Electrons diffusion equation

Most of our ideas about carrier transport in semiconductors are based on tire assumption of diffusive motion. Wlren tire electron concentration in a semiconductor is not unifonn, tire electrons move diffuse) under tire influence of concentration gradients, giving rise to an additional contribution to tire current. In tliis motion, electrons also undergo collisions and tlieir temporal and spatial distributions are described by the diffusion equation. The... 

The Fokker-Planck equation is essentially a diffusion equation in phase space. Sano and Mozumder (SM) s model is phenomenological in the sense that they identify the energy-loss mechanism of the subvibrational electron with that of the quasi-free electron slightly heated by the external field, without delineating the physical cause of either. Here, we will briefly describe the physical aspects of this model. The reader is referred to the original article for mathematical and other details. SM start with the Fokker-Planck equation for the probability density W of the electron in the phase space written as follows ... 

The fundamental theory of electron escape, owing to Onsager (1938), follows Smoluchowski s (1906) equation of Brownian motion in the presence of a field F. Using the Nemst-Einstein relation p = eD/kRT between the mobility and the diffusion coefficient, Onsager writes the diffusion equation as... 

With the advent of picosecond-pulse radiolysis and laser technologies, it has been possible to study geminate-ion recombination (Jonah et al, 1979 Sauer and Jonah, 1980 Tagawa et al 1982a, b) and subsequently electron-ion recombination (Katsumura et al, 1982 Tagawa et al, 1983 Jonah, 1983) in hydrocarbon liquids. Using cyclohexane solutions of 9,10-diphenylanthracene (DPA) and p-terphenyl (PT), Jonah et al. (1979) observed light emission from the first excited state of the solutes, interpreted in terms of solute cation-anion recombination. In the early work of Sauer and Jonah (1980), the kinetics of solute excited state formation was studied in cyclohexane solutions of DPA and PT, and some inconsistency with respect to the solution of the diffusion equation was noted.1... 

So eq. (11.47) can be viewed as a diffusion equation in the spatial coordinates of the electrons with a diffusion coefficient D equal to j. The source and sink term S is related to the potential energy V. In regions of space where V is attractive (negative) the concentration of diffusing material (here the wavefunction) will accumulate and it will decrease where V is positive. It turns out that if we start from an initial trial wavefunction and propagate it forward in time using eq. (11.47),... 

When the motion of electrons and positive ions in a particular system may be described as ideal diffusion, the process of bulk recombination of these particles is described by the diffusion equation. The mathematical formalism of the bulk recombination theory is very similar to that used in the theory of geminate electron-ion recombination, which was described in Sec. 10.1.2 ( Diffusion-Controlled Geminate Ion Recombination ). Geminate recombination is described by the Smoluchowski equation for the probability density w(r,i) [cf. Eq. (2)], while the bulk recombination is described by the diffusion equation for the space and time-dependent concentration of electrons around a cation (or vice versa), c(r,i). 

WolfE and more recently other investigators have applied the Boltzmann diffusion equation to a description of the secondary-electron cascade. This approach is quite satisfying because it has a clearly defined foundation which seems to encompass all of the basic physical processes needed to describe the situation. It also yields an approximate solution in analytic form which is given by Equation 22. 

When electrons are injected as minority carriers into a/>-type semiconductor they may diffuse, drift, or disappear. That is, their electrical behavior is determined by diffusion in concentration gradients, drift in electric fields (potential gradients), or disappearance through recombination with majority carrier holes. Thus, the transport behavior of minority carriers can be described by a continuity equation. To derive the p—n junction equation, steady-state is assumed, so that dnp/dt = 0, and a neutral region outside the depletion region is assumed, so that the electric field is zero. Under these circumstances, the continuity equation reduces to a diffusion equation (eq. 10), where is the lifetime of minority... 

In this section we give a proof of the Kawabata formula (52), following a method due to Kaveh (1984) and Mott and Kaveh (1985a, b). We assume that an electron undergoes a random walk, which determines an elastic mean free path l and diffusion coefficient D. If an electron starts at time t=0 at the point r0 then the probability per unit volume of finding it at a distance r, at time U denoted by n(r, t) obeys a diffusion equation... 

In Chap. 2 and 3, the motion of two reactants was considered and a diffusion equation was derived based upon the equation of continuity and Fick s first law of diffusion (see, for instance, Chap. 2 and Chap. 3, Sect. 1.1). When one reactant (say D) can transfer energy or an electron to the other reactant (say A) over distances greater than the encounter separation, an additional term must be considered in the equation of continuity. The two-body density n (rj, r2, t) decays with a rate coefficient l(r, — r2) due to long-range transfer. Furthermore, if energy is being transferred from an excited donor to an acceptor, the donor molecular excited state will decay, even in the absence of acceptor molecules with a natural lifetime r0. Hence, the equation of continuity (42) becomes extended to include two such terms and is... 

Current densities in the cathode are mainly determined by the respective value of oxide anion conductivity compared to the electronic conductivity (/Co" and ice", coupled to each other in Wagner diffusion). Equation (34) describes the current density limit for coupled transport of oxygen anions and electrons (777) ... 

The diffusion equation for the rate constant, eqn. (4.13), assumes that the molecules must come within van der Waals (hard sphere) contact for reaction to take place. This is of course a reasonable assumption when chemical bonds are made or broken in the reactants, but does not apply to long range processes such as energy transfer and possibly electron transfer. 

By way of illustration, we note that in the recombination problem mentioned above the energies of the electrons are the variables in the diffusion equation, the bottleneck is the region of energies near the boundary of the continuous spectrum, and the slowness of the process is related to the small amount of energy transfer from an electron to a heavy particle in one collision. In the problem of escaping electrons in a plasma, slowness is ensured by the weakness of the electric field, the independent variable in the diffusion equation is the momentum component along the field, and the bottleneck is determined, as in the kinetics of new phase formation, by the saddle point of the integral. 

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