Soft Modes and Structural Phase Transitions

Abstract

Soft Modes were put forward by Cochran and Anderson about 50 years ago as an explanation of structural phase transitions. Extending Landau's theory of phase transitions, their prediction was that the square of the frequency of the soft mode was proportional to τ = |T/T _c-1| where T_c is the transition temperature. Experimental measurements then showed on a variety of materials that this was a reasonable approximation. However, in the early 1970's, there were new developments. On the theoretical side critical phenomena suggested that at continuous phase transitions there was only one length scale and that the exponents could be very different from mean field theory. On the experimental side it has proved difficult to measure the critical exponents reliably while the detailed measurements showed that there were two time scales in the measurements on a variety of different types of transitions. Experiments showed that the new time scale was very long but could be observed for τ less than about 0.3. It was suggested that this long time scale was associated with defects but the particular type of defects were not identified. Measurements were made that have shown that the long time scale is a bulk property, has a long length scale and occurs in a wide range of different materials. In the late 1980's x-ray scattering measurements showed in SrTiO₃ that a second long length scale was also observed. After considerable effort it was shown that this length scale was associated with the surface and was very long and could be observed in a wide variety of different materials. It has also been suggested that this long length scale is associated with defects but it is possibly more likely that it is associated with the elastic strain that occurs at the surface of the crystal as the transition temperature is approached. A more recent development has been that the soft mode consists not of a single wave-vector but of a bunch of similar wave vectors. This is the case for uniaxial ferroelectrics such as lead germinate. In the absence of an applied electric field the macroscopic electric field destroys the long range order and the order parameter causes a compromise between the macroscopic electric field and the minimum energy of the ferroelectric fluctuations. This minimum is a domain structure which cannot be described by a single periodic wave of distortion. Similarly relaxors such as PMN, that do not have long range polarisation although there is a peak in the dielectric constant, do have a distorted structure similar to the domain structure discussed above. For these materials a wide variety of different soft modes and phase transitions have been proposed. However, the soft mode has been clearly identified and the order parameter consists of a range of periodic waves that make up the low temperature structure. Soft modes are still a useful and simple concept that can guide much research on structural phase transitions. The concept has been extended to apply to materials for which the order parameter consists of a range of different wave vectors and there are many unexplained features in simpler structural phase transitions that still need to be explained.

Acknowledgments

This review is a collection of comments on soft modes. It could not have been written without the introduction by W. Cochran who introduced me to the subject of soft modes. I have discussed and argued about soft modes with many of my colleagues and students including in particular: G. Dolling, W.J.L. Buyers, G. Shirane, J. D. Axe, S. M. Shapiro, B. Roessli and S. N. Gvasaliya all of whom have improved my understanding.