- Email: [email protected]
Quantum hopping in metallic polymers
ARTICLE IN PRESS
Physica B 338 (2003) 310–317
Quantum hopping in metallic polymers V.N. Prigodina, A.J. Epsteina,b,* b
a Department of Physics, The Ohio State University, 174 W. 18th Avenue, Columbus, OH 43210-1106, USA Department of Chemistry, The Ohio State University, 100 W. 18th Avenue, Columbus, OH 43210-1173, USA
Abstract Highly conducting polymers such as polyaniline and polypyrrole in a metallic state have unusual frequencydependent conductivity, including multiple zero crossing of the dielectric function. A low-frequency electromagnetic response in terms of a Drude metal is provided by an extremely small fraction of the total number of conduction electrons B0.1% but with extremely high mobility or anomalously long scattering time B1013 s. We show that a network of metallic grains connected by resonance quantum tunneling has a Drude-type response for both the highand 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 the amorphous media. r 2003 Elsevier B.V. All rights reserved. PACS: 72.10Bg; 72.80.Le; 73.20.Jc Keywords: Quantum transport; Conducting polymers; Metallic states; Resonance tunneling; Granular model; Quantum dots
1. Introduction Initially, the electrical conductivity of doped polymers had modest values at room temperature and turned nonmetallic with cooling [1]. In the last several years, a finite residual conductivity at very low temperatures B10 mK was successfully attained for heavily doped polyacetylene, polyaniline, and polypyrrole [2,3]. There is a general consensus [2–5] that the insulator–metal transition (IMT) observed in doped polymers is an Anderson *Corresponding author. Department of Physics, Ohio State University, 174 W. 18th Avenue, Columbus OH 43210-1106, USA. Tel.: 614-292-4443; fax: 614-292-3706. E-mail address: [email protected] (A.J. Epstein).
disorder-driven localization–delocalization transition [5]. Through a small increase of disorder, e.g., with aging, the polymers are driven into the insulator state. The dielectric phase [6] (doped polymer that becomes insulating at low temperatures) can be converted back into the metallic state by, e.g., applying pressure. At the same time, the room-temperature conductivity (sRT ) varies only slightly with pressure [4]. It was widely viewed that the discovery of highly conducting polymers only added another materials class to ‘‘dirty’’ conductors and did not contribute to the fundamental physics of charge transport. Based on recent optical and microwave measurement of electromagnetic response [5,7], we propose that a new mechanism of charge transport,
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.011
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resonance quantum hopping among metallic dots, is present in highly conducting doped polymers. Also, we conclude that they exhibit a new type of IMT. In general, there are two fundamental types of single particle charge transport: hopping and band transport mechanisms. Fig. 1 illustrates the principal difference among them. For band transport (Drude theory) electrons behave as ‘‘free
311
particles’’. They are accelerated by the applied electric field and loose their momentum through scattering by impurities and phonons. As a result, an electron’s motion may be described as quantum diffusion. At low temperature the phonon scattering becomes weak and the conductivity increases with decreasing temperature to its residual value. This conductivity behavior occurs in bulk metals with small and moderate disorder.
Fig. 1. Comparison of band (upper panel) and hopping (lower panel) mechanisms of transport.
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For the hopping mechanism, the zero-temperature electrical conductivity is zero because the charge carriers are localized. At finite temperatures, electrons hop from one localized state to another by absorbing or emitting phonons. The electron’s random walk among localized states is described as classical or incoherent diffusion as in this case electronic transitions are inelastic. In contrast to band transport, hopping conductivity increases with temperature because of increased availability of phonons (see Fig. 1). Hopping transport is a general mechanism for the lowtemperature electrical conductivity of disordered materials with localized electronic states. A conventional example of materials with hopping conductivity is doped semiconductors in which electrons hop among impurity states whose energy levels vary randomly inside the energy gap. Other examples are glassy materials such as amorphous semiconductors and ‘‘dirty’’ metals. In practical situations it is difficult to determine the transport mechanism on the basis of DC conductivity alone. The dielectric constant e (electric field induced polarization divided by the electric field) can be used to identify the mechanism of charge transport [3,5]. For band transport, e is negative because of inertia of ‘‘free electrons’’; for hopping e is positive and proportional to the square of the size of the localized state. Also, hopping and band mechanisms of charge transport have very different frequency dependence of their conductivities. The real part of conductivity sðoÞ is proportional to the absorption of time varying external field. For hopping, this effective absorption is provided by pairs of states (the transition rate between which is less than o (Debye’s losses)) and therefore the hopping absorption increases with increasing frequency (Fig. 1). For band transport, the absorption of the time varying external field is limited by the rate at which the electrons lose their momentum. Therefore, the absorption for band-type conductivity (Fig. 1) decreases with increasing frequency as the electron motion becomes more spatially restricted and less scattering occurs. Hopping should be the basic mechanism of transport for polymers because of their irregular structure and because even for weak disorder
electrons in single chains are localized. This type of conductivity is observed in the dielectric polymers and their behavior qualitatively follows the above dependencies (Fig. 1). However, the behavior of metallic polymers can be understood within neither the above hopping model nor the model of conventional metal with band electrons. The metallic conductivity of doped polymers decreases with decreasing temperature and a finite residual conductivity is within a decade of the roomtemperature value (see inset, Fig. 2(a)). Though the decrease of the DC conductivity for metallic polymers with decreasing temperature can be accounted for within the band model by effects of localization caused by disorder, the experimental optical and low-frequency conductivity and dielectric constant in the metallic state of doped polymers [5,7] in principle cannot be accounted for by band and hopping models with homogeneous disorder. We propose that in highly conducting polymers there is a new mechanism of charge transport, resonance quantum tunneling among metallic domains.
