Macroscopic disorder and the metal–insulator transition in conducting polymers

We describe a simple model for treating the frequency-dependent conductivity σe(ω) and dielectric function Re e(ω) of a quasi-one-dimensional conducting polymer. The polymer is modelled as a composite medium consisting of spherical regions of ordered polymer, randomly distributed in a much more disordered polymer host. Within each spherical region, the polymer chains are highly oriented, but the axis of orientation varies randomly from sphere to sphere. The disordered host is assumed to be isotropic, but with a conductivity which depends on the connectivity of the polymer chains in the host. σe(ω) and Re (ω), as calculated from this model using a suitable effective medium approximation, reproduce all the main experimental features associated with the metal–insulator transition in these polymers. Highly conducting polymers have been the subject of intense recent interest [1–6]. Detailed investigation of the temperature and frequency dependence of the dielectric response in these systems has led to the discovery of metallic behaviour and a related insulator–metal transition in doped polypyrrole and polyaniline [5, 6]. This transition is expressed by a dramatic change in the qualitative behaviour of the dielectric coefficient , depending on the static electrical conductivity σ . In one type of sample, σ decreases sharply at low frequencies, and has a very low static value. At higher frequencies in these samples, there is a peak in the conductivity which moves towards lower frequencies as the d.c. conductivity increases. The dielectric response 1(ω) ≡ Re (ω) of these samples is typical of ordinary dielectric materials, namely, it is positive at all frequencies. Furthermore, the dielectric coefficient increases with decreasing frequency, and this increase is sharper in samples with higher conductivities. The second type of sample is characterized by larger conductivities which do not fall towards zero at low frequencies. Instead, they either stay roughly constant or even increase at very low frequencies. These samples also exhibit a peak in the a.c. conductivity that continues to move towards lower frequencies as the d.c. conductivity increases. The dielectric response of these samples exhibits three zero crossings, leading to two frequency bands of negative dielectric coefficient. One band appears at very low frequencies, where 1(ω) attains large negative values. A smaller and shallower band appears at higher frequencies. Both bands are wider and deeper in highconductivity samples. Between these two bands of negative lies an intermediate frequency range of positive dielectric coefficient. The value of 1(ω) in this regime increases with increasing conductivity. In recent experimental studies, this conductivity-dependent transition, from a dielectric to metallic behaviour, is attributed to percolation in the presence of inhomogeneous 0953-8984/97/450599+07$19.50 c © 1997 IOP Publishing Ltd L599 L600 Letter to the Editor Figure 1. Schematic representation of a random suspension of uniaxial inclusions in an isotropic host. In samples of conducting polymers both the inclusions and the host are made from chains of the same polymer. Inside the inclusions, these chains are aligned in parallel. Their distribution of orientations in the amorphous host is highly disordered and is represented here by the wispy lines around the ordered inclusions. disorder [6]. Specifically, it is conjectured that the observed behaviour may be explained by the existence of three-dimensional metallic regions embedded in disordered quasi-onedimensional regions with strong localization effects. Both regions are made up of the same polymer chains and are distinguished only by having different degrees of threedimensional ordering. In this letter we propose a simple model of macroscopic disorder which incorporates these two basic elements as different components of a composite material. Similar microgeometric models [7] have been used previously to investigate the mechanism for transport in conducting polymers. We use the model to calculate the macroscopic dielectric response of such systems, and to investigate the percolation induced insulator–metal transition. The model leads to a dielectric behaviour that reproduces all the important features of the experimental results. Our model for the microstructure of conducting polymers includes two basic components (see figure 1). The first component is a collection of randomly distributed spheres. Inside each sphere, the polymer chains are assumed to be ordered parallel to each other. Macroscopically, these spheres are viewed as highly anisotropic particles with dielectric axes defined by the direction of the polymer chains. They have a high conductivity in the direction of the principal dielectric axis and low conductivities in the two perpendicular directions. The dielectric tensor ̃s of these spherical inclusions is assumed to be given by ̃s = Rθφ̃R θφ , where ̃ = ( ⊥ 0 0 0 ⊥ 0 0 0 ‖ )