Drift mobility dependence

This cannot be solved analytically, even if the functional forms of the drift mobilities dependence on electric field are known. Providing the inter-ion electric field is directed radially (centro-symmetric) and there is no applied electric field, only the radial component of the tensor equation (158) is of interest for recombination of ion-pairs. Furthermore, if electron—cation recombination is of interest, the electron is much more mobile than the cation generally there are exceptions to this statement [352—354]. Equation (158) becomes... 

There are several techniques for measuring the mobility in a-Si H, most notably the time-of-flight method. All the techniques measure the average motion of the carriers over a time longer than that taken to trap a carrier in the band tail states, so that the drift mobility is always measured, rather than the free carrier mobility. The drift mobility depends on the distribution of traps and the free mobility can only be extracted if the density of states distribution is known. Chapter 3 describes how the time-of-flight experiment is used to determine the shape of the band tail through the analysis of the dispersive transport process. 

The dependence of drift mobility on the external field must be interpreted by a theoretical model, and as such it can elucidate the transport mechanism in a given case. In this section, we will only describe the phenomenological and experimental aspects. Their theoretical significance will be taken up in Sect. 10.2. 

A survey is given of the theoretical and experimental studies of electron-ion recombination in condensed matter as classified into geminate and bulk recombination processes. Because the recombination processes are closely related with the magnitudes of the electron drift mobility, which is largely dependent on molecular media of condensed matter, each recombination process is discussed by further classifying it to the recombination in low- and high-mobility media. 

The study of the dispersion of photoinjected charge-carrier packets in conventional TOP measurements can provide important information about the electronic and ionic charge transport mechanism in disordered semiconductors [5]. In several materials—among which polysilicon, a-Si H, and amorphous Se films are typical examples—it has been observed that following photoexcitation, the TOP photocurrent reaches the plateau region, within which the photocurrent is constant, and then exhibits considerable spread around the transit time. Because the photocurrent remains constant at times shorter than the transit time and, further, because the drift mobility determined from tt does not depend on the applied electric field, the sample thickness carrier thermalization effects cannot be responsible for the transit time dispersion observed in these experiments. 

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 dependence of the drift mobility p on the electric field is represented by formula p (p-E1/2/kTcf) which corresponds to the Pool-Frenkel effect. The good correspondence between experimental and theoretical quantity for Pool-Frenkel coefficient 3 was obtained. But in spite of this the interpretation of the drift mobility in the frame of the Coulombic traps may be wrong. The origin of the equal density of the positive and negative traps is not clear. The relative contribution of the intrinsic traps defined by the sample morphology is also not clear [17,18]. This is very important in the case of dispersive transport. A detailed analysis of the polymer polarity morphology and nature of the dopant molecules on mobility was made by many authors [55-58]. 

Up to now, only hydrodynamic repulsion effects (Chap. 8, Sect. 2.5) have caused the diffusion coefficient to be position-dependent. Of course, the diffusion coefficient is dependent on viscosity and temperature [Stokes—Einstein relationship, eqn. (38)] but viscosity and temperature are constant during the duration of most experiments. There have been several studies which have shown that the drift mobility of solvated electrons in alkanes is not constant. On the contrary, as the electric field increases, the solvated electron drift velocity either increases super-linearly (for cases where the mobility is small, < 10 4 m2 V-1 s-1) or sub-linearly (for cases where the mobility is larger than 10 3 m2 V 1 s 1) as shown in Fig. 28. Consequently, the mobility of the solvated electron either increases or decreases, respectively, as the electric field is increased [341— 348]. 

Doldissen et al. [348] found that the solvated electron mobility may be increased (or decreased) fourfold at fields 107Vm 1 compared with the mobility at low fields (<105 Vm 1). Such electric fields are small compared with those mutual fields when ions approach to within 2 nm or less of each other (>2xl08 Vm 1). No measurements of drift mobilities have been possible at such electric fields. It is not possible to state that the field dependence extends to these large electric fields, though the trend of the experimental results at moderately large electric fields is often extrapolated to large electric fields (Mozumder [349] and... 