2. Electric properties of metallic state in polymers The 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). Experiments [3–5] (Fig. 2(a)) show that the high-frequency (\0.1 eV) conductivity and dielectric constant generally follow a Drude law with the number of electrons B1021 cm3 corresponding to the total density of conduction electrons and conventional scattering time B1015 s in both the metallic and dielectric phases. At decreasing frequency, the polymers in the dielectric phase progressively display insulator properties and e becomes positive for frequency t0.1 eV signaling that charge carriers are now localized. Microwave frequency (B6.6 GHz) e experiments [3,7a,7b] yield localization lengths B5 nm, depending on sample, dopants, and preparation conditions.
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Fig. 2. (a) eðoÞ for polyaniline metallic samples with conductivities in the order of sA > sB > sC > sD (inset: sDC ðTÞ for sample A, after [3]). (b) eðoÞ and sðoÞ for metallic doped polymers (after Ref. [7c]).
A puzzling feature of the metallic phase in polymers is that e is similar to that of dielectric samples with decreasing frequency, also changing sign from negative to positive at approximately the same frequency B0.1 eV. However, for metallic samples e changes again to negative at yet lower frequencies t0.01 eV indicating that free electronic motion is present [3,5]. The parameters of this low-frequency coherent phase are quite anomalous. From the Drude model, the relaxation time is very long \1013 s; also, the new plasma
frequency below which e is again negative is very small B0.01 eV [3,5]. Recently, this second zero crossing of dielectric constant at low frequency and the conclusion about a long relaxation time and a small plasma frequency were confirmed with radiofrequency conductivity (see Fig. 2(b)) [7c]. The results of Ref. [7c] are very important because they were obtained by direct measurement of conductivity. The early frequency dependence of conductivity derived from the reflectance coefficient by using
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Kramers–Kroning procedure requires some assumption on behavior of reflectance in the limit o-0: These experimental findings for low-frequency electromagnetic response are in contrast with the Anderson IMT [6] in which electronic behavior is controlled by disorder. In the dielectric phase electrons are bound by fluctuations of the random potential. On the metallic side of the transition, free carriers have short scattering times. In the metallic phase near the transition e is positive because the disorder causes dynamic polarization due to slowing diffusion by localization effects. When approaching the IMT transition the localization effects increase and e diverges (‘‘dielectric catastrophe’’ [8]). The small plasma frequency and very long t of the metallic state in doped polymers can be explained [3,5,7] assuming that the conductivity is provided by a small fraction B0.1% of the total carriers with long scattering time >1013 s. However, it is difficult to reconcile this conclusion with the behavior for high frequencies which supports that the scattering time is usually B1015 s and all available electrons participate in conduction. To account for these anomalies the possible presence of a collective mode, as in a charge density wave conductor, or superconductor, was suggested [9].
3. Model of resonant transport In explanation of the low-frequency anomaly in doped polymers in the metallic phase, their chain
morphology is very important. These materials are strongly inhomogeneous [10] with ‘‘crystalline’’ regions within which polymer chains are well ordered (Fig. 3(a)). When the IMT is approached delocalization first occurs inside these regions. Outside the crystalline regions, the chain order is poor and the electronic wave functions are strongly localized. Therefore, the crystalline domains can be considered as nanoscale metallic dots embedded in amorphous poorly conducting medium [11]. The metallic grains remain always spatially separated by amorphous regions, and, therefore, direct tunneling between grains is exponentially suppressed. The intergrain tunneling is possible through intermediate localized states in the disordered portion with strong contribution from resonance states whose energy is close to the Fermi level (Fig. 3(b)). The dynamics of resonance tunneling can account for the frequency-dependent anomalies in the conductivity and dielectric constant of the metallic phase of these doped polymers. We assume that grains have N> chains densely packed over NJ repeat units yielding N> NJ units cells in each grain. Neglecting intergrain coupling leaves the electronic levels inside grains quantized with mean level spacing DE ¼ * F ÞðN> NJ Þ1 ; where Nðe * F Þ is the density of ½Nðe states per unit cell. For metallic doped polypyrrole * F Þ ¼ 0:8 PPy(PF6) [10]: N> E3 8; NJ E7; Nðe states/(eV ring) and DEE7:4 meV. For doped metallic polyaniline PAN(HCl) [10], N> E9 12; * F Þ ¼ 1:1 states/(eV 2 rings), and NJ E7; Nðe DEE1 meV.
