
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00029878/00001
Material Information
 Title:
 Optical properties of segmented and oriented polyacetylene
 Creator:
 Woo, HyungSuk, 1955
 Publication Date:
 1990
 Language:
 English
 Physical Description:
 vi, 154 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Conductivity ( jstor )
Doping ( jstor ) Electrons ( jstor ) Iodine ( jstor ) Oscillator strengths ( jstor ) Polarons ( jstor ) Polyacetylenes ( jstor ) Reflectance ( jstor ) Solitons ( jstor ) Temperature dependence ( jstor ) Dissertations, Academic  Physics  UF ( lcsh ) Physics thesis Ph. D ( lcsh ) Polymers ( lcsh )
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1990.
 Bibliography:
 Includes bibliographical references (leaves 148153).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by HyungSuk Woo.
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 University of Florida
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 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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Full Text 
OPTICAL PROPERTIES OF SEGMENTED AND ORIENTED
POLYACETYLENE
BY
HYUNGSUK WOO
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
1990
ACKNOWLEDGMENTS
It is my great pleasure to thank my advisor, Professor D.B. Tanner, for his guidance, patience, valuable advice, and support all throughout my graduate career. I would like to thank my supervisory committee, Professors N.S. Sullivan, S.E. Nagler, H. Monkhorst, G. Ihas, and W.B. Wagener.
I would like to thank Professor A.G. MacDiarmid, Dr. Theophilou, and Dr. Arbuckle at the University of Pennsylvania for their nice and fresh samples, which were essential to complete this dissertation. I would also like to thank Dr. S. Jeyadev at the Xerox Webster Research Center for his valuable discussions and suggestions on all levels of physics.
I would like to thank Drs. X. Q. Yang, Y.H. Kim, I. Hamburg, S.L. Herr, and K. Kamaras for their help during my fiveyear stay in the "Tanner's Group." My fellow graduate students deserve thanks for their friendship and cooperation in development of the equipment and computer software. In particular, I would like to thank C.D. Porter for his help in converting the software from PDP 11 computer to personal IBM computer, which was a great convenience for data analysis.
I would like to thank the staff members of the Physics Department shop and the engineers of the condensed matter physics group for their technical support. I would like to thank my Korean fellow students for sharing friendship and useful conversation.
ii
Special thanks go to my wife, KumOk, my son, SoonBo, daughter, JungYoon, and my parents, whose love, care, understanding, and support were most important during my studies.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ................................ ii
ABSTRACT ..................................... v
CHAPTER
I INTRODUCTION ............................ 1
II INSTRUMENTATION AND EXPERIMENT ........ 9
IR and Optical Technique ..............  9
Instrumentation and Data Acquisition  15
Measurements ............................ 17
Sample Preparation and Doping Technique  19
III BASIC THEORY .............................. 36
ElectronPhonon Interaction and
Peierls Transition ................... 36
Soliton Excitation ........................ 45
Polaron and Bipolaron Excitations .......... 60
Dopinginduced Infrared Active
Vibrational Modes ................... 63
Electronic Excitation ................... 66
Transport ............................ 69
IV SEGMENTED POLYACETYLENE ............... 95
Segmented Polyacetylene: (CDHy)x ......... 95
Infrared Active Vibrational Mode (IRAV)  95 Midgap and Interband Absorptions ......... 102
V ORIENTED NEWPOLYACETYLENE ............ 124
The NewPolyacetylene ................... 124
The dc Conductivity .. ......... 124
Temperature and Polarization Dependence
in Infrared Absorptions .............. 126
Midgap and Interband Absorptions .......... 129
VI SUMMARY AND CONCLUSIONS ................ 145
REFERENCES ...................................... 148
BIOGRAPHICAL SKETCH ............................. 154
iv
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
OPTICAL PROPERTIES OF
SEGMENTED AND ORIENTED POLYACETYLENE
BY
HyungSuk Woo
May 1990
Chairman: David B. Tanner
Major Department: Physics
The dopinginduced infrared and optical absorptions in polyacetylene have been measured. The samples studied were oriented new[CH] and segmented polyacetylene, [CDH ] , in which either 15% or approximately 27% sp3bonded CDH units had been incorporated. All samples were doped with iodine by insitu vapor doping (segmented polyacetylene) and chemical doping (newpolyacetylene) techniques with various dopant levels. We have also studied the temperature and polarization dependence of the ac and dc conductivity of new[CH] .
All doped segmented polyacetylene showed the usual dopinginduced infrared active vibrational (IRAV) modes at 745 cm~1 and 1140 cm' and at midgap. The oscillator strength of these absorptions increased with
v
increasing iodine concentrations, even for the samples with 27% sp3 sites. The interband absorption data suggested that the sp3 defects of [CDH ] were not uniformly distributed over the polymer chain. For a sample with 27% sp3 hybridization, we have found a new broad band around 1.12 eV with relatively lower intensity for lower sp3. We assigned this as an absorption due to the neutral solitons.
In 550% stretched new[CH],, doped with iodine at level 6% (I), the dc conductivity was about 10000 2 cm~1 at 300 K, decreasing as temperature is lowered and reaching 2500 Olcmd at 84 K. In contrast, the farinfrared oscillator strength in the 500 cm' region, where the "pinning" mode is usually seen, increased and shifted to lower frequencies with decreasing temperature. This effect was observed for polarizations parallel and perpendicular to the stretching direction. This sample displayed highly anisotropic behavior in IRAV modes, with substantially greater a 1(w) for polarization parallel to the chain direction. This anisotropic behavior was observed over the range of frequencies including the midgap and interband region.
vi
CHAPTER I
INTRODUCTION
In conducting polymers, the electronic structure is determined by the chain symmetry (e.g., the number and kind of atoms in the repeat unit, etc.) and by the concentration of charge carriers. The result is that such polymers can exhibit semiconducting or even metallic properties. Polyacetylene, (CH),, is a particularly interesting for in this case the 7r band is half filled, implying the possibility of metallic conductivity [1]. In polyacetylene, the 7relectrons are delocalized along the polymer chain due to the strong intrachain bonding and weak interchain interaction, which results in this polymer being a one dimensional electronic system.
Polyacetylene consists of long quasione dimensional chains of
(CH) units aligned in a partially crystalline lattice. Three of the four carbon valence electrons are in sp2 hybridized orbitals, where two of the atype bonds connect neighboring carbons along the polymer backbone and a third forms a bond with the hydrogen side group [1]. Thus, there are two possible types of molecular structure for this polymer with optimal bond angle 1200 between these three abonds, cis(CHI and trans(CH) (see Figure 1.1). There are two and four CH monomers per unit cell in trans(CH) and cis(CH) respectively. The
1
2
remaining valence electron, a electron, has an important role in this conducting polymer forming a Kbond with its chargedensity lobes perpendicular to the plane defined by the abonds. While the abonds form completely filled bands, the Kbond leads to the partially filled energy band.
If the chain lengths were equal, pure trans(CH)x would be quasione dimensional metal with a halffilled band. This configuration is, however, naturally unstable due to the Peierls instability [2], which leads to a dimerization distortion; adjacent CH groups move toward each other forming alternately short (or "double") bonds and long (or "single") bonds. By symmetry, there are two lowestenergy states, A and B (see Figure 1.2). This twofold degeneracy leads to the existence of nonlinear topological excitations [3,4], solitons, which appear to be responsible for many of the remarkable properties of trans(CH)X. In cis(CH) , no such twofold degeneracy exists [1].
The trans(CH) is the thermodynamically stable form. Complete isomerization from cis to trans(CH) can be accomplished after synthesis by heating the film to a temperature above 150'C for a few minutes [1]. It is also known that cis(CH)x can be changed to trans(CH) by doping [5]. Trans(CH) films was first prepared by Shirakawa and Ikeda [6] with film thicknesses varying from <10~5 cm to about 0.5 cm. This films consist of randomly oriented fibrils with average diameter about 200 A [6]. The bulk density of unoriented Shirakawa polyacetylene is about 0.4 g/cm3.
3
Xray studies of trans(CH) [6,7] have shown that the films are highly crystalline (see Figure 1.3). According to these Xray data, the chain structure is dimerized with the distortion parameter (the difference between short and long bonds) u00.03A and the length of a unit cell 2.46 A.
Recently, a new technique has been developed by Naarmann and Theophilou [8] to synthesize polyacetylene with fewer sp3 (< < 1 mol%) defects (CH2) than in Shirakawa(CH)X. The reduction of sp3 [1] defects implies longer conjugation length and fewer crosslinks, which leads to higher dc conductivity. Naarmann and Theophilou [8] have reported the dc conductivity of heavily doped film by iodine was about 20000 Q~1cm'1 which is much higher than one for Shirakawa(CH),, 10103 0~1cm~1. The bulk density of stretched film was about 0.85 and 1.12 g/cm3 before and after doping respectively. Xray studies of these films, however, show no major differences from Shirakawa(CH)X.
The undoped polyacetylene (from now on, "polyacetylene" or "(CH) " will be referred to as tran(CH) through this dissertation) has semiconducting gap (or Peierls gap) of 1.45 eV [7]. Electron spin resonance (ESR) measurements by Davidov et al. [9] show that the neutral solitons which exist in undoped (CH) carry spin 1/2, whereas the charged solitons introduced by doping carry no spin. From optical measurements [7], it is known that the charged solitons have a broad absorption band at midgap and doping induced infrared active vibrational modes (IRAV modes) are shown in the infrared region. The
4
soliton model first introduced by Rice [101 and Su, Schrieffer, and Heeger (SSH) [3,4] and its continuum version by Takayami, LinLiu, and Maki (TLM) [11] have successfully described the ground state and nonlinear excitations in polyacetylene with good agreements with experiments. Horovitz [12] calculated the infrared conductivity from which he obtained the doping induced IRAV modes. Mele and Rice [1315] predicted a "pinning mode" a characteristic infrared activity associated with the oscillation of the charged solitons bound by Coulomb interaction to ionized impurities.
In heavily doped polyacetylene, the specific heat is linear in temperature [16] and there is a somewhat temperature dependent Pauli susceptibility [17,18]. These results are consistent with a metal. The temperature dependence of dc conductivity, however, is not metallike. Instead, it decreases as the temperature is lowered. Kivelson [1921] proposed a intersoliton hopping model to describe the conductivity in lightly doped polyacetylene. Mele and Rice [22] predicted a transition to a metallic state as a consequence of closing the Peierls gap because of increased disorder, which introduces states into the gap. According to this model, the Peierls gap would exist at high doping level but be incommensurate with lattice. Then the introduction of disorder puts states in the gap, leading to a "dirty" metal with a finite density of states at EF.
The rest of this thesis is laid out as follows. Chapter II will describe the optical technique and apparatus including the experimental
5
details. Some of the basic theory for polyacetylene will be given in Chapter III. Chapter IV and V will discuss the data and results obtained from this work and conclusions are in Chapter VI.
6
(a)
H
C=c
H
/
H H C C
H
C== C
/
7
N
H H
C=C
H
(b)
H
I
L;
H
C
C
H
H
C
C
H
H
C
C
H
C
H
Figure 1.1 Structual forms of polyacetylene (a) cis and (b) trans(CH)
7
H H H H H H
C C C
H H H H H H
H H H H H H
I I I I I I
H I I I I
H H H H H H
Figure 1.2 The twofold degenerate ground states of trans(CH)
8
a
Figure 1.3 The crystal structure of transpolyacetylene obtained from Xray scattering. The arrows represent the bondalternation atomic displacement.
(From reference 7 )
b
Al"
''10
o :H
a=4.24 A b=7.32 A c=2.46 A
01_
CHAPTER II
INSTRUMENTATION AND EXPERIMENT
IR and Optical Technique
Fourier Transform Spectroscopy
The Michelson interferometer, which is well known for demonstrating the interference of two mutually coherent light beams, was supplanted years ago by the introduction of higher resolution instrumentsFabryPerot etalon, Michelson echelon, etc. During the 1960s, however, it became more useful for measuring relatively complicated spectra in the infrared region for the following reasons [27]:
i) The availability of computers makes the Fourier transformation of a measured interferogram to a spectrogram relatively easy.
ii) The signal to noise ratio for an interferometer is greater than that of a conventional dispersing spectrometer, due to the Fellgett advantage [28].
In this section, these aspects will be discussed briefly in turn. A more detailed description of the method will be found elsewhere [28].
9
10
Fourier transformation of an interferogram. As can be seen in Figure 2.1, the basic principle is the superposition of two wavetrains. From the Figure 2.1, the addition of the wavetrains which travelled different distances 2xI and 2x2 yields the superposed wave
y = a sin(27r(ft  vxI)) + a sin(27r(ft  vx2))'
= 2a cos(7rv(xI  x2)) sin(27r(ft  v(x1 + x2)/ 2 )), (2.1)
where v is the wavenumber and f=cv the frequency of the light. The sine term represents a wave motion and the cosine term is the amplitude. The intensity S(v) can be obtained from equation 2.1 as
S(v) = 4a2 cos20rV)
= 2a2 + 2a2 cos(27rv3), (2.2)
where x x2 =3, the path difference. If the radiation is not monochromatic but consists of a continuous spectrum each component having intensity S(v), then, for some path difference 3, the last equation must be modified by writing the intensity I(J)
1(4) = J'(v) dv + fS(v) cos(27rv6) dv, (2.3)
0 0
11
This equation will give the appropriate distributions of a source interferograms. When 3  ,
im I(J) = S(v) dv = I() = 1(0). (2.4)
Equation 2.4 holds because the cosine term in equation 2.3 will oscillate rapidly and average to zero as 3 approaches infinity (essentially, cos(27rv3) is more rapidly varying than S(v), which is, in turn, equivalent to saying that once all the structure in S(v) is resolved, I(3)1(co)). So, the interferogram can be written as
I(6)  I(o) = S(v) cos(27rv3) dv . (2.5)
0
The spectral distribution, S(v), is the inverse Fourier transform of interferogram. Using the fact that S(v) = S(v), we obtain
S(v) = c f (I(j)  I(o)) e2nivJ d5 , (2.6)
where c is a constant. The spectrum derived is exact only if the interferogram, in accordance with equation 2.6, covers all path differences from 3 =  to +. In practice the limits must be finite, which causes broadening of the line shape and side lobes, or "feet", to occur in sharp spectral structures [29]. Apodization [28] is necessary to remove these "feet" arising from finite maximum path difference.
12
The resolution of the obtained spectrum is limited to A v=1/3 and the maximum cutoff frequency of the spectrum, v , is given by
S. 1 = 1 . , (2.7)
min 2 vmax  m
where m is a sampling interval. Equation 2.7 requires the sampling interval must be small enough to eliminate aliasing in entire frequency range. Proper filtering is also necessary to remove the frequencies higher than the maximum cutoff frequency [28].
Fellgett advantage. The Fellgett advantage is the term for the enhanced signal to noise level of an interferometer as compared to a dispersing spectrometer. The noise in a detecting arrangement normally used in the infrared arises from statistical fluctuations in the motion of the electrons comprising the electronic current. This fundamental noise is known as "Johnson noise." The fluctuations are in fact a Brownian movement of the electrons.
If the current i is averaged over a time t in a circuit of resistance R, the fluctuation current is given by, setting the energy =thermal energy,
[(6v)2/R] t = (C5i)2 R t ~ k T , (2.8)
where k is the Boltzmann's constant and T is the temperature at the detector. Equation 2.8 implies that the Johnson noise, Ji, is
13
1/2 1/2
proportional to t/2 and to T2. Thus, it is useful to cool the detecting system. Fellgett pointed out that the total number, N, of resolved elements of a spectrum is exposed to the interferometer detector for the total measuring time t. On the other hand with a dispersing spectrometer, the total observing time for N different wavelengths is divided so that the measurement of each element is allowed only a time t/N. In accordance with equation 2.8, the noise 1/2
will be greater for the dispersing instruments by a factor N'
Optical Spectroscopy
The grating monochromator is used at higher frequency range, from near IR to UV, where the Fellget advantage loses its importance due to the increasing photon noise in the detectors used.
In general, the equation for diffraction grating is written as (see Figure 2.2a)
pA = b (sin 4 + sin 0), (2.9)
where p is called the order of the spectrum, 0 is the angle of the incident light and 6 is the angle of the diffracted light both measured from the normal direction to the plane of the grating. In Figure 2.2a, the path difference between the two rays shown is defined as I AB +AC I . For p= 0 (zero order) the path difference is zero for all directions, r0=0 (see Figure 2.2b). For other orders (p = 1, 2, etc.) and for fixed
14
0, the extra path length must equal pA for constructive interference. This leads to equation 2.9 showing that different wavelengths are diffracted into different angles 0. That is, the light is split up into a series of spectra  first order spectrum etc. The dispersion of the spectrum can be obtained by differentiation of equation 2.9.
d, = cos . (2.10)
Thus the dispersion of the spectrum, dO/dA, is greater for higher orders of the spectrum and for smaller values of grating spacing. For an almost normal incident light, from equation 2.10, we note that the cosO changes very slowly with 0, so that the dispersion is approximately constant and we have the convenience of linear interpolation. In most grating spectrometer, the mechanical drive is arranged to give a linear wavenumber output. Thus, for a normal incident, dA is directly determined by do, rotation angle of grating at each step. For a fixed do, the resolution of a spectrometer, AMA, can be determined by the slit width because the smaller angle subtended by the slit at the grating, for example, results in the smaller JA, which leads to higher resolution.
For fixed 0 and 0, several spectra overlap such that the product pA remains constant. We may express this overlapping by a formula. If in a fixed diffracted angle two wavelengths show maxima for successive orders p, p1, we have pA=(p  1)(A + A A), which leads to A A=A/(p  1).
15
This overlapping of orders means that unwanted wavelengths must be removed from the wanted order by absorption with a suitable filter or use of a detector sensitive only to a limited range of wavelengths.
Instrumentation and Data Acquisition
IBMBRUKER IR/98 Interferometer
The rapidscan interferometer used in this work is a IBMBRUKER IR/98, a Fourier transform infrared (FTIR) spectrometer, which covers frequencies ranging from 25 to 5000 cm~1. The system is basically divided into four modules: source housing, interferometer, sample chamber, and detector compartment (see Figure 2.3). The mercury arc and globar lamps were used as sources for farIR (20600 cm') and midIR (4505000 cm) measurements respectively. The interferometer consists of twosided movable mirror, beam splitters, and filter changer. The sample chamber consists of two parts for the transmittance and reflectance measurements. For detectors, a liquid He cooled bolometer (infrared laboratories LN6/C) and pyroelectric deuterated triglycine sulfate (DTGS) were installed in the detector chamber for the farIR and midIR respectively. A diagram of the bolometer is shown in Figure 2.4. All chambers were maintained under vacuum to prevent IR absorption due to water vapor in the air.
In the rapidscan interferometer, the mirror is moved at a constant speed v so that, from the equation 2.5, the optical path
16
difference can be replaced by J = 2vt where t is a sampling time. This implies the signal seen at the detector is modulated to acoustic frequencies, f = 2vv. This modulated signal is fed into a preamplifier which amplifies the detected signal and does the frequency filtering. The signal is digitized by a 16bit analogtodigital converter and recorded in an IBM computer system which came along with the interferometer. More detailed descriptions of the method can be found elsewhere [30].
Grating Monochromator
A PerkinElmer monochromator was used to measure the optical data for the frequencies ranging from midIR to ultraviolet (80045000 cm~%). As shown in Figure 2.5, the monochromator consists of four major parts: sources, grating, detector, and sample area. For sources, globar, tungsten, and deuterium lamps were used for the frequencies of midIR, visible, and UV regions respectively.
The light from a source which can be selected by mirror M2 passes through the chopper and lowpass or bandpass filter which eliminates the unwanted orders of diffraction. The monochromator has two slits called entrance and exit slit and a grating between two slits. A grating can scan from 60 degrees to about 15 degrees corresponding to socalled "drum number" from 0 to 24. This means that a drum number corresponds to about 1.8 degrees of rotation of the grating at each step. We saw, from equation 2.10 that the rotation angles of the
17
grating along with the slit widths can determine the resolution of the monochromator. A polarizer can be placed in front of the exit slit if necessary. Two large spherical mirrors make the fine images at their focal points where we put the samples for the reflectance and transmittance respectively. An analyzer can be placed near the transmittance site. Finally an ellipsoidal mirror makes a very fine image on the detector. The detectors used are thermocouple, PbS, and Siphotocell for midIR, visible, and UV respectively. Table 2.1 shows in more detail the combinations of detectors, sources, grating, and polarizers for each frequency range and Figure 2.6 shows the bias circuit for the Siphotocell.
The signal from a detector along with the chopped reference signal were fed into a lockin amplifier (Ithaco 393). The amplified signal was sent to a pen recorder and a digital voltmeter (Fluke 8520A) where the signal was averaged for a given time interval and digitized. This digitized data were sent through the IEEE488 Bus and a general purpose interface box (GPIB) to a PDP 1123 computer and recorded for the data analysis.
Measurements
General
All optical data were taken by the monochromator and interferometer with a suitable sample holder which will be explained in
18
the next section. The transmittance and reflectance were the major part of the optical measurements. The spectra from the spectrometer were recorded in the computer where all necessary data analysis had been performed
Temperature and Polarization Dependence
Figure 2.7 shows the sample holder used in the temperature dependence measurements.The main body of the sample holder was made of oxygen free copper. Four thin brass bars insulated from each other (A, B, C, and D in Figure 2.7) were used to contact the sample for the 4terminal resistance measurements. To make good ohmic contact, the sample was pressed by the two plastic bars (F and F' in Figure 2.7) and the bar "G" was moved up and down by turning the screw "E" to stretch and flatten the sample. Finally, the whole sample holder along with the silicon diode thermometer (Scientific Ins. Si410A) and the heater were connected to the LN2 cryostat. Figure 2.8 illustrates the LN2 cryostat, with which we took the dc and optical data from liquid N2 temperature to 300 K. The Polyethylene (20600 cm1) and KCl (450 25000 cm' 1) were used as the windows for the frequency ranges indicated during the optical measurements.
For the samples which were highly stretched, we used wire grid polarizers to investigate the anisotropy of the samples by putting the polarizer such that the electric field was parallel and perpendicular to the chain direction of the samples.
19
Sample Preparation and Doping Techniques
All samples used in this work were synthesized by a research group at the University of Pennsylvania. This section presents the sample preparation for deuterated polyacetylene and newpolyacetylene including doping techniques. The synthesis of the deuterated polyacetylene was done by Georgia Arbuckle and that of the "new"polyacetylene by Nicolas Theophilou. It is described here for completeness. The iodine doping was done at Florida just before the infrared measurements.
Deuterated and Partially Hybridized Polyacetylene
In order to introduce the sp3 defects in trans(CD) chain, completely deuterated polyacetylene, (CD)x was polymerized and subsequently partially hydrogenerated to create sp3 hybridized (CDH) units in the otherwise sp2 (CD) chain (Figure 2.9).
