Hopping crystals

Inter-atomic two-centre matrix elements (cp the hopping of electrons from one site to another. They can be described [7] as linear combmations of so-called Slater-Koster elements [9], The coefficients depend only on the orientation of the atoms / and m. in the crystal. For elementary metals described with s, p, and d basis fiinctions there are ten independent Slater-Koster elements. In the traditional fonnulation, the orientation is neglected and the two-centre elements depend only on the distance between the atoms [6]. (In several models [6,... 

The diffusion of H and D atoms in the molecular crystals of hydrogen isotopes was explored with the EPR method. The atoms were generated by y-irradiation of crystals or by photolysis of a dopant. In the H2 crystals the initial concentration of the hydrogen atoms 4x 10 mol/cm is halved during 10 s at 4.2 K as well as at 1.9 K [Miyazaki et al. 1984 Itskovskii et al. 1986]. The bimolecular recombination (with rate constant /ch = 82cm mol s ) is limited by diffusion, where, because of the low concentration of H atoms, each encounter of the recombinating partners is preceded by 10 -10 hops between adjacent 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... 

At very low temperatures, Holstein predicted that the small polaron would move in delocalized levels, the so-called small polaron band. In that case, mobility is expected to increase when temperature decreases. The transition between the hopping and band regimes would occur at a critical temperature T, 0.40. We note, however, that the polaron bandwidth is predicted to be very narrow ( IO Viojo, or lO 4 eV for a typical phonon frequency of 1000 cm-1). It is therefore expected that this band transport mechanism would be easily disturbed by crystal defects. 

In oxide ion conductors, current flow occurs by the movement of oxide ions through the crystal lattice. This movement is a result of thermally activated hopping of the... 

Unlike a geometrical factor, the value of the factor

with composition in a predictable way. To illustrate this, suppose that stoichiometric MO2 is heated in a vacuum so that it loses oxygen. Initially, all cations are in the M4+ state, and we expect the material to be an insulator. Removal of O2- to the gas phase as oxygen causes electrons to be left in the crystal, which will be localized on cation sites to produce some M3+ cations. The oxide now has a few M3+ cations in the M4+ matrix, and thermal energy should allow electrons to hop from M3+ to M4+. Thus, the oxide should be an n-type semiconductor. The conductivity increases until

reduction continues, eventually almost all the ions will be in the M3+ state and only a few M4+ cations will remain. In this condition it is convenient to imagine holes hopping from site to site and the material will be a p-typc semiconductor. Eventually at x = 1.5, all cations will be in the M3+ state and M2C>3 is an insulator (Fig. 7.3). 

Ion exclusion chromatography, of ascorbic acid, 25 760 Ion hopping, 14 469 Ionic aggregates, 14 463—466 Ionically conducting polymers, 13 540 Ionic carbides, 4 647 Ionic compounds, rubidium, 21 822 Ionic conduction, ceramics, 5 587-589 Ionic crystals, 19 185. See also Silver halide crystals... 

Ion hopping is a familiar concept in the chemistry of solid-state conductors, e.g. in the semiconductor industry. In the fluoride electrode, fluoride vacancies in.side the solid LaF lattice allow for conduction of charge (see Figure 3.11), in turn registered by the electrode as a potential. The emf is zero if the internal and external solutions are the same because the same numbers of fluoride ion enter the crystal from either face. 

The activation energy represents the ease of ion hopping, as already indicated above and shown in Fig. 2.5. It is related directly to the crystal structure and in particular, to the openness of the conduction pathways. Most ionic solids have densely packed crystal structures with narrow bottlenecks and without obvious well-defined conduction pathways. Consequently, the activation energies for ion hopping are large, usually 1 eV ( 96 kJ mole ) or greater and conductivity values are low. In solid electrolytes, by contrast, open conduction pathways exist and activation energies may be much lower, as low as 0.03 eV in Agl, 0.15 eV in /S-alumina and 0.90 eV in yttria-stabilised zirconia. 

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