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Fluctuation-assisted transport in polymers
ELSEVIER
Synthetic Metals 78 (1996) 169-176
Fluctuation-assisted transport in polymers Maria C. Anglada a, Nuria Ferrer-Anglada a, Josep M. Ribo ‘, B. Movaghar ’ a Departaarnent
de Quirnica Organica, University of Barcelona, Marti i Franques 1, Barcelona E-82028, b Soms Centre, Department of Chemistry, Universily of Leeds, Leeds LS2 9JT, UK Received
30 June 1995; revised
12 September
1995; accepted 6 November
Spain
1995
Abstract We consider for the first time the explicit role that multi-mode fluctuations play in determining the transport characteristics of inhomogeneous
conductorsandinsulators.Basingourselveson rigorousanalyticalandthermodynamic calculationsandrecentMonte Carlosimulations, we havebeenable to identify experimentallythe novel exp(aT’), k> 0, behaviourin the low-temperature transportregimeof pyrrole-based polymers.The variablerangehoppingandfluctuation-assisted granulartransportregimescanbeclearlydifferentiatedfrom eachotherat very low temperatures usingdifferent pyrrole compounds. The apparentuniversalityof exp[ - ( T,/T)S] -type lawscanbe relatedto the fact that variablerangehoppingandvariablerangetunnellinggive riseto similarlawsin intermediatetemperatureregimes.The excitation modes whichassistthecarriersto crossthe barriersat low temperatures canbeidentifiedasacousticphononmodesratherthanthechargefluctuation modes invoked in doped polyacetylene. Keywords:
Transport; Doping
1. Introduction Ever since the observation that doped polymers of the polyacetylene category exhibited truly metallic conductivities, reaching values of order LTN lo5 S cm-’ [ 1,2], experimentalists and chemists have been strongly motivated and encouragedto synthesize new and better materials with particular emphasison stability and technological exploitability [ 31. Improved stability in air has been achieved with polyaniline and polypyrrole [ 4,5], and the conductivity is lower, with c reaching values of order IO* S cm-‘, but this is good enough for a range of applications. Most of the applications of conducting polymers are basedon materialsor composites which are structurally inhomogeneousgranular or fibrillar in their morphology. Understanding the transport mechanisms in thesedisordered or partially disordered systemscan help us design new materials with novel technological applications. The transport data in doped polymers have exhibited novel universalfeatureswhich we shallbriefly summarizeby taking our own observations on doped polypyrrole [ 5,6] as representative for the entire category of materials [ 51. We shall demonstratethat this is a reasonableapproach. 2. New experimental data on pyrrole derivatives Fig. 1 shows the temperature dependenceof the conductivity of p-toluensulfonate-doped polypyrrole ( PPy,, PPy,,, 0379-6779/96/$15.00 ~~~Tf-l270-67~1
0 1996 Elsevier Science S.A. All rights reserved
and PPy,,,,,,) , poly (3,4-dimethylpyrrole) ( DMPPy) and poly (3-methoxy-4-methylpyrrole) (MMPPy) . These pplymers were obtained by anodic growth and acetonitrile solutions of the correspondingmonomersand their preparations and chemical characterization have been already described[ 61. It hasbeenreported that the structure of polypyrroles obtained at high and low current densitiesdiffer in their structure [ 71, and in this sensetwo films were obtained, one at a current density of 3 mA cm-* (PPy,) and the other one at 0.7 mA cm-* (PPy,,,). From the polymer film obtained at 0.7 mA cm-*, we obtained a pellet (grinding and pressingan evacuated sample, for 3 min at lo3 kg cm-*) ( PPypellet). Thesethree materialsPPy and MMPPy exhibit a finite T+O conductivity and can be said to correspond to systemsin which the Fermi energy is above the mobility edge EC (seeFig. 2). The lowest conductivity can however be seen to besmallin comparisonto the Mott minimum metallic value which is of order about IO3S cm- ‘. It can be further observed that the conductivity, which startsat a smallvalue, risesstrongly with temperatureto reach valuescommon to most dopedpolymers. Similar behaviour is exhibited down to 10 K in the work of Ref. [ 71 (see Figs. 2 and 8 of Ref. [ 41) though the data are not explicitly plotted in this way. The structure of the salts of polypyrrole with polystyrene sulfonate can be changedby the addition of dioxane to the polymerization solutions; this
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10- (a)
I
-10 0.2
I
I
0.3
i 0.4
0.5 T-1’4 K-1’4)
,:’ ,:’
3.0
I
I
4.0
I
I
I
1
5.0 6.0 In [T(K)1 Fig. 1. Temperature dependence of conductivity for several p-toluenesulfonate-doped polypyrroles (see text). (a) Plot of In c vs. T-l“‘. (b) Plot of the activation energy EA( r) (arbitrary scale values to avoid overlapping between the curves), The activation energy is defined by In o = EA( T). The effective Mott exponent y, assuming an exp( - Tey) dependence, is shown in various temperature regimes.
results in the family of curves shown in Fig. 4 of Ref. 141, showing a continuous change from tunnelling metal-like to hopping and insulating-like behaviour. These facts seem to demonstrate that, when~the morphology of the substance is taken into account, one not only has ‘disorder’ but also granularity. Small metallic grains are separated from each other by energy barriers [ 81: the conductivity at low temperature is small because the barriers are limiting the magnitude of the conductivity. This type of behaviour is common to doped inhomogeneous systems. Indeed truns-polyacetylenes exhibit similar curves for intermediate doping levels [ 91. Comparing PPy,,,,,, with PPY,,~ films, from which they are derived, not only is a decrease in conductivity observed, but also a shift of the ‘hopping region’ to’higher temperatures. We believe that this can immediately
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be related to the existence of intergranular barriers: we now have smaller metallic grains and therefore a higher resistance. Again, the difference in conductivity between PPy, and PPy,,, can only be understood as being due to differences in structure. DMPPy exhibits what appears to be standard hopping conduction throughout the measured temperatures. We conclude that transport in DMPl?y is below the mobility edge (see Fig. 2) ; there is no way for the carriers to traverse the entire system without exchanging energy with the phonon bath. Despite these differences apparent in the low-T region, all four curves shown exhibit, in the intermediate temperature regime, what appears to be a universal exponential T-” hopping law. As a consequence of this apparent universal hopping behaviour, no attention has been paid to the actual details of the transport mechanisms, magnitude of the parameters, and in particular no serious attempt has been made to reconcile the inhomogeneous and the homogeneous (hopping) models. The present measurements and those of Ref. [ 41 go down to low temperatures, and give us the possibility of studying more closely the interesting and informative crossover between the tunnelling dominated quantum regime and the thermally assisted inhomogeneous transport regime. The results for this crossover are shown in Fig. 2. The remarkable feature here is the apparently new exp( 0”) behaviour. Naturally, as we go down in temperature, it cannot be totally excluded that the granular metal phases in Ref. [4] and those in the present data, eventually as temperature tends to zero, turn into true insulating states. This could occur as a result of ‘Coulomb glass’ formation, but seems unlikely. The zero-point (quantum) motion of the electronic and phononic system can, in principle, keep a finite conductivity state down to very low temperatures. There is however as yet no rigorous proof of the exact ‘zero-Tbehaviour’ in such inhomogeneous systems. Indeed it may in some circumstances even be possible to obtain the opposite behaviour, namely, superconductivity at extremely low temperatures in highly doped ‘compensated’ samples. In this paper we are concerned with the way ‘real’ excitations enhance the pathways and add to the zero-point motions of the coupled electron-phonon system to increase the conductivity as a function of T. We are concerned with the region above the single-particle electronic mobility edge where a continuum of excited states appears to exist (Fig. 2). We shall argue in this paper that a universal behaviour can be derived, and that the exp( aT”) law is indeed the right way of
Fig. 2. Typical potential profile representing a granular systemprojected in one dimension. The Fermi energy I$ is represented by the dashed lines and can be above (m) or below (I) the mobility edge represented by EC.
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interpreting this transport regime. We shall base ourselves on rigorous mathematical and thermodynamic methods. We shall also demonstrate, in the following theoretical discussion, that transport which is often considered to be Mott hopping is most probably thermally assisted tunnelling over barriers. We shall call this the variable range tunnelling (VRT) mode.
