Second-Order Renormalization and Phase Stability in Strontium Titanate



The temperature dependence of the transverse optic (T.O.) mode with wave number k ∼ 0, and of its contribution to the electric susceptibility, is discussed both in zero and in finite field, in terms of the renormalization of a Hamiltonian proposed by SILVERMAN and JOSEPH. The renormalization proceeds in two steps: (a) a canonical transformation is introduced which removes linear terms and also quadratic and cubic terms nondiagonal in the normal mode coordinates for which k > 0; (b) a thermal average is made of quartic terms proportional to the square of the ferroelectric mode frequency, giving rise to an effective frequency which depends on temperature. In zero field, it is shown that the contributions of T. O. modes to the renormalized frequency become large at low-temperature, preventing this frequency from vanishing at any temperature, and thus rendering impossible a ferroelectric transition. In finite field of sufficient magnitude the optical branch frequencies are all raised, reducing their stabilizing effect. It is shown that if the stabilization is reduced sufficiently, there will exist two phases of which one is metastable, has a ferroelectric mode frequency which vanishes, and is capable of two opposed states of polarization. The second phase is stable, with a frequency which does not vanish and a susceptibility which exhibits a maximum independently of the properties of the first phase. Such a maximum has been observed by HEGENBARTH.