Hopping current

The second aspect of this problem is that the area densities may be different on the two sides of a given barrier. This difference in the area densities would lead to a difference in the forward and reverse components of the hopping current, even if the height of the barrier were the same as viewed from the forward and reverse directions. For example, if the barrier heights are W(f) and W(r) as viewed from the two directions, respectively, and the area density for forward hopping is denoted by nif) and the area density for reverse hopping is denoted by n(r), then the net current density over the barrier would be given by... 

It is informative to examine the limiting cases of the above expressions for the current over a specific local barrier. The thermal equilibrium limit is reached whenever the current J is chosen to be zero. We note that this requires forward and reverse components of the hopping current to be equal in magnitude. From eqn. (74), we obtain... 

In terms of the bulk defect concentration, the hopping current for equal barrier heights in the forward and reverse directions [see eqn. (79)] takes the form... 

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). 

Let us apply expression (317) for the calculation of the proton current density associated with the hopping mobility of protons. Neglecting the dispersion in expression (317) and assuming that the applied electric field is small, we can write the hopping current density as follows ... 

In Situ STM Hopping Current Dependence on Probe Length and Bias Voltage... 

Mauser, H. and Guba, W. (2008) Recent developments in de novo design and scaffold hopping. Current Opinion in Drug Discovery ei Development, 11 (3), 365—374. 

A second wave is also predicted, but it was not observed experimentally due to the large overpotential required to oxidize Fe " at the support electrode. Typically a value of Ir given by 4.5 X 10 "F Acm was obtained. Electron-hopping current were obtained using potential step chronoamperometry (see Chapter 2). Typically is = 2.9 X IQ-io/F Acm The value of is was obtained via an indirect route. The corresponding Fe(III/II) reduction process was examined at the metallo-polymer. The reduction via mediation is thermodynamically unfavorable, so only the direct reduction at the inner interface should be observed. 

Figure 12-4. Dependence of die injection current on the zero Held barrier height A lor variable widths of the density of hopping states (Rel. 2I ).

If the film is nonconductive, the ion must diffuse to the electrode surface before it can be oxidized or reduced, or electrons must diffuse (hop) through the film by self-exchange, as in regular ionomer-modified electrodes.9 Cyclic voltammograms have the characteristic shape for diffusion control, and peak currents are proportional to the square root of the scan speed, as seen for species in solution. This is illustrated in Fig. 21 (A) for [Fe(CN)6]3 /4 in polypyrrole with a pyridinium substituent at the 1-position.243 This N-substituted polypyrrole does not become conductive until potentials significantly above the formal potential of the [Fe(CN)6]3"/4 couple. In contrast, a similar polymer with a pyridinium substituent at the 3-position is conductive at this potential. The polymer can therefore mediate electron transport to and from the immobilized ions, and their voltammetry becomes characteristic of thin-layer electrochemistry [Fig. 21(B)], with sharp symmetrical peaks that increase linearly with increasing scan speed. 

Hence the top grid pattern is usualty widely spaced but not the extent that the electrical contact layer will have difficulty in collecting the current produced by the cell s other active layer. Cleau ly, the silicon disc needs to be heated as well during the process to aid the diffusion process. Note that the surface will be rieh in diffusing species and that the density of species declines within the interior What happens is that once the ion contacts the silicon surface, it "hops from site to site into the interior of the bulk of the silicon matrix. 

In oxide ion conductors, current flow occurs by the movement of oxide ions through the crystal lattice. This movement is a result of thermally activated hopping of the... 

The charge propagates in the film by electron hopping between the polymer Red/Ox couples. This is controlled by the electrode potential only in a close proximity of the electrode in more distant sites, the charge transport is driven by a concentration gradient of reduced or oxidized mediators. The observed faradaic current density, jF, is a superposition of... 

The above mechanistic aspect of electron transport in electroactive polymer films has been an active and chemically rich research topic (13-18) in polymer coated electrodes. We have called (19) the process "redox conduction", since it is a non-ohmic form of electrical conductivity that is intrinsically different from that in metals or semiconductors. Some of the special characteristics of redox conductivity are non-linear current-voltage relations and a narrow band of conductivity centered around electrode potentials that yield the necessary mixture of oxidized and reduced states of the redox sites in the polymer (mixed valent form). Electron hopping in redox conductivity is obviously also peculiar to polymers whose sites comprise spatially localized electronic states. 

Well, there we have it our list of current unsolved/unaddressed problems. Hop to it, readers ... 

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