3. Models of transport in disordered systems 3. I. Hopping conduction
To begin with, it is generally accepted that, at low doping levels and at low enough temperatures, conduction is by hopping, from defect to defect. As shown by Miller and Abrahams [ lo] the phonon-assisted hopping rate Wis of the form:
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171
become important in transport processes where many degrees of freedom participate in achieving the carrier transfer. Examples are complex organic molecules (Marcus theory [ 141) and indeed most materials exhibiting the Meyer-Neldel rule [ 121, We argue that most organic materials exhibiting hopping-like transport with high effective activation energies are also necessarily in this category because they are complex ‘large’ objects in which the atomic and electronic coordinates are strongly coupled. In the Mott approach to conduction, it is the sequential or percolative competition between space and energy which gives rise to the T 1’4 laws . The so-called VRH transport can be given a mathematically rigorous treatment as shown in Ref. [ 151. 3.2. The tunnelling approach
expressing the fact that, in order to reach another point in space, a particle normally has to tunnel and absorb energy from its environment. In principle, all possible paths can be sumed from initial to final state and then the prefactorcontains the total number of ‘paths’ [ 11,121 that the particle can take and this makes v a complex but very important quantity. The analysis in Ref. [ 71 has shown that, for large energy changes compared to kT, it can be approximately written in terms of a thermodynamic sum:
There are however other ways to obtain this type of law. The competition between space and energy need not always be due to percolation, as we shall show below; in many systems, the following picture is much more reasonable. The particles encounter Coulomb-like barriers as shown in Fig. 2. The barrier tunnelling rate T(E) depends on the energy E because, as one can see, for higher energies, the barrier is shorter so higher energies mean shorter barriers. Indeed for a general potential V(r) , the standard quantum mechanical approximation referred to as ‘WKB [ 161 gives us the transmission velocity:
u(E) = I, exp[S(E)];
T(E) = (2Elm) ‘I2 exp{ - (2mlfi’)
WC= v exp( -~cYR,)
exp[ - (Ej-Ei)/kT]
S(E) = [~(E-E,)/cLJ,]~
(1)
(2)
where 4 is related to an interaction volume, o, is a typical dominant mode and y an exponent which depends on the dominant modes. This expresses the fact that, for a large number of paths, or when many degrees of freedom are involved, it is the free energy change of the ‘process’ which is the correct measure for the rate of the transition and not the simple Fermi golden rule quantum mechanical transition probability. Free energy changes however include both enthalpic and entropic changes. The latter is measured by the number of different ways the transition can take place and enters the prefactor in front of the activation term. Transition (1) must be understood as linking many different configurations which are only characterized by the fact that a carrier initially in the vicinity of Ri ends up in the vicinity of Rj in a time of order l/ Wjy In the usual Miller-Abrahams hopping process [lo], where a single or only a few phonons are exchanged, the thermodynamic considerations are obviously not necessary, and we can evaluate the rates in the ‘Fermi golden rule’ approach. The standard Mott hopping [ 131 or variable range hopping (VRH) approach gives rise to the well-known hopping conductivity laws: CT(T) =a,exp[
-(T,,/T)$];
O
(3)
derived with a prefactor Y which is a constant and of the order of a phonon frequency. The sums of the thermodynamic paths
l/2
dr [V(r)
-l$‘z}
(4)
For a pure Coulomb barrier, the energy ‘dependence of the transmission probability is evaluated in Landau Lifshitz quantum mechanics [ 161. Generalizing this expression to allow for different (but similar) types of barriers we write for the energy-dependent velocity: T(E)=(2E/m)“2exp{[E,/IE-EFI
+El(n)]@}:‘,
(5a)
E,>E,(n)
where E,(n) is the mobility edge (see Fig. 2) and we obtain for the linear response conductivity: I’ 1 . cc (T = e2db a(T=O)
dEN(E) =e’N(E,)D,
exp[S(E)]
( - Sf/6E) T(E)“~
exp{ - [f$,lEr’(n)]‘}
.,:
(5b)
(5i)
where N(E) is the one-particle electronic density of states, d,, is a barrier width, f(E) is the Fermi function, and D, is the Mott
diffusivity
(minimum
metallic)
with the. factor
e2N(EF) D, of. order e2/ (Aa), where a is a typical lattice spacing. In the expression for T(E), E,, is a measure of the barrier height,, and El (n) .is a measure of the inverse barrier
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width and depends on the doping level IZ, and is related to the ‘free’ kinetic energy above the mobility edge&(n) . The form of E, (n) can be written as fiiD( n) la*, where D(n) is the true zero-Tdiffusivity and a function of the doping level n. Indeed here, we expect D( n) N D,( n - n,) lnc, where D, is the Mott value and II, is the critical doping concentration for a metalto-nonmetal transition, In a system for which the doping level is below the mobility edge it
(6)
E, = (2m/fi*)g,
El = 0 and p= l/2. A simple steepest descent evaluation for E, = 0 will give rise to a Mott-type law
with s = l/3. Barriers in doped polymers are basically of two types: chain ends and Coulomb-like due to space charge fluctuations. If we took a configuration average of such barriers, then the generalized Coulomb form assumed here is a good representation. Returning to the generalized forms (5), we can see that the total conductivity now includes the path summation factor eScE),which measures the total number of dynamical pathways which the particle can use to cross the barrier. In this model we are no longer talking about a transition between single quantum levels, but rather between groups of levels forming metallic-like reservoirs separated by Coulomb-like barriers. Now this is precisely what is expected when there is a weak localization on a chain, or if the doping concentration is increased so that clusters of states develop in which electron eigenstates are delocalized or only weakly localized. The strongly localized levels and the charge carriers added to the system by doping will initially act as single well-defined quantum levels Ei and gradually, with increasing concentration, form clusters, then metallic islands and, finally even at the highest concentration, when local metallicity has been achieved, we can expect some granularity to remain in the system. The degree and nature of the remnant granularity will depend on the type of polymer and the intrinsic disorder in the system (chain lengths, fibre structure, etc.). We conclude that a Mott-type law can also result from VRT and not just from VRH! In the present category of systems, both can exist and coexist, depending on the concentration of dopants, nature of intrinsic disorder and temperature. Experimentally it is not immediately obvious how to tell them apart. This is why most authors only use the Mott law. Differentiating VRH
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from VRT can only be done through a systematic quantitative analysis, which pays attention to the detail of the transitions and the magnitude of the prefactors. Electric field (F), magnetic field (B) and frequency dependences (w) can also be used to discriminate between these mechanisms. We shall summarize the expected behaviour later in this paper.
4. Fluctuation-assisted transport methods: the temperature dependence of prefactors 4.1. The temperature laws
The mobility and conductivity, as given by Eqs. (5), encompass both hopping and tunnelling mechanisms in situations where the initial and final energies are in the same range (about /CT) and can be considered to be a universal form of describing transport through insulating regions in disordered inhomogeneous materials, when many levels and many degrees of freedom can participate. The above approach bears some similarity, and is basically in spirit, in agreement with Sheng’s ‘fluctuation-assisted tunnelling’ method [ 81. The difference with Sheng’s method is in the way the fluctuations are treated. We have introduced an entropy factor to count the number of paths. Sheng observed that fluctuations are important when there are barriers, irrespective of the system. He considered that the fluctuations in the barrier width result from the fact that fluctuations cause the amount of charge in each reservoir to change in time and, therefore, an internal field FT to develop, which obeys a Gaussian distribution and adds or subtracts to the external field. This observation is true, but the stored ‘plasma-like’ charge density fluctuations, which occur in the metallic grains, constitute only a subset of the total number of degrees of freedom that help to lower the barrier for a particle undergoing tunnelling. Many other single-particle-like modes can modulate the barrier height and shape during the time the particles attempt to cross. The entropy factor can in principle allow for all the phonon and electronic degrees of freedom. The most elegant way to treat it is to first evaluate the energydependent function S(E) as was done in Ref. [ 111. Different types of excitations give rise to different exponents, as shown in Ref. Ill]. One of the mathematically rigorous methods of evaluating fluctuation-assisted transitions through barriers has been given by Bruinsma and Bak [ 171, These authors use the functional integral technique and apply the WKB approximation to tunnelling. They have treated the case when the barrier heights are subject to local fluctuations. Their method can be generalized to allow for barrier width fluctuations. The width fluctuations would then include the charge and internal field changes considered in the Sheng model. Thus Sheng, and Bruinsma and Bak address the same problem, but only include a different subset of the total number of degrees of
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’
-2’
0
’ 100
’
’ 200
’
’ 300
’
’ 400
’
’ 500
T3/2
Fig. 3. Plot of In D vs. T3’*in the low-temperature range for the polypyr~oles of Fig. 1.
