Magnetism and superconductivity in the layered hexagonal transition metal pnictides

We investigate the electronic and magnetic structures of the $122\left({AM}_2{B}_2\right)$ hexagonal transition metal pnictides with $A=(\text{Sr},\text{Ca}),M=(\text{Cr},\text{Mn},\text{Fe},\text{Co},\text{Ni})$, and $B=(\text{As},\text{P},\text{Sb})$. It is found that the family of materials shares critical similarities with those of tetragonal structures that include the famous iron-based high-temperature superconductors. In both families, the next-nearest-neighbor (NNN) effective antiferromagnetic (AFM) exchange couplings reach the maximum value in the iron-based materials. While the NNN couplings in the latter are known to be responsible for the $C$-type AFM state and to result in the extended $s$-wave superconducting state upon doping, they cause the former to be extremely frustrated magnetic systems and can lead to a time-reversal symmetry-broken $d+id$ superconducting state upon doping. Thus, if synthesized, iron-based compounds with hexagonal structure can help us to determine the origin of high-temperature superconductivity.

Figure 1

(a) The crystal structure of the hexagonal ${AM}_2{B}_2$ $(A=$ Sr, Ca; $M$ = Cr, Mn, Fe, Co, Ni; $B$ = As, P, Sb) with the trigonal ${\mathrm{CaAl}}_2{\mathrm{Si}}_2$-type structure (space group $P\overline{3}\mathrm{m}1$, No. 164). (b) The corrugated honeycomb lattice formed only by the $M$ atoms in the $ab$ plane. Exchange interactions between NN $J_1$, NNN $J_2$, and third NN $J_3$ are indicated. (c) The crystal structure of ${AM}_2{B}_2$ $(A$ = Ba; $M$ = Cr, Mn, Fe, Co, Ni; $B$ = As, P) with the body-centered-tetragonal structure (space group $I4/mmm$, No. 139). The in-plane NN magnetic exchange interaction $J_1$, the in-plane NNN exchange interaction $J_2$, and the out-of-plane NN along the $c$ axis exchange interaction $J_z$ are indicated.

Figure 2

Sketch of the four collinear magnetic states, including the FM states, the $G$-type AFM state, the stripe FM state, and the $C$-type AFM state.

Figure 3

The $J_2$ magnetic exchange coupling constants of (a) $\mathrm{Ba}{M}_2{\mathrm{As}}_2$ $(M$ = Cr, Mn, Fe, Co, Ni) and (b) ${\mathrm{BaFe}}_2{B}_2$ $(B$ = P, As), which are extracted from the $\mathrm{GGA}+U$ calculations with the values $U=(0.0,0.5,1.0,1.5,2.0,2.5)$ eV.

Figure 4

Sketch of four collinear magnetic states: the FM states, the AFM state, the zigzag state, and the stripe state. The red frame indicates the magnetic unit cell in the $\mathrm{GGA}+U$ calculations of these magnetic structures.

Figure 5

The $J_2$ exchange coupling parameters in (a) $\mathrm{Ca}{M}_2{\mathrm{As}}_2$ $(M$ = Cr, Mn, Fe, Co, Ni), (b) $\mathrm{Ca}{M}_2{\mathrm{P}}_2$ $(M$ = Cr, Mn, Fe, Co, Ni), and (c) ${\mathrm{CaFe}}_2{B}_2$ $(B$ = P, As), which are extracted from the $\mathrm{GGA}+U$ calculations with the values $U=(0.0,0.5,1.0,1.5,2.0,2.5)$ eV.

Figure 6

(a) The magnetic moments as functions of pressure; (b) the superexchange antiferromagnetic interaction constants $J_1$ and $J_2$ as a function of the pressure for ${\mathrm{CaFe}}_2{\mathrm{As}}_2$.

Figure 7

(a) Band structure and density of states and (b) Fermi surface of ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ with the optimized structural parameters in the paramagnetic state. (c) Sketch of the iron lattice: ${\rho }_1$ is the NN bond, $R_1$ is the NNN bond, $r_1$ is the third-NN bond, ${\delta }_1$ is the fourth-NN bond, and ${\gamma }_1$ is the fifth-NN bond. The white dots denote the A sublattice, and the gray dots denotes the B sublattice.

Figure 8

The overlap between the Fermi surfaces (black lines) and the superconducting gap distribution for ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ in the first Brillouin zone (BZ) in the $k_z=0$ plane for the extended (a) $s$-wave and (a) $(d+id)$-wave cases. The $s$-wave order parameter in $k$ space has a momentum form factor $\mathrm{\Delta }\left(\mathit{k}\right)=\mathrm{\Delta }(2cos\frac{\sqrt{3}}{2}{k}_{x}cos\frac{1}{2}{k}_y+cos{k}_y)$, and the $\left(d+id\right)$-wave order parameter has $\mathrm{\Delta }\left(\mathit{k}\right)=\mathrm{\Delta }(cos{k}_y-cos\frac{\sqrt{3}}{2}{k}_{x}cos\frac{1}{2}{k}_y+i\sqrt{3}sin\frac{\sqrt{3}}{2}{k}_{x}sin\frac{1}{2}{k}_y)$. The color bar labels the amplitude of the SC gap when $\mathrm{\Delta }=0.02$.