitations implies the presence of localized electronic levels (in one dimension). For a breather there is a pair of intragap levels symmetrically about the Fermi level. The lower level is doubly occupied, the upper is empty: the breather is a neutral lattice excitation. From fig. 4.3 it is clear that the dynamics of the electronic levels for the numerical and analytic breathers are very similar. It is important to note that these intragap levels oscillate deeply into the gap (for this numerical breather to 0.55 A ). However the intragap levels spend much of the breather cycle close to or beyond the band edge, thereby increasing the effective band gap. Transitions into and out of these localized intragap levels produce contributions to the optical absorption below the band edge. In the absence of an analytic form for the breather electronic wavefunctions it is not possible to analytically calculate the breather optical absorption. However, useful qualitative insights can be gained from analogies with the exactly calculated (in the continuum limit) absorp tion of the polaron. The breather is a neutral excitation, whereas the polaron is the static groundstate of the system with a single charge added. They do, however, share the same basic topology and both have localized electronic intragap levels, thoi':' differing in their occupancies. (The relative intensities of the polaron transitions are shown in fig. 3.3.) For the transpolyacetylene polaron w! /A 0.7. For the breather the location of the int"'i>ap level varies with time and thus, in a quasiclassical sense, an average in w0 must be taken over a full breather period. (Quantization of the breather levels will be (2) discussed in a later section.) It is important to note that a ( the analog of the strongest transition in the polaron is absent in the breather due to the differing electronic occupancy. For this numerical breather the intragap level moves over the rjn.e 0.55 1.05 A over which a1) is approximately independent of the location of the intrijp;; p Z level. The relative intensity of this contribution to the absorption is, thus, largely governed by the amount of time spent at each gap location. Two dominant contributions are expected. First, a strong contribution close to, or beyond, the band edge, where the intragap level spends most of its period. Second, a contribution when the intragap level is farthest from the band edge (for this numerical breather at 1.1 A ). There should be a weaker, nearly uniform (3) contribution over intermediate energies. The transition a is weak at all relevant energies, with no significant contribution at the lowest energies. For this breather a weak contribution at 1.5 A can thus be expected. The nature of the compensating above band edge bleaching can also be understood by analogy with the polaron. If the polaron is narrow it bleaches over a wide ringe of energiesa narrow excitation requires a large number of Fourier modes, including those at hi 'i energies and wavevectors, for its description. A wide polaron, on the other hand, can be described by fewer, lower wavevector modes. It bleaches over a narrow energy range close to the band edge. The breather is extremely wide and thus can be described by only the low wavevector Fourier components. Its above band edge bleaching should, therefore, be mainly close to the band edge. ''sin the algorithm described in Chapter III we numerically calculated the optical absorption of the numerically created breather over a full breather period (fig. 4.4). Removal of the breather from the environment of the two kinks quantitatively changes the kink dynamics due to the strong modification of the extended wavefunctions by the kink. The expected subband absorption at A can be seen. There are also contributions close to the band edge, which overlap with the interband absorption. Here the sum rule is obeyed to better than 99.9% when averaged over a full breather period, and to 99.6% at each instant. Figure 4.5a shows the change in optical absorption when the numerical breather is added to a 98 site ring, i.e. fig 4.5a = fig. 4.4  fig. 3.4. The breather signature is clear: absorption enhancements at Ao and close to the band edge, which are largely compensated by bleaching of the interband just above band edge. (It should be noted that the optical absorption of the same breather in the presence of a KK pair is a single peak in the energy range 12A This strong modification is due to the effect of the kinks on the continuum state with which the breather interacts.) Figure 4.5b shows the change in optical absorption, Aa(w), when an analytic breather with e=0.75 is added to a 98 site ring. Below the band edge is a single broad peak