Citation for published version (APA): Figge, M. T. G. (2000). Disorder and interchain interactions in Peierls systems Groningen: s.n.

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1 University of Groningen Disorder and interchain interactions in Peierls systems Figge, Marc Thilo Günter IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2000 Link to publication in University of Groningen/UMG research database itation for published version (APA): Figge, M. T. G. (2000). Disorder and interchain interactions in Peierls systems Groningen: s.n. opyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like reative ommons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:

2 1 General Introduction During the past three decades a wide variety of quasi-one-dimensional metals has been discovered that undergo a phase transition into a charge-density-wave (DW) state. This phase transition involves a change in the system's lattice structure below the transition temperature T c and is called Peierls transition. The Peierls transition occurs in quasi-one-dimensional systems if the coupling between the conduction electrons and the lattice dominates other interactions, and is caused by an instability of the Fermi surface. The DW state describes a periodic modulation of the conduction electron density which is accompanied by a modulation of the atomic positions. In a one-dimensional lattice the elastic energy cost, needed to modulate the atomic positions, is, at sufficiently low temperatures, less than the gain in the conduction electron energy. The electron energy is lowered due to the opening of an energy gap at the Fermi surface, which in the case of a strictly one-dimensional system consists of two points that are denoted by the Fermi wave vectors ±k F. The allowed electron energies in a chain with original lattice period a form a band as a function of the electron wave vector k. Only electron states with jkj smaller than k F = ß=(ma) are occupied, where m denotes the inverse filling factor of the electron band. The DW wavelength is given by DW = 2ß=q with q = 2k F, as for this periodic perturbation the response of the electron gas is most pronounced, due to the perfect nesting property of the one-dimensional Fermi surface. The Fermi surface of quasi-one-dimensional systems, which consist of weakly coupled chains, is still almost completely nested. This is due to the highly anisotropic electronic structure, which still makes it possible to map a substantial part of the Fermi surface onto itself by translation over a single wave vector. Thus, also quasi-one-dimensional systems can undergo a Peierls transition. The transverse coupling between one-dimensional chains even is an essential ingredient to obtain a phase transition, since in a strictly one-dimensional system any kind of fluctuation (thermal or disorder) would break the long-range DW order. If the strength of the coupling between neighboring chains becomes comparable to the intrachain coupling, however, the Fermi surface becomes isotropic, and the nesting property as a necessary prerequisite for a Peierls transition is lost. A variety of organic polymers, as well as numerous quasi-one-dimensional inorganic materials, undergo a Peierls transition into a DW state. Among the inorganic linear chain compounds are, for example, the platinum chain complex Krogmann's salt (KP), the transition metalchalcogenides NbSe 3 and TaS 3, and the transition metal bronzes, such as K 0:3 MoO 3 (blue bronze). The transition temperatures T c of these compounds are in the order of hundreds of Kelvin. At temperatures well below T c, the DW system has, similar to a semiconductor, an energy gap at the Fermi energy. owever, if DW = ma is incommensurate with the original lattice period a, a collective charge transport mode exists in the DW material. Upon applying an electric field, the phase of the periodic lattice distortion can move through the lattice, carrying the DW and giving rise to a finite conductivity. This DW transport mechanism has been observed experimentally in the inorganic conductors TaS 3 and K 0:3 MoO 3. istorically, much of the interest in conjugated polymers originated from the hope to synthesize in a sim-

3 2 General Introduction (a) (b) (c) (d) (e) Figure 1-1: (a) Polyethylene consists of ( 2 )-units containing one carbon () and two hydrogen () atoms. (b) One hydrogen atom per unit polyethylene is removed to obtain trans-polyacetylene with one unpaired ß-electron (ffl) at each carbon atom. (c) The electronlattice coupling produces a DW state, resulting in an alternation of long (-) and short (=) bonds (dimerization). The two lattice configurations (c) and (d) are energetically degenerate, as the ground state energy does not depend on whether the even or the odd bonds are short. (e) A neutral soliton is a kink in the lattice dimerization, which acts as a domain wall between the two degenerate lattice configurations (c) and (d).