Sample preparation. Sample preparation followed the method described by Arbuckle [31]: Free standing films of cisrich (CD) were synthesized from C2D2 (99.5% isotopic purity) by the Shirakawa method [32]. The films were isomerized to the trans isomer by heating in vacuo for about 1 hour at about 160C. The average thickness of these films was about 100 pm.
The method of introducing sp3 defects is variation of the technique initially investigated by Pron [33] and also studied by Soga
20
et al. [34]. The reactions involved are
(CD)x + xyNa+Naphth  [Na+ (CD)' ] + xyNaphth (2.11)
[Na1 (CD)'15 + xyCH  (CDH ) + xyCHONa (2.12)
(CDH 015) + xyNa+Naphth  [Na+(CDH )~Y], + xyNaphth (2.13)
[Na+(CDH )~Y] + xyCH OH  (CDH ) + xyCH ONa (2.14)
y 0.15 X 3 0.15+y 3
The apparatus used for both the ndoping (Eq. 2.11) of the trans films and subsequent reaction with CH3OH (Eq. 2.12) is shown in Figure
2.10 [31]. The reactions (Eqs. 2.112.12 ) were performed as follows
(i) The apparatus was evacuated and transferred to the inert atmosphere dry box.
(ii) The CH3OH (about 25 ml) was transferred into the round bottom bulb (1) and 0.32 g of naphthalene and an excess 0.2 g of clean sodium metal was placed in bulb (2) to prepare the about 0.1 M sodium naphthalide doping solution (about 3 hours) most often used to give approximately 15% doping.
(iii) After cleaning tube (3) using this sodium naphthalide solution (this procedure ensured dry and oxygen free conditions), a weighed amount of trans(CD) (1020 mg) was placed in tube (3).
21
(iv) After closing stopcock (4) and opening stopcock (7), the reaction vessel was evacuated on the vacuum line. After pumping, stopcock (7) was closed and stopcock (4) was then opened so that the sodium naphthalide solution was poured through the frit (5) and into the main chamber (3) until the trans(CD) film was submerged. It took about 10 hours for complete doping for a sample with a thickness about 100 pm.
(v) This ndoped sample was washed several times by repeated internal distillation of tetrahydrofuran (THF) to ensure that all the sodium naphthalide solution had been removed from the ndoped (CD),.
(vi) The washed film was dried by the cryopumping technique.
(vii) This dried ndoped film was compensated by CH3OH through vapor phase reaction by opening stopcock (4b) (stopcock (4a) closed) for 40 hours. After compensation, the film was washed by CH3OH/CH3ONa solution about 12 times and then dried by cryopumping technique.
The reactions given by equations 2.11 and 2.12 produced film with 15% incorporation of sp3 defects in trans(CD) (this is denoted by (CDH 015) ). In order to introduce a larger amount of sp3 defects, reactions 2.13 and 2.14 were performed. The composition of rehydrogenerated (CDH. ) film as determined by titration of the CH3OH/CH3ONa wash solution in bulb (1) was (CDH ) with y0.27 because an additional about 12 % doping had occurred.
All these films ((CDH,),, y=0, y=0.15, and y=0.27) were sealed in glass tubes with argon atmosphere and sent to the University of Florida
22
for the electrical and optical measurements with various conditions, e.g., doping and temperature.
Doping techniques of deuterated polyacetylene. In order to see the change in dc conductivity and optical properties of (CDH ) films, iodine was doped to the samples by insitu vapor doping. The sample was placed on a sample holder as described earlier in this chapter (Figure 2.7). The N2 cryostat (Figure 2.8) along with this sample holder was placed in the spectrometer, with the hoses connected to the iodine chamber and vacuum pump (Figure 2.11). A digital voltmeter (KEITHLEY 195 DMM) was connected to the cryostat for 4terminal resistance measurements. After finishing the optical and dc measurements for an undoped sample, the sample was exposed by opening a valve in the iodine chamber for a short time (about 5 seconds) for light doping. While keeping the iodine concentration, a second optical and dc measurements were performed. This procedure was repeated by increasing the exposure time to the iodine vapor until we obtain the maximum doping concentration. The maximum iodine concentration was determined by weight uptake and the intermediate concentrations were estimated by assuming a linear relationship [43] of Ina with concentrations up to about 3% doping, where a is the DC conductivity. Doped (CDH ) is denoted by [CDH (I)z], where z is the iodine concentration. Most of the sample handing, such as doping, weighing, and mounting on the sample holder, was performed in a dry box.
23
NewPolyacetylene: New(CH)x
The Shirakawapolyacetylene [32] has been considerably studied on account of its simple conjugated chain structure and highly conducting behavior with doping. Nevertheless, its degradation in air, which decrease the conductivity, affect its potential application. Naarmann and Theophilou [8] presented a new process for the production of metallike, stable, highly conducting polyacetylene, whose dc conductivity is in range of 10000100000 Y'cm' with iodine doping at room temperature.
Sample preparation. According to Naarmann and Theophilou [8], the newpolyacetylene was prepared in following way. The apparatuses were a 500ml fourneck flask with thermometer, funnel, magnetic stirrer and connection for vacuum and argon. The reactive mixture was 50 ml silicon oil AV 1000 (Th. Goldschmidt), 31 ml triethylaluminiumTEA (C2H5)3Al, and 41 ml tetrabutoxytitanium Ti(C4 H 0)4 freshly distilled (Dynamit Nobel). The silicon oil was stirred and degassed for 20 minutes at 0.05 mbar. The TEA was added in a stream of argon and Ti(C4H90)4 was added, drop by drop, through the inactivated funnel over a period of one hour at 3842*C. This mixture was degassed for one hour at room temperature and subsequently stirred for two hours at 120C in a weak current of argon.
Inside the glove box, an even, homogeneous layer of standard catalyst was applied to a flat carrier, e.g., a highdensity
24
polyethylene film which is a stretchable polymersupporting material. This carrier, coated with catalyst, was sealed by means of a hood fitted with a gas inlet valve. After evacuating the hood, 600 ml of purified acetylene was passed into it over a period of 15 minutes and acetylene polymerized at the surface of catalyst on the carrier to form
(CH),. This new(CH) prepared on the surface of polyethylene was stretched (200500 % longer than its original length) inside the glove box, then removed from the supporting film and washed in the usual fashion with toluene, CH3OH/HCL, and methanol. The amount of catalyst to obtain a black (CH) film with thickness about 15 pm was 7 ml.
Doping procedure. Although the insitu vapor doping, shown in previous sections, had had good results in increasing the dc conductivity, it was known that this newpolyacetylene has had better results with chemical doping [8] for increasing its conductivity.
Thus, these samples were doped in chemically with 12/CC4 saturated solution (0.26 g of iodine with 10 ml of CCl4 at room temperature). Samples were submerged in this solution for one hour. Afterward, they were washed only once with CCl4 for 1 minute, and then dried for 5 to 10 minutes in a weak stream of argon. The dopant level obtained by weight uptake was about 6%.
25
Table 2.1
Grating monochrometor parameters
Wavenumber
Grating
a)
Source Detector
(cm 1)
(lines/mm)
801965 9051458
14031752
16442612 24674191 40155105
47937977
38295105 47937822
751110234 919113545
1290420144 1703324924 2206628059
2570637964 3636845333
a) b) c)
101 101 101
240 240 590 590 590 590 590
1200 1200 2400 2400 2400 2400
A B
Globar Globar Globar Globar Globar Globar Tungsten Tungsten Tungsten Tungsten Tungsten Tungsten Tungsten D2 lamp D2 lamp D2 lamp
Thermo Thermo Thermo Thermo Thermo Thermo Thermo PbS PbS PbS PbS PbS 576 576 576 576
Grid/blank Grid/Grid Grid/Grid Grid/Grid Grid/Grid IR/Grid IR/Grid IR/Grid IR/IR IR/IR IR/IR IR/VIS VIS/VIS VIS/VIS VIS/VIS
Blank/Blank
(Um)
2000 1200 1200 1200 1200 1200 1200 225 75
75
225 225 225 700 700 700
Thermo: Thermocouple, 576: Si photocell A: polarizer, B: analyzer Max. slit width
Polarizers
b) c)
Slit
26
Fixed mirror
X
Source
Beam splitter
X2
6
Movable mirror
Detector
Schematic diagram of Michelson Interferometer
)
Figure 2.1
27
(a)
A
b
incident light diffracted
light
normal
(b) zero  order
zeroorder position
Figure 2.2 ("b" is the
Ray diagram of grating (a) for nonzero order position grating constant) ; (b) for zero order position.
(b)
rh
t I
I
U F
III
!IL J ' l kJ__W
I Source Chamber a Near, mid, or farIR sources b Automated Aperture
II Interferometer Chamber c Optical filter d Automatic beamsplitter changer e Twosided movable mirror f Control interferometer g Reference laser h Remote control alignment mirror
III Sample Chamber
I Sample focus
I Reference focus IV Detector Chamber k Near, mid, or
farIR detectors
tr)
Figure 2.3 Schematic diagram of the IBMBRUKER IR/98 Interferometer
I
IV
k
00
 V_ ...
b
eve
TOP C GR
CASE
NITROGEN CAN
ACUUM ALVE
Figure 2.4 Schematic diagram of Bolometer
29
RADIATION SHIELD SEWUUM CAN
WORK SOLACE
WINDOW HOLDER OPTILAL AXIS
q
ELECTRICAL CONE.CTOR
fi
/
 Ti
q
\CAP
OETECTO HOLDER
30
VACUUM TANK
CHOPPER 2
FILTERS
GRATING
* GLOWBAR
TUNGSTEN weQ LAMP
DE SAMP
0L AMP
SLITS
POLARIZER
SAMPLE ROTATOR
Figure 2.5 Schematic diagram of grating monochromator
DETECTOR
31
200 MO
I OPF + 8BQ IN
0 +
LF355N cc
10Mo
+v
576 IN 0 pF
0
880 OUT
I
576 OUT
V
cc
+20V IN 500 c
7815 +V
I pF T 500pF
V
cc
Figure 2.6 Bias citcuit for the Siphotocells
8
I pF i500 Q T 500 pF
7915  o
20V IN
32
8 SAMPLE (b C D
0 G
A'
D'
A, B, C, D :Cu contacts for 4 probe measurement
A', B', C', D', Leads for 4 probe measurement
E :Screw
F, F, G : Plastic bars
H :Supporting plate
I :Silicon diode thermometer
J :Heater
Figure 2.7 Sample holder used in the temperature dependence
u ntmeasurement
33
Liquid N2
M'
Lea
Electrical feed through
Heater Orings
Thermometer Window
Sample
Figure 2.8 Schematic diagram of liquid N cryostat
Vacuum
Sample holder
IIIIllI Mg 2
34
D D D D D
I I I I I
V \c/ C
D D D D D D
(CH3OH)
D D D H D D I I \/I I
C C S C C C
DC
D D D D D
Figure 2.9
Formation of a sp3 defect in (CD) chain
6
7
4 4
25cm 5
1
2
3
I 25cm
Figure 2.10 Schematic diagram of apparatus for preparing trans(CDH )
(From reference 31)
Liquid N2
Electrical leadin Keithley 195 DMM
t
Cold trap
1 
I chamber
2
Figure 2.11 Schematic diagram for insitu iodine vapor doping and fourterminal resistance measurement during the optical measurement
Pump
('I
sample 4LN2 cryostat Spectrometer
9:;:3
CHAPTER III
BASIC THEORY
ElectronPhonon Interaction and Peierls Transition
In order to consider the Peierls transition [2] in onedimensional electronic system, we start with the tight binding model with electronphonon interaction. A model Hamiltonian, known as SSH model [3] which neglects the electronelectron interactions, is written as
H =H + H +H
SSH e cph ph
=H + Hph ,
(3.1)
where
H =  t(j,j+l){c c+
H=K 2
Hph TE (uj~1 u i) +
J
+ c c },
2+
12
J
Where t(j,j+1) is a hopping integral between sites j and j+1, u. denotes the lattice deformation at the jh site, p. is the momentum conjugate to u., and the operator ct(c.) creates (destroys) an electron
J J
36
(3.2) (3.3)
37
at site j.
The equation 3.1, which is often applied to the system of trans(CH),, is a simple onedimensional model of tightbinding band for the pz electrons coupled to the elastically vibrating (mainly abonded) skeleton of the CH monomers. In (CH),, M in equation 3.3 is the mass of the (CH) group and u is the dimerization coordinates, as shown in Figure 3.1. The Hamiltonian for phonons may be rewritten
p p MW 2
H E ( 2M + 2 qQ ''
Hph= ( + 23QqQ q) (3.4)
q
where
P = p eiq(ja) (3.5)
Q= I u eiq(ja), (3.6)
q N''/2 1
where "Q "and "P " are normal coordinates and momenta, "N" is the number of atoms within length L of the onedimensional system, a is the frequency of the normal modes of vibration of the lattice, and "q" is a wave vector for phonons, which covers only the first Brillouin zone, ,r/a s q s 7r/a.
38
ElectronPhonon Interaction
In order to consider the electronphonon interaction, we recall the equation 3.2. The effect of the lattice vibrations can be calculated by assuming that the motion of the lattice affects only the nearestneighbor hopping integral t(j,j + 1). Thus, for small displacement of the lattice
t(j,j+1) = to + tl')(u  u ) , (3.7)
where to is just hopping integral for the nearest neighbor, t') is the electronphonon coupling constant, which is the first derivative of to with respect to the interatomic distance. The t(') and to are independent of j. The Fourier expansion of c. is
c= E e' j c, (3.8)
N 1
where x. =ja, 7r/a
H= H + H , (3.9)
e eph
where
39
H = 2t E cos kac C (3.10)
c 0 kk k (.0
and
H E(0 c+ c + C c)
cph k q k+q k (3.11)
where
O = 2t' N1/2 (eiqa 1) sin ka Q . (3.12)
Here H and H are Hamiltonians for the tightbinding of the nearest
neighbor and the electronphonon interaction respectively and 0 is the Fourier components of the potential 4(x), which contributes to the electronphonon interactions and is assumed to be sinusoidal. Later in this section, it will be shown that 4 will be identical to the gap parameter for the Peierls distortion [2] at q=2kF (kF: wave vector at Fermi surface).
Kohn Anomaly
Before we go to the details of the Peierls transition [2] which is associated with electrons interacting with lattice vibrations, it is important to consider a fundamental instability in the onedimensional
40
electronic system, which comes out as a result of softening the phonon frequency. This phenomenon is known as the Kohn anomaly [35].
A useful measure of the linear density response of the electronic system to the potential is provided by the density response function, or Lindhard function [36]
X(q) < n >
S f f k+q j(3.13)
k k k +q
where is the mean value of the electron number density operator,
q
n =C c
q k +q k
and f is the equilibrium occupation number for FermiDirac statistics
f [ I + exp{$i(se) k 1 (3.14)
Here "kB" is the Boltzmann constant. In accordance with equations 3.13 and 3.14, X(q) is dependent on temperature. The forms of X(q) for a one dimensional and three dimensional electron gas are shown in Figure 3.2 [37]. In accordance with Figure 3.2(a), X(q) in one dimensional system has a peak at q = 2kF for T * 0 and increases as temperature is
41
lowered, diverging logarithmically as T + 0. For three dimensions (see Figure 3.2(b)) there is no such peak. To understand the origin of this peak, we consider the Fermi surface of a one dimensional free electron gas.
Figure 3.3(a) [37] shows the Fermi surface of a one dimensional metal at low temperature (T * 0). The states with low kinetic energy are filled until we have accounted for all of the electrons in the metal. The momentum of the last electron added is called the Fermi momentum kF, and its energy the Fermi energy eF. States with IkI < kF are filled and those above are empty [38]. In one dimension the energy is determined solely by the momentum in one direction; call it px= hkx. All k states are occupied up to Ikx = kF. For any particular value of "q", the states connected do not lie on the Fermi surface. However, for q = 2kF there are many states that are connected by "q" and lie on the Fermi surface with same energy ek, which in this case is just two parallel planes at k = k, and k = k F [37]. This implies the system at q = 2kF is infinitely degenerate. This degeneracy is known as Fermi surface "nesting", and it causes the infinite value of response function x(q=2kF) as T  0 because the electron number density "n " is strongly related to the degeneracy of the system. If T is not zero but sufficiently low, then the two Fermiplanes are no longer exactly sharp. However, the system can have still very high degeneracy so that we have still high peak of x(2kF) at reasonably low temperature as shown in Figure 3.3(a).
42
By contrast, we look at the situation for a three dimensional free electron metal (see Figure 3.3(b)) in which, for any q s 2k, only two states at zF are connected by 'q" which have the same energy. Slightly different values of "q" will couple other states and give a density change with a different periodicity. Adding random periodicities results in no net contribution to the charge density.
To investigate the kind of instabilities that result from the divergence of the response function in one dimensional system, let's consider the Heisenberg equation of the motion of Q (normal coordinates defined in equation 3.6)
dQ
id q = [Q , H]
or
d 2
(i)2 q  [ Q H], H].
This equation can be rewritten as
d2
(ih)2 = h 2 q q (3.15)
d t 2q q q
where "H" is the total Hamiltonian defined in equation 3.1. By using the harmonic approximation, Q x e9t, and equation 3.13, this equation
3.15 is reduced to
43
22
2 2 _ X(q) (3.16)
q q mQ2
q
where 0 is the new phonon frequency for wave vector "q" after the electronphonon interaction has been taken into account. Before we saw that X(q) becomes large for q = 2kF when the temperature is lowered. In accordance with equation 3.16, there is a certain temperature at which O(q=2kF) tends to zero, which implies that there is no restoring force for q = 2kF mode and the unstable lattice can distort. This phenomenon is known as "Kohn anomaly" and is the origin of the Peierls instability. Figure 3.4 shows the phonon frequencies around 2kF soften as the temperature is lowered, and 0(2kF) goes to zero at the Peierls transition temperature.
At below the Peierls transition temperature, there will be a periodic lattice distortion accompanied by a charge density wave. Details of this phenomenon can be found elsewhere [3942].
Peierls Gap
It can be seen that the instability of the onedimensional free electron system to an applied potential of wave vector 2kF F(q=2k )" is a generalization of the Peierls instability which produces gaps at the Fermi energy. We recall the Hamiltonian in equation 3.9
H =2t E cos ka c c  E (0 c c + c( c).
k k k k q k+q k q kq k
44
The new eigenvalues of energy Ek are obtained by using degenerate perturbation theory as there are states for which zk= ck when
k=q/2. Thus,
k  E q
=0,
q k E
where
(3.17)
(3.18)
ek = 2t cos ka .
Equation 3.17 can be solved in the usual way and gives
E + 4  0 + ] 12
k 2 k kk q
(3.19)
Using the relation e q2 = q  /2), expression 3.19 simplifies to
give the new electron energies
E = e2 [ / + 0 12 . 12
(3.20)
The electron energy levels are split for wave vectors k= q/2 (half filled band) and the potential q5(x) due to the electronphonon interaction, which has the periodicity 7r/kF has created a gap of size
45
2 1 0 (q=2kj) I at Fermisurface in the electronic energy spectrum. This gap is known as the Peierls gap (see Figure 3.5). From equations 3.12 and 3.13, the gap energy is proportional to Q(2kF,T), which implies that it is directly proportional to the lattice deformation and dependent on temperature.
According to molecular orbital theory, polyacetylene has aa* and
7* energy bands [43]. There is one 7r electron per primitive cell in polyacetylene, which makes the ,rn* band a half filled conduction band. Thus, polyacetylene should be a conductor. However, experiments show that trans(cis) polyacetylene has a semiconducting energy gap at
1.45eV (1.9eV), which is a consequence of the Peierls transition.
Soliton Excitation
The soliton model first introduced by Rice [10] and Su, Schrieffer, and Heeger [3] is most successful in polyacetylene with good agreement with experiments. Therefore, this model will be used to discuss polyacetylene in this work.
Due to the dimerization of the carbon atoms, there exists a twofold degenerate ground state in trans(CH)x. This leads to the formation of nonlinear topological excitations, in this case neutral solitons (see Figure 3.6(a)). An infinite, exactly halffilled chain contains no solitons in its ground state. Solitons are found either at finite chains, e.g. by thermally isomerizing cis(CH)x, or in not exactly halffilled chains obtained e.g. by doping polyacetylene, or as
46
excited states. Solitons can only be created by pairs, solitonantisoliton, due to the conservation of the particles. Figure 3.6(b) shows these soliton pairs in trans(CH)x. Doping changes the neutral solitons to the positively or negatively charged solitons (Figure 3.6(c)) depend on the dopants and creates more charged soliton pairs. The neutral solitons carry spin 1/2 and charged solitons (+ or
) are spinless, which is known as "spincharge reversal".
The soliton state is not simply localized at one atom site. Elastic energy would prefer to spread the defect through the whole chain, while the Peierls distortion would favor a narrow kink. The competition between these two effects make the soliton spread over about 15 CH units [43].
In this section, by using the SuSchriefferHeeger(SSH) method [3,4], the roll of solitons in polyacetylene will be discussed including the ground state, soliton excitations, and soliton quntum number. The SSH model is the independentparticle model in which the correlation effects from interactions between the pz electrons are neglected.
Ground State
Already, we have seen that the Peierls transition implied that the ground state of the onedimensional tightbinding metal is spontaneously distorted to open a gap at Fermisurface and to form a charge density wave with # 0 below the transition temperature.
47
Since the strongest instability occurs for a charge density wave of wave number q=2kF =7r/a, we consider the adiabatic ground state energy E0 for the twofolddegenerate ground state in polyacetylene as a function of the mean amplitude of distortion u, where
u.  = (1)" u . (3.21)
J J
By definition, E0 is the total energy of the system, electronic plus elastic energy with Mxo. For u. given by equation 3.21, H in equation 3.2 is invariant under spatial translations and the total Hamiltonian can be diagonalized in k space in the reduced zone, 7r/2a
HSSH  [t + (1) 2 t u] (c+ c + c c ) + 2NKu (3.22)
for a chain of N monomers in a ring geometry. We define the valence and conduction band operators as
(v) 1/2 ikja
ck N ~e c.i
and
(c) i N1/2 ik 2)ja
ck = 1 e c .
j
48
Thus,
(3.23)
= N12 [ c + ic C) ] eika
Ci k k I
By substituting equation 3.23 into 3.22, HSSH can be rewritten as
7rI2a
SSH
7r/2a
I Z k (
c(c) (C) + c(v) C(v)
k k k k
+ ( c) c + c CC ) + 2KNu2, (3.24)
where
(3.25)
e = 2t cos ka,
and
A k = 4tN') u sin ka
(3.26)
Here e is the tight binding energy for nearest neighbor in reduced zone schemes and A is the energygap parameter,which is same as 20(q=2kF) shown in equation 3.12 if we consider the half filled band (2kF = r/a) with Q(2kF) as the normal coordinate of the lattice deformation. Now HSSH can be diagonalized by the transformation
(7r/2a < k < 7r/2a)
49
a (V)= (V)  .8 CC)
ak a~ C f ck)
aC) = .8 ck +(aV clc) ,
k kk C
where ak and $, satisfy Fermi anticommutation relations
la i2 + 1 2 = 1 . Finally, HSSH becomes
7r 2a Hss= SSH
 7r/2a
E (n(C)  n(v)) + 2NKu2
k k k
where n(c)and n(v) are the number density operators in conduction band
k k
and valence band respectively, defined as
(C) =(c) n k a
ac
(3.30)
n(V) =a(V) a(V)
k k k
The quasiparticle energy, E., relative to the Fermi energy is given by
2 2 1/2
E k (k+ A k) ,(3.31)
(3.27a) (3.27b)
(3.28)
(3.29)
50
with
ak (1+ck/Ek)/2 ](,3.32a)
f$ [ (1+sk/E)/2] sgnA k (3.32b)
We note that equation 3.31 is equivalent to equation 3.20 by setting the Fermi energy to be zero, i.e., e 2(q=2kF) = 0.
