$S_4$ Symmetric Microscopic Model for Iron-Based Superconductors

Although iron-based superconductors are multiorbital systems with complicated band structures, we demonstrate that the low-energy physics which is responsible for their high-$T_c$ superconductivity is essentially governed by an effective two-orbital Hamiltonian near half filling. This underlying electronic structure is protected by the $S_4$ symmetry. With repulsive or strong next-nearest-neighbor antiferromagnetic exchange interactions, the model results in a robust $A_{1g}$ $s$-wave pairing which can be mapped exactly to the $d$-wave pairing observed in cuprates. The classification of the superconducting (SC) states according to the $S_4$ symmetry leads to a natural prediction of the existence of two different phases, named the $A$ and $B$ phases. In the $B$ phase, the superconducting order has an overall sign change along the $c$ axis between the top and bottom As (or Se) planes in a single Fe-As (or Fe-Se) trilayer structure, the common building block of iron-based superconductors. The sign change is analogous to the sign change in the $d$-wave superconducting state of cuprates upon 90° rotation. Our derivation provides a unified understanding of iron pnictides and iron chalcogenides, and suggests that cuprates and iron-based superconductors share an identical high-$T_c$ superconducting mechanism.

Popular Summary

The discoveries since 2008 of a large and chemically rather diverse class of high-temperature iron-based superconductors with maximal superconducting temperatures reaching above 50 K have renewed the excitement in superconductivity. In addition to offering the promise of new high-temperature superconductors, these materials have posed some great fundamental scientific challenges: Is there a microscopic theory that unifies our understanding of the superconductivity of these varied materials? What may this theory be? Are there deep connections between the new superconductors and the older, copper-based high-temperature superconductors (cuprates)? In this paper, we present a symmetry-based effective microscopic theory of simplicity that makes an extraordinary stride, in a new direction, toward the ultimate answers to these challenging questions.

The current favorite effective microscopic theory for iron-based superconductors employs all five orbitals associated with the $d$ electrons of an iron atom. While this theory offers flexibility in terms of allowing for fair agreement with fully microscopic (first principles) electronic structure calculations, it is unable, given its vast parameter space, to explain the robustness of the electron-pairing symmetry observed across many different members of the family. In a remarkable contrast, our theory involves only two iron orbitals, but the orbitals are deeply connected through the $S_4$ lattice symmetry characteristic of this family of new superconductors. Despite its simplicity (and very few free parameters), the two-orbital model can explain the big changes seen in the band structures across different materials in the family.

Equally, if not more, remarkably, the apparently different electron-pairing symmetries seen in the new superconductors and the older cuprates turn out to be, within the framework of our model, connected by a mathematical (“gauge”) transformation, which is not unlike a transformation connecting two different reference frames. Our model strongly suggests, therefore, a unifying way to understand both the new superconductors and the older cuprates. The problem of studying the pairing symmetry in the iron-based superconductors becomes then the well-studied problem of pairing symmetry in the cuprates.

We believe that our introduction of the $S_4$-symmetry-related two orbitals provides a new basic platform from which many exciting possibilities of predicting, discovering, and understanding novel properties of iron-based superconductors can be launched.

Figure 1

(a) The square lattice structure of a single iron layer: One cell includes two Fe ions shown as differently filled black balls forming two sublattices. We use $x\mathrm{\text{−}}y$ coordinates to mark the original tetragonal lattices and $x^{\prime }\mathrm{\text{−}}{y}^{\prime }$ to mark the sublattice direction. (b) The gauge transformation is illustrated. The balls with red circles are affected by the gauge transformation. (c) and (d) The mapping from the $s$-wave to the $d$-wave pairing symmetry by the gauge transformation.

Figure 2

Three-orbital [30] and five-orbital [10] models: (a),(e) The Fermi surfaces; (b),(f) the band dispersion along the high-symmetry lines; (c),(g) the Fermi surfaces after the gauge transformation; (d),(h) the band dispersions along the high-symmetry lines after the gauge transformation. The hopping parameters can be found in the two references. The y axis for (b),(d),(f),(h) is in units of E(ev).

Figure 3

A sketch of the $d_{x^{\prime }z}$ and $d_{y^{\prime }z}$ orbitals, their orientations, and their coupling into the two As(Se) layers. The hopping parameters are indicated: The nearest-neighbor hopping is marked by $t_{1x}\mathrm{\text{and}}{t}_{1y}$; the next-nearest-neighbor hoppings are $t_2$ and $t_2^{\prime }$ due to the broken symmetry along two different diagonal directions; and the third NN hopping is marked by $t_{3x}\mathrm{\text{and}}{t}_{3y}$. The coupling between the two layers is marked by the nearest-neighbor hopping $t_c$.

Figure 4

The Fermi surfaces of each component when only parameters $t_{1s}$, $t_{2d}$, and $t_{2s}$ are considered. The layout for (a)–(h) exactly follows that of Fig. 2. The parameters are $t_1=0.24$, $t_2=0.52$, and $\mu =-0.273$. The only difference in parameters between (a) and (e) is that $t_2^{\prime }=-0.1$ in (a) and $t_2^{\prime }=-0.2$ in (e). The y axis for (b),(d),(f),(h) is in units of E(ev).

Figure 5

Typical Fermi surfaces (a) and band dispersions (b) resulting from Eq. (10), with $t_c=0$ and parameters $t_{1s}=0.4$, $t_{1d}=-0.03$, $t_{2s}=0.3$, $t_{2d}=0.6$, $t_{3s}=0.05$, $t_{3d}=-0.05$, and $\mu =-0.3$. (c),(d) show the corresponding results when $t_c=0.02$ is used in Eq. (7) with the same parameter settings. The y axis for (b),(d) is in units of E(ev).

Figure 6

Fermi surfaces and band dispersions in the presence of the $S_4$ symmetry breaking: (a),(e) $t_{b1}=0.005$ in Eq. (22); (b),(f) $t_{bt}=0.05$ in Eq. (23); (c),(g) $t_{bo}=0.05$ in Eq. (24); (d),(h) $t_{bso}=0.05$ in Eq. (25). Other parameters are the same as in Fig. 5. The y axis for (e)–(h) is in units of E(ev).

Figure 7

(a) An illustration of a single Fe-As(Se) layer and the setup for a dc SQUIDS measurement to measure the sign change of the SC phase between top and bottom As(Se) layers. (b) The phase distribution in the $A$ phase of the $A_{1g}$ $s$-wave state in the view of a $d$-wave picture (red for one isospin component and blue for the other). (c) The phase distribution in the $B$ phase of the $A_{1g}$ $s$-wave state.

Figure 8

A sketch of the correlation between the hopping and pairing symmetries for both iron-based superconductors and cuprates. The black (a) and the bronze (b) balls represent Fe and Cu atoms, respectively. The blue and green solid lines indicate that the hoppings between two connected atoms have opposite signs. The red and blue dashed lines indicate that the SC pairings between two connected atoms have opposite signs.