Nature of insulator–metal transition and novel mechanism of charge transport in the metallic state of highly doped electronic polymers

Synthetic Metals 125 (2002) 43±53

Nature of insulator±metal transition and novel mechanism of charge transport in the metallic state of highly doped electronic polymers V.N. Prigodina,b, A.J. Epsteina,c,* a

Department of Physics, The Ohio State University, Columbus, OH 43210-1106, USA b Ioffe Institute, 194021 St., Petersburg, Russia c Department of Chemistry, The Ohio State University, Columbus, OH 43210-1173, USA

Abstract Highly conducting polymers such as polyaniline and polypyrrole in a metallic state have unusual frequency dependent conductivity, including multiple zero crossings of the dielectric function. A low frequency electromagnetic response, when be analyzed by the Drude theory of metals, is provided by an extremely small fraction of the total number electrons 0.1%, but with extremely high mobility or anomalously long scattering time 10 12 s. We show that a network of metallic grains connected by resonance quantum tunneling has a Drude type response for both the high and low frequency regimes and behaves as a dielectric at intermediate frequency in agreement with experimental observations. The metallic grains in polymers represent crystalline domains of well-packed chains with delocalized electrons embedded in the amorphous media of poor chain order. Intergrain resonance tunneling occurs through the strongly localized states in amorphous media. The small concentration of electrons participating in dc-transport is assigned to the low density of resonance states, and the long relaxation time is related to the narrow width of energy levels in resonance. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Conducting polymers; Transport properties; Dielectric spectroscopy; Network model; Resonant quantum tunneling

1. Introduction The Chemistry Nobel Prize of 2000 went to A.J. Heeger, A.G. MacDiarmid, and H. Shirikawa for the discovery that polymers can be made electrically conductive.1 This discovery opened the use of organic chemistry in the development of metals and semiconductors. The Royal Swedish Academy of Sciences wrote, ``The choice is motivated by the important scienti®c position that the ®eld has achieved and the consequences in terms of practical applications and of interdisciplinary development between chemistry and physics''. Polymers are made up of many repeating groups based predominantly on carbon. Polymers generally do not conduct electricity since the electrons of carbon and other atoms that comprise the chains are included in covalent bonding with neighboring atoms. This is in contrast to metal where electrons freely move over the entire sample. The discovery of Heeger, MacDiarmid and Shirikawa demonstrated that the extra electrons chemically or electrochemically can be introduced in or removed from polymer chains, thereby providing the charge transport. In contrast to conventional *

Corresponding author. Tel.: ‡1-614-292-1133; fax: ‡1-614-292-3706. E-mail address: [email protected] (A.J. Epstein). 1 See Nobel lectures in this volume.

semiconductors the electronic gap of polymers principally is attributed to the Peierls dimerization of polymer chains with some contribution due to Coulomb interaction and, therefore, the gap rapidly is shrunken with increasing doping [1,2]. Presently, there is not a widely recognized theory for the suppression of the Peierls phase by doping, but the experiment says that in the heavily doped limit the electronic gap, as well as chain dimerization are entirely absent [3] The present work addresses solely these heavily doped conjugated polymers. Initially, the intense interest in conjugated polymers was due to the profound increase in their room-temperature conductivity, sRT, upon doping. The sRT of the ®rst doped polymers had modest values and turned nonmetallic with cooling [4,5]. The low temperature conductivity was dominated by variable range hopping and vanished at T ! 0. The continuous improvements of synthesis and processing of polymers enabled steady increase in conductivity magnitude and weaker temperature dependence [6±9]. Remarkable progress was made in developing and stabilizing the metallic state in conjugated polymers in the last decade. A ®nite residual conductivity at very low temperatures up to 10 mK was successfully attained for heavily doped polyacetylene, polyaniline, and polypyrrole (for recent review see, e.g. [9±11]).

0379-6779/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 9 - 6 7 7 9 ( 0 1 ) 0 0 5 1 0 - 0

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V.N. Prigodin, A.J. Epstein / Synthetic Metals 125 (2002) 43±53