For the halffilled band of (CH),, the ground state energy E0(u) as a function of u is given by taking n(v)=1 and n()=0. Thus, for a k k
ring of circumference L, we have
EO(u)  2NKu2 = 2L/n E kdk (3.33)
0
or the energy per site is
22
E (u)/N = (4t /7r)E(1_Z2) + 4 Z2 /2 t , (3.34)
where E(1z2) is the complete elliptic integral of the second kind and
z = 2 Pu /t (3.35)
For small z,
51
E(1z2) 1 +  [ln(4/z) '] 9 +    (3.36)
2 2
Thus,
E0(u)/N :  4t /r  2t /r (ln(4/z)  1/2) 9
2
+ Kt2 z2/2 t(,) +  (3.37)
In equation 3.37, as Iz 10, the logarithmic term dominates, and Eo has a maximum at u=0, which is consistent with the Peierls instability. Figure 3.8 shows the total energy as a function of u for parameters characteristic of polyacetylene [3,4]. For an assumed energy gap of 2A =1.4 eV (from experimental data), to =2.5 eV, t*)= 4.1 eV, and K=21 eV/A2 [3,4]. These parameters lead to the minimum energy distortion UO= 0.04 A. Some experimental results for the Xray structural studies by Fincher et al. [7] and for the NMR studies by Yannoni and Clarke [44] show that u03 0.035 A which is good agreement with this estimate.
Soliton Excitations
As mentioned before, there are twofolddegenerate ground state with the same energy E(uo)=E(uo) in trans(CH)x and the system supports nonlinear excitations which act as moving domain walls separating regions having different ground states: A phase (+uo) and B phase (u0). These walls act as topological solitons because they are
52
shapepreserving excitations which alter the medium after they have passed a given point. Figure 3.9(a) shows the A, B phases and the mixture phase where a neutral soliton exists. Figure 3.9(b) illustrates the order parameter "4 " for a soliton and antisoliton, where
4. = (1) u., (3.38)
which is analogous to the staggered magnetization in antiferomagnets. Thus,
r.= u , A phase ]
(3.39)
U, B phase
If . changes suddenly from u0 to uO, say at j=0, the electronic energy will be quite large due to the uncertainty principle. Alternatively, if . changes very slowly from u0 to u0, there will be a large region surrounding j=0 where the condensation energy per site is greatly reduced, again raising the energy. Thus, there is a preferred width 4 of the soliton that minimizes the total energy. Numerical calculations [3,4] show that the form of 0 that minimizes the adiabatic energy with these boundary conditions is
0 1 u0 tanh[ja/ f] ,(
(3.40)
53
where = 7a for the SSH set of parameters, and it is assumed that j =0 is the location of the center of the soliton. With the above parameters, the energy to create a soliton at rest is E ~0.42 eV, which is less than onehalf the single particle gap A. This result shows that a soliton is less costly to create than either an electron or hole because the Fermi energy is midgap for the undoped polyacetylene. This is the reason why solitons, in this case charged solitons, are spontaneously generated when charge carriers are injected by doping, by photoexcitation, or by thermal generation.
The order parameter which is given by equation 3.40 is valid only for the solitons and there must be another solution of 0. for the antisolitons. To understand this, we define a state y of zero energy which is centered at the soliton and falls off on the scale of . By expanding y0 in site basis states,
k %> = Ij> (3.41)
or
V k%> = E V(i) 1 j > . (3.42)
Thus, we require
,(3.43)
or
54
E = E 0
'i
(3.44)
where H is a Hamiltonian given in equation 3.22 excluding the phonon contribution term. The matrix elements of H are
= [t0 + (l) J t ) (0 + 0 )] for /=j+1,(3.45(a)) = [t 0  (1) J t 1 ( + )] for i=j1, (3.45(b))
= 0
otherwise .
(3.45(c))
For a solution of E=O, V (/) in equation 3.44 must satisfy
E V () = 0
'I0
or
0(j+1) = (/) V j1)
SR (3.(47)
and
(3.46)
(3.47)
55
From equation 3.40, as j)w, = . Thus, R. becomes
R > [1(1)3 2t"'O / t 0]/[1 +(1) 2t(, j / t 0 . (3.48)
For a bounded solution, R must be less than 1 as jxo, and greater than 1 as jxo. Note that this condition is satisfied only if V0(j) for even j are nonzero because O. u as j> o. The V (j) for any odd j is not j 0
bounded and is not a solution for the soliton. However, if we interchange the A and B phase in equation 3.47 so that .+u as j* co, j 0
then
= u0 tanh (ja/) (3.49)
represents an antisoliton, and the oddj solution exists with E0=0, and the evenj solution is unbounded in this case. Therefore, for each widely spaced soliton or antisoliton, there exists a normalized singleelectron state in the midgap which can accommodate 0, 1, or 2 electrons due to spin degeneracy.
An important property of a soliton is its effective mass, Ms. If the soliton is translating slowly, #. is given by
0 = u tanh[(ja  v t)/f] , (3.50)
where v is the velocity of the moving domain wall. The increase of
56
energy is given by
Mv2 1 M E(& /dt)2
~ M(u v /(2 E sech4[a/fl for small v . (3.51)
Thus,
M = (4u /3 a)M ~ 6m (3.52)
for the SSH parameters uo~0.04A and for 2A=1.4eV, with the mass of the (CH) group "M" and 'm" the electron mass. This small mass of a soliton indicates a high mobility of the soliton, which arises from the large width of the soliton and the small displacements of the nuclear compared to the lattice constant, so that the nuclei gain little kinetic energy as a soliton passes.
We have mentioned before that the soliton width exists to minimize the total energy at soliton site. Figure 3.10 [3,4] shows that creation energy of a soliton is a function of and the energy gap. Usually the soliton width is quite bigger than the lattice constant so that it is possible to apply a continuum limit from the SSH model adapted to the nearly halffilled band. This was first introduced by Takayama, LinLiu, and Maki [11] (TLM model). Details of this model can be found elsewhere [11,46,47]. However, the main ideas and results of the TLM model will be presented briefly.
57
TLM Model
The TLM model [11] is a further approximation and linearizes the tightbinding band structure, shown in Figure 3.11 [1]. The advantage of this approximation is that the resulting theory is a continuum, or field theory [48], which is generally more amenable to analytic solution than the tightbinding model. The resulting electronic band structure has two branches (see Figure 3.11) for each spin s and wave number k: a rightgoing branch (n=1) with energy Z~hvF (kkF) and a leftgoing branch (n=1) with Z ~hvF(kk F). The electronic wave function can be described by a twocomponent field Vg, where V creates an electron of spin s on branch n, where n =1. For a given gap parameter A (x) the dimerization has wave number 2kF, and it causes scattering from one side of the Fermi surface to the other. By representing particles close to the Fermi surface, the TLM Hamiltonian [11] is written as
H = E dx V (x)[ihv a /Ox + A(x)O'] (x)
TLM F Fz S
3
+ 5 dx [A (x)2 / 0 + A2(x)]/27rhvA , (3.53)
where vF is the Fermi velocity, a1(i=x,z) is the Pauli matrix, A is the dimensionless electronphonon coupling constant, 0 is the bare opticalphonon frequency, and [(dA /dt)/(7rhvFAf)] is the momentum conjugate to A. The SSH model Hamiltonian can be transformed to this by
58
expanding about the Fermi surface, keeping terms only to lowest order in (a/c), where is the electronic correlation length, =hvF1A, or soliton width. If we derive the TLM model in this way, we find that hv~ =2t a, A =(4t~l' ) 2I/2rKt0, 122=4K/M, and [1]
F 02 0
A(ja) = (1)4t(' u . (3.54)
This equation can be compared to the gap parameter in equation 3.26. In polyacetylene, many properties can be approximated by the continuum model. However, some exceptions exist, for instance, the acoustic modes which are certainly present in the SSH model are not shown in the continuum limit. They appear only if terms are kept to next order in (a/c) when the SSH model is transformed to the TLM model [1].
The soliton excitations can also be studied in the TLM model. For given gap parameter, A (x), the u= V+ and v= y have cnumber solutions u (x) and v (x), which correspond to the expansion coefficients for creating or destroying a quasiparticle. These amplitudes satisfy equations of the Bogoliubovde Gennes form [49]
iv au (x)/Ox + A (x)v (x) = E u (x)
F MS MS M MS
(3.55)
iv av (x)/ax + A (x)u (x) =E v (x)
F ms ms m ms
The variation of the system energy, equation 3.53, with respect to A (x)
59
gives the selfconsistency relation
(3.56)
A (x) = (4t' a/K)E [u (x)v (x) + vm(x)u,(x)].
M's
Here the sum over "m" and "s" extends over occupied states and spin components. A nonlinear Schrodingerlike equation can be derived by differentiating equation 3.55 with respect to x and introducing [11]
f +(x) = u(x) + iv(x)
(3.57)
f (x) = u(x)  iv(x) .
Equation 3.55 becomes
[v a 2+ E2 v F 82 n F
aA x) _ 2()] f (x) = 0
(3.58)
For the perfectly dimerized lattice A (x)=A0 , the solutions to equation 3.58 are just plane wave functions with the energy dispersion relation E 2(k) =A +v2 k2, where the gap parameter is written as
A 0= 4t e1/2A
(3.59)
where A is a dimensionless electronphonon coupling constant and 4tO=W is the 7rband width. Equation 3.59 satisfies the selfconsistency
60
relation and it is analog to the BCS gap equation determining A. By introducing the solitons in polyacetylene, the system is no longer homogeneously dimerized. It is remarkable, however, that the nonlinear Schrodingerlike equation can be solved with the ansatz A (x)=A 0tanh(x/) in terms of hypergeometric function [50] and the selfconsistency condition is satisfied. The energy to form a soliton, which defined as the difference between the mean field energies in presence of a soliton and the ground state, is written as [11]
E = 2A/r ~ 0.63A , (3.60)
which is in reasonably good agreement with the SSH value, 0.6A.
Polaron and Bipolaron Excitations
When a conduction electron (hole) exits in a rigid lattice of an crystal, there are coulomb forces on the ions or atoms adjacent to the electron (hole). This forces deform the lattice and produce a polarization. In the potential field resulting from this polarization, the electron (hole) could become bound. This electron (hole) plus the associated potential field is called a polaron [51].
In trans(CH),, by adding one electron or hole, one charged soliton can not be formed because the bond alternation must be changed over half of the chain, which causes infinite potential barrier for infinite chain (see Figure 3.12(a)). Instead, a polaron is formed by
61
the selftrapping of an added electron or hole because the dimerized chain with a polaron has same phase (A or B phase) on both side so that the bond alternation is not required in this case (see Figure 3.12(b)).
Independent of the molecular dynamics studies [3], polaron or baglike solution [5254] were discovered using the relation of the meanfield approximation to the continuum model and the GrossNeveu model [55] of quantum field theory. In the continuum limit, the order parameter 0 (x) (Figure 3.13) describing a polaron centered at the origin is given by [1]
OPx = uO + (u0/V2){tanh[(xx0)/V '
 tanh[(xxO)/V2 f]}, (3.61)
where x0=((/v'2)ln(l +V2) ~0.623 . We note that the order parameter of a soliton located at x0 was written as
O, = uotanh[(xxO)/]. (3.62)
Thus, we can roughly describe an electron (hole) polaron as a bound pair of a negatively (positively) charged soliton and a neutral antisoliton (or a neutral soliton and a negatively (positively) charged antisoliton), with centers separated by 2x0 =. Figure 3.14(a)shows the negatively and positively charged polarons (or electron and hole
62
polaron) in trans(CH)X.
As shown in earlier section, in trans(CH),, the soliton has a single bound state associated with it. For polaron there are two bound states. These two states are symmetric with respect to the gap center and can be thought of as the bonding and antibonding combinations of the two midgap states associated with the bound solitonantisoliton pair that makes up the polaron; in effect, the lower state is split off from the top of the valence band and the upper state is split off from the bottom of the conduction band [1]. As shown in Figure 3.14(b), for the electron polaron case, the lower state is filled with each spin orientation and upper band is half filled. On the other hand, for the hole polaron the lower band is half filled and upper band is empty. Therefore the electron and hole polaron each carry spin 1/2, which is the conventional spincharge relation.
According to the SSH result [3], the formation energy for a polaron in trans(CH) xis written as
E = 0.61 v'2 A, (3.63)
which is greater than one for a soliton, E =0.61A, but still less than half of the gap energy.
To form a polaron in polyacetylene chain, as mentioned before, we need to add only one electron or hole. However, adding only one electron or hole, by doping etc., is not possible in real world. By
63
adding more than two electrons or holes, solitons are less costly to form than polarons because E . Thus, if we add two electron, for example, the first injected electron forms a polaron; the second injected electron breaks apart the polaron to form two negatively charged solitons if the groundstate degeneracy is not broken. However, if the groundstate degeneracy of the polymer is lifted, the resulting confinement of the two solitons leads to a stable bipolaron which lowers its energy by increasing the solitonsoliton spacing 2x0 until a free kinkantikink pair evolves at large separation. Figure 3.15(a) shows the order parameter for two solitons and polarons and formation of two solitons or bipolarons is shown in Figure 3.15(b).
Since a polaron is stable only in the absence of another polaron, the role played by a polaron in trans(CH) where the solitons are most important is rather limited. However, most of the conducting polymers, like polyanilline, polypyrrole, and polythiophene, possess only near groundstate degeneracy so that the polarons and/or bipolarons play the major role in such cases [1].
Doping Induced InfraredActive Vibrational Modes
In a general point of view the skeleton chain (i.e., without 7r electrons) has a few bare normal modes which have the right symmetry to induce dimerization. The coupling of the 7r electron to these normal modes drives the dimerization, and each normal mode becomes a symmetric
64
vibration of the dimerization amplitude, which is called a Ramanactive amplitude mode [5658].
When charge carriers are added to the chain, by doping for example, the translational symmetry of the system is broken and there are dipole moments coupled with bare normal modes. New normal modes, called phase modes, which are related to the translational degree of freedom of the charge, become infrared active and they describe charge oscillations [59].
Rice and Mele [1315] predicted a socalled "pinning mode" which is a characteristic infrared activity associated with the oscillation of the charged soliton bound to an ionized impurity. Each bare mode results in an infrared active vibration with restoring force given by the "pinning" potential which resists the charge translation. In particular, in a perfectly translational invariant system (i.e. no pinning potential) the lowest frequency mode (i.e., pinning mode) shifts down to zero frequency.
Horovitz [12] calculated the infrared conductivity from which he could obtain the infrared active vibrational modes. Consider the charge with dimerization amplitude A (xO(t)), i.e., its position is a timedependent field. At frequencies low compared with the gap 2A0 this lead to a current j(t)=ep8O/at where p is the charge density. Each phonon has an order parameter A (x#n(t)) (n=1, 2, ,N) with n(t) its oscillation degree of freedom. In the ground state [12] A n(x) =A (X)A. n/A where A n is the dimensionless electronphonon coupling
65
constants and A =Enn. The effective Lagrangian to second order in 0. is given by [60]
L M & {E [2 + ( /j + (1a)02}, (3.64)
Lff= Z {S [0 +
where M is the soliton kinetic mass, g 2=E (Ow)2A /A, co are the bare
3 0 n n n n
phonon frequencies, and ct represents pinning parameter. The 42 term in equation 3.64 comes from the electronphonon interaction and if a=0 it reduces the translational mode frequency to zero.
Equation 3.64 yields a set of equations of motion for 4,(t) which determine 4(t) =E4,(t)A /A. This timedependent field 4(t) gives the current j(t) and infrared conductivity
a(w) = e2icopM 0;2D (w)/ [1 + (1a)Dc(w)], (3.65)
where the phonon propagator D0 (w) is given by
D (w) = I [(w/CO)2 yI, /A. (3.66)
0 n n n
From equation 3.65, the solution of D0(w)=(1a)~I gives the infrared active frequencies. All phonons couple in phase, i.e., they all tend to induce a charge density wave centered at the same position. This is indeed the case in polyacetylene, with N=3 phonons favoring bond dimerization [61]. A fourth normal mode that is mainly CH stretching favors site dimerization [62]; however, its coupling is too
66
weak and it can be neglected. Using the parameters a =0.23 and those in Table 3.1 [61], Figure 3.16(a) [61] shows the form of D0(w); given pinning parameter defines the dashed line and its intersections with D0(wo) yield the renormalized frequencies. Figure 3.16(b) [63] shows these three infrared active modes in trans(CH) .
Horovitz argued that the 886 cm'1 mode from Table 3.1 [61] must be a pinning mode since the lowest frequency Raman mode is at 1075 cm~1 [43]. The width of this absorption is about 400 cm' and attributed to a distribution of pinning parameter reflecting variations in the potential between the charged soliton and dopant ion. On the other hand, Rice and Mele [15] have shown that charged solitons in the Peierls model of polyacetylene lead to infrared active modes within the semiconductor gap. They suggest that the experimentally observed [64] modes at 900 and 1370 cm'1 result from two vibrational normal modes while the oscillation of the soliton pinned to an ionized impurity yields a mode at a much lower frequency (300500 cm'1). In the next chapter, all these doping induced infrared active modes in polyacetylene will be compared with the experimental results from this work, including the pinning mode with the temperature dependence.
Electronic Excitation
Just as the soliton leads to new peaks in the infrared absorption spectrum discussed last section, it is expected to lead to new peaks in the optical absorption spectrum. Figure 3.17 shows that the possible
67
optical absorptions for the positively charged solitons (S+), negatively charged solitons (S~), neutral solitons (S0), and interband transition. From Figure 3.17 we see that when charged solitons are introduced, a new absorption edge should onset at midgap. Since the optical sum rule implies conservation of oscillator strength, there will be bleaching (reduction of absorption) above the band edge to compensate for the new peaks at midgap [1].
The calculations of optical absorptions in trans(CH) have been carried out by Horovitz [59], Suzuki et al. [65], and Kivelson et al. [66] in the continuum model. The absorption to the soliton gap state is given by
a(W) = 7r(e2/hce"2)yN 2[(hO/A)2 _ ]"2.
sech2{n/2[ho/)2 _ }]1/2}, (3.67)
where N is the number of carbons per unit volume and y is the density of solitons per carbon. The interband absorption is given by
a (hw) = (2/w) E , 12 J(hW  k*  F,' ), (3.68)
kk'
where the summation is over all allowed transitions from eigenstate I k> to eigenstate I k' > and the superscript "c" and "v" represent for the conduction band and valence band, respectively. The matrix element
68
I M , 12 which is based on the golden rule is given by a rather complicated expression and can be found elsewhere [43]. The change of interband absorption due to the soliton is given by
A a(w) =  y(2t/a)a0(CO). (3.69)
The total absorption satisfies a sum rule [66]
c(w) d(w) = (2/r)(e2/hceZ) Nav F (3.70)
independent of the lattice configuration or doping level. Thus, the increase of absorption in the gap is compensated by a decrease of the interband absorption.
While the general features of the curves from the experimental results [65] are in qualitative agreement with theory, the squareroot singularity in equation 3.67 is considerably smeared out in the experimental data. Grant et al. [67] argued that part of this comes from interchain hopping and showed that the interchain hopping integral 4t, o 0.1eV, a width considerably smaller than required to account for the observed rounding. The nonlinear dynamics of trans(CH) leads to a strong broadening of the opticalabsorption structure due to multiphonon emission [68,22] as well as polaron formation and due to solitonantisoliton creation for (2/7r)A
We should also note that the transition energy might deviate from
69
the midgap at least because of following influences. Electronelectron correlation effects may account for about 0.2eV [71]. Furthermore impurities will introduce states in the gap and perhaps shift the solitonic midgap state [72,73].
Since a polaron has the symmetrically arranged localized gap states, there are three possible transitions apart from the interband transitions. These are transitions between the lower and upper localized polaron states, from the valence band to the lower polaron level (except electron .polaron) or from the upper polaron level to the conduction band, and from the valence band to the upper polaron level or from the lower polaron level to the conduction band. Fesser et al. [74] calculated the absorptions of all of these transitions for a polaron from the golden rule [1]. Polaron absorption in trans(CH),, which is of interest only at extremely low doping, however, never has not yet been observed. More details can be found elsewhere [53,54,74].
Transport
Although high electrical conductivity has been a major interesting in the field of conducting polymers, progress toward understanding of the mechanisms involved has been slow. The difficulties rise principally from the complex morphology and the incomplete crystallinity [7579].
Many of the properties of polyacetylene, after heavy doping (about 6%) to the regime of highest conductivity, are those of a metal [8,80]
70
For example, there is a term in the specific heat that is linear in temperature [16], there is a (temperature dependent) Pauli susceptibility [17,18], and the thermoelectric power is linear in temperature [81]. All these results are consistent with a broadbandwidth metal. There is, however, a exception that the temperature dependence of the resistivity is not metallike. The resistivity increases as the temperature is lowered even for samples with very high conductivity (asl.5x105i'0cm') [8]. For samples with conductivity values in the range of <10 30cm1, the anomalous temperature dependence may be rationalized in terms of phonon assisted transport through localized states and across the interfibrillar contacts in the complex morphology. On the other hand, for values greater than 105 cm', a phonon assisted transport is more difficult to understand [1].
Magnetic Properties
Besides the dramatic increase of the conductivity upon doping different doping regions are distinguished by the magnetic properties of transpolyacetylene. The magnetic susceptibility of undoped transpolyacetylene is observed as a Curielike (1/T behavior) contribution [82].
While doping transpolyacetylene two processes are expected in the soliton picture [83]
(i) neutral solitons may pick up charge from the dopant and
71
transform to charged solitons; the density of spin diminishes.
(ii) the charge introduced to the chain creates a polaron which eventually dissociates into two charged solitons. This process will be the only one at higher doping levels.
There are three doping regions which may be distinguished. First, Tokumoto et al. [83] show that the low levels doping results in a decrease of the Curie susceptibility. Davidov et al. [9] also show that a Curielike electronspin resonance (ESR) line decreases with increased doping. Second, a considerable increase in conductivity is measured at a dopant mole fraction y=0.001. It is however not accompanied by a similar increase in the paramagnetic Pauli susceptibility [17,82] indicating that the doped polymer has not transformed to a simple metal. The charge carrier seem to have no spin in this "intermediate region". Thus, charged solitons, which are created by process (ii), are assumed to be charge carriers in this region. Third, above doping levels of y=0.06, a metallic behavior is observed in the magnetic susceptibility. The Pauli susceptibility X = 2BN(EF) abruptly increases to values corresponding to N(EF) =O.lstates/eV C atom as is expected for a tightbinding band with 10eV bandwidth [82].
