Symmetry-Breaking Lattice Distortion in ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$

The electronic nematic phase of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ is investigated by high-resolution in-plane thermal expansion measurements in magnetic fields close to 8 T applied at various angles $\Theta$ off the $c$ axis. At $\Theta <10°$ we observe a very small (${10}^{-7}$) lattice distortion which breaks the fourfold in-plane symmetry, resulting in nematic domains with interchanged $a$ and $b$ axis. At $\Theta \gtrsim 10°$ the domains are almost fully aligned and thermal expansion indicates an area-preserving lattice distortion of order $2\times {10}^{-6}$ which is likely related to orbital ordering. Since the system is located in the immediate vicinity of a metamagnetic quantum critical end point, the results represent the first observation of a structural relaxation driven by quantum criticality.

Figure 1

In-plane relative length change $\Delta L/L$ (a) and respective thermal expansion coefficient $\alpha =d(\Delta L/L)/dT$ (b) vs temperature for ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$. The black curves represent data at a magnetic field of 7.95 T applied along the $c$ axis, whereas the light gray (green) and gray (red) curves are taken at a field of 7.92 T tilted by 5° from the $c$ axis with $\Delta L\perp B_{\mathrm{in}\mathrm{\text{−}}\mathrm{plane}}$ and $\Delta L\parallel B_{\mathrm{in}\mathrm{\text{−}}\mathrm{plane}}$, as sketched in the inset.

Figure 2

In-plane relative length change (a) and thermal expansion (b) at magnetic fields of 7.75 T (circles) and 7.48 T (triangles) applied 10° and 15° off the $c$ axis, respectively. For each angle, the length change has been determined parallel (gray or red symbols) and perpendicular (light gray or green symbols) to the in-plane component of the magnetic field. The dotted lines indicate $T^2$ behavior.

Figure 3

Isothermal magnetostriction at temperatures below 0.1 K (cf. symbols in Fig. 4) of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ at different angles $\Theta$ of the applied field to the $c$ axis. The solid and dotted curves represent data of the magnetostriction coefficient $\lambda =d(\Delta L/L)/dB$ along and perpendicular to the in-plane field, respectively. The inset shows a sketch of the lattice distortion in the nematic phase as derived from thermal expansion and magnetostriction measurements.

Figure 4

Phase diagram of the nematic phase of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ at different tilt angles (cf. different colors). The squares and circles denote transitions in magnetostriction and thermal expansion, respectively (solid symbols: $\Delta L\parallel B_{\mathrm{in}\mathrm{\text{−}}\mathrm{plane}}$, open symbols: $\Delta L\perp B_{\mathrm{in}\mathrm{\text{−}}\mathrm{plane}}$). The black solid lines indicate first-order transitions ending at tricritical points (cf. filled stars) [8], connected by a (dash-dotted) line of second-order transitions. The colored open stars denote the presumed position of critical points, connected by crossover lines (dotted).

Figure 5

Illustration of the proposed domain configuration (left, on macroscopic scale) and orbital ordering (right, on atomic scale) for the nematic phase of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ [12]. The different orbitally ordered states are indicated by the light gray (green) and gray (red) shading or color. The arrows indicate the directions of the in-plane field, easy and hard transport, as well as expansion and contraction.