Barrier Hopping

The conductivity in the CBH model indicates that ct(co) x Ra- RJ is then expanded in terms of (VWm ) (1/coxo) in order to arrive at a frequency exponent of co. The CBH model gives the exponent p which is (l-j ) as 

Therefore the exponent s in the power law form is [1-6 77 JVm] suggesting that s decreases with increase in temperature, which is the most commonly observed behaviour of s. Another implication from the CBH model is that 

Since Edc EJ2, where Eq is the optical gap, Wm is itself a measure of the optical gap. 

Long (1982) derived an expression for the real part of the a.c. conductivity for the same CBH model, which is given by, 

Similarly the temperature dependence of a has been derived as a oc 7, where a = 1 - 5 In (l/(toTo)). In the case of small polarons there is a strong local polarization and an associated polaron formation energy, Wp. Due to the strongly localized nature of the distortions, there may not be any overlap between small polarons and the corresponding activation energy is simply Wn Wp/2, which is independent of the inter-site separation R. The small polaron transport is therefore characterized by a modified s value, which is given by. 

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For g(x) to be proportional to 1/x, it requires the relaxation time to be an exponential function of some random variable such that x = xq exp ( ), where itself has a flat distribution. It means that ( ) = constant, and rt(x) = n ). (d /dx) oc x. If a, the polarisability, is also a function of then it can lead to a sub-linear frequency dependence of a (co). The functional form given for the variation of x can arise from two different relaxation mechanisms. The first is a classical barrier hopping, in which two energetically favourable sites like in a double well potential are separated by a barrier fV and = W/kT. The second mechanism is a phonon assisted quantum tunneling through a barrier, which separates two equilibrium positions, in which case = 2aR, where a is the localization length and R is the separation between the sites. In the first case, by treating JV as independent of R, it has been shown (Poliak and Pike, 1972) that... 

Figure 12. The PT rate, r in BA as a function of temperature at a pressure of 2 kbar. Solid squares ( ) are data points from QENS and open circles (O) are values derived from NMR Tj measurements. The data asymptotically approach the incoherent quantum-tunneling regime at low temperature and the classical barrier-hopping regime at high temperature. The dashed and solid lines are different fits to theory (see text). (Adapted from Horsewill et al. [158].)...

Figure 8.12. Frequency dependence of the frequency exponent s for various models correlated-barrier hopping (CBH) (WM/kT=75 has been assumed) small polaron (SP), quantum mechanical tunneling (QMT) and overlapping large polaron (OLP)...

In(frequency) curves the value of the exponent is estimated to be 0.9 for the last two cases, which shows a possibility of variable range hopping phenomena [84]. This quantum mechanical tunneling or correlated barrier hopping is based on the pair approximation in which the motion of the carriers is contained within a pair of sites. If the DC and AC conductivities arise from the same hopping mechanism, the pair approximation cannot be applied [85]. Scher and Lax [86] proposed the continuous random network [65]. The frequency-dependent conductivity is given by... 

The molecular dynamics of polyvinyl alcohol (PVA) and carboxymethyl cellulose (CMC) blends was investigated as a function of composition, temperature and frequency using DRS [44]. PVA and CMC were found to be compatible over the range of composition studied. When the dielectric permittivity, loss tangent and a. c. conductivity of all samples were studied as functions of temperature and frequency, the results showed that the dielectric dispersion consisted of both dipolar and interfacial polarization. The frequency dependence of the a.c. conductivity indicated that correlated barrier hopping (CBH) was the most suitable mechanism for conduction. 

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