Intersoliton Hopping Model
The main idea of the intersoliton electron hopping theory is concerned with that transport of charge occurs through energetically
72
equivalent levels that are at midgap. At light doping concentration, where the neutral solitons may still exist, moving a charged soliton far from the accompanying ion requires an activation energy. However, a neutral soliton can move freely along the chain without costing energy. If a neutral soliton moves past a charged soliton which is within a soliton width (=7a) from neutral one, there is a probability that the electron will hop from the neutral one into the empty state of the neighboring charged (positive in this case) soliton, which maintains local charge neutrality and ensures that the states are energetically equivalent.
Thus, Kivelson [1921] proposed a phononassisted hopping conduction utilizing the soliton model. The average dc conductivity is
adc T exp(2BRO/) (3.70)
where R is the mean distance between the hopping centers, =(c j/1)3 is the effective soliton extension (( and _ are the inchain and outofchain wavefunction decay lengths, respectively, , 7a=l0A, 4 C3A), and B=1.39 which is dimensionless number. y(T) determines the transition rate between the initial and final solitons between which the electron hops. In general, this temperature dependence of y(T) can be quite complicated, depending on the dynamical nature of the states between which the solitons hop. However, Kivelson [1921] assumed a power law dependence y(T)=T" and from the phonon
73
density of states estimated n~10 which is in rough agreement with the experimentally determined T3, law for ad [84].
The thermopower in this model is essentially given by
S = (k /e)[(n+2)/2 + ln(y /yh (3.71)
where + or  sign designates p type or n type. The variables y and ych are the concentrations of neutral and charged solitons.
The ac conductivity within this model is
a = K'(co/T)[ln(D' w/T"n I)]4 (3.72)
where K' and D' are constants. It shows a strong temperature and frequency dependence, which is in good agreement with experiment [84] using n=14.7 as obtained from the temperature dependence of a .
The temperature dependence of the electrical conductivity of transpolyacetylene together with the frequency dependence of the hopping conductivity and the electrochemical measurements are able to give strong evidence for the intersoliton hopping mechanism. In lightly doped transpolyacetylene, charge is neither transported among a distribution of bandgap impurity states nor among localized (polaron) states near a band edge. The transport occurs at midgap. The agreement between theory and experimental results indicates that the electrical properties of transpolyacetylene at dilute doping are intrinsic and
74
not determined by the complex morphology [1].
Metal Transition
The Pauli susceptibility of transpolyacetylene rises steeply by increasing dpoing (4%6%), which suggests a transition from the semiconducting state, with a gap ~1.4eV, to a metallic state. In fact, conductivity close to that of copper has already been attained for carefully processed (CH)X with =6% iodine doping [80].
Several ideas have been developed to describe the transition of the system to a metallic behavior at high doping level or in the soliton picture at high soliton densities.
Rice et al. [85] focused upon the lattice distortion and arbitrarily modeled the influence of the electrons by a doublewell #4potential in the continuum approximation. The soliton extension is considered here as a constant and there is a finite soliton density n =I(V n), at which the dimerization u(x) vanishes and the system transforms to a metal. If we set =hvF/z equal to the singlesoliton extension, the soliton density n e0.03 electrons/C atom can be found.
Horovitz [86] derived an effective amplitudephase Hamiltonian intended to describe also the phase of a soliton solution by taking into account some Coulomb interaction. If the phase and amplitude of the order parameter are coupled strongly enough, numerical solutions by Grabowski et al. [87] describe a discontinues transition to an incommensurate conducting phase. This involves the formation of
75
metallic islands growing with increasing doping level. A qualitatively similar model was proposed by Tomkiewicz et al. [88] as deduced from their observation of a gradual increase of the Pauli susceptibility. The existence of the intermediate doping region with very low Pauli susceptibility, however, seems to rule out the metallic islands [82,89].
Mele and Rice [22] calculated the density of states in the SSH model introducing random impurities. They explained the metal transition as a consequence of closing the Peierls gap because of increased disorder and the influence of neighboring chains. In this model, the impurities were treated either as a random diagonal potential (covalent bond model) or as a Coulomb potential. While the undisturbed SSH model shows a Peierls gap even for arbitrarily high soliton concentrations (i.e. high doping), disorder introduces states into the gap which closes at doping level y=O.1 to form a pseudogap with localized states at the Fermi level and persisting bond alternation. In this model, the density of states near the Fermi level is given by
N(E) = N [g + (EE )2/,2A2g5] (3.73)
0 'Z F
where ?I=h/27 T is the Peierlspairbreaking parameter, A is the usual incommensurate order parameter, r is a mean electronic impurity scattering time, g=(12)"2 measures the reduction in the Fermilevel
76
density of states relative to N0, the metallic value. The dc conductivity is reduced from the Drude metallic value a to a =a0g2 and, for 1 < q <31t2, the conductivity increases with increasing impurity scattering. Thus, if impurity scattering is proportional to phonon scattering which has the proportionality with temperature, the dc conductivity could be increasing by increasing temperature.
77
H
I
C
C
I
H
H
C
C
I
H
H
I
C
+000
J
C
I
H
H C
4 ,,
C
H
H
Dimerization coordinate u defined for trans(CH)
Figure 3. 1
78
(a)
d 2F+
Zn
2 TF
2AF
X (q)
N(EF
3d (Free electron)
2F
Figure 3.2 The response function for
(a) Onedimensional electron gas
(b) Threedimensional electron gas
(From reference 37)
X(q)
N(EF
(b)
79
IAI
AX
*F
Degenerate States
F
Figure 3.3 The Fermi surfaces of a free electron gas
(a) For one dimension all states with I k I < I k F are occupied. A perturbation of wave vector q=2kF couples a set of planes of degenerate states on the Fermi surfaces.
(b) For three dimensions all states with I k I
(From reference 37)
(a)
II
q:2
(b)
dP
80
Q(q)
//
/ /
T > T D~~I/\i /
\I
t 4eT=T \ q
0 2k 7r/q
F
Figure 3.4 The solid line represents the natural frequencies of the lattice vibrations of a crystal. The dashed lines represent the lowering of the frequency og the lattice vibration at q=2kF caused by the response of the electrons, known as "Kohn anomaly".
81
kF
I Ek
/
4/
/1
2  1
2F
I
I
0
K q
Illustration of the Peierls gap at q=2kF.
I2A
k
Figure 3.5
82
C C Cerl CCton
Neutral soliton
A
B
soliton f
antisoliton
0
C C C C C C
positively charged soliton (Q = + e, s = 0)
C C ccC) c C C
negatively charged soliton (Q=e, s=0)
Figure 3.6
(a) A neutral soliton in trans(CH) with twofold
degenerate ground states A and B.
(b) Formation of the soliton pair.
(c) Positively and negatively charged solitons (Q: charge, s: spin)
(a)
(b)
(c)
83
oft% I
, C
F c
S
V
;rk
Figure 3.7 The extended and reduced zone schemes.
(From reference 3)
N L
84
E(u)
U0 0
u
Figure 3.8 The total energy (electronic plus lattice distortion) as function of u. The double minimum associated with spontaneous symmetry breaking and the twofold degenerate ground states.
(From reference 1)
85
C C'C 'C 'C lc~~c
Acc , Bc
0
Figure 3.9
soliton
U
0
i
antisoliton
A neutral soliton.
(a) A neutral soliton in mixture phase in trans(CH)x.
(b) The order parameter for a soliton and an antisoliton.
(a)
(b)
0i
86
2 4 8 S 10 12 14 18 16 20 22 24 25 20 30
Assumed halfwidth
Figure 3.10 Soliton energy as a function
for several values of the energy gap E .
g
of an assumed soliton width
(From reference 3, 4)
1
0. 9 f
0.8 0.7 0.68 0.5
CO. 4 0.3 0.2
4)
lap
ES *2.OeV E3 1.4eV
Eq./..Oev. . . . . . . . .
0.1
CL
0
/
11!
Figure 3.11 The transformed
TakayamaLinLiuMaki continuum limit model.
energy bands entering the of the Su, Schrieffer, and Heeger
(From reference 1)
87
Lek
af
m
m
0 T
/
k
\
\~
88
(a)
ikC. C C C
C C C e
C #4., kve k C e  t N
A
B
(b)
A Cixture of A d
A ~mixture of A andBJ A
Figure 3.12 Formation of a particle by adding an electron in infinite chain of trans(CH) . (a) For a soliton which requires to overcome
infinite potential barrier due to the band polaron which does not require to change the has same phase at both side.
alternation. chain at co.
(b) For a Note that it
89
U0
4,(x)
A phase
_____________________________________________ I.
u
0
x
B phase
The order parameter of a polaron centered at origin.
Figure 3.13
90
negatively charged polaron (electron polaron)
Q=e s=1/2
(a)
C C C
positively charged polaron (hole polaron) Q=+e s=1/2
0
C )
(b)
(conduction band)
half filled filled
(valence band)
(conduction band)
empty
half filled (valence band)
C
Figure 3.14
The polarons in trans(CH)x.
(a) Formation of an electron polaron and a hole polaron
in trans(CH)x
(b) Schematic diagram showing the localized energy states.
91
(a)
0 (X)
0 
x
u
0
A
ol
B
p ()
x
U
0
(b)
0
C ,~ K P,; ,C. C
C C C C
0
positively charged two solitons (bipolaron)
~S~c , 4 + ,C ,C% C
C C C C C %IC 10c d
negatively charged two solitons (bipolaron)
Figure 3.15 polarons; (b) trans(CH)x.
Comparison of (a) the order parameter of two solitons and the two positively and negatively charged solitons in
Z
:7

92
Table 3.1
Doping induced infrared active phonon
frequencies (w)
(a=0.23)
w 0 (cm'1) A I(cm') n n n
1234 0.07 886
(CH)x 1309 0.02 1285
2040 0.91 1397
921 0.06 770
(CD). 1207 0.005 1148
2040 0.93 1236
(From reference 61)
(a)
0 400 800 1200 1600 2000 2400 2800 W, cm
(o =886 cm'
02=1285 cm w =1397 cm1
A
H
C C
2H
(2)
C C
(3v)
Figure 3.16 IRAV modes in trans(CH)x.
(a) The function Do(co). The dashed line is (1a)~1 for determining IR modes. (b) Normal coordinates of the infrared active modes (wc, CO2, and c ) in trans(CH) . The arrows indicate the atomic displacement vectors.
(From references 61 and 63)
93
(a=0.23)
 (Ia)'
2
0
2
(b)
H
C
at
C C
H% 
(CO )
3
C
half filled
A ' >A) 2AV
(a)
(b)
(b)
empty
A
V
C
filled
V
Figure 3.17 The optical absorptions for a (a) neutral soliton (note that the energy level is lifted toward the conduction band due to the ee repulsive interaction), (b) positively charged soliton, and (c) negatively charged soliton
94
So
C
2A
J
S~
2A
i

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OPTICAL PROPERTIES OF SEGMENTED AND ORIENTED POLYACETYLENE BY HYUNGSUK WOO 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 1990
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ACKNOWLEDGMENTS It is my great pleasure to thank my advisor, Professor D.B. Tanner, for his guidance, patience, valuable advice, and support all throughout my graduate career. I would like to thank my supervisory committee, Professors N.S. Sullivan, S.E. Nagler, H, Monkhorst, G. Ihas, and W.B. Wagener, I would like to thank Professor A.G. MacDiarmid, Dr. Theophilou, and Dr. Arbuckle at the University of Pennsylvania for their nice and fresh samples, which were essential to complete this dissertation. I would also like to thank Dr. S. Jeyadev at the Xerox Webster Research Center for his valuable discussions and suggestions on all levels of physics. I would like to thank Drs. X. Q. Yang, Y.H. Kim, I. Hamburgh, S.L. Herr, and K. Kamaras for their help during my fiveyear stay in the "Tanner's Group." My fellow graduate students deserve thanks for their friendship and cooperation in development of the equipment and computer software. In particular, I would like to thank CD. Porter for his help in converting the software from PDP 11 computer to personal IBM computer, which was a great convenience for data analysis. I would like to thank the staff members of the Physics Department shop and the engineers of the condensed matter physics group for their technical support. I would like to thank my Korean fellow students for sharing friendship and useful conversation. ii
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Special thanks go to my wife, KumOk, my son, SoonBo, daughter, JungYoon, and my parents, whose love, care, understanding, and support were most important during my studies. iii
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TABLE OF CONTENTS Pa ge ACKNOWLEDGMENTS ii ABSTRACT V CHAPTER I INTRODUCTION 1 n INSTRUMENTATION AND EXPERIMENT 9 IR and Optical Technique 9 Instrumentation and Data Acquisition 15 Measurements 17 Sample Preparation and Doping Technique 19 m BASIC THEORY 36 ElectronPhonon Interaction and Peierls Transition 36 Soliton Excitation 45 Polaron and Bipolaron Excitations 60 Dopinginduced Infrared Active Vibrational Modes 63 Electronic Excitation 66 Transport 69 IV SEGMENTED POLY ACETYLENE 95 Segmented Poly acetylene: (CDHy)x 95 Infrared Active Vibrational Mode (IRAV) 95 Midgap and Interband Absorptions 102 V ORIENTED NEWPOLYACETYLENE 124 The NewPolyacetylene 124 The dc Conductivity 124 Temperature and Polarization Dependence in Infrared Absorptions 126 Midgap and Interband Absorptions 129 VI SUMMARY AND CONCLUSIONS 145 REFERENCES 148 BIOGRAPHICAL SKETCH 154 iv
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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 OPTICAL PROPERTIES OF SEGMENTED AND ORIENTED POLYACETYLENE BY HyungSuk Woo May 1990 Chairman: David B. Tanner Major Department: Physics The dopinginduced infrared and optical absorptions in polyacetylene have been measured. The samples studied were oriented new[CH] and segmented polyacetylene, [CDH ] , in which either 15% or X y X approximately 27% sp^bonded CDH units had been incorporated. All samples were doped with iodine by insitu vapor doping (segmented polyacetylene) and chemical doping (newpolyacetylene) techniques with various dopant levels. We have also studied the temperature and polarization dependence of the ac and dc conductivity of new[CH] , All doped segmented polyacetylene showed the usual dopinginduced infrared active vibrational (IRAV) modes at 745 cm'' and 1140 cm"' and at midgap. The oscillator strength of these absorptions increased with V
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increasing iodine concentrations, even for the samples with 27% sp^ sites. The interband absorption data suggested that the sp' defects of [CDH^]^ were not uniformly distributed over the polymer chain. For a sample with 27% sp hybridization, we have found a new broad band around 1.12 eV with relatively lower intensity for lower sp^. We assigned this as an absorption due to the neutral solitons. In 550% stretched new[CH] , doped with iodine at level 6% (F), the dc conductivity was about 10000 Q'^cm'^ at 300 K, decreasing as temperature is lowered and reaching 2500 Q\m'^ at 84 K. In contrast, the farinfrared oscillator strength in the 500 cm"' region, where the "pinning" mode is usually seen, increased and shifted to lower frequencies with decreasing temperature. This effect was observed for polarizations parallel and perpendicular to the stretching direction. This sample displayed highly anisotropic behavior in IRAV modes, with substantially greater o^((o) for polarization parallel to the chain direction. This anisotropic behavior was observed over the range of frequencies including the midgap and interband region. vi
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CHAPTER I INTRODUCTION In conducting polymers, the electronic structure is determined by the chain symmetry (e.g., the number and kind of atoms in the repeat unit, etc.) and by the concentration of charge carriers. The result is that such polymers can exhibit semiconducting or even metallic properties. Poly acetylene, (CH)^, is a particularly interesting for in this case the n band is half filled, implying the possibility of metallic conductivity [1], In poly acetylene, the 7relectrons are delocalized along the polymer chain due to the strong intrachain bonding and weak interchain interaction, which results in this polymer being a one dimensional electronic system. Polyacetylene consists of long quasione dimensional chains of (CH) units aligned in a partially crystalline lattice. Three of the four carbon valence electrons are in sp^ hybridized orbitals, where two of the <7type bonds connect neighboring carbons along the polymer backbone and a third forms a bond with the hydrogen side group [1]. Thus, there are two possible types of molecular structure for this polymer with optimal bond angle 120Â° between these three <7bonds, cw(CH^ and /ra/w(CH)^ (see Figure 1.1). There are two and four CH monomers per unit cell in /ra/jj(CH) and ci5'(CH) respectively. The 1
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2 remaining valence electron, a ;relectron, has an important role in this conducting polymer forming a 7rbond with its chargedensity lobes perpendicular to the plane defined by the abonds. While the abonds form completely filled bands, the 7rbond leads to the partially filled energy band. If the chain lengths were equal, pure /raw(CH) would be quasione dimensional metal with a halffilled band. This configuration is, however, naturally unstable due to the Peierls instability [2], which leads to a dimerization distortion; adjacent CH groups move toward each other forming alternately short (or "double") bonds and long (or "single") bonds. By symmetry, there are two lowestenergy states, A and B (see Figure 1.2). This twofold degeneracy leads to the existence of nonlinear topological excitations [3,4], solitons, which appear to be responsible for many of the remarkable properties of trans(CH)^. In d5(CH)^, no such twofold degeneracy exists [1]. The trans(Câ€¢[)^ is the thermodynamically stable form. Complete isomerization from cisto trans(CH)^ can be accomplished after synthesis by heating the film to a temperature above 150Â° C for a few minutes [1]. It is also known that ciJ(CH)^ can be changed to transiCH)^ by doping [5]. TransiCH)^ films was first prepared by Shirakawa and Ikeda [6] with film thicknesses varying from < 10"^ cm to about 0.5 cm. This films consist of randomly oriented fibrils with average diameter about 200 A [6]. The bulk density of unoriented Shirakawa polyacetylene is about 0.4 g/cm^.
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Xray studies of rra/i5(CH)^ [6,7] have shown that the films are highly crystalline (see Figure 1.3). According to these Xray data, the chain structure is dimerized with the distortion parameter (the difference between short and long bonds) u^='0.03A and the length of a unit cell 2.46 A. Recently, a new technique has been developed by Naarmann and Theophilou [8] to synthesize poly acetylene with fewer sp' (< < 1 mol%) defects (CH^) than in Shirakawa(CH)^. The reduction of sp^ [1] defects implies longer conjugation length and fewer crosslinks, which leads to higher dc conductivity. Naarmann and Theophilou [8] have reported the dc conductivity of heavily doped film by iodine was about 20000 Q'^cm^ which is much higher than one for Shirakawa(CH) , 10lO' Q\m'\ The bulk density of stretched film was about 0.85 and 1.12 g/cm before and after doping respectively. Xray studies of these films, however, show no major differences from Shirakawa(CH) . The undoped polyacetylene (from now on, "polyacetylene" or "(CH) " will be referred to as tran(CH)^ through this dissertation) has semiconducting gap (or Peierls gap) of 1.45 eV [7]. Electron spin resonance (ESR) measurements by Davidov et al. [9] show that the neutral solitons which exist in undoped (CH)^ carry spin 1/2, whereas the charged solitons introduced by doping carry no spin. From optical measurements [7], it is known that the charged solitons have a broad absorption band at midgap and doping induced infrared active vibrational modes (IRAV modes) are shown in the infrared region. The
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soliton model first introduced by Rice [10] and Su, Schrieffer, and Heeger (SSH) [3,4] and its continuum version by Takayami, LinLiu, and Maki (TLM) [11] have successfully described the ground state and nonlinear excitations in polyacetylene with good agreements with experiments. Horovitz [12] calculated the infrared conductivity from which he obtained the doping induced IRAV modes. Mele and Rice [1315] predicted a "pinning mode" a characteristic infrared activity associated with the oscillation of the charged solitons bound by Coulomb interaction to ionized impurities. In heavily doped polyacetylene, the specific heat is linear in temperature [16] and there is a somewhat temperature dependent Pauli susceptibility [17,18]. These results are consistent with a metal. The temperature dependence of dc conductivity, however, is not metallike. Instead, it decreases as the temperature is lowered. Kivelson [1921] proposed a intersoliton hopping model to describe the conductivity in lightly doped polyacetylene. Mele and Rice [22] predicted a transition to a metallic state as a consequence of closing the Peierls gap because of increased disorder, which introduces states into the gap. According to this model, the Peierls gap would exist at high doping level but be incommensurate with lattice. Then the introduction of disorder puts states in the gap, leading to a "dirty" metal with a finite density of states at E . F The rest of this thesis is laid out as follows. Chapter n will describe the optical technique and apparatus including the experimental
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details. Some of the basic theory for poly acetylene will be given in Chapter III. Chapter IV and V will discuss the data and results obtained from this work and conclusions are in Chapter VI.
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6 Figure 1.1 Structual forms of polyacetylene (a) cisand (b) transiCU)
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7 Figure 1.2 The twofold degenerate ground states of transiCB)
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8 Figure 1,3 The crystal structure of fro/wpoly acetylene obtained from Xray scattering. The arrows represent the bondalternation atomic displacement. (From reference 7 )
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CHAPTER II INSTRUMENTATION AND EXPERIMENT IR and Optical Technique Fourier Transform Spectroscopy The Michelson interferometer, which is well known for demonstrating the interference of two mutually coherent light beams, was supplanted years ago by the introduction of higher resolution instrumentsÂ—FabryPerot etalon, Michelson echelon, etc. During the 1960s, however, it became more useful for measuring relatively complicated spectra in the infrared region for the following reasons [27]: i) The availability of computers makes the Fourier transformation of a measured interferogram to a spectrogram relatively easy. ii) The signal to noise ratio for an interferometer is greater than that of a conventional dispersing spectrometer, due to the Fellgett advantage [28]. In this section, these aspects will be discussed briefly in turn. A more detailed description of the method will be found elsewhere [28]. 9
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10 Fourier transformation of an interferogram . As can be seen in Figure 2.1, the basic principle is the superposition of two wavetrains. From the Figure 2.1, the addition of the wavetrains which travelled different distances 2x^ and 2x^ yields the superposed wave y = a jzÂ«(27r(ft vx^)) + a ji>i(27r(ft vx^)), = 2a cosinvix^ x^) jm(2;r(ft v(Xj + x^/ 2 )), (2.1) where v is the wavenumber and f=cv the frequency of the light. The sine term represents a wave motion and the cosine term is the amplitude. The intensity S(v) can be obtained from equation 2.1 as S(v) = 4a^ cos^nvS) = 2a^ + 2a^ cosilnvd), (2.2) where x^x^=S, the path difference. If the radiation is not monochromatic but consists of a continuous spectrum each component having intensity S(v), then, for some path difference d, the last equation must be modified by writing the intensity I(^) I(^) = S(v) dv + \S(v) cos(27tvd) dv, (2 3)
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11 This equation will give the appropriate distributions of a source interferograms. When J Â» <Â», lim I(^) =f S(v) dv = I(co) = \ 1(0). (2.4) Equation 2.4 holds because the cosine term in equation 2.3 will oscillate rapidly and average to zero as d approaches infinity (essentially, cosi2nvd) is more rapidly varying than S(v), which is, in turn, equivalent to saying that once all the structure in S(v) is resolved, I(^)>I(oo)). So, the interferogram can be written as r" I(^) !(Â«) = S(v) cos(2nvd) dv . (2.5) The spectral distribution, S(v), is the inverse Fourier transform of interferogram. Using the fact that S(v)* = S(v), we obtain S(v) = c J^" did) I(oo)) e'^^^^ dd , (2.6) 00 where c is a constant. The spectrum derived is exact only if the interferogram, in accordance with equation 2.6, covers all path differences from ^ = co to +oo. In practice the limits must be finite, which causes broadening of the line shape and side lobes, or "feet", to occur in sharp spectral structures [29]. Apodization [28] is necessary to remove these "feet" arising from finite maximum path difference.
