Adiabatic nonlinear dynamics in models of quasi-one-dimensional conjugated polymers


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Adiabatic nonlinear dynamics in models of quasi-one-dimensional conjugated polymers
Physical Description:
vii, 219 leaves : ill. ; 28 cm.
Phillpot, Simon Robert, 1959-
Publication Date:


Subjects / Keywords:
Polyacetylenes   ( lcsh )
Polymers -- Electric properties   ( lcsh )
Polymers and polymerization -- Effect of radiation on   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- Uf
bibliography   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1985.
Bibliography: leaves 214-218.
Statement of Responsibility:
by Simon Robert Phillpot.
General Note:
General Note:

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Full Text







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


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

f-lJe-ted 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

T-11 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


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.


ACKNOWLEDGMENTS.................................... ........ ..........iii


I [L TRODUCTION ................... .................................1

II BASIC T.-EORFi ....................................................5

The H Model..............................................7
The TLM Model.............................................10
Limitations of the SSH and TLM models....................15


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 TRANS-POLYACETYLENE. ........................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


Statics in Cis-Polyacetylene.............................94
Dynamics in Cis-Polyacetylene............................97
Breather Dynamics in Cis-Polyacetylene...................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

Kink-Site 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
Kink-Bond 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 Gr-s..-leve- ? Model ................................... 169
Nonlinear Excitations in Polyyne........................171
Adiabatic Nonlinear D,;'rinlics in Polyyne.................. 173

VIII CONCLUSIONS............................ .......................185


A PAR.A' E1T79 OF THE SSH AND TLM MODELS..........................189

B OPTICAL ABSOCRPTION IN THE SSH MODEL...........................191

C THE ANALYTIC BREATHER.........................................199



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



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 H-eger (SSH) model

for polyacetylene. The SSH model is a tight-binding electron-phonon

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 trans-polyacetylene we find that a

kink-antikink 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 trans-polyacetylene. We compare

two suggested Hamiltonians for cis-polyacetylene and compare their

photoexcitation dynamics with those of trans-polyacetylene. 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.

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 non-metallic dopants the

conductivity of polyacetylene can be varied over twelve orders of

magnitude, from that of a :r-J 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


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 self-supporting 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


synthesized u:ing conventional ioni.3n1c techniques. T'-dir conductivities

are only, however, at most 1-10% that of polyacetylene.

The full barrage of standard solid state spectroscopies has been

used to characterize polyacetylene : electron and X-ray 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 c-harce 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 Hartree-Fock 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 Hamiltonians-8 that include strong

electron-electron 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)9-11 proposed a simple tight-binding electron-phonon coupled

model, which naturally explained the reverse spin-charge 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

tight-binding 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 electron-phonon 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 cis-polyacetylene 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




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: cis-polyacetylene and the energetically more

stable trans-polyacetylene. The present discussion will concentrate on

trans-pol;.:e-tyene. (A full discussion of cis-polyacetylene 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 trans-polyacetylene to be a conductor. However,

undoped trans-polyacetylene is a semiconductor with a bandgap

2A 1.4-1.6 eV. This gap is, at least in part, due to the Peierls

effectl7: for a one dimensional system any non-zero electron-phonon

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


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, over-simplified; 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 spin-charge 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 trans-polyacetylene based on a number of reasonable assumptions:

1. All many-body effects can be incorporated in a si-"ile particle


2. The effects of a electrons can be accounted for in the chain

cohesion and lattice dynamics. X-r -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


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 trans-polyacetylene p 0.1.
o o

4. A linear combination of atomic orbitals can be used as a basis


5. Only matrix elements between states with bonded sites need be

considered (tight binding approximation).

6. The system is one-dimensional 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 cross-li-n1:4 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 n-1 n n- n-1 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 electron-phonon

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 phonon-assisted contributions.

Dimerized Groundstate9

The SSH Hamiltonian has two jezenerate uniform solutions

un = (-1)n uo.


