Author Correction: Strain engineering of electro-optic constants in ferroelectric materials

Correction to: npj Computational Materials https://doi.org/10.1038/s41524-018-0141-4, published online 8 January 2019

After publishing this article, we realized that the calculation of the piezoelectric constants d_ij by Density Functional Perturbation Theory (DFPT), as explained in the methods described in the original version of the Supplementary Information, are incorrect in the case of a partially clamped situation. The correct methodology is to inverse the subspace matrix of elastic constants:

$$\left( {\begin{array}{*{20}{c}} {C_{33}} & {C_{34}} & {C_{35}} \\ {C_{43}} & {C_{44}} & {C_{45}} \\ {C_{53}} & {C_{54}} & {C_{55}} \end{array}} \right)$$

This obtains the matrix of elastic compliances (S_ij)_i,j=3,4,5 that applies to the mechanical directions that are unclamped in our geometry (i.e. the strains η₃, η₄ and η₅, corresponding to the relaxation of the out-of-plane axis). The relevant piezoelectric constants d_ij can then be calculated from: (1) the obtained reduced matrix of elastic compliances, and (2) the piezoelectric tensor e_ij obtained from DFPT, following the formula:

$$d_{ij} = \mathop {\sum }\limits_{k = 3,\,4,\,5} e_{ik}S_{kj}\quad{\mathrm{for}}\quad j = 3,\,4,\,5$$

This correction to the methodology changes the d_ij calculated in Fig. 1d and Fig. 4a–d. The general findings of this work, that strain-engineering of PbTiO₃ films can improve electro-optic constants, is unaffected, though the size of the effect is changed.

The following corrections have been applied to the HTML and PDF versions of the article:

• 1. A new Supplementary Information file has been uploaded containing the above methodology in the “ELASTO-OPTIC CONSTANTS” section.

• 2. Following the change in methodology, Fig. 1d and Fig. 4a–d have been replaced with graphs showing the corrected data. The following sections of text have been revised due to the change in the data:

Abstract

“…via compressive strain to obtain extremely large piezoelectric constants.” has been replaced with “…via compressive strain to obtain large piezoelectric constants.”

Results section

“The piezoelectric constant d₃₃ exhibits a strong increase to 618 p CN⁻¹ at this specific strain, as shown in Fig. 1d.” has been replaced with:

”As a result of this effect, the piezoelectric constant d₃₃ exhibits a large increase up to 80 p CN^-1 within the tetragonal phase, as also displayed in Fig. 1d.”

“…and the large increase of the d₃₃ piezoelectric constant at η = −2.5% (see Fig. 1d) induces very large unclamped EO constants r₃₃ ≈ 95.8 pm V⁻¹ and r₁₃ = 127.1 pm V⁻¹ in Fig. 4b, c. In comparison, at zero strain, r₃₃ ≈ 30.5 pm V⁻¹ and r₁₃ = 25.2 pm V⁻¹, and a three and five time increase (with respect to the bulk case) is thus achieved under the compressive strain η = −2.5%!” has been replaced with:

“the increase of the d₃₃ piezoelectric constant at η = −2.5% (see Fig. 1d) induces a plateau in r₃₃ and a local maximum in the r₁₃ which peaks at 24.1 pm V⁻¹. More interestingly, owing to the phase-transition induced divergence of d₃₃ in the monoclinic phase, the unclamped EO constant r₃₃ reaches as high as 124 pm V⁻¹, which is roughly four times its clamped value and represent a four time increase with respect to the zero strain value of the unclamped EO constant in the tetragonal phase (for which r₃₃ ≈ 30.5 pm V⁻¹).”

Discussion section

“In PTO, that difference is small near the P4mm–Cm transition.” has been replaced with:

“In PTO, that difference is small near the P4mm–Cm transition for the in-plane EO constants.”

The following text has been added after “…without significantly affecting the EO constants in this strain region.”:

“On the other hand, at the Cm-Ic2m phase boundary, large out-of-plane EO constants (r₃₃ ≈ 124 pm V⁻¹) can be engineered due to the strain-induced divergence of the piezoelectric constant d₃₃. The latter comes from the flattening of the energy landscape to allow for polarization rotation from the [001] pseudo-cubic direction to [110] pseudo-cubic direction [19,20]. We thus show that strain, through the exploration of phases with different symmetries, allows fine tuning of the EO constants within a single material.”

“On the other hand, the large unclamped r₃₃ and r₁₃ of PbTiO₃ films…” has been replaced with:

On the other hand, the anomalous behaviors of unclamped r₃₃ and r₁₃ of PbTiO₃ films…”

“Rather, they come from the strong peak in the piezoelectric constant d₃₃ seen in Fig. 1d and thus represent an original way to improve EO constants…” has been replaced with:

“Rather, they come from the strong peak in the piezoelectric constant d₃₃ seen in Fig. 1d and thus represent an original way to change EO constants…”

“At such strain, Fig. 4c reveals that the unclamped EO constant r₃₃ is ≈57 pm V⁻¹, which is already twice as much as that of the standard EO material, LiNbO₃.” has been replaced with:

“At such strain, Fig. 4c reveals that the unclamped EO constant r₁₃ is ${\approx}\hskip -2pt 23.4$ pm V⁻¹, which is already more than twice as much as that of the standard EO material, LiNbO₃ [8].”

We thank David Vanderbilt for insightful comments on this matter.