The experimental study indicates that in spite of the large value of sRT and ®nite residual conductivity, transport properties of these highly conducting polymers are still far from being typical traditional metals: the conductivity decreases with decreasing temperature. The ratio of the room temperature to residual conductivity remains appreciable. Depending on sRT this decay of conductivity with decreasing temperature varies from an activation-type to a weak logarithmic behavior [6,9]. At the same time their thermoelectric power [12] and Pauli susceptibility [13] suggest a metallic density of states at the Fermi level, i.e. the absence of any structural or electronic transformations. On the basis of these observations it was suggested [14± 16] that this last generation of conducting polymers is close to the insulator±metal transition (IMT). There is a general consensus that the IMT is an Anderson disorder-driven localization±delocalization transition [6,17±21]. Through a small increase of disorder, e.g. with aging, the polymers are driven into the insulator state. The dielectric phase (doped polymer that becomes insulating at low temperatures) can be converted back into the metallic state by, for example, applying pressure [22]. At the same time the sRT varies only slightly with pressure. This type of IMT is well known and widely studied for strongly doped amorphous semiconductor and in dirty metals [23,24]. The natural question arises: did the discovery of conducting chemically doped polymers contribute to fundamental physics of charge transport? Or simply, did the discovery of conducting chemically doped polymers only add another class to the list of dirty conductors? This last view prevailed in the beginning of study of conducting polymers. Indeed, the transport behavior of poorly conducting polymers is close to that which is observed in dirty metals and semiconductors in the absence of crystal structure [10], although some difference due to the chain structure of polymers was documented [17,18,26±29] even in the dielectric phase [15,16,30,31]. In the present work, we argue that highly conducting polymers demonstrate a new mechanism of charge transport. Also, we conclude that they exhibit a new type of IMT. The article is organized as follows: the arguments for a new mechanism of charge transport in conducting polymers comes from their optical and low frequency measurements of conductivity [9,21,32±35]. The puzzling features of the frequency dependence are described in detail in Section 2. To resolve these puzzles we exploit the structural peculiarities of the polymers. It is well known that these materials (polyaniline and polypyrrole) are strongly inhomogeneous [15,16,36±38]. In cooked ``spaghetti-like'' polymer chain media one can distinguish the ``crystalline'' regions within which polymer chains are well-ordered [36±38]. When we approach the IMT the electron delocalization ®rst happens inside of these regions. Outside the crystalline regions, the chain order is poor and the wave functions of electronic states in the amorphous region may be strongly localized. Therefore, the crystalline domains can be considered like

metallic grains of mesoscopic size embedded in amorphous poorly conducting (non-metallic) media. Metallic grains are electrically connected by single poorly conducting chains. This model of chain linked nanoscale grain network is introduced, and the IMT is discussed in Section 3. In Section 4, we argue that the charge transfer between grains is provided by resonance quatum tunneling through localized states in amorphous media even at room temperature. We show that the direct tunneling between the grains is strongly suppressed because of large spatial intergrain separation. Our estimates point out that for the structural parameters of the polymers: chain length, degree of disorder, and the number of chains connecting grains, the charge transport in the metallic state is provided by intergrain resonance tunneling. In Section 5, we show that the resonance mechanism of transport enables us to understand the origin of low-frequency anomaly in the metallic phase and the nature of the IMT in the polymers. Our concluding remarks on the chain-linked granular model of transport in the polymers are collected in Section 6. 2. Puzzles of the metallic state of conducting polymers In general, there are two fundamental types of charge transport: hopping [23] and band [39] transport mechanisms. Fig. 1 illustrates the principal difference among them. During band transport, which is a the quantum or coherent process, electrons move ballistically, and are occasionally scattered by impurities and phonons. After scattering, the direction of an electron's propagation is random, and, as a result, the electron's motion represents a diffusion, more exactly, a quantum diffusion since in intervals between scattering the electrons propagate in a coherent way. Thus, in band transport phonons are a source of resistance in addition to impurities, and therefore, the conductivity decreases with increasing temperature for more intensive phonon scattering [39]. There is a characteristic temperature above which the phonon scattering dominates over the impurity one. Below this characteristic temperature the phonon scattering rate becomes weaker than the impurity one, and the conductivity approaches a residual conductivity with decreasing temperature. This zero-temperature conductivity is entirely determined by the impurity scattering. A classical example of materials where such a type of conductivity is realized is bulk metals with small and moderate disorder. In the hopping mechanism of transport, zero-temperature conductivity is zero because the carriers are bound [40]. Electron's transitions between localized states are forbidden since they require external energy. At ®nite temperatures electrons can get energy from phonons. By absorbing or emitting phonons, electrons hop from one localized state to another. An electron's hops over localized states is a random walk or classical or incoherent diffusion since the electron's transitions in this case represent inelastic processes described by the probability.

V.N. Prigodin, A.J. Epstein / Synthetic Metals 125 (2002) 43±53

45

Fig. 1. Band vs. hopping mechanism of transport. Insert 3 (left) schematically illustrates the scattering of ballistic electron by phonons. Insert 3 (right) is the schematic illustration of phonon-assisted hopping of electron among localized states.

