VRH model

As the first insulating/semiconducting higher boride series, the electrical transport of these compounds has been carefully investigated (e.g. Slack et al., 1977 Golikova, 1987 Werheit et al., 1991). The temperature dependence of the resistivity p follows the dependency of Mott s variable range hopping (VRH) model for 3 dimensional systems (Mott, 1968 Efros and Shklovskii, 1985), where... 

Charge transport in the accumulation channel is described by the percolation model [24] based on thermally activated tunneling of holes between localized states in an exponential density of states, described in Section 13.2.2. In the accumulation regime this Variable Range Hopping (VRH) model yields a gate-voltage dependent field-effect mobility of the form ... 

Another refinement of the VRH model consists in assuming that the charges are delocalized over segments of length L, instead of being strictly localized on point sites [40]. This is indeed a more realistic picture, leading to better fits with the data, but it has the drawback that an extra parameter has been added. Note that the temperature dependence, log o- -T y, can be found by other approaches, such as the percolation model, the effective medium approximation (EMA), the extended pair approximation (EPA) [41], the random walk theory, and so on. 

The NNH formulas [Eqs. (35) and (36)] have been applied by numerous workers also for hopping in tail states. It should be noted that it is not correct to do so because in a tail with a rapidly rising density-of-states function nearest neighbors cannot be defined. In this case, clearly, a VRH model has to be considered. This has been done by Grunewald and Thomas (1979). The authors find for a density-of-states distribution rising exponentially as g(E) = 0 exp(E/e) that a difference between apparent conductivity and thermopower activation energies of E — E% = 3.5e will result. 

As discussed in Section 6c, Dbhler s model of a soft mobility edge is in principle capable of explaining a difference between E and E% and hence a nonzero slope E. The extreme sensibility of the shape of QiT) on a(E), however, makes this transport model rather unlikely. The different E values observed in the experiment require different o(E) functions yet the observed Q T) data are dways linear if plotted versus l/T, in contrast to what is obtained if we interpolate between different a(E) curves. The same shortcoming is found in the VRH model of Grunewald and Thomas (1979) (see Section 6d). Again, it is unlikely that g E) is always changed by doping in such way as to retain the linearity of the Q versus 1/T curves. 

In summary, we have shown that the experimental results on the o(7) of SWCNT networks, can be explained by the model based on PhAT initiated by electric field. An advantage of this model over the often used VRH model is the possibility to describe the behavior of I-V data measured at both high and low T with the same set of parameters characterizing the material. On the basis of this model, the phenomenon of the crossover from non-metallic to metallic behavior of the conductivity is explained. The decrease of conductivity at T > Tc in the framework of this model is a result of the temperature dependent carrier tunneling process attended by the phonon emission. 

First, at low dopant concentrations, both PPY and PAc act as insulators. At a relatively high temperature, electronic transport can be described by Mott variable-range-hopping (VRH) model. Resistivity is a function of temperature that follows ... 

Figure 12.8b shows the temperature-dependent conductivity (a). The conductivity of pure PPy and nanocomposites with two different particle loadings can be linearly correlated to temperature with the quasi-three-dimensional variable-range hopping (quasi-3D-VRH) model. The variable-range hopping theory analysis indicates that three-dimensional hopping dominates in these samples, and the correlation between conductivity and temperature can be expressed by Equation (12.1),... 

In cr(r) f-oz5 crossover to In cr(r) — T at low temperature [24] characteristic of a 3D VRH model [40,60]. This means the presumed low-dimensional PANI-CSA tubes behave as 3D systems that imply the high degree of disorder of these samples. 

Polymer fibers and tubes under consideration are initially ID conductors thus one can expect that either short-range EEI or LRCI affect the transport in such quasi-ID systems. However, the applicability of some traditional models should be also considered first. Let s analyze the G(T) curves in the framework of ID VRH [40,60,61] and FIT [42] models. The VRH model implies o-(T) in the form of Equation 16.1, i.e., G(T) oc Go exp]—(To/T) ]. The standard derivation of the VRH transport for the d-dimensional hopping predicts power exponent p in Equation 16.1 in the form p=(y+ l)/(y-l- d+1) [40,60,61]. In this expression y is power exponent in a power-law density of states g(x) oc and d is the dimension-ahty of the system. The exponents p of VRH conductivity for ID systems are strongly dependent on the shape of density of states g( ) near the Fermi level p that arises due to 3D CL Thus, for d= 1 one can ohtainp = 0.5,0.67, and 0.75 hy setting y to 0,1, and 2 respectively. However, even FIT with p 0.25 for... 

FIGURE 46.11. o-Dc (7)for crosslinked PAN-ES, PAN-CSA (CHCI3), and PAN-CSA (m-cresol) samples (from Ref. [73]). The dashed lines are based upon the quasi-1 D VRH model. Note here crosslinks refers to physical crosslinks (microcrystalline regions) not chemical crosslinks (covalent bonds). 

The macroscopic conductivity ff T) follows a law ia(T) a or indicating electronic transport is dominated by hopping of the polaronic species. This can be explained by involving models such as variable range hopping (VRH) [228-230], the quasi-ID VRH model [231] or the metallic rods model [232]. 

In the view of the morphology of ER suspensions, the Quasi-ld-VRH model also seems reasonable, as the filament chains may form onedimensional paths for charge carriers. Although the anisotropic network structure could build up in concentrated ER suspensions [62], the hidden chains, spanned between two electrodes, still exist in tlie system [131]. Essentially, the Quasi-ld-VRH model is the specific consequence of strong... 

In summary, the dc current absorption is observed in the oxidized polyacrylonitrile based-ER fluids. The conductive behaviors of ER suspensions with or without an oscillatory mechanical field are confined by the microstructure—the fibrillated chain structure induced by an external electric field. The de current oscillates with the mechanical frequency and strain amplitude, implying that ER suspensions could be used as a mechanical sensor transferring a mechanical signal to an electric one. The conductive mechanism of an oxidized polyacrylonitrile-based ER suspension can be well described by a Quasi-Id-VRH model, where the localized charges hop from one localized state to another along the chains. This conduction model can be used to quantitatively describe the dc current oscillation phenomena. 

The systematic increase of. v from 0.25 to I upon dilution of PANI-CSA (0.012 I) is not expected in the standard VRH model. This increase in x is observed at volume fractions of PANI-CSA well above the percolation threshold (/e = 0.3 0.05%) where the system behaves as an effective medium [175,196,197]. Superlocalization is expected to play a significant role only very near the percolation threshold where the connected structure is fractal. Moreover, the theoretical models of... 

---