Defect hopping

Diffusion in solids occnrs as Schottky or Frenkel defects hop from one lattice site to another by a thermally activated process akin to chemical reactions. 

How strongly a high defect concentration affects the temperature dependence of the mobility can be seen from the comparison of the experimental results for a perylene crystal with a concentration of 0.17% traps of depth Et = 270 meV (Fig. 8.33). Even near room temperature, the mobility ii decreases steeply with decreasing temperature. In this crystal with defects, hopping conductivity occurs even in the temperature region near 300 K. 

In addition to the configuration, electronic stmcture and thennal stability of point defects, it is essential to know how they diffuse. A variety of mechanisms have been identified. The simplest one involves the diffusion of an impurity tlirough the interstitial sites. For example, copper in Si diffuses by hopping from one tetrahedral interstitial site to the next via a saddle point at the hexagonal interstitial site. 

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. 

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. 

The charge transport in a conjugated chain and the interchain hopping is explained in terms of conjugation defects (radical or ionic sites), called solitons and polarons. Several possible conjugation defects are demonstrated in Fig. 5.33 on the example of trans-polyacetylene. 

The electronic band structure of a neutral polyacetylene is characterized by an empty band gap, like in other intrinsic semiconductors. Defect sites (solitons, polarons, bipolarons) can be regarded as electronic states within the band gap. The conduction in low-doped poly acetylene is attributed mainly to the transport of solitons within and between chains, as described by the intersoliton-hopping model (IHM) . Polarons and bipolarons are important charge carriers at higher doping levels and with polymers other than polyacetylene. 

The composition of a slightly nonstoichiometric cobalt oxide is C00.999O. (a) Write a defect formation equation for this phase assuming that electronic compensation occurs, (b) If conductivity takes place by hopping, what is the value of the factor cp in Eq. (7.1) ... 

In practice, the defect structure of the materials LiJCo, M)02 and Lix(Ni, M)02 under oxidizing conditions found at cathodes, is complex. For example, oxidation of Fe3+ substituted lithium nickelate, LL(Ni, Fe)02, under cathodic conditions leads to the formation of Fe4+ and Ni4+. Conductivity can then take place by means of rapid charge hopping between Fe3+, Ni3+, Fe4+, and Ni4+, giving average charges of Fe3+S and Ni3+S. These solids are the subject of ongoing research. 

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