In contrast to band transport, phonons are now a source of electrical conductivity, and hopping conductivity increases with temperature because of more available phonons. At high temperatures hopping conductivity reaches a maximum. At this temperature, the interaction with phonons is so intensive that electron localization is entirely suppressed, and in fact, we deal with the band transport with strong phonon scattering. In general, the magnitude of the hopping conductivity is expected to be less than that of Mott's minimum metallic conductivity [23], smin, while the metallic conductivity associated with the states far above the mobility edge remains higher than this boundary value. The hopping transport is a general mechanism for the lowtemperature electrical conductivity of disordered materials with localized states. A classical representative of materials with hopping conductivity is a doped semiconductor where electrons hop among impurity states whose levels are inside the energy gap [40]. Another example is glassy materials such as amorphous semiconductors and dirty metals. The localization of carries in this case is due to potential ¯uctuations that localize carriers. In general, the hopping motion takes place for electrons with energies below the mobility edge. For polymers, the hopping should present the basic mechanism of transport not only because of their irregular structure and expected high disorder, but also because electron localization occurs in polymer chains even in the presence of weak disorder. In practical situations it is sometimes dif®cult to determine the transport mechanism on the basis of

dc-conductivity only. This is the case for conducting polymers: they demonstrate relatively high conductivities at room temperature, and at the same time they often have a positive ds/dT. The dielectric constant,E, measurements can be used to identify the mechanism of charge transport. For band transport, E is negative. This corresponds to retardation of the current response due to the inertial mass of the carriers. The value of the negative dielectric constant is limited by the relaxation rates and the frequency. In the hopping regime, electrons remain bound within localized states; at low frequencies they follow the variation of the external ®eld. For this case, is positive and proportional to the square of the spatial size of the localized state. Finally, the hopping and band mechanisms of transport have very different frequency dependence of conductivities. We consider the absorptive part of ac-conductivity. For hopping the increase of frequency o of the external electric ®eld involves more electronic states, the transition rate between which is large than o. Each transition is accompanied by absorption of the external ®eld. As the result Re s(o) increases with increasing frequency of the external ®eld [41]. For band transport the absorption of the external ®eld is limited by the rate at which the electronic subsystem loses its momentum, since the electronic gas itself represents a quantum system and cannot itself absorb a varying electric ®eld. The electronic motion becomes more spatially restricted with increasing frequency of the external ®eld. Therefore, the scattering rate of electrons as well as Re s(o) decreases with increasing frequency (see Fig. 1).

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V.N. Prigodin, A.J. Epstein / Synthetic Metals 125 (2002) 43±53

Fig. 2. Room-temperature dielectric function, E(o), for PAN±CSA samples with conductivities increasing in the order of sC > sA > sD > sB (after [9,33]).

We propose that in the highly conducting polymers there is a new mechanism of transport, which is the quantum resonance hopping between metallic grains. On the one hand, it is quantum transport, and at the same time it represents random hops over a network of grains. Randomization of electron motion comes from the internal disorder of grains. Also the thermalization of electron's energy takes place inside grains. This proposal about a new mechanism of transport is based on puzzles which are revealed in the frequency dependence of conductivity of highly conducting polymers through the optical and low frequency measurements of conductivity (see Fig. 2) [9]. The experiments [21,32±35] showed that high frequency (00.1 eV) conductivity follows a Drude law for a standard metal with plasma frequency corresponding to the density of charge carriers n  1021 cm 3 and ``standard'' scattering time 10 15 s. This behavior takes place in both the metallic and dielectric phases. At decreasing frequency the polymers in the dielectric phase progressively display insulator properties. The dielectric constant becomes positive for frequency 0.1 eV signaling that charge carriers are no longer free, but localized. The positive value of the dielectric constant also is found from microwave frequency (6.6 GHz) experiments [21,32,33], and its value enables an estimate of the localization size 5 nm depending upon polymers, dopants, and preparation conditions. It is a surprising and remarkable feature of the metallic phase that the dielectric constant is similar to that of dielectric samples with decreasing frequency, also changing sign from negative to positive at the same frequency 0.1 eV. However, for metallic samples the dielectric constant changes again to negative at lower frequencies 0.01 eV indicating that electronic motion is free [21,32,33]. The parameters of this low frequency coherent phase are anomalous. By applying the Drude formula the relaxation time is found to be very long (10 13 s), and the new plasma frequency is very small 0.01 eV. The small plasma frequency suggests either the concentration of carriers is low or the mass of carriers is very heavy.

