First-principles calculations of elasto-optical properties of $R$Te$_3$

TL;DR

This work provides a quantitative, first-principles framework linking lattice strain to optical response in NdTe3, a representative member of RTe3. By computing the elastic moduli, dielectric tensors, and piezo-optical coefficients, the authors map how strain induces birefringence and anisotropy, enabling elasto-optical probes of symmetry-breaking states. The results reveal strong in-plane stiffness and pronounced dielectric anisotropy, with energy-dependent piezo-optical responses that can drive measurable elasto-birefringence in realistic device geometries. A symmetry analysis shows that while D2h symmetry forbids certain couplings, a symmetry lowering to C2h would activate additional tensor components, offering a pathway to detect hidden CDW-related orders through optical measurements.

Abstract

Rare-earth tritellurides ($R$Te$_3$) exhibit complex charge-density-wave (CDW) phases intertwined with lattice symmetry, offering a platform to explore unconventional symmetry breaking in correlated materials. Elasto-optical probing, which detects strain-induced changes in birefringence, provides a non-invasive approach to visualize anisotropy and emergent order in these quasi-two-dimensional systems. However, the magnitude and symmetry of the expected optical response remain poorly quantified, hindering experimental interpretation. Here, we perform first-principles calculations of the elastic, dielectric, and piezo-optical tensors of NdTe$_3$ to establish a quantitative framework for strain-induced optical anisotropy. These results establish a quantitative link between lattice strain and optical response in $R$Te$_3$, providing a predictive framework for probing symmetry-breaking states via elasto-birefringence.

Figures (4)

• Figure 1: Elasto-modulus Calculations of NdTe$_{3}$. (a) Crystal structure of $R$Te$_{3}$ with specific $R$ to be neodymium. (b) Energy change $\Delta E$ as a function of strain for $\mathrm{NdTe}_3$ under six independent distortions. Cross markers denote first-principles data for the energy difference relative to the ground state, $\Delta E = E(\varepsilon)-E_0$ (Ry) under various external strain, whereas dashed curves represent the quadratic fitting to the data. The elastic constants $C_{11}$, $C_{22}$, $C_{33}$, $C_{44}$, $C_{55}$, and $C_{66}$ are obtained from the zero-strain curvature of the quadratic fits.
• Figure 2: Calculations on Optical Properties of NdTe$_{3}$. (a–c) Real (solid) and imaginary (dashed) parts of the dielectric function $\epsilon_{ij}$ for $ij=xx,yy,zz$. (d–e) Real and imaginary parts of the complex refractive index, $n$ and $k$, along $x$, $y$, and $z$, obtained from the physical branch of the complex refractive index $\tilde{n} = n + i k = \sqrt{\epsilon}$ (chosen such that $k \ge 0$). (f) Normal-incidence reflectivity $R$ for $x$, $y$, and $z$, computed as $R=\left|\tfrac{n-1}{n+1}\right|^{2}$. The tensor form is constrained by $D_{2h}$ symmetry: in our computational frame only three components are symmetry-allowed, while $\epsilon_{xy}=\epsilon_{xz}=\epsilon_{yz}=0$.
• Figure 3: Dispersion of piezo-optical tensor of NdTe$_{3}$. Real (left column) and imaginary (right column) parts of the selected piezo-optical coefficients $p_{\alpha\beta}$ as functions of photon energy, calculated under $\pm 1\%$ strain. Each row corresponds to a different applied strain component: (a,b) $\varepsilon_{1}$-induced response with $p_{11}$, $p_{21}$, $p_{31}$; (c,d) $\varepsilon_{2}$-induced response with $p_{22}$, $p_{12}$, $p_{32}$; (e,f) $\varepsilon_{3}$-induced response with $p_{33}$, $p_{13}$, $p_{23}$; (g,h) shear-strain-induced response with $p_{44}$, $p_{55}$, $p_{66}$, associated with $\varepsilon_{4}$, $\varepsilon_{5}$, $\varepsilon_{6}$, respectively.
• Figure 4: Normal strain $\varepsilon_{xx}$ and elasto-birefringence. (a) Top-view distribution of the normal strain component $\varepsilon_{xx}$ in the bow-tie plate. The strain field is obtained from a finite-element simulation in which one circular hole is fixed while a prescribed in-plane displacement is applied to the opposite hole. The center rectangle lies NdTe$_3$ material. (b,c) Spatial distribution of the elasto-birefringence at photon energy $E \simeq 1.96~\mathrm{eV}$. The elasto-optic response is evaluated from the complex dielectric tensor $\epsilon(\omega)$ and the piezo-optic tensor $p_{\alpha\beta}(\omega)$, combined with the full three-dimensional strain field from finite-element simulations. (b) and (c) are respectively real and imaginary part of the elasto-birefringence $\delta\Delta n + i\delta\Delta k = [(\tilde{n}_x - \tilde{n}_z) - (\tilde{n}^0_x - \tilde{n}^0_z)]$. Here, "surface" indicates the top interface of the material and "mid-plane" indicates the middle cross-section of the material between the bottom interface and the top surface.