Electron , continuity equation

Electron Continuity Equation. The electron continuity equation is... 

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 earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

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,... 

In the MO approach molecular orbitals are expressed as a linear combination of atomic orbitals (LCAO) atomic orbitals (AO), in return, are determined from the approximate numerical solution of the electronic Schrodinger equation for each of the parent atoms in the molecule. This is the reason why hydrogen-atom-like wavefunctions continue to be so important in quantum mechanics. Mathematically, MO-LCAO means that the wave-functions of the molecule containing N atoms can be expressed as... 

Stagnation flows represent a very important class of flow configurations wherein the steady-state Navier-Stokes equations, together with thermal-energy and species-continuity equations, reduce to systems of ordinary-differential-equation boundary-value problems. Some of these flows have great practical value in applications, such as chemical-vapor-deposition reactors for electronic thin-film growth. They are also widely used in combustion research to study the effects of fluid-mechanical strain on flame behavior. 

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... 

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

Formulation of Equations. Discharge structure influences chemistry primarily through electron-impact dissociation and surface ion bombardment. To predict the rate of electron-impact dissociation, local electron number density and energy must be known. These quantities are obtained from equations for electron continuity and electron energy, respectively. 

Ion bombardment rate is determined from ion momentum or continuity equations, depending upon assumptions made in the model. To solve equations for ion and electron momentum and energy balances, the electric field profile must be known. This profile is obtained from the governing Maxwell equation, which is usually Poisson s equation. 

Equation for Electric Field Strength. To include electron-impact source terms in continuity equations for neutral species and to include the effect of ion bombardment on the rate of surface reactions, equations that predict electron and ion densities, momentums, and energy profiles are required. The profiles require an equation for the electric field strength. In general equations for electron and ion continuities (these equations yield electron density and ion density, respectively), electron and ion momentums (for electron and ion net or directed velocity), and electron and ion energies (for electron and ion random or thermal energy) must be solved. Finally, the electric field profile is obtained from Poisson s equation. 

Let us now separately consider the continuity equations for ions and electrons... 

With the initial condition c (t= 0,x) = c °, and the boundary conditions / n = 0 (for an electronic electrode), fam = 0 for an ionic electrode or, dc Idt =0 for a reversible electrode one obtains with the flux equations and the continuity equation the following as solution for the concentration profile ... 

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... 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

Figure 56 uses the example of associate formation between the ionic defect O and the electronic defect h to emphasize that the strict treatment requires the solution of coupled diffusion-reaction relationships, describing the general (electro-)chemical reaction scheme with individual diffusion or rate constants as parameters (cf. Section VI. 2). Source terms (q) must be taken into account in the relevant continuity equations, e.g., for defect B that can be created by... 

In the independent-partlcle-model (IPM) originally due to Bohr [1], each particle moves under the Influence of the outer potential and the average potential of all the other particles in the system. In modem quantum theory, this model was first Implemented by Hartree 12], who solved the corresponding one-electron SchrSdlnger equation by means of an iterative numerical procedure, which was continued until there was no change in the slgniflcant figures associated with the electric fields involved so that these could be considered as self-consistent this approach was hence labelled the Self-Conslstent-Fleld (SCF) method. In order to take the Pauli exclusion principle into account. Waller and Hartree [3] approximated the total wave function for a N-electron system as a product of two determinants associated with the electrons of... 

A fascinating aspect of the sensitized colloidal semiconductor films is that injected electrons created throughout the semiconductor network are collected in the external circuit with high efficiency. This implies that carrier transport through the 10 pm thick film occurs with no measurable recombination loss. The mechanisms of carrier transport have been studied in some detail. Carrier transport in a semiconductor film can be described by the continuity equation [155] ... 

In deriving the structure of a molecule, or distribution of atoms using X-ray crystallography, we do not directly obtain the x, y, z coordinates of the atoms. That is, we don t solve some system of linear equations whose solution is the set of numerical values for jc, y, z. We employ a Fourier transform equation that incorporates the diffraction data, the structure factors, and yields the value of the electron density p(x, y, z) at any and all points x, y, z within the crystallographic unit cell. From the peaks and features of this continuous electron density distribution in the unit cell we then infer the locations of the atoms, and hence their coordinates. This will be described as it is done in practice, in Chapter 10. Following this, the coordinates are improved by applying refinement procedures, as also outlined in Chapter 10. Here, however, our objective is to understand this Fourier transform equation, namely the electron density equation. 

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