Effective ohmic conductivity

Ohm s law relates the electric current (7) of a conducting material directly to the apphed voltage or potential (AE), as described by  

The effective resistance (Reif) of a heterogeneous material, composed by a conductive phase and one or more insulating phases, is a function of the conductive material s resistivity (pm), its effective area (Amefr), and its effective length (Lmeff) as described in the following equation  

From a different angle, the Rgg can be calculated as a function of the effective resistivity (/Omeff) and the input data the area of sample (A ,) and the length of sample as described by  

Using equations (2.26) and (2.27) we can obtain Aneff, which relates the effective resistivity of the whole sample with the resistivity, effective length, and effective area of conduction of the conductive phase  

As the effective resistivity characterizes the material by being an intensive property and being able to be extended to find the resistance of any continuous structure formed by the same material, this equation can be generalized to find the resistivity of an element formed by subdomains of smaller scales  

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In this procedure, the derivation of s denoted as "deriv (2) is used in order to obtain narrow and sharp peaks and to eliminate conductivity effects due to the independence of e from ohmic conductivity ... 

Reference 70 provides the first quantitative test of the random resistor network model. In Ref. 121 the authors employed the random resistor network model to determine the behavior of the low-field Hall effect in a 3D metal-nonmetal composite near the percolation threshold. For the following power laws of effective values of ohmic conductivity a, Hall coefficient R, and Hall conductivity a 12, Bergman et al. 121 have obtained the critical exponents ... 

The non-ohmic conductivity at high electric fields was discussed in crystalline and amorphous semiconductors and dielectrics in terms of such theoretical models as Pool-Frenkel effect [78], ShHovsldi nonlinear hopping conduction in disordered sohds [79,80], and tunneling in granular structures [81]. 

When the NCS is sufficiently thick, the high field effects should be determined by the uniform field in the bulk. The independence of the current on the polarity of the applied voltage is usually taken as an experimental proof that conduction is not determined by nonuniform fields near the contact. This requirement is, however, not sufficient since most processes, discussed in the previous chapter, are insensitive to the parameters of the electrode material. The injection mechanisms are essentially controlled by the properties of the semiconductor too and Queisser et al (1971) showed that ohmic conduction can be space charge controlled over a wide voltage range in a relaxation case semiconductor. Moreover, the rectification ratio of NCS contacts are expected to be small even for blocking... 

This major advance in nanoelectronic device fabrication was accompanied by key new insights into fundamental electronic behaviour at the atomic limit. Remarkably, the heavily doped silicon wires were found to have Ohmic conductance (i.e. their resistivity was independent of wire diameter or length) due to the very small separation between donors ( 1 nm, i.e. less than the Bohr radius). The abrupt doping profile - ranging from 10 cm outside the wire to 10 inside - yields very effective charge confinement. Moreover, and perhaps surprisingly, the atomic wires tolerate extremely high current densities (5 x 10 Acm ), comparable to those in state-of-the-art copper interconnects. 

In Eq. (7), Kj represents the total area specific ohmic resistance. R, is the sum of the cathode, electrolyte, anode, interconnect, and contact ohmic resistances expressed in Q m. Typically, R, is dominated by the electrolyte resistance and decreases with increasing operating temperature. To account for any electronic conductivity in the electrolyte, the effective ohmic resistance should be used in Eq. (7). The effective conductivity depends on the applied voltage and can be expressed as a correction to the ionic conductivity, Oiont by a term involving the electronic conductivity, Gg as follows [11,12] ... 

The high resistivity allows very effective ohmic heating of Si and Ge by direct current Samples with a specific resistivity in the order of 20-500 mS2 cm, which corresponds to a doping level between Nd = 10 —10 cm are used either in scanning tunneling microscopic (STM) experiments at low temperatures down to 80 K or for experiments between room temperature and 800 °C when direct current heating of the sample is used. STM experiments at 5 K, however, require Si samples that are degenerately doped with Nq > 10 cm to ensure metaUic conductivity at these very low temperatures and thus avoid tip crashes. 

Interparticle contact is of critical importance to the behavior of lithium batteries. Most lithium-ion electrodes contain 2 to 15 wt% conductive filler, such as carbon black, in order to maintain contact among aU the particles of active material and in order to reduce ohmic losses in the electrodes. Presently, there are few models available for predicting contact resistance, and the effect of the weight fraction of conductive filler on the overall electronic conductivity of the composite electrode must be determined experimentally. Doyle et al. [35] demonstrate how the fuU-cell-sandwich model can be used to determine what minimum value of effective electronic conductivity is needed to make solid-phase ohmic resistance negligible. Then, one need only measure the effective conductivity of the composite electrode as a function of filler content, and one need not run separate experiments on complete cells to determine the optimum filler content. Modeling techniques for predicting effective electroitic conductivities of composite electrodes are under development, and hold promise to aid in optimizing filler shape and volume fraction [85]. 

The trends of behavior described above are found in solutions containing an excess of foreign electrolyte, which by definition is not involved in the electrode reaction. Without this excess of foreign electrolyte, additional effects arise that are most distinct in binary solutions. An appreciable diffusion potential q) arises in the diffusion layer because of the gradient of overall electrolyte concentration that is present there. Moreover, the conductivity of the solution will decrease and an additional ohmic potential drop will arise when an electrolyte ion is the reactant and the overall concentration decreases. Both of these potential differences are associated with the diffusion layer in the solution, and strictly speaking, are not a part of electrode polarization. But in polarization measurements, the potential of the electrode usually is defined relative to a point in the solution which, although not far from the electrode, is outside the diffusion layer. Hence, in addition to the true polarization AE, the overall potential drop across the diffusion layer, 9 = 9 + 9ohm is included in the measured value of polarization, AE. ... 

The Schottky-Mott theory predicts a current / = (4 7t e m kB2/h3) T2 exp (—e A/kB 7) exp (e n V/kB T)— 1], where e is the electronic charge, m is the effective mass of the carrier, kB is Boltzmann s constant, T is the absolute temperature, n is a filling factor, A is the Schottky barrier height (see Fig. 1), and V is the applied voltage [31]. In Schottky-Mott theory, A should be the difference between the Fermi level of the metal and the conduction band minimum (for an n-type semiconductor-to-metal interface) or the valence band maximum (for a p-type semiconductor-metal interface) [32, 33]. Certain experimentally observed variations of A were for decades ascribed to pinning of states, but can now be attributed to local inhomogeneities of the interface, so the Schottky-Mott theory is secure. The opposite of a Schottky barrier is an ohmic contact, where there is only an added electrical resistance at the junction, typically between two metals. 

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