The Scher-Montroll Formalism

In polymers, it is always observed that a packet of carriers spreads faster with time than predicted by Eq. (30). Thus, the spatial variance of the packet yields an apparent diffusivily that exceeds the zero-field diffusivity predicted by the Einstein relationship. Further, the pholocurrent transients frequently do not show a region in which the photocurrent is independent of time. As a result, inflection points, indicative of the arrival of the carrier packet at an electrode, can only be observed by plotting the time variance of the photocurrent in double logarithmic representation. The explanation of this behavior, as originally proposed by Scher and Lax (1972, 1973) and Scher and Montroll (1975), is that the carrier mean velocity decreases continuously and the packet spreads anomalously with time, if the time required to establish dynamic equilibrium exceeds the average transit time. Under these conditions, the transport is described as dispersive. There have been many models proposed to describe dispersive transport. Of these, the formalism of Scher and Montroll has been the most widely used. 

A key prediction of the Scher-Montroll model is that the photocurrent transients will decay as 

The point of intersection of the two branches described in Eqs. (31) and (32) defines a transit time as 

The Scher-Montroll model has been widely used to describe dispersive transport phenomena in polymers as well as the chalcogenide glasses. For a review, see Scher et al. (1991). 

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This chapter reviews theories proposed to describe charge transport in materials of potential relevance to xerography. The emphasis is on the disorder formalism, polaron arguments, and the Scher-Montroll formalism. These have been the most widely used during the past decade. For reviews, see Silinsh (1980), Movaghar (1987, 1991), Bassler (1993), Silinsh and Capek (1994), and Silinsh and Nespurek (1996). Experimental results are described in the following chapters. 

Schein et al. (1993) measured W for DEH doped PS over a wide range of fields and temperatures. Figure 69 shows a series of photocurrent transients. The DEH concentration was 50% and the temperature 305 K. The slowly decreasing current before the transit time varies as approximately t-0-07 jn the Scher-Montroll formalism (1975), t-0-07 corresponds to a = 0.93. The parameter a decreases with decreasing temperature to approximately 0.8 at 254 K. The results show W is independent of field. For 30% DEH, W was, independent of thickness in the range of 3.8 to 42 pm. These results contrast with those of Yuh and Stolka (1988) for N,N -diphenyl-N,N -bis(3-methylphenyl)-( 1,1 -bipheny l)-4,4 -diamine (TPD) doped PC, where WkL-1/2, For DEH concentrations in excess of 30%, W was independent of concentratioa... 

The results were described by the Scher-Montroll formalism. The authors derived an expression for IF as... 

In the original Scher-Montroll formalism, hopping was assumed to occur through a manifold of spatially disordered isoenergetic hopping sites. In the... 

Pfister (1977) measured hole mobilities of TPA doped PC. Figure 51 shows the temperature dependencies for different concentrations. The field was 7.0 x 105 v/cm. The concentration is expressed as the weight ratio X of TPA to PC. The mobilities were thermally activated with activation energies that increase with decreasing TPA concentration. The concentration dependence was described by the lattice gas model with a wavefunetion decay constant of 1.3 A. Figure 52 shows the field dependencies at different temperatures for X - 0.40. The solid lines were derived from the Scher-Montroll theoiy (1975) using the listed parameters. Pfister concluded that the theoiy provides a self-consistent interpretation of all experimental observations if field-induced barrier lowering and temperature-dependent dispersion are formally introduced into the expression for the transit time. 

Seki (1974) measured electron mobilities in TNF and PVK mixtures. The mobilities were in the range of 10-9 to 10-6 cm2/Vs and strongly field dependent. A key feature of Seki s results is the superimposition of current transients, when normalized to the transit time, for transit times varying by over two orders of magnitude. This is described as universality and a fundamental prediction of the Scher-Montroll (1975) formalism. The room temperature results are illustrated in Fig. 20. Seki argued that the free and trapping lifetimes... 

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