freedom. Naturally each distinct physical transport situation is characterized by particular classes of dominant excitations, but they can vary strongly, have mixed modes and, most of all, there can be complex dimensionality (fractal) behaviour, resulting from a combination of disorder and low dimensionality. Bruisnma and Bak also restrict their analysis to the situation where the tunnelling particle is energetically far from the top of the barrier so that the ‘classical pathways’ are not included. The thermodynamic approach developed in [ 111 is to some extent phenomenological, but it has the advantage of being simple to apply and including in principle the total change in the number of variables with no restrictions on the tunnel energy. It also gives one an intuitive ‘chemists’ way’ of visualizing what is in general a very complex manybody transfer process. It is therefore very satisfying to see that the thermodynamic method roughly agrees with the first principles methods [ 17,181 when considering similar modes. The correctness of the approach has been beautifully demonstrated recently using exact Monte Carlo many-particle simulations of surface diffusion [ 191.
173
dard hopping law. The analysis of the results, reported in Ref. [ 61 for several perchlorate-doped polypyrroles, show that at low temperatures they also follow this behaviour (see Fig. 4). In the case of the polypyrrole polystyrene sulfonate salts exhibiting finite conductivity as T+ 0 in Ref. [ 61, we observe similar behaviour (Fig. 4). These results clearly demonstrate that this novel temperature dependence is connected to the structure of the sample at the nanoscale to microscale. The dependence appears at higher temperatures for more inhomogeneous samples, i.e. compare, in Fig. 3, PPY,,~,,, with PPY~,~ which is more ‘compact’, and PPy,,, with PPy, which shows a different structure on a nanoscale [7]. In all cases observed, the value of k appears to decrease to roughly k N 1 at higher temperatures. We can understand the origin of this behaviour by looking at the fluctuation-assisted model discussed above. At low temperatures, the integral (6) is dominated by the entropy term multiplied by the Fermi function so that a-exp(aT)Y’(l-Y)
(8)
In polypyrrole it follows that y= 3/5. Now we can attempt to relate this exponent to the type of modes which are dominating the fluctuations in this regime and in this particular system. From Ref. [ 181 and thermodynamic entropy calculations, we observe that for acoustic phonons in d dimensions where d can be fractal and in the range l
@a>
optic phonons will give (see Ref. [ 91 for details of how to treat the n-optic phonon rates) : S(E) =Eln(AlG)
(9b)
where h is the coupling and 6 the phonon frequency. For electronic excitations:
4.2. Low-temperature transport in granular structures
The interesting point about granular structures is that almost any value of conductivity can be acheived up to about lo4 S cm-‘, yet the system still has a conductivity which increases with temperature at low temperature because there are barriers which can be thermally overcome [ 17,181. The intermediate temperature dependence cah look like hopping, and this then confuses the mechanism. Our results show that, at low temperatures, systems exhibiting a finite T=O conductivity exhibit a novel temperature dependence which can be written as a(T) =(T,exp[(aT)k]
(7)
where k= 3/2 (see Fig. 3). For comparison the insulating system DMPPy is shown to obey what appears to be a stan-
-6
t t 0
100
200
300
I-= Fig. 4. Plot of In u vs. T3’*in the low-temperature range for several of the polypyrroles of Ref. [4] (data from Figs. 2 and 8 of Ref. [4] ): 0, poly(3methylpyrrole); x, poly(3-nonylpyrrole); @, poly(3-heptadecylpyrrole); 0, poly[3,4-(buta-1,4-diyl)pyrrole perchlorates] ; + , polypyrrole polystyrene saltsobtained from aqueous solution, and A, from 10% dioxaneaqueous solution.
174
y= l/2
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(9c)
At low temperatures in polypyrrole ( 15-35 K), we expect the acoustic modes to dominate over the electronic excitations so that the effective dimensionality of the spectrum is d = 12/ 5, which is indeed a reasonable number for a polymer. Optic modes and electronic fluctuations will contribute at ,higher temperatures. From the change in the curvature at higher temperature (T> 35 K), we conclude that electronic modes and optic modes are determining the magnitude of the fluctuating tunnelling fields in this regime until, eventually, thermal fluctuations are strong enough to allow VRT to take over (when kT> hq+ ). This in principle can be checked by measuring the low-temperature heat capacity of the system which should behave as C, N Td for phonons. Note that twolevel systems associated with glass-like modes can also contribute and these would behave like one-dimensional phonons. Ultra-sound absorption may be able to detect glasslike excitations due to moving chains or clusters and is an interesting experiment to carry out at low temperature. They could tell us about low-energy structural modes. ‘Organized’ space charge fluctuations as opposed to single-particle fluctuations, as considered by Sheng, can also contribute and assist the tunnelling of carriers, but then the direct singleparticle electronic excitations in the reservoirs also give rise to strong local fields, so that the decision on which modes dominate has to be more global, i.e., we have to ask: Is the dominant mechanism electronic or phononic? We are here referring to real excitations caused by temperature. There are also the zero-temperature quantum electronic and quantum phononic (zero-point motion) processes which help assist the carrier motion at T= 0 and cause quantum delocalization and tunnelling, as considered for example by Emtage [20] and one of the present authors [ 211. The quantum manybody processes are part of the determination of the zerotemperature conductivity and will not be discussed further in this paper. They could be the reason for delocalization and, for example, also give rise to superconductivity [ 211, Before turning to a first-principles quantum statistical derivation, it is worth pointing out that, according to thermodynamic fluctuation theory, the fluctuation in ‘local entropy S/ kB’ near the tunnelling transition point is (AS’) = CJk,, which is the specific heat divided by the Boltzmann constant at constant pressure. For acoustic phonons, C, scales as T3 so that the prefactor behaves as exp( T/T,) 3’2, as found in this paper. The same argument would suggest that the temperature dependence observed in highly doped polyacetylene is due to the electronic component of the specific heat, not for the initial increase which again must be due to phonons, but for the later T”2 part of the curves [ 91, i.e., when T> 30 K. The above suggests that the low-temperature transport behaviour in’ inhomogeneous systems might indeed have a very simple’and appealing solution. Though evidently the role of thermodynamic fluctuations has been established, as demonstrated-in several ways above, it is nevertheless important to give rigorous derivations using the actual quantum
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mechanical mechanisms. Thermodynamic theory, as a firstprinciples theory, has a hierarchy of solutions. The fluctuation theory method, for example, assumes that the modes are contributing to the transitions in a completely random phase manner, basically by lowering the ‘heat’ required to effect the ‘transformation’ of going from the initial to the final state. First-principles approaches however begin by describing the actual many-body transitions and then adding them up. This is a very difficult task and should be restricted to a ‘small subclass’ of processes which may or may not be adequate in a given temperature range. Progress in identifying the type of transitions dominating the low-temperature transport in pyrrole will, hopefully, help us to unravel the many-body transport modes influencing other and more complex inhomogeneous systems. Electronic modes, for example, must depend on the doping level. This is certainly consistent with the observation in highly doped polyacetylene, but a detailed correlation between doping level and the appearance of a linear to sublinear value of k has yet to be established. Let us now go back to the first-principles calculations of Ref. [ 141 and the work of Bruinsma and Bak [ 171. 4.3. Comparison of thermodynamic andfirst-principles methods Here we obtain in WKB, the tunnelling amplitude to reach a final energy E’, A( E,E’) where co
A(E,E’) =A,(E)
dt/2nexp[i(E-E’)tlfi-fi-1
The first termA, is the zero-temperature tunnelling amplitude. The remaining factor results from fluctuating fields which modulate the barrier height. These fields are represented by the dynamical coordinates f(t) = Z,x,,( t) , and with d
g(W)
=
dX/2W?lV,(X)K-‘(4 I
0
eXp[
-
I
d.?’ (mw/hK(x”))]
(lla)
Here VI (x) is the coupling strength of the coordinates to the barrier at the point X, and K is the inverse WKB ‘tunnelling length’: K(X)
=h-‘{2m[V(x)
-E]}“2
(lib)
Each coordinate X, represents a possible dynamical mode which changes the barrier height at a point x in space, with coupling strength V,(x) . The model can be generalized to