Now defining a "staggered order parameter",

n = (1) /uo (2.3)


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 :0i-ling 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 7-10a. 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 electron-phonon
coupling is weak: <1 where A = 2a /wt k. For .:'! parameters

X 0.2. The fermion operators C and C can be considered as the sim
n n
of left- and right-going waves.12,13

Cns (n) e iv n) e. (2.6)

For the half-filled 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, Lin-Liu, Maki)



HTLM dxA2(x) + dx Y+(x) a[iv + A(x)o,]V(x) (2.8)


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.

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


+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)]

(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

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 +






and then

Thus the problem has reduced to the solution of a "Schrtdinger-like"

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


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

[ F2 + E 2 2- f (x) = 0.
F 2 x +


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 [ v-K-
n k L v k


N = 1i [ o 1/2,
k 2 v-2 w

These plane wave eigenfunctions have the energy dispersion relation


2 2 2 2
i = A + vk
0 F


where the consistency equation is satisfied by

A =We


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/)


where = vF/A (The minus sign gives the antikink solution.) This

has both plane-wave solutions and a single localized level

at E = 0 with wavefunctions

uo(x) = N sech(x/)

v (x) = -iN sech(x/E)


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)=x-Vt 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 3-6m (2.24)

where me is the electron mass.

Tnis the kink is very light and kink dynamics can be expected to be


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

self-trapping 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 (x-x )]
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 self-consistency 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 trans-polyacetylene are

at = A //2. For an electron polaron the lower level is doubly

occupied and the upper is sinr-ly 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


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.

Electron-Electron Interactions

Experiments show the first excited state of the finite polyenes-4

((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 electron-electron 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 infra-red absorption peaks are correctly predicted.25

A number of studies on electron-electron effects in a Peierls

distorted phase have been undertaken with conflicting conclu-

sions.6-8,26-28 Calculations at the Hartree-Fock level generally show

that the dimerization is decreased by electron-electron effects, whilst

calculations that go beyond Hartree-Fock conclude that the dimerization

is increased.

Al'horuh electron-electron 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

electron-electron effects. However, without four-body operators in the

Hamiltonian some electron-electron 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 electron-electron effects (see Chapter

III) the assumption that the nonlinear dynamics and statistical

thermodynamics of polyacetylene can be considered as mainly arising from

the electron-phonon coupling seems to be justified. All many-body

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

trans-polyacetylene: the '* (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 15-25%.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, non-adiabatic 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,

X-ray 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 3-10K 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


Finite Temperature Effects

Both the SSH and the TLM models are zero temperature models. As

the band gap of trans-polyacetylene is 1.4eV one expects thermal

production of electrons into the conduction band to be negligble.

However, the self-focussing of thermal -:ri-tr-' into nonlinear Lhv-n.rin

wavepackets, breathers, may be important.

Extrinsic Doping

The most striking experimental feature of trans-polacetylene is

that its conductivity can be varied over thirteen orders of .Tir-nitude 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



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 TLM-like equations have proven unsuccessful (the

TLM Hamiltonian is neither Lorentzian nor Gaililean invariant--there are

two velocity scales, the Fermi velocity and the sound velocity).


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 trans-polyacetylene.

c. b.


.' ^ : ^ :^ v.b.
v' ^y .*.*.*>>^ '"
1-:^ *>^ X^X< Al>

1 I

I I '


C (

* *C


-A0 +A0



.- *



0-1 ^

-I z

0- I 0

< z
O-I <


fl "T"

'-- -,









0 *


4 -1
-0 -

S *- r Y

c c
() ) )











.,-4 4--




*O *




0 Q.
U ()

C -)


*O -
*i (
^-1 '-

o 0
3 3




0 n 0

" I ; 0 c Y L0

. 0





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)

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 n-1 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 ,j-1 i,j+1

This has eigenvalues e given by

(T E I) = 0



E = (e ,E E )

Newton's equations of motion are

MU6V(u })
M1 = 6.-({


The time derivative can be evaluated by finite differences as

u (m) = -- [un(m) (m-1)]
n dt n n


where dt is the time between the m-1 and the m iterations. This gives

the equation of motion as

u (m) = 2u (m-1) u (m-2) t2 (.(m-1)}).
n n n M u n


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


6u << .

