Conductivity distribution

The theoretically derived formula (21) relating PMC measurements to the surface concentration of minority carriers and interfacial rate constants contains a proportionality constant, S, the sensitivity factor. This factor depends on both the conductivity distribution in the semiconductor elec-... 

The distribution of the conductance histogram (Fig. 21.8A) could not be described by a normal distribution and showed the presence of several peaks. The SENS conductance distribution was therefore fitted with the sum of four Gaussian distributions that had means of 21.4 2.3 pS (8% area) labelled G25 33.0 + 4.8 pS (31% area) labelled G35 38.1 1.2 pS... 

Fig. 17.7. The effect of electric field perturbations, due to differences in conductivity between the sample- and the buffer electrolyte zone on the shape of the peaks, (a) conductivity distribution, (b) sample ions distribution, (c) electric field strength perturbations, and (d) effect on the peak shapes.

In this section, we study the images of atomic states. Starting with an analytic expression of the tunneling conductivity distribution, we will eompare the theoretical predictions and experimental findings of an easily measurable quantity the radius of the STM image of an atomic state near its peak. 

A convenient quantity for comparing with the STM image profile for a single atomic state is the apparent radius of the image near its peak, R, or the apparent curvature, K = l/R. Considering the w = 0 state only, these quantities are related to the tunneling conductance distribution by... 

Intuitively, we will expect that for p. or d- states on the tip and on the sample, the images should be sharper that is, the apparent radius should be smaller. This is indeed true. For example, for a 4- tip state and an. r-wave sample state, the tunneling conductance distribution is... 

Naturally, we observed the reciprocity principle. By interchanging the tip state and the sample state, the conductance distribution, and consequently, the apparent radius of the image, are unchanged. 

Table 6.1. Conductance distribution and apparent curvature of the STM images of individual atomic states ...

Therefore, the problem of calculating the STM images reduces to the problem of evaluating the Fourier coefficients for the tunneling conductance distribution of a single atomic state, Eq. (6.23). 

Using these equations and the conductance distribution functions listed in Table 6.1, the corrugation amplitudes for a tetragonal close-packed surface with different tip states and sample states can be obtained. For example, for a Is state, using Eq. (6.32), we have... 

Banks, R. J. (1972) The overall electrical conductivity distribution of the Earth. J. Geomagn. Geoelectr., 24,337-51. 

Fig. 32. Sketches of the electrode geometries used (a) for the electrocoloration of the SrTiOj sample and (b) for the microcontact impedance measurements to determine the local conductivity distribution. The sketch does not give the true scale the number of microelectrodes is much larger than shown.

There is a striking similarity between the experimentally observed and the theoretically calculated profiles, and all four characteristic features occur in both. The calculated location of the minimum, which mainly depends on the vacancy mobility, is close to the location observed in the experiment. The computed temperature dependence of the depth of the minimum corresponds with the results of the measurement. Obviously, the stoichiometry polarization model of resistance degradation correctly predicts the conductivity variations. In particular the almost quantitative agreement of the very characteristic shape of the conductivity distribution proves the validity of the existing model described above. It should be noted that in the calculations only the hole mobility is chosen such that the theoretically and the experimentally observed depth of the minimum is similar, but all other parameters used in the simulation are taken from literature [77, 336, 338],... 

Fig. 7a, b Average mass transfer coefficient as a function of a aquifer anisotropy ratio for several variances of the log-transformed hydraulic conductivity distribution b variance of the log-transformed hydraulic conductivity distribution where open circles represent numerically generated data and solid lines represent linear fits. All model parameter values are identical with those used in Fig. 6... 

Let us perturb the conductivity distribution, d x,z), within the region F. The equation for the corresponding perturbed electric field, SEy, can be obtained by perturbing ccination (9.11) ... 

Let us perturb the conductivity distribution a (r). Applying the perturbation operator to both sides of equations (9.46) and (9.47), we obtain the equations for corresponding variations of the electromagnetic field ... 

Note that the arguments in the expressions for the Frechet differentials, F/. //(5,6ct), consist of two parts. The first part, d, is a conductivity distribution, at which we calculate the forward modeling ojicrator variation, the Green s tensors are... 

We assume that the model parameters, anomalous conductivity distribution. Ad (r), belong to some complex Hilbert space Mp of the functions 7(r), 7 6 My, defined in domain D and integrable in D with the inner product ... 

We can discretize the model, in this case the anomalous conductivity distribution, Ad (r), by introducing a set of basis functions, V l (r), V 2 (r), / /v (r) in the finite dimensional Hilbert space M, which is a subspace of the complex Hilbert space Mo M/v C Mo- Let us approximate the anomalous conductivity by its projection over the basis functions ... 

Thus, we have the following representation for the anomalous conductivity distribution ... 

Now we can use also expansion (9.179) of the anomalous conductivity distribution over the boxcar functions. Substituting (9.179) in the last formula, we finally find... 

The argument in the expressions for the Prechet differentials, FE,H b,Aa), consists of two parts. The first part, at, is the background conductivity distribution, for which we calculate the forward modeling operator variation the second part, Aa, is the anomalous conductivity, which plays the role of the background conductivity variation. We will use below the following simplified notations for the Fr6chet differentials... 

In a solution of the inverse problem we assume that the anomalous field, E , and the background conductivity are given. The goal is to find the anomalous conductivity distribution. Ad. In this case, formula (10.16) has to be treated as a linear equation with respect to Ad. 

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