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take into account width fluctuations as well. The probability of crossing the barrier is found by taking the thermodynamic average of the modulus squared amplitude integrated over all final energies [ 151, For acoustic phonons, the coupling parameter, g(w) N w so that for d = 3, Eq. ( 10) would predict an enhancement of theform [16,17]: fl N exp ( UT) 4, acoustic phonons (12) g N exp( bT2), Gaussian random force (13) whereas the thermodynamic methods would give cr w exp( bT3’2) for acoustic phonons, in agreement with experiments in pyrrole. There is therefore a discrepancy in the value of k between the first-principles and the thermodynamic derivation, but this is not surprising in view of the approximations made in both derivations. In the thermodynamic method, the linkage between energy change and transfer of charge is somewhat ‘phenomenological’ and can be termed a ‘random phase approach’ since we are not actually saying how the entropy change is deterministically intervening in the transition. Going back to Eq. (2)) which is one step more fundamental than the simple random phase method, a comparison with the first-principles method also allows us to interpret the meaning of the constant q/w, in Eq. (2). We find, for example, that, in the random Gauss model, q/w, can be interpreted to be truly an inverse typical oscillation (excitation) mode multiplied by a term which is of order (barrier width/inelastic length) [9]. This means that the wider the barrier in comparison to the inelastic length, the stronger is the multiphonon enhancement. We conclude that the low-temperature conductivity of inhomogeneous conductors of the doped polymer category can be analysed in terms of fluctuation-assisted tunnelling. The first-principles exponent k is not always easy to relate to the corresponding simple thermodynamic results, but this is not surprising in view of the model assumptions, temperature range, and the complex nature of many-mode-assisted transitions. Our thermodynamic method (Eq. (2) ) predicts that k> 31’2 for acoustic phonons, 0
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enon of ‘fluctuation-assisted conduction’ at low temperatures in inhomogeneous media exhibiting finite d-c. conductivities. Both first-principles theory and thermodynamic arguments confirm this point of view. From the value of the exponents, the modes which are most effective in assisting the carriers to cross the tunnel barriers in a given temperature range can be determined. Low-temperature specific heat data can tell us the effective dimensionality of the excitation spectra. It is fairly safe to presume that every mode which can be excited in these strongly coupled electron-phonon systems will also help the transfer of charge from one point to another, with a given efficiency coefficient which will depend on the nature of the ‘barriers’. In many such disordered systems, it is often difficult to tell the difference between fluctuation-assisted tunnelling at higher temperatures and Mott-type hopping. With the help of Eqs. (5) and (6) we can now shed some light on this question. It follows that hopping will give rise to zero conductivity as T-, 0, but there are also other differences which may be exploitable. These are related to the high electric and magnetic field behaviours of the two models [ 211. Thus, for example, it is well known how hopping from site to site gives rise to high-field positive magnetoresistance [ 221. Clustered levels or energy levels of reservoirs are more spread out in space and will react ( ‘shrink’) to smaller (than a single localized site) magnetic fields. This can be checked. The barrier potential will be modified by an additional term of the form + ( eBr)2/2mc2. If we are close to a metal-insulator percolation transition, then the magnetic-field-induced upward shift of the Fermi energy E,(B) can lead to ‘magneto-delocalization’ and therefore to a negative magnetoresistance [ 221. The Fermi energy shift is due to the orbital and spin energy redistribution and is of order pgB/4, important at high magnetic fields. Note that the spin shift is isotropic but the orbital contribution is anisotropic in a polymer, so the two can be differentiated experimentally. The negative magnetoresistance effect is assisted by the fact that the quantum diffusivity also increases with the magnetic field, and that consequently the mobility edge is lowered. Indeed in Eq. (5a), El (n,B) increases with B in a way predicted by scaling theory [ 221. The electric field dependence of hopping and the present model are also different and can be used as a test of the mechanisms. In hopping, the electric field lowers the activation energy barrier by a factor exp(eFRqIkT) . In tunnelling, the field lowers the barrier potential by a term of the form - eFr along the tunnelling trajectory [ 211. This gives rise to very different high field behaviour as can be seen from Eq. (4). Finally, there is also the a.c. conductivity. The hopping a.c. behaviour is well known and can be rigorously evaluated theoretically [ 151: wp (0
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ogeneous resistor network model is used. The cluster or metallic reservoir with charge density II, will contribute an a.c. conductance which in the metallic limit will be of the Drude form n,e’/ [ m( iw + l/ 7) 1, and the barrier region will contribute a tunnelling resistance (R) and capacitance C, N qE,/r,,, where r, is the tunnelling length, plus of course the thermal fluctuations as considered by Sheng, but these are negligible in the present context. Constructing the resistor network, it is seen that there will be distribution of tunnelling resistors (R}, P(R), with impedance Z(o) = (l/R + iwC,) so that, as frequency increases, the conductivity will rise up to the point where the smallest barriers have been reached. Then the a.c. conductivity will start to decrease as a result of the Drude response of the metallic domains. The turnover frequency will depend on the conductivity relaxation time of the ‘delocalized’ carriers in the reservoirs, and can be as high as lOI Hz. Finally, we observe that plasma-like modes in the conductivity should also manifest themselves as reqectivity dips due to the small metallic clusters. We conclude that with the present formalism, and the formalism of hopping conductivity, both of which can now be said to form very complete representations of transport in disordered media, we are in a position to discriminate between the various transport models and assign universal laws to the various transport coefficients. In this paper we have, in particular, been able to demonstrate that complex thermodynamic fluctuation processes give rise to systematic observable laws in low-temperature transport experiments on pyrrole, and confirm what is often intuitively suspected but rarely quantified, namely, that “transport in disorderedmedia is a result of an interplay of many degrees of freedom and this is also the reason for the ‘anomalous’ prefactors”. We have shown that it is possible to track down these degrees of freedom and learn something very useful from the detailed low-temperature rise in conductivity even in such highly complex systems as doped organic polymers. We believe that the present analysis can be extended to other systems and in particular for example to the study of inhomogeneous regimes of high T, superconductors. It may eventually be possible to distinguish electronic from phononic quantum fluctuations.
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Acknowledgements The authors are grateful to Professors A. Yelon and L. Lewis for sending a preprint of their manuscript before publication and to Professor G. Wegner (Mainz) for emphasizing the importance of inhomogeneity in the description of the structural properties of conducting polymers. J. Clements of Leeds University is acknowledged for his help in preparing the manuscript. Part of this work was financed by the Spanish Government (DGlCYT PB 93-1257).
References [I] A.J. Heeger, S. Kivelson and J,R. Schrieffer and W. Su, Rev. Mod. Phys., 60 (1988) 781. [2] H. Naarmann and N. Theophilou, Synth. Mer., 22 (1987) 1. [3] S. Roth,Adv. Phys., 36 (1987) 385. [4] G. Wegner and J. Ruhe, Faraday Discuss. ChernSoc., 88 (1989) 333. [5] A.J. Heeger, Synth. Met., 55-57 (1993) 3471. [6] J.M. Ribo, MC. Anglada, J.M. Tura and N. Ferrer-Anglada, Synth. Met., 72 (1995) 173. [7] (a) D. Schmeisser and W. GGpel, Ber. Bunsenges. Phys. Chertr., 97 (1992) 32; (b) G. Paasch, D. Schmeisser, A. Bartl, H. Naarmann, L. Dunsch and W. G8pe1, Synth. Met,, 66 (1994) 135. [8] P. Sheng, Phys. Rev. B, 21 (1980) 2180. [9] Y. Nogami, H. Kaneto, H. Itoh, T. Tsasaki, N. Toyota, H. Ishiguro, A. Takahashi and I. Tsharmoto, Phys. Rev. B, 43 (1991) 829. [lo] A. Miller and H. Abrahams, Phys. Rev., 120 (1960) 745. [ 111 A. Yelon and B. Movaghar, Phys. Rev. L&t., 65 (1990) 618. [ 121 W. Meyer and H. Neldel, Z. Tech. Phys., 12 (1937) 588. [ 131 N.F. Mott, Electronic Processes in Disordered Materials, Clarendon Press, Oxford, 1984. [ 141 R.A. Marcus, Ann. Rev. Phys. Chem., 15 (1964) 55. [ 151 B. Movaghar and W. Schhmacher, J. Phys, C: Solid State Phys., 14 (1981) 859. [ 161 L. Landau and F. Lifshitz, Qlrarilron Mechanics, Pergamon Press, Oxford, 1958. [ 171 R. Bruinsma and P. Bak, Phys. Rev. Left., 56 (1986) 420. [ 181 H. Grabert, U. Weiss and P. Hanggi, Phys. Rev. kw., 52 (1984) 2193. [ 191 G. Boisvert, L. Lewis and A. Yelon, Phys. Rev. Letr., (1995) in press. [20] P.R. Emtage, J. Appl. Phys., 18 (1967) 1820. [21] B. Movaghar, Chem. Phys., 177 (1993) 751. [22] B. Movaghar and S. Roth, Syrlrh. Met., 63 (1994) 163.
