Nematicity in Electron-Doped Iron-Pnictide Superconductors

Abstract

The nature of the nematicity in iron pnictides is studied with a proposed magnetic fluctuation. The spin-driven order in the iron-based superconductor has been realized in two categories: stripe SDW state and nematic state. The stripe SDW order opens a gap in the band structure and causes a deformed Fermi surface. The nematic order does not make any gap in the band structure and still deforms the Fermi surface. The electronic mechanism of nematicity is discussed in an effective model by solving the self-consistent Bogoliubov-de Gennes equations. The nematic order can be visualized as crisscross horizontal and vertical stripes. Both stripes have the same period with different magnitudes. The appearance of the orthorhombic magnetic fluctuations generates two uneven pairs of peaks at ±π0 and 0±π in its Fourier transformation. In addition, the nematic order breaks the degeneracy of dxz and dyz orbitals and causes the elliptic Fermi surface near the Γ point. The spatial image of the local density of states reveals a dx2-y2-symmetry form factor density wave.

1. Introduction

The discovery of Fe-based superconductors with critical temperatures up to 55 K has begun a new era of investigations of the unconventional superconductivity. In common with copper-like superconductors (cuprate), the emergency of superconductivity in electron-doped Fe-pnictides such as $\mathrm{Ba}{\left(\mathrm{F}{\mathrm{e}}_{1-x}\mathrm{C}{\mathrm{o}}_x\mathrm{As}\right)}_2$ is to suppress the magnetic order and fluctuations originated in the parent compound with $x=0$ [1]. The intertwined phases between the superconductivity and stripe spin density wave (SDW) order (ferromagnetic stripes along one Fe-Fe bond direction that is antiferromagnetically aligned along orthogonal Fe▬Fe bond) are of particular interests. In both pnictides and cuprates, the experimentally observed nematicity exists in an exotic phase between the superconductivity (SC) and the stripe SDW [2]. The nematicity occurs in weakly doped iron pnictides with tetragonal-to-orthorhombic structural transition [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], i.e., in a square unit cell, the point-group symmetry is reduced from C₄ (tetragonal) to C₂ (orthorhombic).

At present, there are two scenarios for the development of nematic order through the electronic configurations [18]. One scenario is the orbital fluctuations [19, 20, 21, 22, 23]. The structural order is driven by orbital ordering. The orbital ordering induces magnetic anisotropy and triggers the magnetic transition at a lower temperature. The other scenario is the spin fluctuation [24, 25, 26, 27]. The magnetic mechanism for the structural order is associated with the onset of SDW.

Recently, Lu et al. [28] reported that the low-energy spin fluctuation excitations in underdoped sample ${\text{BaFe}}_{1.915}\mathrm{N}{\mathrm{i}}_{0.085}\mathrm{A}{\mathrm{s}}_2$ change from C₄ symmetry to C₂ symmetry in the nematic state. Zhang et al. [29] exhibited that the reduction of the spin-spin correlation length at $\left(0,\pi \right)$ in ${\text{BaFe}}_{1.935}\mathrm{N}{\mathrm{i}}_{0.065}\mathrm{A}{\mathrm{s}}_2$ happens just below $T_s$, suggesting the strong effect of nematic order on low-energy spin fluctuations. Apparently, these experiments above have provided a spin-driven nematicity picture.

The partial melting of SDW has been proposed as the mechanism to explain the nematicity. The properties of the spin-driven nematic order have been studied in Landau-Ginzburg-Wilson’s theory [18, 24, 25, 26]. Meanwhile, the lack of the realistic microscopic model is responsible for the debates where the leading electronic instability, i.e., the onset of SDW, causes the nematic order. Recently, an extended random phase approximation (RPA) approach in a five-orbital Hubbard model including Hund’s rule interaction has shown that the leading instability is the SDW-driven nematic phase [30]. Although the establishment of the nematicity in the normal state has attracted a lot of attentions, the microscopic description of the nematic order and, particularly, the relation between SC and the nematic order are still missing.

The magnetic mechanism for the structural order is usually referred to the Ising-nematic phase where stripe SDW order can be along the x-axis or the y-axis. The nematic phase is characterized by an underlying electronic order that the Z₂ symmetry between the x- and y- directions is broken above and the O(3) spin-rotational symmetry is preserved [25].

The magnetic configuration in FeSCs can be described in terms of two magnetic order parameters ${\mathrm{∆}}_x$ and ${\mathrm{∆}}_y$. Both order parameters conventionally defined in momentum space are written as

$${\mathrm{∆}}_{\ell }=\sum _k{c}_{k+Q_{\ell },\alpha }^{†}{\sigma }_{\mathit{\alpha \beta }}{c}_{k,\beta },$$E1

where $\ell =x$ or $y$. Here the wave vectors $Q_x=\left(\pi ,0\right)$ and $Q_y=\left(0,\pi \right)$ correspond to the spins parallel along the y-axis and antiparallel along the x-axis and the spins parallel along the x-axis and antiparallel along the y-axis, respectively.

In the stripe SDW state, the order parameters are set to $⟨{\mathrm{∆}}_x⟩\ne 0$ or $⟨{\mathrm{∆}}_y⟩\ne 0$, i.e., $⟨{\mathrm{∆}}_{\ell }⟩\ne 0.$ This implies to choose an ordering vector either $Q_x$ or $Q_y$. The Z₂ symmetry indicating to the degenerate of spin stripes along the y-axis (corresponding to $Q_x$) or x-axis (corresponding to $Q_y$) is broken. In addition, the O(3) spin-rotational symmetry is also broken. In the real space configuration, the magnetic ground state is an orthorhombic uniaxial stripe state. The stripe order reduces the point-group symmetry of a unit cell from C₄ (tetragonal) to C₂ (orthorhombic). In the nematic state, the order parameters are set to $⟨{\mathrm{∆}}_x⟩=⟨{\mathrm{∆}}_y⟩=0$ and $⟨{\mathrm{∆}}_x^2⟩\ne ⟨{\mathrm{∆}}_y^2⟩$. This implies that the magnetic fluctuations associated with one of the ordering vectors are stronger than the other $⟨{\mathrm{∆}}_x^2⟩>⟨{\mathrm{∆}}_y^2⟩$ or $⟨{\mathrm{∆}}_y^2⟩>⟨{\mathrm{∆}}_x^2⟩$. Therefore, the Z₂ symmetry is broken, but the O(3) spin-rotational symmetry is not. In the real space configuration, the x- and y-directions of the magnetic fluctuations are inequivalent.