Anomalously long t's also were reported in microwave frequency conductivity (10 11 to 10 12 s) and dielectric constant studies [21,25,26,32,33], time resolved spectroscopy ((5±9†  10 12 s) [42], magnetoresistance [43], and NMR [44] studies. Recently, the above experimental results for frequency dependence of conductivity in-dependently were con®rmed by Martens et al. [45] with radio frequency measurement of ac-conductivity. They also have found that in the low frequency range the dielectric constant is negative, and the conductivity of doped polyaniline and polypyrrole follows the Drude behavior with low plasma frequency and long relaxation time. The results by Martens et al. are very important, since they were obtained by direct measurement of conductivity. The early frequency dependence of conductivity [9] was derived from the re¯ectance coef®cient by using the Kramers± Kroning transformation. This procedure requires some assumption on behavior of re¯ectance in the limit o ! 0. Therefore, the ®rst report on the low frequency anomaly of conductivity [9] was objected to by other authors [10]. Measurement of Martens et al. unambigously con®rmed this anomaly [9]. In particular, for consistence with the negative dielectric constant and the Drude behavior, it is crucial that the conductivity decreases with increasing frequency in the low frequency range. Fig. 3 reproduces the experimental result of Martens et al. [45] which supports such a dependence. These ®ndings for low-frequency electromagnetic response are in contrast to those that are expected for the conventional IMT [24]. In the Anderson IMT model, the electronic behavior is controlled by disorder. In the dielectric phase, electrons are bound by ¯uctuations of random potential. On the metallic side of the IMT there are free carriers with a reduced diffusion coef®cient or with shortened scattering time. The dielectric constant in the metallic phase near the transition is positive since the disorder leads to dynamic polarization because of slowing diffusion. When approaching the transition the dielectric constant diverges ``dielectric catastrophe'' [46]. The small plasma frequency and very long relaxation time of the metallic state in doped polymers can be explained [9,45] by assuming that the conductivity in polymers is provided by a small fraction 0.1% of the total carriers with long scattering time 10 12 s. However, it is dif®cult to reconcile such an assumption with the behavior for high frequencies [21,32±35] which supports that the scattering time is usual 10 15 s. As an alternative, a collective mechanism of transport, like superconducting ¯uctuations, was suggested [47]. It is noted that similar long scattering times also are reported in the heavy fermions and organic compounds [48]. We anticipate that the granular model with the resonance coupling enables us to understand the nature of the IMT and the origin of low frequency anomaly in the metallic phase of the doped polymers.

V.N. Prigodin, A.J. Epstein / Synthetic Metals 125 (2002) 43±53

47

Fig. 3. Symbols E(o) (left) and Re s(o) (right) for doped polypyrrole and doped polyaniline (after [45]).

3. A chain-linked granular network and the insulator±metal transition In explanation of the charge transport in doped polymers their chain structure is very important. Electrons move primarily along polymer chains hopping sometimes between neighboring chains. In polyaniline and polypyrrole single chains are arranged in the space in a very complicated way. Structural study [36±38] points out that there are crystalline regions within which the chains are regularly and densely packed. Outside these regions the order in chain arrangement is poor, and here, the chains form amorphous media. A schematic structure is illustrated in Fig. 4. These metallic islands are coupled into the network with the twisted and tangled polymer chains. Because of its relatively high molecular weight, a single chain can cross a few crystalline regions. If the interchain

Fig. 4. Schematic view on the structure of polyaniline and polypyrrole. The lines represent polymer chains. The dashed squares mark the regions where polymer chains demonstrate the crystalline order.

coupling within the crystalline domain is weak, the existence of the crystalline ordering is not essential for the transport properties of the materials as a whole. The system can be treated like a network of randomly coupled chains that was discussed in [49,50]. One can, however, expect that inside the crystalline domains there is good interchain overlap, and the electron wave function is extended over the whole domain volume. In this case, the domains can be considered like metallic grains embedded in the amorphous poorly conducting matrix. Thus, we come to the model of a granular metal (for a general review see, e.g. [51]) for the description of transport in the polymers. Before discussing this model a number of peculiarities of this granular metal should be mentioned: (i) the polymer grain has essentially quantum size, and therefore, the energy levels inside the grain are appreciably quantized. The scale of this quantization is about a few meV; (ii) the shape of grains cannot be described by a simple geometric form. As a result the energy spectrum and eigenfunctions are expected to obey chaotic statistics [52]; (iii) the coupling between the metallic dots is provided by single chains, therefore, the electric connection is not only between the nearest grains, but between the remote ones as well; (iv) the shape and size of grains as well as intergrain coupling ¯uctuate strongly over the system. Depending on the route of synthesis the fraction of crystalline regions varies from a few percent to approximately 50%. It should be stressed that the standard percolation analysis [53] which predicts that 30% of metallic ®lling guarantees the metallic phase, is not adequate for the polymer granular metal, as even with 50% of crystallinity the materials can remain a dielectric. Presumably, the grains