Synthetic Metals 78 (1996) 169-176
Fluctuation-assisted transport in polymers Maria C. Anglada a, Nuria Ferrer-Anglada a, Josep M. Ribo ‘, B. Movaghar ’ a Departaarnent
de Quirnica Organica, University of Barcelona, Marti i Franques 1, Barcelona E-82028, b Soms Centre, Department of Chemistry, Universily of Leeds, Leeds LS2 9JT, UK Received
30 June 1995; revised
12 September
1995; accepted 6 November
Spain
1995
Abstract We consider for the first time the explicit role that multi-mode fluctuations play in determining the transport characteristics of inhomogeneous
conductorsandinsulators.Basingourselveson rigorousanalyticalandthermodynamic calculationsandrecentMonte Carlosimulations, we havebeenable to identify experimentallythe novel exp(aT’), k> 0, behaviourin the low-temperature transportregimeof pyrrole-based polymers.The variablerangehoppingandfluctuation-assisted granulartransportregimescanbeclearlydifferentiatedfrom eachotherat very low temperatures usingdifferent pyrrole compounds. The apparentuniversalityof exp[ - ( T,/T)S] -type lawscanbe relatedto the fact that variablerangehoppingandvariablerangetunnellinggive riseto similarlawsin intermediatetemperatureregimes.The excitation modes whichassistthecarriersto crossthe barriersat low temperatures canbeidentifiedasacousticphononmodesratherthanthechargefluctuation modes invoked in doped polyacetylene. Keywords:
Transport; Doping
1. Introduction Ever since the observation that doped polymers of the polyacetylene category exhibited truly metallic conductivities, reaching values of order LTN lo5 S cm-’ [ 1,2], experimentalists and chemists have been strongly motivated and encouragedto synthesize new and better materials with particular emphasison stability and technological exploitability [ 31. Improved stability in air has been achieved with polyaniline and polypyrrole [ 4,5], and the conductivity is lower, with c reaching values of order IO* S cm-‘, but this is good enough for a range of applications. Most of the applications of conducting polymers are basedon materialsor composites which are structurally inhomogeneousgranular or fibrillar in their morphology. Understanding the transport mechanisms in thesedisordered or partially disordered systemscan help us design new materials with novel technological applications. The transport data in doped polymers have exhibited novel universalfeatureswhich we shallbriefly summarizeby taking our own observations on doped polypyrrole [ 5,6] as representative for the entire category of materials [ 51. We shall demonstratethat this is a reasonableapproach. 2. New experimental data on pyrrole derivatives Fig. 1 shows the temperature dependenceof the conductivity of p-toluensulfonate-doped polypyrrole ( PPy,, PPy,,, 0379-6779/96/$15.00 ~~~Tf-l270-67~1
0 1996 Elsevier Science S.A. All rights reserved
and PPy,,,,,,) , poly (3,4-dimethylpyrrole) ( DMPPy) and poly (3-methoxy-4-methylpyrrole) (MMPPy) . These pplymers were obtained by anodic growth and acetonitrile solutions of the correspondingmonomersand their preparations and chemical characterization have been already described[ 61. It hasbeenreported that the structure of polypyrroles obtained at high and low current densitiesdiffer in their structure [ 71, and in this sensetwo films were obtained, one at a current density of 3 mA cm-* (PPy,) and the other one at 0.7 mA cm-* (PPy,,,). From the polymer film obtained at 0.7 mA cm-*, we obtained a pellet (grinding and pressingan evacuated sample, for 3 min at lo3 kg cm-*) ( PPypellet). Thesethree materialsPPy and MMPPy exhibit a finite T+O conductivity and can be said to correspond to systemsin which the Fermi energy is above the mobility edge EC (seeFig. 2). The lowest conductivity can however be seen to besmallin comparisonto the Mott minimum metallic value which is of order about IO3S cm- ‘. It can be further observed that the conductivity, which startsat a smallvalue, risesstrongly with temperatureto reach valuescommon to most dopedpolymers. Similar behaviour is exhibited down to 10 K in the work of Ref. [ 71 (see Figs. 2 and 8 of Ref. [ 41) though the data are not explicitly plotted in this way. The structure of the salts of polypyrrole with polystyrene sulfonate can be changedby the addition of dioxane to the polymerization solutions; this
170
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10- (a)
I
-10 0.2
I
I
0.3
i 0.4
0.5 T-1’4 K-1’4)
,:’ ,:’
3.0
I
I
4.0
I
I
I
1
5.0 6.0 In [T(K)1 Fig. 1. Temperature dependence of conductivity for several p-toluenesulfonate-doped polypyrroles (see text). (a) Plot of In c vs. T-l“‘. (b) Plot of the activation energy EA( r) (arbitrary scale values to avoid overlapping between the curves), The activation energy is defined by In o = EA( T). The effective Mott exponent y, assuming an exp( - Tey) dependence, is shown in various temperature regimes.
results in the family of curves shown in Fig. 4 of Ref. 141, showing a continuous change from tunnelling metal-like to hopping and insulating-like behaviour. These facts seem to demonstrate that, when~the morphology of the substance is taken into account, one not only has ‘disorder’ but also granularity. Small metallic grains are separated from each other by energy barriers [ 81: the conductivity at low temperature is small because the barriers are limiting the magnitude of the conductivity. This type of behaviour is common to doped inhomogeneous systems. Indeed truns-polyacetylenes exhibit similar curves for intermediate doping levels [ 91. Comparing PPy,,,,,, with PPY,,~ films, from which they are derived, not only is a decrease in conductivity observed, but also a shift of the ‘hopping region’ to’higher temperatures. We believe that this can immediately
Metals
78 (1996)
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be related to the existence of intergranular barriers: we now have smaller metallic grains and therefore a higher resistance. Again, the difference in conductivity between PPy, and PPy,,, can only be understood as being due to differences in structure. DMPPy exhibits what appears to be standard hopping conduction throughout the measured temperatures. We conclude that transport in DMPl?y is below the mobility edge (see Fig. 2) ; there is no way for the carriers to traverse the entire system without exchanging energy with the phonon bath. Despite these differences apparent in the low-T region, all four curves shown exhibit, in the intermediate temperature regime, what appears to be a universal exponential T-” hopping law. As a consequence of this apparent universal hopping behaviour, no attention has been paid to the actual details of the transport mechanisms, magnitude of the parameters, and in particular no serious attempt has been made to reconcile the inhomogeneous and the homogeneous (hopping) models. The present measurements and those of Ref. [ 41 go down to low temperatures, and give us the possibility of studying more closely the interesting and informative crossover between the tunnelling dominated quantum regime and the thermally assisted inhomogeneous transport regime. The results for this crossover are shown in Fig. 2. The remarkable feature here is the apparently new exp( 0”) behaviour. Naturally, as we go down in temperature, it cannot be totally excluded that the granular metal phases in Ref. [4] and those in the present data, eventually as temperature tends to zero, turn into true insulating states. This could occur as a result of ‘Coulomb glass’ formation, but seems unlikely. The zero-point (quantum) motion of the electronic and phononic system can, in principle, keep a finite conductivity state down to very low temperatures. There is however as yet no rigorous proof of the exact ‘zero-Tbehaviour’ in such inhomogeneous systems. Indeed it may in some circumstances even be possible to obtain the opposite behaviour, namely, superconductivity at extremely low temperatures in highly doped ‘compensated’ samples. In this paper we are concerned with the way ‘real’ excitations enhance the pathways and add to the zero-point motions of the coupled electron-phonon system to increase the conductivity as a function of T. We are concerned with the region above the single-particle electronic mobility edge where a continuum of excited states appears to exist (Fig. 2). We shall argue in this paper that a universal behaviour can be derived, and that the exp( aT”) law is indeed the right way of
Fig. 2. Typical potential profile representing a granular systemprojected in one dimension. The Fermi energy I$ is represented by the dashed lines and can be above (m) or below (I) the mobility edge represented by EC.