Recently, the reentrant C₄ symmetry magnetic orders have been reported in hole-doped Fe-pnictide [27, 31, 32]. A double-Q order (choose both $Q_x$ and $Q_y$) has been proposed to change the ground state from striped to tetragonal [33, 34]. Two stripe orders with the ordering vectors $Q_x$ and $Q_y$ are superposed to preserve the tetragonal symmetry. As a matter of fact, the nematic phase is characterized by an underlying electronic order that spontaneously breaks tetragonal symmetry. Since the double-Q order does not break the C₄ symmetry, it is not suitable to explain the nematicity.

The magnetic fluctuations trigger a transition from the tetragonal-to-orthorhombic phase. At very high temperature, $T>T_S$, $⟨{\mathrm{∆}}_x⟩=⟨{\mathrm{∆}}_y⟩=0$, and the fluctuations of all order parameters have equal strength, i.e., $⟨{\mathrm{∆}}_x^2⟩=⟨{\mathrm{∆}}_y^2⟩$. As the temperature lowers, $T_N<T<T_S$, the thermodynamic average of order parameters still remains $⟨{\mathrm{∆}}_x⟩=⟨{\mathrm{∆}}_y⟩=0$ but $⟨{\mathrm{∆}}_x^2⟩\ne ⟨{\mathrm{∆}}_y^2⟩$. The fluctuations of one of the orders ${\mathrm{∆}}_x$ are on average different from the fluctuations of the other order ${\mathrm{∆}}_y$ implying a broken $Z_2$ symmetry and a preserved O(3) symmetry. When the temperature is below $T_N$, the magnetic ground state is a stripe SDW state, i.e., $⟨{\mathrm{∆}}_x⟩=0$ or $⟨{\mathrm{∆}}_y⟩=0$.

In this chapter, we will exploit a two-orbital model to study the interplay between SC and nematicity in a two-dimensional lattice. The two-orbital model has been successfully used in many studies such as quasiparticle excitation, the density of states near an impurity [35, 36] and the magnetic structure of a vortex core [37].

2. Model

Superconductivity in the iron-pnictide superconductors originates from the FeAs layer. The Fe atoms form a square lattice, and the As atoms are alternatively above and below the Fe-Fe plane. This leads to two sublattices of irons denoted by sublattices A and B. Many tight-binding Hamiltonians have been proposed to study the electronic band structure that includes five Fe 3d orbitals [38], three Fe orbitals [39, 40], and simply two Fe bands [41, 42, 43]. Each of these models has its own advantages and range of convenience for calculations. For example, the five-orbital tight-binding model can capture all details of the DFT bands across the Fermi energy in the first Brillouin zone. However, in practice, it becomes a formidable task to solve the Hamiltonian with a large size of lattice in real space even in the mean-field level. Several studies used five-orbital models in momentum space to investigate the single-impurity problem for different iron-based compounds such as ${\text{LaFeAsO}}_{1-\mathrm{x}}{\mathrm{F}}_{\mathrm{x}}$, LiFeAs, and ${\mathrm{K}}_{\mathrm{x}}\mathrm{F}{\mathrm{e}}_{2-\mathrm{y}}\mathrm{S}{\mathrm{e}}_2$. These studies confirmed that the detail of electronic bands strongly influences on the magnitude and location of the in-gap resonant states generated by the scattering of quasiparticles from single impurity [44, 45, 46].

On the other hand, the two-orbital models apparently have a numerical advantage dealing with a large size of lattice while retaining some of the orbital characters of the low-energy bands. Among the two-orbital ($d_{\mathit{xz}}$ and $d_{\mathit{yz}}$) models, Zhang [47] proposed a phenomenological approach to take into account the two Fe atoms per unit cell and the asymmetry of the As atoms below and above of the Fe plane. Later on, Tai and co-workers [48] improved Zhang’s model by a phenomenological two-by-two-orbital model (two Fe sites with two orbitals each). The obtained low-energy electronic dispersion agrees qualitatively well with DFT in LDA calculations of the entire Brillouin zone of the 122 compounds.

The multi-orbital Hamiltonian of the iron-pnictide superconductors in a two-dimensional square lattice is described as

$$\begin{array}{c}H={\displaystyle \sum }_{ijuv\alpha }{t}_{ijuv}{c}_{iu\alpha }^{†}{c}_{jv\alpha }-\mu {\displaystyle \sum }_{iu\alpha }{n}_{iu\alpha }+U{\displaystyle \sum }_{iu}{n}_{iu\uparrow }{n}_{iu\downarrow }+\left(U^{\prime }-\frac{J_H}{2}\right){\displaystyle \sum }_{i,u<v,\alpha \beta }{n}_{iu\alpha }{n}_{iv\beta }\\ -2J_H{\displaystyle \sum }_{i,u<v}{S}_{iu}·S_{iv}+J^{\prime }{\displaystyle \sum }_{i,u\ne v}{c}_{iu\uparrow }^{†}{c}_{\text{i}u\downarrow }^{†}{c}_{iv\downarrow }{c}_{iv\uparrow },\end{array}$$E2

where

$$n_{\mathit{iu\alpha }}=c_{\mathit{iu\alpha }}^{†}{c}_{\mathit{iu\alpha }},$$E3

$$S_{\mathit{iu}}=\frac{1}{2}\sum _{\mathit{\alpha \beta }}{c}_{\mathit{iu\alpha }}^{†}{\sigma }_{\mathit{\alpha \beta }}{c}_{\mathit{iu\beta }},$$E4

with ${\sigma }_{\mathit{\alpha \beta }}$ the Pauli matrices. The operators $c_{\mathit{iu\alpha }}^{†}$ ($c_{\mathit{iu\alpha }}$) create (annihilate) an ectron with spin $\alpha ,\beta =\uparrow ,\downarrow$ in the orbital $u,v=1,2$ on the lattice site $i$; $t_{\mathit{\text{ijuv}}}$ is the hopping matrix element between the neighbor sites, and $\mu$ is the chemical potential. $U$ ($U^{\prime }$) is the intraorbital (interorbital) on-site interaction. Hund’s rule coupling is $J_H$ and the pair hopping energy is $J^{\prime }$. The spin-rotation invariance gives $U^{\prime }=U-2J_H$ and $J^{\prime }=J_H$ [49]. Repulsion between electrons requires $J_H<U∕3$.