4 3 ilar way plastics that conduct". owever, pristine conjugated polymers quite generally show insulating or, at best, semiconducting behavior at all accessible temperatures. The terminology conducting polymers" originates from the near-metallic conductivities that conjugated polymers achieve upon chemical doping, making them very attractive for potential technological applications in plastic electronics". Among the most promising candidates for technological applications are conjugated polymers involving aromatic units, such as poly-para-phenylene and poly-para-phenylene vinylene, and polymers involving heteroatoms (e.g., polythiophene and polypyrrole) or both aromatic units and heteroatoms (e.g., polyaniline). One of the most extensively studied conjutaged polymers is polyacetylene, which has a relatively simple structure consisting of connected carbon-hydrogen units. Polyacetylene exists in the two different varieties, cis and trans, of which the thermodynamically more stable form is trans-polyacetylene. Trans-polyacetylene is a DW system with a half-filled electron band and a dimerized lattice. Figure 1-1 shows how trans-polyacetylene is conceptually obtained from a polyethylene chain. Removing one hydrogen atom per carbon atom in the polyethylene chain, leaves one ß-electron per site which occupies the carbon 2p z -orbital perpendicular to the plane of the chain. The ß-electron states form a half-filled electron band and trans-polyacetylene could be conducting if the electron-lattice interaction would be negligible. owever, the finite coupling between the electrons and the lattice deformation with wave vector q = ß=a (optical phonon) leads to the opening of a Peierls gap. The Peierls phase of pristine trans-polyacetylene is characterized by a DW with wavelength DW = 2a, which is twice the original lattice period. This DW is more precisely called a bond order wave (BOW), as it describes the dimerization of the bond lengths due to an alternation in the bond charges. While the undimerized lattice in Figure 1-1 (b) has a reflection symmetry with respect to a plane perpendicular to the chain direction at any carbon atom site in the chain, this Z 2 symmetry is broken in the dimerized lattice below the transition temperature T c. Obviously, the ground state is doubly degenerate, as the total energy is independent of the two possible sequences in the bond length alternation shown in Figure 1-1 (c) and (d). Trans-polyacetylene is in the BOW state at all accessible temperatures, as T c is in the order of several thousands of Kelvin. The typical atomic displacement u 0 associated with the instability has been experimentally determined to be of the order u 0 =a ο 3%. By far most of the theoretical studies on the conjugated polymer trans-polyacetylene have focused on models of idealized, strictly one-dimensional, and infinite chains. owever, especially in the light of its quasi-one-dimensional nature, disorder must be taken seriously even in crystalline trans-polyacetylene, as it affects the long-range order in the bond length alternation. Many different sources and length scales of disorder, such as partial crystallinity, cross linked chains, cis segments within a trans chain, impurities, and even the particular sample preparation method, make it extremely difficult to obtain a quantitative description of this polymer. Another equally important aspect, is the coupling between the polymer chains, originating from interchain electron hopping, elastic forces, and oulomb interactions. Interchain interactions impose a coherence in the bond alternation of neighboring chains and, in this way, act to sustain long-range order in a chain's bond alternation. This thesis deals with the effect of disorder and interchain interactions in Peierls systems with a broken Z 2 symmetry. The main motivation for this study is the desire to obtain a better understanding of the competing interplay between the Peierls instability, disorder, and interchain interactions. We take into account disorder in the electron hopping amplitudes (off-

5 4 General Introduction diagonal disorder) that appears, for example, as a consequence of conformational disorder (random twists of the polymer chain). We start by considering an isolated Peierls chain and address the question: What is the effect of off-diagonal disorder on the long-range order in the bond length alternation of an isolated Peierls chain?" We find that allowing the lattice dimerization to adjust to the electronic disorder fluctuation, results in a finite density of neutral solitons in the minimal-energy lattice configuration of a single chain. istorically, the interest in these topological excitations motivated much of the work on trans-polyacetylene. In particular their spin-charge relations aroused a lot of interest: neutral solitons have spin 1=2, while charged solitons do not carry a spin. A neutral soliton is schematically shown in Figure 1-1 (e) and corresponds to a kink in the lattice dimerization that can be viewed as a domain wall between the two degenerate lattice configurations Figure 1-1 (c) and (d). This topological excitation does not occur in the ground state of an ideal Peierls chain, because it has a creation energy in the order of several thousands of Kelvin and is thus very unlikely to be thermally induced. In a disordered Peierls chain, however, neutral solitons are induced by the disorder with a density that is proportional to the strength of the disorder. Located at random positions in the disordered chain, they destroy the long-range bond order in the one-dimensional system at any temperature. As neutral solitons carry a spin 1/2, they should clearly contribute to the magnetic susceptibility of trans-polyacetylene. Experiments on trans-polyacetylene reveal a usual urie behavior of the magnetic susceptibility at sufficiently high temperatures and a deviation from the urie behavior below a sample dependent temperature in the order of several tens of Kelvin. The typically observed low density of one free spin per 3000 carbon atoms in trans-polyacetylene, however, can hardly be reconciled with a neutral soliton density proportional to the disorder strength as expected for a single polymer chain. Taking into account that experiments are performed on three-dimensional samples, we are therefore led to pose the question: Does the magnetic susceptibility of trans-polyacetylene display the competition between disorder and interchain interactions in Peierls systems?" It is well-known that solitons are confined into pairs due to the interaction between neighboring chains. We find that due to this confinement the density of disorder-induced neutral solitons is exponentially suppressed by the ratio of the strength of the interchain coupling to the disorder. Moreover, the confinement may re-establish a bond-ordered phase, where the disorder-induced neutral solitons are bound into pairs and neighboring chains are ordered with respect to their bond length alternation. A confined soliton-antisoliton pair is schematically shown in Figure 1-2 (a). The strong suppression of the soliton density by the interchain interactions may explain the observed low density of free spins. The deviation of the magnetic susceptibility from urie behavior can be understood within the picture of confined spin pairs as well: A spin pair is characterized by a pair-size dependent antiferromagnetic exchange coupling and is bound in the nonmagnetic singlet state at temperatures that are smaller than this coupling. Upon increasing the strength of the interchain interactions, weak disorder effects become negligible and the occurence of dimerization kinks becomes very rare. In fact, by removing the hydrogen atom from each carbon-hydrogen unit in trans-polyacetylene, a strong covalent bond