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12 The resolution of the obtained spectrum is limited to Av = l/d max and the maximum cutoff frequency of the spectrum, v . is given by ^ . < Â— = iA . , (2.7) min 2 V 2 min V^V max where <5^.^ is a sampling interval. Equation 2.7 requires the sampling interval must be small enough to eliminate aliasing in entire frequency range. Proper filtering is also necessary to remove the frequencies higher than the maximum cutoff frequency [28]. Fellgett advantage . The Fellgett advantage is the term for the enhanced signal to noise level of an interferometer as compared to a dispersing spectrometer. The noise in a detecting arrangement normally used in the infrared arises from statistical fluctuations in the motion of the electrons comprising the electronic current. This fundamental noise is known as "Johnson noise." The fluctuations are in fact a Brownian movement of the electrons. If the current i is averaged over a time t in a circuit of resistance R, the fluctuation current is given by, setting the energy = thermal energy, [(Swf/R] t = (dif R t Â« k T , (2.8) where k is the Boltzmann's constant and T is the temperature at the detector. Equation 2.8 implies that the Johnson noise, <5i, is
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13 proportional to t"*'^ and to T*'^. Thus, it is useful to cool the detecting system. Fellgett pointed out that the total number, N, of resolved elements of a spectrum is exposed to the interferometer detector for the total measuring time t. On the other hand with a dispersing spectrometer, the total observing time for N different wavelengths is divided so that the measurement of each element is allowed only a time t/N. In accordance with equation 2.8, the noise will be greater for the dispersing instruments by a factor n''^. Optical Spectroscopy The grating monochromator is used at higher frequency range, from near IR to UV, where the Fellget advantage loses its importance due to the increasing photon noise in the detectors used. In general, the equation for diffraction grating is written as (see Figure 2.2a) pA = b (sin + sin 6) , (2.9) where p is called the order of the spectrum, 0 is the angle of the incident light and 9 is the angle of the diffracted light both measured from the normal direction to the plane of the grating. In Figure 2.2a, the path difference between the two rays shown is defined as  AB+AC  . For p= 0 (zero order) the path difference is zero for all directions, 4> = d (see Figure 2.2b). For other orders (p = 1, 2, etc.) and for fixed
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14 , several spectra overlap such that the product pA remains constant. We may express this overlapping by a formula. If in a fixed diffracted angle two wavelengths show maxima for successive orders p, p1, we have pA=(p 1)(A + AX), which leads to JA=A/(p 1). (2.10)
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15 This overlapping of orders means that unwanted wavelengths must be removed from the wanted order by absorption with a suitable filter or use of a detector sensitive only to a limited range of wavelengths. Instrumentation and Data Acquisition IBMBRUKER IR/98 Interferometer The rapidscan interferometer used in this work is a IBMBRUKER IR/98, a Fourier transform infrared (FTIR) spectrometer, which covers frequencies ranging from 25 to 5000 cm"^ The system is basically divided into four modules: source housing, interferometer, sample chamber, and detector compartment (see Figure 2.3). The mercury arc and globar lamps were used as sources for farIR (20600 cm"^) and midIR (4505000 cm'^) measurements respectively. The interferometer consists of twosided movable mirror, beam splitters, and filter changer. The sample chamber consists of two parts for the transmittance and reflectance measurements. For detectors, a liquid He cooled bolometer (infrared laboratories LN6/C) and pyroelectric deuterated triglycine sulfate (DTGS) were installed in the detector chamber for the farIR and midIR respectively. A diagram of the bolometer is shown in Figure 2.4. All chambers were maintained under vacuum to prevent IR absorption due to water vapor in the air. In the rapidscan interferometer, the mirror is moved at a constant speed v so that, from the equation 2.5, the optical path
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16 difference can be replaced by d = 2vt where t is a sampling time. This implies the signal seen at the detector is modulated to acoustic frequencies, f = 2vv. This modulated signal is fed into a preamplifier which amplifies the detected signal and does the frequency filtering. The signal is digitized by a 16bit analogtodigital converter and recorded in an IBM computer system which came along with the interferometer. More detailed descriptions of the method can be found elsewhere [30]. Grating Monochromator A PerkinElmer monochromator was used to measure the optical data for the frequencies ranging from midIR to ultraviolet (80045000 cm'^). As shown in Figure 2.5, the monochromator consists of four major parts: sources, grating, detector, and sample area. For sources, globar, tungsten, and deuterium lamps were used for the frequencies of midIR, visible, and UV regions respectively. The light from a source which can be selected by mirror M2 passes through the chopper and lowpass or bandpass filter which eliminates the unwanted orders of diffraction. The monochromator has two slits called entrance and exit slit and a grating between two slits. A grating can scan from 60 degrees to about 15 degrees corresponding to socalled "drum number" from 0 to 24. This means that a drum number corresponds to about 1.8 degrees of rotation of the grating at each step. We saw, from equation 2.10 that the rotation angles of the
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17 grating along with the slit widths can determine the resolution of the monochromator. A polarizer can be placed in front of the exit slit if necessary. Two large spherical mirrors make the fine images at their focal points where we put the samples for the reflectance and transmittance respectively. An analyzer can be placed near the transmittance site. Finally an ellipsoidal mirror makes a very fine image on the detector. The detectors used are thermocouple, PbS, and Siphotocell for midIR, visible, and UV respectively. Table 2.1 shows in more detail the combinations of detectors, sources, grating, and polarizers for each frequency range and Figure 2.6 shows the bias circuit for the Siphotocell. The signal from a detector along with the chopped reference signal were fed into a lockin amplifier (Ithaco 393). The amplified signal was sent to a pen recorder and a digital voltmeter (Fluke 8520A) where the signal was averaged for a given time interval and digitized. This digitized data were sent through the IEEE488 Bus and a general purpose interface box (GPIB) to a PDP 1123 computer and recorded for the data analysis. Measurements General All optical data were taken by the monochromator and interferometer with a suitable sample holder which will be explained in
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18 the next section. The transmittance and reflectance were the major part of the optical measurements. The spectra from the spectrometer were recorded in the computer where all necessary data analysis had been performed Temperature and Polarization Dependence Figure 2.7 shows the sample holder used in the temperature dependence measurements. The main body of the sample holder was made of oxygen free copper. Four thin brass bars insulated from each other (A, B, C, and D in Figure 2.7) were used to contact the sample for the 4terminal resistance measurements. To make good ohmic contact, the sample was pressed by the two plastic bars (F and F' in Figure 2.7) and the bar "G" was moved up and down by turning the screw "E" to stretch and flatten the sample. Finally, the whole sample holder along with the silicon diode thermometer (Scientific Ins. Si410A) and the heater were connected to the LN^ cryostat. Figure 2.8 illustrates the LN^ cryostat, with which we took the dc and optical data from liquid temperature to 300 K. The Polyethylene (20600 cm" ^) and KCl (45025000 cm" ^) were used as the windows for the frequency ranges indicated during the optical measurements. For the samples which were highly stretched, we used wire grid polarizers to investigate the anisotropy of the samples by putting the polarizer such that the electric field was parallel and perpendicular to the chain direction of the samples.
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19 Sample Preparation and Doping Techniques All samples used in this work were synthesized by a research group at the University of Pennsylvania. This section presents the sample preparation for deuterated polyacetylene and newpolyacetylene including doping techniques. The synthesis of the deuterated polyacetylene was done by Georgia Arbuckle and that of the "new^'polyacetylene by Nicolas Theophilou. It is described here for completeness. The iodine doping was done at Florida just before the infrared measurements. Deuterated and Partially Hybridized Polyacetylene In order to introduce the sp^ defects in trans(CD)^ chain, completely deuterated polyacetylene, (CD)ji was polymerized and subsequently partially hydrogenerated to create sp^ hybridized (CDH) units in the otherwise sp^ (CD) chain (Figure 2.9). Sample preparation . Sample preparation followed the method described by Arbuckle [31]: Free standing films of cwrich (CD) were synthesized from C^D^ (99.5% isotopic purity) by the Shirakawa method [32]. The films were isomerized to the trans isomer by heating in vacuo for about 1 hour at about 160Â° C. The average thickness of these films was about 100 fim. The method of introducing sp^ defects is variation of the technique initially investigated by Pron [33] and also studied by Soga
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20 et al. [34]. The reactions involved are (CD)x + xyNa'Naphth> [NaJ + xyNaphth (2.11) [Na^ jj(CD)Â°^']^ + xyCH^OH > + xyCH^ONa (2.12) (CDH ) + xyNa'Naphth" > [Na"'(CDH V^] + xyNaphth (2.13) [Na J(CDH ^p'']^ + xyCH^OH > (CDH ^^^^)^ + xyCH ONa (2.14) The apparatus used for both the ndoping (Eq. 2.11) of the trans films and subsequent reaction with CH^OH (Eq. 2.12) is shown in Figure 2.10 [31]. The reactions (Eqs. 2.112.12 ) were performed as follows (i) The apparatus was evacuated and transferred to the inert atmosphere dry box. (ii) The CH^OH (about 25 ml) was transferred into the round bottom bulb (1) and 0.32 g of naphthalene and an excess 0.2 g of clean sodium metal was placed in bulb (2) to prepare the about 0.1 M sodium naphthalide doping solution (about 3 hours) most often used to give approximately 15% doping. (iii) After cleaning tube (3) using this sodium naphthalide solution (this procedure ensured dry and oxygen free conditions), a weighed amount of trans(CD)^ (1020 mg) was placed in tube (3).
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21 (iv) After closing stopcock (4) and opening stopcock (7), the reaction vessel was evacuated on the vacuum line. After pumping, stopcock (7) was closed and stopcock (4) was then opened so that the sodium naphthalide solution was poured through the frit (5) and into the main chamber (3) until the trans(CD)^ film was submerged. It took about 10 hours for complete doping for a sample with a thickness about 100 //m. (v) This ndoped sample was washed several times by repeated internal distillation of tetrahydrofuran (THF) to ensure that all the sodium naphthalide solution had been removed from the ndoped (CD) . (vi) The washed film was dried by the cryopumping technique. (vii) This dried ndoped film was compensated by CH^OH through vapor phase reaction by opening stopcock (4b) (stopcock (4a) closed) for 40 hours. After compensation, the film was washed by CH OH/CH ONa Â•^3 3 solution about 12 times and then dried by cryopumping technique. The reactions given by equations 2.11 and 2.12 produced film with 15% incorporation of sp^ defects in trans(CD)^ (this is denoted by (CDH^^^)^). In order to introduce a larger amount of sp^ defects, reactions 2.13 and 2.14 were performed. The composition of rehydrogenerated (CDH^ film as determined by titration of the CH OH/CH ONa wash solution in bulb (1) was (CDH ) with y==0.27 because an additional about 12 % doping had occurred. All these films ((CDH ) , y=0, y=0.15, and y=0.27) were sealed in glass tubes with argon atmosphere and sent to the University of Florida
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22 for the electrical and optical measurements with various conditions, e.g., doping and temperature. Doping techniques of deuterated polyacetylene . In order to see the change in dc conductivity and optical properties of (CDH ) films, y ^ iodine was doped to the samples by insitu vapor doping. The sample was placed on a sample holder as described earlier in this chapter (Figure 2.7). The cryostat (Figure 2.8) along with this sample holder was placed in the spectrometer, with the hoses connected to the iodine chamber and vacuum pump (Figure 2.11). A digital voltmeter (KEITHLEY 195 DMM) was connected to the cryostat for 4terminal resistance measurements. After finishing the optical and dc measurements for an undoped sample, the sample was exposed by opening a valve in the iodine chamber for a short time (about 5 seconds) for light doping. While keeping the iodine concentration, a second optical and dc measurements were performed. This procedure was repeated by increasing the exposure time to the iodine vapor until we obtain the maximum doping concentration. The maximum iodine concentration was determined by weight uptake and the intermediate concentrations were estimated by assuming a linear relationship [43] of Ina with concentrations up to about 3% doping, where a is the DC conductivity. Doped (CDH ) is y 3t denoted by [CDH^(I^)J^, where z is the iodine concentration. Most of the sample handing, such as doping, weighing, and mounting on the sample holder, was performed in a dry box.
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23 NewPolyacetylene: New(CH)Â» The Shirakawapolyacetylene [32] has been considerably studied on account of its simple conjugated chain structure and highly conducting behavior with doping. Nevertheless, its degradation in air, which decrease the conductivity, affect its potential application. Naarmann and Theophilou [8] presented a new process for the production of metallike, stable, highly conducting polyacetylene, whose dc conductivity is in range of 10000100000 Q'^cm'^ with iodine doping at room temperature. Sample preparation . According to Naarmann and Theophilou [8], the newpolyacetylene was prepared in following way. The apparatuses were a 500ml fourneck flask with thermometer, funnel, magnetic stirrer and connection for vacuum and argon. The reactive mixture was 50 ml silicon oil AV 1000 (Th. Goldschmidt), 31 ml triethylaluminiumTEA ^^2^3^3^^' tetrabutoxytitanium Ti(C^H^O)^ freshly distilled (Dynamit Nobel). The silicon oil was stirred and degassed for 20 minutes at 0.05 mbar. The TEA was added in a stream of argon and Ti(C^H^O)^ was added, drop by drop, through the inactivated funnel over a period of one hour at 3842 Â°C. This mixture was degassed for one hour at room temperature and subsequently stirred for two hours at 120Â° C in a weak current of argon. Inside the glove box, an even, homogeneous layer of standard catalyst was applied to a flat carrier, e.g., a highdensity
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24 polyethylene film which is a stretchable polymersupporting material. This carrier, coated with catalyst, was sealed by means of a hood fitted with a gas inlet valve. After evacuating the hood, 600 ml of purified acetylene was passed into it over a period of 15 minutes and acetylene polymerized at the surface of catalyst on the carrier to form (CH)^. This new(CH)^ prepared on the surface of polyethylene was stretched (200500 % longer than its original length) inside the glove box, then removed from the supporting film and washed in the usual fashion with toluene, CH^OH/HCL, and methanol. The amount of catalyst to obtain a black (CH)^ film with thickness about 15 fi^ was 7 ml. Doping procedure . Although the insitu vapor doping, shown in previous sections, had had good results in increasing the dc conductivity, it was known that this newpolyacetylene has had better results with chemical doping [8] for increasing its conductivity. Thus, these samples were doped in chemically with Ij/CCl^ saturated solution (0.26 g of iodine with 10 ml of CCl at room 4 temperature). Samples were submerged in this solution for one hour. Afterward, they were washed only once with CCl^ for 1 minute, and then dried for 5 to 10 minutes in a weak stream of argon. The dopant level obtained by weight uptake was about 6%.
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25 Table 2.1 Grating monochrometor parameters a) b) c) Wavenumber Grating Source Detector Polarizers Slit (cm ) (lines/mm) A B Cum) lUl (jlooar Thermo Gnd/blank 2000 101 Globar Thermo Gnd/Gnd 1200 140J1752 101 Globar Thermo Gnd/Gnd 1200 1044ZOlZ 240 Globar Thermo Gnd/Gnd 1200 24674101 viiuuar I iicriiiu vjria/ vjnu IZUU 40155105 590 Globar Thermo IR/Grid 1200 47937977 590 Tungsten Thermo IR/Grid 1200 38295105 590 Tungsten PbS IR/Grid 225 47937822 590 Tungsten PbS m/IR 75 751110234 590 Tungsten PbS IR/IR 75 919113545 1200 Tungsten PbS IR/IR 225 1290420144 1200 Tungsten PbS IR/VIS 225 1703324924 2400 Tungsten 576 VIS/VIS 225 2206628059 2400 Eh lamp 576 VIS/VIS 700 2570637964 2400 D2 lamp 576 VIS/VIS 700 3636845333 2400 Da lamp 576 Blank/Blank 700 a) Thermo: Thermocouple, 576: Si photocell b) A: polarizer, B: analyzer c) Max. slit width
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26 Fixed mirror Beam splitter Source Movable mirror O Detector Figure 2.1 Schematic diagram of Michelson Interferometer
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27 (a) incident light diffracted liffht normal (b) zeroorder position w'?c^tK ^f^ "^'^^'^^ Â°/ (a) for nonzero order position ( b IS the grating constant) ; (b) for zero order position.
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28
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29 Figure 2.4 Schematic diagram of Bolometer
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30 SAMPLE ROTATORÂ—^ Figure 2.5 Schematic diagram of grating monochromator
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31 200 ViQ 8BQ IN LF355 V +20V IN 1 ixYd^t 7815 500 Q o 8BQ OUT 576 OUT O +V 500 X 20V IN 7915 15V 500 r3 Iztl 500 //F ^ V^^^^ ^ O V Figure 2.6 Bias citcuit for the Siphotocells
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32 A, B, C, D : Cu contacts for 4 probe measurement A' , B' , C , D' Leads for 4 probe measurement E : Screw F, F' , G : Plastic bars H : Supporting plate I : Silicon diode thermometer J : Heater Figure 2.7 Sample holder used in the temperature dependence measurement
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33 Liquid N2 Vacuum Heater Thermometer Sample holder iiini II . n r Electrical feed through 0rings Window Sample Figure 2.8 Schematic diagram of liquid cryostat
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34 D D D D D VV'VVVv 1 I T I I I D D D D D D n (CH OH) D D D H D D T I I I I I D D D D D D Figure 2.9 Formation of a sp^ defect in (CD)^ chain Figure 2.10 Schematic diagram of apparatus for preparing transiCDB ) (From reference 31) ^
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35
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CHAPTER in BASIC THEORY ElectronPhonon Interaction and Peierls Transition In order to consider the Peierls transition [2] in onedimensional electronic system, we start with the tight binding model with electronphonon interaction. A model Hamiltonian, known as SSH model [3] which neglects the electronelectron interactions, is written as H Â„ = H + H ^ + H ^ Son e eph ph = H + H eph ' (3.1) where H'=i:taj+i){Cjc;^, 0.2) Â«pb=f !:(Â°j + ,V +TMpJ(3.3) Where t0,j + l) is a hopping integral between sites j and j + 1, u. denotes the lattice deformation at the j'*' site, p is the momentum j conjugate to u., and the operator cj(c.) creates (destroys) an electron 36
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37 at site j. The equation 3.1, which is often applied to the system of /ra/w(CH)^, is a simple onedimensional model of tightbinding band for the p^ electrons coupled to the elastically vibrating (mainly CTbonded) skeleton of the CH monomers. In (CH)^, M in equation 3.3 is the mass of the (CH) group and u. is the dimerization coordinates, as shown in Figure 3.1. The Hamiltonian for phonons may be rewritten p p M(o^ H,.=i:(^ +^QQ ), (3.4) q ^ ^ where P = N 1/2 ^ Q, = N 1/2 ^ j ' (3.5) (3.6) where "Q^^and "P^" are normal coordinates and momenta, "N" is the number of atoms within length L of the onedimensional system, co is
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38 ElectronPhonon Interaction In order to consider the electronphonon interaction, we recall the equation 3,2. The effect of the lattice vibrations can be calculated by assuming that the motion of the lattice affects only the nearestneighbor hopping integral tO',j + l). Thus, for small displacement of the lattice tOj + 1) = + t^>.^iu.) , (3.7) where t^ is just hopping integral for the nearest neighbor, t^^^ is the electronphonon coupling constant, which is the first derivative of t 0 with respect to the interatomic distance. The t^^^ and t are 0 independent of j. The Fourier expansion of c is j _ 1 ikx ^ " P ^ ' (3.8) where x.=ja, n/a
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39 H = 2tÂ„ z cos ka c*c^ _ ^3 and H = Z E (0 c"^ c + <6 c"^ c ) k q ^'^'^ ^ % kq V , (3.11) where 0q = 2t('> N'^'Ce^^^ 1) Jta . (3.12) Here and H^ ^^ are Hamiltonians for the tightbinding of the nearest neighbor and the electronphonon interaction respectively and is the Fourier components of the potential (x), which contributes to the electronphonon interactions and is assumed to be sinusoidal. Later in this section, it will be shown that will be identical to the gap parameter for the Peierls distortion [2] at q=2k (k : wave vector at F F Fermi surface). Kohn Anomaly Before we go to the details of the Peierls transition [2] which is associated with electrons interacting with lattice vibrations, it is important to consider a fundamental instability in the onedimensional
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40 electronic system, which comes out as a result of softening the phonon frequency. This phenomenon is known as the Kohn anomaly [35]. A useful measure of the linear density response of the electronic system to the potential is provided by the density response function, or Lindhard function [36] < n > q = S [ l" '. y ] (3.13) k ^ k k+q J where is the mean value of the electron number density operator. n = c"*" c q k+q k and f is the equilibrium occupation number for FermiDirac statistics 1 f^ = [ 1 + txp{fi(e^ . Â£j} ] ,fi= , (3.14) Here "k^" is the Boltzmann constant. In accordance with equations 3.13 and 3.14, x(q) is dependent on temperature. The forms of xiq) for a one dimensional and three dimensional electron gas are shown in Figure 3.2 [37]. In accordance with Figure 3.2(a), ;f(q) in one dimensional system has a peak at q = 2k^ for T :;fc 0 and increases as temperature is
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41 lowered, diverging logarithmically as T ^ 0. For three dimensions (see Figure 3, 2(b)) there is no such peak. To understand the origin of this peak, we consider the Fermi surface of a one dimensional free electron gas. Figure 3.3(a) [37] shows the Fermi surface of a one dimensional metal at low temperature (T 0). The states with low kinetic energy are filled until we have accounted for all of the electrons in the metal. The momentum of the last electron added is called the Fermi momentum k , and its energy the Fermi energy e . States with Ikl < k r F ' ' F are filled and those above are empty [38]. In one dimension the energy is determined solely by the momentum in one direction; call it p = hk . X X All k states are occupied up to k  = k . For any particular value of A F "q", the states connected do not lie on the Fermi surface. However, for q = 2kp there are many states that are connected by "q" and lie on the Fermi surface with same energy s^, which in this case is just two parallel planes at k = k^ and k = k^ [37]. This implies the system at q = 2kp is infinitely degenerate. This degeneracy is known as Fermi surface "nesting", and it causes the infinite value of response function x((l=2k^) as T ^ 0 because the electron number density "n " is strongly related to the degeneracy of the system. If T is not zero but sufficiently low, then the two Fermiplanes are no longer exactly sharp. However, the system can have still very high degeneracy so that we have still high peak of ;e(2kp) at reasonably low temperature as shown in Figure 3.3(a).
