1. Extensions of the Stochastic Model of the Overdamped Oscillator Applied to AC Ionic Conductivity in Solids Juan Bisquert Departament de Ciències Experimentals Universitat Jaume I, Spain Salamanca, 20 setember 2005

2. Background: ac conductivity of disodered solids

3. AC conductivity in solids In many different kinds of solids with disordered structures such as glasses, structurally disordered crystals, and polymers the ionic ac conductivity displays the same qualitative features. At low frequencies, the long range displacement of carriers gives the frequency independent (dc) conduction. At increasing frequencies, the conductivity increases approximately as a power law Ac conductivity of 50LiF–30KF– 20Al(PO3)3 glass

4. AC conductivity in solids The features of ac conductivity can be explained in terms of energy disorder. Long range transport at low frequencies is determined by the higher barriers in the percolation cluster Displacement in a restricted cluster of sites at intermediate frequencies. Hopping between a pair of sites at large frequencies.

5. Jump relaxation model Funke and coworkers explained the ac conductivity in terms of local dynamic effects K. Funke, Solid State Ionics 18-19, 183 (1986). a) Ions in a sublattice b) Effective single ion potential c) Develoment of the potential after an ion hop If at t = 0, an ion hops to a neighbouring site, then a mismatch occurs. After the hop, (1) the ion may hop back to its original site or 2) the neighbouring mobile ions rearrange during their own hopping motion. Situation (2) results in a shifting of the cage effect potential. A new ‘global’ minimum is formed for the given ion at its new site. The initial forward hop has succeeded and d.c. conduction occurs.

6. Jump relaxation model The jump relaxation model provides a quantitative description of ac conduction in ionic glasses at the cost of a very complex set of equations with phenomenological parameters K. Funke, R. D. Banhatti, S. Brückner, C. Cramer, C. Krieger, A. Mandacini, C. Martiny, and I. Ross, Phys. Chem. Chem. Phys. 4, 3155 (2002).

7. Modified overdamped oscillator

8. We present a model for the ac conduction in ionically conducting solids that takes into account, in a simple way, the interaction between carriers. The Coulomb force forms an “ionic atmosphere” that exerts a restoring force on a central ion, whose motion corresponds to an overdamped oscillator. We consider the effect of the relaxation of the ionic atmosphere by introducing an additional equation for the displacement of the potential towards the particle position. J. Bisquert, V. Halpern, and F. Henn, J. Chem. Phys. 122, 151101 (2005).

9. Overdamped oscillator Dynamics of a particle in a potential well Since the carrier is bound, long range displacement of the particle (dc conductivity) is not possible At low frequencies than the displacement is slowed down by the restoring force In the high frequency regime the effect of the elastic force can be neglected, and the particle behaves as if it were moving freely around the potential minimum under the influence of the viscous force only

10. Displacement of the potential cage We consider the rearrangement of the potential surface during the motion of the particle. When the hopping carrier is displaced from the equilibrium position in the ionic atmosphere, either it may return to the equilibrium position, as in oscillator model, or a rearrangement of the surrounding charge towards the new position of the carrier may occur.

11. Displacement of the potential cage We obtain a domain of constant conductivity at low frequencies that can be associated to the dc conductivity Dc conductivity

12. Displacement of the potential cage First the particle moves uphill in the potential well by the action of the external force. When the potential well relaxes towards the particle’s position, the particle loses the potential energy that it had in the previous configuration, and so the work that was provided by the external force in the uphill displacement is dissipated into heat. When the process is repeated, there is a sustained advancement of both the particle and its confinement potential.

13. dc conductivity Dc conductivity is governed by the combination of the restoring force and the potential relaxation The particle motion loses the information about the bound particle mobility that is determined by the microscopic friction.

14. Experimental results on nanoscale TiO2 Limitation of the mobility of charge carriers in a nanoscaled heterogeneous system by dynamical coulomb screening V. Kytin, T. Dittrich, J. Bisquert, E. A. Lebedev, and F. Koch, Phys. Rev. B 68, 195308 (2003). ionic conductivity Electron drift mobility TOF Impedance spectroscopy

15. Memory effects

16. GLE The generalized Langevin equation (GLE) describes temporal correlations in the interaction between the particle and the medium. The particle is coupled to a thermal bath represented by a set of oscillators. Solving the equation of motion of these oscillator modes, it is obtained that motion of the particle depends on its past history through a memory kernel

17. GLE We calculate the ac conductivity from the GLE, with an additional time constant representing the relaxation of the friction The additional relaxation of the friction displaces to lower frequencies the microscopic constant conductivity (high frequency), but the general pattern of the overdamped oscillator is not modified

18. Memory in the elastic force We assume an equation similar to the generalized Langevin equation (GLE) but with the memory effect in the elastic force At low frequencies the relaxation of the force introduces a negative friction which overcomes the positive component. This negative friction supplies power to the external circuit and causes a negative conductivity

19. Conclusions Modifications of the overdamped oscillator provide simple frameworks for ac conductivity with a rich dynamics Relationship with stochastic models will be further investigated Homepage: www.elp.uji.es/jb.htm E-mail: bisquert@uji.es