Fig. 3. (a) Schematic view on structure of polyaniline and polypyrrole. The lines represent polymer chains. The dashed squares mark the regions where polymer chains demonstrate crystalline order. (b) The electrical coupling between metallic grains is provided by resonance tunneling through localized states in the amorphous region.
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We assume that the electronic states are delocalized inside grains and electron’s dynamics is diffusive with diffusion coefficient D ¼ v2F t=3: This diffusive behavior is restricted to times smaller than the time, tT ; for a charge carrier to cross the grain (Thouless time), tT ¼ L2J =D; where LJ B5 nm is taken for the size of a typical grain [10]. At frequency otT b1; the system should show bulk metal behavior and sðoÞ is given by the Drude formula sðoÞ ¼ ðb=ð4pÞÞ ðO2p tÞ=ð1 iwtÞ; O2p ¼ 4pe2 n=m;
ð1Þ
where Op is the unscreened plasma frequency and b is the degree of crystallinity (usually, t50% crystallinity). The Drude dielectric constant is eðoÞ ¼ 1 bO2p t2 =ð1 þ ðotÞ2 Þ
ð2Þ
and is negative for o5Op ; eðoÞB bðOp =oÞ2 : Optical data [3,5] show that at high-frequencies b0.1 eV the system indeed is metal-like (Fig. 2(a)) which we attribute to the above metallic island response (1,2) (Fig. 4). For PAN(CSA) metallic samples [3,5] (Fig. 2(a)), the corresponding parameters are Op E2 eV; tB1015 s and tT is estimated to be B5 1014 s. We note that the condition for applicability of the grain model, t 5tT 51=DEB1013 s, is fulfilled. The high-frequency Drude response [3,5] transforms with decreasing frequency into dielectric behavior at a frequency 1=tT : For otT 51;
electrons follow an external field and the conductivity is purely capacitive: sðoÞ ¼ ioe2 bNðeF ÞL2J
ð3Þ
with positive dielectric constant given by the polarization of grains: eðop1=tT ÞBbðOp LJ =vF Þ2 :
ð4Þ
The behavior (3,4) is in good agreement with indistinguishable experimental results for both dielectrics near the IMT and conductive phases at high and intermediate frequencies (0.01–0.1 eV) (Fig. 2). Whether the lower-frequency behavior is metallic or dielectric depends on the intergrain coupling. Each grain is coupled to other grains by 2N> independent chains through amorphous media. For simplicity, we assume the two nearest grains are electrically connected by BN> =z chains, where z is the number of nearest neighboring grains. In the metallic phase, the intergrain coupling leads to broadening of quantized levels in the grains, dE ¼ 2N> g DE; where g is the transmission coefficient between grains through a single chain. The IMT occurs when dEBDE (Thouless criterion of IMT [12]) and the critical chain-link coupling gc satisfies 2N> gc ¼ 1:
ð5Þ
For PAN(HCl) and similar PAN(CSA) this yields gc B102 : If gogc ; the system is a dielectric and behavior (4) is retained for all otT 51: However, on the metallic side (g > gc ) electrons are delocalized over network of grains and their low-frequency motion is a random hopping among the grains. The hopping between the grains is a quantum process and can be described by mean transition frequency W : Introducing the mean distance between the centers of neighboring grains, R ðbBðLJ =RÞ3 ), the corresponding diffusion coefficient D3 and the macroscopic conductivity of network are D3 ¼ R2 W ;
Fig. 4. eðoÞ for the chain-linked granular model in the metallic phase. Inset: Re sðoÞ:
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sðoB0Þ ¼ be2 NðeF ÞD3 :
ð6Þ
On approaching the IMT from the metallic side W tends to 0 as [13]: W ¼ ðDE=ð2zÞÞ exp½22pðgc =ðgFgc ÞÞ1=2 :
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In the metallic phase the hopping frequency W is related to the above model parameters as W ¼ dE=ð2zÞ ¼ ðN> =zÞg DE;
ð7Þ
and the whole system can be represented as a network of random conductors. The nodes represent the grains where randomization of electronic motion happens. Combining all together, the conductivity (6) also can be written in the simple form sð0Þ ¼ ðe2 gÞ ðN> =zÞ ð1=RÞ:
ð8Þ
Here, the first brackets represent the conductance of a single intergrain chain-link, the second factor in brackets is a number of chains connecting neighboring grains, and R in last brackets mimics the period of the grain network.
