Temperature hopping rate

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. 

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... 

Fig. 6. The hop rate of muonium in KC1 as a function of temperature. The crossover from stochastic to quantum diffusion occurs at about 70 K, as evidence by a dramatic increase in the hop rate at lower temperatures. From Kiefl et al. (1989a).

The ion hopping rate is an apparently simple parameter with a clear physical significance. It is the number of hops per second that an ion makes, on average. As an example of the use of hopping rates, measurements on Na )3-alumina indicate that many, if not all the Na" ions can move and at rates that vary enormously with temperature, from, for example, 10 jumps per second at liquid nitrogen temperatures to 10 ° jumps per second at room temperature. Mobilities of ions may be calculated from Eqn (2.1) provided the number of carriers is known, but it is not possible to measure ion mobilities directly. 

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.

Extensive discussions of this problem are given in pertinent monographs (e.g., [A. R. Allnatt, A. B. Lidiard (1993)]). We will instead present Vineyard s version and add a few comments which are relevant to diffusional transport in crystals. This version yields for the vacancy (VA) hopping rate in crystal A at a given temperature... 

The structure of Eq. (7.22) is the same as that of Eq. (2.1). It determines the statistically averaged probability of incoherent transition at T>TC. Note that it is the interference of many waves that renders the transition incoherent. Formally below the crossover temperature this formula should describe the rate of the incoherent transitions from the ground state. However, the transition under these conditions should be coherent. This means that the incoherent hopping rate should vanish at 7 = 0. To take this fact into account Clough et al. [1982] calculated 1 statistically averaging (mn) - (m0) rather than mn). Their results show that this correction is significant only for low barriers and T TC. 

The main point of the argument set out in Refs. 101 and 102 is whether < ) increases or decreases in the low-temperature limit at high frequencies (i.e., fh0 > 1). Note that the reversed characteristic intrawell time x01, which remains finite at T —> 0, is the only natural frequency scaling parameter here, since the interwell hopping rate xj"1, which is exponential in T, may not be used for this purpose. 

The transverse charge motion is incoherent for a quasi-one-dimensional conductor as long as the condition h hiT is fulfilled (where the intrachain scattering time provides a broadening to the quasi-one-dimensional Fermi surface). This situation is encountered at high temperature. The interchain hopping rate is thus determined by the one-dimensional Fermi golden rule [60] ... 

The model predicts that below E and at elevated temperatures, the conductivity is given by the usual thermally activated hopping rate. 

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