Unified minimum effective model of magnetic properties of iron-based superconductors

Iron-based superconductors exhibit many different antiferromagnetically ordered ground states. We construct a minimum effective magnetic model that displays all magnetic phases. This model also captures three incommensurate magnetic phases as well, two of which have been observed experimentally. The model characterizes the nature of phase transitions between the different magnetic phases and explains a variety of magnetic properties, such as spin-wave spectra and electronic nematism. Most importantly, by unifying the understanding of magnetism, we cast insight on the key ingredients of magnetic interactions that are critical to the occurrence of superconductivity.

Figure 1

(a) The collinear-antiferromagnetic state (CAF) in ironpnictides, for example, ${\text{CaFe}}_2{\text{As}}_2$. (b) The bicollinear-antiferromagnetic state (BCAF) in FeTe. (c) the block-antiferromagnetic state (BAF) without iron vacancies. (d) The block-antiferromagnetic state with $\sqrt{5}\times \sqrt{5}$ vacancy ordering (BAF${}_v$) in K${}_{0.8}$Fe${}_{1.6}$Se${}_2$. The exchange couplings are indicated in (a).

Figure 2

The classical phase diagram of the $J_1-J_2-J_3-K$ model at $K=0.2J_1$. There are total four commensurate and three incommensurate magnetic phases. Their labels and the ordered wave vectors are specified in the figure.

Figure 3

(a) and (b) Spin waves in the $(\pi ,q)$ incommensurate phases along $(1,\delta )\pi$ and $(\delta ,1)\pi$, respectively. (c) Hour-glass-like spin waves along ${(1/2+\delta ,1/2-\delta )}_{\mathrm{T}}$ in the $(\pi ,q)$ phase. The parameters are taken as $S=1$, $(J_1,J_2/|J_1|,J_3/|J_1|,K{S}^2/|J_1\left|\right)=(1,0.6,0.06,0.02)$.

Figure 4

The three-dimensional $J_3/\left|J_1\right|$-$J_2/\left|J_1\right|$-$K{S}^2/\left|J_1\right|$ phase diagram with $S=1$. The exchange coupling $J_2$ is fixed to be antiferromagnetic, namely, positive.

Figure 5

The spin-wave excitations along $(k_x,k_y)=(1,\delta )\pi$ and along $(\delta ,1)\pi$ in the $(\pi ,q)$ phase near CAF phase. The parameters are chosen to be $S=1$ and $(J_1,{\tilde{J}}_2,{\tilde{J}}_3,\tilde{K})=(1,0.7,0.15,0)$ in (a) and (b), and $(1,0.7,0.15,0.05)$ in (c) and (d). Increasing $\tilde{K}$ (biquadratic interactions) opens a gap between the acoustic and optical modes near $(\pi ,0)$ in the $(1,\delta )\pi$ direction.

Figure 6

The spin-wave excitations along $(k_x,k_y)=(1,\delta )\pi$ and along $(\delta ,1)\pi$ in the $(\pi ,q)$ phase near AFM phase. The parameters are chosen to be $S=1$ and $(J_1,{\tilde{J}}_2,{\tilde{J}}_3,\tilde{K})=(1,0.4,0.15,0)$ in (a) and (b), and $(1,0.4,0.15,0.05)$ in (c) and (d). Increasing $\tilde{K}$ has a smaller effect in the $(1,\delta )\pi$ direction but opens a gap near $(\pi /2,\pi )$ in the $(\delta ,1)\pi$ direction.

Figure 7

The INS intensity along $(k_x,k_y)=(1,\delta )\pi$ and along $(\delta ,1)\pi$ in the $(\pi ,q)$ phase. (a) and (b) are close to AFM phase with parameters chosen to be $(J_1,{\tilde{J}}_2,{\tilde{J}}_3,\tilde{K})=(1,0.4,0.15,0)$, (c) and (d) are near CAF with parameters equal to $(1,0.6,0.07,0.01)$. The thin lines label the boundaries of zone with vanishing intensities.

Figure 8

The spin-wave excitations along $(k_x,k_y)=(\delta ,\delta )\pi$ and along $(\delta ,-\delta )\pi$ in the $(q,q)$ phase. The parameters are chosen to be $S=1$ and $(J_1,{\tilde{J}}_2,{\tilde{J}}_3,\tilde{K})=(-1,0.7,0.4,0.1)$ in (a),(b), and $(-1,0.8,0.5,0)$ in (c),(d).

Figure 9

The INS intensity along $(k_x,k_y)=(\delta ,\delta )\pi$ and along $(\delta ,-\delta )\pi$ in the $(q,q)$ phase. The parameters are chosen to be $S=1$ and $(J_1,{\tilde{J}}_2,{\tilde{J}}_3,\tilde{K})=(-1,0.7,0.4,0.1)$ in (a) and (b), and $(-1,0.8,0.5,0)$ in (c) and (d). The thin lines label the boundaries of zone with vanishing intensities.

Figure 10

The spin-wave excitations along $(k_x,k_y)=(0,\delta )\pi$ and along $(\delta ,0)\pi$ in the $(0,q)$ phase. The parameters are chosen to be $S=1$ and $(J_1,{\tilde{J}}_2,{\tilde{J}}_3,\tilde{K})=(-1,0.8,0.2,0.)$ in (a) and (b), and $(-1,0.8,0.2,0.05)$ in (c) and (d).

Figure 11

The scattering intensity along $(k_x,k_y)=(0,\delta )\pi$ and along $(\delta ,0)\pi$ in the $(0,q)$ phase. The parameters are chosen to be $S=1$ and $(J_1,{\tilde{J}}_2,{\tilde{J}}_3,\tilde{K})=(-1,0.8,0.2,0.)$ in (a) and (b), and $(-1,0.8,0.2,0.05)$ in (c) and (d). The thin lines label the boundaries of zone with vanishing intensities.

Figure 12

The phase diagram of the quantum model that breaks the degeneracy of the BCAF and BAF phases for $S=1$, $\tilde{K}=0.2$.