Vibrational properties of ${\mathrm{LiNbO}}_3$ and ${\mathrm{LiTaO}}_3$ under uniaxial stress

Structural strain severely impacts material properties, such as the linear and nonlinear optical response. Moreover, strain plays a key role, e.g., in the physics of ferroelectrics and, in particular, of their domain walls. $\mu$-Raman spectroscopy is a well-suited technique for the investigation of such strain effects as it allows to measure the lattice dynamics locally. However, quantifying and reconstructing strain fields from Raman maps requires knowledge on the strain dependence of phonon frequencies. In this paper, we have analyzed both theoretically and experimentally the phonon frequencies in the widely used ferroelectrics lithium niobate and lithium tantalate as a function of uniaxial strain via density functional theory and $\mu$-Raman spectroscopy. Overall, we find a good agreement between our ab initio models and the experimental data performed with a stress cell. The majority of phonons show an increase in frequency under compressive strain, whereas the opposite is observed for tensile strains. Moreover, for $E$-type phonons, we observe the lifting of degeneracy already at moderate strain fields (i.e., at $\pm 0.2\%$) along the $x$ and $y$ directions. This paper, hence, allows for the systematic analysis of three-dimensional strains in modern-type bulk and thin-film devices assembled from lithium niobate and tantalate.

Figure 1

(a) $\overline{Z}$(XX)$Z$ Raman spectra of unstressed (0 MPa) and $x$ compressed ($-85.7$ MPa) $5\%$ MgO-doped LN (5 Mg-LN-$x$). (b) $\overline{Z}$(XY)$Z$ Raman spectra of unstressed (0 MPa) and $y$ compressed ($-140.7$ MPa) $5\%$ MgO-doped LN (5 Mg-LN-$y$). (c) $\overline{X}$(ZZ)$X$ Raman spectra of unstressed (0 MPa) and $z$ compressed ($-175$ MPa) $z$ axis of $5\%$ MgO-doped LN (5 Mg-LN-$z$). The difference curve in all (a)–(c) graphs represents the difference spectra between the stressed and unstressed states of the sample.

Figure 2

Measured response of the $E({\mathrm{TO}}_1$) phonon mode (a) and (b) in the samples sLN compressed along the $x$ and $y$ axes, respectively and (c) and (d) in the sample 5 Mg-LN compressed along the $x$ and the $y$ axes, respectively. The stoichiometric and $5\%$ MgO-doped LN behave similarly.

Figure 3

DFT calculated relative frequency shifts of the $E({\mathrm{TO}}_4$) mode of stoichiometric LN as a function of strain in $z$ direction in (a) steps of $0.4\%$ and (b) steps of $0.02\%$. The linear fit and the standard deviation of both are plotted and their parameters displayed. The slopes for strain are defined as $\left(\mathrm{\Delta }\omega \right)/(\left|\mathrm{\Delta }\epsilon \right|)$ and displayed with the fitting error.

Figure 4

The DFT calculated relative frequency shifts of (a) $E({\mathrm{TO}}_9$) (first row), (b) $E({\mathrm{TO}}_5$) (second row), and (c) $A_1({\mathrm{TO}}_2$) (third row) modes of stoichiometric LN as a function of strain in steps of $0.4\%$ along the $x, y$, and $z$ directions. The slopes for strain are defined as $\left(\mathrm{\Delta }\omega \right)/(\left|\mathrm{\Delta }\epsilon \right|)$ and displayed with the fitting error.

Figure 5

The eigenvectors of the Raman-active vibrations of unstrained stoichiometric LN and LT in the rhombohedral unit cell of the transversal phonon modes with $A_1$ and $E$ symmetries. Thereby light gray is denoted as Li, dark gray is denoted as Nb/Ta, and red is denoted as an O ion, whereas the green arrows represent the displacement direction.

Figure 6

Comparison of the experimental and theoretical frequency shifts (at lower strains): (a) and (b) $E({\mathrm{TO}}_6$) phonon mode under compression along the $x$ and $y$ axes of stoichiometric LN. The shaded region around all the curves is the confidence interval of the linear fit.

Figure 7

Comparison of the theoretical stoichiometric and experimental 5 Mg-LN-$z$ sample when compressing along the $z$ direction for frequency shifts of (a) $E({\mathrm{TO}}_7$) and (b) $A_1({\mathrm{TO}}_4$). $E({\mathrm{TO}}_7$) shows a positive slope whereas $A_1({\mathrm{TO}}_4$) shows a negative slope for $z$ compression.