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 Permanent Link:
 http://ufdc.ufl.edu/AA00004883/00001
Material Information
 Title:
 Adiabatic nonlinear dynamics in models of quasionedimensional conjugated polymers
 Creator:
 Phillpot, Simon Robert, 1959
 Publication Date:
 1985
 Language:
 English
 Physical Description:
 vii, 219 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Analytics ( jstor )
Conceptual lattices ( jstor ) Electromagnetic absorption ( jstor ) Electrons ( jstor ) Impurities ( jstor ) Phonons ( jstor ) Photoexcitation ( jstor ) Polarons ( jstor ) Polyacetylenes ( jstor ) Velocity ( jstor ) Dissertations, Academic  Physics  Uf Physics thesis Ph. D Polyacetylenes ( lcsh ) Polymers  Electric properties ( lcsh ) Polymers and polymerization  Effect of radiation on ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1985.
 Bibliography:
 Bibliography: leaves 214218.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Simon Robert Phillpot.
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 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 029609954 ( ALEPH )
14972005 ( OCLC ) AEH8284 ( NOTIS ) AA00004883_00001 ( sobekcm )

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Full Text 
ADIABATIC NONLINEAR DYNAMICS IN MODELS OF
QUASIONEDIMENSIONAL CONJUGATED 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
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/adiabaticnonlineOOphil
ACKNOWLEDGMENTS
It is with great pleasure that I thank my mentor and friend,
Pradeep Kumar, who has provided both inspiration and guidance throughout
my graduate career. I also owe a great debt to Alan Bishop, who
suggested 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 benefited 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 suggestions 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.
Although 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.
iii
TABLE OF CONTENTS
ACKNOWLEDGMENTS iii
ABSTRACT vi
CHAPTER
I INTRODUCTION 1
II BASIC THEORY 5
TransPol yacetyl ene 5
The SSH Model 7
The TLM Model 10
Limitations of the SSH and TLM models 15
III DYNAMICS 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 PHOTOEXCITATION 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
IV
KinkSite Imparity Interactions 125
Photoexcitation in the Presence
of a Single Site Impurity 128
Photoexcitation with Many Site Impurities 1 31
The Single Bond Impurity 132
KinkBond Impurity Interactions 134
Photoexcitation in the Presence
of a Single Bond Impurity 134
VII STATICS AND DYNAMICS IN POLYYNE 163
Formalism 163
The Continuum Model 167
The GrossNeveu Model 169
Nonlinear Excitations in Polyyne 171
Adiabatic Nonlinear Dynamics in Polyyne 173
VIII CONCLUSIONS 185
APPENDIX
A PARAMETERS OF THE SSH AND TLM MODELS 189
B OPTICAL ABSORPTION IN THE SSH MODEL 191
C THE ANALYTIC BREATHER 199
D CLASSICAL DYNAMIC STRUCTURE FACTOR OF THE BREATHER 204
E CONTINUED FRACTION SCHEME FOR IMPURITY LEVELS 208
BIBLIOGRAPHY 21 4
BIOGRAPHICAL SKETCH 219
v
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
QUASIONEDIMENSIONAL CONJUGATED 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 Heeger (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
vx
photoexcitation dynamics with those of transpolyacetylene. We farther
investigate the role of coherent anharmonicity in simple models of a
finite polyene and of polyyne. As the observed transport properties of
polyacetylene depend critically on the presence of external imparities
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 trapped
kinks may be produced.
CHAPTER I
INTRODUCTION
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 good insulator to that of a fair metal, has
not suprisingly, produced a great deal of interest. Most of the ensuing
O
activity has been in four broad areas: technological applications,
synthesis, physical testing and characterization, and theoretical
modeling.
p
Amongst the suggested technological applications for polyacetylene
and related materials are Schottky diodes, solar cells, lightweight
metals, and high power storage batteries.
Ito et al.3 first synthesized selfsupporting films of
polyacetylene by the direct polymerization of acetylene gas in the
presence of catalysts. This method is essentially uncontrolled and
produces samples of widely varying structure and physical properties.
p
Since then much effort 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 conjugated polyaromatic
heterocyclic polymers and some conjugated block copolymers can be
1
synthesized using conventional organic techniques. Their conductivities
are only, however, at most 110% that of polyacetylene.
The full barrage of standard solid state spectroscopies has been
p
used to characterize polyacetylene: electron and Xray diffraction
have been used to study its structure; Raman scattering to determine its
conjugation length; 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 responsible for the widely differing experimental data and
interpretation. However one particularly significant conclusion has
ii
emerged: ESR experiments have shown that the charge carriers in
polyacetylene are spinless and therefore are not electrons.
The theoretical work has been in three general areas. Ab initio
calculations,^ 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
/TO
excited system. Phenomenological Hamiltonians0 0 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)^'1 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
1 O 10
continuum limit J proved to be analytically tractable and their
3
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.
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
4
finite response time of the lattice. In Chapter IV a simple
photoexcitation experiment in transpolyacetylene is modelen. 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 transport properties discussed.
Chapter VII introduces the polyynes and suggests that nonlinear
excitations may be important here, also. Chapter VIII presents our
conclusions.
CHAPTER II
BASIC THEORY
TransPoiyacetylene
Polyacetylene is a linear chain polymer consisting of a spine of
carbon atoms with a single hydrogen atom bonded to each.liJ1^ It exists
in two isomeric forms: cispolyacetylene and the energetically more
stable transpolyacetylene. The present discussion will concentrate on
transpolyacetyene. (A full discussion of cispolyacetylene will be
undertaken in Chapter V.) Of the four carbon valence electrons, three
are in a bonds to a hydrogen atom and to the two neighboring carbon
atoms. The remaining electron is in the it band, which is half filled.
One thus expects transpolyacetylene to be a conductor. However,
undoped transpolyacetylene is a semiconductor with a bandgap
2Aq ~ 1.41 .6 eV. This gap is, at least in part, due to the Peierls
effect1"^: 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 = ir/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
6
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 single
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
energy 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 2E, 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. There 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 midgap 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
7
explanation for the simultaneous observation of high electrical
conductivity1 and low Curie susceptibility11: the charge carriers in
doped polyacetylene are charged, zero spin, kinks.
The SSH Model
The SSH Hamiltonian
Su, Schrieffer and Heeger^ (SSH) have proposed a simple Hamiltonian
for transpolyacetylene based on a number of reasonable assumptions:
1. All manybody effects can be incorporated in a single particle
Hamiltonian.
2. The effects of o electrons can be accounted for in the chain
cohesion and lattice dynamics. Xray1 and NMR1^ data show that
a 1.22A and Uq 0.03 0.01 A (a is the lattice constant and Uq
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 it 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
?n
is 0.05. Brazovskn and Dzyaloshinskii have shewn rigorously
that the adiabatic approximation requires y << 1,
where y = Kwq/Ao. For transpolyacetylene y 0.1.
8
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 crosslinking may,
however, be important in many materials.
The SSH Hamiltonian is
, M *2 K
^SSH 2ldn+ 2 Z(an Jn1 }
n
z[t +ct(u u )1[C,C +C+ ,C
o n n1 JL n n1 n1 n
n
(2.1)
where M is the mass of a (CH) unit, K is the force constant, tg is the
intrinsic electron hopping matrix element, a is the electronphonon
coupling constant. 0^(0^) creates (annihilates) an electron on site 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 o electrons); the third is the it electronic
energy, consisting of intrinsic and phononassisted contributions.
Q
Dimerized Groundstate7
The SSH Hamiltonian has two degenerate uniform solutions
= (1)n
(2.2)
9
Now defining a "staggered order parameter",
u = (1)n u /u
n no
(2.3)
then
= 1 (2.4)
It is interesting to note that with two degenerate groundstates and with
a local energy maximum at u=0 the SSH model is topologically identical
4
to the $ model. This similarity will be exploited at various times,
4
but it must be remembered that 4> 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 coupling limit that the band
gap, 2Aq, is given by
2A = 8au (2.5)
o o
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) = AQtanh(x/^) (the exact kink
solution), the kink half width, E_, is 7~10a. This suggests that a
continuum approximation to the SSH Hamiltonian may provide a useful
basis for further analytic studies.
10
The TLM Model
The TLM Hamiltonian
The continuum approximation is valid if the electronphonon
2
coupling is weak: X <<1 where X = 2a /irt^k. For SSH parameters
X 0.2. The fermion operators C* and C can be considered as the sum
n n
ip i j
of left and rightgoing waves. J
1
C = [u (n) e F
ns / s
/a
ik^na . ik^nai
f lv (n) e F I.
(2.6)
For the halffilled band the system dimerizes to give a lattice period
2a. Thus we define
u =
Met
[A+(n)
2ikna
+ A(n) e
2ik^na
]
(2.7)
To order a, and dropping the rapidly varying terms of the type (1)n the
SSH Hamiltonian becomes the classical TLM (Takayama, LinLiu, Maki)
Hamiltonian.
a)2
HTLM = "~2 idxA'^x^ + Jdx 'i'+(x) [_ivFa + A (x)o 1 ]t(x) (2.8)
2g J
where
T(x)
f u ( X )
V ( X )
)
2
WQ
4K
M
g =
ai 1/2
2t a
o
and is the ith Pauli matrix.
(2.9)
11
Varying eq.
# Â£
2.8 with respect to u (x) and v'(x)
gives
i v
F
3 a (x)
3x
+ A(x)v(x)
e u(x)
v
(2.10)
+ i v + A ( X ) u (x) = Â£ v(x).
F dX 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 tt band an artificial energy
cutoff W is introduced to simulate the band edges. This is expected
only to alter the energy scale.
+
Varying eq. 2.8 with respect to A gives the consistency equation
A(x) = g2/w^ X' [u (x) v(x) + v (x) u(x)] (2.11)
V ,s
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(x) (2.12)
equations 2.10 become
3f_
e f (x) = iv (x) iA(x) f_(x)
v F 3x +
and then
(2.13)
2 3
3x
2
e
v
3A(x)
3x
 A (x)] f (x) = 0.
(2.14)
12
Thus the problem has reduced to the solution of a "SchrOdinger1ike"
equation, with the potential being a function of the order
parameter A(x). Of course the consistency conditions (2.11) must also
be obeyed. Solutions are particularly easy to obtain for the class of
potentials with
Two particular solutions that fall into this class are the purely
dimerized lattice and the kink solution.
21
The Dimerized Lattice
For the purely dimerized lattice: A(x) = Aq one gets
(2.16)
This has solutions
u (x) = N. e
n k
o
v (x) = N, e
n k
(2.17)
with
These plane wave eigenfunctions have the energy dispersion relation
13
2 2 2 2
to = A + v"k (2.18)
O r
where the consistency equation is satisfied by
Aq = W e~'/2X. (2.19)
Here one sees that W merely sets the scale of energies.
The Kink Soliton^^^1
There is also a kink solution (fig. 2.3a) with order parameter
A(x) = AQtanh(x/5) (2.20)
where F = v_/A (The minus sign gives the antikink solution.) This
F o
has both planewave solutions and a single localized level
at e =0 with wavefunctions
o
and
u (x) = NQsech(x/5)
vq(x) = iNQsech(x/5)
A /H
o
(2.21)
This electronic level is localized on the kink with the same coherence
length as the lattice distortion.
14
The kink creation energy
15,22
can also be calculated as
E. = 2A /tt.
k o
(2.22)
Although the TLM model is neither Galilean nor Lorentzian invariant, the
kink mass can be calculated in the ansatz x(t)=xVt by equating the mass
O
to the coefficient of the (1/2)V term. This gives the mass of the
kink, Mk (where M is the mass of a (CH) unit)
(2.23)
for SSH parameters
M, 3~6m
k e
(2.24)
where mg is the electron mass.
Thus the kink is very light and kink dynamics can be expected to be
important.
The Polaron21,2^
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) = A k v_ [tanh k (x+x ) tanh k (xx )1
o o F L o o o o J
15
where j2 = k2v2 + A" (2.25)
o o F o
and tanh k x = (A iu )/k v.
o o o o OP
This selfconsisteney condition is, however, only satisfied in trans
polyacetylene if
k v = A //2. (2.26)
o F o
This lattice deformation is supported by a pair of intragap levels
symmetrically about the Fermi level, which in transpolyacetylene are
at u = A //2. For an electron polaron the lower level is doubly
o o
occupied and the upper is singly occupied. For the hole polaron the
lower level is singly occupied and the upper is empty.
The polaron creation energy is
Ek = 2/2 Aq/tt (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
o
(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.
16
ElectronElectron Interactions
p ii
Experiments show the first excited state of the finite polyenes
((CH)n, n=2,3...) has 1Ag 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
analogy, 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
pc
energies of the infrared absorption peaks are correctly predicted.J
A number of studies on electronelectron effects in a Peierls
distorted phase have been undertaken with conflicting conclu
sions >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.
Although 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 "fermion 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
17
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.1 Here two parameterizations are chosen. "SSH
parameters"^ are chosen to reproduce three experimental observables of
transpolyacetylene: the bandgap (2 1.4eV), the dimerization
(uQ 0.04A) and the full bandwidth (W 10eV). The spring constant
2
(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
Â£ ~ 7a. For many of the simulations performed the parameterization W =
10eV, 2Aq = 3.8eV, uQ = 0.1A, Â£ = 2.7a will be used. These "rescaled
parameters"2^ 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
18
transition rate between them. Explicit calculations show that the
dimerization is reduced by Quantum fluctuations also
reduce the kink creation energy by 25%.One can consider these
effects as being included in the renormalized SSH 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 J show that there is three dimensional ordering in trans
polyacetylene with the antiparallel alignment of dimerization patterns
on neighboring chains. A kink introduces a defect^ in this alignment
with an energy that increases linearly with the separation of kinks.
Thus one expects a KK pair to be confined (though 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 TOO lattice sites. (As will be seen in Chapter V this
confinement is similar to the intrinsic confinement in cis
polyacetylene.) Interchain hopping^ of charge between kinks has also
been suggested as the dominant conduction mechanism for electrical
conductivity at low dopant concentrations.
19
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 energy into nonlinear phonon
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 magnitude by
extrinsic doping.^ 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 deficiences of the basic model need to be
addressed. A first logical step is to study the dynamics of the basic
3SH model. This is the subject of Chapter III.
The two degenerate phases of transpolyacetylene.
22
C.b.
v.b.
c
I
H
H
I
C
H
I
C
c
I
H
('Aq)
H
l
C
H
i
C
H,
C
c
I
H
GROUND STATE
H
I
C
c
I
H
C
l
H
H
I
C
C
I
H
C
I
H
KINK SOLI TON
H
i
C
\ / \ / K./ \/ H/ \ / \ / \
C
l
H
( + Aq )
c
I
H
GROUND STATE
NJ
UJ
Figure 2.2
Kink interpolating between the two phases of transpolyacetylene.
To the left of the kink the double bonds slope upwards; to the right of
the kink the double bonds slope downward.
Figure 2.3
Order parameter (solid line) and wavefunction (dashed line) and
electronic spectrum of (a) the kink (b) the polaron. Note the reversed
spincharge relations for the kinks.
A(y)
(a)
(b)
y/c
0
'm.
*
11
m!
I
u
m
Q = + e
Q = 0
Q = e
S = 0
S = 1/2
S = 0
OJQ
rvl.^Tr
oj0
Q = + e
Q= e
S= 1/2 S = 1/2
hO
Ln
2
2
CHAPTER III
DYNAMICS OF A SINGLE KINK IN THE SSH AND TLM MODELS
In this chapter a systematic study of the dynamics of a single kink
in both the TLM and SSH models will be pursued. 3efore 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.
pq
The Numerical Technique 7
The energy of the SSH Hamiltonian can be formally written as
HSSH
2
u
n
+
v(iunn
(3.1)
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
v = !z
n
v, s
m e
v,s v ,s
(3.2)
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
26
27
direct diagonalization of the hopping matrix T, which has components
{t + a [ u.
1 o L 1
u.
J
+ 6 .
l
(3.3)
This has eigenvalues
e given by
v
(T E I) = 0
where
(3.4)
Newton's equations of motion are
MU
n
5V({un})
6un
(3.5)
The time derivative can be evaluated by finite differences as
u
n
(m) = ^ [un(rn) un(m1)]
(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)
n n n
(dt)2
M
({un(m1)}).
(3.7)
The functional derivative of the potential can also be calculated by
finite differences as
28
6v(iunn
5a 'V(U1 u2 Un+U" *UN) ;(J1U2
*un *UM ; /Su
(3.3)
n
where
u << a .
o
(3.9)
Energy conservation is ased as a check on the stability of the
algorithm. In general energy is conserved to better than 99.99%.
At some points in this study spatially homogeneous velocity
dependent damping is applied by the addition of the term
to the eqaations of motion. This gives evolation eqaations
u
n
Also in some studies instantaneous damping is applied by setting
the velocity to zero at every timestep, i.e.,
u (m1) = a (m2).
n n
(3.12)
This modifies the equations of motion, eq. 37, to
(3.13)
29
Boundary Conditions
Two types of boundary conditions are used in this numerical
study. ^
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
(3.14)
u
u.
N+n
n
and
(315)
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
(1)
n+N
u
u
n+N*
n
30
There is also a more subtle distinction between systems of length
4n and 4n+2. For an undimerized system (u = 0) the eigenstates of the
SSH Hamiltonian are Bloch states with energy
E = 2t cos k a (317)
non
where the wavevector,kn, satisfies the quantization condition
kn = 2;rn/N ^ < n S ^ (3.18)
Thus for N=4n and a half filled band the Fermi level lies at E=0. 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 (tt/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, \.)
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
31
lengths is small. Here two different types of chain boundary 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"
4a v
(u, u.J.
it 1 N
2. In "fixed boundary conditions" each of the end sites is bound
to its nearest neighbor by the constraint
U1 u2
un jnt
(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=0 lattice configuration and
minimize the energy of the system by varying the dimerization uQ 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).
32
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 staggered 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)n n + n.a (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
def ine^
r
n
= i M)
[2l
n+
, u ,1
1 n1J
[ 2u
L r
n+1
+ u
n1
(3.21)
(3.22)
Then rn Un and sn n.$a. In these variables the kink profile is more
easily seen (fig 3.1b). Note that the dip in sn corresponds to a
33
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*3 The
optical absorption is defined within the linear response dipole
approximation as
A
o(u) = tt f A E I^JoJhvI 5(E, ~ Ei ~ u) (3.23)
Lio 1 p 2' 3 I 2 1
where is the current operator, the 5function ensures the
conservation of energy and A is an unimportant constant. Figure 32
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
aD(u) = a(2Aq/w)(io2 4A2)~/. (3.24)
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."^ 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
34
phase.!) This alters the phase relations in the valence and conduction
bands, causing diagonal transitions (Ak=0) to be explicitly forbidden.
For the midgap transition
*r/2
it
2A
o
*2r,/2i
(3.25)
whilst for the interband absorption
a*3() = otD(ai) [l ~ 2c,/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.
