We outline a modern theory of the spontaneous polarization P in pyroelectric and ferroelectric materials. Although P itself is not an observable, the difference ΔP 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 ΔP 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 KNbO3 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 O ions. Only the highest nine valence-band states (O 2p) contribute to P, while all the other states behave as rigid core states.
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Resta, R., Posternak, M. & Balderescrt, A. First-Principles Theory of Polarization in Ferroelectrics. MRS Online Proceedings Library 291, 647–652 (1992). https://doi.org/10.1557/PROC-291-647