About the existence of a percolation threshold in solid solution of N-isopropylcarbazole/polycarbonate mixtures.

INTRODUCTION

It is now established that "molecularly doped polymers" form ideal systems for studying hopping transport and thermal charge generation, Mort and Pfister (1). These types of binary systems are composed of a glass polymer matrix and photoelectrically active molecules dispersed in it, forming organic photoconductors of great technological application, particularly in electrophotography, Kanemitsu (2). In molecularly doped polymers the hopping energy is likely to be well defined, but the separation between molecules of dopant (R) may be subjected to strong fluctuation instead of random hopping between neighbors. Consequently, the carrier may choose the shortest hopping distance, i.e. under go R-percolation. In this context, any electrical characterization of these systems must consider a possibility of the percolative character of the transport.

In a typical percolation problem, one has randomly distributed sites connected by percolation channels. A carrier can move only if there is a suitable channel available. Each site has a probability p to be occupied and 1 - p to be vacant. In a percolation model there exists a critical value [p.sub.c] below which the percolation probability P(p) falls sharply to zero. [p.sub.c] is known as the percolation threshold. If there is an energy E associated which each hop, p becomes a function of E. The relation between conductivity and the percolation ideas in first approximation is given, according to Ziman (3), by [Sigma](E)= [[Sigma].sub.o]P[p(E)], so the behavior of conductivity is governed by that of P[p(E)]. Kirkpatrick (4), has studied the behavior of conductivity close to the percolation threshold and found that the conductivity is described by the relation:

[Sigma](p) [is proportional to] [(p - [p.sub.c]).sup.1.8] (1)

On the other hand, Addler, et al. (5), found:

[Sigma](p) [is proportional to] [(p - [p.sub.c]).sup.2] (2)

More recently, Silver, et al. (6), replotted the data by Pfister (7) for drift mobility ([Mu]) in the system triphenylamine/polycarbonate (TPA/PC) as a function of the weight ratio (x) in log [Mu] vs. log x representation and reported a relationship of the form:

[Mu] [is proportional to] [(x - [x.sub.c]).sup.1.6] (3)

where [x.sub.c] = 0.17, which suggests a classical percolation mechanism with a concentration threshold about 17% by weight of TPA in PC. Below [x.sub.c] carriers become increasingly localized and longer jumps necessarily lead to a dispersive transport Sober and Montroll (8).

In this paper, we are reporting a study of DC dark conductivity in solid solution of N-isopropylcarbazole dispersed in amorphous polycarbonate, as a function of the concentration of the dopant molecules, to see if a percolation threshold can be detected at low dopant concentration which, according to Silver, et al. (6), is expected at 15 to 20% molecular concentration of N-isopropylcarbazole in polycarbonate.

EXPERIMENTAL DETAILS

Weight amounts of NIPC were incorporated into PC, namely, 5 x [10.sup.-4] kg of PC was fixed and 1 x [10.sup.-5] kg to 4 x [10.sup.-4] kg of NIPC (2 to 80% of PC weight) were mixed together by dissolving in chloroform with continuous magnetic stirring. Films [10.sup.-5]m thick were deposited onto aluminized glass substrates by spreading the mixture under a sliding blade. The films were top coated with aluminum by thermal evaporation forming parallel plate capacitor samples. DC dark conductivity was estimated from I-V curves obtained with the help of a digital electrometer (Keithley Model 600 B).

EXPERIMENTAL RESULTS AND DISCUSSION

Figure 1 depicts the DC dark electrical conductivity at room temperature as a function of NIPC content expressed by the weight ratio. The conductivity is observed to increase with increasing content of NIPC with a tendency towards saturation at high doping level. This strong dependence of the conductivity on the content of NIPC was taken as the starting point to verify the hopping nature of the transport. The figure includes the DC conductivity value of undoped polycarbonate ([[Sigma].sub.DC] = 4 x [10.sup.-11][[Omega].sup.-1][M.sup.-1]) deduced from AC conductivity experiments Santos, (9).

Figure 2 shows the dependence of conductivity on average separation between dopant molecules (R) plotted in ln [Sigma]/[R.sup.2] vs. R at room temperature. The molecular separation was estimated, assuming uniform dispersion of the molecules of NIPC into the PC matrix, by the relation R = [N.sup.-1/3] where N is the concentration of NIPC molecules, calculated assuming that the volume of the film remains unchanged by the incorporation of the NIPC molecules. It is observed that for the content of NIPC higher than 20%, the experimental points fit the relation describing the hopping mobility, Mott and Davis, (10)

[Mu] = [R.sup.2](e/6kT) [v.sub.Ph]exp(-2 [Alpha] R)exp(-[Delta][Epsilon]/kT) (4)

where the term [v.sub.Ph]exp(-[Delta][Epsilon]/kT) represents the probability of transition between sites, with [Delta][Epsilon] as the activation energy for hopping and [v.sub.Ph] being of the order of the phonon frequency. The term exp(-2 [Alpha] R) describes the overlap of the wave functions on neighboring hopping sites, where a [[Alpha].sup.-1] = p is a measure of the spatial decay of the wave function; so, in this range of concentration the DC conductivity would be described by

[Sigma] [Alpha] [R.sup.2]exp(-2 [Alpha] R) (5)

with [[Alpha].sup.-1] [is approximately equal to] 1.45 x [10.sup.-10] m.

