Spontaneous Lattice Distortion in the Spin-Triplet Superconductor Cu$_{x}$Bi$_2$Se$_3$
The doped topological insulator Cu$_x$Bi$_2$Se$_3$ has attracted considerable attention as a new platform for studying novel properties of spin-triplet and topological superconductivity. In this work, we performed synchrotron x-ray diffraction measurements on Cu$_x$Bi$_2$Se$_3$ (0.24$\leq x\leq$ 0.46) to investigate the coupling between the superconducting order parameter and crystal lattice. In the crystals in which the vector order parameter (${\boldsymbol d}$ vector) is tilted from the crystal high-symmetry directions as evidenced by nematic diamagnetic susceptibility, we find a sizable lattice distortion ($\sim$100 ppm) associated with the onset of superconductivity. In contrast, in crystals with the ${\boldsymbol d}$ vector aligned along the high-symmetry directions, we find no appreciable change in lattice constant. Together with a pronounced vestigial behavior of the distortion, the results are clear evidence for an odd-parity $E_u$ order parameter that couples with trigonal lattice. Furthermore, in the crystal with $x$ = 0.46 where diamagnetic susceptibility is isotropic in the plane, no lattice distortion accompanying the superconducting transition is found, which is in line with a chiral superconducting state in the highly doped region. Our work shows that lattice distortion can be a powerful diagnosing quantity for nematic superconductivity with two-component order parameter.