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interpreting this transport regime. We shall base ourselves on rigorous mathematical and thermodynamic methods. We shall also demonstrate, in the following theoretical discussion, that transport which is often considered to be Mott hopping is most probably thermally assisted tunnelling over barriers. We shall call this the variable range tunnelling (VRT) mode.
3. Models of transport in disordered systems 3. I. Hopping conduction
To begin with, it is generally accepted that, at low doping levels and at low enough temperatures, conduction is by hopping, from defect to defect. As shown by Miller and Abrahams [ lo] the phonon-assisted hopping rate Wis of the form:
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171
become important in transport processes where many degrees of freedom participate in achieving the carrier transfer. Examples are complex organic molecules (Marcus theory [ 141) and indeed most materials exhibiting the Meyer-Neldel rule [ 121, We argue that most organic materials exhibiting hopping-like transport with high effective activation energies are also necessarily in this category because they are complex ‘large’ objects in which the atomic and electronic coordinates are strongly coupled. In the Mott approach to conduction, it is the sequential or percolative competition between space and energy which gives rise to the T 1’4 laws . The so-called VRH transport can be given a mathematically rigorous treatment as shown in Ref. [ 151. 3.2. The tunnelling approach
expressing the fact that, in order to reach another point in space, a particle normally has to tunnel and absorb energy from its environment. In principle, all possible paths can be sumed from initial to final state and then the prefactorcontains the total number of ‘paths’ [ 11,121 that the particle can take and this makes v a complex but very important quantity. The analysis in Ref. [ 71 has shown that, for large energy changes compared to kT, it can be approximately written in terms of a thermodynamic sum:
There are however other ways to obtain this type of law. The competition between space and energy need not always be due to percolation, as we shall show below; in many systems, the following picture is much more reasonable. The particles encounter Coulomb-like barriers as shown in Fig. 2. The barrier tunnelling rate T(E) depends on the energy E because, as one can see, for higher energies, the barrier is shorter so higher energies mean shorter barriers. Indeed for a general potential V(r) , the standard quantum mechanical approximation referred to as ‘WKB [ 161 gives us the transmission velocity:
u(E) = I, exp[S(E)];
T(E) = (2Elm) ‘I2 exp{ - (2mlfi’)
WC= v exp( -~cYR,)
exp[ - (Ej-Ei)/kT]
S(E) = [~(E-E,)/cLJ,]~
(1)
(2)
where 4 is related to an interaction volume, o, is a typical dominant mode and y an exponent which depends on the dominant modes. This expresses the fact that, for a large number of paths, or when many degrees of freedom are involved, it is the free energy change of the ‘process’ which is the correct measure for the rate of the transition and not the simple Fermi golden rule quantum mechanical transition probability. Free energy changes however include both enthalpic and entropic changes. The latter is measured by the number of different ways the transition can take place and enters the prefactor in front of the activation term. Transition (1) must be understood as linking many different configurations which are only characterized by the fact that a carrier initially in the vicinity of Ri ends up in the vicinity of Rj in a time of order l/ Wjy In the usual Miller-Abrahams hopping process [lo], where a single or only a few phonons are exchanged, the thermodynamic considerations are obviously not necessary, and we can evaluate the rates in the ‘Fermi golden rule’ approach. The standard Mott hopping [ 131 or variable range hopping (VRH) approach gives rise to the well-known hopping conductivity laws: CT(T) =a,exp[
-(T,,/T)$];
O
(3)
derived with a prefactor Y which is a constant and of the order of a phonon frequency. The sums of the thermodynamic paths
l/2
dr [V(r)
-l$‘z}
(4)
For a pure Coulomb barrier, the energy ‘dependence of the transmission probability is evaluated in Landau Lifshitz quantum mechanics [ 161. Generalizing this expression to allow for different (but similar) types of barriers we write for the energy-dependent velocity: T(E)=(2E/m)“2exp{[E,/IE-EFI
+El(n)]@}:‘,
(5a)
E,>E,(n)
where E,(n) is the mobility edge (see Fig. 2) and we obtain for the linear response conductivity: I’ 1 . cc (T = e2db a(T=O)
dEN(E) =e’N(E,)D,
exp[S(E)]
( - Sf/6E) T(E)“~
exp{ - [f$,lEr’(n)]‘}
.,:
(5b)
(5i)
where N(E) is the one-particle electronic density of states, d,, is a barrier width, f(E) is the Fermi function, and D, is the Mott
diffusivity
(minimum
metallic)
with the. factor
e2N(EF) D, of. order e2/ (Aa), where a is a typical lattice spacing. In the expression for T(E), E,, is a measure of the barrier height,, and El (n) .is a measure of the inverse barrier
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width and depends on the doping level IZ, and is related to the ‘free’ kinetic energy above the mobility edge&(n) . The form of E, (n) can be written as fiiD( n) la*, where D(n) is the true zero-Tdiffusivity and a function of the doping level n. Indeed here, we expect D( n) N D,( n - n,) lnc, where D, is the Mott value and II, is the critical doping concentration for a metalto-nonmetal transition, In a system for which the doping level is below the mobility edge it
(6)
E, = (2m/fi*)g,
El = 0 and p= l/2. A simple steepest descent evaluation for E, = 0 will give rise to a Mott-type law
with s = l/3. Barriers in doped polymers are basically of two types: chain ends and Coulomb-like due to space charge fluctuations. If we took a configuration average of such barriers, then the generalized Coulomb form assumed here is a good representation. Returning to the generalized forms (5), we can see that the total conductivity now includes the path summation factor eScE),which measures the total number of dynamical pathways which the particle can use to cross the barrier. In this model we are no longer talking about a transition between single quantum levels, but rather between groups of levels forming metallic-like reservoirs separated by Coulomb-like barriers. Now this is precisely what is expected when there is a weak localization on a chain, or if the doping concentration is increased so that clusters of states develop in which electron eigenstates are delocalized or only weakly localized. The strongly localized levels and the charge carriers added to the system by doping will initially act as single well-defined quantum levels Ei and gradually, with increasing concentration, form clusters, then metallic islands and, finally even at the highest concentration, when local metallicity has been achieved, we can expect some granularity to remain in the system. The degree and nature of the remnant granularity will depend on the type of polymer and the intrinsic disorder in the system (chain lengths, fibre structure, etc.). We conclude that a Mott-type law can also result from VRT and not just from VRH! In the present category of systems, both can exist and coexist, depending on the concentration of dopants, nature of intrinsic disorder and temperature. Experimentally it is not immediately obvious how to tell them apart. This is why most authors only use the Mott law. Differentiating VRH
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from VRT can only be done through a systematic quantitative analysis, which pays attention to the detail of the transitions and the magnitude of the prefactors. Electric field (F), magnetic field (B) and frequency dependences (w) can also be used to discriminate between these mechanisms. We shall summarize the expected behaviour later in this paper.