4. Average vs. typical transmission Thus the problem of transport in the metallic phase far from the IMT is reduced to the study of the average transmission coefficient g in (7) for a chain of finite length. For direct tunneling between grains g ¼ exp½22L=x; where L is the length of chain connecting neighboring grains and x the localization length. For 50% crystallinity L is of the same order as a grain size LJ ; i.e., LB5 nm. For weak localization, x ¼ 4l, where l ¼ vF t is the mean free path. For PAN(HCl) parameters assuming vF B3 107 cm/s [9] and tB1015 s; xB1:2 nm. Therefore, gB104 and direct tunneling is essentially suppressed. However, the transmission coefficient is unity for resonance tunneling (Fig. 3(b)). The probability of finding a resonance state is proportional to the width of the resonance level g: For constant x the resonance state needs to be in the center of chain, therefore, gBð1=tÞexp½2L=x: As a result, the average transmission coefficient is determined by the probability of finding the resonance state at the center, i.e., /gS ¼ ðgtÞBexp½2L=x: For PAN(HCl) we have /gSB102 which is close to the critical value for IMT gc ¼ 1=ð2N> ÞB102 : Thus, the probability of a resonance state is small B102, and resonance tunneling is not taken into account for a single chain. However, as the
number of interconnecting chains of a given grain with others is large B 100, resonance coupling occurs between grains. A principal difference between direct and resonance tunneling is the time for tunneling. Direct tunneling that occurs in a conventional granular metal is an almost instantaneous process, i.e., its characteristic time is the scattering time t: Resonance tunneling that is anticipated to be in the metallic polymers shows a delay determined by the level width g: The frequency-dependent transmission coefficient g ¼ gðoÞ for resonance tunneling is given by a generalization of the Bright–Wigner formula [14] gðoÞ ¼ ½1 io=g1 :
ð9Þ
With (6–8), we find that in the region o5W ; sðoÞ can be written in the standard Drude form sðoÞ ¼ ð1=ð4pÞÞop2 t1 =ð1 iot1 Þ
ð10Þ
with op being ‘‘the plasma frequency’’ determined by the frequency /W S of intergrain hops ðop =Op Þ2 ¼ ðb DEtÞ ð/gSÞ2 ðR=lÞ2
ð11Þ
and the relaxation time t1 ¼ ð1=gÞ ¼ t/gSB t exp½L=x determined by the Wigner transmission time. The overall frequency dependence of dielectric function and conductivity in the metallic state of polymers are shown in Fig. 4.
5. Conclusions The dependencies (1–4,10) (Fig. 4) agree with experimental observations (Fig. 2). In the metallic phase at low frequencies (o0.01 eV) the studies [3,5,7] show that the polymers behave again like a Drude metal with anomalous long relaxation times and with very small plasma frequency (Fig. 2). Microwave experiment and optical data yield t1 B(1013–1012) s. Being compared with (1,10,11) effectively corresponds to the decrease of electron concentration by a factor ðop =Op Þ2 B 103 for R=lB30; DEtB102 and the increase of ‘‘mean time of free path’’ by a factor
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t1 =t ¼ 1=/gSB102 to be t1 B(1013–1012) s. Further, op B1=t1 ; which was observed in Ref. [7]. The temperature dependence of the DC conductivity in the metallic phase follows from the resonant tunneling through the strongly localized states in the amorphous regions of the metallic polymer. With increasing temperature, phonons increase the localization length of the states in the disordered regions thereby increasing the resonant transmission rate between grains and increasing the conductivity. As a result, the low-frequency part of electromagnetic response is shifted with increasing temperature to a range of higher frequencies as it experimentally is observed in Ref. [5]. In summary, we propose that resonant quantum tunneling among the chain-linked network of nanoscale grains is the mechanism of charge transport in metallic doped polymers. The model explains key experimental findings of eðoÞ; sðoÞ; very long relaxation time t at low frequencies, and low fraction of electrons contributing to DC-transport.
Acknowledgements Supported in part by the Office of Naval Research and Army Research Office.
References [1] C.K. Chiang, C.R. Fincher, Y.W. Park, A.J. Heeger, H. Shirakawa, E.L. Louis, S.C. Gau, A.G. MacDiarmid, Phys. Rev. Lett. 39 (1977) 1098.
