Charge transport theory

Instead of the usual Boltzmann transport equation approach, we will present a more phenomenological method of attaining the desired results (Bube, 1974). The hope here is that the physics will not so easily be obscured in the 

We will first consider only electrons (q = —e) and later generalize to include holes. By letting the external magnetic field define the z axis of our coordinate system, i.e., = Bz, we can write the components of Eq. (Al) as 

To solve the above equations we can take the derivative of Eq. (A2a), and then substitute Eq. (A2b), getting 

Equations (A9) show that the electron exhibits the expected cyclotron motion in the presence of the magnetic field. However, collisions must also be taken into account. Let N(t) be the number of particles that have not experienced a collision for time t (after some arbitrary beginning time, t = 0). Then it is reasonable to assume that the rate of decrease of N(t) will be given by dN oc —Ndt = —Ndt/1. The solution of this equation is N(t) = N0 exp (—t/ ), where N0 is the total number of particles. It can easily be shown that x is simply the mean time between collisions. The probability of having not experienced a collision in time t is, of course, N(t)/N0 = exp(—t/ ). The 

Similar expressions can be generated for holes simply by letting coc - — oc. The use of a simple relaxation time xB needs justification, which will not be attempted here. Suffice it to say that this assumption is not bad for elastic scattering processes, which include most of the important mechanisms. A well-known exception is polar optical-phonon scattering, at temperatures below the Debye temperature (Putley, 1968, p. 138). We have further assumed here that t is independent of energy, although this condition will be relaxed later. 

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For route 2, charge transport mechanism is similar to the hopping mechanism of small molecules. According to the charge transport theory developed hy Bredas et al. [22], the charge carrier transport in organic materials can be described by Marcus electron transfer theory (Eq. 1.3). In organic ciystals, the AG is 0 because the electron transfer happens in a same kind of molecules. Thus the Eq. 1.3 formula can be simplified into Eq. 1.4 as follows ... 

Flowever, the charge carrier motions in many organic semiconductors are between these two limits. It is thus expected that more sophisticated microscopic charge transport theories need to be developed to unify the concepts of the band-like and hopping transport. Indeed, many work along this line have been performed." It is known, however, that most of those rigorous quantum approaehes are limited to tens of sites because of the numerical convergence problem and computer memory limitations. 

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