? 1
The polaron absorption has similarly been calculated and consists
of three contributions plus the interband. Transitions between the two
intragap polaron levels give a'(to). There is a second contribution,
P
2
a(co), 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 (to). The relative intensities of each of these is shown
p
in fig. 33 as a function of the location of the intragap polaron
level. (For the polaron in transpolyacetylene ooo = 1//2. For cis
polyacetylene see Chapter V.) For most gap locations cJ (to) is much
P
v
stronger than a (to) even though it involves only one transition,
P
35
3
whilst cxd(io) involves many transitions. This suprising 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.^ A review is given here (full details are
given in Appendix C). Starting from the definition of the charge
density, p(n) = eC^C^, 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[t + ctfu u J1[C+C C+ ,C 1. (327)
J L o n n+1;jL n n+1 n+1 nJ
Then in the linear response dipole approximation
i(u) = ire2t2 1 l M O2o(e0 e. u)
o wN 1 ,21 2 1
' iÂ£
(3.28)
where
,2
= 2] [1
T (u 
t n
o
n+1
)][f1(n)f*(n+1) fi(n+1)f*(n)] (3.29)
and f is the wavefunction of the a level,
a
He further showed that a(w) obeys a sum rule
36
J a(u)>do) = ~ prr
1 2 .
2 116
(3.30)
where H is the electronic energy of the configuration. This algorithm
for the absorption is easily implemented. It is important to note that
all these results remain unchanged when electronelectron effects are
added at the Hubbard level, i.e. H = E V p(n) p(m) is added to the
ee nm
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 5function 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?.
Figure 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(to), 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 (
P P o
3
and A^/4 respectively. The weak contrbution, a^(a)), is present but
only 2? of that of cip(u). 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 (Â£ 2.7), the
polaron is is narrow and requires a large number of Fourier modes for
its description.
37
These 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 dynamical 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 changes in its dimerization, the kink has a maximum free
propagation velocity as can be seen from the following simple
argument.37> "9,40 por a ^ink Qf velocity v the lattice changes from the
A to B phase (say) over a distance Â£(v) and in time Â£(v)/v. The
minimum response time of the lattice is the inverse of the renormalized
1/2
phonon frequency, u), where wn = (2A) w (w being the bare phonon
a a O O
frequency). This gives the maximum kink propagation velocity (vm)
as v oj s(v ) Assuming that the kink width is not strongly velocity
m K m
dependent, then
vm/vs ~ (2A)I/2 U/a> (3.3D
For SSH parameters (Â£ 7a, A ~ 0.2) this gives (vm/vs)2 19the
maximum velocity is greater than the sound velocity.
The conclusion of the above argument can be confirmed from an
approximate analytic calculation. Using the ansatz for the kink profile
38
A(x,t) = Aq tanh (x vt)/Â£(v)
(3.32)
it is straightforward to show that the lattice kinetic energy of the
kink (T) is
M r 2
T = j A (x)dx = AqY
(v/v )'
s
(333)
where
r = 5(v)/5(0)
(3.34)
and
Y = A2/24ir\t2.
o o
(3.35)
It has been shown numerically^ 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
E
D
BAq(1 r)2
(3.36)
where numerically we find
6(5 = 7) ~ 0.20 (337)
6(5 = 2.7) ~ 0.10.
Then the Langragian of a kink of finite velocity is
39
L = Aq{b(1 r) YqVr} .
(3.33)
Thence the momentum (pq) conjugate to the displacement q (q = v/v ) is
2Yq
p = 
q r
(339)
and the momentum, p conjugate to the width r is
pr = 0.
(3.^0)
Now extremizing the Lagrangian with respect to r and q gives
?Br (1 r)
(3.41)
q qr
r r
0.
(3.42)
Then for consistency
r = 0, 5} = 0. (3.43)
Thus from eq. 3.41 the velocity has an extremum at r=2/3 independent of
the parameters Y and 8.
Substituting back for r the kink has a maximum propagation velocity
(vm) given by
(v /v =
v m s'
()
v27
B/Y
(3.44)
40
which gives the momentum at the maximum kink velocity as
2 8
Pm = 3 YB
(3.45)
Thus the energy at maximum kink velocity is
Ek = 9 SAo*
(3.46)
The kink energy can also be written
E (p) = Aq [p2/4Y p4/64Y23]
(3.47)
This gives a maximum in the energy at
pm 8T6'
(3.18)
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
m*
d2E
dp2
(3.49)
gives
1 3 2
= 2 T6 P /Y8
m*
(3.50)
41
Thus in the low momentum limit the kink behaves like a Newtonian
particle of mass 2 a. However, the effective mass diverges at the
critical value of the momentum
2
P
c
(3.51 )
This is the same as the kink momentum for free propagation at the
maximum velocity (eq. 3.^5). Using appropriate value for B and '< one
gets for SSH parameters
(v /v )2 11, m'
m s
5.6 m
e
E
m
0.1 A
o
(3.52)
and for rescaled parameters
(v /v ) ~ 1.2
m s
80 m
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 Model^
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
42
conditions and for chains of differing lengths.) In fig 3.9 the short
time kink velocity squared is plotted as a function of the energy
input. Both axes are in dimensionless units, where unity corresponds to
2
the theoretical value of v and the theoretically calculated energy, Em,
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 single curve. Indeed by
treating g and y as free parameters and force fitting, better agreement
may be obtained. At low energy inputs the data are in good agreement
with the continuum theory. The maximum kink velocity is also well
predicted by the continuum calculation. However the continuum theory
strongly underestimates the energy 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/Â£) whilst acoustic phonons couple to order (a/Â£) Thus for
rescaled parameters phonon effects should be much stronger than for the
SSH parameters. Figure 39 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
43
the phonon effects reduce the maximum kink velocity by a factor of 2,
for E,~ 7, but by a factor of 10 for the more discrete kink
with c,~ 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
m
and E2.0 respectively. At low energy inputs the kink propagates
essentially uniformly, whilst at higher energy inputs kinetic energy is
lost by the emission of an optical phonon package.^ This may be viewed
as the classical analog to the emission of Cerenkov radiation by a
particle traveling faster than its maximum propagation velocity in the
medium.
Although, as we shall see in Chapter VI cnarge conduction by the
ballistic transport of kinks is unlikely, kink propagation is essential
to the understanding of the short time dynamics of the photoexcited
system.
44
Ufl
Figure 31
The Single Kink
(a) unsmoothed order parameter, u for single kink.
(b) Smoothed order parameter, r for the same single kink and sn
(dashed line).
Figure 3.2
Electronic transitions of (a) dimerized groundstate (b)
the single kink (c) the polaron. The valence bond is v.b. and is full.
The empty conduction bond is c.b. The dotted line in (c) gives the Fermi level.
46
o
47
Figure 3.3
Relative strength of transitions in the polaron as a function
of the location, too/Ao, of the intragap levels. For
transpolyacetylene w /A 0.7.
00
a (a
48
E/A0
Figure 31)
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 = 4tQ = 10 eV. The
5functions 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.
49
Figure 35
Optical absorption, a(u), of two widely spaced kinks on a 98 site ring.
Aa(oi)
50
E/A0
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).
51
Figure 37
Optical absorption, a(w), of a polaron on a 98 site ring.
52
3
a
E/A0
Figure 3.8
Difference in absorption of a single polaron
and the purely dimerized lattice of a 98 site ring.
(Figure 3.8 = Figure 3.7 Figure 3.4)
Figure 3.9
Output velocity squared versus input energy for 5 = 7a (triangles) and
5 = 2.7a (circles) for period < 1 phonon period (open markers) and ~ 48
phonon periods (solid markers). The solid line comes from the continuum theory.
54
S
>
Figure 3.10
Single Kink Dynamics.
(a) Time evolution of a single kink of input energy E 0.3 Effl.
Note the smooth propogation of the kink. This evolution is essentially
independent of the size of the system.
56
O
Figure 310
Single Kink Dynamics
(b) Time evolution of single kink of input energy E 2.0 Em.
Note the production of an optical phonon tail to the breather.
58
CHAPTER IV
PHOTOEXCITATION IN TRANSPOLYACETYLENE
We begin this study of photoexcitation in transpolyacetylene with
a simple experiment:^ ^ 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 "rescaled parameters"
will be used as they allow smaller systems to be studied, thus reducing
the computation time. (The dynamics are expected to be similar for the
more realistic SSH parameters.) At t=0 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 2Aq 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, vm, (fig. 4.1). Each kink has creation energy 2AQ/ir 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
60
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 may be
obtained from the known effective Lagrangian^ for the halffilled band
'2 *2
L = {1/2 A2[i,n(2E /A) + 1/2] AV4X v2 + A ] (4.1)
v_ c F 2 2
F 24A 2iod
K
where EQ 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 A+0.) From eq.
4.1 the equation of motion is
T 2 It
A/24 A In 2E /A VF ~ a,A b/J. (4.2)
0 F 12 A3 R
To find spatially localized time periodic solutions for small deviations
from the dimerized lattice define^112
A(x,t) = Aq[1 + 6(x.t)] (4.3)
where
61
r *i
(x,t)=e[A(X,T)exp icot+A (X,T)expiuitJ +
:251 (x,t)+e262(x,t). (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)
5(x,t)=2e sech(x/d)cos
w t+j e secii (x/d)[cos 2u t3] (4.5)
D J D
where
d = E/2/2 e
(4.6a)
and
u^b = (1 3 e2) V
(4.6b)
It is easy to show^1*2 that the effective energy density is
E =
2An r2 ,2 .3 4
EL j_^ i2 Â§_ 2_ 0_]
itv L 2 24 6 2 6 24
F 2w
R
(4.7)
and, thus, that the classical breather energy is
2/2 A,
E =
B tr
r, 10 2 ,~4.i
M 27 e + 0(e
(4.8)
62
Breather Dynamics and Optical Absorption
The approximate analytic form for the breather (eq. 4.5) was used
as an initial condition and allowed to evolve under the adiabatic SSH
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)) a^e 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 Aq 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
63
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, though 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 intragap level varies with time and
thus, in a quasiclassical sense, an average in cq 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 ctp 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 range 0.55 1.05 Ao, over
(1)
which cip is approximately independent of the location of the intragap
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 is weak at
all relevant energies, with no significant contribution at the lowest
energies. For this breather a weak contribution at 1.5 Ao 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
64
narrow it bleaches over a wide range of energiesa narrow excitation
requires a large number of Fourier modes, including those at high
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.
Using 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
A 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 2A0. This strong
modification is due to the effect of the kinks on the continuum state
with which the breather interacts.)
65
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 average of the optical
absorption over the periods 0.02 0.04 psecs and 0.30 0.34 psecs after
photoexcitation. There are three important differences to note. First,
the "midgap 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 splitting 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 midgap. Third,
and perhaps most suprisingly, the intensity of transitions between
levels which will eventually evolve to the breather is large at all
times. Thus, the electronic properties of the breather fully evolve
very rapidly (<0.04 psecs), although 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.)
66
Photoexcitation in the Presence of Intrinsic Gap States
To this point only photoexcitation across the full band gap has
ll oc
been considered. However ESR and electrical conductivity^ 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 /it 0.9 A Photoexcitation adds energy A to the system,
o o o
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
Aq//2 from midgap. Thus the photoexcited polaron has energy 2.3 AQ
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 AQ. However, a simple calculation shows that
the maximum energy of the classical approximate analytic breather
67
is 0.63 Aq. Thus this breather is not well described by eq. 4.5,
reflecting the need for higher order terms in e expansion for the
4
breather (c.f. results for the breather in, e.g. <Â¡> equation).
Since in this case three levels oscillate into the gap, 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 arising from transitions between
o
the breather and continuum levels. The intragap bleaching arise from
the evolution of the two polaron levels to midgap, 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 high 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,
1 4
10 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 emisssion of acoustic phonons,
_ Q
which typically have a time constant of 10 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,
68
breathers and excitons. The choice of channel will depend on, e.g.
correlation and impurity effects.
A fully quantum 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.
Baeriswyl 3 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. Figure 4.9 shows
that when the electron and hole are excited the system tries to
equilibriate by changing 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.
69
In both cases there are contributions to the structure factor at oj ~ 0,
as well as responses associated with the internal vibratory modes.
Here, as has been done previously for the sineGordon system,'*1* 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
S(q,oj)
) dx]dx2dt1 dt ,exp[ ico(11 t ,)iq(x x,) ]
(2 it )
< exp iqu(x^t^) expiqu(x^t?)>
(4.9)
where u 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
S(q,oi)
Z'A nB(B;T)
r 2 r 2 2
JdvP(v)q [Jdx dt exp iY (q+ir/ava)/cn)
(4.10)
2 __
exp iY"(wqv)t. [u(x11^)] ][ x x2>t +t ]
2W
where e is the DebyeWaller factor, cQ is the velocity scale and
u(x,t)=na+(1)n (x,t). For simplicity we use the MaxwellBoltzmann
R
distribution, P1 (v). After a little algebra one finds
70
S(q,>e2 [2(f'(k*,2?(!77?)f1(k)2?aR(^2)
(2tt )
(4.11)
% {f2(k;)2pNR(^)Hf2(^)2PNR(^)36f2(kn)PNR(a)/q)}]
0'
where
_i (jjno)D
k = d [q + Tr/a + ] (4.12a)
c 4
o
kQ = d 1 [q + ir/a idVc^q] (4.12b)
and
1(q) = ird sech (nqd/2) (4.13a)
f2(q) = irqd^ cosech (iTqd/2) (4.13b)
For simplicity we look at the behavior of S(q,w) in the T=0 limit,
i.e. when PNR (v) > 5(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,u)=0)/S( q,u)=2w ) = 36.
D
(4.14)
This zero frequency inelastic scattering can be observed separately from
71
the elastic scattering of the phonons. Further contributions at w=0 are
expected from other nonlinear excitations. These, however, may have
differing temperature and Intensity dependences.
Also
S (q, uj=a)g) /S (q ,w=2tg)
9a 1
2Â£2 (1q/ir)
tanh^[3*~~( 1 ~q/a) ]
e
(4.15)
For nearly all values of q (except q it) this gives almost
independently of e
s(q,iij = oi )/s(q = tt to = 2u)_) ~ 9 x 10 ^ (4.16)
D D
At the zone boundary q=ir one finds
S(q = t m = u )/S(q = a, cj = 2ui ) = ^ 38 (4.17)
B B 2 t/i
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.^ 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
72
the classical extended states and the intragap polaron levels are
essentially time independent and thus the locations of the quantized
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 energy of the breather. This can easily be quantized at the Bohr
Sommerfeld level by demanding that
(4.18)
where J is the action and p^ is the conjugate momentum to the order
parameter, A. Thus
2nir
2
TrV^
j p dx dt
(4.19)
which gives
2nTr
[1
(4.20)
Inverting and substituting into eq. 4.8 for the breather energy gives
2 2 2
r n V
E = nw 1 iL. + 0(ii ) (4.21)
724 2
o
Equation (4.21) shows explicitly that a breather of quantum number n is
energetically favored over n incoherent phonons.
73
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 Lagrangian41 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 formalism4 4 or by direct numerical
quantization of equation 4.19. Even if this can be achieved there
remains the conceptual problem of quantizing the electronic spectrum,
from a knowledge of the breather energy spectrum.
At present we can only say that on quantization, the location of
the breather electronic levels and thence the energy 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 .
Comparison between Theory and Experiment
In this chapter a simple scenario has been proposed 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 lOK.4'''^ 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.
74
Clearly we mast identify the 0.45eV line with the kink "midgap"
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 daring photoexcitation ,4^ ^ ^ In
experiments, in which the temperature, laser intensity and chopping
C 1
frequency 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, 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 extended^
and it has been shown that there are additional contributions to the
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 of the kind of dopant and thus must arise from intrinsic
excitations of the system. Further, on doping a midgap absorption peak
75
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
h
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.
The 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
zhczf.
decaying 0.1 psecs after photoexcitation. J This short timescale for
production and the estimate of the quantum production efficiency 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
57
with increasing temperature and entirely disappears at 150K. 1
Second, at approximately the same temperature there is a sharp increase
in the photocurrent Third, the energy of the peak decreases by
approximately 3$ on deuteration.^9 Fourth, the excitation responsible
Q
for the !.35eV line is dipole forbidden from the groundstate.
76
Let us examine each of these in turn. First, the breather is a
lattice excitation and thus one expects it to be strongly temperature
dependent. Indeed the temperature at which the breather becomes
unstable can be estimated by a simple argument. A typical breather may
have energy ~0.5 Aq (=0.35eV) and be localized over ~2Â£=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 disappearence 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) ] 1 ,/2 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 A ; (c) we have assumed a breather
o
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.
Figure 4.1
Single electron photoexcitation for transpolyacetylene with
rescaled parameters. The time (t) is in fsecs, n is the site number
and rn is the smoothed order parameter (rn=1 = 0.1A). Lines are drawn
at alternate lattice sites and at 15 fsec intervals. Note the initial
transient in the kink velocity, followed by an approximately constant velocity
regime. The breather is evident between the two kinks.
79
O
CO
O)
\
W
Figure 4.2
Breather Dynamics
(a) Evolution of the numerical breather created in the photoexcitation.
The period is approximately 40 fsecs. (The long time oscillations in the
dimerization result from the removal of the breather from a 98 to 42 site ring.)
81
O
o
CO
\
w
\
(M
Breather Dynamics
(b) As figure H.2a but for analytic breather with e = 0.75
(chosen to match the frequency of the numerical breather).
83
\
Figure M.3
The location of the lower intragap impurity level for the
numerical breather (solid line) and the analytic breather (dashed line).
85
Figure 4.4
Average optical absorption, a(io), over one phonon period and
over a 98 length chain for the numerical breather. For the
dimerized lattice see fig 3.4.
Aa(cu)
86
E/A
O
Figure H.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.
87
Figure 4.6
Change in optical absorption, Aa(ai), on photoexcitation for
transpolyacety1ene for (a) t~21~43 fsecs (aolid line)
(b) t=301~343 fsecs (dashed line)
6.0
Figure 4.7
Photoexcitation of electron pelaron. Note the strong breather
excitation between the kinks.
(m)DV
90
E/A0
Figure 4.8
The change, Aa(io), 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 edge.
(b) Location, with respect to the Fermi level, of electron and
hole (solid line). Level occupied by photoexcited electron
and hole (dashed line), where the band edge state are levels 49 and 50.
o.o a.0 lo oa.0 bo.o ioo.o ix.o io o ico.o 1*0.0
TIME
level r i
M , , r
f>>rrfPPrrP
^1 O IP O M O (T O l/l O
ENERGY
I GTa
vO
to
ir

Full Text 
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FILES
ADIABATIC NONLINEAR DYNAMICS IN MODELS OF
QUASIONEDIMENSIONAL CONJUGATED 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
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/adiabaticnonlineOOphil
ACKNOWLEDGMENTS
It is with great pleasure that I thank my mentor and friend,
Pradeep Kumar, who has provided both inspiration and guidance throughout
my graduate career. I also owe a great debt to Alan Bishop, who
suggested 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 benefited 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 suggestions 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
Tâ€”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
gratitude.
Although 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.
iii
TABLE OF CONTENTS
ACKNOWLEDGMENTS iii
ABSTRACT vi
CHAPTER
I INTRODUCTION 1
II BASIC THEORY 5
TransPolyacetylene 5
The SSH Model 7
The TLM Model 10
Limitations of the SSH and TLM models 15
III DYNAMICS 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 PHOTOEXCITATION 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
IV
125
KinkSite Impurity Interactions
Photoexcitation in the Presence
of a Single Site Impurity 128
Photoexcitation with Many Site Impurities 1 31
The Single Bond Impurity 132
KinkBond Impurity Interactions 134
Photoexcitation in the Presence
of a Single Bond Impurity 1 34
VII STATICS AND DYNAMICS IN POLYYNE 163
Formalism 163
The Continuum Model 167
The GrossNeveu Model 169
Nonlinear Excitations in Polyyne 171
Adiabatic Nonlinear Dynamics in Polyyne 173
VIII CONCLUSIONS 185
APPENDIX
A PARAMETERS OF THE SSH AND TLM MODELS 189
B OPTICAL ABSORPTION IN THE SSH MODEL 191
C THE ANALYTIC BREATHER 199
D CLASSICAL DYNAMIC STRUCTURE FACTOR OF THE BREATHER 204
E CONTINUED FRACTION SCHEME FOR IMPURITY LEVELS 208
BIBLIOGRAPHY 21 4
BIOGRAPHICAL SKETCH 219
v
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
QUASIONEDIMENSIONAL CONJUGATED 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 Heeger (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
vx
photoexcitation dynamics with those of transpolyacetylene. We further
investigate the role of coherent anharmonicity in simple models of a
finite polyene and of polyyne. As the observed transport properties of
polyacetylene depend 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 trapped
kinks may be produced.