For the concentrations smaller than 20% the experimental points in Fig. 2 deviate from the straight line. In this concentration range the probability of hopping from one molecule of NIPC to another given by Eq 5 seems to be slowed down by hopping between them and localized states eventually present in the polycarbonate matrix. The role played by the polymer matrix in the carriers' transport will be reported in a future paper.

Figure 3 depicts the dependence of the DC conductivity on the NIPC molecules content in a plot of log [Sigma] vs. log x. For concentrations larger than x = 0.2, the behavior of the conductivity obeys an empirical relation

[Sigma] [Alpha] [x.sup.3] (6)

For lower concentration the conductivity deviates from this power law in a manner, that the sharp discontinuity in the conductivity predicted by percolation theory, Kirkpatrick (4), is not observed.

As it was mentioned before, if the intermolecular separation between molecules of NIPC were subject to a wide fluctuation, transport would be expected to be percolative. The carriers preferentially choose the shortest hopping distance available and a percolation threshold would appear at a certain mean concentration of NIPC. This critical concentration according to Silver, et al. (6), would be around a weight ratio between 0.15 and 0.20. At first glance our experimental results are in complete disaccord with the percolation theory because no percolation threshold can be detected in Fig. 1. However, the data plotted in Fig. 2 suggest that the standard hopping model is applicable only for NIPC concentrations higher than 20%. At smaller concentrations we observe a deviation from the law given by Eq 5 which is probably due to the participation of matrix in transport process. Consequently, we propose the empirical relation given by Eq 6 is a result of Kirkpatrick-type conductivity (see Eq 1)

[Sigma] [Alpha] [(x - [x.sub.c]).sup.3] (7)

for x [is greater than] [x.sub.c]. To verify this idea we replot in Fig. 4 our experimental data from Fig. 1 in the representation [[Sigma].sup.1/3] vs. x. We observe that a critical concentration [x.sub.c] = 0.16 can be estimated by extrapolating the straight line behavior described by Eq 7. This value of [x.sub.c] agrees with the theoretical predictions of Silver, et al. (6), for a percolation threshold.

CONCLUSIONS

No discontinuity in the behavior of the DC conductivity with respect to the concentration of molecules of N-isopropylcarbazole was observed at the concentration expected for the theoretical percolation threshold, or anywhere in the range studied, which included molecular concentration lower than have been previously reported for any system of this kind. For high concentration of NIPC (x [is greater than] 0.2) our results support a hopping character of conductivity with power law dependence on x. The analysis of these data by extrapolating the relations found at high concentrations gives the percolation threshold [x.sub.c] = 0.16. However, experimental data for concentration x [is less than] 0.2 deviate from simple percolative picture, pointing to the increasing influence of hops involving localized states present in the polymer matrix.

ACKNOWLEDGMENT

The author would like to thank Dr. V. M. Yartsev for the invaluable discussions during the revision of this paper.

NOMENCLATURE

[Sigma] = DC dark conductivity.

NIPC = N-isopropylcarbazole dopant molecules.

PC = Polycarbonate polymer matrix.

R = Average separation between NIPC molecules.

N = NIPC molecules concentration.

PC = Polycarbonate matrix.

x = Weight ratio defined as: (weight of NIPC/weight of PC).

REFERENCES

1. J. Mort and G. Pfister, Polym.-Plast. Technol. Eng., 12, 89 (1979).

2. S. Kanemitsu, Solid St. Comm. 68, 701 (1988).

3. J. M. Ziman, J. Phys. C., 1, 1532 (1968).

4. S. Kirkpatrick, Phys. Rev., 27, 1722 (1971).

5. D. Addler, I. P. Flora, and S. Satura, Solid St. Comm., 12, 9 (1973).

6. M. Silver, K. Risko, and H. Bassler, Phil. Mag. B40, 247 (1979).

7. G. Pfister, Phys. Rev., B16, 3676 (1977).

8. H. Scher and E. Montroll, Phys. Rev., B12, 2455 (1975).

9. S. Santos Lemus, Proc. 2nd International Conference On Conduction and Breakdown in Solid Dielectric, Erlangen, Germany (1986).

10. N. F. Mott and E. A. Davis, in Electronic Processes in Non-Crystalline Materials, Clarendon Press, Oxford (1979).