Here, we adopt Tai’s phenomenological two-by-two-orbital model because it is able to deal with a large size of lattice in many aspects and the details of low-energy bands are similar to the results from DFT + LDA. In a two-orbital model, the hopping amplitudes are chosen as shown in Figure 1 [48] to fit the band structure from the first-principle calculations:

$$t_1=t_{i,i\pm x\left(y\right),u,u}=-1,$$E5

$$\left\{\begin{array}{c}\frac{1+{\left(-1\right)}^{x+y+u}}{2}{t}_2+\frac{1-{\left(-1\right)}^{x+y+u}}{2}{t}_3=t_{i,i\pm \left(x+y\right),u,u}=0.08\\ \frac{1+{\left(-1\right)}^{x-y+u}}{2}{t}_3+\frac{1-{\left(-1\right)}^{x-y+u}}{2}{t}_2=t_{i,i\pm \left(x-y\right),u,u}=1.35\end{array},\right.$$E6

$$t_4=t_{i,i\pm \left(x\pm y\right),u,v\ne u}=-0.12,$$E7

$$t_5=t_{i,i\pm x\left(y\right),u,v\ne u}=0.09,$$E8

$$t_6=t_{i,i\pm 2x\left(2y\right),u,u}=0.25,$$E9

where $u\ne v$ indicates two different orbitals.

Figure 1.

(color online) two-dimensional square lattice of the iron-based superconductors. There are two Fe atoms (green and gray color) in a unit cell, and each atom has two orbitals. The bright color circle represents the first orbital, and the faded color circle represents the second orbital. Solid lines indicate the hopping between the atoms in the same orbital and dashed lines indicate the hopping between the atoms in the different orbitals.

Figure 1 shows the hopping parameters between unit cells and orbitals. For the same orbital, the hopping parameters $t_2$ and $t_3$ are chosen differently along the mutually perpendicular directions. The C₄ symmetry on the same orbital between different sublattices is broken. However, $t_2$ and $t_3$ are twisted for the different Fe atoms on the same sublattice which restore the C₄ symmetry of the lattice structure.

In the mean-field level

$$\mathrm{H}={\mathrm{H}}_0+{\mathrm{H}}_{\mathrm{∆}}+{\mathrm{H}}_{\mathrm{int}},$$E10

the Hamiltonian is self-consistently solved accompanied with $s^{+-}$-wave superconducting order. The mean-field scheme is the same as Ref. [48]:

$${\mathrm{H}}_0=\sum _{\mathit{\text{ijuv}}}{t}_{\mathit{\text{ijuv}}}{c}_{\mathit{iu\alpha }}^{†}{c}_{\mathit{jv\alpha }}-\mu \sum _{\mathit{iu\alpha }}{n}_{\mathit{iu\alpha }},$$E11

$${\mathrm{H}}_{\mathrm{∆}}=\sum _{\mathit{\text{iju}}}\left({\mathrm{∆}}_{\mathit{iju}}{c}_{\mathit{iu\alpha }}^{†}{c}_{\mathit{ju\beta }}^{†}+\mathrm{h}.\mathrm{c}.\right),$$E12

$${\mathrm{H}}_{\mathrm{int}}=U\sum _{\mathit{iu},\alpha \ne \beta }⟨n_{\mathit{iu\beta }}⟩n_{\mathit{iu\alpha }}+U^{\prime }\sum _{i,u<v,\alpha \ne \beta }⟨n_{\mathit{iu\beta }}⟩n_{\mathit{iv\alpha }}+\left(U^{\prime }-J_H\right)\sum _{i,u<v,\alpha }⟨n_{\mathit{iu\alpha }}⟩n_{\mathit{iv\alpha }}.$$E13

The next nearest-neighbor intraorbital attractive interaction $V$ is responsible for the superconducting order parameter ${\mathrm{\Delta }}_{\mathit{\text{ijuu}}}=V⟨c_{i\uparrow }{c}_{j\downarrow }⟩$ [50, 51, 52]. According to the literatures where $U$ is chosen to be 3.2 or less [48, 52], the magnetic configuration is a uniform SDW order. In order to make the stripe SDW order and the nematic order by changing electron doping, i.e., the chemical potential $\mu$, we choose the parameters of interactions $U=3.5$, $J_H=0.4$, and $V=1.3$ to induce a nematic order within a small doping range.

In momentum space, the spin configuration is determined by the order parameters ${\mathrm{∆}}_x$ and ${\mathrm{∆}}_y$. The combination of these order parameters lacks the visualization of the magnetic structure in real space. Therefore, we choose the staggered magnetization $M_i$ in a lattice to study the states driven by the magnetic mechanism. The magnetic configuration is described as

$$M_i=M_{1}cos\left(q_y·r_i\right)e^{i{Q}_x·r_i}+M_{2}sin\left(q_x·r_i\right)e^{{\mathit{iQ}}_y·r_i},$$E14

where the wave vectors $q_x=\left(2\pi /\lambda ,0\right)$ and $q_y=\left(0,2\pi /\lambda \right)$ correspond to a modulation along the x-axis and the y-axis with wavelength $\lambda$. $M_1$ and $M_2$ are the amplitude of the modulation.

In the case of the absence of both $q_x$ and $q_y$, $M_i$ becomes

$$M_i=M_1{e}^{i{Q}_x·r_i}+M_2{e}^{{\mathit{iQ}}_y·r_i}.$$E15

As $M_1=0$ or $M_2=0$ is chosen, i.e., choosing the ordering vector either $Q_x$ or $Q_y$, the state is a stripe SDW state. The existence of $q_x$ and $q_y$ does not affect the stripe SDW state. As two values of $M_1$ and $M_2$ are arbitrarily chosen, the spin configuration forms a stripe SDW along the $\left(1,1\right)$ direction. The existence of $q_x$ and $q_y$ has a lot of influences on the nematic state.