CHAPTER I
INTRODUCTION
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 good insulator to that of a fair metal, has
not suprisingly, produced a great deal of interest. Most of the ensuing
O
activity has been in four broad areas: technological applications,
synthesis, physical testing and characterization, and theoretical
modeling.
p
Amongst the suggested technological applications for polyacetylene
and related materials are Schottky diodes, solar cells, lightweight
metals, and high power storage batteries.
Ito et al.3 first synthesized selfsupporting films of
polyacetylene by the direct polymerization of acetylene gas in the
presence of catalysts. This method is essentially uncontrolled and
produces samples of widely varying structure and physical properties.
p
Since then much effort 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 conjugated polyaromatic
heterocyclic polymers and some conjugated block copolymers can be
1
synthesized using conventional organic techniques. Their conductivities
are only, however, at most 1â€”10% that of polyacetylene.
The full barrage of standard solid state spectroscopies has been
p
used to characterize polyacetylene: electron and Xray diffraction
have been used to study its structure; Raman scattering to determine its
conjugation length; 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 responsible for the widely differing experimental data and
interpretation. However one particularly significant conclusion has
emerged: ESR experiments have shown that the charge carriers in
polyacetylene are spinless and therefore are not electrons.
The theoretical work has been in three general areas. Ab initio
calculations,0 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
/TO
excited system. Phenomenological Hamiltonians0 0 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)^'1 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
1 O 10
continuum limit â€™ J proved to be analytically tractable and their
3
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
O
these is separately difficult to include and studies continue.
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
4
finite response time of the lattice. In Chapter IV a simple
photoexcitation experiment in transpolyacetylene is modelen. 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 transport properties discussed.
Chapter VII introduces the polyynes and suggests that nonlinear
excitations may be important here, also. Chapter VIII presents our
conclusions.
CHAPTER II
BASIC THEORY
TransPoiyacetylene
Polyacetylene is a linear chain polymer consisting of a spine of
carbon atoms with a single hydrogen atom bonded to each.liJ1^ It exists
in two isomeric forms: cispolyacetylene and the energetically more
stable transpolyacetylene. The present discussion will concentrate on
transpolyacetyene. (A full discussion of cispolyacetylene will be
undertaken in Chapter V.) Of the four carbon valence electrons, three
are in a bonds to a hydrogen atom and to the two neighboring carbon
atoms. The remaining electron is in the it band, which is half filled.
One thus expects transpolyacetylene to be a conductor. However,
undoped transpolyacetylene is a semiconductor with a bandgap
2Aq ~ 1.41 .6 eV. This gap is, at least in part, due to the Peierls
effect1"^: 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 = ir/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
6
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 single
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
energy 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 2E,  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. There 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 midgap 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
7
explanation for the simultaneous observation of high electrical
conductivity1 and low Curie susceptibility11: the charge carriers in
doped polyacetylene are charged, zero spin, kinks.
The SSH Model
The SSH Hamiltonian
Su, Schrieffer and Heeger^ (SSH) have proposed a simple Hamiltonian
for transpolyacetylene based on a number of reasonable assumptions:
1. All manybody effects can be incorporated in a single particle
Hamiltonian.
2. The effects of o electrons can be accounted for in the chain
cohesion and lattice dynamics. Xray1Â® and NMR1^ data show that
a  1.22A and Uq  0.03Â± 0.01 A (a is the lattice constant and Uq
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 it 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 "realisticâ€ polyacetylene parameters this ratio
on
is  0.05. Brazovskn and Dzyaloshinskii have shown rigorously
that the adiabatic approximation requires y << 1 ,
where y = Kwq/Ao. For transpolyacetylene y  0.1.
8
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
i Ã±
neglected. Experiments Â° suggest that the interchain overlap
is ~ 100 times less than the intrachain overlap in well oriented
samples. The effects of interchain coupling and crosslinking may,
however, be important in many materials.
The SSH Hamiltonian is
, M *2 K
^SSH 2 + 2 Z(an Jn1 }
n
z[t +ct(u u . )1[C,â€˜C +C+ ,C
o n n1 JL n n1 n1 n
n
(2.1)
where M is the mass of a (CH) unit, K is the force constant, tp is the
intrinsic electron hopping matrix element, a is the electronphonon
coupling constant. C+(C ) creates (annihilates) an electron on site 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 o electrons); the third is the it electronic
energy, consisting of intrinsic and phononassisted contributions.
Q
Dimerized Groundstate7
The SSH Hamiltonian has two degenerate uniform solutions
= Â±(1)n
(2.2)
9
Now defining a "staggered order parameter",
u = (1)n u /u
n no
(2.3)
then
= Â±1 (2.4)
It is interesting to note that with two degenerate groundstates and with
a local energy maximum at u=0 the SSH model is topologically identical
4
to the $ model. This similarity will be exploited at various times,
4
but it must be remembered that 4> 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 coupling limit that the band
gap, 2Aq, is given by
2A = 8au . (2.5)
o o
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
4
variational kink profile, A(x) = AQtanh(x/^) (the exact kink
solution), the kink half width, E_, is  7~10a. This suggests that a
continuum approximation to the SSH Hamiltonian may provide a useful
basis for further analytic studies.
10
The TLM Model
The TLM Hamiltonian
The continuum approximation is valid if the electronphonon
2
coupling is weak: X <<1 , where X = 2a /irt^k. For SSH parameters
X  0.2. The fermion operators C* and C can be considered as the sum
n n
ip i j
of left and rightgoing waves. â€™ J
1
C = â€” [u (n) e F
ns / s
/a
ik^na . . ik^nai
p  lv (n) e F I.
(2.6)
For the halffilled band the system dimerizes to give a lattice period
2a. Thus we define
u =
Ma
[A+(n)
2ikâ€žna
+ A(n) e
2ik^na
]â€¢
(2.7)
To order a, and dropping the rapidly varying terms of the type (~l)n the
SSH Hamiltonian becomes the classical TLM (Takayama, LinLiu, Maki)
Hamiltonian.
a)2
HTLM = ~~2 ÃdxA'â€™^x) + Jdx 'i'+(x) ["iVpO, + A (x)o 1 ]t(x) (2.8)
2g J
where
T(x)
( u ( X )
V ( X )
)
2
WQ
if. s = Mg)1/2,
2t a
o
and o^ is the ith Pauli matrix.
(2.9)
11
Varying eq.
# Â£
2.8 with respect to u (x) and v'(x)
gives
i v
F
3 a (x)
3x
+ A(x)v(x)
e u(x)
v
(2.10)
+ i v *â–  + A ( X ) U ( X ) = Â£ V ( X ) .
F dX 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 tt band an artificial energy
cutoff W is introduced to simulate the band edges. This is expected
only to alter the energy scale.
+
Varying eq. 2.8 with respect to A gives the consistency equation
A(x) = g2/w~ Z' [u (x) v(x) + v (x) u(x)] (2.11)
V , s
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(x) (2.12)
equations 2.10 become
3f_
e f (x) = iv (x) Â± iA(x) f_(x)
v Â± F 3x +
and then
(2.13)
2 3
3x
2
e Â±
v
3A(x)
3x
 A (x)] f (x) = 0.
(2.14)
12
Thus the problem has reduced to the solution of a "SchrOdinger1ike"
equation, with the potential being a function of the order
parameter A(x). Of course the consistency conditions (2.11) must also
be obeyed. Solutions are particularly easy to obtain for the class of
potentials with
Two particular solutions that fall into this class are the purely
dimerized lattice and the kink solution.
21
The Dimerized Lattice
For the purely dimerized lattice: A(x) = Aq one gets
(2.16)
This has solutions
u (x) = N. e
n k
o
v (x) = N, e
n k
(2.17)
with
03  A^ 1
oi 1 / Â¿
i.\ *
GO
These plane wave eigenfunctions have the energy dispersion relation
13
2 2 2 2
a) = A + v k (2.18)
O r
where the consistency equation is satisfied by
Aq = W e~'/2A. (2.19)
Here one sees that W merely sets the scale of energies.
The Kink Soliton^^^1
There is also a kink solution (fig. 2.3a) with order parameter
A(x) = Â±AQtanh(x/5) (2.20)
where F = v_/A . (The minus sign gives the antikink solution.) This
F o
has both planewave solutions and a single localized level
at e =0 with wavefunctions
o
and
u (x) = NQsech(x/5)
vq(x) = iNQsech(x/5)
A /lJ
o
(2.21)
This electronic level is localized on the kink with the same coherence
length as the lattice distortion.
14
The kink creation energy
15,22
can also be calculated as
E, = 2A /tt .
k o
(2.22)
Although the TLM model is neither Galilean nor Lorentzian invariant, the
kink mass can be calculated in the ansatz x(t)=xVt by equating the mass
O
to the coefficient of the (1/2)V term. This gives the mass of the
kink, Mk (where M is the mass of a (CH) unit)
(2.23)
for SSH parameters
M,  3~6m
k e
(2.24)
where mg is the electron mass.
Thus the kink is very light and kink dynamics can be expected to be
important.
The Polaron21
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) = A  k v_ [tanh k (x+x )  tanh k (xx )1
o o F L o o o o J
15
where to2 = k2v2 + A (2.25''
o o F o
and tanh k x = (A  tu )/k vâ€ž.
o o o o o F
This selfconsistency condition is, however, only satisfied in transÂ¬
polyacetylene if
k v = A //2. (2.26)
o F o
This lattice deformation is supported by a pair of intragap levels
symmetrically about the Fermi level, which in transpoiyacetylene are
at u = Â±A //2. For an electron polaron the lower level is doubly
o o
occupied and the upper is singly occupied. For the hole polaron the
lower level is singly occupied and the upper is empty.
The polaron creation energy is
Ek = 2/2 Aq/tt (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
o
(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.
16
ElectronElectron Interactions
p ii
Experiments show the first excited state of the finite polyenes
((CH)n, n=2,3>...) has 1Ag 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
analogy, 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
pc
energies of the infrared absorption peaks are correctly predicted.
A number of studies on electronelectron effects in a Peierls
distorted phase have been undertaken with conflicting concluÂ¬
sions >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.
Although 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 "fermion quasiÂ¬
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
17
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.1â€™ Here two parameterizations are chosen. "SSH
parameters"^ are chosen to reproduce three experimental observables of
transpolyacetylene: the bandgap (2 ~ 1.4eV), the dimerization
(uQ  0.04A) and the full bandwidth (W  10eV). The spring constant
2
(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
Â£ ~ 7a. For many of the simulations performed the parameterization W =
10eV, 2Aq = 3.8eV, uQ = 0.1A, Â£ = 2.7a will be used. These "rescaled
parameters"2^ 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
18
transition rate between them. Explicit calculations show that the
dimerization is reduced by  Quantum fluctuations also
reduce the kink creation energy by  25%.One can consider these
effects as being included in the renormalized SSH 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 Â° show that there is three dimensional ordering in transÂ¬
polyacetylene with the antiparallel alignment of dimerization patterns
on neighboring chains. A kink introduces a defect^ in this alignment
with an energy that increases linearly with the separation of kinks.
Thus one expects a KK pair to be confined (though 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  TOO lattice sites. (As will be seen in Chapter V this
confinement is similar to the intrinsic confinement in cis
polyacetylene.) Interchain hopping^ of charge between kinks has also
been suggested as the dominant conduction mechanism for electrical
conductivity at low dopant concentrations.
19
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 energy into nonlinear phonon
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 magnitude by
extrinsic doping.^ 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 invariantâ€”there are
two velocity scales, the Fermi velocity and the sound velocity).
20
All of the above deficiences of the basic model need to be
addressed. A first logical step is to study the dynamics of the basic
3SH model. This is the subject of Chapter III.
The two degenerate phases of transpolyacetylene.
CNJ
CN
o_ xi
vu >
ox
X~O
1
X
Oâ€”X
xo
X
ox
xo
X
ox
xo
X
OX
xâ€”o
V
X
ox
xâ€”o
X
oâ€”X
xâ€”o
X
Oâ€”X
X â€”o
X
ox
OX
o
<1
I
E/2
c
I
H
H
I
C
H
I
C
c
I
H
('Ao)
H
I
C
H
i
C
H,
C
c
I
H
GROUND STATE
H
I
C
c
I
H
H
l
C
j
c
I
H
C
I
H
KINK SOLI TON
H
i
C
\ / \ / K./ Xâ– / H/ \ / \ / \
C
I
H
C
I
H
CA0)
C
I
H
GROUND STATE
NJ
UJ
Figure 2.2
Kink interpolating between the two phases of transpolyacetylene.
To the left of the kink the double bonds slope upwards; to the right of
the kink the double bonds slope downward.
Figure 2.3
Order parameter (solid line) and wavefunction (dashed line) and
electronic spectrum of (a) the kink (b) the polaron. Note the reversed
spincharge relations for the kinks.
(b)
y/e
o
2
2
Wi
VM
YM
\
Q = + e
Q = 0
Q = e
S = 0
S = 1/2
S = 0
W/A
â€œtâ€œ w0
^ '"o
&\X\â€œ
Q = + e
Q=e
S= 1/2
S = 1/2
Ni
Ln
CHAPTER III
DYNAMICS OF A SINGLE KINK IN THE SSH AND TLM MODELS
In this chapter a systematic study of the dynamics of a single kink
in both the TLM and SSH models will be pursued. 3efore 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.
pq
The Numerical Technique 7
The energy of the SSH Hamiltonian can be formally written as
HSSH
â€¢2
u
n
+
v({unn
(3.1)
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
V = !Z (un~ V/
n
v, s
m e
v,s v ,s
(3.2)
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
26
27
direct diagonalization of the hopping matrix T, which has components
{t + a [ u.
1 o L 1
u.
J
+ 6 .
l
(3.3)
This has eigenvalues
e given by
v
(T  E I) = 0
where
(3.4)
Newton's equations of motion are
MU
n
5V({un})
6un
(3.5)
The time derivative can be evaluated by finite differences as
u
n
(m) = ^ [un(rn)  un(m1)]
(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)
n n n
(dt)2
M
({un(m1)}).
(3.7)
The functional derivative of the potential can also be calculated by
finite differences as
28
6V({un})
5a " ^V(U1 u2â€™â€¢Un+5U" *UN) â€™/(u1u2â€
*unâ€ â€˜ *UN ; /6u
(3.3)
n
where
Su << a .
o
(3.9)
Energy conservation is used as a check on the stability of the
algorithm. In general energy is conserved to better than 99.99%.
At some points in this study spatially homogeneous velocity
dependent damping is applied by the addition of the term
to the eqaations of motion. This gives evolation eqaations
a
n
Also in some stadies instantaneous damping is applied by setting
the velocity to zero at every timestep, i.e.,
u (m1) = a (m2).
n n
(3.12)
This modifies the equations of motion, eq. 37, to
(3.13)
29
Boundary Conditions
Two types of boundary conditions are used in this numerical
study. ^
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
(3.14)
u
u.
N+n
n
and
(315)
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 Â±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
(1)
n+N
u
u
n+N*
n
30
There is also a more subtle distinction between systems of length
4n and 4n+2. For an undimerized system (u = 0) the eigenstates of the
SSH Hamiltonian are Bloch states with energy
E = 2t cos k a (317)
non
where the wavevector,kn, satisfies the quantization condition
kn = 2;rn/N ^ < n S ^ (3.18)
Thus for N=4n and a half filled band the Fermi level lies at E=0. 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 (tt/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, X.)
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
31
lengths is small. Here two different types of chain boundary 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"
4a , v
â€” (u,  u.J.
it 1 N
2. In "fixed boundary conditions" each of the end sites is bound
to its nearest neighbor by the constraint
U1 u2â€™
un jnt
(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=0 lattice configuration and
minimize the energy of the system by varying the dimerization uQ 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).
32
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 staggered lattice distortion as a "sawÂ¬
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)n Ã¼n + n.Ã³a (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
def ine^
r
n
= i M)
[2l
n+
â€ž  u .1
1 n1J
[ 2u
L r
n+1
+ u
n1
(3.21)
(3.22)
Then rn _ Un and sn  n.$a. In these variables the kink profile is more
easily seen (fig 3.1b). Note that the dip in sn corresponds to a
33
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*3Â° The
optical absorption is defined within the linear response dipole
approximation as
A
o(u) = tt f â€” A E r 5(E,  E.  ai) (3.23)
Lio 1 o ' 2' 3 I 2 1
where is the current operator, the Â¿function ensures the
conservation of energy and A is an unimportant constant. Figure 32
shows the available electronic transitions for the dimerized lattice,
the kink and the polaron. Using the analytic form of the wavefunutions
(eq. 2.17) the optical absorption of the purely dimerized lattice can
easily be calculated as
aD(u) = A(2Ao/w)U2 ~ 4A2)~/Â¿. (3.24)
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."^ *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
34
phase.') This alters the phase relations in the valence and conduction
bands, causing diagonal transitions (Ak=0) to be explicitly forbidden.
For the midgap transition
0, . IT
a. (ai) = j
ft
2',1/2
sechâ€˜â€˜[
2A
U2 *
,/2l
(3.25)
whilst for the interband absorption
a*3(Â«) = otD(ai) [l ~ 2^/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.
21
The polaron absorption has similarly been calculated and consists
of three contributions plus the interband. Transitions between the two
intragap polaron levels give a'(to). There is a second contribution,
P
2
aâ€œ(to), 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 (to). The relative intensities of each of these is shown
p
in fig. 33 as a function of the location of the intragap polaron
level. (For the polaron in transpolyacetylene Ã¼>o = 1//2. For cis
polyacetylene see Chapter V.) For most gap locations cJ (
P
2
stronger than a (to) even though it involves only one transition,
P
35
3
whilst cxd(io) involves many transitions. This suprising 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.^ A review is given here (full details are
given in Appendix C). Starting from the definition of the charge
density, p(n) = eC^C^, 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[t + ctfu  u J1[C+C  C+ ,C 1. (327)
J L o n n+1;jL n n+1 n+1 nJ
Then in the linear response dipole approximation
a(w) = Tre^tjl 1 E M. J2Ã³(e,  e  u)
o wN 1 1 ,21 2 1
' tÂ£
(3.28)
where
,2
= 2 [1
T (u 
t n
o
n+1
)][f1(n)f*(n+1)  f^(n+1)f*(n)] (3.29)
and f is the wavefunction of the a level,
a
He further showed that a(w) obeys a sum rule
36
J a(u)>du)
1 2 .
2Ã‘ 116
(3.30)
where H is the electronic energy of the configuration. This algorithm
for the absorption is easily implemented. It is important to note that
all these results remain unchanged when electronelectron effects are
added at the Hubbard level, i.e. H = E V p(n) p(m) is added to the
ee nm
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  Aq/10). The sum rule is obeyed to better than 99.92?.
