Electronic conduction theory hopping

While the field-dependent hopping conductivity at low temperatures was always a challenge for theoretical description, the theories for the temperature dependence of the hopping conductivity at low electric fields were successfully developed for all transport regimes for the dark conductivity [28, 43], for the drift mobility [29], and for the photoconductivity [30]. In all these theories, hopping transitions of electrons between localised states in the exponential band-tails play a decisive role, as described above. 

However, on the one hand, low ESR signals alone are a weak argument for the assumption of hole bipolarons. On the other hand, several experimental results are in contradiction of this model. For example, (a) the electrical conductivity of boron carbide is maximum at the minimum concentration of BnC icosahedra in the homogeneity range (b) polaron-type effects are restricted to one electron per icosahedron and no corresponding electron-phonon interaction with holes, in particular not with hole pairs in icosahedra, has been proved experimentally (c) the distortion of the icosahedra in boron carbide depends to only a small degree on electron-phonon interaction and (d) the electronic transport in boron-rich solids is due to classical band-type conduction and hopping processes side by side. Hence, the hole bipolaron theory for boron-rich solids can hardly be maintained. 

This notion of occasional ion hops, apparently at random, forms the basis of random walk theory which is widely used to provide a semi-quantitative analysis or description of ionic conductivity (Goodenough, 1983 see Chapter 3 for a more detailed treatment of conduction). There is very little evidence in most solid electrolytes that the ions are instead able to move around without thermal activation in a true liquid-like motion. Nor is there much evidence of a free-ion state in which a particular ion can be activated to a state in which it is completely free to move, i.e. there appears to be no ionic equivalent of free or nearly free electron motion. 

Eq. (18) is usually attributed to the variable-range hopping conductivity in presence of the Coulomb gap [34]. However, the analysis [72,75,76] shows that it is unrealistic explanation for the case of nanocomposites, because to fit experimental value on the basis of this theory one has to assume that the length of a single hop is less than the size of granules D and the electron... 

Thus, when a particle jumps, it leaves behind a hole. So then, instead of saying that a transport process occurs by particles hopping along, one could equally well say that the transport processes occur by holes moving. The concept is commonplace in semiconductor theory, where the movement of electrons in the conduction band is taken as being equivalent to a movement of so-called holes in the valence band. It has in fact already been assumed at the start of the viscosity treatment (Section 5.7.1) that the viscous flow of fused salts can be discussed in terms of the momentum transferred between liquid layers by moving holes. 

In the usual space-charge limited theory, electrons are injected into the insulator conduction band, and some of these electrons are immobilized in localized defect states. We have considered an alternate mechanism more appropriate to the polymer structure. Contact charge transfer studies in Polyethylene Terephthalate (PET) and other polymers (15-16) suggest that the electronic states accessible from metal contacts are localized molecular-ion states located deep in the forbidden energy gap. Charge transport is by hopping between localized states. 

Soviet scientists have been particularly interested in impurity effects in ID metals. It was shown (INV 8) that impurities in a half-filled band give a singular enhancement in the density of states at the Fermi surface. This may be another manifestation of the well known impurity localization of states in ID. This latter implies (INV 13) that at T=0, o(to)-K) as u>-K). With increasing temperature, phonons allow a hopping type transport from one localized site to another, with increasing conductivity. At still higher temperatures, phonons scatter the electrons with a corresponding decrease in a(co). The theory developed fits quantitatively with experiments on TCNQ salts with structural disorder. 

The theory of hopping conduction described above differs greatly in its fundamental structure from typical theories of conduction which start with electrons in... 

The conventional hopping theory of the phonon-assisted (relaxation) AC conductivity for noninteracting electrons predicts the dependence with exponent s slightly below 1 [3]. A similar frequency dependence (with a somewhat weaker temperature dependence) is also predicted in the presence of Coulomb interaction [4],... 

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