Fig. 28. Electric field dependence of the solvated electron drift mobility in liquid ethane at various temperatures (K). After Doldissen et al. [348].

Baird et al. [350]). In the following analysis, the functional forms, p(E), which have been proposed (see below) to represent the field-dependence of the drift mobility are used for electric fields up to 1010Vm 1. The diffusion coefficient of ions is related to the drift mobility. Mozumder [349] suggested that the escape probability of an ion-pair should be influenced by the electric field-dependence of both the drift mobility and diffusion coefficient. Baird et al. [350] pointed out that the Nernst— Einstein relationship is not strictly appropriate when the mobility is field-dependent instead, the diffusion coefficient is a tensor D [351]. Choosing one orthogonal coordinate to lie in the direction of the electric field forces the tensor to be diagonal, with two components perpendicular and one parallel to the electric field. 

The other source of an effective electric field dependence of the diffusion coefficient is due to hydrodynamic repulsion. As the ions approach (or recede from) one another, the intervening solvent has to be squeezed out of (or flow into) the intervening space. The faster the ions move, the more rapidly does the solvent have to move. A Coulomb interaction will markedly increase the rate of approach of ions of opposite charge and so the hydrodynamic repulsion is correspondingly larger. It is necessary to include such an effect in an analysis of escape probabilities. Again, the force is directed parallel to the electric field and so the hydro-dynamic repulsion is also directed parallel to the electric field. Perpendicular to the electric field, there is no hydrodynamic repulsion. Hence, like the complication of the electric field-dependent drift mobility, hydro-dynamic repulsion leads to a tensorial diffusion coefficient, D, which is similarly diagonal, with components... 

Consequently, while the effect of an electric field dependence of both drift mobility and diffusion coefficient and also hydrodynamic repulsion decreases, the recombination probability, dielectric saturation and relaxation effects increase the recombination probability. 

To show this connection, consider an ion-pair as above (Sect. 2.1). Not only may the ion-pair diffuse and drift in the presence of an electric field arising from the mutual coulomb interaction, but also charge-dipole, charge-induced dipole, potential of mean force and an external electric field may all be included in the potential energy term, U. Both the diffusion coefficient and drift mobility may be position-dependent and a long-range transfer process, Z(r), may lead to recombination of the ion-pair. Equation (141) for the ion-pair density distribution becomes... 

The electric field dependence of the hole mobility in a series of PVK TNF films of varying composition is shown in Fig. 5(b)17. The carrier drift mobilities are extremely low and strongly dependent on the electric field. Similar electric field dependence was observed for electron drift mobilities. The mobilities obey the empirical relation... 

Fig. 5. Field dependence of hole drift mobilities for a range of TNF PVK molar ratios. Data taken at T = 24 °C17 ...

The drift mobility in this dispersive regime has an unusual electric field and thickness dependence. Fig. 3.13 shows the field dependence of the electron and hole mobility at different temperatures (Marshall et al. 1986, Nebel, Bauer, Gom and Lechner 1989). The electron drift... 

Fig. 3.13. Temperature dependence of the (a) electron and (b) hole drift mobility at different applied fields ranging from 5 x 10 V cm" to 5 X 10 V cm". The field dependence of is caused by the dispersion (Marshall et al. 1986, Nebel et al. 1989).

The low defect density in compensated material is apparent from the optical data in Fig. 5.18, which show a much reduced defect absorption band. The same result is deduced from time-of-flight and ESR data. Although the drift mobility is low, the mobility-lifetime product is comparable with the best undoped material, confirming the low defect density (see Fig. 8.24). The dangling bond density in the dark ESR experiment is about 4x10 cm" , with little dependence on the doping... 

Fig. 5.19. Temperature dependence of the electron drift mobility of compensated a-Si H. Data for a range of applied fields from 3 x 10 to 5 X 10 V cm" are shown (Marshall el al. 1984).

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