[@mywang11]. In spite of dramatically different dispersions for various iron-based materials, their spin wave overall energy scales are similar and about 220 meV, less than that of the insulating copper oxides. Twinning is considered.](Fig1R2)

Four years after the initial discovery of superconductivity in LaFeAsO$_{1-x}$F$_x$ [@kamihara], there are now three major families of iron-based superconductors: the iron pnictides [@johnston; @stewart], iron chalcogenides [@mawkuen2; @mhfang08], and alkaline iron selenides [@jgguo; @mhfang]. The parent compounds of the pnictides, such as $A$FeAsO ($A=$ La, Ce, Sm, Pr, etc.), $A$Fe$_2$As$_2$ ($A=$ Ba, Sr, Ca), and NaFeAs, all have the same collinear AF structure as shown in Fig. 1b, with a small ordered moment ($<1\ \mu_B$/Fe) and N$\rm \acute{e}$el temperature $T_N\le 200$ K [@cruz; @qhuang; @cwchu]. The AF spin moments are aligned along the weak orthorhombic unit cell $a$-axis direction ($a\approx5.62$, $b\approx 5.57$ Å). In reciprocal space, the AF Bragg peaks occur at in-plane ordering wave vectors ${\bf Q}_{AF}=(\pm 1+m, n)$ and at ${\bf Q}_{AF}\approx (m, \pm 1+n)$ due to twinning (red circles in Fig. 1f), consistent with the $\Gamma(0,0)\leftrightarrow M(1,0)/M(0,1)$ FS nesting picture [@mazin2011n]. However, although the calculated FS of the chalcogenides Fe$_{1+y}$Te$_{1-x}$Se$_x$ is similar to that of iron pnictides [@subedi], surprisingly its parent compound 