In the nematic state, the presence of both $Q$ and $q$ ordering vectors is necessary. Unlike the double-Q model, with the choice $M_1\ne M_2$, the magnitudes of the modulated antiparallel spins along the x-axis and the y-axis are different due to the existence of $q$ vectors. The magnetic configuration is attributed to two inequivalent stripes interpenetrating each other and formed a checked pattern. The checked pattern has the same period along the x- and y-directions. The period is determined by the value of $q$ in the periodic boundary conditions. The value of $q$, therefore, cannot be arbitrary and must commensurate the lattice to stabilize the modulation and lower the energy of the system. As two modulated stripes have no phase difference, the checked patter shows a $s$-wave-like symmetry. In addition, as the modulations have a phase shift of $\pi /2$, the checked pattern shows a d-wave-like symmetry.

Figure 2 displays the Fermi surface and the band structure in the absence of SDW at the normal state, i.e., the superconductivity is set to zero. In the absence of SDW $\left(U=0\right)$, the Hamiltonian in momentum space is

$${\mathrm{\Psi }}^{†}=\left(c_{A1\mathit{k\sigma }}^{†}{c}_{A2\mathit{k\sigma }}^{†}{c}_{B1\mathit{k\sigma }}^{†}{c}_{B2\mathit{k\sigma }}^{†}\right),$$E16

$${\mathrm{H}}_0=\sum _{\mathit{k\sigma }}{\mathrm{\Psi }}^{†}\left(\begin{array}{cccc}{\epsilon }_{A1}& {\epsilon }_{12}& {\epsilon }_{\mathit{AB}}& {\epsilon }_c\\ {\epsilon }_{12}& {\epsilon }_{A2}& {\epsilon }_c& {\epsilon }_{\mathit{AB}}\\ {\epsilon }_{\mathit{AB}}& {\epsilon }_c& {\epsilon }_{B1}& {\epsilon }_{12}\\ {\epsilon }_c& {\epsilon }_{\mathit{AB}}& {\epsilon }_{12}& {\epsilon }_{B2}\end{array}\right)\mathrm{\Psi },$$E17

where

$${\epsilon }_{A1}=-2t_{3}cos{k}_x-2t_{2}cos{k}_y-4t_{6}cos{k}_{x}cos{k}_y,$$E18

$${\epsilon }_{A2}=-2t_{2}cos{k}_x-2t_{3}cos{k}_y-4t_{6}cos{k}_{x}cos{k}_y,$$E19

$${\epsilon }_{B1}=-2t_{2}cos{k}_x-2t_{3}cos{k}_y-4t_{6}cos{k}_{x}cos{k}_y,$$E20

$${\epsilon }_{B2}=-2t_{3}cos{k}_x-2t_{2}cos{k}_y-4t_{6}cos{k}_{x}cos{k}_y,$$E21

$${\epsilon }_{12}=t-2_{4}cos{k}_x-2t_{4}cos{k}_y,$$E22

$${\epsilon }_{\mathit{AB}}=-4t_{1}cos\frac{k_x}{2}cos\frac{k_y}{2},$$E23

$${\epsilon }_c=-4t_{5}cos\frac{k_x}{2}cos\frac{k_y}{2}.$$E24

Figure 2.

(color online) (a) and (b) are, respectively, the band structure and the Fermi surface without SDW. The Fermi energy (red dashed line) corresponds to the electron filling $n=\mathit{2.1}$.

The eigenvalues are

$$E_{1,k}={\overline{\epsilon }}_{+}-\sqrt{{\overline{\epsilon }}_{-}^2+{\left({\epsilon }_{12}-{\epsilon }_{\mathit{AB}}\right)}^2}-{\epsilon }_c-\mu ,$$E25

$$E_{2,k}={\overline{\epsilon }}_{+}+\sqrt{{\overline{\epsilon }}_{-}^2+{\left({\epsilon }_{12}-{\epsilon }_{\mathit{AB}}\right)}^2}-{\epsilon }_c-\mu ,$$E26

$$E_{3,k}={\overline{\epsilon }}_{+}-\sqrt{{\overline{\epsilon }}_{-}^2+{\left({\epsilon }_{12}+{\epsilon }_{\mathit{AB}}\right)}^2}+{\epsilon }_c-\mu ,$$E27

$$E_{4,k}={\overline{\epsilon }}_{+}+\sqrt{{\overline{\epsilon }}_{-}^2+{\left({\epsilon }_{12}+{\epsilon }_{\mathit{AB}}\right)}^2}+{\epsilon }_c-\mu ,$$E28

where

$${\overline{\epsilon }}_{\pm }=\frac{{\epsilon }_{A1}\pm {\epsilon }_{B1}}{2}.$$E29

Figure 2(a) shows that two hole bands are around the $\mathrm{\Gamma }$ point and two electron bands are around the $\mathrm{M}$ point. There is no gap in the band structure where the superconductivity is able to open a gap. The nature of Fermi surface is revealed by the line at the Fermi energy crossing the band dispersion through $\mathrm{\Gamma }-\mathrm{X}-\mathrm{M}-\mathrm{\Gamma }$ points. Figure 2(b) displays that the Fermi surface contains two hole pockets that are centered at $\mathrm{\Gamma }$ point and two electron pockets that are centered at $\mathrm{M}$ points. In addition, the Fermi surface also exhibits the C₄ symmetry of the lattice structure.