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[2] T. Ishiguro, H. Kaneko, Y. Nogami, H. Ishimoto, H. Nishiyama, A. Tsukamoto, A. Takahashi, M. Yamaura, T. Hagiwara, K. Sato, Phys. Rev. Lett. 69 (1992) 660. [3] R.S. Kohlman, A. Zibold, D.B. Tanner, G.G. Ihas, T. Ishuguro, Y.G. Min, A.G. MacDiarmid, A.J. Epstein, Phys. Rev. Lett. 78 (1997) 3915. [4] R. Menon, C.O. Yoon, D. Moses, A.J. Heeger, In: T. Skotheim, R. Elsenbaumer, J. Reynolds (Eds.), Handbook of Conducting Polymers, 2nd Ed., Marcel Dekker, New York, 1998, pp. 27–84. [5] R.S. Kohlman, A.J. Epstein, In: T. Skotheim, R. Elsenbaumer, J. Reynolds (Eds.), Handbook of Conducting Polymers, 2nd Ed., Marcel Dekker, New York, 1998, pp. 85–121. [6] P.A. Lee, T.R. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 287. [7] (a) H.H.S. Javadi, A. Chakraborty, C. Li, N. Theophilou, D.B. Swanson, A.G. MacDiarmid, A.J. Epstein, Phys. Rev. B 43 (1991) 2183; (b) J. Joo, Z. Oblakowski, G. Du, J.P. Pouget, E.J. Oh, J.M. Wiesinger, Y. Min, A.G. MacDiarmid, A.J. Epstein, Phys. Rev. B 49 (1994) 2977; (c) H.C.F. Martens, J.A. Reedijk, H.B. Brom, D.M. de Leuw, R. Menon, Phys. Rev. B 63 (2001) 073203. [8] N.F. Mott, M. Kaveh, Adv. Phys. 34 (1985) 329. [9] K. Lee, M. Reghu, E.L. Yuh, N.S. Sariciftci, A.J. Heeger, Synth. Met. 68 (1995) 287. [10] J.P. Pouget, M.E. Jozefowicz, A.J. Epstein, X. Tang, A.G. MacDiarmid, Macromolecules 24 (1991) 779; J.P. Pouget, Z. Oblakowski, Y. Nogami, P.A. Albouy, M. Laridjani, E.J. Oh, Y. Min, A.G. MacDiarmid, J. Tsukamoto, T. Ishiguro, A.J. Epstein, Synth. Met. 65 (1994) 131. [11] B. Bean, J.P. Travers, Synth. Met. 65 (1999) 101; F. Zuo, M. Angelopoulos, A.G. MacDiarmid, A.J. Epstein, Phys. Rev. B (1987) 3475. [12] D.J. Thouless, Phys. Rep. 13 (1974) 93. [13] O. Viehweger, K.B. Efetov, Phys. Rev. B 45 (1992) 11546. [14] C.S. Bolton-Heaton, C.J. Lambert, V.I. Falko, V.N. Prigodin, A.J. Epstein, Phys. Rev. B 60 (1999) 10569.
Physica B 338 (2003) 310–317
Quantum hopping in metallic polymers V.N. Prigodina, A.J. Epsteina,b,* b
a Department of Physics, The Ohio State University, 174 W. 18th Avenue, Columbus, OH 43210-1106, USA Department of Chemistry, The Ohio State University, 100 W. 18th Avenue, Columbus, OH 43210-1173, USA
Abstract Highly conducting polymers such as polyaniline and polypyrrole in a metallic state have unusual frequencydependent conductivity, including multiple zero crossing of the dielectric function. A low-frequency electromagnetic response in terms of a Drude metal is provided by an extremely small fraction of the total number of conduction electrons B0.1% but with extremely high mobility or anomalously long scattering time B1013 s. We show that a network of metallic grains connected by resonance quantum tunneling has a Drude-type response for both the highand 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 the amorphous media. r 2003 Elsevier B.V. All rights reserved. PACS: 72.10Bg; 72.80.Le; 73.20.Jc Keywords: Quantum transport; Conducting polymers; Metallic states; Resonance tunneling; Granular model; Quantum dots
1. Introduction Initially, the electrical conductivity of doped polymers had modest values at room temperature and turned nonmetallic with cooling [1]. In the last several years, a finite residual conductivity at very low temperatures B10 mK was successfully attained for heavily doped polyacetylene, polyaniline, and polypyrrole [2,3]. There is a general consensus [2–5] that the insulator–metal transition (IMT) observed in doped polymers is an Anderson *Corresponding author. Department of Physics, Ohio State University, 174 W. 18th Avenue, Columbus OH 43210-1106, USA. Tel.: 614-292-4443; fax: 614-292-3706. E-mail address: [email protected] (A.J. Epstein).
disorder-driven localization–delocalization transition [5]. Through a small increase of disorder, e.g., with aging, the polymers are driven into the insulator state. The dielectric phase [6] (doped polymer that becomes insulating at low temperatures) can be converted back into the metallic state by, e.g., applying pressure. At the same time, the room-temperature conductivity (sRT ) varies only slightly with pressure [4]. It was widely viewed that the discovery of highly conducting polymers only added another materials class to ‘‘dirty’’ conductors and did not contribute to the fundamental physics of charge transport. Based on recent optical and microwave measurement of electromagnetic response [5,7], we propose that a new mechanism of charge transport,
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.011
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resonance quantum hopping among metallic dots, is present in highly conducting doped polymers. Also, we conclude that they exhibit a new type of IMT. In general, there are two fundamental types of single particle charge transport: hopping and band transport mechanisms. Fig. 1 illustrates the principal difference among them. For band transport (Drude theory) electrons behave as ‘‘free
311
particles’’. They are accelerated by the applied electric field and loose their momentum through scattering by impurities and phonons. As a result, an electron’s motion may be described as quantum diffusion. At low temperature the phonon scattering becomes weak and the conductivity increases with decreasing temperature to its residual value. This conductivity behavior occurs in bulk metals with small and moderate disorder.
Fig. 1. Comparison of band (upper panel) and hopping (lower panel) mechanisms of transport.
