First-Principles Calculations for Insulators at Constant Polarization

We develop an exact formalism for performing first-principles calculations for insulators at fixed electric polarization. As shown by Sai, Rabe, and Vanderbilt (SRV) [Phys. Rev. B 66, 104108 (2002)], who designed an approximate method to tackle the same problem, such an approach allows one to map out the energy landscape as a function of polarization, providing a powerful tool for the theoretical investigation of polar materials. We apply our method to a system in which the ionic contribution to the polarization dominates (a broken-inversion-symmetry perovskite), one in which this is not the case (a III–V semiconductor), and one in which an additional degree of freedom plays an important role (a ferroelectric phase of ${\mathrm{KNO}}_3$). We find that while the SRV method gives rather accurate results in the first case, the present approach provides important improvements to the physical description in the latter cases.

Figure 1

Sketch of energy as a function of polarization for a double-well potential. The value of the electric field needed to impose the polarization is the same at the three indicated points, as their slopes are the same.

Figure 2

Energy versus polarization along the $c$ axis for the broken-inversion-symmetry perovskite described in the text, for different values of $\delta$. The solid (dashed) line is a polynomial fit of the values obtained using our (the SRV) method.

Figure 3

Energy versus magnitude of polarization in the line joining two nearest Al and As neighbors. The solid (dashed) line is a polynomial fit of the values obtained using our (the SRV) method.

Figure 4

(a) Side view of the ferroelectric phase of ${\mathrm{KNO}}_3$ (phase III), with the $c$ hexagonal lattice axis running vertically. (b) Top view of the same structure. (c) Energy versus polarization $\mathbf{P}=(0,0,P)$. (d) Angle of rotation of the ${\mathrm{NO}}_3$ group as a function of polarization $P$.