near $x=0.092$. The inset shows the transverse incommensurate AF order. Superconductivity in the electron-doped materials only extends to $x\approx 0.25$. (c) Schematics of FS’s correspond to 35% hole-doped BaFe$_2$As$_2$ [@chenglinzhang] with possible nesting vectors marked with arrows. The $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals for different Fermi surfaces are colored as red, green, and blue, respectively. (d) FS’s of BaFe$_2$As$_2$ with orbital characters [@graser10]. (e) Fermi surfaces of 8% electron-doped BaFe$_2$As$_2$ [@chenglinzhang]. For all three cases, FS’s are plotted with zero wave vector transfers along the $c$-axis. (f) Longitudinally elongated spin excitations at $E=20$ meV seen in the optimally hole-doped Ba$_{0.67}$K$_{0.33}$Fe$_2$As$_2$ [@chenglinzhang]. (g) Transversely elongated spin waves at $E=20$ meV for BaFe$_2$As$_2$ [@lharriger]. (h) Transversely elongated spin excitations at $E=20$ meV for BaFe$_{1.9}$Ni$_{0.1}$As$_2$ [@jtpark; @msliu12]. (i) Energy dependence of $\chi^{\prime\prime}(\omega)$ for BaFe$_2$As$_2$ (blue solid line) and BaFe$_{1.9}$Ni$_{0.1}$As$_2$ below (red dashed line) and above (red solid circles) $T_c$ in absolute units of $\mu_B^2$eV$^{-1}$f.u.$^{-1}$. The sharp peak near $E\approx 8$ meV below $T_c$ is the neutron spin resonance coupled directly to superconductivity [@lumsden; @schi09; @dsinosov09; @jtpark; @clester; @hfli; @hqluo12]. ](Fig2R2)

In the case of insulating $A$Fe$_{1.6+x}$Se$_2$, spin waves have an acoustic mode and two optical modes separated by spin gaps (Fig. 1l) [@mywang11]. 