Phonon-induced quadrupolar ordering of the magnetic superconductor ${\mathrm{TmNi}}_2{\mathrm{B}}_2\mathrm{C}$

We present synchrotron x-ray diffraction studies revealing that the lattice of thulium borocarbide is distorted below $T_Q\simeq 13.5\mathrm{K}$ at zero field. $T_Q$ increases and the amplitude of the displacements is drastically enhanced by a factor of 10 at 60 kOe when a magnetic field is applied along [100]. The distortion occurs at the same wave vector as the antiferromagnetic ordering induced by the $a$-axis field. A model is presented that accounts for the properties of the quadrupolar phase and explains the peculiar behavior of the antiferromagnetic ordering previously observed in this compound.

Figure 1

(Color online) Integrated x-ray scattering intensity at (0.484, 0, 8) as a function of temperature in the presence of an applied field along [100]. The colored symbols used for the different values of the field (in units of kOe) are specified in the legend box. The two figures include the same results, but the intensity scales differ by a factor of 10. The intensity is proportional to the square of the displacement of the Tm ions, and the solid curves show the results derived from the model. The inset shows the field dependence of the strain amplitude $E_{13}$ at 1.7 K derived from the experiments in comparison with the calculated results.

Figure 2

(Color online) Temperature dependence of the integrated neutron scattering intensity at ${\mathit{Q}}_A=(0.484,0,0)$ at various values of the applied field along [100] (see Ref. 12). The different values of the field (in units of kOe) are specified in the box. The results were obtained in two different experimental setups (circles and squares). The repetition of the experiment at 40 kOe was used for determining a common scale for the two sets. The intensity is proportional to the squared amplitude of the transverse magnetic moment at the wave vector ${\mathit{Q}}_A$, and the solid lines are the calculated results.