Hopping temperature dependence

Figure 12-2. Temperature dependent j l0 characteristics for injection into a hopping system characterized by A=0.4eV and a=0.08eV (Ref. 1211).

Figures 12-17 and 12-18 show the temperature dependencies of the mobility in a hopping system with a Gaussian DOS of variance <7=0.065 eV as a function of the relative concentration c of traps of average depth ,=0.25 eV and as a function of the trap depth E, at a fixed concentration < =0.03, respectively. For c=0...

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... 

Besides its temperature dependence, hopping transport is also characterized by an electric field-dependent mobility. This dependence becomes measurable at high field (namely, for a field in excess of ca. 10d V/cm). Such a behavior was first reported in 1970 in polyvinylcarbazole (PVK) [48. The phenomenon was explained through a Poole-ITenkel mechanism [49], in which the Coulomb potential near a charged localized level is modified by the applied field in such a way that the tunnel transfer rale between sites increases. The general dependence of the mobility is then given by Eq. (14.69)... 

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. 

Fig. 11 63CuMAS-NMR at 9.0 kHz spinning speed, partial spectra of y-Cul (ZB structure) diluted in an inert matrix, showing broadening of first three STs as temperature is increased. The spectra shift to the right due to the temperature dependence of the chemical shift. Quantitative analysis of the broadening yields an activation energy for Cu+ hopping of 0.64 eV. Reprinted from [122]...

Hines T, Diez-Perez I, Hihath J, Liu HM, Wang ZS, Zhao JW, Zhou G, Muellen K, Tao NJ (2010) Transition from tunneling to hopping in single molecular junctions by measuring length and temperature dependence. J Am Chem Soc 132 11658-11664... 

Finally, recently depolarized light scattering spectra [191] display an additional process that shows a much faster characteristic time and a much weaker temperature dependence than the dielectric j0-relaxation (more than three orders of magnitude faster time at -200 K and an activation energy of 0.16 eV, about half of the dielectric value). Also atomistic simulations on PB have indicated hopping processes of the frans-double bond [192,193] with an associated activation energy of -0.15 eV. Whether these observations may be related with the discrepancy in the apparent time scale of the NSE and dielectric experiments remains to be seen. 

Figure 4 Left Distribution of hopping times for an adatom at a solid-liquid interface at 600 K for conventional molecular dynamics and for superstate parallel-replica dynamics. Right Temperature dependence of the hopping rate for an adatom at a dry interface and at a wet interface as obtained using superstate parallel-replica dynamics.

Usually it is assumed that tc is the only temperature-dependent variable in Eq. 9. This might be the case for an order-disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square fluctuation amplitudes are reduced by the so-called depopulation factor (l - and become in this way temperature-dependent also. The temperature dependence of tc in this model is given by Eq. 8, i.e. r would be zero at Tc, proportional to (T - Tc) above Tc and proportional to (Tc - T) below Tc. 

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