Hopping conductivity in a quasi-one-dimensional lattice gas with three-dimensional ordering

The dc ionic conductivity is calculated for one-dimensional (1-D) classical hopping with effects of nearest-neighbor repulsion included. Repulsion between ions in different channels which leads to three-dimensional ordering of the ions is accounted for in a mean-field manner. The intrachannel repulsion $U$ is treated exactly by using results for the equivalent Ising antiferromagnet in a staggered field. It is shown that several choices can exist for the dependence of transition probabilities on nearest-neighbor occupation numbers and still satisfy detailed balance. In almost any case, however, the activation energy increases by $\frac{U}{2}$ as the temperature $T$ goes through the ordering temperature $T_c$ from above. An appreciable change in activation energy should then be observed in a 1-D superionic conductor which undergoes an order-disorder transition, provided this transition is triggered by interactions between the mobile ions. The dependence of activation energy upon $U$ above $T_c$ depends on the range of the forces and whether the hopping is purely classical or involves tunneling. We find that Kikuchi's result of a decrease in activation energy by $\frac{U}{2}$ from the noninteracting value is reproduced if very-short-range forces and classical activation over a barrier are assumed. On the other hand, we get Mahan's result of an increase by $\frac{U}{2}$ if the transition rate is governed by tunneling through a barrier.