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V.N. Prigodin, A.J. Epstein / Synthetic Metals 125 (2002) 43±53

remain always spatially separated by amorphous regions of disordered chains. To clarify the question of whether the system is a metal or an insulator requires quantum-mechanical analysis. In the limit of weak intergrain coupling, the system is an insulator. We will demonstrate that there is critical coupling between grains at which the system undergoes the transition into a metallic state. The grain model, which includes the above features was studied by a number of authors [44±56], and we will use their results to establish the critical coupling for the IMT. Afterwards we will discuss the electrical properties of metallic and dielectric phases. The chain coupling between grains manifests itself in the speci®c frequency dependence of conductivity. Let us specify the model in more detail. We will assume that the grains are formed by N? chains which are densely packed over the length N|| in cell units. The whole volume of the grain is (N?  Njj ) cells and, hence, the mean level spacing is estimated to be [52] DE ˆ

1 ; ~ N…EF †Njj N?

(1)

~ F † is the density of states per the unit cell. As an where N…E example, the corresponding parameters for PPy(PF6) are [33,36±38] N? ' 3  8;

Njj ' 7;

~ F † ˆ 0:8…eV ring† 1 ; N…E (2)

and DE ' 7:4 meV. In the case of PAN±CSA, we have [9,36±38]

grains. Since chains in the amorphous region are considered to be independent, the level broadening, dE, can be estimated as dE ˆ 2N? g DE;

(4)

where g is the transmission coefficient [57] or the dimensionless conductance (in units of 2e2/h) of chain-link between the grains. If we denote with J the charge transfer integral between the grain, the transmission coefficient can be written as g ˆ …J=DE†2 . As it ®rst was noted by Thouless [58], the IMT in electronic disordered systems happens when the level broadening dE is of the order of level spacing DE, and then the critical coupling in terms of the transmission coef®cient, gc, of chain-link satis®es 2N? gc ˆ 1:

(5)

For PAN±CSA, this yields [36±38] gc  10 2 . If the chainlink transmission g is less than gc, the system is a dielectric. However, on the metallic side (g > gc ) the electrons are delocalized, and the network represents the metal with finite macroscopic conductivity. On the metallic side electron's motion in the granular network represents random hops over the grains. The randomization of electron's motion is provided by intragrain disorder. The hopping between metallic grains is a quantum process and can be described by the mean hopping frequency, W. Far away from the IMT, W is W0 ˆ

DE N? g ; h z

(6)

states ; eV  2 rings (3)

and, when approaching to the IMT, W? tends to 0 as [59]  r  W gc ˆ exp 2p : (7) W0 g gc

and, therefore, DE ' 1 meV. We assume that the electronic states are delocalized within grains, and the electron's motion inside the grain is diffusive with the diffusion coef®cient D ˆ v2F t=3, where t is the scattering time and vF is the Fermi velocity. The scattering rate can be extracted from optical measurement (see Section 5) [21,32±35,45] t  10 15 s. Roughly, vF can be estimated within the simple free electron picture. For charge density n  1021 cm 3, vF is expected to be vF  107 cm s 1. The diffusive behavior inside grains is restricted to times smaller than the Thouless time, tT , to cross a grain, tT  L2jj =D  5 10 14 s, where Ljj' 5 nm is the size of the grain. We note that the condition for applicability of the grain model t  10 15 s ! tT  10 14 s !  h=DE  10 13 s is ful®lled. Each grain is coupled to other grains by 2N? chains. This coupling extends to remote grains, but we for a simplicity will assume that the coupling is restricted to the nearest 2z grains, and the chains providing this coupling are equivalent. Thus, two nearest grains are electrically connected by approximately N?/z chains. In the metallic phase the intergrain coupling leads to broadening of quantized levels in the

The hopping distance in the metallic phase is the mean distance between the centers of neighboring grains, R. The length R is related to the size of grain and the degree of crystallinity c by the equation  3 Ljj cˆ (8) R

N? ' 9  12;

Njj ' 7;

~ F † ˆ 1:1 N…E

with R and W, the macroscopic diffusion coefficient D3 can be determined for a random walk over the granular network as D3 ˆ R2 W:

(9)

