Linear scaling relationship of N\'{e}el temperature and dominant magnons in pyrochlore ruthenates

We present a systematic Raman spectroscopy study on a series of pyrochlore ruthenates, a system which is not yet clearly settled on its magnetic origin and structure. Apart from the Raman-active phonon modes, new peaks that appear in the energy range of 15 - 35 meV below the N\'{e}el temperature are assigned as one-magnon modes. The temperature evolution of one-magnon modes displays no significant thermal dependence in mode frequencies while the intensities decrease monotonically. Remarkably, one-magnons from all compounds show similar characteristics with a single dominant peak at lower energy and weaker side peaks at a couple of meV higher energy. Most importantly, we uncover a striking proportionality between the dominant magnon mode energies and the N\'{e}el temperatures. Our results suggest the Ru ions may have similar or the same magnetic phase in all pyrochlore ruthenates of our study. We have thus found an avenue for directly tuning the magnetic exchange interaction by the selection of the $A$-site ion.