Semiconductor modeling equations

The equations used to describe dopant incorporation are identical to those used to describe the deposition of the semiconductor. Thus equations 12-14 are applicable to a diflusion-limited model, with the number of components, n, increased by the number of dopants added. The equilibrium distribution coefficient, ki9 is defined as... 

One of the key steps in the chip-making process is the deposition of different semiconductors and metals on the surface of the chip. This step can be achieved by CVD. CVD mechanisms were discussed in Chapter 10 consequently, this section will focus on CVD reactors. A number of CVD reactor types have been used, such as barrel reactors, boat reactors, and horizontal and vertical reactors. A description of these reactors and modeling equations are given by Jensen. 

The semianalytical model by Roichman et al. [42] is based on the mean medium approximation (MMA) and is probably the only one that shares the assumptions found in standard semiconductor models (for better or worse). A percolation-type model, which predicts that there are bottlenecks or small regions in which the current density is much larger than average, do not agree with the assumptions behind the semiconductor device model equations. The assumptions behind the MMA model include... 

In Electrical Engineering, the semiconductor diode provides one of the most fundamental nonlinear devices. The current in die forward or conduction region is approximately an exponential function of diode voltage. Data for such a semiconductor diode is shown in Figure 9.48, along with a typical model equation of the form ... 

Many physical models can only be expressed in terms of several equations that provide good estimates of a set of measurements over different ranges of the independent variable - a piecewise definition of model equations. The semiconductor diode is one such example. In terms of theoretical operation of a diode, it is known that there are three fimctional forms for the diode current depending on the range of the voltage. These can be expressed in equation form as ... 

Modem theoretical treatments of defects in semiconductors usually begin with an approximate solution of the Schrodinger equation appropriate to an approximate model of the defect and its environment (Pantelides, 1978 Bachelet, 1986). Both classes of approximation are described in the following subsection as they pertain to the computational studies addressed in this Chapter. If it were not necessary to make approximations, the computational simulation would faithfully reproduce the experimental result. This would be ideal, but unfortunately, it is not possible. As a consequence, contact with experiment is not always so conclusive or satisfying. A successful theory, however, may still extract from the computational results the important essential features that lead to simple and general models for the fundamental phenomena. 

We proceed now to describe some of the most common approximations to the defect environment and the many-body Schrodinger equation and some simple models relating to defects in semiconductors that have been deduced from them. 

Extensions of this model in which the atomic nuclei and core electrons are included by representing them by a potential function, V, in Equation (4.1) (plane wave methods) can account for the density of states in Figure 4.3 and can be used for semiconductors and insulators as well. We shall however use a different model to describe these solids, one based on the molecular orbital theory of molecules. We describe this in the next section. We end this section by using our simple model to explain the electrical conductivity of metals. 

We know that not all solids conduct electricity, and the simple free electron model discussed previously does not explain this. To understand semiconductors and insulators, we turn to another description of solids, molecular orbital theory. In the molecular orbital approach to bonding in solids, we regard solids as a very large collection of atoms bonded together and try to solve the Schrodinger equation for a periodically repeating system. For chemists, this has the advantage that solids are not treated as very different species from small molecules. 

Equations (1.194) and (1.195) can be accepted, within reason, because both the chemical equilibrium constants and the hole mobility for semiconductors have an Arrhenius-type temperature dependence. It has been shown, by a least-square fitting of the electrical conductivity data of Maruenda et al. to eqn (1.193), that 85 per cent of the data points are within 1.5 per cent of the calculated values, as shown in Fig. 1.58. This indicates that the model proposed here gives an accurate description of the data. The fitting parameters are listed in Table 1.5. 

The above relationship between 0 and the rate constants is derived based on the conventional formulation of the rate equations. The unit to measure the amount of electrons and holes in the particle is density, the same as in bulk semiconductors. When the particle size is extremely small or the photon density is very low, only a few pairs of electron and hole are photogenerated and recombine with each other in the particle. This means that photon density does not take continuous values as suitably used in the conventional rate equations, but takes some series of values whose unit is the inverse of the particle volume. Taking into account this deviation, we proposed a new model in which particles are assigned by two integers, n and m, which represent the numbers of... 

The sensing mechanisms of the tin oxide based sensors have been discussed in many publications (9,10,11). The most widely accepted model for tin oxide based sensors operated at temperatures <400°C is based on the modulation of the depletion layer width in the semiconductor (sensor) due to chemisorption as illustrated schematically in Figure 6. For C2H 0H and Sn0x (or PdAu/Sn0x) interaction, the possible reaction steps may be expressed by the following equations ... 

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