The concentration of the electron participating in charge transport is proportional to cN(EF)kBT and their mobility is given by the Enstein relation m3 ˆ D3 =…kB T†. Thus, the macroscopic conductivity is found to be s…0† ˆ ce2 N…EF †D3 ;

(10)

with D3 from Eq. (9). In particular, far from the IMT by combining all together, the conductivity also can be

V.N. Prigodin, A.J. Epstein / Synthetic Metals 125 (2002) 43±53

rewritten in the simple form    N? 1 s…0† ˆ …e2 g† R z

(11)

Here, the first brackets represent the conductivity of a single chain-link, the second brackets are the number of chain links between the neighboring grains, and R mimics the period of the grain network. As we said before the size and form of crystalline grain, as well as the coupling between grains vary over the system. The parameters which were introduced to describe the conductivity should be understood as the average characteristics. 4. Resonance quantum tunneling between grains The speci®c feature of the granular model for polymers is the coupling between the grains (see Figs. 4 and 5). In standard granular metals, the grains are assumed to have mechanical contacts, and the electric coupling is given by the direct tunneling between grains. In polymers, the crystalline domains are spatially separated, and therefore, direct tunneling should be essentially suppressed. Indeed, assuming the electronic states of amorphous chains are strongly localized and chains are noninteracting in the amorphous part, the transmission coef®cient for tunneling between grains is [60]   2L g ˆ exp ; (12) x where L is the length of chain and x the localization length. For samples of 50% crystallinity L approximately is of the same order as a grain size L||, e.g. L  5 nm. If we accept a picture of weak localization [61], the localization length is x ˆ 4l, where l ˆ vF t is the mean free path. For PAN±CSA, we use [32] vF  3  107 cm s 1 and for t  10 15 s, the

49

localization length is found to be x  1:2 nm. Therefore, g  10 4 , i.e. the direct tunneling is essentially suppressed. However, the intergrain transmission coef®cient is equal to the unity for resonance tunneling. The probability of ®nding a resonance state is proportional to the width of resonance level [60]. The resonance state has to be located in the center of a chain, therefore, its width g is g ˆ …1=T† exp‰ L=xŠ. As a result the average transmission coef®cient g is determined not by the direct tunneling exp[ 2L/x], but by the probability of ®nding the resonance state, i.e. g  exp‰ L=xŠ. For PAN±CSA, thus, we have gc  10 2 , which is close to the critical value for the IMT gc ˆ 1=…2N? †  10 2. It is a very important statement that near the IMT the intergrain coupling in polymers occurs through single resonant chains out of the whole bundle of chains connecting grains. Indeed, the probability of ®nding the resonance state is very small 10 2, since they are nontypical. Therefore, similar resonance transitions usually are not taken into account. However, due to the large number of chains 100 connecting one grain with the rest, the resonance coupling is realized at least for one chain, and this resonance tunneling dominates the entire transport through a grain. For simplicity we omitted the ¯uctuations of the localization radius [61], so that the resonance tunneling with the unit transmission coef®cient happens only for a state in the middle of a chain. Including these ¯uctuations (which allow resonance tunneling through states at locations other than the middle of the chain) does not change our main conclusions about the central role of resonance conductivity. The principal difference between the direct and resonance tunneling is the time for tunneling. Direct tunneling is an almost instantaneous process, i.e. its characteristic time is t. Resonance tunneling shows a delay determined by the level width g. This difference in relaxation time can be detected in the low frequency conductivity. 5. Frequency-dependent conductivity In this section, we discuss the frequency-dependent conductivity of the polymer granular model and compare it with the experimental dependencies. At higher frequencyotT @ 1, where tT is the Thouless time of diffusive spreading over a grain, the system should show bulk metal behavior and the conductivity is given by the Drude formula [39] s…o† ˆ

Fig. 5. Electric connection between the metallic grains (well-packed and well overlapping chain regions) embedded in amorphous media of poor ordered chains: (a) the localization radius of electronic states in metallic grains is of the grain size and is of the scale of the polymer repeat unit in the amorphous media; (b) intergrain charge transfer effectively is provided by tunneling through resonance states in the amorphous regions.

2 a Op t ; 4p 1 iot

O2p ˆ

4pe2 n 8p 2 ˆ e N…EF †v2F : m 3

(13) Here, a factor a changes with frequency from c at ot < 1 to 1 for ot > 1. The dielectric constant according to Eq. (13) is E…o† ˆ 1

a

O2p t2

1 ‡ …ot†2

;

and being negative for o ! Op ; E…o†  a…Op =o†2.

