Band tails hopping conduction

Although carriers cannot conduct in localized states at zero temperature, conduction by hopping from site to site is possible at elevated temperatures. Hopping conduction in the band tail is given by. 

Conductivity sufficiently far down the band tails is by hopping and goes to zero at T = 0 K. The hopping may or may not have a polaron behavior depending on the size of the phonon interaction. 

Figure 7. LFCM volt-ampere characteristic at T = 294 K. One can identify non-ohmic regime connected with hopping conductivity in the band tails.

The temperature dependence of the total conductivity of As2Te3 is shown in Figure 5.27 for two frequencies. These measurements by Rockstad (1972) show that the a.c. conductivity of the small band gap materials may be composed of three contributions. The low temperature part is proportional to T as described by Eq. (5.33). If this portion is extrapolated to higher T and added to the d.c. conductivity, one obtains less than the total a.c. conductivity. The additional component 02 is also proportional to but rises more rapidly with T than Eq. (5.33). Rockstad (1971, 1972) attributes 02 to hopping conduction in localized tail states. The insert in Figure 5.27 shows that 02 is thermally activated. The slope is always smaller... 

In all cases, ac conductivity for hopping mechanisms follows a law dependent on frequency o-ac = < > (with. v 1) (Austin and Mott f79J) this law agrees also with polaronic tunneling considered sometimes (see p. 346 of Ref. 69). Moreover [26,27], as a function of temperature, this law is either thermally activated (conduction process in band tails) or linear (conduction in levels near Ey). 

While the field-dependent hopping conductivity at low temperatures was always a challenge for theoretical description, the theories for the temperature dependence of the hopping conductivity at low electric fields were successfully developed for all transport regimes for the dark conductivity [28, 43], for the drift mobility [29], and for the photoconductivity [30]. In all these theories, hopping transitions of electrons between localised states in the exponential band-tails play a decisive role, as described above. 

The study of the nature of states in band tails (the Anderson problem, Section 7.4) has not yet progressed to the point where detailed quantitative comparison with experiment are meaningful. So far the emphasis has been on the extraction of qualitative and semiquantitative consequences of the localization of such states. Mott concluded that, if conduction was due to phonon-assisted hopping among such localized states, it should vary with temperature... 

Conduction can also operate in the band tails below the mobility edge when such tails are wide and have a sufficiently high density of states to allow rapid hopping of a carrier from one site to another via thermally-activated processes. In this case the mobility of a carrier depends exponentially upon the amount of energy it must gain to leave a given state and hop to the next. Thus, it was proposed that the mobility will have the form [5]... 

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