In the stripe SDW state, the spin configuration is shown as Figure 3. The stripe SDW order enlarges the two-Fe unit cell to four-Fe unit cell as denoted by the blue dashed square in Figure 3. The antiferromagnetic order is along the ${\mathrm{X}}^{\prime }$-axis and the ferromagnetic order is along the ${\mathrm{Y}}^{\prime }$-axis. The magnetic unit cell (four-Fe unit cell) with lattice spacing $\sqrt{2}a$ is oriented at a 45° angle with respect to the nonmagnetic unit cell (two-Fe unit cell). The magnetic Brillouin zone is a square oriented at a 45° angle with respect to the crystal Brillouin zone. The size of the magnetic unit cell is twice of the nonmagnetic unit cell, and the size of the magnetic Brillouin zone is half of the crystal Brillouin zone. The Hamiltonian in momentum space is

$${\stackrel{\sim }{\mathrm{\Psi }}}_k^{†}=\left(\begin{array}{cccccccc}{c}_{A_1^{\left(1\right)}k}^{†}& c_{A_1^{\left(2\right)}k}^{†}& c_{B_1^{\left(1\right)}k}^{†}& c_{B_1^{\left(2\right)}k}^{†}& c_{B_2^{\left(1\right)}k}^{†}& c_{B_2^{\left(2\right)}k}^{†}& c_{A_2^{\left(1\right)}k}^{†}& c_{A_2^{\left(2\right)}k}^{†}\end{array}\right),$$E30

$$\mathrm{H}=\sum _k{\stackrel{\sim }{\mathrm{\Psi }}}_k^{†}·\mathbf{h}·{\stackrel{\sim }{\mathrm{\Psi }}}_k,$$E31

$$\mathbf{h}=\left(\begin{array}{cccccccc}{\epsilon }_{A_1^1}& 0& {\epsilon }_{t_1^x}& {\epsilon }_{t_5^x}& {\epsilon }_{t_1^y}& {\epsilon }_{t_5^y}& {\epsilon }_{t_{23}}& {\epsilon }_{t_4}\\ 0& {\epsilon }_{A_1^2}& {\epsilon }_{t_5^x}& {\epsilon }_{t_1^x}& {\epsilon }_{t_5^y}& {\epsilon }_{t_1^y}& {\epsilon }_{t_4}& {\epsilon }_{t_{32}}\\ {\epsilon }_{t_1^x}& {\epsilon }_{t_5^x}& {\epsilon }_{B_1^1}& 0& {\epsilon }_{t_{32}}& {\epsilon }_{t_4}& {\epsilon }_{t_1^y}& {\epsilon }_{t_5^y}\\ {\epsilon }_{t_5^x}& {\epsilon }_{t_1^x}& 0& {\epsilon }_{B_1^2}& {\epsilon }_{t_4}& {\epsilon }_{t_{23}}& {\epsilon }_{t_5^y}& {\epsilon }_{t_1^y}\\ {\epsilon }_{t_1^y}& {\epsilon }_{t_5^y}& {\epsilon }_{t_{32}}& {\epsilon }_{t_4}& {\epsilon }_{B_2^1}& 0& {\epsilon }_{t_1^x}& {\epsilon }_{t_5^x}\\ {\epsilon }_{t_5^y}& {\epsilon }_{t_1^y}& {\epsilon }_{t_4}& {\epsilon }_{t_{23}}& 0& {\epsilon }_{B_2^2}& {\epsilon }_{t_5^x}& {\epsilon }_{t_1^x}\\ {\epsilon }_{t_{23}}& {\epsilon }_{t_4}& {\epsilon }_{t_1^y}& {\epsilon }_{t_5^y}& {\epsilon }_{t_1^x}& {\epsilon }_{t_5^x}& {\epsilon }_{A_2^1}& 0\\ {\epsilon }_{t_4}& {\epsilon }_{t_{32}}& {\epsilon }_{t_5^y}& {\epsilon }_{t_1^y}& {\epsilon }_{t_5^x}& {\epsilon }_{t_1^x}& 0& {\epsilon }_{A_2^2}\end{array}\right),$$E32

where

$${\epsilon }_{t_1^x}=-2t_{1}cos{k}_x=-2t_{1}cos\left(\frac{k_{x^{\prime }}}{\sqrt{2}}\right),$$E33

$${\epsilon }_{t_1^y}=-2t_{1}cos{k}_y=-2t_{1}cos\left(\frac{k_{y^{\prime }}}{\sqrt{2}}\right),$$E34

$${\epsilon }_{t_5^x}=-2t_{5}cos{k}_x=-2t_{5}cos\left(\frac{k_{x^{\prime }}}{\sqrt{2}}\right),$$E35

$${\epsilon }_{t_5^y}=-2t_{5}cos{k}_y=-2t_{5}cos\left(\frac{k_{y^{\prime }}}{\sqrt{2}}\right),$$E36

$${\epsilon }_{t_{23}}=-2t_{2}cos\left(k_x+k_y\right)-2t_{3}cos\left(k_x-k_y\right)=-2t_{2}cos\left(\frac{k_{x^{\prime }}}{\sqrt{2}}+\frac{k_{y^{\prime }}}{\sqrt{2}}\right)-2t_{3}cos\left(\frac{k_{x^{\prime }}}{\sqrt{2}}-\frac{k_{y^{\prime }}}{\sqrt{2}}\right),$$E37

$${\epsilon }_{t_{32}}=-2t_{3}cos\left(k_x+k_y\right)-2t_{2}cos\left(k_x-k_y\right)=-2t_{3}cos\left(\frac{k_{x^{\prime }}+k_{y^{\prime }}}{\sqrt{2}}\right)-2t_{2}cos\left(\frac{k_{x^{\prime }}-k_{y^{\prime }}}{\sqrt{2}}\right),$$E38

$${\epsilon }_{t_4}=-2t_{4}cos\left(k_x+k_y\right)-2t_{4}cos\left(k_x-k_y\right)=-4t_{4}cos\frac{k_{x^{\prime }}}{\sqrt{2}}cos\frac{k_{y^{\prime }}}{\sqrt{2}},$$E39

$${\epsilon }_{t_6}=-2t_{6}cos2k_x-2t_{6}cos2k_y=-2t_{6}cos\sqrt{2}{k}_{x^{\prime }}-2t_{6}cos\sqrt{2}{k}_{y^{\prime }}$$E40

$${\epsilon }_{A_1^1}={\epsilon }_{t_6}+U⟨n_{A_1^{1}k\downarrow }⟩+U^{\prime }⟨n_{A_1^{2}k\downarrow }⟩+\left(U^{\prime }-J_H\right)⟨n_{A_1^{2}k\uparrow }⟩-\mu ,$$E41

$${\epsilon }_{A_1^2}={\epsilon }_{t_6}+U⟨n_{A_1^{2}k\downarrow }⟩+U^{\prime }⟨n_{A_1^{1}k\downarrow }⟩+\left(U^{\prime }-J_H\right)⟨n_{A_1^{1}k\uparrow }⟩-\mu ,$$E42

$${\epsilon }_{B_1^1}={\epsilon }_{t_6}+U⟨n_{B_1^{1}k\downarrow }⟩+U^{\prime }⟨n_{B_1^{2}k\downarrow }⟩+\left(U^{\prime }-J_H\right)⟨n_{B_1^{2}k\uparrow }⟩-\mu ,$$E43

$${\epsilon }_{B_1^2}={\epsilon }_{t_6}+U⟨n_{B_1^{2}k\downarrow }⟩+U^{\prime }⟨n_{B_1^{1}k\downarrow }⟩+\left(U^{\prime }-J_H\right)⟨n_{B_1^{1}k\uparrow }⟩-\mu .$$E44

Figure 3.