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For the hopping mechanism, the zero-temperature electrical conductivity is zero because the charge carriers are localized. At finite temperatures, electrons hop from one localized state to another by absorbing or emitting phonons. The electron’s random walk among localized states is described as classical or incoherent diffusion as in this case electronic transitions are inelastic. In contrast to band transport, hopping conductivity increases with temperature because of increased availability of phonons (see Fig. 1). Hopping transport is a general mechanism for the lowtemperature electrical conductivity of disordered materials with localized electronic states. A conventional example of materials with hopping conductivity is doped semiconductors in which electrons hop among impurity states whose energy levels vary randomly inside the energy gap. Other examples are glassy materials such as amorphous semiconductors and ‘‘dirty’’ metals. In practical situations it is difficult to determine the transport mechanism on the basis of DC conductivity alone. The dielectric constant e (electric field induced polarization divided by the electric field) can be used to identify the mechanism of charge transport [3,5]. For band transport, e is negative because of inertia of ‘‘free electrons’’; for hopping e is positive and proportional to the square of the size of the localized state. Also, hopping and band mechanisms of charge transport have very different frequency dependence of their conductivities. The real part of conductivity sðoÞ is proportional to the absorption of time varying external field. For hopping, this effective absorption is provided by pairs of states (the transition rate between which is less than o (Debye’s losses)) and therefore the hopping absorption increases with increasing frequency (Fig. 1). For band transport, the absorption of the time varying external field is limited by the rate at which the electrons lose their momentum. Therefore, the absorption for band-type conductivity (Fig. 1) decreases with increasing frequency as the electron motion becomes more spatially restricted and less scattering occurs. Hopping should be the basic mechanism of transport for polymers because of their irregular structure and because even for weak disorder
electrons in single chains are localized. This type of conductivity is observed in the dielectric polymers and their behavior qualitatively follows the above dependencies (Fig. 1). However, the behavior of metallic polymers can be understood within neither the above hopping model nor the model of conventional metal with band electrons. The metallic conductivity of doped polymers decreases with decreasing temperature and a finite residual conductivity is within a decade of the roomtemperature value (see inset, Fig. 2(a)). Though the decrease of the DC conductivity for metallic polymers with decreasing temperature can be accounted for within the band model by effects of localization caused by disorder, the experimental optical and low-frequency conductivity and dielectric constant in the metallic state of doped polymers [5,7] in principle cannot be accounted for by band and hopping models with homogeneous disorder. We propose that in highly conducting polymers there is a new mechanism of charge transport, resonance quantum tunneling among metallic domains.
2. Electric properties of metallic state in polymers The 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). Experiments [3–5] (Fig. 2(a)) show that the high-frequency (\0.1 eV) conductivity and dielectric constant generally follow a Drude law with the number of electrons B1021 cm3 corresponding to the total density of conduction electrons and conventional scattering time B1015 s in both the metallic and dielectric phases. At decreasing frequency, the polymers in the dielectric phase progressively display insulator properties and e becomes positive for frequency t0.1 eV signaling that charge carriers are now localized. Microwave frequency (B6.6 GHz) e experiments [3,7a,7b] yield localization lengths B5 nm, depending on sample, dopants, and preparation conditions.
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Fig. 2. (a) eðoÞ for polyaniline metallic samples with conductivities in the order of sA > sB > sC > sD (inset: sDC ðTÞ for sample A, after [3]). (b) eðoÞ and sðoÞ for metallic doped polymers (after Ref. [7c]).
A puzzling feature of the metallic phase in polymers is that e is similar to that of dielectric samples with decreasing frequency, also changing sign from negative to positive at approximately the same frequency B0.1 eV. However, for metallic samples e changes again to negative at yet lower frequencies t0.01 eV indicating that free electronic motion is present [3,5]. The parameters of this low-frequency coherent phase are quite anomalous. From the Drude model, the relaxation time is very long \1013 s; also, the new plasma
frequency below which e is again negative is very small B0.01 eV [3,5]. Recently, this second zero crossing of dielectric constant at low frequency and the conclusion about a long relaxation time and a small plasma frequency were confirmed with radiofrequency conductivity (see Fig. 2(b)) [7c]. The results of Ref. [7c] are very important because they were obtained by direct measurement of conductivity. The early frequency dependence of conductivity derived from the reflectance coefficient by using
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Kramers–Kroning procedure requires some assumption on behavior of reflectance in the limit o-0: These experimental findings for low-frequency electromagnetic response are in contrast with the Anderson IMT [6] in which electronic behavior is controlled by disorder. In the dielectric phase electrons are bound by fluctuations of the random potential. On the metallic side of the transition, free carriers have short scattering times. In the metallic phase near the transition e is positive because the disorder causes dynamic polarization due to slowing diffusion by localization effects. When approaching the IMT transition the localization effects increase and e diverges (‘‘dielectric catastrophe’’ [8]). The small plasma frequency and very long t of the metallic state in doped polymers can be explained [3,5,7] assuming that the conductivity is provided by a small fraction B0.1% of the total carriers with long scattering time >1013 s. However, it is difficult to reconcile this conclusion with the behavior for high frequencies which supports that the scattering time is usually B1015 s and all available electrons participate in conduction. To account for these anomalies the possible presence of a collective mode, as in a charge density wave conductor, or superconductor, was suggested [9].
