A. P. Levanyuk

We account for inhomogeneous strains, while calculating two characteristic thicknesses arising in the problem of critical thickness for ferroelectric memory. One of them marks the stability limit of the metastable single-domain state under zero bias voltage with respect to small fluctuations (the spinodal point of the single-domain state). The second one appears when the free energies of the single and multidomain states become equal while the latter is considered within a ‘one sinusoid’ approximation. At this thickness, the single-domain state remains metastable but one may hope that the lifetime of this state becomes considerable for memory applications. We use the Landau–Ginzburg–Devonshire approach for an elastically isotropic solid with a single electrostriction constant to illustrate this general behavior. It is found that the effect of the elastic strains is qualitatively different for free films compared to films on substrates.

We discuss ferroelectric phase transitions into single- and multidomain states in very thin films using continuous theory. We calculate a critical thickness of the homogeneous state and its stability with respect to domain formation for strained thin films of BaTiO₃ on SrRuO₃/SrTiO₃ substrate within the Landau theory. While the former (2.5 nm) is the same as given by ab-initio calculations, the actual critical thickness is set by the domains at 1.6 nm. There is a large Merz's activation field for polarization relaxation. Remarkably, the results show a negative slope of the “actual” hysteresis loops, a hallmark of the domain structures in ideal thin films with imperfect screening. The conclusion that domains always form in thin films is valid almost irrespective of the nature of electrodes (metallic or semiconducting) and whether or not the screening carriers may be present in the ferroelectric itself. In particular, crossover from second to first order phase transition with thickness, envisaged by Batra et al. (1972), is not possible, since it is being preempted by formation of domain structure.

We comment on zero- and low-temperature structural phase transitions, expecting that these comments might be relevant not only for this structural case. We first consider a textbook model whose classical version is the only model for which the Landau theory of phase transitions and the concept of ``soft mode'' introduced by Ginzburg are exact. Within this model, we reveal the effects of quantum fluctuations and thermal ones at low temperatures. To do so, the knowledge of the dynamics of the model is needed. However, as already was emphasized by Ginzburg et al. in eighties, a realistic theory for such a dynamics at high temperatures is lacking, what also seems to be the case in the low temperature regime. Consequently, some theoretical conclusions turn out to be dependent on the assumptions on this dynamics. We illustrate this point with the low-temperature phase diagram, and discuss some unexpected shortcomings of the continuous medium approaches.

It is shown that any defect gives an Ohmic contribution to the damping of any normal mode of the crystal lattice with nonzero wavevector which does not vanish at zero temperature. This explains the large phason damping observed at low temperatures in incommensurate phases, and might be a key factor to understand the linear-in-$T$ specific heat observed in a number of real dielectrics at low enough temperatures.

A Reply to the Comment by K. Biljaković.

Specific features of the low-frequency response of a phason and an amplitudon in the region of small wave vectors are considered in incommensurate displacive-type phases at high and low temperatures. The corresponding anomalies of spin-lattice relaxation (SLR) rate and attenuation of the longitudinal sound in the vicinity of the normal phase transformation into an incommensurate phase are analyzed. It is shown that the divergence of the amplitudon response at small wave vectors and low frequencies induced by the interaction with phasons clearly manifests itself only at low temperatures T and that the pronounced amplitudon contribution proportional to T ³ to the SLR rate is associated with this divergence. The phason-response divergence at low temperatures leads to the frequency-independent contribution proportional to T to the SLR rate.

Striction-mediated attraction of domain walls, solitons in incommensurate phases, and Abrikosov vortices in superconductors are considered. It is shown (a) that it is this type of attraction that can be responsible for a soliton-density jump in lock-in transitions and (b) that the strain-induced vortex interaction in superconductors with a high Ginzburg-Landau parameter is higher by one or two orders of magnitude than was assumed earlier.