Range Hopping

In highly defective amorphous semiconductor, there is a significant DOS in the gap and electron transport can also occur among such states. 

The conductivity is calculated using the relation, p = eD/kT, where D = v )/6, so that 

Substituting this back into the previous expression for A, one gets Kptimum-The two, Ropiimum and AopUmum can together be substituted in the expression for conductivity so that the expression for a becomes 

---

More recently, a comprehensive model has been developed by Vissenberg and Matters [120 to account for these data. The model is based on a variable range-hopping system with an exponential distribution of localized slates (Eq. (14.71)). The... 

Another famous hopping model is Mott s variable range hopping [23], in which it is assumed that the localized sites are spread over the entire gap. At low temperatures, the probability to find a phonon of sufficient energy to induce a jump to the nearest neighbor is low, and hops over larger distances may be more favorable. In that case, the conductivity is given by... 

We note a temperature dependence of the zero field mobility as exp[—( F()/F)2], a behavior which is indeed encountered in real organic semiconductors, and differs from both Millers-Abrahams fixed range and Moll s variable range hopping models. 

Figure 24. Plot of the particle-size distribution versus the transition temperature Tross, which describes the crossover point between an activated transport mechanism (ln(A) oc EJT and variable range hopping (VRH) (ln(R)ocT ). Note that Tdoss has a OK value at a finite (3%) particle-size distribution. (Reprinted with permission from Ref. [56], 2002, American Chemical Society.)...

Evidence on this question may be taken by the behavior of the electrical conductivity CT as a function of temperature. A thermally activated process T dependence on log(CT), Arrhenius plot) is expected if doping takes place, whereas j -i/4 dependence, characteristic of a variable range hopping at the Fermi level is expected for a nondoping situation. 

In view of the complexity of the material, it is difficult to unequivocally assign a particular mechanism for electronic conduction in the polymer, although some evidence exists to suggest that it involves three-dimensional variable-range hopping as found for other polymers [213], It has also been suggested that the conductivity of polyaniline is a combination of both ionic and electronic conductivity [207], and is... 

Yu, D. Wang, C. Wehrenberg, B. L. Guyot-Sionnest, P. 2004. Variable range hopping conduction in semiconductor nanocrystal solids. Phys. Rev. Lett. 92 216802-216806. 

This distance is far beyond the equilibrium region of primary chemical interest, so we do not consider the long-range-hopping process further. 

Fig. 14 A comparison of different approaches to describe the charge carrier mobility in a Gaussian-type hopping system as a function of the normalized concentration of the charge carriers, (a) Full curves are the result of effective medium calculations [100] while symbols are computer simulations [96], (b) Full curves are calculated using the variable range hopping concept [101], symbols are the computer simulations. From [100] with permission. Copyright (2007) by the American Institute of Physics...

The tail states associated with F"-ion substitutions also introduce the possibility of variable-range hopping and hence a In a versus dependence. Indeed, Graener et... 

At sufficiently low temperatures, under all circumstances where N(EF) is finite but states are localized near the Fermi energy, we expect the phenomenon of variable-range hopping to set in. If interaction between electrons is not taken into account then the conductivity will follow the formula... 

Fig. 130 Temperature dependence of variable-range hopping in a sample of Si P plotted against T 1/2 ----------------, plotted against T 1/4 (Ionov et al 1985).

The Hall effect in the temperature range where conduction is by variable-range hopping is not well understood. Evidence from the early work by Fritzsche is discussed by Shklovksii and Efros (1984), who come to the conclusion that the Hall mobility must be small. Hopkins et al (1989) have investigated the behaviour of heavily doped Ge Sb, pushed into the non-metallic regime by magnetic fields up to 7 T, at temperatures down to 100 mK. Below 1 K the... 

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... 

---