Crystal structure and phonon softening in ${\mathrm{Ca}}_3{\mathrm{Ir}}_4{\mathrm{Sn}}_{13}$

We investigated the crystal structure and lattice excitations of the ternary intermetallic stannide ${\mathrm{Ca}}_3{\mathrm{Ir}}_4{\mathrm{Sn}}_{13}$ using neutron and x-ray scattering techniques. For $T>T^{*}\approx 38$ K, the x-ray diffraction data can be satisfactorily refined using the space group $Pm\overline{3}n$. Below $T^{*}$, the crystal structure is modulated with a propagation vector of $\vec{q}=(1/2,1/2,0)$. This may arise from a merohedral twinning in which three tetragonal domains overlap to mimic a higher symmetry, or from a doubling of the cubic unit cell. Neutron diffraction and neutron spectroscopy results show that the structural transition at $T^{*}$ is of a second-order, and that it is well described by mean-field theory. Inelastic neutron scattering data point towards a displacive structural transition at $T^{*}$ arising from the softening of a low-energy phonon mode with an energy gap of $\mathrm{\Delta }\left(120\text{K}\right)=1.05$ meV. Using density functional theory, the soft phonon mode is identified as a “breathing” mode of the ${\mathrm{Sn}}_{12}$ icosahedra and is consistent with the thermal ellipsoids of the Sn2 atoms found by single-crystal diffraction data.

Figure 1

Crystal structure of ${\mathrm{Ca}}_3{\mathrm{Ir}}_4{\mathrm{Sn}}_{13}$ measured by single-crystal neutron diffraction and refined in $Pm\overline{3}n$ for $T>T^{*}$. Color code: red Ca, green Ir, blue Sn. The plotted ellipsoids denote the anisotropic mean-square displacements of the atoms (with a magnification factor of two for clarity).

Figure 2

Two cuts of the reciprocal space of ${\mathrm{Ca}}_3{\mathrm{Ir}}_4{\mathrm{Sn}}_{13}$: (a) the ($h,k$, 0) and (b) the $(h,h,l)$ plane as measured on BM01A at $T=21.5$ K. The $(h,k,l)$ indexing corresponds to the cubic $Pm\overline{3}n$ lattice. The fundamental reflections originating from the space group $Pm\overline{3}n$ are emphasized in blue. Additional (superstructure) reflections are represented as red circles and can be indexed with the propagation vector $\vec{q}=(1/2,1/2,0)$. (Bottom) Reciprocal lattice for (c) the double cubic unit cell description proposed for ${\mathrm{Sr}}_3{\mathrm{Ir}}_4{\mathrm{Sn}}_{13}$ [8] and (d) the alternative description with three tetragonal domains.

Figure 3

Temperature dependence of the superstructure Bragg reflection at $\vec{Q}=(7/2,7/2,1)$ with a critical exponent of $\beta =0.50\left(2\right)$ and $T^{*}=38.5\left(2\right)$ K. The inset shows diffraction patterns for $T<T^{*}$ measured at selected temperatures.

Figure 4

(a) Energy scans at $\vec{Q}=(7/2,7/2,1)$ and $T=120$, 70, 45, and 20 K (scans at different temperatures are plotted with a relative offset). The solid line denotes the total fit, the dotted line the elastic coherent contribution and the dashed line the phonon. (b) Softening of the $\vec{Q}=(7/2,7/2,1)$ phonon above $T^{*}$. The dashed line correspond to a mean-field behavior $\mathrm{\Delta }\left(T\right)\propto {(T-T^{*})}^{\beta }$ with $\beta =1/2$.

Figure 5

(a) Illustration of the breathing of ${\mathrm{Sn}}_{12}$ icosahedra in (Ca, ${\mathrm{Sr})}_3{\mathrm{Ir}}_4{\mathrm{Sn}}_{13}$. The arrows represent the atomic replacements of the Sn2 atoms on the surface of the icosahedra. (b) Thermal ellipsoids of the Sn1 and Sn2 atoms at $T=50$ K with the suggested “breathing” mode. (Inset) Orientations of the thermal ellipsoid of Sn2 atom and of the Sn1–Sn2 bond.