Multiple trapping model

Concerning the nature of electronic traps for this class of ladder polymers, we would like to recall the experimental facts. On comparing the results of LPPP to those of poly(para-phenylene vinylene) (PPV) [38] it must be noted that the appearance of the maximum current at 167 K, for heating rates between 0.06 K/s and 0.25 K/s, can be attributed to monomolecular kinetics with non-retrapping traps [26]. In PPV the density of trap states is evaluated on the basis of a multiple trapping model [38], leading to a trap density which is comparable to the density of monomer units and very low mobilities of 10-8 cm2 V-1 s l. These values for PPV have to be compared to trap densities of 0.0002 and 0.00003 traps per monomer unit in the LPPP. As a consequence of the low trap densities, high mobility values of 0.1 cm2 V-1 s-1 for the LPPPs are obtained [39]. 

Traditionally, charge-carrier transport in pure and doped a-Se is considered within the framework of the multiple-trapping model [17], and the density-of-state distribution in this material was determined from the temperature dependence of the drift mobility and from xerographic residual measurements [18] and posttransient photocurrent analysis. 

The multiple trapping model of transport in an exponential band tail is described by Eq. (3.20) in Section 3.2.1 and a fit to this expression is given in Fig. 7.8. TTie free carrier mobilities are 13 and 1 cm V s" for electrons and holes respectively, with the band tail slopes of 300 °C and 450 C (Tiedje el al. 1981). Implicit in the analysis is the assumption that the exponential band tail extends up to the mobility edge, but the density of states model developed in Fig. 3.16 shows that this is a poor approximation. The band taU changes slope below E. and this may change the estimated values of the mobility. 

The physics of this multiple-trapping process is clear, but how do we quantify it The multiple trapping model has been solved exactly for a number of special distributions of localized states by several authors, including Noolandi (1977) and Rudenko and Arkhipov (1982). The approximate analysis that we give here is due to Tiedje and Rose (1981) and Orenstein and Kastner (1981). 

Several completely different experiments support our interpretation of the time-of-flight transport process and the conclusions we have drawn about the distribution of band-tail states. The time-resolved photoinduced absorption experiments of Ray etal. 9% ) support the view that the photogenerated holes are concentrated in the vicinity of an energy E, which moves deeper into the localized state distribution, linearly with temperature and logarithmically with time. Furthermore, the time decay of the photoinduced absorption, which is controlled by the more mobile of the two carriers (electrons), has the t form expected from the multiple trapping model (see, for example, Orenstein eta/., 1982). Thea = r/300°K temperature dependence for a reported by Tauc (1982) is in excellent agreement with the electron time-of-flight results. 

The multiple-trapping model can also be solved analytically for the transit time by the method of Laplace transforms. The one-dimensional transport equations for the fiee-electron density n(x, t) in a semiconductor with a distribution of discrete trapping levels are... 

Schwarz etal. (1995) Mesoporous membrane Al/PS coplanar (CP) Transport in nearly extended states and process similar to multiple trapping model... 

Fig. 5 Schematic illustration of the multiple trapping model. The traps are shown as localized states in the bandgap, where filled states are black lines and empty states are grey lines. The conduction band c. valence band y and quasi Fermi level E-p are also shown. The path of an electron trapping into empty states and subsequent detrapping by thermal excitation into the conduction band is shown by the arrcws...

The dispersion parameter a takes a value between 0.1 and 0.3 and is obtained from optical density measurements which vary with time t as exp[—(t/r) ] for a characteristic time t. For an exponential distribution of trap states of width Tq, a = TITq. In a development of these ideas, the multiple trapping model was used to explain the variation of experimental cation absorption decay data as a function of applied bias using spectral information to obtain the initial electron number density per nanoparticle [31]. 

Experimental data on the variation of with electron quasi Fermi level [34] suggests that nonlinear recombination occurs in DSSC and so a continuum implementation of the multiple trapping model has been extended to this case [35]. Whilst there is insufficient experimental evidence to be certain about the origin of the nonhnear recombination, it is likely that it is due to recombination from trap states [17]. 

It is important to note that (97) indicates that the conductivity is determined exclusively by the transport level and is completely independent of the presence and distribution of traps, in the context of the multiple trapping model that we have used herein. The steady-state conduction is not affected by the trapping process because the traps remain in equilibrium. Alternatively, one can view conduction as the result of the displacement of the whole electron density, n, with a smaller jump diffusion coefficient see (96). 

Arkhipov V, lovu M, Rudenko A, Shutov S (1987) Multiple trapping model approximate and exact solutions. Solid State Commun 62(5) 339-340... 

---