Figure 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(co), 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 (
P P o
â– 3
and  A /4 respectively. The weak contrbution, a (oj), is present but
o p
only  2? of that of . Figure 3.8 shows Aa(tu) for the polaron.
Note that the polaron removes intensity over a wide energy range above
the band edge. This is because, for these parameters (Â£  2.7), the
polaron is is narrow and requires a large number of Fourier modes for
its description.
37
These 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 dynamical 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 changes in its dimerization, the kink has a maximum free
propagation velocity as can be seen from the following simple
argument.37>"9,40 por a ^ink Qf velocity v the lattice changes from the
A to B phase (say) over a distance Â£(v) and in time Â£(v)/v. The
minimum response time of the lattice is the inverse of the renormalized
1/2
phonon frequency, a>_, where wn = (2A) w (w being the bare phonon
frequency). This gives the maximum kink propagation velocity (vm)
as v  oj s(v ) . Assuming that the kink width is not strongly velocity
m K m
dependent, then
vm/vs ~ (2A)I/2 U/a> (3.3D
For SSH parameters (Â£  7a, A ~ 0.2) this gives (vm/vs)2  19â€”the
maximum velocity is greater than the sound velocity.
The conclusion of the above argument can be confirmed from an
approximate analytic calculation. Using the ansatz for the kink profile
38
A(x,t) = Aq tanh (x  vt)/5(v)
(3.32)
it is straightforward to show that the lattice kinetic energy of the
kink (T) is
M r â€¢ 2
T =  j A (x)dx = AqY
(v/v )'
s
(333)
where
r = 5(v)/5(0)
(3.34)
and
Y = A2/24ir\t2.
o o
(3.35)
It has been shown numerically^ 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
E
D
BAq(1  r)2
(3.36)
where numerically we find
6(5 = 7) ~ 0.20 (337)
6(5 = 2.7) ~ 0.10.
Then the Langragian of a kink of finite velocity is
39
L = Aq{ 8 (1  r)â€œ  YqVr} .
(3.33)
Thence the momentum (pq) conjugate to the displacement q (q = v/v ) is
2Yq
p = â€”
q r
(339)
and the momentum, p , conjugate to the width r is
pr = 0.
(3.^0)
Now extremizing the Lagrangian with respect to r and q gives
?Br (1  r)
(3.41)
q _ qr
r r
0.
(3.42)
Then for consistency
r = 0, 5} = 0. (3.43)
Thus from eq. 3.41 the velocity has an extremum at r=2/3 independent of
the parameters Y and 8.
Substituting back for r the kink has a maximum propagation velocity
(vm) given by
(v /v =
v m s'
(â€”)
v27
B/Y
(3.44)
40
which gives the momentum at the maximum kink velocity as
2 8
Pm = 3 YB
(3.45)
Thus the energy at maximum kink velocity is
Ek = 9 SAo*
(3.46)
The kink energy can also be written
E (p) = Aq [p2/4Y  p4/64Y23]
(3.47)
This gives a maximum in the energy at
pm  8T6'
(3.18)
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
m*
d2E
dp2
(3.49)
gives
1 3 2
= *  T6 P /Y8
m*
(3.50)
41
Thus in the low momentum limit the kink behaves like a Newtonian
particle of mass 2 a. However, the effective mass diverges at the
critical value of the momentum
2
P
c
(3.51 )
This is the same as the kink momentum for free propagation at the
maximum velocity (eq. 3.^5). Using appropriate value for 6 and y one
gets for SSH parameters
(v /vJ 11,
m s
 5.6 m.
E  0.1 A
m o
(3.52)
and for rescaled parameters
(v /v )  1.2
m s
80 m
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 Model^
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
42
conditions and for chains of differing lengths.) In fig 3.9 the short
time kink velocity squared is plotted as a function of the energy
input. Both axes are in dimensionless units, where unity corresponds to
2
the theoretical value of v and the theoretically calculated energy, Em,
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 single curve. Indeed by
treating g and y as free parameters and force fitting, better agreement
may be obtained. At low energy inputs the data are in good agreement
with the continuum theory. The maximum kink velocity is also well
predicted by the continuum calculation. However the continuum theory
strongly underestimates the energy 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 â€”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/Â£) whilst acoustic phonons couple to order (a/Â£) . Thus for
rescaled parameters phonon effects should be much stronger than for the
SSH parameters. Figure 39 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
43
the phonon effects reduce the maximum kink velocity by a factor of 2,
for E,~ 7, but by a factor of 10 for the more discrete kink
with c,~ 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
m
and E2.0 E^ respectively. At low energy inputs the kink propagates
essentially uniformly, whilst at higher energy inputs kinetic energy is
lost by the emission of an optical phonon package.^ This may be viewed
as the classical analog to the emission of Cerenkov radiation by a
particle traveling faster than its maximum propagation velocity in the
medium.
Although, as we shall see in Chapter VI cnarge conduction by the
ballistic transport of kinks is unlikely, kink propagation is essential
to the understanding of the short time dynamics of the photoexcited
system.
44
Un
Figure 3*1
The Single Kink
(a) unsmoothed order parameter, u . for single kink.
(b) Smoothed order parameter, r , for the same single kink and sn
(dashed line).
Figure 3.2
Electronic transitions of (a) dimerized groundstate (b)
the single kink (c) the polaron. The valence bond is v.b. and is full.
The empty conduction bond is c.b. The dotted line in (c) gives the Fermi level.
46
o
47
Figure 3*3
Relative strength of transitions in the polaron as a function
of the location, too/Ao, of the intragap levels. For
transpolyacetylene w /A  0.7.
00
a (aÂ»
48
E/A0
Figure 3Â»1)
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 = 4tQ = 10 eV. The
5functions 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.
49
Figure 35
Optical absorption, a(u), of two widely spaced kinks on a 98 site ring.
Aa(oi)
50
E/A0
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).
51
Figure 37
Optical absorption, a(w), of a polaron on a 98 site ring.
52
3
a
E/A0
Figure 3.8
Difference in absorption of a single polaron
and the purely dimerized lattice of a 98 site ring.
(Figure 3.8 = Figure 3.7  Figure 34)
Figure 3.9
Output velocity squared versus input energy for 5 = 7a (triangles) and
5 = 2.7a (circles) for period < 1 phonon period (open markers) and ~ 48
phonon periods (solid markers). The solid line comes from the continuum theory.
54
>
Figure 3.10
Single
Kink Dynamics.
Note
(a) Time evolution of a single kink of input energy E  0.3 Em.
the smooth propogation of the kink. This evolution is essentially
independent of the size of the system.
56
O
Figure 310
Single Kink Dynamics
(b) Time evolution of single kink of input energy E  2.0 Em.
Note the production of an optical phonon tail to the breather.
58
CHAPTER IV
PHOTOEXCITATION IN TRANSPOLYACETYLENE
We begin this study of photoexcitation in transpolyacetylene with
a simple experiment:^ 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 "rescaled parameters"
will be used as they allow smaller systems to be studied, thus reducing
the computation time. (The dynamics are expected to be similar for the
more realistic SSH parameters.) At t=0 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 2Aq 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, vm, (fig. *1.1 ). Each kink has creation energy 2AQ/Tr and
kinetic energy <0.1 Aq. Thus the two kinks have a total energy
of <1.5 Aq. Figure 4.1 shows that a substantial fraction of the
remaining energy from the photoexcitation is localized as a nonlinear
lattice excitationâ€”an amplitude breather. The breather dynamics were
59
60
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 may be
obtained from the known effective Lagrangian^ for the halffilled band
'2 *2
L = â€” {1/2 A2[Â£n(2E /A) + 1/2]  AV4X  v2 A_ + A_ } (4.1)
v_ c F . 2 â€ž 2
F 24A 2ujd
K
where EQ 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 A+0.) From eq.
4.1 the equation of motion is
T 2 It
A/24 â€¢ I In 2E /A  VR â€” ~ a,A '  b/J. (4.2)
0 F 12 A3 R
To find spatially localized time periodic solutions for small deviations
from the dimerized lattice define^â€™112
A(x,t) = AjÃ + 6 (x.t)] (4.3)
where
61
r * *i
Ã³(x,t)=e[A(X,T)exp icot+A (X ,T )expiuit J +
:251(x,t)+e262(x,t). (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)
5(x,t)=2e sech(x/d)cos
aj_t+4 e secii (x/d)[cos 2u t3] (4.5)
D _j O
where
d = E/2/2 e
(4.6a)
and
u^b = (1  3 e2) â€œR *
(4.6b)
It is easy to show^Â»1*2 that the effective energy density is
E =
2An ;2 cr2 O ,2 r 3 .4
. Â° \A + i_ 6.2 + Â§ + !L  L]
itv 2 24 Â° 2 6 24
F 2w
R
(4.7)
and, thus, that the classical breather energy is
2/2 A,
Eâ€ž =
B TT
fl 10 2
M ~ 27 e + 0(e )J*
(4.8)
62
Breather Dynamics and Optical Absorption
The approximate analytic form for the breather (eq. 4.5) was used
as an initial condition and allowed to evolve under the adiabatic SSH
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)) a^e 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 Aq 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
63
analogies with the exactly calculated (in the continuum limit) absorpÂ¬
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, though differing in their
occupancies. (The relative intensities of the polaron transitions are
shown in fig. 3.3) For the transpolyacetylene polaron m /A  0.7.
For the breather the location of the intragap level varies with time and
thus, in a quasiclassical sense, an average in cÃ¼q 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 , 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 range 0.55  1.05 Ao, over
(1)
which cip is approximately independent of the location of the intragap
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 Aq). There should be a weaker, nearly uniform
(3)
contribution over intermediate energies. The transition is weak at
all relevant energies, with no significant contribution at the lowest
energies. For this breather a weak contribution at 1.5 Ao 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
64
narrow it bleaches over a wide range of energiesâ€”a narrow excitation
requires a large number of Fourier modes, including those at high
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.
Using 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
A 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.)
65
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 average of the optical
absorption over the periods 0.02 0.04 psecs and 0.30  0.34 psecs after
photoexcitation. There are three important differences to note. First,
the "midgap levels" are moved away from midgap at short timesâ€”the KK
pair is still evolving and the intragap levels associated with them have
not yet fully evolved. Second, because of this splitting 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 midgap. Third,
and perhaps most suprisingly, the intensity of transitions between
levels which will eventually evolve to the breather is large at all
times. Thus, the electronic properties of the breather fully evolve
very rapidly (<0.04 psecs), although 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.)
66
Photoexcitation in the Presence of Intrinsic Gap States
To this point only photoexcitation across the full band gap has
ii oc
been considered. However ESR and electrical conductivity^ 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 /it  0.9 A . Photoexcitation adds energy A to the system,
o o o
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
Aq//2 from midgap. Thus the photoexcited polaron has energy 2.3 Aq
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 AQ. However, a simple calculation shows that
the maximum energy of the classical approximate analytic breather
67
is 0.63 Aq. Thus this breather is not well described by eq. 4.5,
reflecting the need for higher order terms in e expansion for the
4
breather (c.f. results for the breather in, e.g. <Â¡> equation).
Since in this case three levels oscillate into the gap, 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 arising from transitions between
o
the breather and continuum levels. The intragap bleaching arise from
the evolution of the two pelaron levels to midgap, 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 high 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,
1 4
10 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 emisssion of acoustic phonons,
_ Q
which typically have a time constant of 10 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,
68
breathers and excitons. The choice of channel will depend on, e.g.
correlation and impurity effects.
A fully quantum 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.
Baeriswyl 3 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. Figure 4.9 shows
that when the electron and hole are excited the system tries to
equilibriate by changing 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.
69
In both cases there are contributions to the structure factor at oj ~ 0,
as well as responses associated with the internal vibratory modes.
Here, as has been done previously for the sineGordon system,'*1* 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
S(q,u>) = â€”â€”â€”rp Jdx1dx2dt1dt2exp[iu(t1t2)iq(x x2)]
(2tt)
(4.9)
< exp iqu(Xjt.j) expiqu(x.,t0)>
where u 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
S(q,cÃº)
32W nB(Â¿Â°B;T)
(2tt)4
r 2 r 2 2
JdvP(v)q [Jdx dt exp iY (q+ir/avm/c^)
(4.10)
2 __
exp iY"(wqv)t. [u(x112) ] ][ x *x2>t *t ]
2W
where e is the DebyeWaller factor, c0 is the velocity scale and
u(x,t)=na+(1)n Ã¼(x,t). For simplicity we use the MaxwellBoltzmann
R
distribution, P1 (v). After a little algebra one finds
70
s(tl,â€ž)=e2Â«!B^[j2lf,(k.)2?NR(^)tf,(k)2pNR(^2)
(2tt )â€˜
(4.11)
â™¦ {f2(^)VJR(^)Hf2(kÂ¡)2PNR(^) + 36f2('Kn)PNR(a)/q)l]
0'
where
_i (jjÂ±no)D
k = d 1 [q + Tr/a + ] (4.12a)
c 4
o
kQ = d 1 [q + ir/a  idVc^q] (4.12b)
and
f1(q) = ird sech (nqd/2) (4.13a)
f2(q) = irqd^ cosech (iTqd/2) (4.13b)
For simplicity we look at the behavior of S(q,w) in the T=0 limit,
i.e. when PNR (v) > 5(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,a)=0)/S(q,u)=Â±2m ) = 36.
D
(4.14)
This zero frequency inelastic scattering can be observed separately from
71
the elastic scattering of the phonons. Further contributions at w=0 are
expected from other nonlinear excitations. These, however, may have
differing temperature and intensity dependences.
Also
S(q,a)=Â±(i)g)/S(q ,w=Â±2i0g)
9a 1
2Â£.2 (1q/ir)
tanh^[â€”3*~~( 1~q/a) ]
e
(4.15)
For nearly all values of q (except q  it) this gives almost
independently of e
s(q,m = Â±oi )/s(q = Â±tt , to = Â±2u)n) ~ 9 x 10 ^ (4.16)
D D
At the zone boundary q=Â±a one finds
S(q = Â±a, m = Â±w )/S(q = Â±a, a = Â±2ui ) = ' ^ ~ 38 (4.17)
B B 2 t/i
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.^ 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
72
the classical extended states and the intragap polaron levels are
essentially time independent and thus the locations of the quantized
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 energy of the breather. This can easily be quantized at the Bohr
Sommerfeld level by demanding that
(4.18)
where J is the action and p^ is the conjugate momentum to the order
parameter, A. Thus
2nir
2
TrV^
j â€”p dx dt
(4.19)
which gives
2nTr
[1
(4.20)
Inverting and substituting into eq. 4.8 for the breather energy gives
2 2 2
r n V Ãœ
E = nw 1  iL. + 0(ii ) (4.21)
â€ 724 2
o
Equation (4.21) shows explicitly that a breather of quantum number n is
energetically favored over n incoherent phonons.
73
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 Lagrangian41 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 formalism4â€™ â€™4Â° or by direct numerical
quantization of equation 4.19. Even if this can be achieved there
remains the conceptual problem of quantizing the electronic spectrum,
from a knowledge of the breather energy spectrum.
At present we can only say that on quantization, the location of
the breather electronic levels and thence the energy 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 4 .
Comparison between Theory and Experiment
In this chapter a simple scenario has been proposed 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 ICK.4^â€™4^ 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.
74
Clearly we mast identify the 0.45eV line with the kink "midgap"
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 daring photoexcitation ,4^ â€™ ^ ^ In
experiments, in which the temperature, laser intensity and chopping
C 1
frequency 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, 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 extended^
and it has been shown that there are additional contributions to the
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 Â¡he features
are independent of the kind of dopant and thus must arise from intrinsic
excitations of the system. Further, on doping a midgap absorption peak
75
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
h
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.
The 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. SubÂ¬
picosecond spectroscopy shows that the 1.35eV line is strong and already
zhâ€”czf.
decaying 0.1 psecs after photoexcitation. J This short timescale for
production and the estimate of the quantum production efficiency as
being of order unity strongly suggest that it arises from an intrinsic
excitation of the lattice. All of the above are in qualitative agreeÂ¬
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
57
with increasing temperature and entirely disappears at  150K.
Second, at approximately the same temperature there is a sharp increase
in the photocurrent Third, the energy of the peak decreases by
approximately 3$ on deuteration.^9 Fourth, the excitation responsible
ÃœQ
for the !.35eV line is dipole forbidden from the groundstate.
76
Let us examine each of these in turn. First, the breather is a
lattice excitation and thus one expects it to be strongly temperature
dependent. Indeed the temperature at which the breather becomes
unstable can be estimated by a simple argument. A typical breather may
have energy ~0.5 Aq (=0.35eV) and be localized over ~2Â£=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 disappearence 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) ] 1 ,/2  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 A ; (c) we have assumed a breather
o
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.
Figure 4.1
Single electron photoexcitation for transpolyacetylene with
rescaled parameters. The time (t) is in fsecs, n is the site number
and rn is the smoothed order parameter (rn=1 = 0.1A). Lines are drawn
at alternate lattice sites and at 15 fsec intervals. Note the initial
transient in the kink velocity, followed by an approximately constant velocity
regime. The breather is evident between the two kinks.
79
O
CO
O)
\
W
Figure 4.2
Breather Dynamics
(a) Evolution of the numerical breather created in the photoexcitation.
The period is approximately 40 fsecs. (The long time oscillations in the
dimerization result from the removal of the breather from a 98 to 42 site ring.)
81
O
o
CO
\
w
\
(M
Breather Dynamics
(b) As figure A.2a but for analytic breather with e = 0.75
(chosen to match the frequency of the numerical breather).
83
O
CV
\
W
\
3^
Figure 4.3
The location of the lower intragap impurity level for the
numerical breather (solid line) and the analytic breather (dashed line).
85
Figure 4.4
Average optical absorption, a(to), over one phonon period and
over a 98 length chain for the numerical breather. For the
dimerized lattice see fig 3.4.
Aa(cu)
86
E/A
O
Figure H.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.
87
Figure 4.6
Change in optical absorption, Aa(ai), on photoexcitation for
transpolyacety1ene for (a) t~21~43 fsecs (aolid line)
(b) t=301~343 fsecs (dashed line)
Figure 4.7
Photoexcitation of electron pelaron. Note the strong breather
excitation between the kinks.
(m)DV
90
E/A0
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 edge.
(b) Location, with respect to the Fermi level, of electron and
hole (solid line). Level occupied by photoexcited electron
and hole (dashed line), where the band edge state are levels 49 and 50.
levelr
92
EMERGÃ
{AT/T ) x 10
93
Figure 4.10
Experimentally observed change in absorption on photoexcitation of
(a) cispolyactylene and (b) transpolyacetylene at 10 K.
From data of Orenstein et al.
CHAPTER V
DYNAMICS IN CISPOLYACETYLENE AND RELATED MATERIALS
Unlike transpolyacetylene the two isomers of cispolyacetylene are
not degenerate; the transcisoid configuration being energetically
preferred over the cistransoid configuration (fig. 5.1). This nonÂ¬
degeneracy can be viewed as arising from the differing orientation of
the CH bonds with respect to the CC bonds in the two isomers.
Alternatively one may view the nondegeneracy as arising from the
differing thirdnearest neighbor interactions. As the transcisoid
configuration has 6E energy per site less than the cistransoid
configuration a KK pair at a distance i apart are confined by an
energy 5E.8,. Single kinks may, however, still stabilize at chain
ends. Polarons and breathers, which do not involve a change of phase
are still stable.