arising from transitions between the lower breather level and the conduction band. This enhancement is compensated by a bleaching over a narrow energy range above the band edge. To better understand the dynamics of photoexcitation, it is interesting to follow the optical absorption on the subpicosecond timescale. Figure 4.6 contrasts the time a:ege of the optical absorption over the periods 0.02 0.04 psecs and 0.30 0.34 psecs after photoexcitation. There are three :1cLrtant differences to note. First, the "nidgap levels" are moved away from midgap at short timesthe KK pair is still evolving and the intragap levels associated with them have not yet fully evolved. Second, because of this splitti.ic there are transitions between the putative midgap levels. This is characterized by a low energy contribution to the absorption, which rapidly decreases in energy and intensity as the intragap levels approach midzap. Third, and perhaps most surprisingly, the intensity of transitions between levels which will eventually evolve to the breather is l1rge at all times. Thus, the electronic properties of the breather fully evolve very rapidly (<0.04 psecs), altho:jh the characteristic lattice distortion of the breather does not fully evolve until t 0.15 psecs. (The above possibility of "coexisting" breather and KK complexes is important in cislike materials where the K and K are confined.) nr :. t 'citation in the Presence of Intrinsic Gap States To this point only photoexcitation across the full band gap has been considered. However ESR and electrical conductivity35 experiments show that, even for "pristine" samples of transpolyacetylene, there is a significant density of spinless charge carrierskinks. It is therefore interesting to study photoexcitation in the presence of intrinsic intragap levels. (Photoexcitation in the presence of extrinsic intragap levels will be the subject of Chapter VI.) First consider a single neutral kink on a chain. Photoexcitation of an electron from the midgap level to the bottom of the conduction band produces the electronic configuration of a negatively charged polaron and a positively charged kink. The rest energy of the polaron is 2/2 A /T 0.9 A Photoexcitation adds energy A0 to the system. Thus polaron production is possible. However, for a pure system there is symmetry about the kink and, in the absence of an alternative seeding center the system evolves to an oscillatory boundstate of the kink and polaron. No breather is produced. Now consider photoexcitation of an electron from the lower to the upper intragap level of the polaron. The polaron intragap levels are A //2 from midgap. Thus the photoexcited polaron has energy 2.3 A greater than the groundstate. Consequently this excited polaron can be expected to be unstable to the formation of a KK pair and a breather. Indeed numerically a KK pair is rapidly produced in <0.05 psecs and a breather is also created (fig. 4.7). By energy conservation this breather has energy 0.8 A However, a simple calculation shows that the maximum energy of the classical approximate analytic breather is 0.63 A o. Thus this breather is not well described by eq. 4.5, reflecting the need for higher order terms in E expansion for the breather (c.f. results for the breather in, e.g. D equation). Since in this case three levels oscillate into the .:1, subband edge absorption can be expected over a wide energy range. Indeed, fig. 4.8 shows that, as well as the strong midgap absorption from the two kinks, there is a low energy shoulder from kinkbreather transitions. There is also an enhancement at 1.5 A arisi'; from transitions between a the breather and continuum levels. The intra pp bleachii; arise from the evolution of the two polaron levels to midgp, forming the KK pair. Above Band Edge Photoexcitation In most laboratory experiments photoexcitation is performed with pumping energy well above band edge, thus exciting electrons from deep in the valence band to hih' in the conduction band. Such highly excited electrons (and holes) can be expected to decay towards the edges of the band gap by optical phonon emission with a time constant of, typically,  1014 seconds. However close to the band edge there may not be an optical phonon of sufficiently low energy to mediate further decay. Further decays must take place via the emission of acoustic phonons, _Q which typically have a time constant of 108 secs. In the absence of a preferred nonradiative decay route, via e.g coherent electronhole scattering, this "bottleneck" may mean that the system stays in a metastable excited state that can then evolve adiabatically to form nonlinear excitations. Amongst these nonlinear excitations one may expect to find KK