# Analysis of Phase Transitions in Ferroelectrics

In recent years, ferroelectric compounds based on cubic perovskite structures ABO3 have attracted much interest because of their promise for a series of technological applications.

For the better understanding of structural phase transitions in perovskite oxides, including chemical trends in transition temperatures, first-order vs second-order character of transitions, the relationship between local distortions, and the stability of intermediate temperature phases, first-principles calculations give insight into valuable microscopic information.

With advances in algorithms and computational capabilities, the challenge of achieving the high accuracy necessary for studying these distortions has been largely met, and ground-state distortions have been well reproduced.

Following the soft-mode theory of structural transitions, one can construct a simple effective Hamiltonian acting in the subspace defined by the branches containing unstable phonons, with an explicit form determined from the first-principles calculations. This effective Hamiltonian is sufficient for reproducing the finite-temperature structural transition behavior of an individual material.

Kintech Lab developed a code for the first-principles-based analysis of finite-temperature properties of ferroelectrics using the effective Hamiltonian approach. An example of studying finite temperature properties of PZT perovskite using this code is given below

##### PZT Ferroelectrics

PbZr_{1-x}Ti_{x}O_{3} (also called PZT) solid solutions with x about 0.5 have been used for decades in virtually all piezoelectric devices from ultrasound generators to micropositioners, which is due to their outstanding electromechanical performance.

All PZT compositions have the cubic perovskite structure at high temperatures, but they undergo a phase transition to a ferroelectric rhombohedral phase at about 650 K.

##### Effective Hamiltonian for PZT

Parameters of effective Hamiltonian were determined from fitting to the results of first-principles calculations of displacement along different vibrational modes in PZT, as shown in Fig. 1. The first-principles calculations were done using ABINIT code and virtual crystal approximation (VCA) for the disordered lattice.