Hopping conductivity

For an adequate description of the temperature behavior of electric resistivity of complex salts [N-CH3-Pz](MTCNQ)2 and [N-C2H5-Pz](MTCNQ)2 we use a model based on a conductivity hopping mechanism possibly conditioned by the structural peculiarities of the ARS ... 

Vuilleumier R, Borgis D (2012) Proton conduction Hopping along hydrogen bonds. Nat Phys 119 432... 

In solid state materials, single-step electron transport between dopant species is well known. For example, electron-hole recombination accounts for luminescence in some materials [H]. Multistep hopping is also well known. Models for single and multistep transport are enjoying renewed interest in tlie context of DNA electron transfer [12, 13, 14 and 15]. Indeed, tliere are strong links between tire ET literature and tire literature of hopping conductivity in polymers [16]. 

As is to be expected, inherent disorder has an effect on electronic and optical properties of amorphous semiconductors providing for distinct differences between them and the crystalline semiconductors. The inherent disorder provides for localized as well as nonlocalized states within the same band such that a critical energy, can be defined by distinguishing the two types of states (4). At E = E, the mean free path of the electron is on the order of the interatomic distance and the wave function fluctuates randomly such that the quantum number, k, is no longer vaHd. For E < E the wave functions are localized and for E > E they are nonlocalized. For E > E the motion of the electron is diffusive and the extended state mobiHty is approximately 10 cm /sV. For U <, conduction takes place by hopping from one localized site to the next. Hence, at U =, )J. goes through a... 

Semiconductivity in oxide glasses involves polarons. An electron in a localized state distorts its surroundings to some extent, and this combination of the electron plus its distortion is called a polaron. As the electron moves, the distortion moves with it through the lattice. In oxide glasses the polarons are very localized, because of substantial electrostatic interactions between the electrons and the lattice. Conduction is assisted by electron-phonon coupling, ie, the lattice vibrations help transfer the charge carriers from one site to another. The polarons are said to "hop" between sites. 

In the above consideration it has been tacitly assumed that the charge carrier mobility docs not depend on the electric field. This is a good approximation for molecular crystals yet not for disordered systems in which transport occurs via hopping. Abkowitz et al. [37] have solved that problem for a field dependence of ft of the form p-po (FIFU) and trap-free SCL conduction. Their treatment predicts... 

An important property of the electron Hamiltonian (Eq. (3.3)) is that for arbitrary hopping amplitudes the spectrum of the single-electrons slates is symmetric with respect to c=0 if is the electron amplitude on site n of an eigenstate with energy c, then the state with amplitudes —)"< > is also an eigenstate, with energy -c. In particular, in the uniformly dimerized stale, the gap between the empty conduction and the completely filled valence bands ranges from -A, to A(). 

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... 

Electronic conductivity arises from delocalized electrons, tunneling or hopping processes. 

Figure 4. (A) Cyclic voltammograms over a range of scan rates for a redox polymer (poly-[Fe 5-amino-1,10-phenanthrotme)3]3+/>)91 and (B) p-doping and undoping of a conducting polymer (polypyrrole) (B). [(A) Reprinted from X. Ren and P. O. Pickup, Strong dependence of the election hopping rate in poly-tris(5-amino-1,10-phenan-throline)iron(HI/II) on the nature of the counter-anion J. Electroanal. Chem. 365, 289-292,1994, with kind permission from Elsevier Sciences S.A.]...

If the film is nonconductive, the ion must diffuse to the electrode surface before it can be oxidized or reduced, or electrons must diffuse (hop) through the film by self-exchange, as in regular ionomer-modified electrodes.9 Cyclic voltammograms have the characteristic shape for diffusion control, and peak currents are proportional to the square root of the scan speed, as seen for species in solution. This is illustrated in Fig. 21 (A) for [Fe(CN)6]3 /4 in polypyrrole with a pyridinium substituent at the 1-position.243 This N-substituted polypyrrole does not become conductive until potentials significantly above the formal potential of the [Fe(CN)6]3"/4 couple. In contrast, a similar polymer with a pyridinium substituent at the 3-position is conductive at this potential. The polymer can therefore mediate electron transport to and from the immobilized ions, and their voltammetry becomes characteristic of thin-layer electrochemistry [Fig. 21(B)], with sharp symmetrical peaks that increase linearly with increasing scan speed. 

Apart from the problems of low electrocatalytic activity of the methanol electrode and poisoning of the electrocatalyst by adsorbed intermediates, an overwhelming problem is the migration of the methanol from the anode to the cathode via the proton-conducting membrane. The perfluoro-sulfonic acid membrane contains about 30% of water by weight, which is essential for achieving the desired conductivity. The proton conduction occurs by a mechanism (proton hopping process) similar to what occurs... 

Point defects in solids make it possible for ions to move through the structure. Ionic conductivity represents ion transport under the influence of an external electric field. The movement of ions through a lattice can be explained by two possible mechanisms. Figure 25.3 shows their schematic representation. The first, called the vacancy mechanism, represents an ion that hops or jumps from its normal position on the lattice to a neighboring equivalent but vacant site or the movement of a vacancy in the opposite direction. The second one is an interstitial mechanism where an interstitial ion jumps or hops to an adjacent equivalent site. These simple pictures of movement in an ionic lattice, known as the hopping model, ignore more complicated cooperative motions. 

Solid mixed ionic-electronic conductors (MIECs) exhibit both ionic and electronic (electron-hole) conductivity. Naturally, in any material there are in principle nonzero electronic and ionic conductivities (a i, a,). It is customary to limit the use of the term MIEC to those materials in which a, and 0, 1 do not differ by more than two orders of magnitude. It is also customary to use the term MIEC if a, and Ogi are not too low (o, a i 10 S/cm). Obviously, there are no strict rules. There are processes where the minority carriers play an important role despite the fact that 0,70 1 exceeds those limits and a, aj,i< 10 S/cm. In MIECs, ion transport normally occurs via interstitial sites or by hopping into a vacant site or a more complex combination based on interstitial and vacant sites, and electronic (electron/hole) conductivity occurs via delocalized states in the conduction/valence band or via localized states by a thermally assisted hopping mechanism. With respect to their properties, MIECs have found wide applications in solid oxide fuel cells, batteries, smart windows, selective membranes, sensors, catalysis, and so on. 

---