Breakdown of ion-polarization-correspondence and born effective charges: Algebraic formulas of accurate polarization under field

Polarization, especially of ferroelectrics FEs, is conventionally described by ion positions, e.g., by Born effective charges, where the complete entanglement of electron polarization with that of ions is implicitly assumed. We find that such descriptions or Born effective charge polarization-type approaches break down partially in the presence of high field, owing to the partial disentanglement of electrons with ions. To overcome this, we propose a correction (non-Born effective charge polarization) that calculates both macroscopic and unit-cell-by-unit-cell total polarization accurately. The accuracy of this method is demonstrated in prototypical situations of depolarization field $E_d$ that exists in finite-size or inhomogeneous insulating FEs: paraelectric/FE, FE capacitors, and FE/vacuum. Here, FE/vacuum are shown to be electrically identical to encountering domains. This method provides simple algebraic formulas to calculate total polarization $P_{\mathrm{S}}$ and $E_d$ using conventionally estimated polarizations that are obtained from local ion positions. Therefore, it can be easily used in experimental estimations of $P_{\mathrm{S}}$ and $E_d$, including 3D cases. For example, this method reveals that $P_{\mathrm{S}}$ varies across ferroelectric/insulator far less than the conventional estimate, which explains substantially reduced $E_d$ and the absence of metallicity. In addition, vortexlike domains are discussed in view of $E_d$. The partial disentanglement of ion and electron polarization would imply limitation of Ginzburg-Landau framework of ferroelectrics under high field.

Figure 1

Images of bulk and real FEs [(a)–(c)], the corresponding unit-cells [(d)–(f)] explaining extracted bulk unit-cell, ${P_{\mathrm{S}}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ and $P_{\mathrm{el}}^{\mathrm{Exra}}$. (a) bulk, i.e., infinite-size FE. (b), (c) real, i.e., finite-size FEs: FE/vacuum (b) and FE heterostructures (c), where $E_d$ emerges owing to the imperfect screening of polarization charges. (d)–(f) FE unit-cells showing ions (red) and electron clouds (orange): (d) short-circuit slab, (e) open-circuit slab, and (f) bulk having the same ion positions as (e).

Figure 2

Atomic model of (a) FE/vacuum and (b) FE/$I_{\mathrm{adj}}$ superlattice, and (c) corresponding macroscopic potential $\varphi$. (d) Example of the estimation of ab initio $E_d$ ($E_d^{abinitio}$) from the highest planar averaged atomic potential $\varphi$ by the ab initio calculation of 10-unit-cell BTO/5-unit-cell STO. (e) DOS of STO.9999/vacuum, where $l_f$ in the unit of unit-cells and $l_V$ in Å are shown on the right.

Figure 3

Planar averaged electron density $\rho$ for estimating the FE thickness $l_f$. All the eight $\rho$’s behave the same near the surface, supporting the use of the same smearing length $\sim 0.8\AA$ for different slabs. Inset shows the enlarged view near the surface.

Figure 4

(a) Layer-by-layer plot of the distance between Ti ion and O ion in ${\mathrm{TiO}}_2$ plane along $c$-axis of $n$-unit-cell BTO/5-unit-cell STO superlattices $(n=7,10)$. (b) ${P_{\mathrm{S}}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ of each unit-cell having the ion positions shown by filled pink circles in (a) is represented by filled pink circles in (b). The corresponding $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ is shown by a small red circle. The corrected polarization $P_{\mathrm{S}}$ is shown by filled light-blue squares.

Figure 5

Permittivity of electrons ${\varepsilon }^{\mathrm{el}}$ along $c$ axis of STO1.005, STO.9999, cubic STO and BTO by ab initio calculations with PBEsol, $\mathrm{PBE}+U$, and a hybrid functional HSEsol, while ${\varepsilon }^{\mathrm{el}}$ of STO1.005 is overlapped with that of cubic STO and almost invisible. Experimental optical ${\varepsilon }^{\mathrm{el}}$’s [42] are also shown. In the PBEsol calculations of STO and STO.9999, the agreements with experiment were improved by RPA, which is shown by thin red and scarlet lines that are terminated near 1.8 eV. ${\varepsilon }^{\mathrm{el}}$ along $a$ axis by each functional is very close to ${\varepsilon }^{\mathrm{el}}$ along $c$ axis by each functional.

Figure 6

Bars show the ratio of “Berry phase polarization $P_{\mathrm{S}}$ of a whole slab” to $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ [$P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$: Berry phase polarization of extracted bulk (Fig. 1, Table 1)]. Open circles show the ratios of “$P_{\mathrm{S}}$ calculated by Eq. (1) using $E_d^{\mathit{\text{ab}}\mathit{\text{initio}}}$” to $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$. Filled circles show the ratios of “$P_{\mathrm{S}}$ calculated by Eq. (1) using Eq. (2a)” to $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$. Numbers $n$ and $m$ in $n\text{−}m$ at the bottom represent the thicknesses of STO (in unit-cell) and vacuum (in Å), respectively. Triangles show the ratios of $P_{\mathrm{S}}$ to ${P_{\mathrm{S}}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$, where $P_{\mathrm{S}}$ is calculated by using Eq. (1), Eq. (2a), and ${P_{\mathrm{S}}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ given by ion-position based empirical formula [12].

Figure 7

(a) Bars show the ratio of “$P_{\mathrm{S}}$ calculated by Eq. (1)” using $E_d^{abinitio}$ to $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$. Filled circles show the ratio of “$P_{\mathrm{S}}$ by Eq. (1) using Eq. (2a)” to $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$. (b) Filled blue circles show the ratio of “$E_d$ calculated with $P_{\mathrm{S}}$ and $P_I$ (corrected by Eq. (2a))” to $E_d^{abinitio}$. Open black circles show the ratio of “$E_d$ calculated with $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ and $P_{I\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ (uncorrected)” to $E_d^{abinitio}$. The supercell geometry $\mathrm{BT}n\mathrm{ST}m$ at the bottom represents the number of unit-cells in BTO and STO. Light blue diamonds show the ratios of “$E_d$ by Eq. (2a)” to $E_d^{abinitio}$, where Eq. (2a) used ${P_{\mathrm{S}}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ and ${P_I}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ estimated by ion-position based empirical formula [12].

Figure 8

Ratios of “$P_{\mathrm{S}}$ by Eq. (1)” and $E_d^{abinitio}$ to $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ shown by bars. Ratios of “$P_{\mathrm{S}}$ by Eqs. (1) and (2c)” to $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$ are shown by filled dark-green circles. Asterisks show $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}$. Inset: Estimation of ${\varepsilon }_1/2l_1$ by liner fitting to $P_{\mathrm{S}\mathrm{Berry}}^{\mathrm{iso}\text{−}\mathrm{bulk}}/{\varepsilon }_0{E}_d$ vs. $l_f^{\mathrm{eff}}$ plot.

Figure 9

(a) Relationship between Ti-${\mathrm{O}}_2$ distortion $u$ ($\mathrm{\Delta }{z}_{\mathrm{Ti}-\mathrm{O}2}$) and Berry phase $P_{\mathrm{S}}$ of BTO and STO calculated ab initio using specific exchange correlations extracted from Ref. [12], indicating that exact $P_{\mathrm{S}}$ can be obtained by Born effective charges. The curves by the polynomial in (a) are plotted by solid lines, where $Z$, $b$, and $c$ are constants. (b) Example of the thickness of the surface layer of FE/vacuum and the estimation of $E_d^{abinitio}$ from the highest planar averaged $\varphi$ by the ab initio calculation.