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 (m-1) u (m-2) M 6u



Also in some studies instantaneous damping is applied by setting

the velocity to zero at every timestep, i.e.,

un(m-1) = u (m-2).

This modifies the equations of motion, eq. 3.7, to

(dt)2 n
u (m) = u (m-1) -(dt2 6Vu
n n M 6u





Boundary Conditions

Two types of boundary conditions are used in this numerical


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)


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

electron-hole 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 staBgere-i 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


r = (-1)n 2u u u (3.21)
n 4 n n+1 n-1

s = 1 [2un + + u1]. (3.22)
n 4 n n+1 n-1

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(w) = T A E |<4 203| I ,>2 6(E2 E W)


where a is the current operator, the 6-function 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


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

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 semi-infinite 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,
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
in fig. 3.3 as a function of the location of the intragap polaron

level. (For the polaron in trans-polyacetylene 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,

whilst a (3) involves many transitions. This uprising intensity ratio

is again a result of a fortuitous phase relation in the continuum


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


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


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)


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.
He further showed that ado) obeys a sum rule

fJ a()du = e (3.30)
2N ez

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 electron-electron effects are

added at the Hubbard level, i.e. H = E V p(n) p(m) is added to the
ee rm
SSH Hamiltonian. This is because [H p(n)] = 0.

Figure 3.4 shows the numerically calculated optical absorption of

the purely dimerized lattice. The 6-function 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
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

v /v (2X) (E/a). (3.31)
r S

For SSH parameters (7 7a, A 0.2) this gives (vy/vs)2 19--the

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 r *2 s
T M j A (x)dx = AY ) (3.33)



2 -
Y = A /24AnXt'.
o 0


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


where numerically we find

B(E = 7) 0.20
8(E = 2.7) 0.10.


Then the Langragian of a kink of finite velocity is


L = A (1 r)2 Yq_/r}.

Thence the momentum (Pq) conjugate to the displacement q (q = v/vs) is

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



Then for consistency

r = 0, q = 0.


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





which gives the momentum at the maximum kink velocity as

2 8
P 2- Y
m 3

Thus the en-rr.g, at maximum kink velocity is

(3. 46)

Ek 9 o

The kink energy can also be written

Ek(p) = A [p2/4Y p 24/64Y2].


This gives a maximum in the energy at

pM = 8Y8.


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



1 1 3 2
m 2a 16 p /YB




Thus in the low momentum limit the kink behaves like a Newtonian

particle of mass 2 a. However, the effective mass div-r':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 1-r;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 corr-3sp;-njL to
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
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 0.15-0.30 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 E-0.3 E

and E-2.0 E respectively. At low energy inputs the kink propagates

essentially uniformly, whilst at higher energy inputs kinetic n-rs, 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


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 r-ridi q of the short time jr-inrinics of the photoexcited




-2.0 III
30 n 60


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).



-9 -9






C-a C









H 0




I- -


j *






OI ( 3)

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
trans-polyacetylene w /A 0.7.




2.0 4.0 6.0


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
6-functions 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.






2.0 4.0

E /A0

Figure 3.5

Optical absorption, a(w), of two widely spaced kinks on a 98 site ring.


2.0 4.0


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).













2.0 4.0


Fi 'n're 3.7

Optical absorption, a(w), of a polaron on a 98 site ring.


2.0 4.0


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)










CO e

4O -X

C 4-


C 0


O'2 O


CC r-l
-4 C


0- 0 .0

c .0


4a a) -,

C- Q.- 0
3 0

0 '-- E

0 *-

0 0

Q .