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42 By contrast, we look at the situation for a three dimensional free electron metal (see Figure 3.3(b)) in which, for any q < 21^, only two states at are connected by "q" which have the same energy. Slightly different values of "q" will couple other states and give a density change with a different periodicity. Adding random periodicities results in no net contribution to the charge density. To investigate the kind of instabilities that result from the divergence of the response function in one dimensional system, let's consider the Heisenberg equation of the motion of Q (normal coordinates defined in equation 3,6) or (ihf a= [ [Q H], H] . dt^ 0 ' This equation can be rewritten as '''^ "77^ = " ^< ^ <^ '^> where "H" is the total Hamiltonian defined in equation 3.1. By using the harmonic approximation, Q Â« e"^*, and equation 3.13, this equation 3.15 is reduced to
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43 ^ ;i:(q) . (3.16) q where Q is the new phonon frequency for wave vector "q" after the electronphonon interaction has been taken into account. Before we saw that xiPL> becomes large for q = 2k when the temperature is lowered. In accordance with equation 3.16, there is a certain temperature at which i2(q=2kp) tends to zero, which implies that there is no restoring force for q = 2kp mode and the unstable lattice can distort. This phenomenon is known as "Kohn anomaly" and is the origin of the Peierls instability. Figure 3.4 shows the phonon frequencies around 2k soften F as the temperature is lowered, and i3(2k ) goes to zero at the Peierls transition temperature. At below the Peierls transition temperature, there will be a periodic lattice distortion accompanied by a charge density wave. Details of this phenomenon can be found elsewhere [3942]. Peierls Gap It can be seen that the instability of the onedimensional free electron system to an applied potential of wave vector 2k "0(q=2k )" F F is a generalization of the Peierls instability which produces gaps at the Fermi energy. We recall the Hamiltonian in equation 3.9
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44 The new eigenvalues of energy E^^ are obtained by using degenerate perturbation theory as there are states for which e = e when k kq k=q/2. Thus, E = 0 . (3.17) where \ = 2t cos ka . (3.18) Equation 3.17 can be solved in the usual way and gives \ = ^ ^^k + Â«k Â± H \J ^"^^if } Â• (3.19) Using the relation \\,2^'^\^'\n^' expression 3.19 simplifies to give the new electron energies \ = \n^^(\\J ^\Jf^ ' (3.20) The electron energy levels are split for wave vectors k=Â±q/2 (half filled band) and the potential 0(x) due to the electronphonon interaction, which has the periodicity n/k^ has created a gap of size
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45 2 I ^^(q=21^) I at Fennisurface in the electronic energy spectrum. This gap is known as the Peierls gap (see Figure 3.5). From equations 3.12 and 3.13, the gap energy is proportional to Q(2kp,T), which implies that it is directly proportional to the lattice deformation and dependent on temperature. According to molecular orbital theory, polyacetylene has aa* and TTTT* energy bands [43]. There is one n electron per primitive cell in polyacetylene, which makes the nn* band a half filled conduction band. Thus, polyacetylene should be a conductor. However, experiments show that trans(cis) polyacetylene has a semiconducting energy gap at 1.45eV (1.9eV), which is a consequence of the Peierls transition. Soliton Excitation The soliton model first introduced by Rice [10] and Su, Schrieffer, and Heeger [3] is most successful in polyacetylene with good agreement with experiments. Therefore, this model will be used to discuss polyacetylene in this work. Due to the dimerization of the carbon atoms, there exists a twofold degenerate ground state in trans(CH)x. This leads to the formation of nonlinear topological excitations, in this case neutral solitons (see Figure 3.6(a)). An infinite, exactly halffilled chain contains no solitons in its ground state. Solitons are found either at finite chains, e.g. by thermally isomerizing cis(CH)x, or in not exactly halffilled chains obtained e.g. by doping polyacetylene, or as
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46 excited states, Solitons can only be created by pairs, solitonantisoliton, due to the conservation of the particles. Figure 3.6(b) shows these soliton pairs in trans(CH)x. Doping changes the neutral solitons to the positively or negatively charged solitons (Figure 3.6(c)) depend on the dopants and creates more charged soliton pairs. The neutral solitons carry spin 1/2 and charged solitons (+ or ) are spinless, which is known as "spincharge reversal". The soliton state is not simply localized at one atom site. Elastic energy would prefer to spread the defect through the whole chain, while the Peierls distortion would favor a narrow kink. The competition between these two effects make the soliton spread over about 15 CH units [43]. In this section, by using the SuSchriefferHeeger(SSH) method [3,4], the roll of solitons in polyacetylene will be discussed including the ground state, soliton excitations, and soliton quntum number. The SSH model is the independentparticle model in which the correlation effects from interactions between the p^ electrons are neglected. Ground State Already, we have seen that the Peierls transition implied that the ground state of the onedimensional tightbinding metal is spontaneously distorted to open a gap at Fermisurface and to form a charge density wave with gt 0 below the transition temperature.
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47 Since the strongest instability occurs for a charge density wave of wave number q=2k^=n/a, we consider the adiabatic ground state energy for the twofolddegenerate ground state in polyacetylene as a function of the mean amplitude of distortion u, where Â»j > = ("ly u . (3.21) By definition, is the total energy of the system, electronic plus elastic energy with M^. For u. given by equation 3.21, h' in equation 3.2 is invariant under spatial translations and the total Hamiltonian can be diagonalized in k space in the reduced zone, 7r/2a < k < 7r/2a, for the valence and conduction bands (Figure 3.7) [3]. From the relations in equations 3.2, 3.3, 3.7, and 3.21, we have "ssH t^o (1)' 2 t^'> u] (c;^^ c. + c; c.^p + 2NKu^ (3.22) for a chain of N monomers in a ring geometry. We define the valence and conduction band operators as c^^> = N^'' I e^j' c 7 ^ and
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Thus, 48 Cj = N^'^ I [ c';' + icf ] e^^* . (3.23) By substituting equation 3.23 into 3.22, H can be rewritten as SSH nn* + \ [ ^fc';' + ) ] + 2KNu^ (3.24) where Cj, = 2t^ cos (;r/2a
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49 = \ fi. Â«r . (3.27a) <" = ^ + \ (3.27b) where and fi^ satisfy Fermi anticommutation relations IÂ«J' + = 1 . (3.28) Finally, H becomes tSSH (3.29) k = n/2Â» where n^'^^and n^"^ are the number density operators in conduction band and valence band respectively, defined as k k *k ' (3.30) Â«r = Â• k r k The quasiparticle energy, E^, relative to the Fermi energy is given by + K Â• (3.31)
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50 with \ = [ (i+\f\)f2 , (3.32a) fi^ = [ (l+Â£^/E^)/2 sgn . (3.32b) We note that equation 3.31 is equivalent to equation 3.20 by setting the Fermi energy to be zero, i.e., e (q=2k ) = 0. F For the halffilled band of (CH) , the ground state energy E (u) as a function of u is given by taking n^''^=l and n^*'^=0. Thus, for a ring of circumference L, we have E (u) 2NKu' = 2L/n \ E dk , (3.33) 0 or the energy per site is E^(u)/N = (4t^/:;r)E(l2') + 4tJ z'/2 t^'^' , (3.34) where E(lz ) is the complete elliptic integral of the second kind and z = 2 Pu /t^ . (3.35) For small z.
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51 E(lz') = I + [ln(4/z) + ... (3.36) Thus, E^(u)/N s 4tjn lijn (ln(4/z) 1/2) ^ + KtJ z^/2 t<'> + . . . (3.37) In equation 3.37, as zÂ»0, the logarithmic term dominates, and has a maximum at u=0, which is consistent with the Peierls instability. Figure 3.8 shows the total energy as a function of u for parameters characteristic of polyacetylene [3,4]. For an assumed energy gap of 2J = 1.4 eV (from experimental data), t^=2.5 eV, t^'^=4.1 eV, and K=21 eV/A [3,4]. These parameters lead to the minimum energy distortion u^s 0.04 A. Some experimental results for the Xray structural studies by Fincher et al [7] and for the NMR studies by Yannoni and Clarke [44] show that u^s 0.035 A which is good agreement with this estimate. Soliton Excitations As mentioned before, there are twofolddegenerate ground state with the same energy E(u^)=E(u^) in trans(CH)^ and the system supports nonlinear excitations which act as moving domain walls separating regions having different ground states: A phase flu ) and B 0 phase (u^). These walls act as topological solitons because they are
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52 shapepreserving excitations which alter the medium after they have passed a given point. Figure 3.9(a) shows the A, B phases and the mixture phase where a neutral soliton exists. Figure 3.9(b) illustrates the order parameter for a soliton and antisoliton, where (3.38) which is analogous to the staggered magnetization in antiferomagnets. Thus, ^ = , A phase = u^ , B phase (3.39) If . changes suddenly from u^ to u^, say at j=0, the electronic energy will be quite large due to the uncertainty principle. Alternatively, if ^. changes very slowly from u^ to u^, there will be a large region surrounding j=0 where the condensation energy per site is greatly reduced, again raising the energy. Thus, there is a preferred width ^ of the soliton that minimizes the total energy. Numerical calculations [3,4] show that the form of that minimizes the adiabatic energy with these boundary conditions is 6 u^ tanh[/a/
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53 where ^ 7a for the SSH set of parameters, and it is assumed that j=0 is the location of the center of the soliton. With the above parameters, the energy to create a soliton at rest is Â£==0.42 eV, which s is less than onehalf the single particle gap A. This result shows that a soliton is less costly to create than either an electron or hole because the Fermi energy is midgap for the undoped poly acetylene. This is the reason why solitons, in this case charged solitons, are spontaneously generated when charge carriers are injected by doping, by photoexcitation, or by thermal generation. The order parameter which is given by equation 3.40 is valid only for the solitons and there must be another solution of ^. for the antisolitons. To understand this, we define a state xff^ of zero energy which is centered at the soliton and falls off on the scale of ^. By expanding xjf^ in site basis states. I \\><\\w ^ , (3.41) or E I j> (3.42) Thus, we require , (3.43) or
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54 E = E^ , (3.44) where H is a Hamiltonian given in equation 3.22 excluding the phonon contribution term. The matrix elements of H are = [t^ + (1) j t('> + <^^^ )] for /=j + 1,(3.45(3)) = [t^ (1)^ t('> (,^ + =0 otherwise . (3.45(c)) For a solution of E=0, \i/ (/) in equation 3.44 must satisfy E (^ = 0 , (3.46) or + = (/) V'^Ol) y^Q^^^) Â• (3.47)
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55 From equation 3.40, as j^, <^ ^^^. Thus, R. becomes R. ^ 2t<^>0. / tJ/[l+(iy 2t<'>.^Â±Uq as j^Â±oo. The y/^ (j) for any odd j is not bounded and is not a solution for the soliton. However, if we interchange the A and B phase in equation 3.47 so that ^.^Tu^ as j^Â±oo, then 0. = u^ tanh (ja/0 (3.49) represents an antisoliton, and the oddj solution exists with E^=0, and the evenj solution is unbounded in this case. Therefore, for each widely spaced soliton or antisoliton, there exists a normalized singleelectron state in the midgap which can accommodate 0, 1, or 2 electrons due to spin degeneracy. An important property of a soliton is its effective mass, M . If s the soliton is translating slowly, 0. is given by 0. = u^tanh[(/a v^t)/(f] , (3,50) where v^ is the velocity of the moving domain wall. The increase of
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56 energy is given by ^ M^v^ = ^ M E (d0. /dtf =' \ M(u^v^ /O^ E sech^Da/^] for small . (3.51) Thus, = (4u^ /3^a)M ^ 6m^ (3.52) for the SSH parameters u^==0.04A and
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57 TLM Model The TLM model [11] is a further approximation and linearizes the tightbinding band structure, shown in Figure 3.11 [1]. The advantage of this approximation is that the resulting theory is a continuum, or field theory [48], which is generally more amenable to analytic solution than the tightbinding model. The resulting electronic band structure has two branches (see Figure 3.11) for each spin s and wave number k: a rightgoing branch (n=l) with energy e hv (kk ) and a leftgoing branch (n=l) with Â£^=^hv (kk ). The electronic wave k F F function can be described by a twocomponent field if/ , where >i>8 a , s creates an electron of spin s on branch n, where n=Â±l. For a given gap parameter Aix) the dimerization has wave number 2k , and it causes F scattering from one side of the Fermi surface to the other. By representing particles close to the Fermi surface, the TLM Hamiltonian [11] is written as "tlm ' ^
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58 expanding about the Fermi surface, keeping terms only to lowest order in (a/(J), where ^ is the electronic correlation length, (f=hv I A, or soliton width. If we derive the TLM model in this way, we find that hVp=2t^a, A=(4t^'y/27rKt^, Q^=4K/M, and [1] JO'a) = (iy4t('\ . (3.54) This equation can be compared to the gap parameter in equation 3.26. In polyacetylene, many properties can be approximated by the continuum model. However, some exceptions exist, for instance, the acoustic modes which are certainly present in the SSH model are not shown in the continuum limit. They appear only if terms are kept to next order in (a/(f) when the SSH model is transformed to the TLM model [1]. The soliton excitations can also be studied in the TLM model. For given gap parameter, J(x), the "^^^^^ and v= have cnumber solutions "m,s^^^ \.s^^^' ^^^^^ correspond to the expansion coefficients for creating or destroying a quasiparticle. These amplitudes satisfy equations of the Bogoliubovde Gennes form [49] "^V"Â„Â„W/^^ + (x) = E u (x) , r ms ms m ms (3.55) ^V^Â„,,W/^^ + (x) = E V (x) . The variation of the system energy, equation 3.53, with respect to J(x)
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59 gives the selfconsistency relation 2 Aix) = (4t^*> a/K)5:[u (x)v (x) + v (x)u (x)]. (3.56) ms ms ms ms m,s Here the sum over "m" and "s" extends over occupied states and spin components. A nonlinear Schrodingerlike equation can be derived by differentiating equation 3.55 with respect to x and introducing [11] f^(x) = u(x) + iv(x) , (3.57) f (X) = u(x) iv(x) . Equation 3.55 becomes iv^ ^ + El Â± y/^M . ^^wj f^(,) = 0 . (3.58) For the perfectly dimerized lattice A(x)=A^, the solutions to equation 3.58 are just plane wave fiinctions with the energy dispersion relation E^(k)=jJ+v^pk^, where the gap parameter is written as ^o^^o^ ' (3.59) where A is a dimensionless electronphonon coupling constant and 4t sW is the TTband width. Equation 3.59 satisfies the selfconsistency
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60 relation and it is analog to the BCS gap equation determining A. By introducing the solitons in poly acetylene, the system is no longer homogeneously dimerized. It is remarkable, however, that the nonlinear Schrodingerlike equation can be solved with the ansatz J(x)=J^tanh(x/(J) in terms of hypergeometric function [50] and the selfconsistency condition is satisfied. The energy to form a soliton, which defined as the difference between the mean field energies in presence of a soliton and the ground state, is written as [11] = 2J/7r 0.63J , (3.60) which is in reasonably good agreement with the SSH value, 0.6J. Polaron and Bipolaron Excitations When a conduction electron (hole) exits in a rigid lattice of an crystal, there are coulomb forces on the ions or atoms adjacent to the electron (hole). This forces deform the lattice and produce a polarization. In the potential field resulting from this polarization, the electron (hole) could become bound. This electron (hole) plus the associated potential field is called a polaron [51]. In trans(CH)^, by adding one electron or hole, one charged soliton can not be formed because the bond alternation must be changed over half of the chain, which causes infinite potential barrier for infinite chain (see Figure 3.12(a)). Instead, a polaron is formed by
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61 the selftrapping of an added electron or hole because the dimerized chain with a polaron has same phase (A or B phase) on both side so that the bond alternation is not required in this case (see Figure 3.12(b)). Independent of the molecular dynamics studies [3], polaron or baglike solution [5254] were discovered using the relation of the meanfield approximation to the continuum model and the GrossNeveu model [55] of quantum field theory. In the continuum limit, the order parameter ^(x) (Figure 3.13) describing a polaron centered at the origin is given by [1] = + (uy^{tanh[(xx^)/V2 ^] tanh[(xx^)/^
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62 polaron) in trans(CH)^. As shown in earlier section, in trans(CH)^, the soliton has a single bound state associated with it. For polaron there are two bound states. These two states are symmetric with respect to the gap center and can be thought of as the bonding and antibonding combinations of the two midgap states associated with the bound solitonantisoliton pair that makes up the polaron; in effect, the lower state is split off from the top of the valence band and the upper state is split off from the bottom of the conduction band [1]. As shown in Figure 3.14(b), for the electron polaron case, the lower state is filled with each spin orientation and upper band is half filled. On the other hand, for the hole polaron the lower band is half filled and upper band is empty. Therefore the electron and hole polaron each carry spin 1/2, which is the conventional spincharge relation. According to the SSH result [3], the formation energy for a polaron in trans(CH)^is written as E^ = 0.61 VIA, (3.63) which is greater than one for a soliton, E^=0.61J, but still less than half of the gap energy. To form a polaron in polyacetylene chain, as mentioned before, we need to add only one electron or hole. However, adding only one electron or hole, by doping etc., is not possible in real world. By
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63 adding more than two electrons or holes, solitons are less costly to form than polarons because E
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64 vibration of the dimerization amplitude, which is called a Ramanactive amplitude mode [5658]. When charge carriers are added to the chain, by doping for example, the translational symmetry of the system is broken and there are dipole moments coupled with bare normal modes. New normal modes, called phase modes, which are related to the translational degree of freedom of the charge, become infrared active and they describe charge oscillations [59]. Rice and Mele [1315] predicted a socalled "pinning mode" which is a characteristic infrared activity associated with the oscillation of the charged soliton bound to an ionized impurity. Each bare mode results in an infrared active vibration with restoring force given by the "pinning" potential which resists the charge translation. In particular, in a perfectly translational invariant system (i.e. no pinning potential) the lowest frequency mode (i.e., pinning mode) shifts down to zero frequency. Horovitz [12] calculated the infrared conductivity from which he could obtain the infrared active vibrational modes. Consider the charge with dimerization amplitude ^(x0(t)), i.e., its position is a timedependent field. At frequencies low compared with the gap 2A this 0 lead to a current j(t)=epd/dt where p is the charge density. Each phonon has an order parameter J^(x0_^(t)) (n = l, 2,,N) with ^(t) its oscillation degree of freedom. In the ground state [12] A^ix)=A(x)XJX where is the dimensionless electronphonon coupling
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65 constants and ^=Y^^j^The effective Lagrangian to second order in 0^ is given by [60] Leff = I ^.^0 K + + (364) where M is the soliton kinetic mass, i3"^=2' (co^fX Ik, {X)=ZJ>JS)XJX. This timedependent field ^(t) gives the current 7Y^>) and infrared conductivity a{(o) = t'icopM^'i2^\ico)/ [1 + (la)D^(cy)], (3.65) where the phonon propagator D^((y) is given by D^ico) = 2[(co/coy. 1]\,X. (3.66) From equation 3.65, the solution of D^(w)=(la)"^ gives the infrared active frequencies. All phonons couple in phase, i.e., they all tend to induce a charge density wave centered at the same position. This is indeed the case in polyacetylene, with N=3 phonons favoring bond dimerization [61]. A fourth normal mode that is mainly CH stretching favors site dimerization [62]; however, its coupling is too
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66 weak and it can be neglected. Using the parameters a=0.23 and those in Table 3.1 [61], Figure 3.16(a) [61] shows the form of D^(a>); given pinning parameter defines the dashed line and its intersections with D^(
PAGE 73
67 optical absorptions for the positively charged solitons (S"*"), negatively charged solitons (S ), neutral solitons (S^, and interband transition. From Figure 3.17 we see that when charged solitons are introduced, a new absorption edge should onset at midgap. Since the optical sum rule implies conservation of oscillator strength, there will be bleaching (reduction of absorption) above the band edge to compensate for the new peaks at midgap [1]. The calculations of optical absorptions in trans(CH) have been carried out by Horovitz [59], Suzuki et al. [65], and Kivelson et al. [66] in the continuum model. The absorption to the soliton gap state is given by a^io)) = 7r(e^/hcÂ£''^)yN(f^[(ha>/J)^ 1] 11/2. sech^{n/2[hco/Af 1]^'^}, (3.67) where N is the number of carbons per unit volume and y is the density of solitons per carbon. The interband absorption is given by a. (3.68) where the summation is over all allowed transitions from eigenstate  k> to eigenstate  k' > and the superscript "c" and "v" represent for the conduction band and valence band, respectively. The matrix element
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68 I P which is based on the golden rule is given by a rather complicated expression and can be found elsewhere [43]. The change of interband absorption due to the soliton is given by Aa^ico) = y(2^/a)a^jM. (3.69) The total absorption satisfies a sum rule [66] 00 J a(co) d(a>) = (2/;r)(e^/hcÂ£^'^) Nav , (3.70) 00 ^ independent of the lattice configuration or doping level. Thus, the increase of absorption in the gap is compensated by a decrease of the interband absorption. While the general features of the curves from the experimental results [65] are in qualitative agreement with theory, the squareroot singularity in equation 3.67 is considerably smeared out in the experimental data. Grant et al. [67] argued that part of this comes from interchain hopping and showed that the interchain hopping integral 4t^Â«0.1eV, a width considerably smaller than required to account for the observed rounding. The nonlinear dynamics of trans(CH) leads to a strong broadening of the opticalabsorption structure due to multiphonon emission [68,22] as well as polaron formation and due to solitonantisoliton creation for i2/7t)A
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69 the midgap at least because of following influences. Electronelectron correlation effects may account for about 0,2eV [71]. Furthermore impurities will introduce states in the gap and perhaps shift the solitonic midgap state [72,73], Since a polaron has the symmetrically arranged localized gap states, there are three possible transitions apart from the interband transitions. These are transitions between the lower and upper localized polaron states, from the valence band to the lower polaron level (except electron polaron) or from the upper polaron level to the conduction band, and from the valence band to the upper polaron level or from the lower polaron level to the conduction band. Fesser et al. [74] calculated the absorptions of all of these transitions for a polaron from the golden rule [1]. Polaron absorption in trans(CH) , which is of interest only at extremely low doping, however, never has not yet been observed. More details can be found elsewhere [53,54,74]. Transport Although high electrical conductivity has been a major interesting in the field of conducting polymers, progress toward understanding of the mechanisms involved has been slow. The difficulties rise principally fiom the complex morphology and the incomplete crystallinity [7579]. Many of the properties of polyacetylene, after heavy doping (about 6%) to the regime of highest conductivity, are those of a metal [8,80]
PAGE 76