pairs, polarons, polaronpolaron bound states, breathers and excitons. The choice of channel will depend on, e.g. correlation and impurity effects. A fully qjant.rim nonadiabatic model Is not yet available to investigate this problem in detail. Yet useful insights can be gained from a simple model, which mimics some of the nonadiabatic effects. At t=0 an electron was excited vertically (Ak=0) from the fourth highest level in the valence band to the fourth lowest level in the conduction band. The excited electron and hole were then allowed to decay towards the band edge independently and randomly with time constant T. Baeriswyl43 has estimated that for transpolyacetylene T 0.015 psecs. We simulated only the simplest model in which the electron and hole can decay rapidly all the way to the band edge. Figre 4.9 shows that when the electron and hole are excited the system tries to equilibriate by hanrginrig its dimerization. In this simulation the electron and hole had both reached the band edge by 0.1 psecs after which a KK pair were rapidly produced. Neutron Scattering CrossSection of the Breather Inelastic scattering of low energy neutrons should provide a direct probe of the lattice structure of polyacetylene. In this section a simple phenomenology is used to elucidate the basic features of the classical dynamic structure factor of the breather. It is assumed that any lattice distortion can be considered as a superposition of independent collections of phonons, kinks, polarons, breathers etc. Although the focus here is on breathers, similar phenomenologies can be developed for the kink and the polaron. In both cases there are contributions to the structure factor at w 0, as well as responses associated with the internal vibratory modes. Here, as has been done previously for the sineGordon system, we treat the collection of breathers as an ideal gas, in which all the internal structure is reflected in a qdependent form factor. An outline of the derivation is given here; full details are in Appendix D. Define the classical structure factor as n3 (B;T) S(q,w) jdx dx dt d t2exp[iW(t1 t2)iq(x x2 (4.9) < exp iqu(x It expiqu(xt 2)> where j is the lattice displacement at location x and time t. Keeping only one phonon processes and assuming that the thermal average can be implemented by integrating over a classical distribution P(v), then 2W "B(B;T) 2 2 S(q,w) = e 5 JdvP(v)q [j dx1dt1 exp i2Y (q+r/avaw/co (4.10) 2W where e is the DebyeWaller factor, co is the velocity scale and u(x,t)=na+(1)n jl(x,t). For simplicity we use the MaxwellBoltzmann distribution, P (v). After a little algebra one finds S(qw)=e2W Bn B'T) 1r (k+) 2NR a + If(k) 2 ) (27r) 2 q (4.11) ~4 +2w aw2( B + : {f2(k) 2pNR( B )+f (k) 2NR( B +36f2 pN (/q)R l where + 1 r b .+n) k =d1 [q + /a + ( j)] n 2 q c o k = d1 [q + T/a w2/c2 q] f (q) = aid sech (jrqd/2) f 2(q) = iqd2 cosech (irqd/2) (4.12a) (4.12b) (4.13a) (4.13b) For simplicity we look at the behavior of S(q,w) in the T=0 limit, i.e. when PNR (v) 6(v). Taking this limit is not, of course, strictly meaningful for a classical calculation but does help display the essential features of equation 4.11. We find S(q,w=0)/S(q,w=r2 B) = 36. (4.14) This zero frequency inelastic scattering can be observed separately from and the elastic scattering of the phonons. Further contributions at W=O are expected from other nonlinear excitations. These, however, may have differing temperature and intensity dependence. Also S(q,w=+w )/S(q,w=2Bw) = 1 tanh2 U2/ 1q/i)1 (4.15) 25 E For nearly all values of q (except q r) this gives almost independently of e S(q,w = W ]/s(q = T, w = .?,J 9 x 103 (4.16) At the zone boundary q=ir one finds S(q = T, W = WB )/S(q = = 2 = 3.8 (4.17) 2 E Quantization of the Breather One of the major assumptions of the work has been that the lattice degrees of freedom can be treated as classical variables. One of the effects of a quantum treatment is to renormalize the mass of the nonlinear excitations: e.g. the mass of the kink is reduced by 25% by quantum effects.32 In addition to the lattice being quantized the locations of the electronic levels are quantized, lying within the bounds of the range of values of their classical analogs. Due to the electron hole symmetry of the formalism, the kink intragap level is at midgap for both the classical and quantum treatments. The locations of the classical extended states and the intragap polaron levels are essentially time independent and thus the locations of the qjantized levels are well defined. For the breather the situation is entirely different: it is an inherently dynamic excitation, whose lattice distortion and electronic spectrum are time dependent. In a previous section we derived