Electric Polarization from Nonpolar Phonons

Abstract

Born effective charge (BEC), a fundamental quantity in lattice dynamics and ferroelectric theory, provides a quantitative measure of linear polarization response to ionic displacements. However, it does not account for higher-order effects, which can play a significant role in certain materials, such as fluorite HfO$_2$. In this letter, we extend the BEC framework by introducing the concept of second-order dynamical charge and mode effective charge. Using first-principles calculations, we demonstrate that specific combinations of nonpolar phonon modes in many oxides can induce substantial second-order polarizations, reaching magnitudes comparable to those of intrinsically polar modes. Through a symmetry-based analysis of the charge density, we elucidate the microscopic origin of these effects, tracing them to variations in bond covalency and local electronic rearrangements. We also demonstrate large second-order mode effective charge in well-studied perovskites such as SrTiO$_3$, highlighting the generality of these phenomena. Our results reveal a previously unrecognized mechanism that drives polarization in crystalline solids, offering new insights into the design principles of next-generation ferroelectric, piezoelectric and multifunctional materials.

Figures (3)

• Figure 1: Second-order dynamical charge in HfO$_2$. (a) Illustration of O1 displacement in $+z$ direction and relative position of Hf1 ion. (b) Change of Born effective charge components by ion displacement in cubic HfO$_2$. For a single oxygen displacement in the $z$ direction, largest changes are observed for $xy$ components for the Born effective charge of itself and adjacent Hf ions. (c) Illustration of $\Gamma^-_{4}, \mathrm{X}^+_{5,x}$ and $\mathrm{X}^-_{2,x}$ modes in HfO$_2$. (d) Linear interpolation of polar structure resulting from mode effective charge of polar $\Gamma^-_{4}$ mode in cubic HfO$_2$. (e) Linear interpolation of polar structure resulting from 2nd-order mode effective charge of nonpolar modes $\mathrm{X}^+_{5,x}$ and $\mathrm{X}^-_{2,x}$ in HfO$_2$. Significant polarization can be induced from higher-order contribution of nonpolar modes.
• Figure 2: Isosurfaces of charge density projections of (a) $\mathrm{X}^+_{5,z}\mathrm{X}^-_{5,z:\mathrm{Hf}}$ (b) $\mathrm{X}^+_{5,x}\mathrm{X}^-_{2,x}$ hybrid nonpolar mode pairs onto the $\Gamma^-_4$ polar mode. Panels (c) and (d) show the inversion-odd components of the projected charge density on oxygen sites for the $\mathrm{X}^+_{5,z}\mathrm{X}^-_{5,z:\mathrm{Hf}}$ and $\mathrm{X}^+_{5,x}\mathrm{X}^-_{2,x}$ hybrid modes, respectively. In all plots, yellow and cyan isosurfaces represent regions of charge accumulation and depletion. The isosurface values are set to 12, 35 $\upmu|e|$/Å$^3$ for panels (a), (b), and 3.5 $\upmu|e|$/Å$^3$ for panels (c), (d).
• Figure 3: (a) $\mathrm{X}^+_{1,z:\mathrm{Ti}}\mathrm{X}^-_{3,z:\mathrm{O}}$ and (b) $\mathrm{X}^+_{5,z:\mathrm{O}}\mathrm{X}^-_{5,z:\mathrm{O}}$ hybrid modes in SrTiO$_3$. The second-order mode effective charges in the $z$ direction are 5.17 and --0.05 |e|Å, respectively.