70 For example, there is a term in the speciftc heat that is linear in temperature [16], there is a (temperature dependent) Pauli susceptibility [17,18], and the thermoelectric power is linear in temperature [81]. All these results are consistent with a broadbandwidth metal. There is, however, a exception that the temperature dependence of the resistivity is not metallike. The resistivity increases as the temperature is lowered even for samples with very high conductivity (cr='1.5xlO^Q'^cm'^) [8]. For samples with conductivity values in the range of < 10 i2"'cm"\ the anomalous temperature dependence may be rationalized in terms of phonon assisted transport through localized states and across the interfibrillar contacts in the complex morphology. On the other hand, for values greater than 10'i2'^cm"\ a phonon assisted transport is more difficult to understand [1]. Magnetic Properties Besides the dramatic increase of the conductivity upon doping different doping regions are distinguished by the magnetic properties of transpoly acetylene. The magnetic susceptibility of undoped transpolyacetylene is observed as a Curielike (1/T behavior) contribution [82]. While doping transpolyacetylene two processes are expected in the soliton picture [83] (i) neutral solitons may pick up charge from the dopant and
PAGE 77
71 transform to charged solitons; the density of spin diminishes. (ii) the charge introduced to the chain creates a polaron which eventually dissociates into two charged solitons. This process will be the only one at higher doping levels. There are three doping regions which may be distinguished. First, Tokumoto et al. [83] show that the low levels doping results in a decrease of the Curie susceptibility. Davidov et al. [9] also show that a Curielike electronspin resonance (ESR) line decreases with increased doping. Second, a considerable increase in conductivity is measured at a dopant mole fraction y=^0.001. It is however not accompanied by a similar increase in the paramagnetic Pauli susceptibility [17,82] indicating that the doped polymer has not transformed to a simple metal. The charge carrier seem to have no spin in this "intermediate region". Thus, charged solitons, which are created by process (ii), are assumed to be charge carriers in this region. Third, above doping levels of y=^0.06, a metallic behavior is observed in the magnetic susceptibility. The Pauli susceptibility X^Â—fi BN(Ep) abruptly increases to values corresponding to N(Ep)0.1states/eV C atom as is expected for a tightbinding band with lOeV bandwidth [82]. Intersoliton Hopping Model The main idea of the intersoliton electron hopping theory is concerned with that transport of charge occurs through energetically
PAGE 78
72 equivalent levels that are at midgap. At light doping concentration, where the neutral solitons may still exist, moving a charged soliton far from the accompanying ion requires an activation energy. However, a neutral soliton can move freely along the chain without costing energy. If a neutral soliton moves past a charged soliton which is within a soliton width (=^7a) from neutral one, there is a probability that the electron will hop from the neutral one into the empty state of the neighboring charged (positive in this case) soliton, which maintains local charge neutrality and ensures that the states are energetically equivalent. Thus, Kivelson [1921] proposed a phononassisted hopping conduction utilizing the soliton model. The average dc conductivity is ^dc " ^ exp(2BR^/0 (3.70) where is the mean distance between the hopping centers, 2 1/3 ^=(^j/^j) is the effective soliton extension ((^  and ^ are the inchain and outofchain wavefunction decay lengths, respectively, ^jp'7a='10A, ^^=i3A), and B = 1.39 which is dimensionless number. ^(T) determines the transition rate between the initial and final solitons between which the electron hops. In general, this temperature dependence of y(T) can be quite complicated, depending on the dynamical nature of the states between which the solitons hop. However, Kivelson [1921] assumed a power law dependence yCOT^ and from the phonon
PAGE 79
73 density of states estimated n10 which is in rough agreement with the experimentally determined T^^'^ law for a^^ [84]. The thermopower in this model is essentially given by S = Â±(k^/e)[(n+2)/2 + InCy^y^^)], (3.71) where + or sign designates p type or n type. The variables y^ and y are the concentrations of neutral and charged solitons. ch The ac conductivity within this model is = K'(w/T)an(D'a>/T"'^^)]'* (3.72) where K' and D' are constants. It shows a strong temperature and frequency dependence, which is in good agreement with experiment [84] using n=14.7 as obtained from the temperature dependence of
PAGE 80
74 not determined by the complex morphology [1]. Metal Transition The Pauli susceptibility of transpolyacetylene rises steeply by increasing dpoing (4%6%), which suggests a transition from the semiconducting state, with a gap 1.4eV, to a metallic state. In fact, conductivity close to that of copper has already been attained for carefully processed (CH)^ with =^6% iodine doping [80]. Several ideas have been developed to describe the transition of the system to a metallic behavior at high doping level or in the soliton picture at high soliton densities. Rice et al. [85] focused upon the lattice distortion and arbitrarily modeled the influence of the electrons by a doublewell ^potential in the continuum approximation. The soliton extension ^ is considered here as a constant and there is a finite soliton density n^=l/(v^ nl^,), at which the dimerization u(x) vanishes and the system transforms to a metal. If we set ^=hVp/J equal to the singlesoliton extension, the soliton density n ^0.03 electrons/C atom can be found. s Horovitz [86] derived an effective amplitudephase Hamiltonian intended to describe also the phase of a soliton solution by taking into account some Coulomb interaction. If the phase and amplitude of the order parameter are coupled strongly enough, numerical solutions by Grabowski et al. [87] describe a discontinues transition to an incommensurate conducting phase. This involves the formation of
PAGE 81
75 metallic islands growing with increasing doping level. A qualitatively similar model was proposed by Tomkiewicz et al. [88] as deduced from their observation of a gradual increase of the Pauli susceptibility. The existence of the intermediate doping region with very low Pauli susceptibility, however, seems to rule out the metallic islands [82,89]. Mele and Rice [22] calculated the density of states in the SSH model introducing random impurities. They explained the metal transition as a consequence of closing the Peierls gap because of increased disorder and the influence of neighboring chains. In this model, the impurities were treated either as a random diagonal potential (covalent bond model) or as a Coulomb potential. While the undisturbed SSH model shows a Peierls gap even for arbitrarily high soliton concentrations (i.e. high doping), disorder introduces states into the gap which closes at doping level y0.1 to form a pseudogap with localized states at the Fermi level and persisting bond alternation. In this model, the density of states near the Fermi level is given by N(E) = NJg + \ (EE//;/'^V] (3.73) where rj=\il2nAT; is the Peierlspairbreaking parameter, A is the usual incommensurate order parameter, t is a mean electronic impurity scattering time, g=(l;7'^)*'^ measures the reduction in the Fermilevel
PAGE 82
76 density of states relative to N^, the metallic value. The dc conductivity is reduced from the Drude metallic value
PAGE 83
77 ..V, r '.^ , v', .Â» f 1 l..~T Cv. C\ Cn^^ 0 C Cu Cu H H H H Figure 3.1 Dimerization coordinate u. defined for transiCR)
PAGE 84
78 Figure 3.2 The response function for (a) Onedimensional electron gas (b) Threedimensional electron gas (From reference 37)
PAGE 85
79 Figure 3.3 The Fermi surfaces of a free electron gas (a) For one dimension all states with  k  <  k  are occupied. A perturbation of wave vector q=2kp couples a set of planes of degenerate states on the Fermi surfaces. (b) For three dimensions all states with I k 
PAGE 86
80 Figure 3.4 The solid line represents the natural frequencies of the lattice vibrations of a crystal. The dashed lines represent the lowering of the frequency og the lattice vibration at q=2k^ caused by the response of the electrons, known as "Kohn anomaly".
PAGE 87
81 Figure 3.5 Illustration of the Peierls gap at q=2k .
PAGE 88
82 (Â«) ^ Neutral soliton A B (b) ^ soliton ^ antisoliton positively charged soliton (Q=+e, s=0) negatively charged soliton (Q=e, s=0) Figure 3.6 (a) A neutral soliton in transiCH) with twofold degenerate ground states A and B. (b) Formation of the soliton pair. (c) Positively and negatively charged solitons (Q: charge, s: spin)
PAGE 89
83
PAGE 90
84 Figure 3.8 The total energy (electronic plus lattice distortion) as function of u. The double minimum associated with spontaneous symmetry breaking and the twofold degenerate ground states. (From reference 1)
PAGE 91
I Figure 3.9 A neutral soliton. (a) A neutral soliton in mixture phase in transiCH)^. (b) The order parameter for a soliton and an antisoliton.
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86 9 0.0 ae a? 0.B as &4 0.3 0.2 a 1 ' I I I I I I I I I I I Ol 1 L. * ' ' 0 2 4 e e 10 12 14 le IB 20 22 24 26 28 90 Assumed halfwidth ^ Figure 3.10 Soliton energy as a function of an assumed soliton width C tor several values of the energy gap E . (From reference 3, 4)
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87 Figure 3.11 The transformed energy bands entering the TakayamaLinLiuMaki continuum limit of the Su, Schrieffer, and Heeger model. (From reference 1)
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88 Figure 3.12 Formation of a particle by adding an electron in infinite chain of trans(CH)^. (a) For a soliton which requires to overcome infinite potential barrier due to the band alternation, (b) For a polaron which does not require to change the chain at Â±00. Note that it has same phase at both side.
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89 p Â— """^ A phase 0 \ B phase 1 Figure 3.13 The order parameter of a polaron centered at origin.
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90 negatively charged polaron (electron polaron) Q = e s = l/2 positively charged polaron (hole polaron) Q=+e s=l/2 (a) (b) (conduction band) (conduction band) half filled filled empty half filled (valence band) (valence band) Figure 3.14 The polarons in transiCH)x. (a) Formation of an electron polaron and a hole polaron m trans(CU)x (b) Schematic diagram showing the localized energy states.
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91 (a) (X) B Â— J^wi" 0 (b) positively charged two solitons (bipolaron) negatively charged two solitons (bipolaron) Figure 3.15 Comparison of (a) the order parameter of two solitons and polarons; (b) the two positively and negatively charged solitons in transiCH)x.
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92 Table 3.1 Doping induced infrared active phonon frequencies (o) ) (a =0.23) (wÂ° (cm"') n A /A n CO (cm'') n 1234 0.07 886 (CH)x 1309 0.02 1285 2040 0.91 1397 921 0.06 770 (CD). 1207 0.005 1148 2040 0.93 1236 (From reference 61)
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H H c c (W3) Figure 3.16 IRAV modes in rrfl/w(CH)x. (a) The function D^(ct>). The dashed line is (la) ' for determining IR modes, (b) Normal coordinates of the infrared active modes (w^, o)^ and (a^ in /raAw(CH)^. The arrows indicate the atomic displacement vectors. (From references 61 and 63)
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94 (a) half filled (A' >A) V (b) I Â— empty (b) filled Figure 3.17 The optical absorptions for a (a) neutral soliton (note that the energy level is lifted toward the conduction band due to the ee repulsive interaction), (b) positively charged soliton, and (c) negatively charged soliton
PAGE 101
CHAPTER IV SEGMENTED POLYACETYLENE Segmented Polyacetylene: (CDHx)y Solitons, defects in the bondalternation pattern characteristic of a conjugated polymer chain, are central in determining the properties of ?raÂ«jpolyacetylene and they have been verified experimentally in many ways. Recently, in order to investigate the effects on transport and optical properties in fran^polyacetylene from specific defects, a controlled number of sp^ defects (see Figure 2.9 in chapter II) have been introduced onto the polyacetylene chain. These sp defects, which can be called "conjugationinterrupting" defects have an average spacing considerably less than the length of a soliton. In this section, we will focus on the optical properties of deuterated polyacetylene with sp defects including the comparison with ordinary polyacetylene, (CH)^. Infrared Active Vibrational Mode (IRAV) The IBM BRUKER FTIR/98, which was described in chapter H, was used to measure the transmittance data in the infrared range, from 450 cm'^ to 3000 cm ^ From these transmittance data, the absorption coefficients of [CDH (I ) ] were obtained, where "z" denotes the y 3 z X 95
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96 dopant level in sample. The optical transmittance of a thin film is given by 1 R(a)f e ^"^^^f^ where "d" is the thickness of film, co is the wavenumber, "R" is the reflectance, and a is the absorption coefficient. The average thickness of the films studied was about lOO^m and the reflectance of all these films were very small (less than 10%). Thus, equation 4.1 can be approximately rewritten by aico) = lln(T((w)) + ^ln(l R(a>)) , (4.2) where T(a>) is the measured transmittance and R(
PAGE 103
97 during the normal mode, then the observed ratio of CH/CD frequencies would be approximately 1.4, the square root of the ratio of the masses. As can be seen in Table 4.1, the modes at 2216, 915, and 745 cm"^ in (CD)^ were assigned to CD sp^ stretching, CD in plane bending, and CD out of plane bending modes in trans(CD)^ respectively (for transiCU)^, 3013, 1292, and 1015 cm"'). Using those frequencies, the average ratio of CH/CD frequencies was found to be 1.38 (Table 4.2), which is remarkably close to the assumed one, 1.4. In fact, many of the (CH)^ modes are mixed and depending on the percentage contribution of the various motions one can expect the CH/CD frequency ratio to fall between 1.0 (pure CC stretching) and 1.4 (pure CH stretch and/or bend) [90]. Thus, this result suggests that the contributions of CC stretch to those modes were very small (about 3%). This shifting of frequencies upon deuteration has been observed in all samples, (CDH ) , with y X different y values and different doping concentrations. It is a general feature for doped (CH)^ to show the so called "doping induced infrared active vibrational (IRAV) modes" at 900 cm'\ 1260 cm"\ and 1380 cm'\ which increase in their oscillator strength by increasing doping, but frequencies are essentially independent of the dopant species [64,90]. These IRAV modes are in excellent agreement with the theory by Horovitz et al. [59] (see chapter3). The modes, especially, at 900 cm" 'and 1380 cm"' are very broad and remarkably intense [43,64,90].
PAGE 104
98 As can be seen in Figures 4.14.3, the doping induced IRAV modes were also observed in (CDH ) (with all y values) at lower frequencies, y ^ 745 cm"' and 1140 cm '. These modes correspond to those in (CH)^ at 900 cm ' and 1380 cm"'. Early study by Fincher et al. [64] for doping induced IRAV modes in (CH)^ showed that the absorption is polarized to the parallel to the polymer backbone (this polarization dependence will be discussed later in this chapter in detail with new(CH)^). This implies the doping induced IRAV modes are characteristic of the doped poly acetylene chain. In fact, the average CH/CD frequency ratio of doping induced IRAV modes is about 1.2, which implies there is a large contribution of CC stretch in this modes. Tanaka et al. [91] interpreted that the modes at 745, and 1140 cm ' as the C=C and CC vibrations respectively. Rabolt et al. [90] argued that Raman bands [63] observed at 1352 and 854 cm ' in (CD)^, which have been assigned as the CD in and out of plane bending modes with a small contribution from CC stretching, couple strongly with collective charge oscillations parallel to the polymer chain. If this is the case, then they become IR active thereby contributing to the broad 1140 and 745 cm"' bands. In contrast, Etemad et al. [92] and Mele and Rice [13,14] argued that the absorption arises from localized rather than collective modes of the doped polymer chain because the experimental results [64] show that the frequencies are independent of doping concentrations as well as dopant species. In the collective vibronic picture, frequency shifts would be expected in
PAGE 105
99 proportion to the doping due to a combination of bondweakening and coupling to the n electrons (thus changing the effective masses). In fact, the absorption data (Figure 4.14.3) show no dependence of the frequencies of IRAV modes on doping concentration. On the other hand, the localized mode picture originated from the bond distortions associated with the structure of the soliton. These bond distortions seriously perturb the n electron states and consequently create new vibrational modes which are not simply related to the modes of the perfect chain but coupled to the charged soliton state. As pointed by Horovitz et al. [59] (chapter 3), the broad and intense absorption at 745 cm'^ from Figure 4.14.3 is presumably the pinning mode in trans(CDll ) , shifted up from zero frequency by y ^ binding of the positively charged soliton (ptype doping by iodine) around the negatively charged counter ion, I^. The breadth of this mode together with the large enhancement upon doping can be understood from the spatial extent of the soliton (2
PAGE 106
100 In particular, the y=0.0 sample showed a strong enhancement of oscillator strength for these modes, beginning at dopant levels as low as about 0.4%. In contrast, the y=0.15 and 0.27 samples required higher dopant levels for the same amounts of enhancement. This difference simply indicates that the absorption is smaller for sample with higher sp^ defects at same dopant level due to the loss of 7relectrons by introducing these defects. For each doping induced IRAV mode, the integrated oscillator strength can be characterized as function of dopant level and sp concentration by using the partial conductivity sum rule [93], given by N^^^ (m/m^^P = (m^nc^/TT Ne^) ^ a(w) d
PAGE 107
n = l/(2dJco). 101 (4.4) Thus, using d = 125//m, id
PAGE 108
102 required to generate a charged solitonantisoliton pair with iodine doping because of the necessity of accommodating two ions, each of length 10 A [95,96] (note that the lattice constant for poly acetylene, which contains two (CD) units, is 2.46 A from the Xray study by Fincher et al. [7]) and the soliton length, 2^, believed as long as 14 (CD) units. This implies that there might be region where the separation between two defects is smaller than 7 sites to have such a large separation in another region of the chain. Pairs of defects with smaller separation could not accommodate charged solitons (and ions) by doping, which would limit the maximum doping level below what is possible for pristine samples [95]. In fact, the maximum doping level for y=0.27 sample was about 3%, whereas it was about 6% for pristine film. More discussion of sp^ defects, in connection with the band gap, will be given in next section. Midgap and Interband Absorptions The PerkinElmer monochromator, which was described in chapter H, was used to measure the reflectance data from 5000 cm * to 45(X)0 cm'\ where the transmittance data could not be taken due to the highly absorbing samples. The absolute reflectance was determined by measuring the reflectance of the sample and normalizing that by the reflectance of sample coated with aluminum (about 2000 A). This normalizing procedure was necessary due to the surface roughness.
PAGE 109
103 From this reflectance data, other optical constants such as optical conductivity, absorption coefficient, dielectric constant, etc., can be obtained using KramersKronig (or KK) analysis [93] which will be described as follow. The KK relation, based on causality, relates the real and imaginary part of the dielectric function, allowing one (real or imaginary part) to be calculated if the other is known over a sufficiently wide frequency spectrum. The complex dielectric function is written as Â£(cy) = Â£j(co) + ie^((o). (4.5) Then the KK relation for e and e are 1 2 00 CO' Â£ (cO' ) e^ico) = (2/7r) f / . dco' , (4.6) 0 CO' CO CO e(ca') e(co) = (2/n) i Â—dco', (4.7) ^ Â•'o co'^ where e is the real part of the dielectric constant at ==45000 cm which allows us to replace e
PAGE 110
104 approximately by 1. For a normal incident light, the measured reflectance R((co) = (2/7r) f In Rico')'"' 4co' , (4.11) 'o (O^ Â— 0)'^ where, again, R(co' ) is the measured reflectance data. Once we calculate this phase, from tan0=2K/(l V+K^) and equation 4.8, we can obtain n(cy) and K(
PAGE 111
105 4.9 and 4.10. The real part of conductivity is a^ico) = ico/4n)s^(co), (4.12) and the absorption coefticient is a(co) = (2), (4.13) where c is the speed of light. All these calculations were performed by a PDP 11 computer. The absorption data for [CDH^(I^)^]^ obtained from KK analysis are shown in Figure 4.64.8. From these measurement, the energy gaps of undoped samples with y=0, 0.15, and 0.27 were estimated to be 1.42 eV, 1.55 eV, and 1.75 eV respectively. Table 4.3 [43] shows the relationship between the conjugation length "L" and energy gap obtained by Bredas et al. [97]. It shows that energy gap decreases as increasing "L." According to this results and using the estimated energy gaps, L=Â«, 50, and 30 (CD) units were estimated for undoped (CDH^^ with y=0, 0.15, and 0.27 respectively. Thus, as mentioned in last section, the beginning assumption, L=l/y, that the sp^ defects are uniformly distributed over the chain is not correct. Introducing a theory called "bimodal distribution of conjugation length," Mulazzi et al. [98] pointed that there is a region where the two defects in chain are so close each other (L<6). This is basically
PAGE 112
106 same argument with theory proposed by Jeyadev et al. [95]. They pointed that the sp^ defects tend to form clusters to minimize the total energy of system, which implies there are other region in which the spacing of defects are not much smaller than in pristine sample. Figure 4.9 shows the absorptions in optical region for y=0.15 and 0.27 with about same doping level, 3.5%. Both samples show a broad mid gap absorption (at 0.74 eV) by the charged solitons which were generated by doping. This is other evidence, as mentioned at the end of last section, that longer conjugation length ( L>20 sites) is needed to generate the charged solitons. We see, however, that this mid gap band for y=0.27 is substantially lower in intensity than for y=0.15. The oscillator strength for y=0.27 is about 40% of the y=0.15 sample. This can be understood, again, by considering the relatively shorter conjugation chain of y=0.27 sample (1^30) than of y=0.15 sample (1^50), which leads to localization of charged solitons, with a corresponding increase in effective mass. The optical absorption for undoped (CDH^^ with y=0.15 and 0.27 (Figure 4.7 and 4.8) shows an interesting broad band at about 9000 cm"* (1.12 eV) which is greater than the usual mid gap edge (0.7eV) and smaller than the interband edge (1.45). This band is new, never having been observed before in polyacetylene by optical measurements. This band has greater oscillator strength for y=0.27 than for y=0.15. We can attribute this band to absorption by neutral solitons for following reasons
PAGE 113
107 (i) As discussed in chapter III, neutral solitons in undoped samples can be easily ionized by doping to form charged solitons, depending on the initial number of neutral solitons. Figure 4.7 and 4.8 show good agreement with this prediction, in which this band disappeared with about 2% doping for both samples. In fact, with this 2% doping, the mid gap charged soliton band starts to grow, which implies there were enough number of neutral solitons to observe before doping. (ii) The SSH model [3] (a single particle approximation which does not consider interactions other than electronphonon interactions) predicts that a soliton occupies an energy level at mid gap. It should be noted, however, that the repulsive electronelectron interactions will push this mid gap state up towards the conduction band. Thus, for a neutral soliton, the energy could be below the bottom of the conduction band, whereas the energies are pulled down to mid gap again for a charged soliton due to the attractive "charged solitoncounter ion" interaction by doping. Our measurement does find a peak at 1.12 eV, near the interband edge 1.45 eV, in agreement with this prediction. (iii) According to the theory of Jeyadev et al. [95], there must be at least one neutral soliton between two sp^ defects separated from the first by an odd number of carbon atoms because the introduction of an sp defect in a perfectly dimerized chain changes the phase of the bond alternation pattern in segment of polyene. Figure 4.10 shows an
PAGE 114
108 example, with two sp' defects separated by 5 and 6 carbon atoms. Thus, the introduction of sp^ defects in the conjugated chain, might give more chances to have neutral solitons. Finally, the results for dc conductivity for [CDH^(I^)^]^, measured during optical measurements, are shown in Table 4.4, where we see that the dc conductivity is decreased by increasing the sp defects as expected. This can be viewed as (i) The population of 7relectrons is reduced by introducing the sp^ defects. (ii) As can be seen from Table 4.5, the effective charge carrier masses are increasing by increasing the sp^ defects, which results reducing the mobility of charge carriers. (iii) The sp^ defects might act as the scattering center of charge carriers.