Statics in CisPolyacetylene
The nondegeneracy of cistransoid and transcisoid can be modeled,
in the continuum limit by the addition of an extrinsic gap,DÂ° A , to the
intrinsic gap produced by the electronphonon coupling. This gives the
total gap, A(x), as
A(x) = A.(x) + A .
i e
(5.1)
94
95
This does not alter the structure of the electronic spectrum or its
eigenfunctions. It does, however, modify the selfconsistency gap
relation to
2
A. (x) = A(x)  A =  â– â€” â€” E â€™ [u (x) v(x) + v (x) u(x)] . (5.2)
1 wCT v, s
Substitution of the eigenfunctions of the purely dimerized lattice (eq.
2.17) gives the implicit relation for Ap,
A^ = XA In (A /A ). (5.3)
e o o o
It can easily be shown rigorously (or by direct substitution) that the
kink is no longer a solution of eq. 5.2. One now finds that, as well as
the polaron solution to eq. 5.2 there are also bipolaron solutions, with
either zero or two electrons in each intragap state. There is also an
"exciton" solution with a single electron in each intragap level. In
transpolyacetylene both the bipolaron and the exciton are unstable to a
kinkantikink pair.
In the SSH model the necessary symmetry breaking can be modeled by
a term, which in the continuum limit, reduces to the extrinsic gap.
Here two possible additions to the SSH Hamiltonian are considered
= t E (~1)n [C+ , C + C+ C J
1 L n+1 n n n+1J
(5.4)
and
h3 = t3
E
n
even
r + â€¢+â– i
c _c + c c J.
L n+3 n n n+3J
(5.5)
96
It is straightforward to show that in the continuum limit
t1 = 2Ae = 2t . (5.6)
Although both H and modify the continuum equations in an identical
manner, they may behave qualitatively differently in the discrete
system. In fig. 5.2 the potential energy (electronic plus lattice) is
plotted as a function of the dimerization for and H^. The parameters
tj and t^ were chosen to numerically give the same total groundstate
gap. Close to the absolute minimum the two potentials are very similar
and the dynamics within them should be the same. Although the potential
around the metastable minimum is different in the two cases, this is
less important for the dynamics of confined "kinks". As the relevant
dynamics should be qualitatively similar for the two models only will
be considered here.
It has not yet been possible to derive unique values for the
parameters of cispolyacetylene from the experimental data. In this
study, therefore, the rescaled parameters of transpolyacetylene are
used. The dynamics are then studied as a function of the confinement
strength r = t^/t . (From resonant Raman scattering studies Vardeny et
al.^1 have estimated, assuming that the electronphonon coupling
constant (A) is the same for cispolyacetylene as for transÂ¬
polyacetylene, that the extrinsic gap is  5? of the total gap. Such a
gap increase corresponds to r  0.01, for the parameter set used here.
As we shall see this suggests that the kinks in cispolyacetylene may be
weakly confined.)
Dynamics in CisPolyacetylene
The potential energy of the dimerized lattice was found to vary
linearly with r over the range of values studied (0< r < 0.06).
(Although this appears to be a narrow range of values, we shall see that
for r=0.06 the kinks are strongly confined.) As free kink production is
forbidden we expect that photoexcitation will lead to the production of
a KK bound state. We found that there are two regimes in r, with
qualitatively different behavior during photoexcitation:
1. In the "weak confinement regime" (r < 0.015) a KK pair is
formed. The kinks separate (fig. 5.3a) to some maximum
separation. They then evolve to a dynamic cycle of quasiÂ¬
independent kink and antikink which, after a long time is
expected to produce either (a) a dynamic complex involving
kinkantikink creation and recombination, or (b) a static
configuration with the kink and antikink at some finite
equilibrium separation.
2. In the "strong confinement regime" (r > 0.015) it is not
possible to separately identify a kink and an antikink (fig.
5.3b): the system evolves to a single complex lattice
excitation. The electronic signature of this complex is a pair
of singly occupied intragap states, that oscillate over a
narrow energy range around ~Aq from midgap. This is clearly
the exciton, predicted by the continuum theory. What is more
suprising however, is that the new band edge states also
oscillate periodically into the gap. This breatheiâ€”like
98
electronic signature cannot be identified with a specific
lattice excitation, but appears to be an integral part of the
photoexcited complex.
The optical absorption was calculated for the above two examples.
Figure 5.4 shows the difference, Aa(to), between the long time evolved
photoexcited system and the groundstate. For r=0.002 the absorption
(fig. 5.4a) is qualitatively similar to that of transpolyacetylene
(fig. 4.6b'). The major difference is that the kink midgap and
therefore, compensating above band edge absorption is strongly
diminished. This makes the high energy shoulder, arising from the
breather, much more noticeable. It is clear that for r=0.06 the
absorption spectrum is qualitatively different. First the high value of
T increases the band gap to 2.5 A . Second the contributions
at  Aq/2 and  5Aq/4 arise from transitions between the putative midgap
levels and between the midgap levels and the continuum states
respectively. The broad peak at 1.52.0 Aq arises from transitions
within the breatherlike modes of the photoexcited system.
We have seen numerically that in cispolyacetylene there is an
excitation analogous to the breather of transpolyactylene. Here for
high r, however, it is part of an "excitonbreather" complex. As the
breather is coupled to an electronic excitation we do not expect its
optical signature to be strongly temperature dependent. Indeed, in the
real system (fig. 4.10a) there is a large, temperature independent
peak4^â€™11Â® just below the band edge, which has previously, with little
experimental support, been ascribed to an excitonic breather. In this
simulation we also see contributions to the optical absorption close to
midgap. There is no analog of these in the experimentally observed
99
optical spectra. (There are small enhancements in the absorption close
to midgap bat they are due to remnants of transpolyacetylene in the
system.) This qualitative disagreement may be due to a bad choice of
parameters for cispolyacetylene: with some other suitable choice the
dynamics may be different. It is also possible that the Hamiltonian
used to describe the symmetry breaking is inappropriate. It is,
however, possible that the scenario described above is qualitatively
incorrect for this system.
Breather Dynamics in CisPolyacetylene
In fig. 5.5 the period of an e=0.75 breather is plotted as a
function of r, for breathers in both the absolute and metastable
minima. As r increases the potential around the absolute minimum
becomes deeper and narrower, thereby decreasing the period of the
breather. The metastable minimum becomes shallower and wider with
increasing r. The period of the breather, therefore, increases. Indeed
for r>0.05 the metastable minimum becomes so shallow that it is no
longer able to support the breather.
Dynamics in Finite Polyenes
p li
The finite polyenes ((CH)n, n=2,4,6,...) are the short chain
analogs of polyacetylene. As the polyenes (and polyacetylene) belong to
the C2^ symmetry group their single particle spectrum consists of
alternating 'A and 1B.. states. Since in the groundstate all of the
orbitals are doubly occupied the groundstate should have 1Ag symmetry
100
and the first excited state should have 13.j symmetry with a dipole
allowed transition between them. In studies of octatraene it has been
shown that the optical absorption band edge is above the band edge for
fluorescence. These, and other data, strongly suggest that there is a
reversal of level ordering and that the first excited state is 'a^, and
o
thus dipole forbidden from the groundstake. In pnotoexoitation the next
highest level, which has '3U symmetry is excited. This promptly decays
by a radiationless route to the excited A_ state which then fluoresces
o
to the groundstate. This reversal of levels cannot be explained in the
single electron theory, but is well explained by theories which include
electron correlations e.g. the PariserPariâ€”Pople theory. Â¿
In this study polyacetylene has been viewed exclusively in the
single particle picture. As mentioned in Chapter II this approximation
is satisfactory for the understanding of, at least some of, the
experimental data. Of course it would be more satisfactory if the
polyenes and polyacetylene could be understood within the same
theoretical framework. Although the natural, and qualitatively
successful, approach to the polyenes remains the strongly correlated
models, it is instructive to consider the nonlinear dynamics of the
single particle theory for the polyenes. Although it is unlikely that
nonlinear dynamics alone can adequately describe polyene, this study may
elucidate the role nonlinear excitations in the short chain system.
There are two competing effects: on the one hand a short system has
fewer phonon modes to act as energy sinks for energy removed from
nonlinear excitations; on the other hand in a system with so few modes
the idea of a nonlinear lattice excitation may not be meaningful.
101
The Dimerized Lattice
Figure 5.6 shows that the lattice energy per bond, E(N), varies
inversely with the chain length; and that the configuration with a
double bond at each chain end is lower in energy. Thus the finite
polyene is intrinsically cislike. Indeed for N<20 the configuration
with single bonds at each chain end is absolutely unstable to the
configuration with double bonds at each end. As in cispolyacetylene,
therefore, single kinks are not free and single kink dynamics are not
expected to be important.
Breather Dynamics in Finite Polyenes
The lifting of the groundstate degeneracy does not qualitatively
affect the breather. A numerical study of the dynamics of a e =0.75
analytic breather on a chain of 10 atoms (with SSH parameters) showed
that the breather is persistent to at least 1 psec. Figure 5.7 shows
the difference between the optical absorption of the chain with and
without a breather. It is evident that the optical absorption is very
similar to that of transpolyacetyleneâ€”a single subband edge absorption
peak compensated by an above band edge bleaching.
Photoexcitation in the Presence of Damping
Although emission of optic and acoustic phonons from a breather during
photoexcitation is small, nonradiative decay routes via e.g. vibronic
modes of the CH bond or a carbonimpurity bond may provide an efficient
sink of energy. Unfortunately, due to difficulties in the
interpretation of the experimentally measured phonon dispersion
102
curves,no estimate of the strength of dissipation in polyacetylene
has been made. As a crude approximation we model all forms of
dissipation by the addition of a spatially uniform velocity dependent
damping term to the SSH equations of motion (eq 3.11). This is, of
course, highly phenomenological as dissipative effects are probably only
spatially uniform over macroscopic length scales. Further, rather than
simply providing an energy sink, vibronic modes of the CH bond may add
a crystal phonon field to the problem.
Figure 5.8 shows the result of single electron photoexcitation in
the presence of damping of strength y = 0.15 u . A kink and antikink
are formed. Breather production is suppressed.
103
Figure 5.1
Schematic of lattice  upper is transcisoid, lower is
Total potential energy of cispolyacetylene. Note
transcisoid isomer is lower in energy.
cistransoid.
that the
Figure 5.2
Shape of potential as a function of lattice dimerization for transÂ¬
polyacetylene (solid line) and cispolyacetylene (H^  dashed lines,
 dotted line). The total electronic band gaps were choosen to be the
same for the two formulations of cispolyacetylene.
â€¢ \ /
/
/
â– / ^ ^
105
Figure 5.3
Photoexcitation for eispolyacetylene.
(a) r = 0.002
107
O
CO
O
<^L
\
W
3^
Figure 5.3
Photoexcitation for cispolyacetylene.
(b) r = 0.06.
109
O
CO
0)
\
AX
110
Figure 5.4
Change in optical absorption, Aa(to), on photoexcitation for
r = 0.002 (solid line) and r = 0.06 (dashed line).
Figure 5.5
Period versus confinement strength for breathers in absolute minimum
(solid line), metastable minimum (dashed line).
50
45
h
40
35
0.03
r
âœ“
/
/
/
/*
/
/
0.06
112
Figure 5.6
Energy per unit length as a function of the inverse of the
length for finite polyene: double bond at each chain end (solid line),
single bond at each chain end (dashed line).
E(N)/N
0.1
l/N
0.2
114
a (cu)
115
Figure 5.7
Change in absorption, Aa(u) on photoexeitation of
polyene with 10 sites.
6.0
Figure 58
Photoexcitation of transpolyacetylene in the presence of
damping of strength p = 0.15 .
117
CHAPTER VI
DYNAMICS IN DEFECTED SYSTEMS
It was noted in Chapter I that the electrical conductivity of
transpolyacetylene can be varied over twelve orders of magnitude by
adding small quantities of dopants.35 Clearly, therefore, any model
must include impurity effects for a full understanding of the transport
properties of the real system. Experiments show that the conductivity
can be varied reversibly over its full range of values indicating that
the dopant impurities are nonsubstitutional and nonchemically reactive
with the o electrons of the polymer backbone. It can thus be assumed
that extrinsic defects only affect tr electron dynamics.
Here two qualitatively different types of defect are modeled.^
The site defect, which may arise from an extrinsic impurity or from
morphological defects, such as crosslinking, hybridizes an orbital in
the defect to the it orbital of a single carbon atom. The bond impurity
hybridizes an impurity in the defect to the tt orbital of the bond
joining two neighboring carbon atoms. It can be thought to model
extrinsic impurities or cislike and amorphous regions in transÂ¬
polyacetylene .
118
119
Static Model for the Single Site Impurity
An impurity of strength VQ at site M can be modeled by the addition
to the SSH Hamiltonian of the term0^
H
s
(6.1)
This breaks the electronhole symmetry of the SSH equations,
modifying a single diagonal element of the hopping matrix. The
locations of the impurity induced levels can be calculated using a
simple continued fraction scheme.Â°Â°â€™0^
The electronic density of states is given by
p (E) =   Lim Im G (E+ip) (6.2)
nn 7T â€ž nn
p>0
where Gnn is the single particle Green's funtion. Here the Green's
function is calculated in the real space representation, with Gnn being
the propagator for all paths starting at and returning to site n. From
Gnn the local density of states can then be determined.
As a simple illustration of the continued fraction scheme the band
structure of the purely dimerized infinite chain is calculated here.
From Dyson's equation Gqq can be written as (see Appendix E)
Gqq(z) = (z  XL(z)  XR(z)) (6.3)
where Z is the complex energy E+ip and ED(z) and E,(z) are the self
n L
energies for propagation to the right and left of the original site
120
(0). Now for tA = tQ + Aq/2 and = ~ ^o2 t!ieri
Er(z) = t2k /(z  Zn(z)) (5.4)
where
EI1(z) = tg/(z  E22(z)]. (6.4)
This series is infinite. However as Z~â€ž(z) = ED(z) on an infinite line
the continued fraction has a periodic form and is readily surnmable as
Er(z) = t^ [z  tg(z  ER(z))_1]_l. (6.5)
Solving for ED(z) gives
K
â€ž â€ž , , , 2 t2 .2. \ , 2 2 .2.2 ,lt.2 2i1/2
2zEr(z) = (z +t  tg) Â± [(z + tg  tR)  4tA z J lo.6)
The appropriate sign of the discriminant can be determined from the
condition that the density of states be positive in the bands.
Likewise
El(z) = tR (z  ER(z)) '. (6.7)
Substituting for E (z) yields
Li
G00(z)   VZ)f1  V2'!"'
(6.8)
121
There can only be positive contributions to the density of states when
GQ0(z) is complex. GQ0(z) is complex when E^(z) or E (z) is complex.
After a little algebra this yields bands in the regions
(5.9)
E<2t,E>A;E<A,
o o o
E > 2t .
o
This shows the expected band structure: namely, two bands separated by
the band gap of width 2Aq. It is straightforward to show that there are
tne expected square root singularities in the density of states at the
band edges.
The single site impurity is modeled by adding energy Vg at a single
site (taken here to be the origin"). This gives
=oo(z) â– tz * vo  el(z) 
(6.10)
with Ed and E. as above. The band edges have the same locations as in
R L
the pure system, but the density of states at the band edges now has a
square root convergence. There can be further contributions to the
density of states at the poles of GQO(z). After a little algebra
(Appendix E) one finds impurity levels at
E
+
where
D2 = [4t
2
A
(6.11b)
122
and
Y
2
O
(6.11c)
with the +() sign giving the localized level above (below) the Fermi
level energy. Thus for Vq > 0 a level from the top of the valence band
moves up into the gap, whilst the level at the top of the conduction
band moves beyond the band edge into the "ultraband" region (i.e. the
ultraband state lies in the gap between the ir band and the o band). For
Vq < 0 levels are removed from the bottom of the conduction and valence
bands. It can be shown that these solutions agree with those of
Baeriswyl. J
This study will focus on donor site impurities (Vq < 0); all
results are identical for acceptor sites with electrons and holes
interchanged.
Dynamics in the Presence of a Single Site Impurity
The above analytic calculations do not treat the electronphonon
coupling selfconsistently since the lattice and electronic degrees of
freedom are not allowed to interact to produce a fully relaxed lattice
deformation and electronic band structure. A fully selfconsistent
solution can, in general, only be found numerically. The equations of
motion, derived from the SSH equations, are integrated and energy is
removed at each timestep (by setting the instantaneous velocity to zero
at every site at every timestep) until the total energy reaches a
minimum. It is assumed that this is the fully selfconsistent solution
123
to the defectmodified SSH Hamiltonian. At t=0 the lattice is purely
dimerized and thus this numerical calculation has same level of
consistency as the analytic calculation above. As expected, there is
quantitative agreement (better than \%) between the analytically and
numerically calculated locations of the impurity levels. The
equilibrium electronic structure and lattice deformation are typically
achieved within 0.10.2 psecs. The intragap level selftraps into the
lattice and moves closer to midgap (it still does not, however, reach
midgap for a finite value of Vq). These localized electronic levels are
accompanied by a localized lattice distortion, which consists of a sharp
step on the weak bond neighboring the impurity and a localized tail on
the strong bond neighboring the impurity. With the sign of Uq used
here, this means that for an impurity on an odd site there is an
antikinklike step, followed by a kinklike tail (fig. 6.1a). Such an
impurity will be referred to as being in the "antikink topological
sector". If the impurity is on an even site the impurity is in the
"kink topological sector". In this chapter all site impurities are in
the antikink sector unless otherwise stated. Assignment of the
impurityinduced defect as the bound state of two excitations may seem
suprising but is justified by the numerical observation that the
ultraband level is localized around the step, whilst the intragap level
is localized around the tail. This being so, under certain
circumstances, it should be possible to separate the kink from its tail
to produce a topological kinkantikink pair with one of the kinks
supported, not by a midgap, but by an ultraband level. Indeed this
possibility is realized and it has important consequences for the
dynamics of site defected systems.
124
In the case of the undoped system the addition of an extra electron
to the purely dimerized lattice forms a polaron. In the presence of a
site impurity the addition of an electron into the intragap impurity
level, i.e. a negatively charged donor impurity, increases the
localization of the impurityinduced lattice deformation. The electron
then further selftraps into the lattice (fig. 6.1b). However there is
still only a single level in the gap, which derives its parentage from
the intragap impurity level rather than from a polaronic level.
The addition of two electrons to the pure system creates a
bipolaron which rapidly decays to a separating kinkantikink pair. In
the presence of a site impurity the step evolves to a trapped antikink
supported by the ultraband level, whilst the tail evolves to a free
kink, supported by the intragap impurity level, which drops to midgap.
This free kink is expelled at the maximum free kink propagation velocity
by the antikink trapped at the impurity.
As the trapped kink is maintained by a low lying ultraband level,
optical transitions to the valence band are well above band edge making
such a kink extremely difficult to detect optically. The bound state
modes of the trapped kink may, however, be observable in i.r.
experiments. As the trapped kink is a topological excitation its bound
state spectrum should be only weakly dependent on the impurity type.
Indeed the i.r. modes usually ascribed to the pinned kink are observed
at the same frequency for all impurity types. It should also be
remembered that such a deep lying ultraband level may hybridize with
the o bands, the effects of which cannot be calculated without more
detailed band structure input.
125
Clearly one can expect such a strongly localized selftrapped state
to qualitatively alter the dynamics of the system. In the next section
we investigate this further by looking at the interaction of a kink with
an impurity.
Kink  Site Impurity Interactions
Statics
We first consider the evolution of a static kink and an impurity on
nearby sites. A simple continuum calculation has shown that the kink
and impurity evolve independently J but we have seen that the lattice
defect of the even and odd site impurity are topologically different so
they must be considered separately.