an approximate analytic form for the energ of the breather. This can easily be quantized at the Bohr Sommerfeld level by dmanding that J I P dA = 2nr (4.18) where J is the action and pA is the conjugate momentum to the order parameter, A. Thus 2 .2 2n7r = f J dx dt (4.19) F 2R which gives 4V2 21 2nr = e [1 e (4.20) Inverting and substituting into eq. 4.8 for the breather energy gives 2 22 E = nwR [ + 0(n)] (4.21) 720 Equation (4.21) shows explicitly that a breather of quantum number n is energetically favored over n incoherent phonons. To calculate the optical absorption spectrum of the quantized breather it is necessary to calculate the energy of the excited breather. One might hope to do this using an effective L.jr miian of the type used for the groundstate breather: there are however technical difficulties in extending this to excited states. Other approaches might be a generating functional formalism 45,60 or by direct numerical quantization of equation 4.19. Even if this can be achieved there remains the conceptual problem of quanti::irn:, the electronic spectrum, from a knowledge of the breather en'r:; spectrum. At present we can only say that on quantization, the location of the breather electronic levels and thence the n=rr,'{ of contributions to the optical absorption spectrum are bounded by the limits of their classical analogues. Thus for the quantized breather we expect a single sharp absorption peak somewhere in the range 12 A . Comparison between Theory and Experiment In this chapter a simple scenario has been pro'p s..d for the short time evolution of the photoexcited system. Before proceeding to further detailed studies of the dynamics in transpolyacetylene and related materials it is important to ascertain what, if any, experimental evidence there is to support the validity of this scenario. Figure 4.10 shows the experimentally measured change in absorption during photoexcitation of transpolyacetylene at 10K. ^' The spectrum shows three basic features: a broad peak at 0.45eV, a narrow peak at 1.35eV and bleaching over a wide energy range above the band edge. Clearly we must identify the 0.45eV line with the kink "nili:'" absorption and the 1.35eV line with the breather. The 0.45eV Line.49 As well as the above mentioned high energy peaks, a large number of i.r. active modes have been observed during photoexcitation.47'48,50 In experiments, in which the temperature, laser intensity and chopping frequency51 have been varied some of these peaks have been shown to be directly correlated with the 0.45eV peak. This demonstrates that they arise from the same center, and that the center is charged. Further certain of these i.r. modes have been convincingly identified, in theoretical studies by Horovitz,52 as arising from the pinned modes of an excitation, which breaks the lattice translational symmetry. This does not uniquely identify the excitation as a kink: it might equally well arise from a polaron. Recently Horovitz's work has been extended53 and it has been shown that there are additional contributions to Lhe i.r. absorption unique to the kink, arising from its internal modes. This may allow for the unique identification of the origin of the 0.45eV line from optical data alone. From the intensity ratios of certain of the i.r. lines, the mass of the charge center has been estimated at 10me. This is in good agreement with estimates for both the kink and polaron masses. The determination of the spin of the charge carriers will differentiate between the polaron and the kink. Infrared studies have seen the same features in doping as in photoexcitation.35 The features are independent )f the kind of dopant and thus must arise from intrinsic excitations of the system. Further, on doping a midgap absorption peak appears correlated with some of the i.r. peaks associated with the midgap absorption produced during photoexcitation. This shows that the spectroscopic features arising during photoexcitation and on doping are associated with the same lattice excitation. Flood and Heeger showed in ESR experiments that the number of spin carriers does not increase during doping. This strongly supports the identification of the 0.45 eV line as arising from charged spinless kinks. 