Small polaron mechanism

Similar investigations have been carried out for the system LaCr xMnx03 [106]. A significant improvement in sinterability appears when Mn is substituted for Cr. For example, densities above 95% of theoretical were achieved at 1475 °C in air for La0.9Sr0 ]Cr03Mn07O3. Electrical conductivity and Seebeck coefficient results are interpreted by a small polaron mechanism for all compositions. This is illustrated for conductivity in Fig. 33. It was also demonstrated that the carrier (electron hole) mobility rather than carrier concentration governs the electronic transport. 

Lor other oxides it has been suggested that the interactions between the electronic defects and the surrounding lattice can be relatively strong and more localised. If the dimension of the polaron is smaller than the lattice parameter, it is called a small polaron or localised polaron, and the corresponding electronic conduction mechanism is called a small polaron mechanism. 

As for pure GeOg-, electronic conductivity is introduced in doped ceria by increasing temperature and decreasing pOg, due to the reduction of Ge to Ge and the formation of electronic defects. The n-type conductivity takes place via a small polaron mechanism. According to Eqs (12.30), (12.31), and (12.32), the electronic conductivity of doped ceria is a function of the degree of reduction x, the electronic migration enthalpy and the preexponential term of the electron mobility l . 

Because polarons are localized species, their natural transport mechanism is hopping. We shall now briefly describe the small polaron model, as developed by Holstein and Emin [26, 29, 46]. 

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. 

The experiments just discussed made it clear that the motion of the hole on the series of As represents a different mechanism of transport than tunneling. Giese [13] and Bixon and Jortner [18] suggested that this mechanism is incoherent hopping of the hole between neighboring bases. This means that the hole wavefunction is Hmited to one base. The wavefunctions of the remaining electrons on that base would of course be distorted by the presence of the hole. Thus in this view of the transport process the base on which the hole sits could be called a molecular polaron, or a small polaron because it is limited to one site. 

The disappearance of the sharp Verwey transition was discussed by Mott (1979), who suggested that at low temperatures the material is a Wigner glass , the electrons (Fe2 + ions) being frozen into random sites and the whole system stabilized by the fluorine. Discussion of the thermopower measurements show, according to Mott (1979), that a hopping mechanism is operative at low T. Ihle and Lorenz (1985), however, consider that the electrons in the wrong sites move by a small polaron band mechanism. 

Another mechanism that has been proposed is that the carriers move as small polarons20. A small polaron is a carrier that is self trapped in a well created by the lattice distortion. This lattice distortion is formed when a carrier stays sufficiently long in a position to polarize the medium around it. The applied field can lower the polaron barrier in a PF fashion and increase the mobility. The polaron transport model is attractive in that the mobilities in this mechanism are not critically dependent on the sample preparation. 

Electronic conduction in inorganic melts can occur when atoms of the same kind in different oxidation states are present. Such systems exhibit an increased electrical conductivity with an exponential character of its temperature dependence, caused by a diffusion-like motion called hopping mechanism. The hopping mechanism is characterized by low mobility at elevated temperatures and the charge carrier is termed as a small-polaron. The mobility of the small-polaron is much lower compared to that of the carrier in a broad semi-conductor band. 

At low temperatures, the small-polaron moves by Bloch-type band motion, while at elevated temperatures it moves by thermally activated hopping mechanism. Holstein (1959), Friedman and Holstein (1963), Friedman (1964) performed the theoretical calculations of small-polaron motion and showed that the temperature dependencies of the small-polaron mobility in the two regimes are different. In the high-temperature hopping regime, the electrical conductivity is thermally activated and it increases with increasing temperature. As shown by Naik and Tien (1978), its temperature dependence is characterized by the following equation... 

The carrier transport characteristics were analyzed using Bassler s disorder formalism and Holstein s small polaron theory however, the temperature range in which the carrier mobility was measured precisely was not so wide that the transport mechanism could be determined clearly [108]. 

A number of recent calculations have compared the classical result with quantum mechanical calculations. In many cases, the results from the latter techniques confirm those from classical calculations with a gratifying accuracy. However, one topic on which there is continuing controversy is the nature of the polarons in transition metal oxides. Since the classical method subsumes all the quantum mechanics of the problem into the potential function, it can only tackle problems of electronic structure in a few specific cases, the most common example of which is in non-stoichiometric oxides. Here the question is the location of the electronic hole when the system is metal deficient. The only way such a problem can be tackled by classical methods is to use the small polaron approximation and assume that the hole resides on an ion to produce a new (in effect substitutional) ion with an extra positive charge. This can be successful and the use of the small polaron approximation in crystals is discussed in detail by Shluger and Stoneham (1993). However, all calculations on the first-row transition metal oxides have assumed that the extra charge resides on the metal ion. Recent quantum calculations (Towler et al., 1994) have thrown doubt on this assumption, suggesting that the hole is on the oxide ion. Moreover, the question of whether the hole is a small polaron for all these oxides is, at present, quite uncertain. Further discussion is given in Chapter 8. 

Note term comes from the pre-exponential term in Eq. 7.51, i.e. from the density of states. This result is applicable to an intrinsic semiconductor for which phonon scattering is responsible for the temperature dependence of the electronic mobility, i.e., one in which the mobility decreases with increasing temperature. Other possibilities exist, however two of the more important ones are the small and large polaron mechanisms discussed below. A polaron is a defect in an ionic crystal that is formed when an... 

Small polaron. In this mechanism, conduction occurs by the hopping of the electronic defects between adjacent ions of, usually but not necessarily, the same type but with varying oxidation states. Because of the ease by which transition-metal ions can vary in oxidation states, this type of conduction is most often observed in transition metal oxides. For example, if the charge carrier is an electron, the process can be envisioned as... 

Figure 8.12. Frequency dependence of the frequency exponent s for various models correlated-barrier hopping (CBH) (WM/kT=75 has been assumed) small polaron (SP), quantum mechanical tunneling (QMT) and overlapping large polaron (OLP)...

Electronic hole will be formed on Cr" " sites, and the conduction mechanism is a small polaron hopping process via Cr" " sites. The electronic conductivity is about 10-100 Scm at 1,273 K in air [22,23]. The electronic conductivity increases with increasing temperature, suggesting the semiconductor temperature dependence. An increasing of the Ca concentration in Lai xCaxCr03 8 enhanced the electronic conductivity due to the increase of Cr" " concentration. There are some deviations of the electrical conductivity among the examined alkaline earth elements Ca-doped LaCr03 shows the... 

The transport of small polarons in an ionic solid may take place by two different mechanisms. At low temperatures small polarons may tunnel between localised sites in what is referred to as a narrow band. The temperature dependence of the mobility is determined by lattice scattering and the polaron mobility decreases with increasing temperature in a manner analogous to a broad band semiconductor. 

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