Electron transport delocalized state

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. 

Many ferromagnets are metals or metallic alloys with delocalized bands and require specialized models that explain the spontaneous magnetization below Tc or the paramagnetic susceptibility for T > Tc. The Stoner-Wohlfarth model,6 for example, explains these observed magnetic parameters of d metals as by a formation of excess spin density as a function of energy reduction due to electron spin correlation and dependent on the density of states at the Fermi level. However, a unified model that combines explanations for both electron spin correlations and electron transport properties as predicted by band theory is still lacking today. 

The ratio Vo/B determines the transition from coherent diffusive propagation of wavefunctions (delocalized states) to the trapping of wavefunctions in random potential fluctuations (localized states). If I > Vo, then the electronic states are extended with large mean free path. By tuning the ratio Vq/B, it is possible to have a continuous transition from extended to localized states in 3D systems, with a critical value for Vq/B. Above this critical value, wave-functions fall off exponentially from site to site and the delocalized states cannot exist any more in the system. The states in band tails are the first to get localized, since these rapidly lose the ability for resonant tunnel transport as the randomness of the disorder potential increases. If Vq/B is just below the critical value, then delocalized states at the band center and localized states in the band tails could coexist. 

The conditions for validity of the NFE model break down for conductivity values near the loffe-Regel limit, that is, o-(O) 07. As discussed in Sec. 2.3.2, electronic transport in this situation is best described as diffusion of electrons from atom to atom, even though the system is stiD metallic with delocalized electron states. In this connection, it is interesting that near the range where cr(0) 07, the temperature coefiScient d n.(r(Q)/d nT)y is clearly positive (Freyland, 1981 Freyland et al., 1974). 

In this section we shall discuss simple models of electron transport. Some more detailed considerations will be presented in Chapter 7. Basically, three cases will be described here (1) electrons moving in delocalized states (motion in the conduction band) (2) trap-modulated motion in delocalized states and (3) hopping mobility. The three modes of transport are depicted schematically in Figure 35. 

While in pure TMSi, the electron mobility is almost independent of temperature in the interval concerned, the mobility in the solution shows a strong dependence of temperature, although the absolute values are quite high and are indicative of transport in the delocalized state (see Table 14). According to Equation 75, the electron... 

If Vq < 0, transfer of delocalized electrons from the liquid into the vapor is a thermally activated process. Delocalized electrons in the vapor phase can transfer easily into liquids with Vq < 0. If Vq > 0, delocalized electrons can in principle transfer with no activation energy from the liquid into the vapor, while delocalized electrons in the vapor phase encounter the barrier I Vq I. In most cases, in liquids with Vq > 0 the electron transport proceeds via localized states. When a localized electron comes close to the liquid surface, temporary trapping occurs due to the image force. Emission into the vapor phase is then a thermally activated process, too. 

Electrons in these liquids spend most of their time in localized states. One model of electron transport, derived from semiconductor theory, is that each electron is from time to time thermally excited into the delocalized state (conduction band), where it migrates relatively freely until it becomes de-excited into a localized state again. 

The mobility of the localized state is ionlike and is several orders of magnitude lower than that in the delocalized state. In alcohols the traps are so deep that the electrons are not thermally excited out of them to an appreciable extent the measured electron mobilities are ionlike. However, in liquid alkanes or alkenes the traps are shallow enough that most of the transport occurs by excitation to the delocalized state. 

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