PAGE 115
109 Table 4.1 IR Absorption Bands of (CH)x and Their Assignment to the Molecular Structure Frequency (cm"^) Assignment^ Assignment^ vran^ high cw cisltran^ 3057 3057.6 3057 3044 3045 3045 3013 2983 3011 2963 29272924 28862875 2633 2505 2505 2488 2390 2295 1924 1800 1800.5 1800.5 1724 1719 1690 1690 1672 1473/1665 1378 1329 1328.5 1328.5 1292 1291.8 1249 1253 1248 (1177) 1248 1136+1105 1118 + 1083 1118.2 1118 1015 1015 980 975 940 938 938 892 740 743 (814818) 743 446 445.6 445.6 2487 2420 2200 2136 + 2113 Â— Biu \)CH cir Â— B3u\)CH ciy 3011 B3u\)CHiraÂ«j mCR sp\ (next to cis) \)CH sp, \)CH sp^ Â— as (1328+ 1118) od.(2x 1248) cis (2 X 1252) ? vans trans as as trans trans as 1907.6 (1016 + 894) ?rranj 1329 + 446= 1785 cij \)C=0 1248 + 446= 1694 ciy ^ 1670 vC=0(C=C)xCÂ„ , 6CH3/CH2 &svm%^ 6 CH3 sym sp Â— B 1 u CH in plane def . cw 1291.8 BiuCH in plane def.jrow 1252 CliÂ±L trans B3u CH in plane def. cis (1 170) v(CQ cis V(CC) trans'! vKCO) or (CHCO)? (1118) B3uCCcw(?) 1016 B2u CH out of plane froftj B2u (CH=CH)2 out of plane trans between cis Â— B2u(CH=CH) out of plane < ' trans between cis 894 CH2=Ccx trans (cis between trans) ? 745.5 B2uCH out of plane ci5 BiuCOCdef.cw (From reference 31) I
PAGE 116
no Table 4.2 Typical absorption bands in transiCDH ) and their assignments y ^ Frequency ( CM'^ ) Assignment a (CD)^ (CDH Â„) 2216 2216 2218 CD stretch sp^ 2150 2150 CD stretch sp^ 1295 1295 CHD bend 1140 ^ 1140 1140 CD stretch sp^, CC stretch (dopantactivated) 915 915 915 CD inplane bend 745 ^ 745 745 CD out of plane bend (dopantactivated) a From reference 31 b Doping induced IRAV modes
PAGE 117
Ill Table 4.3 Energy Gap as a function of Chain Length for Polyacetylene a Chain length (CD unit) Energy gap (eV) 1 7.85 2 5.29 3 4.09 4 3.42 6 2.70 10 2.11 00 1.45 b (CDH ) 00 1.42 , y=0.0 50 1.55 , y=0.15 30 1.75 , y=0.27 a From reference 12 b Estimatied values for (CDHy)x
PAGE 118
112 Table 4.4 DC conductivity of [CDH (I^J^ a (i2"^cm'^) z (%) (y=o.O) 1.6 X 10*^ 3.9 X 10"^ 3.9 X 10"* 4.9 X 10'* 0.0 0.0034 0.0096 0.0187 (y=0.15) 2.1 X 10"^ 0.0 3.8 X 10"^ 0.0267 3.8 X 10"* 0.0367 (y=0.27) 1.2 X 10"^ 7.2 X 10"^ 5.9 X 10"^ 0.0 0.0169 0.0334
PAGE 119
113 Table 4.5 Comparison of m /m Â„ and anisotropy * f. ATT Sample m /m (X lU ) C CII w avenumDer Anisoiropy NewlCHCyj^ 21.5 DC z=0.06 308.0 pinning 32 550% stretched 61.7 784 cm"^ 11 57. J inn's /*m'^ luuj cm /u 8.3, (3.0)^ 1283 cm"^ 10 42.0, (40.0)^ 1380 cm"^ 26 [CDcyj^ 1.1 1140 cm'' 0.54 1140 cm'' [CDH (I ) ] 0.27^ Z^z^x 0.39 1140 cm'' z=0.01, unstretched 6.0^ 1400 cm"' tCHD,Â„(l3)J. 0.8^ 1400 cm ' z=0.01, unstretched a values for unstretched sample b From reference 43
PAGE 120
114 Figure 4.1 IR absorption coefficient for [CDH^(I^)J^ with y=0.0 and various doping levels.
PAGE 121
115 Figure 4.2 IR absorption coefficient for [CDH (1^)^^^ with y=0.15 and various doping levels.
PAGE 122
116 Figure 4.3 IR absorption coefficient for [CDH^(yj^ with y=0.27 and various doping levels.
PAGE 123
117 Figure 4.4 Integrated oscillator strength for 1140 cm"^ mode in [CDH^apjj^ with y=0.0, y=0.15, and y=0.27.
PAGE 124
118 Figure 4.5 Integrated oscillator strength for 1370 cm"^ mode [CHD^apj^ with y=0.0 and y=0.16. (From reference 43)
PAGE 125
119 Figure 4.6 Absorption coefficient in midinfrared and visible frequency region for [CDH^(I^)J^ with y=0.0 and various doping levels.
PAGE 126
120 Figure 4.7 Absorption coefficient in midinfrared and visible frequency region for [CDH^(I^)J^ with y=0.15 and various doping levels.
PAGE 127
Figure 4,8 Absorption coefficient in midinfrared and visible frequency region for [CDH (I^J^ with y=0.27 and various doping levels.
PAGE 128
122 Figure 4.9 Absorption coefficient in midinfrared and visible frequency region for [CDH (yj^ with y=0.15 (2=0.0367) and y=0.27 (2=0.0334). ^
PAGE 129
123 (a) c c c c c (b) \/ ^ ^ sp ^ V ^ sp v Figure 4.10 Schematic diagram of a /ran jpoly acetylene chain with two sp^ defects separated by (a) an even number of sites 6; (b) an odd number of sites 5, requiring a soliton between the defects. (From reference 95)
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CHAPTER V ORIENTED NEWPOLY ACETYLENE The NewPolyacetylene This chapter presents the electrical and optical properties of highly oriented (200500% stretched) newpolyacetylene with temperature and polarization dependence. For newpolyacetylene, on account of its extremely high conductivity, the dc conductivity and optical properties under various conditions, e.g., doping and temperature, etc., will be major points. Most of the properties of this sample will be compared with those of conventional Shirakawapoly acetylene. The DC Conductivity Using the fourterminal method, the dc conductivity of the undoped and chemical doped newpolyacetylene, as described in chapter II, was measured from 84 K to 300 K during the optical measurements. The results are shown in Table 5.1. Table 5.2 shows a comparison with the conductivity of Shirakawapolyacetylene and some of metals. As mentioned in chapter in, many transport properties of heavily doped new(CH)^ are metallike except the dc conductivity which decreases with decreasing the temperature (Table 5.1). Table 5.1 shows that the conductivity of doped sample (6%, iodine) increased by factor 124
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125 of 10*Â° from undoped sample at room temperature. The temperature dependence of conductivity of doped sample is shown in Figure 5.1. More detailed measurements for the dc conductivity were done by Naarmann and Theophilou [8]. They measured the conductivity as function of temperature and polarization, as shown in Figures 5.2 and 5.3. Figure 5.3 shows the temperature dependence of conductivity parallel (all) and perpendicular (aX) to the chain direction of a fresh sample. The reduction factor of (t\\ by cooling samples 300 K to 3 K varies between 2.8 and 4.
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126 Thummes et al. [99] pointed that o(T) of highly doped (CH)^ can be fitted with cr(T)=a1^'^+b (it can be compared with cr(T) in Figure 5.2). Schimmel et al. [23], however, argued that this behavior does not extented to very low temperature (T<0.4K), where much steeper temperature dependence was observed. In summary, the dc conductivity of highly conducting new(CH)^ is not metallike because it does decrease with lowering the temperature. This may, as pointed by Heeger et al. [1], indicate that phonon scattering does not contribute significantly to the sample resistance even at room temperature. Temperature and Polarization Dependence in Infrared Absorptions In order to investigate the the anisotropy of samples, wire grid polarizers were used to orient the electric field parallel to the chain direction (E) and perpendicular to the chain direction (EÂ±). For undoped samples, transmittance data were talcen to get the absorption spectrum because this film was highly transparent. The reflectance data for 6% doped new(CH)^ were measured with temperature and polarization dependence. The normalized reflectance data were obtained using a sample coated with aluminum as described earlier in this chapter. The optical data were obtained by KK analysis. Figure 5.4 and 5.5 show the absorption data for doped and undoped samples respectively. As can be seen from the infrared absorption of doped samples (Figure 5.4), a strong broad band was found around 500 cm"' at 250 K
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127 with polarization E. The oscillator strength of this band was drastically increased by lowering the temperature down to 84 K reaching its peak at 400 cm'V Same effect for this band was found for polarization EÂ± with substantially smaller change of oscillator strength. It should be note, however, that the oscillator strength of other IRAV modes (787 cm\ 1005 cm\ 1250 cm\ 1283 cm\ and 1380 cm^) were not influenced by this temperature change. Tanaka et al. [91] have also obtained similar results with doped Shirakawapoly acetylene, where a broad band appears around 600 cm"' with increasing its intensity by lowering the temperature from 300 K to 230 K, whereas the doping induced IRAV modes were unchanged. Strong polarization dependence implies that this broad band is a characteristic of polymer chain. We assumed this broad band as a pinning mode, oscillation of the charged solitons by trapping due to Coulomb interactions with counterdopant ions (1^ and by other interactions. The reasons are follows (i) It is expected that the pinning mode increases its intensity on lowering the temperature because of reducing the screening of Coulomb interactions by charged solitons and counterions, which causes large dipole oscillation of the charged solitons. (ii) Undoped samples (see Figure 5.5) do not show this mode even at 84 K. Thus, we may assume that the charged solitons introduced by doping are strongly involved in this mode.
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128 As can be seen from Figures 5.4 and 5.5 and comparing with Table 4.1, there was a large amount of residual cis(CH)^ in undoped sample, e.g., 740 cm"' for cis CH out of plane bending mode. By doping up to 6%, however, all of these cis IRAV modes were not seen except the modes at 740 cm'' (very weak in doped sample) and 1250 cm"'. Thus, as pointed by Hoffman et al. [100], the cistrans isomerization was almost completed by heavy doping. A new IRAV mode was found upon doping at 787 cm"', which was not shown in undoped sample. Thus, this must be a doping induced IRAV mode. Because of the strong anisotropy, we supposed this mode as a CC vibrational mode in cis/trans mixture phase with small contribution from CH bending. We also note that the doping induced modes at 1003 cm'' and 1283 cm"' in doped sample shifted from those in undoped sample at 1016 cm'' and 1292 cm"' respectively. This might imply that the collective vibronic picture, as argued by Rabolt et al. [90] (see earlier section in this chapter), can be applicable in this case. Another possibility is that the doping screens the Raman frequencies
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129 oscillator strengths at 1380 cm'' was found to be about 28. This large anisotropy indicates that the doping induced IRAV modes are characteristic of the doped polyacetylene chain. Thus, as interpreted by Tanaka et al. [91], CC vibrations are strongly related to this modes. From Figure 5.6, the optical conductivity at zero frequency was found to be 12000 D'^cm ', which is in good agreement with measured dc conductivity, 10000 Q'^cm\ The doping induced IRAV modes for an unstretched sample are also shown in Figure 5.7 with several dopant levels and the integrated oscillator strengths at 1283 cm"' and 1380 cm ' modes are shown in Figure 5.8 as function of dopant levels. From Figure 5.8, the ratios of effective charge carrier mass, vajm^^^, with z=0.01 are also shown in Table 4.5, where we see that m Â„ of 1380 cm ' mode is 7 times smaller ' eff than one for 1283 cm'' mode at 1% dopant level. Midgap and Interband Absorptions The midgap absorption is a universal feature of doped polyacetylene [1]. Thus, it appears in newpolyacetylene as well as Shirakawa and segmented polyacetylene. This dopantinduced nearIR electronic transition has been observed to be independent of dopant species and of whether that dopant is a donor or an acceptor. Tanaka et al. [91] pointed that the mid gap absorption of lightly doped sample (6000 cm"') shifted to lower frequency (4000 cm ') and enhanced in oscillator strength on increasing the dopant level, whereas
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130 the interband transitions shifted to higher frequencies with a substantially decrease in oscillator strength. This reflects the conservation of oscillator strength, or sum rule (equation 4.3). Our results agree with this: Figures 5.9 and 5.10 show that the mid gap absorption in the 5% doped sample has a peak at 5000 cm'\ whereas a peak occurs at 3000 cm'^ for 6% doped sample. Figure 5.11 shows that the interband absorption of doped sample shifted to higher frequency (15000 cm'^) while decreasing in oscillator strength, whereas it occurred at 11500 cm"' for undoped sample. Thus, the interband absorption seems to be smeared out with increasing dopant level. Earlier work by Feldblum et a/., [101] in Shirakawa polyacetylene shows that the interband absorptions are essentially not observed with dopant level >8%. This might be explained with theory by Mele and Rice [22] (see chapter III), in which they explained the metal transition upon doping as a consequence of filling the Peierls gap because of increased disorder and the influence of neighboring chains. The disorder introduces states into the gap to form a pseudogap with localized states at the Fermi level and persisting bond alternation. Thus, in fact, the Peierls gap still exists at high doping level although it is incommensurate with lattice. According to this model, further doping (>6%) will remove the interband transition and the broad absorption band at 3000 cm"' in Figure 5.5 can not be regarded as a usual midgap (in fact, it is no longer a midgap which appears usually around 6000 cm"' for medium doping) but can be regarded as a pseudogap
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131 with an onset around 1500 cm"^ (0.19 eV), The polarization dependence shows that the mid gap and interband transitions are strongly polarized along the polymer chain (in Figures 5.4 and 5.5, E and EÂ± stand for polarization parallel and perpendicular to the polymer chain direction respectively). The ratio of oscillator strength at midgap for 550% and 200% stretched samples were found to be about 20 and 17 respectively, which can be comparable to those of doping induced 1380 cm * mode (28 and 20 respectively). These values are remarkably close to the anisotropy of dc conductivity, 2026, as shown earlier in this chapter. Thus, we conclude, as suggested by Mele and Rice [15], that the enhancement of doping induced IRAV modes are due to coupling of these vibrations with the charged soliton state thus reducing the effective masses of charge carriers, which leads to increasing the dc conductivity.
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132 Table 5.1 DC conductivity of new(CH) as function of temperature Conductivity cm)' Temperature (K) Undoped (CH)^ 6 % iodine doped (CH)^ 300 1.4 X 10"^ 10000 250 2.0 X 10^ 8696 200 3.7 X 10* 6666 150 1.1 X 10* 4950 100 1.0 X 10* 3333 84 8.3 X lo** 2500
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133 Table 5.2 dc conductivity of metals and of polyacetylene Conductivity (Q cm) Density (gr/cm ) 10365 13.564 25575 6.618 101522 21.450 30030 5.720 102986 7.870 146113 8.910 470588 19.300 671140 10.491 645000 8.940 before after doping (Q 520 0.4 1.23 16001800 0.5 1.26 18000^ 10000Â® 0.85 1.12 Hg Sb Pt As Fe Ni Au Ag Cu S(CH)^ (unstretched) S(CH)^^ (stretched) new(CH) (stretched) a, d From reference 8 b From reference 97 c From reference 6 e This measurement
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134 12000 8000 4000 1 1 ' nQ*[CH(ip,], z0.06 1 550 X strotchad B B B Q , 1 1 100 200 TgrnpgraturQ (K) 300 E o o E [CH(ip,], z"0.06 S 550 X strotchad o Â° 1 0 0 100 200 300 TQinperature (K) Figure 5,1 Temperature dependence of dc conductivity (o]) for oriented newpolyacetylene between 84 K and 300 K in two different plots.
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135 60 120 180 T CK) 240 300 Figure 5.2 o'll(T) of oriented newpolyacetylene between 14 mK and 300 K. (From reference 8)
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136 Figure 5.3 Temperature and polarization dependence of dc conductivity for oriented newploy acetylene. (From reference 8)
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137 500000 400000 E 300000 J2 t. g 200000 < 100000 Â— I 1 I I 1 1 1 1 1 1 1 rT ^ r550Z strotched namXHll^ ^1 ^ z=0. 06 300 K _ _ 250 K 200 K 150 K . 100 K 84 K _l I I l_ 500 ' * ' 1 1 Â—I 1000 1500 2000 2500 3000 FrequQncy (cm ) Figure 5.4 Temperature and polarization dependence of the IR absorption coefficient of oriented new[CH(I,) ] with 550 % stretching ^ 3 Z 3t ratio and z=0.06.
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138 Figure 5.5 Temperature and polarization dependence of the IR absorption coefficient of oriented new[CH(I^)J^with 550 % stretching ratio and z=0.0.
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139 Figure 5.6 Polarization dependence of infrared conductivity for oriented new[CH(I^)J^ with 550 % stretching ratio and z=0.06
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140 2500 2000 E U o >^ 1500 4J U 3 T3 C (S 1000 500 Unstretched new[CH (1 3) , z=0. 0 z=0.014 z=0. 021 z=0. 06 1000 A. / / \, \ 1200 1400 1600 1800 Frequency (cm ) Figure 5.7 Doping induced IRAV modes in unstretched new[CH(I^)J with various dopant levels.
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141 Figure 5.8 Integrated oscillator strength at 1283 cm"' and 1380 cm ' modes with various dopant levels in unstretched new[CH(I ) ] .
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142 Figure 5.9 The midgap and interband transitions in 550 % stretched new[CH(Ij)J^ with z= 0.0502 and polarization dependence.
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143 Figure 5.10 The midgap and interband transitions in 200 % stretched new[CH(ipj^ with z=0.06 and polarization dependence.
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144 4000] Â— 1 Â— Â— Â— > Â— r 2000 20000 25000 Frequency (cm ) Figure 5.11 The midgap and interband transitions in 550 % stretched new[CH(I,)] with z=0.0, z=0.0502, and polarization Ell. 3 z X "
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CHAPTER VI SUMMARY AND CONCLUSIONS This dissertation has involved the study of the optical properties of segmented poly acetylene, [CDH ] , and highly oriented "new" y ^ poly acetylene. The interband transition was characterized by the Peierls instability which produces a semiconducting energy gap at the Fermi surface. All samples were doped with iodine to investigate the doping induced IRAV modes in the infrared region as well as the positively charged soliton state at midgap. All doped [CDH^(Ij)^]^ samples showed the doping induced IRAV modes at 745 and 1140 cm'' which are in good agreement with the theoretical results of Horovitz [61]. For a sample without sp^ defects (y=0.0), the strong enhancement of oscillator strength for these modes was observed with dopant level as low as z=0.4%, whereas the samples with y=0.15 and 0.27 required a higher dopant level for the same amounts of enhancement. We estimated the conjugation length "L" of [CDH ] as L=oo, 50 and y ^ 30 for y=0.0, 0.15 and 0.27 samples respectively. These results imply that the beginning assumption, L=l/y, that the sp^ defects are uniformly distributed over the chain is not correct. We conclude, according to the theory, "bimodal distribution of conjugation length," 145
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146 that there must be a region where the separation between two defects is smaller than 1/y CD sites to have such a large separation in another region of the chain. We found that the midgap band for y=0.27 was substantially lower in oscillator strength than for y=0.15, which can be understood by considering the smaller number of charged solitons in the y=0.27 sample where the conjugation length is relatively shorter than one for the y=0.15 sample. A new broad band in undoped [CDH^]^, which was more dominant for the greater y value, was found around 1.12 eV. This was assumed as a absorption band due to neutral solitons. The dc conductivity of segmented polyacetylene was decreased by increasing the sp defects, which implies the population of 7relectrons was reduced and the effective masses of charge carrier were increased by introducing the sp' defects. It has been shown that the dc conductivity of highly oriented new(CH)^ can be close to that of metals. However, the temperature dependence of dc conductivity was not metallike, decreasing with lowering of temperature, which may imply that phonon scattering does not contribute significantly to the sample resistance even at room temperature as indicated by Heeger et al. [1]. A model proposed by Sheng et al. [2426], which describes the dc conductivity of highly conducting regions separated by a potential barrier, indicates that the charge transfer is caused by tunneling, enhanced by thermal voltage fluctuation. The anisotropy of dc conductivity was between 20 and 26.
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147 The optical data for this new(CH)^ show remarkable similarity to those of conventional Shirakawapolyacetylene. For the 6% iodine doped sample, the anisotropy of the oscillator strength of a doping induced IRAV mode at 1380 cm"^ was found to be 2028 depending on the stretching ratio of the sample. This implies CC vibrations were strongly involved in doping induced IRAV modes. The broad absorption by charged solitons was found at 3000 cm"* with substantially smaller interband absorption. Assuming that more than 6% doping will remove the interband transition, this band at 3000 cm"* can be regarded as a pseudogap as proposed by Mele and Rice [22]. For 6% doped mew(CH)^, a pinning mode was found around 500 cm"* at 250 K with polarization E. The intensity of this band was drastically increased by lowering the temperature, reaching its peak around 400 cm"* at 84 K. The same effect for this band was found for polarization EÂ± with substantially smaller changes in oscillator strength. It can be viewed as decreasing the screening of Coulomb interactions between charged solitons and counterions by lowering temperature. Thus, large dipole oscillation of the charged solitons took place. The oscillator strengths of the IRAV modes except the pinning mode, however, were not changed by changing temperature.
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BIOGRAPHICAL SKETCH HyungSuk Woo was born in Seoul, South Korea, on November 22, 1955. He is the son of KyuIl Woo and Youngja Woo (Seo). He was raised and educated in Seoul. After high school, he went to the Yonsei University in 1974, where he was awarded a B.S. in physics in 1978. He joined the graduate school to continue his studying in physics and was awarded an M.S. in 1980. After his military service from 1980 to 1981, he taught general physics at the Soonjun University, Seoul, Korea, until he decided to go abroad for further studies. He went to the University of Pittsburgh in 1982 to join the Ph.D. program in physics and was awarded an M.S. in 1984 due to transferring to the University of Florida. In 1985, he began his research on the optical properties of conducting polymers under Professor David B. Tanner in the Physics Department at the University of Florida. He married KumOk in 1980 and had a son, SoonBo, in 1981 and a daughter, JungYoon, in 1988. 154
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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, Chairman 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. Stephen Nagler ' 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. Hendrick Monkhout 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. Neil Sullivan Professor of Physics
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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. Gary Ihas 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. Â— Kenneth B. Wagener Associate Professor of Chemistry 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 1990 Dean, Graduate School