(14)

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V.N. Prigodin, A.J. Epstein / Synthetic Metals 125 (2002) 43±53

Optical data [21,32±35] shows that at high-frequencies 00.1 eV, the system indeed has the metal-like behavior (see Figs. 2 and 3) which we ascribe to the above response from the metallic islands. For PAN metallic samples reported in [32], the corresponding parameters are Op  2 eV, t  10 15 s and the Thouless time is estimated to be tT  5  10 14 s. The behavior Eqs. (13) and (14) should transform with decreasing frequency into the dielectric one at the frequency 1/tT . For otT ! 1 electrons adiabatically follow an external ®eld, and the conductivity proves purely capacitive s…o† ˆ

ioe2 cN…EF †L2jj :

(15)

We note that the sign of dielectric constant now is positive, and its value is given by polarization of the grains:     Op Ljj 2 o01 E : (16) c tT vF The behavior (15) and (16) is in good agreement with that experimentally observed in both dielectric and conductive phases at high and intermediate phase frequencies. Experiments show that in this frequency interval (00.01 eV) the metallic and dielectric phases are not distinguishable (Figs. 2 and 3). If the chain-link transmission g is less than gc, the system is a dielectric and the behavior (15) is retained for all otT ! 1. However, on the metallic side (g > gc ) the electrons are delocalized and their low frequency response is governed by the dynamics of tunneling between grains. Since as we have shown, the electron tunneling happens through the resonance states, the transmission coef®cient demonstrates a delay determined by the level width. This fact should be revealed in frequency dependent conductivity. Indeed, for resonance tunneling the frequency dependent transmission coef®cient g ˆ g…o† is given by generalization of the Bright±Wigner formula [60,62]   io 1 g…o† ˆ 1 : (17) g

This conclusion (18) is consistent with the experimental observations. In the metallic phase at low frequencies (90.01 eV) the optical studies [9], as well as the microwave experiments [25,26] show that the system behaves again like a Drude metal although now with anomalous long relaxation times and with very small plasma frequency. Microwave experiments and optical data [25,26] yield t1  10 12 s Being compared with (13), Eq. (18) effectively corresponds to the decrease of electron concentration by a factor …op =Op †2  10 3 for R=l  30, DEt  10 2 and the increase of mean time of free path by a factor t1 =t ˆ 1=g  102 to be t1  10 12 s. Let us note the obvious correlation op  1=t1 , which was experimentally identified in [45]. For the metallic phase there is a large interval of intermediate frequency 1=t1  1012 s 1 9o < W  1013 s 1, where the behavior (15) transforms into (18). Schematically, the o-dependence of conductivity and dielectric constant are shown on Fig. 6. It qualitatively follows the experimental dependence [33] shown in Figs. 2 and 3. In the metallic phase at ®nite temperatures, in addition to coherent tunneling along the chains, an inelastic channel for intergrain transitions through the chains opens. Therefore, the total intergrain hopping rate becomes W…o; T† ˆ W…o† ‡ Win …o; T†:

(21)

The temperature and frequency dependence of phonon contribution Win …o; T† in Eq. (21) can be established within a framework of the model discussed in [63]. Avoiding the

With (17), we find that in the region o ! W the frequencydependent conductivity can be written in the form of the standard Drude formula s…o† ˆ

1 o2p t1 ; 4p 1 iot1

(18)

with op being ``the plasma frequency'' determined by the frequency of quantum intergrain hops o2p O2p

ˆ c…DEt†g2

R2 : l2

(19)

The low frequency relaxation time t1 ˆ 1=g is determined the Wigner transmission time between grains and reads   L t1 ˆ t exp  tg: (20) x

Fig. 6. The frequency dependence of conductivity (a) and dielectric constant (b) for the chain-linked granular network in the metallic phase: t  10 15 s is the mean time of free path for the intragrain and intrachain disorder; tT ˆ L2jj =D  10 14 s is the Thouless time for diffusively crossing a grain, Ljj  5 nm. t1  10 12 s is the transit time between neighboring grains for resonance tunneling; W ˆ …N? =z†…1=t1 †  1013 (1/ s) is the intergrain hopping rate, where N?/z  10 is the number of chainlinks. The electromagnetic response has Drude behavior for both high (otT @ 1) and low (o=W ! 1) frequencies with the relaxation times t and t1, and the plasma frequencies Op and op, respectively. Note that t1 @ t and op ! Op , so that op t1 @ Op t, but o2p t1 ! O2p t.