(a) Impurity step and kink in opposite topological sectors.
The kink midgap and the intragap impurity levels evolve
independently, in agreement with the continuum calculation.^
(b) Impurity step and kink in the same topological sector.
At t=0 the kinklike step is supported by a midgap level. The
two topologically equivalent distortions can be supported by a
single localized electronic level. Thus the intragap impurity
level remerges with the conduction band, whilst the kink midgap
level drops below midgap. The long time evolution is sensitive to
initial conditions, particularly the presence of other nonlinear
excitations. There are at least two branches:
126
1 . The kink is expelled by the impurity and the system evolves
to an independent impurity and kink, widely separated from
each other.
2. A "trapped kink" forms around the impurity supported by an
ultraband state. The kink intragap state rejoins the
valence band. The trapped kink also affects the continuum
states; each continuum state now being localized only over
the part of the chain either to the right or the left of
the kink.
Dynamics
The above study of static kink and impurity interactions is clearly
not the whole story; important new effects can be expected in the
dynamics. To examine these a single kink was boosted towards an
impurity such that before collision the kink reached terminal velocity
and the impurityinduced lattice defect was substantially evolved.
(a) Impurity step and kink in opposite topological sectors.
At small Vq  , (0.25 A^) the impurity induced lattice distortion
is sufficiently small that the kink can "ride over" it. At large VqÂ¡
(2.5 a ) we saw that, depending on the velocity the kink could be
either transmitted or reflected by the impurity. We do not expect the
outcome of such a collision to be a simple function of velocity, as e.g.
resonances of the kinkimpurity system and the presence (or absence) of
phonons may also be important. Indeed there may be windows of
reflection within the transmission region, before a critical value of
 Vq I above which the kink is always reflected. (In 4>4 dynamics1^ it
has been shown that for KK collisions, there are windows of reflection
127
within the transmission region. Sven more complex dynamics are observed
for kinkbreather collisions.)
(b) Impurity step and kink in same topological sector.
For small Â¡Vq (0.25 A ) again the kink can ride over the
impurityinduced lattice defect (fig. 6.2a). Here, however, as the kink
approaches the impurity the kink intragap level drops below midgap.
After the kink has passed through it returns to midgap and the kink and
impurity evolve independently, in agreement with the continuum result.
For large Vq (2.5 Aâ€ž) the kink initially accelerates to a high
velocity (greater than the maximum uniform propagation velocity in the
free system) (fig. 6.2b). The propagation is smoother, however, with no
evidence for a kink tail. At a large distance from the impurity (~5a)
the kink is accelerated to a very high velocity (~250v ) and trapped by
the impurity. This kink appears to remain trapped for all times and is
supported by the ultraband level of the impurity. For a neutral kink
the midgap level is singly occupied and thus does not fall back into the
valence band but, together with the intragap impurity level, forms a
hole polaron.
The trapped kink also affects the extended states, each now being
delocalized only over the part of the chain either to the right or to
the left of the impurity. This decreases the effective chain length,
which is, as far as transport properties are concerned, is limited by
the impurity spacing rather than the physical size of the system.
From the above studies we conclude that the ballistic transport of
charge in kinks through the system is unlikely to be an important
conduction mechanism in site defected systems. The above studies do,
however, contain a hint of an alternate mechanism. The trapping of a
128
neutral kink at an impurity results in the production of a hole polaron
that might transport charge by hopping through the system. Such hopping
mechanisms are well known in amorphous and disordered systems1^ and have
previously been suggested as the conduction mechanism in lightly doped
polyacetylene.3^
Photoexcitation in the Presence of a Single Site Impurity
In Chapter IV it was shown that breathers are produced in the
photoexcitation of defect free transpolyacetylene. Clearly if the
breather mechanism is to have any validity, a plausible production
mechanism for their production in the defected system must be
identified, and their robustness established. In this section
photoexcitation in the presence of a single site impurity is considered,
laying particular stress on the effects on the subband edge optical
absorption spectrum. As representative examples four simple cases of a
single impurity on a 98 site ring are considered. In each case the
initial condition is the fully selfconsistent ("relaxed") lattice
distortion and band structure.
(a) Empty intragap impurity level. Photoexcitation into the intragap
level.
Here (fig. 6.3) the impurity traps a negatively charged kink
supported by the ultraband level. The intragap impurity level drops to
midgap forming a free neutral antikink which is expelled by the trapped
kink. The singly occupied level at the top of the valence band and the
empty level at the bottom of the conduction band move into the gap to
form a hole polaron on the opposite side of the ring from the
129
impurity. (This apparent action at a distance should likely be
attributed to the adiabatic dynamics employed in these calculations, and
is probably nonphysical.)
(b) Empty intragap impurity level. Photoexcitation into the bottom of
the conduction band.
In this case a negatively charged trapped kink and a positively
charged antikink are formed in 0.1 psecs. The singly occupied levels
at the top of the valence band and the top of the conduction band move
into the gap creating an excitonlike structure far from the impurity.
The exciton is dressed by a breather.
(c) Singly occupied intragap impurity level. Photoexcitation into the
intragap level.
Here (fig. 6.4) a negatively charged free antikink is expelled by
the negatively charged trapped kink. A dynamic hole polaron is formed
at the impurity and is expelled from it. There is no evidence for an
internal oscillatory mode in the polaron. This is an interesting route
to creating a dynamic polaron as direct attempts to boost a non
oscillatory polaron have failed.
(d) Singly occupied intragap impurity level. Photoexcitation into the
bottom of the conduction band.
A negatively charged trapped kink and a neutral free antikink form
within 0.1 psecs. A dynamic exciton is expelled by the impurity and a
breather forms a long way from the impurity.
With these complicated dynamics it is not easy to predict the
effects on the photoinduced photoabsorption. Using the techniques
described in Chapter III, the optical absorption of the system was
130
calculated after it had evolved for a substantial period (0.3 psecs)
for the cases (a) and (c) above.
As preliminary input the optical absorption of the purely dimerized
lattice in the presence of a single charged or neutral site impurity of
strength 2.5 A was calculated. The presence of the intragap level
o
allows for subband edge photoabsorption. Figures 6.5 show the
difference, Aa(w), between the absorption of a ring with an impurity and
the absorption of the same ring without an impurity.
For the neutral impurity there is a single subband edge peak,
arising from transitions from the valence band into the impurity level
(fig 6.5a). This is compensated by an above band edge bleaching in the
energy range 23 A . For the charged impurity the situation is similar
o
(fig 6.5b). Here, however, there is a low energy shoulder to the below
band edge absorption peak, arising from transitions from the impurity
level into the conduction band. Again there is above band edge
bleaching, but it is over a larger energy range. Due to the very sharp
lattice distortion around the step, a large number of Fourier modes
(including high wavevector contributions) are required for its
description. In both cases the bleaching at the full band width is due
to the removal of the square root singularity in the density of states
at the band edges.
The optical absorption was calculated for the cases (a) and (c)
above: photoexcitation into the impurity level for both neutral and
charged impurities. Figures 6.6 shows the change, Act(w), in the optical
absorption on photoexcitation. Below the band edge the two cases give
similar spectra: a strong midgap absorption from transitions into and
out of the free kink midgap level. The peak at Aq/2 arises from
131
transitions into the polaronic level. The below band edge bleaching is
due to the loss of transitions into the intragap impurity level, which
has now evolved to midgap. The sharp bleaching above the band edge
results from the loss of the low wavevector states needed to produce
the polaron. For the neutral impurity there is further bleaching over a
wide energy range, arising from the loss of wavevector states needed to
describe the trapped kink. (This bleaching is largely absent in the
case of the charged impurity, as the Fourier descriptions of the step
and the trapped kink are similar.)
Photoexcitation with Many Impurities
In each of the above experiments the intragap impurity level
evolves to midgap. In a general photoexcitation experiment this need
not be so. In particular for a system with high impurity density, but
pumped at a low optical intensity, insufficient electrons will be
excited to allow all the impurity levels to evolve to midgap. Thus the
impurity band will remain largely unaffected. As a model of such an
experiment we photoexcited a single electron in the presence of 3 and 5
site impurities, placed randomly on both even and odd sites.
As the initial condition we used the fully relaxed groundstate in
which, if the impurities are widely spaced (> E, apart) the intragap
levels are degenerate and if closely spaced (< Â£ apart) the impurity
wavefunctions overlap to form an impurity band of the finite width. In
each case the impurity induced lattice distortions evolve independently,
(a) Empty intragap impurity levels. Photoexcitation into the bottom of
the conduction band.
132
The singly occupied levels at the top of the conduction band and
the bottom of the valence band are prevented from forming a KK pair
(with their associated midgap levels) by the presence of the intragap
impurity levels. Instead the system evolves to a broadened impurity
band and an exciton, accompanied by a highenergy breather. It is
interesting to note that all of the important dynamics occurs between
the two most widely spaced impurities, the rest of the chain acting as a
spectator. Here, again, the effective chain length is essentially
limited, not by the physical size of the system, but rather by the
impurity spacing.
(b) Empty intragap impurity levels. Photoexcitation into the lowest
intragap state.
Here the singly occupied intragap impurity level drops below
midgap. A second impurity level forms a polaronic excitation between
the two most widely spaced impurities. The qualitative change from
impurity levels to polaronic levels is allowed by the collapse of the
kink and antikink tails of a pair of closely spaced impurities, which
form a pair of trapped kinks. As in case (a) there are a large number
of intragap levelsâ€”two associated with the polaronic excitation, three
associated with inert impurities, and up to six others associated with
breathers.
The Single Bond Impurity
In the remainder of this chapter an impurity that couples to a
single CC bond is considered. As a model the SSH Hamiltonian with the
addition of the term
133
HB  "o ^ CM +1S CMS
CMS Â°M+1S'
(6.12)
is used, where WQ is the impurity strength on the bond between sites M
and M+1. This alters the value of a single hopping matrix element; e.g.
for the bond connecting sites 0 and 1 , tft Â» tF=tA +WQ. Using the
continued fraction scheme impurity levels are found at (see Appendix E)
E
+
* N Utf
(6.13)
where the +() sign gives the states above (below) the Fermi level. The
constant N takes the values +1(1) for the intragap and ultraband
levels. Of the four impurity levels predicted by eq. 6.13, a simple
argument shows that some are unphysical. In particular at tE=0 eq. 6.13
predicts that the location of the ultraband levels diverges to
infinity. Clearly a finite impurity strength cannot result in an
infinite change in the energy of the system. Thus the ultraband branch
for 0 < tE < tA is unphysical. Consider now an impurity on a weak bond
(tA
system locally more translike and increases the local gap: therefore
there should not be any intragap states. On the other hand
strengthening the weak bond creates a configuration in which there are
three strong bonds in a row. This is topologically identical to a pair
of kinks on neighboring sites. Thus one expects intragap states. By
the same token, intragap states are not expected when a strong bond is
strengthened but are expected when a strong bond is weakened. Figure
6.7 shows the numerically calculated location of the intragap levels.
134
This is in qualitative agreement with the above physical arguments and
in quantitative agreement with eq. 6.13.
As in the case of the site impurity the above calculations do not
treat the electronphonon coupling consistently. Vie find that the
impurity selftraps into the lattice and at any value of WQ the
impurity levels become more localized.
As the impurity is on a single bond, it changes the local
environment symmetrically. The induced lattice defect is thus
topologically trivial (fig. 6.8).
Kink3ond Impurity Interactions
Although, due to the electronhole symmetry of the system,
interaction can only take place through the lattice degrees of freedom,
one can still expect the outcome of a kinkimpurity collision to be a
complicated function of a number of parameters. In figs. 6.9 we compare
the dynamics of a single kink boosted with the same initial velocity
towards defects on strong bonds of strength  aq/2 and   Aq
respectively. We see that for the weaker impurity the lattice defect is
sufficiently small the the kink can ride over it, whilst the large
lattice distortion of the stronger impurity sets up a high potential
barrier and the kink is reflected.
Photoexcitation in the Presence of a Single Bond Impurity
We performed simple photoexcitation experiments with an impurity of
strength  a on a strong bond. The initial condition was the fully
135
relaxed system, in which there is a pair of intragap impurity levels.
(a) Neutral impurity. Photoexcitation from the lower to the upper
level to the bottom of the conduction band.
Here the electronic occupancy is that the same as that for
photoexcitation of the undefected system. Although only 1.1 Aq (less
than the energy required to produce a KK pair) is added to the system,
additional energy is gained by the creation of ultraband levels. The
system is then able to create a KK pair and a breather (fig. 6.10).
(b) Neutral impurity level: photoexcitation from the lower impurity
level to lowest level in conduction band.
A pair of oppositely charged polarons is formed on either side of
the imparity (fig. 6.11).
The optical absorption for the neutral impurity was calculated,
both before and after photoexcitation for case (a) above. Figure 6.12
shows the difference, Aot(co), between the optical absorption with and
without an impurity. There is a strong enhancement at a A arising
from transitions between the impurity levels. There is also a weaker
peak at ~3Aq/2 arising from transitions from the lower impurity level to
the conduction band. This is compensated by a weak bleaching over a
wide energy range above the band edge.
Figure 6.13 shows the difference, Aa(m), between the optical
absorption after and before photoexcitation. There is a sharp bleaching
at A A due to the loss of transitions between the impurity levels. A
strong midgap enhancement is produced by transitions into and out of the
kink midgap levels. It has a high energy shoulder arising from
transitions within the breather.
136
In conclusion, the bond impurity only results in
for a limited range of values of the impurity strength
impurity reduces the mobility of a kink the outcome of
experiment is largely unaffected.
intragap levels
. Although the
a photoexcitation
Figure 6.1
Unsmoothed order parameter, u , for (a) neutral (solid line) and
(b) charged (dashed line1? site impurity on an odd site.
138
O
O
c
o
LO
Figure 6.2
Kinksite impurity collision.
(a) VQ  0.25 Aq. Impurity and kink in same topological sector.
140
(b) vQ 
Figure 6.2
Kinksite impurity collision.
Kink and impurity in same topological sector.
2.5 Aq.
142
O
\
Figure 6.3
Photoexcitation into the intragap level of a neutral site impurity.
144
O
O
cu
\
w
9 Q
Figure 6.^
Photoexcitation into the intragap level of charged site impurity.
146
84
147
E/A0
Figure 6.5
Difference between absorption, Aa(to), of 98 ring with and without (a) a
neutral impurity (solid line) and (b) a charged impurity (dashed lineâ€™).
Aa (tu)
143
Figure 6.6
Change in absorption, Aa(oi), for photoexcitation of electron
into impurity level of neutral impurity (solid line)
charged impurity (dashed line).
Figure 6.7
Schematic of the location of the bond impur
function of the impurity strength, for (;
bond and (b) impurity on a strong bond,
the conduction bond and the energy zero
ity induced levels as a
i) impurity on a weak
The shaded region is
is the Fermi level.
150
Figure 6.8
Unsmoothed order parameter of strong bond impurity of strength WQ  A .
152
O
O
o
LO
c=
Figure 6.9
Kinkbond impurity interaction.
(a) Impurity on strong bond WQ ~ A^/2.
154
Figure 6.9
Kinkbond impurity interaction.
(b) Impurity on strong bond Wo  Aq.
156
O
O ^
\

Figure 6.10
Photoexcitation of electron from lower to upper intragap level
of strong bond impurity of strength WQ  ~Ao.
158
9 Q
Figure 6.11
Photoexeitation of electron from lower intragap impurity level to
lowest state in conduction band for strong bond impurity of strength WQ  A .
W
\
ro
o
0
09T
A a (cu)
161
E/A0
Figure 6.12
Difference in absorption, AaCw), between 98 site ring
with and without strong bond impurity of strength WQ  _AQ.
Aa(oj)
162
E/Ac
Figure 6.13
Change in absorption, Aa(oi) on photoexcitation of electron
from lower to upper intragap level strong bond impurity
of strength WQ  a
CHAPTER VII
STATICS AND DYNAMICS IN POLYYNE71
A polyyne is a linear chain of carbon atoms with alternating single
7274
and triple bonds i.e. fC^C} . It has been suggested that the long
chain polyyne, carbyne, is the stable allotrope of carbon over the
temperature range 26003800 K and the pressure range 4 x 10~J  6 x 10^
atmospheres.7' It has been further suggested that polyynes may be been
seen in the infrared spectra of carbon rich stars and may also be a
constituent of interstellar dust.7Â® 7Â®
Ab initio HartreeFock calculations72,7^Â®1 have shown that the
dimerized polyyne structure fC^C} is energetically favored over the
cumulene structure 4C=C4 . It is clear that the major difference
between polyyne and polyacetylene is that polyyne has two irelectron
bands, which couple instantaneously to the neighboring carbon atoms in
an identical way. It is therefore natural to formulate a single
electron theory of polyynes in the spirit of the SSH and TLM models for
polyacetylene.
Formalism71
Within the framework of a one electron tightbinding description,
the iTelectron states of polyyne are determined by the linearchain
Hamiltonian
163
164
H = H 
E t C ,C ,+h.c.
, 1 n+1,n n+1,o ,A n,o,A 1
n, o , A
(7.1)
The chain consists of Nq (Nq + Â®) sp1hybridized carbon atoms labeled by
n, (n=1,2 Nq). Each carbon atom has two degenerate, orthogonal,
atomic p orbitals transverse to the chain axis. The latter two orbitals
are labeled by A(A=1,2). In (7.1), C + , and C , are fermion
n,o ,A n,a ,A
operators which create or destroy, respectively, an electron with
spin o in the Ath orbital of the nth carbon atom. To allow for the
dependence of the hopping integral tn+1 n on the atomic locations, we
adopt the form used in polyacetylene
(7.2)
Hl describes the lattice energy in the absence of the overlap of the
atomic p orbitals
H, = Jr M E Ãš2 + 4 K E (u u )â€˜
L 2 nn 2 n n+1 n
(7.3)
The linear chain Hamiltonian (7.1) describes two degenerate half
filled it bands which couple to the instantaneous positions of the carbon
nuclei in an identical manner. It follows, therefore, that if allowance
is made for the twofold degeneracy of the electronic spin and we treat
the atomic displacements {unl as a classical field, the ground state of
H is a halffilled Peierls insulator in which the electrons possess an
effective internal degeneracy N equal to 4. For the halffilled Peierls
insulator ordinarily encountered, e.g., polyacetylene, N = 2. Then for
165
the uniformly dimerized configuration, un = Â±(1)nUQ
H = 2NKu^ E [t  (1)n2auJ[c + 1 C + h.c.]
0 n y Â® 0J L n+1 ,p n,p J
where p is an index describing both the spin index, o, and the
index, X and can take as the four values p = ( + ,1), ( + ,1), (t,
Now Fourier transforming to momentum space by
c
kp
/N k
ikna
e
c
nu
gi ves
el
E [ 2t_ cos ka c.f c. + 4iaUSin ka c. c, 1.
0 k,p k.p 0 k+iT,p k,pJ
Hel can be diagonalized by defining
r, = a. c. + 3, c,
k, p k k, p k k+ir, p
and demanding
[r, , H n 1 = E
k' ,p elJ k,
rk',p'
This yields
el
= E
k'
(7.4)
orbital
0, ( + ,2).
