First-Principles Theory of Polarization in Ferroelectrics
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FIRST-PRINCIPLES THEORY OF POLARIZATION IN FERROELECTRICS R. RESTA,- M. POSTERNAK, and A. BALDERESCHI** *SISSA, via Beirut 4, 1-34014 Trieste, Italy *-IRRMA, PHB Ecublens, CH-1015 Lausanne, Switzerland
ABSTRACT We outline a modern theory of the spontaneous polarization P in pyroelectric and ferroelectric materials. Although P itself is not an observable, the difference AP between two crystal states can indeed be measured and calculated. We define P as the difference between the polar structure and a suitably chosen nonpolar prototype structure. We previously proposed and implemented a supercell scheme in order to evaluate P in pyroelectric BeO; here we adopt an approach recently developed by King-Smith and Vanderbilt, where AP is obtained from the computation of Berry's phases, with no use of supercells. We apply this novel approach, which is numerically very convenient, in order to revisit our previous work on BeO. We then perform a first-principles investigation of the spontaneous polarization P of KNb0 3 in its tetragonal phase, which is a well studied perovskite ferroelectric. Our calculated P value confirms the most recent experimental data. The polarization is linear in the ferroelectric distortion; the Born effective charges show strong variations from nominal ionic values, and a large inequivalence of the 0 ions. Only the highest nine valence-band states (0 2p) contribute to P, while all the other states behave as rigid core states.
INTRODUCTION The concept of spontaneous polarization has long escaped even a precise microscopic definition. In fact, the polarization of a ferroelectric cannot be measured as an intrinsic equilibrium property: the genuine physical observable is only the difference AP between two enantiomorphous metastable states of the crystal [1], which is usually measured via hysteresis cycles. More generally, any kind of electric polarization is observable only as a differential quantity, as it is well known for the cases of dielectric susceptibility, piezoelectricity, and pyroelectricity. The differential concept is a basic one for a theoretical formulation as well; surprisingly enough, this feature seems to have been realized only recently [2, 3]. We define the spontaneous polarization P of a crystal in a given low-symmetry structure through the difference AP with respect to a prototype nonpolar structure of the same material. Once this definition assumed, the spontaneous polarization is a physical observable within the reach of first-principle theory of materials, as it will be shown throughout this work. We stress that-on the contrary-no information about the value of P can be extracted from the periodic charge density of a crystalline solid in its ferroelectric structure. No quantum-mechanical calculation of the spontaneous polarization of a crystal existed until the work of Ref. [2], dealing with wurtzite BeO, and which remains so far the only published study of this kind. Nonpolar structures of this material do not exist in nature: we have chosen zincblende BeO as the "computational
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