(color online) the schematic lattice structure of the Fe layer in the stripe SDW state. Blue dashed square denotes the four-Fe unit cell in the stripe SDW state.

According to the itinerant picture, the interactions between two sets of pockets give rise to a SDW order at the wave vector connecting them with $Q_x=\left(\pi ,0\right)$ or $Q_y=\left(0,\pi \right)$. For example, as choosing the ordering vector $Q_x$, the spin configuration has antiparallel spins along the ${\mathrm{X}}^{\prime }$-direction and parallel spins along the ${\mathrm{Y}}^{\prime }$-direction [35]. The antiferromagnetism causes the gapless structure along the ${\mathrm{X}}^{\prime }$-direction, and the ferromagnetism opens a gap along the ${\mathrm{Y}}^{\prime }$-direction, as shown in Figure 4(a). There are four pockets centered at the $\mathrm{\Gamma }$ point, and two pockets along the $\mathrm{\Gamma }-{\mathrm{Y}}^{\prime }$ direction are inequivalent to other two pockets along the $\mathrm{\Gamma }-{\mathrm{X}}^{\prime }$ direction. The C₂ symmetry of the Fermi surface resulting from the SDW gap is shown in Figure 4(b).

Figure 4.

(color online) (a) and (b) are, respectively, the band structure and the Fermi surface in the stripe SDW state. The Fermi energy (red dashed line) corresponds to the electron filling $n=\mathit{2.04}$. ${\mathit{X}}^{\prime }$ and ${\mathit{Y}}^{\prime }$ indicate the magnetic Brillouin zone of the stripe SDW state.

In the nematic state, the antiparallel spins are along both the ${\mathrm{X}}^{\prime \prime }$ and ${\mathrm{Y}}^{\prime \prime }$ directions. The nematic unit cell is oriented at a 45° with respect to the nonmagnetic unit cell. The nematic Brillouin zone is also a square oriented at a 45° angle with respect to the crystal Brillouin zone. Since the antiferromagnetism is along both the ${\mathrm{X}}^{\prime \prime }$ and ${\mathrm{Y}}^{\prime \prime }$ directions, there is no gap in the band structure. In addition, as $M_1<M_2$, the bands along the ${\mathrm{Y}}^{\prime \prime }$ direction are lifted higher and cause the bands to be asymmetric with respect to the $\mathrm{\Gamma }$ point. The asymmetric bands result in deformed Fermi surfaces near the $\mathrm{\Gamma }$ point. As shown in Figure 5(a), the blue curve in the band structure forms an elliptic hole-pocket Fermi surface. Furthermore, the elliptic Fermi surface results from the unequal contribution of two orbitals $d_{\mathit{xz}}$ and $d_{\mathit{yz}}$. The mechanism behind the fluctuations of $d_{\mathit{xz}}$ and $d_{\mathit{yz}}$ orbitals can be understood from an extended RPA approach where $d_{\mathit{xz}}$, $d_{\mathit{yz}}$, and $d_{\mathit{xy}}$ orbitals equally contribute to the SDW instability, and in particular the $d_{\mathit{xy}}$ orbital plays a strong role in the nematic instability [30]. The nematic instability breaks the degeneracy of two orbitals $d_{\mathit{xz}}$ and $d_{\mathit{yz}}$ and causes the unequal charge distributions $n_{\mathit{xz}}\left(k\right)$ and $n_{\mathit{yz}}\left(k\right)$. The C₂-symmetry of the Fermi surface results from the broken degeneracy of two orbitals $d_{\mathit{xz}}$ and $d_{\mathit{yz}}$ (blue and red curves in Figure 5(b)).

Figure 5.

(color online) (a) and (b) are, respectively, the band structure and the Fermi surface in the nematic state. The asymmetric band (blue color) is responsible for the elliptic Fermi surface around the $\mathit{\Gamma }$ point. The blue and red solid curves represent the major contributions from the orbitals $d_{\mathit{xz}}$ and $d_{\mathit{yz}}$, respectively. The chemical potential is chosen as the electron filling $n=\mathit{2.1}$. ${\mathit{X}}^{\prime \prime }$, and ${\mathit{Y}}^{\prime \prime }$ indicate the magnetic Brillouin zone of the nematic state.

Recently, Qureshi et al. [53], Wang et al. [54], Steffens et al. [55], and Luo et al. [56] pointed out that in-plane spin excitations exhibit a large gap and indicating that the spin anisotropy is caused by the contribution of itinerant electrons and the topology of Fermi surface. These experiments indicate that the elliptic spin fluctuations at low energy in iron pnictides are mostly caused by the anisotropic damping of spin waves within FeAs plane and the topology of Fermi surface. The degeneracy of orbitals will introduce the single-ion anisotropy in spin fluctuations.