6 5 (a) R * (b) (c) Figure 1-2: (a) Sketch of weakly interacting (dashed lines) trans-polyacetylene chains (zigzagchains) in the bond-ordered phase with in-phase bond order in neighboring chains. The hydrogen atoms are not shown. A disorder-induced soliton-antisoliton pair (shaded box) is depicted which is confined to a typical pair size R Λ due to the energy loss in the interchain interactions. (b) By removing the hydrogen atom from each carbon-hydrogen unit in trans-polyacetylene (see Figure 1-1), a strong covalent bonding arises between neighboring zigzag-chains, resulting in a two-dimensional graphene sheet. (c) Rolling up a graphene sheet perpendicular to the direction of the zigzag-chains produces a carbon nanotube, which is characterized by the number of zigzag-chains around its circumference.

7 6 General Introduction can be formed with the neighboring zigzag-chain. Thus, we obtain the hexagonal lattice of a two-dimensional graphene sheet, as indicated in Figure 1-2 (b).the infinitely extended graphene sheet is clearly no candidate for a DW material, as it is a semimetal, where the density of electron states vanishes at the Fermi surface. Moreover, it does not have the important nesting property which is characteristic for the Fermi surface of quasi-one-dimensional systems. owever, if we consider a graphene sheet that consists of a finite number of zigzag-chains and impose periodic boundary conditions along the direction perpendicular to the chains, we obtain a carbon nanotube (Figure 1-2 (c)), which electronically is a one-dimensional system. arbon nanotubes are well-known for their remarkable electronic properties. From the point of view of technological applications, they are considered as prototypes for one-dimensional quantum wires in nanometer-sized electronics. The carbon nanotube in Figure 1-2 (c) is composed of a two-leg ladder repeat-unit which consists of two strongly coupled" zigzag-chains directed along the nanotube axis. This suggests that the effective low-energy tight-binding model of the carbon nanotube corresponds to a two-leg ladder model itself, containing two electron bands that intersect at two Fermi points. Such a single-particle picture does not account for the electron-lattice interaction and the oulomb interaction between the electrons. On the one hand, due to the electron correlation, the one-dimensional carbon nanotube is a typical candidate for an experimental realization of a Luttinger liquid. On the other hand, focusing on the electron-lattice interaction, a metallic carbon nanotube can be considered as a Peierls system. The first point of view has recently drawn much attention, but the following two aspects make it of special interest to also study the Peierls instability more carefully: (i) Two electron bands with linear dispersion intersect at each Fermi point, which offers the possibility for the opening of an energy gap involving phonon modes with small phonon momenta. (ii) Due to their cylindrical lattice geometry, carbon nanotubes have a phonon spectrum that is relatively rich compared to chain-like systems. We therefore address the question: What is the nature of the Peierls state in a carbon nanotube, and what are the properties of a topological excitation corresponding to the usual soliton?" We suggest that a Peierls transition can occur in this quasi-one-dimensional system, involving both optical and acoustic phonon modes with small phonon momenta q! 0. The considered optical mode, with finite frequency! o, describes a relative shift between the two triangular sublattices of which the hexagonal lattice is composed, while the acoustic mode with frequency! a / q corresponds to a long wavelength twist deformation of the nanotube lattice. Particularly interesting is the fact that the acoustic mode contributes to the opening of the Peierls gap and governs the value of the transition temperature. Below the transition temperature, a topological excitation can exist in the deformed nanotube lattice. It is best described as a solitwiston, because it represents a combination of a dimerization kink and a change in the sign of the twist angle, with properties, such as its creation energy and its mass, that deviate from those of the usual soliton in a trans-polyacetylene chain.