149 : 1.35eV Line. No i.r. active mode has been found to be correlated with the 1.35eV line, implying that this peak arises from a neutral excitation. Sub picosecond spectroscopy shows that the 1.35eV line is strong and already decaying 0.1 psecs after photoexcitation.545 This short timescale for production and the estimate of the 'j'itri production efficin:'ie;, as being of order unity strongly suggest that it arises from an intrinsic excitation of the lattice. All of the above are in qualitative agree ment with the breather model: the breather is indeed neutral and its electronic signature is well developed after one phonon period (0.04 psecs). The observed intensity decrease may be attributable to energy loss to e.g. the vibronic modes of the CH bond. There are four other important pieces of data. First, the 1.35eV line decreases in intensity with increasing temperature and entirely disappears at 150K.57 Second, at approximately the same temperature there is a sharp increase in the photocurrent.58 Third, the energy of the peak decreases by approximately 3% on deuteration.59 Fourth, the excitation responsible for the 1.35eV line is dipole forbidden from the groundstate.49 Let us examine each of these in turn. First, the breather is a lattice excitation and thus one expects it to be stronrily temperature dependent. Indeed the temperature at which the breather becomes unstable can be estimated by a simple argument. A typical breather ;.i have energy 0.5 A (=0.35eV) and be localized over 25=15 lattice sites. Making the plausible assumption that the breather becomes unstable when the thermal energy is equal to the breather energy gives an estimate of 300K for the temperature at which the breather becomes unstable. Second, the increase in the photocurrent at 150K cannot be understood within the breather model. Although it has not been shown that the onset of the photocurrent and the disappearance of the breather are correlated, this must be regarded as a failing of the breather model. Third, assuming that the breather in the hydrogenated and deuterated systems has the same quantum number, one expects the energy of the peak to decrease by [1M(CH)/M(CD)]112 4%. Fourth, the breather has the electronic configuration of the groundstate and can decay to it only by phonon emission. A quantitative comparison between the experimentally observed and numerically calculated optical absorption data is difficult for a number of reasons: (a) the presence of electronelectron interactions shifts the "midgap" absorption to 0.45eV; (b) in the absence of a full quantum theory of the breather one can only say that the breather absorption should consist of a sharp peak at an energy within the bonds of its classical energy; i.e., 1.02.10 A0; (c) we have assumed a breather density of 1%: the actual breather density is unknown; (d) detailed structure arising from vibronic modes of the CH unit are not included in this model. 77 Although a quantitative comparison is not possible the qualitative consistency of the kinkbreather photoexcitation scenario with data of many different types clearly justifies further study of this model. In the next chapter we discuss the modifications to the SSH Hamiltonian needed to study cispolyacetylene. We then explore the dynamics of photoexcitation in cispolyacetylene. C C 4, 0 c 3 L O . 0 w .  CL Q) C 4 > . V) 0 o) + Y co a) C) 4 bo cc i a) b D oo  O 0 .0 Co 0 a .co, c *H c 34 0) 0c 4  * cc ac , 43:~,; O Co \1 As 10 C0 C C 43 CC' 0) 0 4) C U 1) tqM 0 *M ) CO 4 4 M a. r C 0 S00 C4 . 43 Q) t 0 Z: Sa) c o a> a) CL 4) a) v Cd i 4l a) i 0 0 o *l C I 0 0 .0 *4 0 0 L I 0 0 co uN \ L o e II4 0 a) > 0 *^ rA 0 L .i1 S r 0) 4) s Ocr 75 L3 bO 40 4 .* 0 .0 CM O u/ / ^ / r <^   4 0) :5, C)9 0 ' LI o 3 i C ot ^ u "^2 4' 0 0) *r 0) C) L 0) E C *II II *0 0~ 3, c 300 200 100 0 2.0 4.0 6.0 E/AO Figure 4.4 Average optical absorption, a(w), over one phonon period and over a 98 length chain for the numerical breather. For the dimerized lattice see fig 3.4. 20 0 20 40 60 2.0 4.0 E/A, Figure 4.5 Change in optical absorption when (a) the numerical breather (solid line) or (b) the analytic breather (dashed line) is added to the 98 site ring. d K 6.0 80 40 0 40 80 120 p II I' I' 2.0 4.0 E/AO Figure 4.6 Change in optical absorption, Aa(w), on photoexcitation for transpolyacetylene for (a) t=2143 fsecs (solid line) (b) t=301343 fsecs (dashed line) KZI 6.0 L (1 C 4) r4 aC A 0 o c 0 . c) .i !.! 0 O 00 CO 0 oco NI, 2.0 4.0 E/AO Figure 4.8 The change, Aa(w), in the optical absorption on photoexcitation of the electron polaron. 80 40 0 40 80 120 K 6.0 Figure 4.9 Above Band Edge Excitation (a) Dynamics of system with electron hole pair excited above band edJje. (b) Location, with respect to the Fermi level, of electron and hole (solid line). Level .:.;'jxied by photoexcited electron and hole (dashed line), where the band edge state are levels 49 and 50. 2.5 1.5 1.0 0.5 0.0 Li TIE 93 (0) 08 CIS(CH)X T=10 K 06 04  02 0 \ 1E 1 K0 K 2 O0 04 06 08 10 12 14 16 18 20 22 ENERGY (eV) Figure 4.10 Experimentally observed change in absorption on photoexcitation of (a) cispolyactylene and (b) transpolyacetyl,ne at 10 K. From data of Orenstein et al. 474d 