V.N. Prigodin, A.J. Epstein / Synthetic Metals 125 (2002) 43±53

details, we point out only the principal features of this dependence. According to [63], the conductivity increases with increasing temperature. This increase is signi®cant and obeys 1-D Mott's law in the temperature interval, …x=R†T0 < T < T0 , where kB T0 ˆ hp=…4t†. At T0 < T, the conductivity is determined by the nearest-neighbor hopping, and Win …T† is hWin …T† N? t x  ˆ ; D z tin …T† R

(22)

where tin …T† is the time of electron-phonon scattering. The time of single chain intergrain transitions, t1 decreases with increasing temperature. It is important to stress that the character of the low frequency dependence of conductivity does not change at incoherent propagation. As an example, in the regime of nearest-neighbor hopping the frequency-dependent transmission coef®cient reads Win …o; T† ˆ

1

Win …T† ; iot1 …T†

(23)

where Win …T† is given by Eq. (22) and the transit time between the grains, t1 …T†, is determined by the equation t1 …T† ˆ

R2 ; D…T†

D…T† ˆ

x2 : tin …T†

(24)

As a result the low-frequency part of electromagnetic response shown on Fig. 6 is shifted with temperature to a range of higher frequencies as it experimentally is observed [9]. 6. Concluding remarks In summary, we considered the chain-linked granular model to take into account the in-homogeneous nature of polymer materials such as doped polyaniline and doped polypyrrole. The inhomogeneity is in the existence of partially crystalline regions in the amorphous media formed by disordered chains. The electronic coupling between these crystalline regions is produced by the chains that pass through the disordered regions. The inhomogeneity leads us to the new scenario of the insulator±metal transition in decreasing intrachain disorder or increasing chain ordering. Naturally, the delocalization happens ®rst in the crystalline domains due to good-overlap between the chains inside the domains. It takes place at a degree of disorder which corresponds to the 3-D IMT, i.e. at t ˆ tc   h=EF  10 16 to 10 15 s. We assume, and it is also in agreement with conclusions by other authors [10], that in the new generation of conducting polymers the requirement on disorder for 3-D delocalization, t > tc is ful®lled. Thus, at t > tc we have a system of metallic mesoscopic islands coupled to each other by single chains. This chain linkage is essentially different from the simple pointlike contacts of a standard granular metal, since the chain represents an extended object with internal states. In general,

51

the intergrain chain-link is described with the transmission coef®cient, which is itself frequency and temperature dependent. Our basic idea is that the low frequency transport properties of the polymer granular network are originated from these chain links. In the region of parameters, when the delocalization in metallic islands occurs and the system approaches the metallic state, the electron transitions along the chain are determined by tunneling. Therefore, the chain coupling is mostly controlled by a degree of intrachain disorder. On the basis of these observations, it follows that the metallic phase can be induced by increasing the crystallinity through the volume of grains and fraction of crystalline regions. Another way to arrive at the metallic state is to transform from coillike chain coupling into rod-like chain coupling, e.g. by stretch alignment of chains. Within the present model, we are able to pursue, at least, schematically, the frequency dependence of electromagnetic response in the whole region of frequency. Optical data shows us that at high-frequencies (00.1 eV) the system demonstrates metallic behavior. In the present model such a response comes from the metallic islands. In the intermediate frequency (0.01±0.1 eV), the experimental system shows us a featureless optical spectrum with a positive dielectric constant. We have explained that the dielectric background in this region of frequency can arise from the polarization of the metallic islands. As another possible argument for this behavior we mention that in the case of polyacetylene, where the metallic islands are large with high percentage of crystallinity [36±38], and shorten amorphous regions, a similar bump for a dielectric constant is absent [64,65]. In the metallic phase at low frequency (90.01 eV) the optical study [9,10], as well as the microwave experiment [33] suggest that the system behaves again like the Drude metal with very long relaxation times and with very small plasma frequency. Within the present model this type of behavior results from the chain-links between the grains. The metallic type of response (negative dielectric constant) is explained by the retardation during chain transmission. This delay in response is associated with processes of coherent scattering by impurities at electron's transit through the chain and the long relaxation time is related to the time of intergrain transitions. We have, however, found that this behavior is robust in respect to the character of electron propagation along the chain. Therefore, with increasing temperature one expects that the relaxation time would become shorter, but the Drude type of lowfrequency response does not change, as it is experimentally observed [33]. Thus, we developed a chain linked network of mesoscopic grains as a model for charge transport in polymers. Because of a complex polymer morphology, the electron transport involves a broad hierarchy of relaxation times. It manifests itself in a very rich frequency dependence of conductivity and dielectric constant and a wide variety of dependencies

52

V.N. Prigodin, A.J. Epstein / Synthetic Metals 125 (2002) 43±53

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