Temperature hopping models

A celebrated derivation of the temperature dependence of the mobility within the hopping model was made by Miller and Abrahams 22. They first evaluated the hopping rate y,y, that is the probability that an electron at site i jumps to site j. Their evaluation was made in the case of a lightly doped semiconductor at a very low temperature. The localized states are shallow impurity levels their energy stands in a narrow range, so that even at low temperatures, an electron at one site can easily find a phonon to jump to the nearest site. The hopping rate is given by... 

Another famous hopping model is Mott s variable range hopping [23], in which it is assumed that the localized sites are spread over the entire gap. At low temperatures, the probability to find a phonon of sufficient energy to induce a jump to the nearest neighbor is low, and hops over larger distances may be more favorable. In that case, the conductivity is given by... 

We note a temperature dependence of the zero field mobility as exp[—( F()/F)2], a behavior which is indeed encountered in real organic semiconductors, and differs from both Millers-Abrahams fixed range and Moll s variable range hopping models. 

As in the case of other polymers containing saturated cyclic side chains the a relaxation is obscured by low-frequency conductive effects. Moreover in the case of PCBuMM, dielectric activity is also observed. Therefore both polymers show important conductivity contributions at high temperatures and low frequency. The conductivity analysis in this case is also performed using the hopping model... 

Experimental dependences of resistivity p(T) obtained by the procedure [5], are presented in Fig. 1. As can be seen from Fig. 1, the value of resistivity and the character of its temperature dependence essentially differ for different samples. Sample I is the ensemble of the disordered carbon particles, sample II consists of the particles of disordered carbon with an insignificant amount of CNT (see Table 1). It was stated earlier the conductivity of such materials is described by the hopping model with the variable length of jump (Eq. (1)), that is confirmed experimentally (Fig. 2). Sample HI consists of the particles of disordered carbon up to 60 nm in size and CNT s ropes 2.4 mn in diameter, i.e., this sample is non uniform, and its conductivity can be described within the proposed model taking into account the first and the terms (Eq. (4)). 

The importance of the measurements that we have presented so far for the diffusion of embedded tracer atoms becomes evident when we now use these measurements and the model discussed in Section 3 to evaluate the invisible mobility of the Cu atoms in a Cu(00 1) terrace. The results presented in Section 2 imply that not just the tracer atom, but all atoms in the surface are continuously moving. From the tracer diffusion measurements of In/Cu(0 0 1) we have established that the sum of the vacancy formation energy and the vacancy diffusion barrier in the clean Cu(0 01) surface is equal to 717 meV. For the case of self-diffusion in the Cu(0 01) surface we can use this number with the simplest model that we discussed in Section 3.2, i.e. all atoms are equal and no interaction between the vacancy and the tracer atom. In doing so we find a room temperature hop rate for the self-diffusion of Cu atoms in a Cu(00 1) terrace of v = 0.48 s-1. In other words, every terrace Cu atom is displaced by a vacancy, on average, about once per two seconds at room temperature and about 200times/sec at 100 °C. We illustrate this motion by plotting the calculated average displacement rate of Cu terrace atoms vs. 1 /kT in Fig. 14. 

For the band model phonons are not needed to cause energy migration as in the hopping model and instead exciton-phonon scattering limits the mean free path of the excitons and, thus, inhibits the migration at high temperatures. The time between scattering events for diffusion in the ith direction can be calculated from... 

The temperature-dependent spectra were interpreted in terms of a two-site hop model, in which the deuterons undergo jumps through a dihedral angle of 103°. This type of motion is consistent with gauche-trans conformational transitions. At -88"C these motions appear static on the time scale of the deuterium NMR experiment, and at +85 °C the motions are in the fast exchange limit. The rate constants for these motions were obtained from the calculated spectra. An Arrhenius plot of these data show that the apparent activation energy is 5.8 kcal/mol. (Dynamic mechanical data (20 Hz) fall on the Arrhenius plot.) The transitions have an intermediate rate on the deuterium NMR time scale at 20 °C, with the correlation time for the motion being 7 x 10 6 s at this temperature. 

The frequency and temperature dependent dielectric losses in lightly doped poly-3 methylthiophene have been studied by Pameix [44b]. The frequency dependence (S) of ac conductivity (UacCxw ) was found to decrease linearly with temperature in agreement with a hopping model. 

---