3. Visualize nematicity in a lattice

To visualize the nematicity in a lattice, we self-consistently solve the Bogoliubov-de Gennes (BdG) equations for the nematic state in a two-dimensional square lattice:

$$\sum _{\mathit{jv}}\left(\begin{array}{cc}{h}_{\mathit{\text{ijuv}}\uparrow }& {\mathrm{∆}}_{\mathit{\text{ijuv}}}\\ {\mathrm{∆}}_{\mathit{\text{ijuv}}}^{\ast }& h_{\mathit{\text{ijuv}}\downarrow }^{\ast }\end{array}\right)\left(\begin{array}{c}{u}_{\mathit{jv}\uparrow }^n\\ v_{\mathit{jv}\downarrow }^n\end{array}\right)=E_{n\uparrow }\left(\begin{array}{c}{u}_{\mathit{iu}\uparrow }^n\\ v_{\mathit{iu}\downarrow }^n\end{array}\right),$$E45

where

$${\mathrm{h}}_{\mathrm{iju\alpha }}={\mathrm{t}}_{\mathrm{iju\alpha }}+\left\{-\mathrm{\mu }+\mathrm{U}⟨{\mathrm{n}}_{\mathrm{iu\beta }}⟩+{\mathrm{U}}^{\prime }⟨{\mathrm{n}}_{\mathrm{iv\alpha }}⟩+\left({\mathrm{U}}^{\prime }-{\mathrm{J}}_{\mathrm{H}}\right)⟨{\mathrm{n}}_{\mathrm{iv\beta }}⟩\right\}{\mathrm{\delta }}_{\mathrm{ij}},$$E46

and $U^{\prime }=U-2J_H$. The self-consistency conditions are

$$⟨n_{\mathit{iu}\uparrow }⟩=\sum _n{\left|u_{\mathit{iu}\uparrow }^n\right|}^{2}f\left(E_n\right),$$E47

$$⟨n_{\mathit{iu}\downarrow }⟩=\sum _n{\left|v_{\mathit{iu}\downarrow }^n\right|}^{2}f\left(1-E_n\right),$$E48

$${\mathrm{\Delta }}_{\mathit{\text{ijuu}}}=\frac{V}{2}\sum _n{u}_{\mathit{iu}\uparrow }^n{v}_{\mathit{iu}\downarrow }^{n\ast }tanh\left(\frac{\beta E_n}{2}\right).$$E49

Here, $f\left(E_n\right)$ is the Fermi distribution function.

In Figure 6, we show the magnetic configuration in the coexisting state of the nematic order and SC. To view the detail of the structure, the slided profile along the peaks along the x- or y-direction is made (as shown on the sides of $M_i$ in Figure 6). There are two sinusoidally modulated magnetizations on each panel. The warm color and cold color modulations represent the spin-up modulation and the spin-down modulation, respectively. The amplitude of each modulation on the left and right panels corresponds to the value of $M_1$ and $M_2$. On the left panel, we have $M_{1}cos\left(q_y·r_i\right)=0.04cos\left(\frac{2\pi r_i}{28a}\right)$, and on the right panel, we have $M_{2}cos\left(q_x·r_i\right)=0.06sin\left(\frac{2\pi r_i}{28a}\right)$. Both modulations have the same period 28a. Since $M_1<M_2$, the configuration is orthorhombic and breaks the ${90}^{°}$ rotational symmetry.

Figure 6.

(color online) the real space configurations of the magnetization $M_i$ are plotted on a $56\times 56$ square lattice. The left and the right panels are the sliced profile along the peaks along the y- and x-directions, respectively. Two curves are shown in both panels. The upper and lower curves represent the spin-up and spin-down configurations, respectively.

Figure 7 shows the Fourier transformation of the spatial configuration of the nematic fluctuations. Two peaks appearing at $\left(\pm \pi ,0\right)$ correspond to the ordering vector $Q_x$ and two peaks exhibiting at $\left(0,\pm \pi \right)$ correspond to the ordering vector $Q_y$. We found that the intensities of peaks associated with the ordering vector $Q_x$ are greater than peaks associated with the ordering vector $Q_y$, which is due to the unequal value of $M_1$ and $M_2$. The intensities of two peaks along the $k_x\left(k_y\right)$-direction have the same magnitude indicating $⟨{\mathrm{∆}}_x⟩=⟨{\mathrm{∆}}_y⟩=0$. Moreover, the nonequivalence of the intensities between the $k_x$- and $k_y$-directions indicates $⟨{\mathrm{∆}}_y^2⟩>⟨{\mathrm{∆}}_x^2⟩$. Therefore, the modulated antiparallel spin configuration is the nematic state. These features are preserved even as the SC order is equal to zero. This result is in agreement with the neutron scattering experiments [3, 10].

Figure 7.

(color online) the Fourier transformation of the $56\times 56$ spatial magnetic configuration.

We further illustrate the electronic charge density $n_i=\left(n_{i\uparrow }+n_{i\downarrow }\right)$ and the $s^{+-}$-wave SC order parameter ${\mathrm{\Delta }}_i$ as shown in Figure 8(a), (b). Particularly, the nematicity of the spin order induces a modulated charge density wave (CDW) which does not occur in the stripe SDW state. The CDW consists of crisscrossed horizontal and vertical stripes. The amplitudes of the vertical stripes are larger than the horizontal stripes. Therefore, the CDW forms a checked pattern, instead of a checkerboard pattern. The stripes on both x- and y- directions have the same period 14a which is the half period of the magnetization.

Figure 8.

(color online) (a) the spatial configuration of the electronic charge density n_i. (b) the spatial configuration of the $s^{+-}$-wave superconducting order parameter ${\mathit{\Delta }}_i$.

Moreover, although the checked pattern of the CDW is twofold symmetry, the CDW exhibits a $d_{x^2-y^2}$-symmetry, instead of a $d_{\mathit{xy}}$-symmetry (diagonal stripes crisscrossed pattern), form factor density wave. The space configuration of the SC order parameter ${\mathrm{\Delta }}_i$ shows the same features as the CDW order, as shown in Figure 8(b).

4. The local density of states

The local density of states (LDOS) proportional to the differential tunneling conductance as measured by STM is expressed as

$${\rho }_i\left(E\right)=-\frac{1}{N_x{N}_y}\sum _{\mathit{nu}}\left[{\left|u_{\mathit{iu}\uparrow }^n\right|}^2{f}^{\prime }\left(E_n-E\right)+{\left|v_{\mathit{iu}\downarrow }^n\right|}^2{f}^{\prime }\left(E_n+E\right)\right],$$E50

where $N_x\times N_y=24\times 24$ is the size of supercells.

In the striped SDW state, spins are parallel in the y-direction and antiparallel in the x-direction and cause the gap and gapless features in the band structure, respectively. The SDW gap shifts toward negative energy, and the coherence peak at the negative energy is pushed outside the SDW gap and enhanced. The coherence peak at the positive energy is moved inside the SDW gap and suppressed. This is a prominent feature caused by the magnetic SDW order that the intensities of superconducting coherence peaks are obvious asymmetry (as shown in Figure 9(a)) [57].

