Berry flux diagonalization: Application to electric polarization

The switching polarization of a ferroelectric is determined by the current that flows as the system is switched between two variants. Computation of the switching polarization in crystal systems has been enabled by the modern theory of polarization, where it is expressed in terms of a change in Berry phase from the initial state to the final state. It is straightforward to compute this change of phase modulo $2\pi$, thus requiring a branch choice to specify the predicted switching polarization. The measured switching polarization depends on the actual path along which the system is switched, which in general involves nucleation and growth of domains and is therefore quite complex. In this work we present a first-principles approach for predicting the switching polarization that requires only knowledge of the initial and final states based on the empirical observation that for most ferroelectrics, the observed polarization change is the same as that for a path involving minimal evolution of the state. To compute the change along a generic minimal path, we decompose the change of Berry phase into many small contributions, each much less than $2\pi$, allowing for a natural resolution of the branch choice. We show that for typical ferroelectrics, including those that would have otherwise required a densely sampled path, this technique allows the switching polarization to be computed without any need for intermediate sampling between oppositely polarized states.

Figure 1

Sketch of the joint $(k,\lambda )$ space for computing a change in polarization between ${\lambda }_A$ and ${\lambda }_B$. Blue circles represent points where Bloch wave functions have been computed. The light gray box represents the surface $S$ that is integrated over in Eq. (2). Dotted green lines represent the plaquettes $i$, and the solid green lines represent the path on which the parallel transport procedure is performed around the green plaquette it encloses to obtain its contribution to $P^B-P^A$.

Figure 2

Histogram of Wilson loop eigenvalues [${\varphi }_n^p$ from Eq. (27)] for the plaquettes highlighted in the insets following the form of Fig. 1. In each of the two middle plots, the two highlighted plaquettes have identical contributions due to time reversal symmetry. Values for pure ${\mathrm{PbTiO}}_3$ are shown on the right. The occurrences of values for the primitive and supercell systems differ only by a factor of 4, as indicated by the two axis scales at the top and bottom of the figure. Values for ${\mathrm{PbZr}}_{0.25}{\mathrm{Ti}}_{0.75}{\mathrm{O}}_3$ are shown on the left.

Figure 3

Singular values throughout the Brillouin zone for ${\mathrm{PbTiO}}_3$, sampled on a $12\times 12\times 12$ $\mathrm{\Gamma }$-centered $k$ mesh.

Figure 4

Evolution of formal polarization of ${\mathrm{PbTiO}}_3$ along a linearly interpolated switching path for the primitive cell (left) and a $2\times 2\times 1$ supercell (right). Ticks and horizontal lines mark the polarization quantum. The blue arrow indicates the change in polarization, which, with the Berry flux diagonalization method, requires only calculations in the initial state and symmetry-related final state.