4. Fluctuation-assisted transport methods: the temperature dependence of prefactors 4.1. The temperature laws
The mobility and conductivity, as given by Eqs. (5), encompass both hopping and tunnelling mechanisms in situations where the initial and final energies are in the same range (about /CT) and can be considered to be a universal form of describing transport through insulating regions in disordered inhomogeneous materials, when many levels and many degrees of freedom can participate. The above approach bears some similarity, and is basically in spirit, in agreement with Sheng’s ‘fluctuation-assisted tunnelling’ method [ 81. The difference with Sheng’s method is in the way the fluctuations are treated. We have introduced an entropy factor to count the number of paths. Sheng observed that fluctuations are important when there are barriers, irrespective of the system. He considered that the fluctuations in the barrier width result from the fact that fluctuations cause the amount of charge in each reservoir to change in time and, therefore, an internal field FT to develop, which obeys a Gaussian distribution and adds or subtracts to the external field. This observation is true, but the stored ‘plasma-like’ charge density fluctuations, which occur in the metallic grains, constitute only a subset of the total number of degrees of freedom that help to lower the barrier for a particle undergoing tunnelling. Many other single-particle-like modes can modulate the barrier height and shape during the time the particles attempt to cross. The entropy factor can in principle allow for all the phonon and electronic degrees of freedom. The most elegant way to treat it is to first evaluate the energydependent function S(E) as was done in Ref. [ 111. Different types of excitations give rise to different exponents, as shown in Ref. Ill]. One of the mathematically rigorous methods of evaluating fluctuation-assisted transitions through barriers has been given by Bruinsma and Bak [ 171, These authors use the functional integral technique and apply the WKB approximation to tunnelling. They have treated the case when the barrier heights are subject to local fluctuations. Their method can be generalized to allow for barrier width fluctuations. The width fluctuations would then include the charge and internal field changes considered in the Sheng model. Thus Sheng, and Bruinsma and Bak address the same problem, but only include a different subset of the total number of degrees of
M.C. Angladaet al. /SyntheticMetals 78 (1996) 169-176
’
-2’
0
’ 100
’
’ 200
’
’ 300
’
’ 400
’
’ 500
T3/2
Fig. 3. Plot of In D vs. T3’*in the low-temperature range for the polypyr~oles of Fig. 1.
freedom. Naturally each distinct physical transport situation is characterized by particular classes of dominant excitations, but they can vary strongly, have mixed modes and, most of all, there can be complex dimensionality (fractal) behaviour, resulting from a combination of disorder and low dimensionality. Bruisnma and Bak also restrict their analysis to the situation where the tunnelling particle is energetically far from the top of the barrier so that the ‘classical pathways’ are not included. The thermodynamic approach developed in [ 111 is to some extent phenomenological, but it has the advantage of being simple to apply and including in principle the total change in the number of variables with no restrictions on the tunnel energy. It also gives one an intuitive ‘chemists’ way’ of visualizing what is in general a very complex manybody transfer process. It is therefore very satisfying to see that the thermodynamic method roughly agrees with the first principles methods [ 17,181 when considering similar modes. The correctness of the approach has been beautifully demonstrated recently using exact Monte Carlo many-particle simulations of surface diffusion [ 191.
173
dard hopping law. The analysis of the results, reported in Ref. [ 61 for several perchlorate-doped polypyrroles, show that at low temperatures they also follow this behaviour (see Fig. 4). In the case of the polypyrrole polystyrene sulfonate salts exhibiting finite conductivity as T+ 0 in Ref. [ 61, we observe similar behaviour (Fig. 4). These results clearly demonstrate that this novel temperature dependence is connected to the structure of the sample at the nanoscale to microscale. The dependence appears at higher temperatures for more inhomogeneous samples, i.e. compare, in Fig. 3, PPY,,~,,, with PPY~,~ which is more ‘compact’, and PPy,,, with PPy, which shows a different structure on a nanoscale [7]. In all cases observed, the value of k appears to decrease to roughly k N 1 at higher temperatures. We can understand the origin of this behaviour by looking at the fluctuation-assisted model discussed above. At low temperatures, the integral (6) is dominated by the entropy term multiplied by the Fermi function so that a-exp(aT)Y’(l-Y)
(8)
In polypyrrole it follows that y= 3/5. Now we can attempt to relate this exponent to the type of modes which are dominating the fluctuations in this regime and in this particular system. From Ref. [ 181 and thermodynamic entropy calculations, we observe that for acoustic phonons in d dimensions where d can be fractal and in the range l
@a>
optic phonons will give (see Ref. [ 91 for details of how to treat the n-optic phonon rates) : S(E) =Eln(AlG)
(9b)
where h is the coupling and 6 the phonon frequency. For electronic excitations:
4.2. Low-temperature transport in granular structures
The interesting point about granular structures is that almost any value of conductivity can be acheived up to about lo4 S cm-‘, yet the system still has a conductivity which increases with temperature at low temperature because there are barriers which can be thermally overcome [ 17,181. The intermediate temperature dependence cah look like hopping, and this then confuses the mechanism. Our results show that, at low temperatures, systems exhibiting a finite T=O conductivity exhibit a novel temperature dependence which can be written as a(T) =(T,exp[(aT)k]
(7)
where k= 3/2 (see Fig. 3). For comparison the insulating system DMPPy is shown to obey what appears to be a stan-
-6
t t 0
100
200
300
I-= Fig. 4. Plot of In u vs. T3’*in the low-temperature range for several of the polypyrroles of Ref. [4] (data from Figs. 2 and 8 of Ref. [4] ): 0, poly(3methylpyrrole); x, poly(3-nonylpyrrole); @, poly(3-heptadecylpyrrole); 0, poly[3,4-(buta-1,4-diyl)pyrrole perchlorates] ; + , polypyrrole polystyrene saltsobtained from aqueous solution, and A, from 10% dioxaneaqueous solution.
174
y= l/2
M.C. Anglada
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(9c)
At low temperatures in polypyrrole ( 15-35 K), we expect the acoustic modes to dominate over the electronic excitations so that the effective dimensionality of the spectrum is d = 12/ 5, which is indeed a reasonable number for a polymer. Optic modes and electronic fluctuations will contribute at ,higher temperatures. From the change in the curvature at higher temperature (T> 35 K), we conclude that electronic modes and optic modes are determining the magnitude of the fluctuating tunnelling fields in this regime until, eventually, thermal fluctuations are strong enough to allow VRT to take over (when kT> hq+ ). This in principle can be checked by measuring the low-temperature heat capacity of the system which should behave as C, N Td for phonons. Note that twolevel systems associated with glass-like modes can also contribute and these would behave like one-dimensional phonons. Ultra-sound absorption may be able to detect glasslike excitations due to moving chains or clusters and is an interesting experiment to carry out at low temperature. They could tell us about low-energy structural modes. ‘Organized’ space charge fluctuations as opposed to single-particle fluctuations, as considered by Sheng, can also contribute and assist the tunnelling of carriers, but then the direct singleparticle electronic excitations in the reservoirs also give rise to strong local fields, so that the decision on which modes dominate has to be more global, i.e., we have to ask: Is the dominant mechanism electronic or phononic? We are here referring to real excitations caused by temperature. There are also the zero-temperature quantum electronic and quantum phononic (zero-point motion) processes which help assist the carrier motion at T= 0 and cause quantum delocalization and tunnelling, as considered for example by Emtage [20] and one of the present authors [ 211. The quantum manybody processes are part of the determination of the zerotemperature conductivity and will not be discussed further in this paper. They could be the reason for delocalization and, for example, also give rise to superconductivity [ 211, Before turning to a first-principles quantum statistical derivation, it is worth pointing out that, according to thermodynamic fluctuation theory, the fluctuation in ‘local entropy S/ kB’ near the tunnelling transition point is (AS’) = CJk,, which is the specific heat divided by the Boltzmann constant at constant pressure. For acoustic phonons, C, scales as T3 so that the prefactor behaves as exp( T/T,) 3’2, as found in this paper. The same argument would suggest that the temperature dependence observed in highly doped polyacetylene is due to the electronic component of the specific heat, not for the initial increase which again must be due to phonons, but for the later T”2 part of the curves [ 91, i.e., when T> 30 K. The above suggests that the low-temperature transport behaviour in’ inhomogeneous systems might indeed have a very simple’and appealing solution. Though evidently the role of thermodynamic fluctuations has been established, as demonstrated-in several ways above, it is nevertheless important to give rigorous derivations using the actual quantum
Metals