3. Model of resonant transport In explanation of the low-frequency anomaly in doped polymers in the metallic phase, their chain
morphology is very important. These materials are strongly inhomogeneous [10] with ‘‘crystalline’’ regions within which polymer chains are well ordered (Fig. 3(a)). When the IMT is approached delocalization first occurs inside these regions. Outside the crystalline regions, the chain order is poor and the electronic wave functions are strongly localized. Therefore, the crystalline domains can be considered as nanoscale metallic dots embedded in amorphous poorly conducting medium [11]. The metallic grains remain always spatially separated by amorphous regions, and, therefore, direct tunneling between grains is exponentially suppressed. The intergrain tunneling is possible through intermediate localized states in the disordered portion with strong contribution from resonance states whose energy is close to the Fermi level (Fig. 3(b)). The dynamics of resonance tunneling can account for the frequency-dependent anomalies in the conductivity and dielectric constant of the metallic phase of these doped polymers. We assume that grains have N> chains densely packed over NJ repeat units yielding N> NJ units cells in each grain. Neglecting intergrain coupling leaves the electronic levels inside grains quantized with mean level spacing DE ¼ * F ÞðN> NJ Þ1 ; where Nðe * F Þ is the density of ½Nðe states per unit cell. For metallic doped polypyrrole * F Þ ¼ 0:8 PPy(PF6) [10]: N> E3 8; NJ E7; Nðe states/(eV ring) and DEE7:4 meV. For doped metallic polyaniline PAN(HCl) [10], N> E9 12; * F Þ ¼ 1:1 states/(eV 2 rings), and NJ E7; Nðe DEE1 meV.
Fig. 3. (a) Schematic view on structure of polyaniline and polypyrrole. The lines represent polymer chains. The dashed squares mark the regions where polymer chains demonstrate crystalline order. (b) The electrical coupling between metallic grains is provided by resonance tunneling through localized states in the amorphous region.
ARTICLE IN PRESS V.N. Prigodin, A.J. Epstein / Physica B 338 (2003) 310–317
We assume that the electronic states are delocalized inside grains and electron’s dynamics is diffusive with diffusion coefficient D ¼ v2F t=3: This diffusive behavior is restricted to times smaller than the time, tT ; for a charge carrier to cross the grain (Thouless time), tT ¼ L2J =D; where LJ B5 nm is taken for the size of a typical grain [10]. At frequency otT b1; the system should show bulk metal behavior and sðoÞ is given by the Drude formula sðoÞ ¼ ðb=ð4pÞÞ ðO2p tÞ=ð1 iwtÞ; O2p ¼ 4pe2 n=m;
ð1Þ
where Op is the unscreened plasma frequency and b is the degree of crystallinity (usually, t50% crystallinity). The Drude dielectric constant is eðoÞ ¼ 1 bO2p t2 =ð1 þ ðotÞ2 Þ
ð2Þ
and is negative for o5Op ; eðoÞB bðOp =oÞ2 : Optical data [3,5] show that at high-frequencies b0.1 eV the system indeed is metal-like (Fig. 2(a)) which we attribute to the above metallic island response (1,2) (Fig. 4). For PAN(CSA) metallic samples [3,5] (Fig. 2(a)), the corresponding parameters are Op E2 eV; tB1015 s and tT is estimated to be B5 1014 s. We note that the condition for applicability of the grain model, t 5tT 51=DEB1013 s, is fulfilled. The high-frequency Drude response [3,5] transforms with decreasing frequency into dielectric behavior at a frequency 1=tT : For otT 51;
electrons follow an external field and the conductivity is purely capacitive: sðoÞ ¼ ioe2 bNðeF ÞL2J
ð3Þ
with positive dielectric constant given by the polarization of grains: eðop1=tT ÞBbðOp LJ =vF Þ2 :
ð4Þ
The behavior (3,4) is in good agreement with indistinguishable experimental results for both dielectrics near the IMT and conductive phases at high and intermediate frequencies (0.01–0.1 eV) (Fig. 2). Whether the lower-frequency behavior is metallic or dielectric depends on the intergrain coupling. Each grain is coupled to other grains by 2N> independent chains through amorphous media. For simplicity, we assume the two nearest grains are electrically connected by BN> =z chains, where z is the number of nearest neighboring grains. In the metallic phase, the intergrain coupling leads to broadening of quantized levels in the grains, dE ¼ 2N> g DE; where g is the transmission coefficient between grains through a single chain. The IMT occurs when dEBDE (Thouless criterion of IMT [12]) and the critical chain-link coupling gc satisfies 2N> gc ¼ 1:
ð5Þ
For PAN(HCl) and similar PAN(CSA) this yields gc B102 : If gogc ; the system is a dielectric and behavior (4) is retained for all otT 51: However, on the metallic side (g > gc ) electrons are delocalized over network of grains and their low-frequency motion is a random hopping among the grains. The hopping between the grains is a quantum process and can be described by mean transition frequency W : Introducing the mean distance between the centers of neighboring grains, R ðbBðLJ =RÞ3 ), the corresponding diffusion coefficient D3 and the macroscopic conductivity of network are D3 ¼ R2 W ;
Fig. 4. eðoÞ for the chain-linked granular model in the metallic phase. Inset: Re sðoÞ:
315
sðoB0Þ ¼ be2 NðeF ÞD3 :
ð6Þ
On approaching the IMT from the metallic side W tends to 0 as [13]: W ¼ ðDE=ð2zÞÞ exp½22pðgc =ðgFgc ÞÞ1=2 :
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In the metallic phase the hopping frequency W is related to the above model parameters as W ¼ dE=ð2zÞ ¼ ðN> =zÞg DE;
ð7Þ
and the whole system can be represented as a network of random conductors. The nodes represent the grains where randomization of electronic motion happens. Combining all together, the conductivity (6) also can be written in the simple form sð0Þ ¼ ðe2 gÞ ðN> =zÞ ð1=RÞ:
ð8Þ
Here, the first brackets represent the conductance of a single intergrain chain-link, the second factor in brackets is a number of chains connecting neighboring grains, and R in last brackets mimics the period of the grain network.