(7.5)
(7.6)
(7.7a)
(7.7b)
(7.8)
where
166
e
k ,p
(7.9a)
wi th
e = 2tâ€ž cos ka
k 0
(7.9b)
and
Aq = 4a'JQ. (7.9c)
The total ground state energy is then
E = (Nq/2)(K/4Ã¼)Aq  Z* Ek (7.10)
k,u
where the sum is over occupied levels only. Minimizing this with
respect to Aq gives
1 = (4y2/KNq) Â£ sin2ka E 1 . (7.11)
k,u
For A/2tQ << 1 , which is the criterion for the validity of a continuum
description, equation 7.11 then may be evaluated analytically to yield
A0 = (8t/2.7l8)exp(~1/2X ). Note that the dimensionless electron
0 0 ep
2
phonon coupling constant X = 4a /uKtâ€ž is twice that of the case N =
ep 0
2. If we take as representative values for polyyne the values tg=3 eV,
K = 68.6 eV A 1, and a = 3 ev A~1 we obtain 2AQ = 5.0 eV. Clearly the
condition AQ/2t0 << 1 is not well satisfied, implying strong electron
167
phonon interaction for polyyne. However as we shall see numerically
2Aq = 5.88 eV, which indicates that the weak coupling limit is not
prohibitively bad.
The Continuum Model
We have seen in the previous section that the electronphonon
coupling constant is large (A ~ 0.4). Although a continuum
approximation cannot be expected to give a quantitatively good
description of polyyne, it should be able to describe qualitatively the
spectrum of nonlinear excitations. We therefore proceed in the spirit
of the TLM calculation for polyacetylene by linearizing the electronic
spectrum around the Fermi vector Â±kp. â€™ J Thus
ik na ik na
c = /a [u (n) e '  iv (n)e 1 ] (7.12)
n,u y y
where a is the lattice constant. As the lattice is expected to dimerize
we expand the lattice coordinate about Â±2kv.
2ikâ€žna 2ik na
un = â€” [A+(n)e r + A(n) ] (7.13)
where the factor l/4a is put in for future convenience. The discrete
Hamiltonian can now be linearized to order a/Â£. Terms of higher order
in a/r are dropped as are the rapidly varying terms of type (~1)n. This
eliminates acoustic phonons from this model.
168
We then find
H
â€”5 { 3 A (x) + Z ) dx T (x)[iv a, 3â€” + A(x)a. V (x)
2 â€˜ 2a 1 p F 3 3x 1 J p
4a p
(7.14)
where
V (x) = (u (x), v (x)) (7.15)
y y y
and
VF * 2t0a
(7.16)
and 0^ is the ith Pauli matrix. This is the TLM equation with the
single modification that p can take on 4 values rather than 2 as in
polyacetylene. As we shall see shortly this difference is crucial. By
X X
varying (7.14) with respect to u (x) and v (x) we get
p P
e u
n,p n,a
(x)
i v
3_
F 8x
u
n, p
(x)
+ A(x)v (x)
n,p
(7.17a)
E V
n,p n,p
(x)
+iv 3 V (x) + A(x) u (x)
F dx n,p n,p
(7.17b)
where e is the nth eigenvalue of the subscript p. Varying (7.14)
n, p
*
with respect to A (x) gives
A(x) = (4ct2/K) Z' [u (x)
n .. n>4
7T i
V (x) + cc.
n,p
(7.17c)
where the sum n is over occupied levels only. Defining
169
fÂ±(x) = u (x) Â± iv (x) (7.18)
PM P
then one easily obtains
r 2 3 2, x 3A(x)iâ€žÂ±, <
_Vp â€”5 + â€ž  A (x) Â± v? â€”â€”â€”Jf (x) = o.
F 3x2 k,lJ
(7.19)
The groundstate of this is A(x) = AQ(x) with eigenvalues given by
2 2 2. 2
e. = A_ + v_k
k,u 0 F
(7.20)
where k is measured relative to the zone edge rr/2a and the groundstate
gap for the dimerized system in 2A^.
The GrossNeveu Model
Further progress in determining the particle spectrum of equations
(7.14) is most easily made by a consideration of the GrossNeveu field
P P Q p
theory in 1+1 dimensions, â€™ whose Lagrangian is given by
N
LGN(x) = E TV(x) {iY  gâ€žMo(x)} 7V(x)  Â± 02(x) (7.21)
, u dx GN 2
v= 1 p
where TV(x) is a fermion field, 7J(x) = (t^(x), T^(x)) and o(x) is a
scalar field and gQN is the GrossNeveu coupling constant. The
sum v labels the particle type. Y^(p = 0,1) are the Dirac matrices
(Yq = o^, Y i = iOj). Varying (7.21) with respect to yv(x) gives (xQ,x1
= t ,x)
170
(7.22)
Now looking for static solutions of the type
ie t
4,V(x,t) = e n fv(n;x)
(7.23)
gives
(7.24)
where is the nth eigenvalue. Again noting that H'v(n;x) is a fermion
function and that there are a pair of equations in T^(n;x) and 'i'^CnÃx)
for each value of v we see that if N = 4 (ie. v = 1,2,3,4) then (7.24)
has the same formal structure as (7.17a) and (7.17b). This equivalence
Then (7.22) becomes exactly (7.14).
Clearly however, if the two models are equivalent then their
consistency conditions must match. Here a complication arises the
GrossNeveu model is ultraviolet divergent and requires a cutoff
frequency, A, for renormalization. This may seem to be different from
the TLM model, but there a cutoff was introduced in a natural way by the
finite width of the irelectron band. The two consistency conditions can
be shown to be equivalent
171
Nonlinear Excitations in Polyyne
Having shown that the TLM model for polyyne and the statics of the
semiclassical N= 4 GrossNeveu model are equivalent the well known
results for the particle spectrum can be directly carried over from the
GrossNeveu model to polyyne.
There are kink soliton solutions with A(x) = AQ tanh(x/5). This
has a corresoonding eigenvalue soectrum with e =0. As u can take
n,u
four different values the midgap level can be occupied by up to 4
electrons. Thus 4 x ^ = 2 states are removed from the filled valence
band. There is thus a loss of two electrons from the valence band which
screen two electrons in the midgap levels. Therefore a kink can have
charge +2e, +1e, 0, 1e, 2e depending on whether there are 0,1,2,3,4
electrons in the midgap levels respectively. The characteristics of
these kink states are shown in table 7.1. It is important to note that
the spincharge relation of each of these kinks is entirely
conventional. This should be compared with polyaceytiene, in which the
spincharge relations are reversed. The energy of a kink in the Gross
Neveu models is
E =
(7.25)
As topological constraints require that a kink (K) and an
antikink (K) can only be created in pairs, the minimum production energy
of a KK pair is 2Ek = 2(4Aq/tt).
172
In analogy with polyacetylene we again expect there to be polaron
solutions. From the GrossNeveu particle spectrum we find
A(x) = A k v {tanh[k (x+x ) ]  tanh[k (xx )]} (7.26)
0 0 F 0 0 0 0
where
x
0
sin0
Â£ ' sinG
tanh 1 [tan0/2]
(7.27)
(7.28)
and the angle 0 is constrained to be in the regime 0 Ã¡ 0 < tt/2 and is
determined by
0 = [(n+h)/4]ir/2 (7.29)
The energy of formation of the polaron state is
E (0) = 2E sinÂ©. (7.30)
P k
The polaron eigenspectrum has localized levels at e = Â±e (k*0).
Â± Â»M Â¿
In (7.29) n is the number of electrons in the upper localized state
which h is the number of holes in the lower localized state. The
definition (7.29) above and the constraint that 0 Ã¡ 0 < ir/2 means that n
and h can take values 0,1,2,3 as long as (n+h) < 4. The particle
spectrum is easily calculated and is shown in table 7.2. (Note the n =
h = 0 is missing from this table as it simply corresponds to the ground
173
state.) The principle polaronic excitations are the pelaron (Q =
Â±e, e+ = Â±0.92AQ, 5 = 1.07SqÂ» E = 0.38 h ) the bipolaron (Q = Â±2e,
e + = Â±0.71 Aq, g = 1.235q. E2 = 1.80 AQ) and the tripolaron (Q = Â±3e,
e + = 0.38 Aq, 5 = 1 .76Â£q, E^ = 0.92 Ag). Note that the polaron also has
a stable excited state with Q = Â±9, e+ = O.38 A^, E, = 1.76Â£Q, E'
0.92 aq which is in every way, but charge, identical with the
tripolaron.
It is important to note that there is a neutral polaron (n = h =
1). This "polarexciton" (Q = 0, e+ = 0.92AQ, Â£ = 1.155Q, EQ = 1.80AQ)
can be considered as arising from the self trapping of both an electron
and a hole into the lattice. Since the polarexciton has energy
/2E < 2 i it is the lowest energy excitation of the polymer. Thus one
s u
expects single electron photoexcitation of polyyne to result in the
production of a polarexciton rather than a KK pair. The photoexcitation
of two electrons, on the other hand gives the electronic configuration
of a neutralneutral kinkantikink pair, as in polyacetylene.
Adiabatic Nonlinear Dynamics in Polyyne
To check the validity of the above calculations and particularly to
investigate the predicted scenarios during photoexcitation, the discrete
Hamiltonian was integrated directly. Figure 7.1 shows the evaluation of
a 98 site system with 196 electrons during the photoexcitation of a
single electron from the valence to the conduction band. We see that,
in agreement with the continuum calculation, a localized polarexciton is
produced. Figure 7.2 shows that two electron photoexcitation results in
the production of a KK pair which rapidly separate. By comparing with
174
figure 4.1 for the single electron photoexcitation of polyacetylene we
see that, as expected, the kink propagation velocity is higher in
polyyne that for polyacetylene. We also see in fig. 7.2 that a breather
is left behind by the separating kinks.
Using the formalism described in Chapter III we numerically
calculated the optical absorption on the photoexcited system, in the
linear response, dipole approximation. As a preliminary we calculated
the optical absorption of the purely dimerized configuration. Figure
7.3 should be compared with fig. 33 which shows the optical absorption
of the dimerized configuration of polyactylene. In figure 7.3 the
bandgap is 2A^ = 5.88 eV and the full band width is 4tQ = 12 eV. Figure
7.4a and 7.4b shows the optical absorption of the singly and doubly
photoexcited systems over the time period t  0.275  0.315
psecs (~ 1011 phonon periods). Figures 7.5a and 7.5b show the
difference, AaCml, between the absorption after and before
photoexcitation (ie. Fig. 7.5a = Fig. 7.4a  Fig. 73).
The optical absorption of the polarexciton can be understood by
comparing with the absorption of the polaron in polyacetylene. We find
that for the polarexciton Wq/Aq  0.63; this differs from the continuum
value due to the discreteness of the system. We expect, therefore that
the transition between the two localized a*'1'1 levels will be seen
P
at 1.2 Aq! this line should be strong. Transitions between the valence
band and the lower polaron level and between the upper polaron level and
(2)
the conduction band, ctp should be strong and at 0.37 AQ. Here we
find that otp1"* is stronger than , presumably because there are a
larger number of available transitions between the intragap levels of
(2)
the polarexciton than of the polaron. The absorption should be
175
weak and observable at 1.63 A^. Figure 7.5a indeed shows these general
features. The large width of these peaks is due to the oscillation of
the intragap levels of the polarexciton during photoexcitation. The
above band edge bleaching is over a large energy range due to the
narrowness of the polaronâ€”a narrow excitation requires a large number
Fourier modes, including those at high energy for its description.
Figure 7.5b shows a strong midgap absorption arising from
transitions into and out of the kink midgap levels. It has a high
energy shoulder arising from transitions between the localized breather
levels. Again this subband edge enhancement is compensated by a strong
above band edge bleaching over a wide energy range.
176
Table 7.1
Characteristics of soliton states of polyyne.
charge
gapstate spin internal
occupancy states degeneracy
2e
S = 0
i
e
*Â»*â€”
S = 1/2
4
0
S = 1 ,0
6
e
Â«
S = 1/2
4
2e
GO
II
O
1
177
Table 7.2
Characteristics of polaron states of polyyne.
charge gapstate spin internal stable
occupancy states degeneracy excited state
3e
2e
e
0
e
S = 1/2 4
"â€¢ S = 1 ,0 6
S = 1/2 4
â– 4 â™¦
S = 1,0 16
â€¢ â€¢ â™¦
S = 1/2 4
Â«
*
2e
â™¦â€”â™¦
S = 1 ,0
6
3e
S = 1/2
4
Figure 7.1
One electron photoexcitation of 98 site ring of polyyne
179
O
CO
Q
\
w
Figure 7.2
Two electron photoexcitation of 98 site ring of polyyne
181
O
CO
Q
\
3.
{CO)TO
182
Figure 7.3
Optical absorption of
2A = 5.88 eV, W
o
a 98 site dimerized ring of polyyne where
= 12 eV and the Lorentzian lineshape
has width  A /10.
o
183
E/A0
Figure 7.4
Optical absorption, a(w), of one electron photoexcited polyyne
(solid line) and two electron photoexcited polyyne (dashed line).
184
E/A0
Figure 7.5
Change in absorption, Aa(to) on photoexcitation of one electron
(solid line) and two electrons (dashed line) in polyyne.
CHAPTER VIII
CONCLUSIONS
It is generally believed that the unusual transport and optical
properties of the conjugated polymers arise from the presence of
nonlinear excitations in the system.^Previous work, however, has
dealt almost exclusively with the static properties of the various
theoretical models. Here we have undertaken the first systematic study
of the dynamics of the nonlinear excitations of one of these models.
In Chapter III we established the stability of the dimerized
lattice and the single kink. We further saw that the dynamics of a
single kink can be well understood by a continuum calculation. This and
the agreement between the continuum and discrete calculations of the
optical absorption of the kink, breather and polaron show the relevance
of the static calculations for the properties of the dynamical system.
The dynamical studies have not only validated static calculations
but have also introduced a number of important new ideas. In particular
the dynamics provide a unified picture of the short time evolution of
the photoexcited system. The basic theme appears in the photoexcitation
of transpolyacetylene in which a kink and an antikink are produced;
they separate at high speed leaving behind a coherent optical phonon
packageâ€”a breather. Variations on this theme appear throughout these
studies. We find that the pnotoexcitation of a polaron also results in
a KK pair and a breather as does two electron photoexcitation of
polyyne; whilst in site defected system one of the kinks traps and
185
186
additional nonlinear excitations may be produced. We also saw that in
cispolyacetylene strong confinement may result in the production of an
excitonbreather dynamical complex.
The breather has long been known to be an exact solution to the
sineGordon scalar field theory, in which it contributes to the
or
thermodynamics of the system through its lattice degrees of freedom."0
Here the concept of the breather has been extended to models with
fermionic degrees of freedom also, where the breather contributes to the
thermodynamics of the system not only through its lattice degrees of
freedom (as in the calculation of the dynamic structure factor) but also
through its electronic degrees of freedom (as in the optical absorption
calculations). We also saw that the breather is remarkably long lived,
with little tendency to disperse into incoherent phonons; whilst other
studies have shown that the breather only interacts weakly with a
kink.40 These suggest that, although the breather is only the
approximate solution of an effective Lagrangian, it may be an elementary
excitation of the discrete system.
At the outset of this work we stressed the interest aroused in
conjugated polymers by the remarkable conductivity properties of trans
polyacetylene. Although we showed in Chapter III that the kink is light
and highly mobile, in Chapter VI we saw that disorder may pin a kink to
a single lattice site or confine it over a short length of chain. This
strongly suggests that ballistic kink transport is not the conduction
mechanism in the defected system. The extension of the present work to
include more than one coupled chain may provide useful insights into the
charge conduction mechanisms. (Indeed, experiments suggest2 that the
187
conductivity of the lightly doped system may be via a variable range
hopping mechanism, similar to that in amorphous materials.^0)
One of the other directions in which the present work may be
extended is to add an additional crystal field to model the CH vibronic
modes. Although this may not be particularly interesting in
87
polyacetylene ' there may be interesting interactions between the bond
order wave and the charge density wave in the third and quarter filled
O O
band systems.00 The addition of damping and A.C. driving to the equation
of motion may result in, not only spatial and temporal chaos of the
lattice, but also chaotic dynamics of the density of states. (Such an
observation of semiclassical chaos may provide useful insights into the
illunderstood area of quantum chaos.
As mentioned in Chapters I and II this interest in dynamics is only
one thrust in the continuing efforts to understand the conjugated
polymers. It is likely that for a quantitative understanding of their
thermodynamics and transport properties a unified theory including
dynamics, electronelectron effects, quantum effects and morphological
effects will have to be constructed. Although this formidable task is
unlikely to be achieved in the near future, Baeriswyl and Maki^ have
recently used a generating functional scheme to perturbatively calculate
the effects of electronelectron correlations which, it is hoped, may be
integrated with the dynamical studies.
One of the other paths of fruitful analytic study may lie in
collective coordinate schemes. Although a partial differential equation
has an infinite number of degrees of freedom it may be possible to
identify approximate variables e.g. the translational and boundstate
modes of the nonlinear excitations, with which soluble dynamical
188
equations can be constructed; an example or which is the calculation in
Chapter III of the maximum velocity of a single kink.
APPENDIX A
PARAMETERS OF THE SSH AND TLM MODELS
"SSH parameters"
t_ = 2.5 eV
U
a = 4.1 eV A~1
K =21.0 eV A~2
M = 13 a.m.u.
a = 1.22 A
uQ = 0.042 A
A = 0.7 eV
o
Then using the relations
X = 2a2/irtQK
cu = Ak/m
"rescaled parameters"
tQ = 2.5 eV
a = A.8 eV A_1
K = 17.3 eV A2
M = 13 a.m.u.
a = 1.22 A
u0 = 0.1 A
A = 1.9 eV
o
(A. 1 )
(A.2)
c, =
vâ€ž/A0
F o
v = 2t a
F o
v = a) a/2
s o
(A.3)
(A.4)
(A.5)
189
190
one gets the parameters of the TLM Hamiltonian
A =
0.19
A =
0.34
*0
= 2.48
X
1rj4 1
10 s
*0
= 2.25
X
1014
1
3
4 =
7a
5 =
2.7a
VF
= 9.26
X
5 1
10 ms
VF
= 9.26
X
103
1
ms
V
3
= 1 .51
X
4 1
10 ms
V
s
= 1.37
X
104
1
ms
APPENDIX B
OPTICAL ABSORPTION IN THE SSH MODEL
This appendix outlines Horovitz's calculation^ of the optical
absorption in the SSH model.
Formalism
The change in the expectation value, 5<0(t)>, of an
operator 0(t) when coupled to an external Hamiltonian Hex, is, in the
linear response approximation,
6<0(t) >
'dt'<Â¥
iHt â€™
H e
ex
iHt
iHt
0(t) e
iHt
]lve
nt'
(B.1 )
where H in the zeroth order Hamiltonian and n * 0+.
For a system in an electromagnetic field Hp^ =  J.A(t) where J is
the current operator and A(t) is the vector potential.
Thus
. Tv . ft..,.m i iHt'T iH(t't), ~iHt i v nt' .....
= i dt' e A(t') +
J â€”oo O' 1 O
(3.2)
ft.t, , iHt iH(tt')T iHt' v nt' .
i dt' e A(t'
1 oo o' 1 n
For
E(t) = E e
o
iait
(3.3)
191
192
then
. /, > O â€”i cot
A (t) = â€” e
ico
(3.4)
Inserting a complete set of states 'F > with eigenvalues Sâ€ž and then
Ill â€œI
integrating over tâ€™ one gets
= i E
~i tot
iiot
i (s E loin) i(E L
mom mo
tâ€”t} â€”^  <â€™F J Â¥ > Â¡ 2 (3.5)
c, u)in) iio 1 o1 1 ra 1
Thus the conductivity, a(
Reo(m) = â€”
l Re
N
h e
o
iuit
E <Â¥ IJll > I2 [ 6 (E E iol
Nio 1 o1 1 m 1 L v m o 1
m
Ã³ (E E (o)1
o m J
(3.6)
For T = 0, Em 2 Eq and 0 so
Re a (to) = E k'f Ijl'F > IÂ¿ Ã³(e E io]
Nio 1 o1 1 m 1 v m o 1
(B.7)
which is the conductivity in the linear response dipole approximation.