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mechanical mechanisms. Thermodynamic theory, as a firstprinciples theory, has a hierarchy of solutions. The fluctuation theory method, for example, assumes that the modes are contributing to the transitions in a completely random phase manner, basically by lowering the ‘heat’ required to effect the ‘transformation’ of going from the initial to the final state. First-principles approaches however begin by describing the actual many-body transitions and then adding them up. This is a very difficult task and should be restricted to a ‘small subclass’ of processes which may or may not be adequate in a given temperature range. Progress in identifying the type of transitions dominating the low-temperature transport in pyrrole will, hopefully, help us to unravel the many-body transport modes influencing other and more complex inhomogeneous systems. Electronic modes, for example, must depend on the doping level. This is certainly consistent with the observation in highly doped polyacetylene, but a detailed correlation between doping level and the appearance of a linear to sublinear value of k has yet to be established. Let us now go back to the first-principles calculations of Ref. [ 141 and the work of Bruinsma and Bak [ 171. 4.3. Comparison of thermodynamic andfirst-principles methods Here we obtain in WKB, the tunnelling amplitude to reach a final energy E’, A( E,E’) where co
A(E,E’) =A,(E)
dt/2nexp[i(E-E’)tlfi-fi-1
The first termA, is the zero-temperature tunnelling amplitude. The remaining factor results from fluctuating fields which modulate the barrier height. These fields are represented by the dynamical coordinates f(t) = Z,x,,( t) , and with d
g(W)
=
dX/2W?lV,(X)K-‘(4 I
0
eXp[
-
I
d.?’ (mw/hK(x”))]
(lla)
Here VI (x) is the coupling strength of the coordinates to the barrier at the point X, and K is the inverse WKB ‘tunnelling length’: K(X)
=h-‘{2m[V(x)
-E]}“2
(lib)
Each coordinate X, represents a possible dynamical mode which changes the barrier height at a point x in space, with coupling strength V,(x) . The model can be generalized to
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take into account width fluctuations as well. The probability of crossing the barrier is found by taking the thermodynamic average of the modulus squared amplitude integrated over all final energies [ 151, For acoustic phonons, the coupling parameter, g(w) N w so that for d = 3, Eq. ( 10) would predict an enhancement of theform [16,17]: fl N exp ( UT) 4, acoustic phonons (12) g N exp( bT2), Gaussian random force (13) whereas the thermodynamic methods would give cr w exp( bT3’2) for acoustic phonons, in agreement with experiments in pyrrole. There is therefore a discrepancy in the value of k between the first-principles and the thermodynamic derivation, but this is not surprising in view of the approximations made in both derivations. In the thermodynamic method, the linkage between energy change and transfer of charge is somewhat ‘phenomenological’ and can be termed a ‘random phase approach’ since we are not actually saying how the entropy change is deterministically intervening in the transition. Going back to Eq. (2)) which is one step more fundamental than the simple random phase method, a comparison with the first-principles method also allows us to interpret the meaning of the constant q/w, in Eq. (2). We find, for example, that, in the random Gauss model, q/w, can be interpreted to be truly an inverse typical oscillation (excitation) mode multiplied by a term which is of order (barrier width/inelastic length) [9]. This means that the wider the barrier in comparison to the inelastic length, the stronger is the multiphonon enhancement. We conclude that the low-temperature conductivity of inhomogeneous conductors of the doped polymer category can be analysed in terms of fluctuation-assisted tunnelling. The first-principles exponent k is not always easy to relate to the corresponding simple thermodynamic results, but this is not surprising in view of the model assumptions, temperature range, and the complex nature of many-mode-assisted transitions. Our thermodynamic method (Eq. (2) ) predicts that k> 31’2 for acoustic phonons, 0
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enon of ‘fluctuation-assisted conduction’ at low temperatures in inhomogeneous media exhibiting finite d-c. conductivities. Both first-principles theory and thermodynamic arguments confirm this point of view. From the value of the exponents, the modes which are most effective in assisting the carriers to cross the tunnel barriers in a given temperature range can be determined. Low-temperature specific heat data can tell us the effective dimensionality of the excitation spectra. It is fairly safe to presume that every mode which can be excited in these strongly coupled electron-phonon systems will also help the transfer of charge from one point to another, with a given efficiency coefficient which will depend on the nature of the ‘barriers’. In many such disordered systems, it is often difficult to tell the difference between fluctuation-assisted tunnelling at higher temperatures and Mott-type hopping. With the help of Eqs. (5) and (6) we can now shed some light on this question. It follows that hopping will give rise to zero conductivity as T-, 0, but there are also other differences which may be exploitable. These are related to the high electric and magnetic field behaviours of the two models [ 211. Thus, for example, it is well known how hopping from site to site gives rise to high-field positive magnetoresistance [ 221. Clustered levels or energy levels of reservoirs are more spread out in space and will react ( ‘shrink’) to smaller (than a single localized site) magnetic fields. This can be checked. The barrier potential will be modified by an additional term of the form + ( eBr)2/2mc2. If we are close to a metal-insulator percolation transition, then the magnetic-field-induced upward shift of the Fermi energy E,(B) can lead to ‘magneto-delocalization’ and therefore to a negative magnetoresistance [ 221. The Fermi energy shift is due to the orbital and spin energy redistribution and is of order pgB/4, important at high magnetic fields. Note that the spin shift is isotropic but the orbital contribution is anisotropic in a polymer, so the two can be differentiated experimentally. The negative magnetoresistance effect is assisted by the fact that the quantum diffusivity also increases with the magnetic field, and that consequently the mobility edge is lowered. Indeed in Eq. (5a), El (n,B) increases with B in a way predicted by scaling theory [ 221. The electric field dependence of hopping and the present model are also different and can be used as a test of the mechanisms. In hopping, the electric field lowers the activation energy barrier by a factor exp(eFRqIkT) . In tunnelling, the field lowers the barrier potential by a term of the form - eFr along the tunnelling trajectory [ 211. This gives rise to very different high field behaviour as can be seen from Eq. (4). Finally, there is also the a.c. conductivity. The hopping a.c. behaviour is well known and can be rigorously evaluated theoretically [ 151: wp (0
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ogeneous resistor network model is used. The cluster or metallic reservoir with charge density II, will contribute an a.c. conductance which in the metallic limit will be of the Drude form n,e’/ [ m( iw + l/ 7) 1, and the barrier region will contribute a tunnelling resistance (R) and capacitance C, N qE,/r,,, where r, is the tunnelling length, plus of course the thermal fluctuations as considered by Sheng, but these are negligible in the present context. Constructing the resistor network, it is seen that there will be distribution of tunnelling resistors (R}, P(R), with impedance Z(o) = (l/R + iwC,) so that, as frequency increases, the conductivity will rise up to the point where the smallest barriers have been reached. Then the a.c. conductivity will start to decrease as a result of the Drude response of the metallic domains. The turnover frequency will depend on the conductivity relaxation time of the ‘delocalized’ carriers in the reservoirs, and can be as high as lOI Hz. Finally, we observe that plasma-like modes in the conductivity should also manifest themselves as reqectivity dips due to the small metallic clusters. We conclude that with the present formalism, and the formalism of hopping conductivity, both of which can now be said to form very complete representations of transport in disordered media, we are in a position to discriminate between the various transport models and assign universal laws to the various transport coefficients. In this paper we have, in particular, been able to demonstrate that complex thermodynamic fluctuation processes give rise to systematic observable laws in low-temperature transport experiments on pyrrole, and confirm what is often intuitively suspected but rarely quantified, namely, that “transport in disorderedmedia is a result of an interplay of many degrees of freedom and this is also the reason for the ‘anomalous’ prefactors”. We have shown that it is possible to track down these degrees of freedom and learn something very useful from the detailed low-temperature rise in conductivity even in such highly complex systems as doped organic polymers. We believe that the present analysis can be extended to other systems and in particular for example to the study of inhomogeneous regimes of high T, superconductors. It may eventually be possible to distinguish electronic from phononic quantum fluctuations.
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Acknowledgements The authors are grateful to Professors A. Yelon and L. Lewis for sending a preprint of their manuscript before publication and to Professor G. Wegner (Mainz) for emphasizing the importance of inhomogeneity in the description of the structural properties of conducting polymers. J. Clements of Leeds University is acknowledged for his help in preparing the manuscript. Part of this work was financed by the Spanish Government (DGlCYT PB 93-1257).
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