Spin Waves and magnetic exchange interactions in insulating Rb$_{0.89}$Fe$_{1.58}$Se$_2$

The discovery of alkaline iron selenide $A$Fe$_{1.6+x}$Se$_2$ ($A=$ K, Rb, Cs) superconductors has generated considerable excitement in the condensed matter physics community because superconductivity in these materials may have a different origin from the sign reversed s-wave electron pairing mechanism, a leading candidate proposed for all other Fe-based superconductors. Although $A$Fe$_{1.6+x}$Se$_2$ are isostructural with the metallic antiferromagnetic (AF) iron pnictides such as (Ba,Ca,Sr)Fe$_2$As$_2$, they are insulators near $x=0$ and form a $\sqrt{5}\times\sqrt{5}$ blocked AF structure (Fig. 1a) completely different from the iron pnictides. If magnetism is responsible for superconductivity of all iron-based materials, it is important to determine their common magnetic features. Here we use neutron scattering to map out spin waves in the AF insulating Rb$_{0.89}$Fe$_{1.58}$Se$_2$. We find that although Rb$_{0.89}$Fe$_{1.58}$Se$_2$ has a N$\rm \acute{e}$el temperature ($T_N=475$ K) much higher than that of the iron pnictides ($T_N\leq 220$ K), spin waves for both classes of materials have similar zone boundary energies. A comparison of the fitted effective exchange couplings using a local moment Heisenberg Hamiltonian in Rb$_{0.89}$Fe$_{1.58}$Se$_2$, (Ba,Ca,Sr)Fe$_2$As$_2$, and iron chalcogenide Fe$_{1.05}$Te reveals that their next nearest neighbor (NNN) exchange couplings are similar. Therefore, superconductivity in all Fe-based materials may have a common magnetic origin that is intimately associated with the NNN magnetic exchange interactions, even though they have metallic or insulating ground states, different AF orders and electronic band structures.