Figure 9.

(color online) the LDOS in the (a) stripe SDW state and (b) nematic state. The dashed (blue) line represents the LDOS without magnetization ($M_i=0$).

In the nematic state, spins are antiparallel in the x- and y-directions leading to a gapless feature in the band structure. The superconducting gap is the only gap that appears in the LDOS. Moreover, comparing to the state without SDW, the competition between the nematic order and the superconducting order causes the slightly suppression of the coherence peaks. The feature of the suppression results in a dip at the negative energy outside the coherence peaks (as shown in Figure 9(b)).

Furthermore, Figure 10 displays a spatial distribution of LDOS, also known as LDOS map, at $E=0.14$. The LDOS map shows the same features as the charge density distribution at the energy within the coherence peaks. The LDOS map exhibits a checked pattern, twofold symmetric configuration, and $d_{x^2-y^2}$-symmetry form factor density wave. These features have not yet been reported by STM experiments.

Figure 10.

(Color online) The LDOS map at $E=0.14$ with $M_i\ne 0$.

It is worth to note that STM measurements by Chuang et al. [5] and Allan [58] reported that the dimension of the electronic nanostructure is around $8a$ and the nanostructure aligns in a unidirectional fashion. The highly twofold symmetric structure of the QPI patterns is represented by using the Fourier transformation of the STS imaging. Moreover, in cuprate, the more advanced measurement of the atomic-scale electronic structure has shown a d-wavelike symmetry form factor density wave [59, 60]. There are four peaks that appear around the center of the momentum space in the QPI patterns. Such an atomic-scale feature in cuprates has not yet been reported in iron pnictides and in the nematic state.

5. Phase diagram

To further verify the spin configuration of the nematic order, a phase diagram is presented in Figure 11. In the phase diagram, the stripe SDW order, nematic order, and $s^{+-}$-wave superconducting order as a function of doping are obtained from the self-consistent calculation to solve the BdG equations.

Figure 11.

(color online) the phase diagram of the stripe SDW order (blue), nematic order (green), and superconducting order (red) as a function of doping.

In the hole-doped region, the magnetization exhibits the stripe SDW order and drops dramatically around $n=1.85$ and vanishes at $n=1.80$. In the meantime, the $s^{+-}$-wave superconducting order reaches its maximal value at $n=1.85$ and then gradually decreases.

In the electron-doped region, the stripe SDW order (green curve) has its maximal value at $n=2.00$, then rapidly diminishes in a small region of doping, and finally reaches zero at $n=2.09$. The superconductivity (blue curve) swiftly increases in a small region from $n=2.02$ to $n=2.04$, then reaches its maximal value at $n=2.10$, and finally gradually decreases to almost zero around $n=2.40$. The nematic state (red region) is in a small region next to the stripe SDW where the nematic transition line (red dashed curve) tracks closely the stripe SDW transition line. The nematic order is favored to appear in the electron-doped regime, but not the hole-doped regime.

There are two regions where the stripe SDW coexist with the SC and the nematic order coexists with the SC. In the region where the stripe SDW coexist with the SC, the magnetic structure is an orthorhombic uniaxial stripe state. The ordering vector is either $Q_x$ or $Q_y$ implying $⟨{\mathrm{∆}}_x⟩=0$ or $⟨{\mathrm{∆}}_y⟩=0$. In the region where the nematic order coexists with the SC, the magnetic structure is a crisscrossed stripe state with twofold symmetry. The ordering vectors are $Q_x$ and $Q_y$ associated with two modulating vectors $q_x$ and $q_y$ implying $⟨{\mathrm{∆}}_x⟩=⟨{\mathrm{∆}}_y⟩=0$ and $⟨{\mathrm{∆}}_x^2-{\mathrm{∆}}_y^2⟩\ne 0$ [61].

It is worth to note that the phase diagram of the electron-doped region is consistent with Figure 1.3 of Kuo’s thesis on $\mathrm{Ba}{\left(\mathrm{F}{\mathrm{e}}_{1-x}\mathrm{C}{\mathrm{o}}_x\mathrm{As}\right)}_2$ [62]. Both figures show the same behavior of the nematic phase. Near the optimally doped region under the superconducting dome, it is mentioned that the long-range nematic order coexists with the superconductivity. However, such results still have not been reported from experiments. In addition, the magnetoresistivity of ${\mathrm{Ba}}_{0.5}{\mathrm{K}}_{0.5}\mathrm{F}{\mathrm{e}}_2\mathrm{A}{\mathrm{s}}_2$ reported a nematic superconducting state recently and suggested that the hole-doped superconductor is the mixture of $s$-wave and $d$-wave superconducting orders [63]. These results provide a different path to further researches to understand the mechanism of the nematic state in the superconductivity.

6. Conclusions

The two-orbital Hamiltonian used in the iron-based superconductors has always been questioned for its validity. Many studies have approved that a lot of phenomena are attributed to $d_{\mathit{xz}}$ and $d_{\mathit{yz}}$ orbitals. In particular, $d_{\mathit{xz}}$ and $d_{\mathit{yz}}$ orbitals are responsible for the SDW instability.

The stripe SDW order opens a gap in the band structure and deforms the Fermi surface. However, the band structure of the nematic order is gapless, and the Fermi surface is deformed to an ellipse. The mechanism can be understood from the instability of SDW. The nematic order has visualized as a checked pattern formed by a crisscrossed modulated horizontal and vertical stripes. The inequivalent strengths of the horizontal and vertical stripes break the degeneracy of two orbitals d_xz and d_yz and cause an elliptic Fermi surface. The Fourier transformation of the orthorhombic structure of the magnetization shows two uneven pairs of peaks at (±π,0) and (0,±π). Moreover, the LDOS map shows a d_x2-y2-symmetry form factor density wave.

Finally, the nematic order is favored to exist in the electron-doped regime, but not the hole-doped regime.

Acknowledgments

HYC was supported by MOST of Taiwan under Grant MOST 107-2112-M-003-002 and National Center for Theoretical Science of Taiwan.