PHYSICAL REVIEW 8 VOLUME 24, NUMBER 4 15 AUGUST 1981 Configurational excitations and low-temperature specific heat of the Frenkel-Kontorova model L Pietronero, W. R. Schneider, and S. Strassler Brown Boueri Research Center, CH-5405 Baden, Switzerland (Received 9 December 1980) We study the specific heat due to configurational changes in a Frenkel-Kontorova model with a finite density of defects. This model provides an example of intermediate disorder related only to the diffusing ions. The results show that it is possible to obtain excitations with very low energy due to quasidegenerate configurations, but it is not possible to obtain a smooth distribution for these energies. The present model applied to the one-dimensional ionic conductor hollandite can very well describe the peak structure recently observed in the low-temperature specific heat. I. INTRODUCTION The long-standing problem of the low- temperature extra specific heat of glasses and amor- phous materials' receives further interest from the measurements of analogous effects in the ionic con- ductor P-alumina and very recently in hollandite where a peaked contribution is observed instead of the usual pseudolinear behavior. The phenomeno- logical model that seems to be the most consistent with various observations (specific heat, sound ab- sorption, etc. ) is based on the assumption that two- level systems exist in disordered systems and that the occurrence probability of various energy gaps between the two levels is nonzero and smooth for small gaps. ' This smooth distribution is necessary to reproduce the pseudolinear behavior of the specif- ic heat as a function of temperature. ' In order to say something about the microscopic origin of these excitations one has to consider specif- ic models of disordered systems. In this respect the observation in ionic conductors of anomalies similar to those of other amorphous systems is of particular interest because these materials are rather well characterized microscopically. The basic model is that of a periodic potential (due to the host lattice) whose pots are partly filled by the interacting diffus- ing ions. In particular, in hollandite ionic diffusion is along channels so that the corresponding model is one-dimensional and it is essentially the Frenkel- Kontorova model with a finite density of defects. We have therefore a precise formulation for a prob- lem of intermediate disorder, the disorder being only that due to the diffusing ions while the cage ions are assumed to give rise to the periodic poten- tial. The questions of interest are the following: (i) Does this model give rise to low-energy ( — I K) configurational excitations (two-level systems with low gap)? (ii) Do these excitations have a distribu- tion that can give rise to a pseudolinear specific heat? As we will see the answer is yes to the first question and no to the second one. In fact it is pos- sible to have very low excitations because of quasidegenerate configurations but the spectrum of these excitations is discrete and we conjecture this discreteness not to be removed by going to two or three dimensions. The present model applied to hollandite is in very good agreement with observa- tions. On the other hand, from this model we see that the disorder due to the diffusing ions in a periodic potential does not give rise to a linear extra specific heat. From this we conclude that in those ionic conductors where the linear part is actually observed there must be additional sources of disor- der. In P-alumina this additional disorder could be due, for example, to the field of the randomly placed " compensating ions. " In Sec. II we define the model and recall the transformation to a spin system defined in Refs. 8 and 10. In Sec. III we give simple arguments for the characteristic energies of the configurational ex- citations. In Sec. IV we study the partition function and derive analytical expressions for the specific heat. In Sec. V we discuss the application of the present model to the case of hollandite. In Sec. VI the main results are summarized. II. THE MODEL AND A USEFUL TRANSFORMATION The model we consider is that of Frenkel- Kontorova generalized to an arbitrary density of 2187 1981 The American Physical Society