Hole , continuity equation

In addition to the electron and hole continuity equations, Poisson s equation must be satisfied. 

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 = 0, and a neutral region outside the depletion region is assumed, so that the electric field is zero. Under these circumstances,... 

The device model describes transport in the organic device by the time-dependent continuity equation, with a drift-diffusion form for the current density, coupled to Poisson s equation. To be specific, consider single-carrier structures with holes as the dominant carrier type. In this case,... 

The flow in the die hole is predominantly in the axial direction with the axial velocity, Vz, a function of both r and z position. The radial component of velocity, Vr, is significant only in the conical section of the die hole. However, Vj- is about two orders of magnitude smaller compared with the axial velocity, vz. Therefore, Vj- is estimated by forcing the continuity equation to be... 

Example 5-5 The Toricelli Problem. Consider an open vessel with diameter Di containing a fluid at a depth h that is draining out of a hole of diameter D2 in the bottom of the tank. We would like to determine the velocity of the fluid flowing out of the hole in the bottom. As a first approximation, we neglect the friction loss in the tank and through the hole. Point 1 is taken at the surface of the fluid in the tank, and point 2 is taken at the exit from the hole, because the pressure is known to be atmospheric at both points. The velocity in the tank is related to that through the hole by the continuity equation... 

Like the performance of chemical reactors, in which the transport and reactions of chemical species govern the outcome, the performance of electronic devices is determined by the transport, generation, and recombination of carriers. The main difference is that electronic devices involve charged species and electric fields, which are present only in specialized chemical reactors such as plasma reactors and electrochemical systems. Furthermore, electronic devices involve only two species, electrons and holes, whereas 10-100 species are encountered commonly in chemical reactors. In the same manner that species continuity balances are used to predict the performance of chemical reactors, continuity balances for electrons and holes may be used to simulate electronic devices. The basic continuity equation for electrons has the form... 

A continuity equation similar to equation 15 can be derived for holes ... 

In the most general situation, the current density in semiconductors and in solar cells is composed of electron and hole contributions jq = jh+ Je The relevant carrier concentrations ne(x) and rih(x) are subject to generation and recombination and have to obey continuity equations... 

More carrier types A continuity equation should be constructed for each of the species involved in charge transfer electrons, reduced and oxidized ions in the hole conductor. In addition, cations in the hole conductor are mobile and will affect electrostatics, and therefore they should be included, although these do not contribute to charge transfer. 

Impedance spectroscopic investigations on PPV-based LEDs are reported in the literature by several groups with different results and interpretations. Impedance spectra on ITO/PPV/Al devices were described by two semicircles within a Schottky model representing bulk and junction region [122] or by the presence of an interfacial oxide layer at the PPV/Al contact [123]. Equivalent circuits using three RC elements suggested a spatial variation of the conductivity in the PPV film [124]. A more complex equivalent circuit was proposed by analyzing the Poisson s and the hole and electron continuity equations [125]. 

The continuity equation for the minority carrier holes when trapping is negligible is... 

Recombination is either characterized using steady state measurements, for instance in the dark or under open-circuit conditions, or transient methods where the decay of the concentration of charges is used to analyze the strength and type of recombination. To determine the recombination rate itself is not useful because the continuity equations are solved in terms of the charge densities of electrons and holes on which the recombination rate is strongly dependent. Therefore, we need to measure directly a recombination lifetime r or an effective recombination prefactor k, which is usually defined as k = R/rP, where n is the average excess electron and hole concentration. Typically, these measurements are done by transient photovoltage measurements [31, 42, 148-152], by transient absorption measurements [148, 153] or by impedance measurements [154—156]. 

The equations for hole current density and the corresponding continuity equation are obtained in the equivalent manner. 

According to the continuity equation of minority carriers from (3.79), again neglecting the diflhision term, and using the conditions that electron and hole concentration gradients are equal and that p = n - (No - Na) we obtain for the majority carrier concentration gradient... 

The flux of charge carriers across the semiconductor/electrolyte interface under illumination is dependent on the potential (E), the electron density (n), and the hole density (p) throughout the semiconductor. The calculation of electron and hole current flow requires the use of the Poisson, steady-state carrier continuity equations and carrier transport equations ... 

The drift-difiusion model captures the drift of charge carrier concentrations in an electric field and their diffusion due to gradients in the concentrations. In organic solar cells, it is necessary to solve the continuity equations for electron (n), hole [p], and exciton (x) concentrations. " These are updated along with the Poisson s equation for the electrostatic potential ((/>). These four equations can be written as... 

As the pressure diminishes far below that of the stoichiometric point, the number of metal vacancies and holes will continue to fall and the electroneutrality equation chosen, Eq. (7.16), will no longer be representative. A more appropriate form of the electroneutrality equation for the low-pressure region ignores the minority defects, which are now metal vacancies and holes, to give... 

Another of our present aims is to continue investigating the theoretical properties of the 2- and 3-body terms. Thus, there are many exact relations linking the different 3-body terms arising from each of the 36 options of equation (9) as well as their holes counterparts [16]. These relations are interesting by themselves, because they widen our understanding of the problem and may be helpful for improving our approximations. Another important theoretical question is, as mentioned above, to investigate the reason why one Ai corrects the VCP error which in some cases is only due to the 3-body contribution but which may also involve other kind of correlation effects. 

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