4. Average vs. typical transmission Thus the problem of transport in the metallic phase far from the IMT is reduced to the study of the average transmission coefficient g in (7) for a chain of finite length. For direct tunneling between grains g ¼ exp½22L=x; where L is the length of chain connecting neighboring grains and x the localization length. For 50% crystallinity L is of the same order as a grain size LJ ; i.e., LB5 nm. For weak localization, x ¼ 4l, where l ¼ vF t is the mean free path. For PAN(HCl) parameters assuming vF B3 107 cm/s [9] and tB1015 s; xB1:2 nm. Therefore, gB104 and direct tunneling is essentially suppressed. However, the transmission coefficient is unity for resonance tunneling (Fig. 3(b)). The probability of finding a resonance state is proportional to the width of the resonance level g: For constant x the resonance state needs to be in the center of chain, therefore, gBð1=tÞexp½2L=x: As a result, the average transmission coefficient is determined by the probability of finding the resonance state at the center, i.e., /gS ¼ ðgtÞBexp½2L=x: For PAN(HCl) we have /gSB102 which is close to the critical value for IMT gc ¼ 1=ð2N> ÞB102 : Thus, the probability of a resonance state is small B102, and resonance tunneling is not taken into account for a single chain. However, as the
number of interconnecting chains of a given grain with others is large B 100, resonance coupling occurs between grains. A principal difference between direct and resonance tunneling is the time for tunneling. Direct tunneling that occurs in a conventional granular metal is an almost instantaneous process, i.e., its characteristic time is the scattering time t: Resonance tunneling that is anticipated to be in the metallic polymers shows a delay determined by the level width g: The frequency-dependent transmission coefficient g ¼ gðoÞ for resonance tunneling is given by a generalization of the Bright–Wigner formula [14] gðoÞ ¼ ½1 io=g1 :
ð9Þ
With (6–8), we find that in the region o5W ; sðoÞ can be written in the standard Drude form sðoÞ ¼ ð1=ð4pÞÞop2 t1 =ð1 iot1 Þ
ð10Þ
with op being ‘‘the plasma frequency’’ determined by the frequency /W S of intergrain hops ðop =Op Þ2 ¼ ðb DEtÞ ð/gSÞ2 ðR=lÞ2
ð11Þ
and the relaxation time t1 ¼ ð1=gÞ ¼ t/gSB t exp½L=x determined by the Wigner transmission time. The overall frequency dependence of dielectric function and conductivity in the metallic state of polymers are shown in Fig. 4.
5. Conclusions The dependencies (1–4,10) (Fig. 4) agree with experimental observations (Fig. 2). In the metallic phase at low frequencies (o0.01 eV) the studies [3,5,7] show that the polymers behave again like a Drude metal with anomalous long relaxation times and with very small plasma frequency (Fig. 2). Microwave experiment and optical data yield t1 B(1013–1012) s. Being compared with (1,10,11) effectively corresponds to the decrease of electron concentration by a factor ðop =Op Þ2 B 103 for R=lB30; DEtB102 and the increase of ‘‘mean time of free path’’ by a factor
ARTICLE IN PRESS V.N. Prigodin, A.J. Epstein / Physica B 338 (2003) 310–317
t1 =t ¼ 1=/gSB102 to be t1 B(1013–1012) s. Further, op B1=t1 ; which was observed in Ref. [7]. The temperature dependence of the DC conductivity in the metallic phase follows from the resonant tunneling through the strongly localized states in the amorphous regions of the metallic polymer. With increasing temperature, phonons increase the localization length of the states in the disordered regions thereby increasing the resonant transmission rate between grains and increasing the conductivity. As a result, the low-frequency part of electromagnetic response is shifted with increasing temperature to a range of higher frequencies as it experimentally is observed in Ref. [5]. In summary, we propose that resonant quantum tunneling among the chain-linked network of nanoscale grains is the mechanism of charge transport in metallic doped polymers. The model explains key experimental findings of eðoÞ; sðoÞ; very long relaxation time t at low frequencies, and low fraction of electrons contributing to DC-transport.
Acknowledgements Supported in part by the Office of Naval Research and Army Research Office.
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