Evidence for ferroaxial order in 1T-TiSe$_2$ via elastoresistivity measurements

Abstract

The study of spontaneous symmetry breaking and electronic order is fundamental in condensed matter physics. Hidden order, symmetry-breaking states that elude conventional probes, potentially plays a crucial role in understanding complex quantum phases in a wide range of materials. Ferroaxial order, a state characterized by broken mirror symmetries while maintaining time-reversal and inversion symmetries, is one of the hidden orders that have proven most challenging to detect experimentally. Here, we demonstrate a new approach for investigating both the ferroaxial order parameter and ferroaxial susceptibility using elastoresistivity measurements. We do this for 1T-TiSe$_{2}$, a material that exhibits charge density wave order that has eluded comprehensive understanding for a long time. These measurements reveal an anomalous off-diagonal linear elastoresistivity in the CDW state. We discuss why this provides a smoking gun for ferroaxial order. Furthermore, we construct an appropriate combination of the symmetry-breaking strains $ε_{x^2-y^2}$ and $ε_{xy}$ that acts as an effective conjugate field for the ferroaxial order, and demonstrate how sweeping this effective field in the CDW state results in a hysteretic behavior of the elastoresistivity, associated with the movement of ferroaxial domain walls. Finally, we reveal a divergence of certain nonlinear elastoresistivity coefficients above the critical temperature, and discuss how this is consistent with a divergence of the ferroaxial susceptibility near T$_{\rm{CDW}}$ $\sim$ 200K. Our study also includes detailed elastocaloric measurements, which reveal the presence of an additional phase transition several tens of Kelvin below T$_{\rm{CDW}}$. Our results provide new insight into the symmetry of the ordered state in 1T-TiSe$_2$ and establish elastoresistivity as a powerful probe of hidden order and its symmetry.

Figures (5)

• Figure 1: Illustrating how strain can act as an effective conjugate field to couple to (a) nematic and (b) ferroaxial order. (a) The case of nematic order in a tetragonal lattice. (i) Schematics of a hypothetical Fermi surface in a tetragonal material for temperatures above (left) and below (right) the nematic transition, which breaks the C$_{4}$ rotational symmetry of the tetragonal lattice, here with an $x^2-y^2$ ($B_{1g}$) symmetry. Below, schematic illustration of in-plane anisotropic strain $\epsilon_{xx}-\epsilon_{yy}$, which has the same symmetry as the nematic order parameter, and is therefore an effective conjugate field for the nematic order parameter. (ii) Schematic showing the temperature-dependence of the nematic order parameter, $\psi_{B_{1g}}$, which is proportional to the resistivity anisotropy $\rho_{xx} -\rho_{yy}$ close to the critical temperature, $T_c$. (iii) Schematic showing the temperature-dependence of the nematic susceptibility $\chi_{B_{1g}}$, which can be probed by the linear elastoresistivity $\frac{d(\rho_{xx}-\rho_{yy})/\rho_{0}}{d(\epsilon_{xx}-\epsilon_{yy})}$ and which shows divergent behavior above the nematic transition. (b) The case of ferroaxial order in a trigonal lattice. (i) Schematics of a hypothetical Fermi surface for temperatures above (left) and below (right) the ferro-axial transition, which breaks the vertical mirror symmetry $\sigma_d$. Only a cubic combination of the deviatoric strains $\epsilon_{xy} \left[ 3(\epsilon_{xx}-\epsilon_{yy})^2-4\epsilon_{xy}^2\right]$ has the correct symmetry to serve as a conjugate effective field for the ferroaxial order. (ii) Data for TiSe$_2$ showing the elastoresitivity coefficient $\frac{d\rho_{xy}/\rho_{0}}{d\epsilon_{xx}-\epsilon_{yy}}$ as a function of temperature, which reveals the temperature-dependence of the ferroaxial order parameter $\psi_{A_{2g}}$. (iii) Data for TiSe$_2$ showing the nonlinear elastoresistivity $\frac{d^{2}(\rho_{xx}-\rho_{yy})/\rho_{0}}{d^{2}\epsilon_{xy}}$, which for temperatures above the ferroaxial transition reveals the temperature-dependence of the ferro-axial susceptibility $\chi_{A_{2g}}$.
• Figure 2: Thermodynamic evidence for two sequential phase transitions in 1T-TiSe$_{2}$. (a) The elastocaloric effect as a function of temperature on a sample whose $x$-axis is oriented $22^\circ$ off the crystal's $a$-axis, measured under different DC offset strains. Uniaxial strain is applied along the $x$-axis. The shaded area indicates the transition ranges for $T_{\rm{CDW}}$ and $T^{*}$ under varied strain offsets. Inset of (a) represents the sample configuration. (b) Strain-temperature phase diagram extracted from the first derivative of the EC signal with respect to temperature. Dashed lines serve as visual guides, indicating $T_{\rm{CDW}}$ (black) and $T^{*}$ (gray) transitions.
• Figure 3: Evidence for non-chiral orders for both transitions in 1T-TiSe$_2$ (a) Comparison of SHG signal strengths from GaAs (black) and TiSe$_{2}$ (blue) using the same experimental parameters. The orders-of-magnitude stronger signal from GaAs indicates that the signal measured in TiSe$_{2}$ stems exclusively from its surface. (b) Measurement of the SHG in TiSe$_{2}$ as a function of temperature shows only minimal changes, consistent with a centrosymmetric order. (c) Elastoresistivity measurements on a sample whose $x'$-axis is oriented $10^\circ$ off the crystal $a$-axis for various temperatures. Strain is applied along the $x'$-axis. Inset of (c) shows the schematics of the sample geometry and strain applied. (d) Hysteresis in $\rho_{x'y'}$ normalized by zero-strain resistivity $\rho_{0}$ as a function of temperature.
• Figure 4: Evidence for ferroaxial order and a divergent ferroaxial susceptibility approaching the CDW transition in 1T-TiSe$_{2}$. (a) Elastoresistivity measurements on a sample whose $x'$-axis is oriented $45^\circ$ off the crystal $a$-axis for various temperatures. (b) Linear and (c) quadratic elastoresistivity coefficient as a function of temperature, extracted by fitting the elastoresistivity data in panel (a). Inset of (a) shows the quadratic elastoresistivity coefficient as a function of temperature above $T_{\rm{CDW}}$ and the schematics of the sample geometry. The red curve represents a fit to the Curie–Weiss equation, yielding a Weiss temperature of 199.3 $\pm$ 0.2 K, very close to $T_{CDW}$. Note the negative sign for the y-axis labels in panels (a) and (b).
• Figure 5: Evidence for a ferroaxial order parameter across the CDW transition in 1T-TiSe$_{2}$ (a) Elastoresistivity measurements on a sample whose $x'$-axis is oriented parallel to the crystal axis $a$ for various temperatures. Inset of (a) shows the schematics of the sample geometry and strain applied. (b) Linear and (c) quadratic elastoresistivity coefficients $\frac{\partial(\Delta \rho_{xy}/\rho_{0})}{\partial\epsilon_{xx-yy}}$ and $\frac{\partial^{2}(\Delta \rho_{xy}/\rho_{0})}{\partial\epsilon_{xx-yy}^{2}}$, respectively, as a function of temperature and extracted by fitting the elastoresistivity data in panel (a).