The current operator
The charge density, p(n), is given by
p(n) = eC C
n n
(B.8)
where e is the electronic charge. The charge density is related to the
current by the continuity equation
193
  = j(n)  j (nâ€” 1 )
(3.9)
where the RHS is the net flux of current out of site n. For the SSH
Hamiltonian, the electronic part of the Hamiltonian, Ho1, is
Now
el
= I
[t +
L o
afu  u J][C+C ,
n n+ r JL n n+1
+ C , C
n+1 n
(B.10)
9C
ST'
(3.11)
and so
k ÃœP
e at
3C
r n
at Cn
+
+ c
n
(3.12)
Thus
i lÂ£
e at
+ a u
(u
u J]
n+1; J
C .C
n+1 n
[to + aK1
n1
C
n
(B. 1 3)
[t 
L o
a (i
n+1
)] c;c
. ,  [t +
n n+1 L o
afu ~u)]cc .
^ n1 n; J n n1
Using
J(n)
a p (m)
at
(B.14)
then
194
j(n)  ie [tD * a(un  V,) ] [c^,  C^cJ (B.15)
which is eq. 3.27.
Note that for
(B.16)
(B.17)
and thus (B.15) is correct even in the presence of this electron
electron interaction term.
The Conductivity
The eigenfunction a has amplitude f (n) on site a. Define
a
eigenoperators
ee
E V(nm) p(m) p(n)
nm
p , H 1=0
â– n eeJ
f
a
Z f (n) C .
a n
Because f (n) is a complete basis set, then
a
Expanding
C
n
= Z f (n) f .
a cc
a
Â¥ > = n f*0>
m B
(B.18)
(B.20)
(B.21)
then
195
%!JIV = _ie ln [ValWi)][fa(n)Vn+l) â€™ fa(n+1)fB(n)] (B*22)
where a(B) is an unoccupied (occupied level).
Thus
Re o(u))
ire2t2 1 E M
O coN 1 ^
otB
Ã³ (e.e oj)
3 a
(B.23)
wi th
M n
ct, 3
Z [1
n
u â€”u
n n+1
)][
f (n)f (n+1)
ot 3
f (n+1)f (n)]
a 3
(B .24)
when e and eâ€ž
a 3
are the single particle energies and
E 
m
(B.25)
Equation B.23 and B.24 (eq. 3.28 and 329) are again unchanged by the
addition of Hge.
Sum Rule
The sum rule can be established by evaluating a commutator in two
different ways. Consider
Â±2 Z [p(m) Cp(n), H]] eiq(n m). (B.26)
e nm
Now the commutator,
196
[A, BC] = [A,B]C + B[A,C]
(B.27a)
and the anti commutator,
[A, BC ] = {A , B} C  3 {A , C}
(B.27b)
Then using
{C ,C } = 5
n m n,m
(B.28a)
{C ,C } = 0
n m
(B.28b)
one finds
^2 z Up(n) ,H]] eiq(n ml
eÂ¿ nm
E [t + afu u J][C+C 6 ,  C+ ,C Ã³
1 L o v n n+1;jL m n n+1,m n+1 m n,mJ
nm
[t + a[u .u )][C C 6 , ~ C C 6
L o â€™â– n1 n'JL m n n1,m n1 m n,m
(B.29)
+ [t + afu u J1[C+C 6  C+C 5 , 1
L o n n+1 J m n+1 n,m n m n+1,mJ
ft + afu u )][C+C ,6  C+C 6 , ]} e
L o n1 n'JL m n1 n,m n m n1 ,mJi
iq(nm)
4 sin^(q/2) H ^.
(3.30)
Alternatively the commutator can be evaluated directly from the
197
conservation law.
[ p (n) ,h] = i  j (nâ€” 1) ] (B.3D
which gives

_ j j <3j(n) . J(n.)a>[elq(n,n)
ran S 6 a
(B.32)
= 4sin2() Z Z <8  j (n) a> elq(n 'n'1 + [q Â»â– q] .
nm 6 "a
(3.33)
And thus
2 Z
nm
<81j(n)a>
E E
6 a
cos[q(nm)] = e
(3.34)
So for q = 0
J"Rea(a))da) = J Z 1 = g <0Hgl0>e2 (3.35)
and thus
C 00 7T 0
JoReo(u))dÃ¼j = 
(B.36)
198
which is eq. 330. This is also correct in the presence of Hge.
numerically the sum rule is checked by
E
aS
I /(VEJ
el
>
So
(3.37)
APPENDIX C
THE ANALYTIC BREATHER
The Lagrangian for the dimerized half filled band is
L
A_
Â¿U
2, ' 2
VF A
24a'
(C.1 )
It is important to note that in question (C.1) a term of the
â€¢ 2 2
type A /A is omitted. This arises from dynamics generated by the
valence band electrons. This is not included in the calculated
numerical studies and will, therefore, be dropped here.
The equation of motion from (C.1) is then
2A
2Ec 2 (aâ€™2Aa") l
= Ain â€”  vF â€”  â€”
12A
to
R
(C. 2)
We are looking for a solution of (C.2) consisting of a nonlinear package
of optical phonons, which are small deviations from the purely dimerized
lattice. If an optical phonon has wavevector  e then it has
2 q?
frequency  e . Now use multipletimescale perturbation theory7 to
extract the part that varies slowly with both space and time. So let
A(x,t) = A 0(1 + 6(x,t))
(C. 3)
r * i 2
<5(x,t) = e[A(X,T) exp iwDt + A (X,T) exp iDtJ + e 5,(x,t)
a K
199
200
+ Â£362(x,t) + 0(e4) (C.H)
where the variables X and T are
X = EX
T = e2t.
3
Now expand (C.2) to order e . It is important
differentiation is with respect to t and x.
Thus e.g. râ€” A (X ,T) exp ioint = sAv(X,T) exp itont. (C . 6
d X K a K
It is straightforward to show that substituting eq. 0.4 into eq. C.2
gives
(C.5a)
(C.5b)
to note in (C.2) that
0 = e0^  In 2Eg/Aq] + e1 (0)
+ Â£
!u.
(1)
2 12â€œ,,XX eXÂ» (21V> * lSi2
R
+ A*2 exp (~2ia)pjt)] +
e3[Â«2 * i2>xx * UTÂ®xpÂ«Â»Bt)  4 expiiÂ»Rt)}
â€œr r
2 2
f , * i AJ 1 i 12
Axxexp(iwRt) + Axx exp(~io)Rt)}  â€” exp(3iwRt)   A  Aexp iu)Rt
R
*3
 iA2A%xp U,Rt  â€”gâ€”exp(3iÃ¼)Rt) * (Â«, * 4 6,iXX).
r * i i 4
(A exp iÃ¼) t +â– A exp iu>ntH + 0(e ).
rv K
(C.7)
201
Now each order of the expansion in e can be set to zero individually
__ Q
which gives for the e term
= 2E exp
c
ikh
(C.3)
Now if we choose
i p % o o
6 ^ (x,t) = ^ (A exp 2iooRt + A exP _ 2iuÃ¼Rt}  A  (C.9)
2
and look at the z term, then the 5, term is of higher order in z and
1 ,xx Â°
2
can be formally dropped. Thus the e terms cancel exactly.
v
Terms in 6p(x,t) appears only linearly to order z and thus merely
represents optical phonon modes. For the pure breather solution,
therefore, set 3 0.
Now substituting for 5^(x,t) from equation (C.9) and again noting
that the term 6^ xx is of higher order in e and can be dropped then we
get
0 = e3[ exp(ia) t) (â€” At  A  Â¿  A  2a) + c.c.]. (C.10)
R ai T 12 xx 3
n
Now equation (C.10) is the Nonlinear SchrOdinger Equation, whose
solutions are well known. In particular there is a spatially localized
solution
A(X,T)
p/py p
b sech  exp ia)Db T/3
t, K
(C.11)
then using relations (C.5) this gives
202
A(x,t) = c sech
2/2ex . 2, ,0
â€”â€” exp iu_e t/3
t, ri
(C.12)
where e = be
Substituting back into equation (C.4) gives
t ,,  , 2/2 ex r , 1  2 â–
5 (x,t) = 2e sech â€” cos [ujDt(1  ttÂ£ )
5 R 3 â–
12 ,2 2/2ex r <â€¢ 1 â€” 2
+ sech â€”â€” [cos 2o)^t[1  â€”e J â€” 3J + 0(e )
(G. 1 3)
'lote that there is a time independent part that reduces the effective
band gap.
Expanding to appropriate order in 5 the energy density is
n
(C.14)
3
To order e only the first two terms contribute. They are easily
evaluated by remembering that energy is conserved and choosing a
convenient instant to evaluate the integrals. This gives the classical
breather energy as
E
2/2A
0 
[1
10 2
Â£. +
27
o(Ã4)]
The Bohiâ€”Sommerfeld quantization condition is
(C.15)
J = J pÂ¿dA dx = 2mr
(C. 1 6)
203
2
1TV
dx dt
This gives
2nn
V2A
0 ~r 1
 e 1
7_ ~2i
27 ^
(C.17)
(C . 18)
Inverting this to and substituting into (C.15) gives
2 2 2
n TT uJ
E = nwR [l 
72A
0
r
L
(C.19)
APPENDIX D
CLASSICAL DYNAMIC STRUCTURE FACTOR OF THE BREATHER
Define the classical dynamic structure factor by
( 2tt )
I dx1dx2dtldt2exp [iu^t^  iq^x^]
(D. 1 )
where u(x,t) is the lattice displacement at location x on the chain and
t is the time. The breather density is n (w ;T) and < > is a thermal
D D
average. (A thermal distribution may, however, not be relevant due to
the presence of nonequilibrium creation processes.)
Applying the BakerHausdorf therorem
= e 2Wexp (D.2)
where e is the DebyeWaller factor.
Now looking at one phonon processes only
(D. 3)
But
un(x) = na + (1)n un(x)
(D.4)
where n is the lattice site and u (x) is the staggered lattice
n
204
205
displacement. Then
u(x2tx)> â™¦ (1)n1 (1)n2 Ã¼n(xtt,) Sâ€ž2
(D.5)
The other terms can be absorbed into the DebyeWaller factor. Now in
, , , . \n i it x/a
the continuum limit (1) = e , so
S(q,oj) = e
2W nB(
(2ir)â€˜
Jdx 1 dx^dt 1 dt^exp[ ioi(t1t?)i(qi;/a) (x^x,,)]
1 2'
1 2'
(D.6)
Assuming that the breather behaves as a relativistic particle, i.e.,
x *â– Y(xvt) (D.7)
t > YÃtvx/c')
v o'
where Y is the Lorentz factor and cQ is the velocity scale. We also
assume that the thermal average can be implemented by integrating over a
classical velocity distribution, P(v).
Thus
_ py r O o
S(q,u) = e â€”â€” q^JdvP(v)[ J dx dt exp[iYÂ¿(q+iT/avto/c')x exp iâ€™
(2 ttT 11 Â° 1
(ujqv) 11 ]u (x 111 ) ] [ x 1 *â– x2, t1 *â– t2]
(D.8)
For simplicity assume the MaxwellBoltzmann distribution can be used so,
206
Y *â– 1
where a = ED/kT.
D
Now for the breather
u(x,t) = 2e(wa) sech(x/d) cos
1
+ â€”
3
__ ^ ^
e (w0) sech (x/d)[cos 2uQt
D D
where
 \ ;21
and
d = 5/2/2 e.
Define the qdependent from factors
f1 (q) = Jdye 1 qysech(y/d)
ird sech(irqd/2)
(D.9)
(D.10)
~ 3]
(D.11)
(D.12)
(D.13)
f2(q) s Jdye iqysech2(y/d) = irqd"co3ech(Trqd/2)
(D.14)
207
Now performing the integrates over x and t1 in (D.8) gives
o(q,to; = â€”
(2it ;
i 1 â€¢i
6 (Ã¼)(jjgqv) ] + â€” e f (k) [â€” 6 (w+2wg~qv) + â€” S(oj2a>2qv) (D.15)
35 (uqv) ]Jdx2dt2exp[i(q+ir/avio/c^)x2]exp[ i (<Ãºqv)t2]n(x2t2)
2
with k = q+ir/a  vm/câ€œ (D.16)
Now perform the v integral followed by the integrals over x2 and t2 (the
integral over t? introduces a 5function which forbids terms coupling
e.g. Ã¼j+iÃº with (jÃ¼)D).
D D
One finds
2W1
JdvP'JR(v) {
P1
ef
(k )[5 (to+u) qv)
D
: (q, to)
2W VVT) r~2riâ€ž1r,^,2 â€žNRrâ€œ+uB
(2ir)"
[^{ifâ€™Cicpr pr,Kir)
1 r 1,2 â€žNRrâ€œ_
M P hf
!?2(kÂ¡) I2 PNR(^) + 36 f2(ko)2pNR(m/q)}]
(D.17)
wnere
. ii)Â±nu_.
^ = d [q + ir/a  ( ]]
n L M 2 v q 1 J
c
o
(D.18)
kQ = d 1 (q + it/a .
Co q
(D.19 )
APPENDIX E
CONTINUED FRACTION SCHEME FOR IMPURITY LEVELS
Formalism
The local electronic density of states at site n is
p (E) = â€” Lim
n it . A+
Im G (E+in)
nn
(E. 1 )
where Gnn is the single particle Green's function. For the Hamiltonian
H
X
n
E
n
C
n
+
v
n
t
n,n+1
C + h.c.)
n
(E .2)
Dyson's equation may be written
G(z) = (zH)
(E  3)
7 + 7 HG(z)
z z
(E.U)
Taking matrix elements between the basis of Wannier functions i>}
gives
z < i IG(z) I j> =
+ e ^ +
E
k
t kXk G j >
n,n+11 11
(E.5)
208
209
which may be written
z G..(z) =6.. + e.
ij ij i
Vz)
(E .6)
Iterative elimination of offdiagonal terms gives
Gii(:
(E .7)
where
t2
i ,11
(E.8a)
(E8b)
i + 1 â€œi + 1 , i + 2
Undefected Lattice
V
'i1
and
t2 .
i,i + 1
 t
Let us calculate the density of states at site zero for the
undefected lattice. Let t2^ 2i+l = â€™ ^2i 2i1 = tB
then
ZR(z)
(z)
(E .9)
210
and
E,,U) 
2  Â£22(z)
(3.10)
Since E^ (z) = Eâ€ž(z) on an infinite line then the continued fraction is
Â¿Â¿ H
periodic and readily summable
Â£r(z) â– [z 11 (z  VzÂ»'r'.
(E.11)
Solving for ED(z) gives
H
2zEr(z) = (z2 + t2
,.2, r, 2 , 42 42s2 ,..2 2i 1/2 , 10>
tg) Â± [(z +  tg)  4t^z J (E.12)
Using
ZL(:
;) =
 V:
(E. 13)
gives
G00(z) = ^z ~ fcB^Z ' Zr(z)) ' â€œ ZR(z)i ' * (E.14)
From equation (E.1) there are contributions to the density of states if
Gqq(z) is complex. Combining (E.12) and (E.14) yields contributions to
the density of states at
, 2 .2 .2.2 ,..2 2 . .
(E.15a)
211
, 2 .2 .2,2
(Z â€˜ lB * â€˜aâ€™
0 ?
4t,z < 0
A
(E.15b)
Assuming > tB this gives
E < t. + t_
A B
(E.16a)
E > t.  t_
A B
(E.l6b)
E < fcA + S
(E. I6e)
E > t  tâ€ž
A B
(E. 16d)
Now for tA = t0 + Aq/2 and tg = tQ  AQ/2 the bands are in the region
A < E < 2t and A > E > 2t .
o o o o
(E.17)
That is we have a full band width of W = 4tn and a band gap of 2A
The Site Defected System
For an impurity at site 0 we add to the SSH Hamiltonian the term
H
s
"Os
(E.18)
then
G
00
(z)
z â€¢ v0  Â£l(z)  ZR(z>
(E.I9)
212
where I. (z) and ZD(z) are given by equations E.12 and E.13 above. There
L n
can be contributions to pq(E) at the poles of Gqq(E)
2 * vo  V2)  V2> â– 0
(E.20)
It is easy to show that this gives
E =
Â± 4(Vq Â± Y ) L A v 0 ~ 0
[Ht2 + (V Â± Y )2 + MD] (E.21 a)
wnere
D2  [lit2 
(VQ t Y0)2]2  16(V,
0 1
(E . 21 b)
ana
2 2 2
Y0 = 4tA + V0
(S.21c)
where L and M can take the values of Â±1.
Expanding eq. E.21 to lowest order gives
E  L(t# * MtB)(l  V0/t#)
(E.22)
This identifies the various band edges with particular signs of L and M,
and shows that for VQ < 0 states are removed from the bottom edges of
the conduction and valence bands, whilst for VQ > 0 state are removed
from the top edges of the valence and conduction bands.
213
Bond Impurity
The bond impurity modifies the transfer integral between two single
sites e.g. site 0 and site 1.
So
t
01
t^
R
+ w
o
(E.23)
A little algebra then yields
E â€˜ 1 5T * 4 > * " tÃtf  t2)2 * 4t2t2 ]1/2 CE.24)
E
Using the physical arguments of Chapter VI we find
N = 1 tE < tA , tA > tQ (E.25a)
tE > tA , tA < tB (E.25b)
and
N = +1
II > tA
(E.26)
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3I0GRAPHICAL SKETCH
Simon Robert Phillpot was born in Kettering, England, on January 3,
1959. He is the son of John H. Phillpot and the late Elizabeth E.
Phillpot. He was reared in London and after attending Merchant Taylors'
School, Northwood, he went up to Lincoln College, Oxford, to read
physics. A college exhibitioner in his schoolsman year he was awarded a
B.A. in 1930. He took up the President's Scholarship at the University
of Florida in the Physics Department and on passing the qualifying
examination in 1931 he began research under Dr. Pradeep Kumar. Whilst
doing his doctoral research he spent extensive periods working under Dr.
Alan Bishop in the Theoretical Division and the Center For Nonlinear
Studies at Los Alamos National Laboratory. On graduation he will take
up a postdoctoral position at Xerox Webster Research Center, working
with Dr. Michael Rice.
219
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy. r\
.Ukxjf/
/
Pradeep Kumar, Chairman
Associate Professor of Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of_
Doctor of philosophy.
Charles L. Beatty
Professor of Materials Science
and Engineering
I certify that I have read thi
conforms to acceptable standards of
adequate, in scope and quality, as
Doctor of Philosophy.
s study and that in my opinion it
scholarly presentation and is fully
a dissertation for the degree of
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
David B. Tanner
Professor of Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy. %
\
v
A
Hi.;
Samuel B. Trickey
Professor of Physic
This dissertation was submitted to the Graduate Faculty of the
Department of Physics in the College of Liberal Arts and Sciences and to
the Graduate School, and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
May 1985
Dean, Graduate School
UNIVERSITY OF FLORIDA
3 1262 08554 1356
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