Order-disorder transition in the prototypical antiferroelectric PbZrO$_3$
The prototypical antiferroelectric PbZrO$_3$ has several unsettled questions, such as the nature of the antiferroelectric transition, possible intermediate phase and the microscopic origin of the Pbam ground state. Using first principles, we show that no phonon becomes truly soft at the cubic-to-Pbam transition temperature, and the order-disorder character of this transition is clearly demonstrated based on molecular dynamics simulations and potential energy surfaces. The out-of-phase octahedral tilting is an important degree of freedom, which can collaborate with other phonon distortions and form a complex energy landscape with multiple minima. Candidates of the possible intermediate phase are suggested based on the calculated kinetic barriers between energy minima, and the development of a first-principles-based effective Hamiltonian. The use of this latter scheme further reveals that specific bi-linear interactions between local dipoles and octahedral tiltings play a major role in the formation of the Pbam ground state, which contrasts with most of the previous explanations.