Field limiting current

In contrast to a direct injection of dc or ac currents in the sample to be tested, the induction of eddy currents by an external excitation coil generates a locally limited current distribution. Since no electrical connection to the sample is required, eddy current NDE is easier to use from a practical point of view, however, the choice of the optimum measurement parameters, like e.g. the excitation frequency, is more critical. Furthermore, the calculation of the current flow in the sample from the measured field distribution tends to be more difficult than in case of a direct current injection. A homogenous field distribution produced by e.g. direct current injection or a sheet inducer [1] allows one to estimate more easily the defect geometry. However, for the detection of technically relevant cracks, these methods do not seem to be easily applicable and sensitive enough, especially in the case of deep lying and small cracks. 

One result of the field-dependent mobility is that the space charge-limited current (SCLC, the maximum current that can How in the bulk of the sample) does no longer follow a simple V2H scaling [132] on the voltage Land sample thickness L. Muigatroyd [133] was able to show that, for a mobility as in Eq. (13.4), the monopolar SCLC current could be well approximated by ... 

Currem field characteristics measured wiih conjugated polymers sandwiched between an indium-tin oxide (ITO) anode and an aluminum cathode are usually hole dominated and are, consequently, appropriate for testing injection/lransport models for the case of unipolar current How. Data shown in Figure 12-1 refer to injection-limited currents recorded on typically 100 nm thick spin-coated films of derivatives of poly(y d/"fi-phenylenevinylene) (PPV) and a planarized poly(/ /" -pheny-leue) employing a Keilhley source measure unit. The polymers were ... 

Figure 28. Semiconductor interfaces with increasing electric fields in the space charge layer (from top to bottom) compared with tubes of different diameters through which an equivalent amount of water is pressed per unit time (equivalent to limiting current).

In binary solutions, for example, CuS04 in H20, the limiting current exceeds that due to convective diffusion alone by a factor of about two. The excess mass transfer is caused by migration of the reacting ion in the electric field. In both forced and free convection it is important to know the ion flux contributed by migration, which can never be suppressed completely. 

Migration of the reacting ion in the electric field, briefly referred to in Section II,B, is usually suppressed by the addition of excess inert electrolyte. Incorrect values for mass-transfer rates are obtained if migration contributes more than a negligible fraction of the total limiting current. 

While initial attempts of quantification of i-Az traces obtained in (single) molecule conductance studies suffered from rather poorly defined data selection strategies and limited current ranges [64, 116], most of the leading groups in the field focus now on the construction of all-data point conductance and plateau length... 

M. V. Williams, H. R. Kunz, and J. M. Fenton. Influence of convection through gas-diffusion layers on limiting current in PEM FCs using a serpentine flow field. Journal of the Electrochemical Society 151 (2004) A1617-A1627. 

Fig. 8 Temperature dependence of the zero field hole mobility in the low carrier density limit in a polyfluorene copolymer. The data are inferred from space-charge-limited current experiments and analyzed in terms of the extended Gaussian disorder model (see Sect. 4.1). From [90] with permission. Copyright (2008) by the American Institute of Physics...

Perhaps not so very well suitable Thus, semiconductor electrodes exhibit limiting currents that arise from the transport of the charge carrier inside the semiconductor. In practice, this means that it is difficult to get current densities above -1 mA cm-2 at moderately doped semiconductors. No such limitation occurs with metals that have roughly l electron per atom free to move under an electron field gradient. The limiting currents that arise with metals are due to the limitation in supply of materials in the solution. 

Multiplication of P by the number of electrons arriving at unit surface in unit time should then give the field emission current density J. The argument just presented has limited itself to electrons at the top of the Fermi sea. It is apparent from Equation (1) and the nature of the electron energy distribution that this assumption is good. 

Finding rigorous analytical expressions for the single potential step voltammograms of these reaction mechanisms in a spherical diffusion field is not easy. However, they can be found in reference [63, 64, 71-73] for the complete current-potential curve of CE and EC mechanisms. The solutions of CE and EC processes under kinetic steady state can be found in references [63, 64] and the expression of the limiting current in reference [74], Both rigorous and kinetic steady state solutions are too complex to be treated within the scope of this book. Thus, the analysis of these processes in spherical diffusion will be restricted to the application of diffusive-kinetic steady-state treatment. 

The effects of the catalytic reaction on the CV curve are related to the value of dimensionless parameter A in whose expressions appear variables related to the chemical reaction and also to the geometry of the diffusion field. For small values of A, the surface concentration of species C is scarcely affected by the catalysis for any value of the electrode radius, such that r)7,> —> c c and the current becomes identical to that corresponding to a pseudo-first-order catalytic mechanism (see Eq. (6.203)). In contrast, for high values of A and f —> 1 (cathodic limit), the rate-determining step of the process is the mass transport. In this case, the catalytic limiting current coincides with that obtained for a simple charge transfer process. 

Ohmic contacts. An ohmic contact is defined as one which supplies a particular crystal with an infinite supply of either electrons or positive holes. Under an applied field these charge carriers are drawn into the material setting up a space charge. The subsequent currents are thus termed space-charge limited currents. In general the activation energy required to inject a positive hole from an electrode of work function W into a crystal is Ic — W and that to inject an electron W — Ac. Thus for ohmic contacts the conditions to be satisfied for holes and electrons are respectively (12a) and (12b). Although... 

At an appreciable fraction of the limiting current, it is usually not justified to neglect concentration variations— and resulting overpotential—near the electrode. In a general model we need to consider the electric field, kinetic limitations, and concentration variations. The problem is rendered more difficult by the need to know the system hydrodynamics, which, in turn, influence the concentration... 

Under these circumstances, then, the rate of the photoelectrochemical reaction is determined by the rate of transport of charge carriers to the interface. For p-typc photocathodes, this would refer to the photoactivated electrons that will have been produced from the valence band and are now in the conduction band. The electrons there are impelled to the surface both by diffusion (dependent on the concentration gradient of electrons) and by means of the electric field resulting from the potential gradient near the surface (Fig. 10.8), that depends on the electrode potential that shifts the Fermi level. For a given photoillumination intensity, there will be a fixed limiting current where (at best) the rate of transport to the surface becomes equal to the maximum rate of electron production due to photoactivation (Fig. 10.9). In reality, the numerical value of the limiting current will be also determined by various accidents (e.g., collisional deactivations) that destroy photoelectrons on their way to the interface. 

If, at places within the oxide layer, the electric field is too large for the exponential expansion utilized above in deriving the linear diffusion equation to be a valid approximation, then the more exact hopping current expression given by eqn. (88) should be utilized instead. Numerical computations are usually easier to carry out in any electric field limit by using the exact microscopic hopping expression (88). 

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