0 <



0 0







0 '4

to a

c -


r: 0





cd 2



o -













We begin this study of photoexcitation in trans-polyacetylene 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 kink-antikink (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 excitation--an amplitude breather. The breather dynamics were


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 half-filled 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 kink-like solutions as it is ill-defined as A-0.) 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 (4x-t)



and t and T are the two time scales. Using multiple time-scale

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 t-3] (4.5)


d = c/2/2



= 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 7-E + 0( )].



(xt)c[(XT)exp iwt0T)exp-iw +251 (xt)
6(x,t)=e[A(X,T)exp int+A (X,T)exp-iuit]+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


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 trans-polyacetylene 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
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
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 energies--a 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 wave-vector 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

wave-vector 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 1-2A 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


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:e-ge 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 times--the 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 cis-like 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 trans-polyacetylene, there is

a significant density of spinless charge carriers--kinks. 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 kink-breather transitions.

There is also an enhancement at -1.5 A arisi'; from transitions between

the breather and continuum levels. The intra pp bleachi-i; 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,
which typically have a time constant of -10-8 secs. In the absence of

a preferred non-radiative decay route, via e.g coherent electron-hole

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, polaron-polaron bound states,

breathers and excitons. The choice of channel will depend on, e.g.

correlation and impurity effects.

A fully qjant.rim non-adiabatic 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 trans-polyacetylene 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. Fig-re 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 Cross-Section 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 sine-Gordon system, we treat

the collection of breathers as an ideal gas, in which all the internal

structure is reflected in a q-dependent 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


< exp iqu(x It exp-iqu(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/a-vaw/co


where e is the Debye-Waller factor, co is the velocity scale and

u(x,t)=na+(-1)n jl(x,t). For simplicity we use the Maxwell-Boltzmann

distribution, P (v). After a little algebra one finds

S(qw)=e-2W Bn B'T) 1r (k+) 2NR a + If(k-) 2 )
(27r) 2 q


~-4 -+2w aw-2( B
+ : {f2(k) 2pNR( B )+f (k) 2NR( B +36f2 pN (/q)R l


+ -1 r b .+n)
k =d-1 [q + /a +- ( j)]
n 2 q

k = d1 [q + T/a w2/c2 q]

f (q) = aid sech (jrqd/2)

f 2(q) = iqd2 cosech (irqd/2)





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.


This zero frequency inelastic scattering can be observed separately from


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.


S(q,w=+w )/S(q,w=2Bw) = 1 tanh2 U2/ 1-q/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 en-erg of the breather. This can easily be quantized at the Bohr-

Sommerfeld level by d-manding 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

4V-2 -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)

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.j-r 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 1-2 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 trans-polyacetylene 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 trans-polyacetylene 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.

: 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.54-5 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 C-H 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 [1-M(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 electron-electron 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., 1.0-2.10 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 C-H unit are not included

in this model.


Although a quantitative comparison is not possible the qualitative

consistency of the kink-breather 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 cis-polyacetylene. We then explore the dynamics of

photoexcitation in cis-polyacetylene.



c -3

. 0 w .- -

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
3-4 0)

0c 4- -

* cc ac ,








C 4-3
0) 0

4-) C U
1) tqM

0 *M

) CO-
4 4 M

a. -r C

C4 .

4-3 Q)

t 0 Z:
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





o e


-0 a)
> 0
*^ r-A

L .i-1

S r-




-40 4-





/ ^

/ r-









o 3

^ u

4-' 0














2.0 4.0 6.0


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.









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.










2.0 4.0


Figure 4.6

Change in optical absorption, Aa(w), on photoexcitation for
trans-polyacetylene for (a) t=21-43 fsecs (solid line)
(b) t=301-343 fsecs (dashed line)

















2.0 4.0


Figure 4.8
The change, Aa(w), in the optical absorption on
photoexcitation of the electron polaron.









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.








08 CIS-(CH)X
T=10 K

04 -



\ 1E

1 K0 K

O0 04 06 08 10 12 14 16 18 20 22

Figure 4.10

Experimentally observed change in absorption on photoexcitation of
(a) cis-polyactylene and (b) trans-polyacetyl,-ne at 10 K.
From data of Orenstein et al. 474d