8 7 Outline of this thesis In hapter 2 we introduce a continuum model that describes weakly disordered Peierls chains and which accounts for electron correlations as well as interchain interactions. This model represents the starting point of our studies in hapter 3 and hapter 5. Furthermore, hapter 2 is also meant to review characteristic features of Peierls chains, such as the soliton concept. The isolated weakly disordered Peierls chain is considered in hapter 3. We show that the free energy of disordered Peierls chains can be calculated analytically, because the statistical properties of the disorder-induced kinks are described by the random-field Ising model. In this mapping, the Ising variables describe the local lattice dimerization, while the random field corresponds to the disorder in the electron hopping amplitudes. The spin-flips induced by the random field then correspond to the disorder-induced neutral solitons in Peierls chains. The mapping allows us to take also electron correlation effects on the kink creation energy into consideration. We perform numerical calculations for the random-field Ising model using the transfer-matrix approach and compare the results to the analytically obtained free energy and density of dimerization kinks as a function of the temperature and the disorder strength. Weak interchain interactions between Peierls chains are described by the anisotropic randomfield Ising model, which consists of weakly interacting Ising-spin chains. We devote hapter 4 to this model, which is believed to capture the essential physics of many disordered systems and is, therefore, interesting in its own right. The free energy of the anisotropic random-field Ising model is calculated within the chain mean-field approach by a mapping on an exactly solvable Brownian motion model which describes a flying feather under the influence of the earth gravitation, the air friction, and the random forces of wind. In the context of a disordered Peierls system, the interesting properties of the anisotropic random-field Ising model are the temperature of the three-dimensional phase transition, the order parameter, and the density of spin-flips as a function of the random-field strength and (or) the temperature. These properties depend strongly on the degree of anisotropy of the interactions, and correspond to, respectively, the temperature of the three-dimensional Peierls transition, the average dimerization, and the density of neutral solitons in the system of interacting disordered Peierls chains. In hapter 5, we study the magnetic response due to disorder-induced neutral solitons in crystalline trans-polyacetylene. We first perform a numerical calculation of the Peierls transition temperature as a function of the disorder strength. This simulation is done for the anisotropic random-field Ising model using the transfer-matrix approach and is compared to the analytical expression of the critical temperature obtained in hapter 4. The thus obtained phase diagram contains a bond-ordered phase, in which the neutral solitons are bound into pairs by the interchain interactions. We analytically calculate the corresponding pair size distribution and describe the two spins of a soliton-antisoliton pair by an exchange coupled spin pair model. Within this model we calculate the temperature dependence of the magnetic susceptibility in the ordered phase and compare our results to the observed magnetic susceptibility of transpolyacetylene. hapter 6 deals with metallic carbon nanotubes and contains a general introduction to the interesting electronic properties of these systems. We calculate the long wavelength phonon spectrum of the nanotube and identify an optical and an acoustic phonon mode that result in the backscattering of electrons. The electron-lattice interaction is expressed within a continuum

9 8 General Introduction model and we calculate in a self-consistent way the renormalization and mixing of the bare phonon modes due to this coupling. To study the nature of topological excitations in the Peierls phase, we perform numerical calculations in which the minimal-energy lattice configuration is calculated as a function of the number of electrons that are added to the half-filled system. For temperatures above the transition temperature, we calculate the electrical conductivity due to one-phonon scattering processes and discuss the results in the context of experimental data for carbon nanotubes. To make this thesis attractive for the reader who does not want to be confronted with all the details of the calculations in a first reading process, each chapter is provided with a reading advice that is announced by the symbol. The corresponding sections introduce and summarize the most important aspects of that chapter and, as far as possible, draw a qualitative physical picture using simple estimates. The symbol at the beginning of a chapter refers to a scientific journal where the main results have been published. General introductory literature For a review of density wave systems: ffl G. Grüner, Density waves in solids (Addison-Wesley, Reading, 1994); ffl. Kamimura (Ed.), Theoretical Aspects of Band Structures and Electronic Properties of Pseudo-One-Dimensional Solids (D. Reidel Publishing ompany, 1985). For a review of conjugated conducting polymers: ffl.g. Kiess (Ed.), onjugated onducting Polymers (Springer-Verlag, 1992); ffl P. Bernier, S. Lefrant, and G. Bidan (Ed.), Advances in Synthetic Metals - Twenty years of Progress in Science and Technology (Elsevier, 1999). For a review of carbon nanotubes: ffl R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Physical Properties of arbon Nanotubes (Imperial ollege Press, 1998).