Spin-orbital frustrations and anomalous metallic state in iron-pnictide superconductors

We develop an understanding of the anomalous metal state of the parent compounds of recently discovered iron-based superconductors starting from a strong-coupling viewpoint, including orbital degrees of freedom. On the basis of an intermediate-spin $\left(S=1\right)$ state for the ${\text{Fe}}^{2+}$ ions, we derive a Kugel-Khomskii spin-orbital Hamiltonian for the active $t_{2g}$ orbitals. It turns out to be a highly complex model with frustrated spin and orbital interactions. We compute its classical phase diagrams and provide an understanding for the stability of the various phases by investigating its spin-only and orbital-only limits. The experimentally observed spin-stripe state is found to be stable over a wide regime of physical parameters and can be accompanied by three different types of orbital orders. Of these the orbital-ferro and orbital-stripe orders are particularly interesting since they break the in-plane lattice symmetry—a robust feature of the undoped compounds. We compute the magnetic excitation spectra for the effective spin Hamiltonian, observing a strong reduction in the ordered moment, and point out that the proposed orbital ordering pattern can be measured in resonant x-ray diffraction.

Figure 1

(Color online) (a) Fe square lattice (circles) and relative positions of the As ions. The latter are located in adjacent layers above (filled squares) and below (empty squares) the Fe plaquettes. (b) Schematic illustration of a ground-state $d^6$ configuration of the Fe ions corresponding to an intermediate $S=1$ spin state. (c) Multiplet structure of the $d_i^6{d}_j^6\rightleftharpoons d_i^7{d}_j^5$ charge excitations for localized $e_g$ electrons.

Figure 2

Illustration of the effective hopping parameters $t_{\alpha \beta }$, between (a) the $d_{xz}$ and $d_{yz}$ orbitals and (b) those involving the $d_{xy}$ orbitals. The projections of the $d_{xz}$ and $d_{yz}$ orbitals on the Fe plane are depicted in white and light gray, respectively, and the $d_{xy}$ orbitals are shown in dark gray.

Figure 3

(Color online) Various hopping parameters ${\alpha }_i≔t_i/t$ as illustrated in Fig. 2 as a function of the ratio $\gamma =\left(pd\pi \right)/\left(pd\sigma \right)$ for the lattice parameter $\lambda =0.7$.

Figure 4

(Color online) $\eta \text{−}\gamma$ phase diagram for $\lambda =0.7$ and $\delta =0.01$. $\eta =J_H/U$ and $\gamma =\left(pd\pi \right)/\left(pd\sigma \right)$. The phases are denoted by their ordering wave vectors in the spin and orbital variables. $Tz$ or $Tx$ refers to the component of the orbital pseudospin that is saturated in the ordered state.

Figure 5

(Color online) Schematic pictures of the three ground-state orbital ordering patterns that accompany the spin-stripe phase. (a) Orbital ferro, (b) orbital stripe, and (c) orbital antiferro.

Figure 6

(Color online) $\eta \text{−}\gamma$ phase diagram for $\lambda =0.8$ and $\delta =0.01$. Note that the orbital-ordered states that break the orthorhombic symmetry do not exist for this choice of $\lambda$.

Figure 7

(Color online) $\eta \text{−}\delta /\eta$ phase diagram for $\lambda =0.7$ and $\gamma =\left(pd\pi \right)/\left(pd\sigma \right)=-0.05$. This phase diagram illustrates the point that $\delta$ is not a crucial parameter in the Hamiltonian.

Figure 8

(Color online) Relevant structure factors as a function of temperature for different values of $\eta$. The lattice parameter and the relative strength of $\sigma$ and $\pi$ hopping are fixed as $\lambda =0.7$ and $\gamma =-0.05$, respectively.

Figure 9

(Color online) $T\text{−}\eta$ phase diagram for $\delta =0.01$, $\lambda =0.7$, and $\gamma =-0.05$.

Figure 10

(Color online) The coupling constants as a function of $\eta =J_H/U$ for the orbital-only model with frozen spin correlations for $\delta =0.01$, $\gamma =-0.05$, and $\lambda =0.7$. The couplings along $x$, $y$, and diagonal directions are plotted in panels (a), (b), and (c), respectively. The single site term is plotted in (b) to indicate that this term arises due to a ferromagnetic bond along $y$ direction. The solid lines correspond to the ground-state spin order and the dashed lines are for a paramagnetic spin state. The vertical dashed line indicates the location in $\eta$ of the phase transition from spin-stripe to spin-antiferro state as seen in Fig. 9.

Figure 11

(Color online) Effective exchange couplings $J_x$ and $J_y$ for nearest-neighbor and $J_d$ for next-nearest-neighbor spins as a function of $\eta =J_H/U$ for $\delta =0.01$, $\gamma =-0.05$, and $\lambda =0.7$. The solid lines correspond to the couplings resulting for the corresponding orbital ground states, whereas the dashed lines correspond to the orbitally disordered case. Note that for the orbitally disordered case $J_x=J_y$ for all values of $\eta$. The vertical dashed line indicates the location in $\eta$ of the various phase transition as seen in Fig. 9.

Figure 12

(Color online) Spin-wave excitation spectra for different values of $\eta$. Top: $\left(\pi ,0\right)$ magnet for disordered orbitals. The spectral weights are coded by line thickness and color; high intensity corresponds to red, low intensity to blue. Middle: $\left(\pi ,0\right)$ magnet for orbital-ferro $\left(\eta =0.03,0.07,0.11\right)$ and orbital-stripe $\left(\eta =0.12,0.13,0.14\right)$ orders. Bottom: $\left(\pi ,\pi \right)$ magnet with orbital-stripe order $\left(\eta =0.15,0.17,\dots \right)$.