Orbital ordering and unfrustrated $\left(\pi ,0\right)$ magnetism from degenerate double exchange in the iron pnictides

The magnetic excitations of the iron pnictides are explained within a degenerate double-exchange model. The local-moment spins are coupled by superexchanges $J_1$ and $J_2$ between nearest and next-nearest neighbors, respectively, and interact with the itinerant electrons of the degenerate $d_{xz}$ and $d_{yz}$ orbitals via a ferromagnetic Hund exchange. The latter stabilizes $\left(\pi ,0\right)$ stripe antiferromagnetism due to emergent ferro-orbital order and the resulting kinetic-energy gain by hopping preferably along the ferromagnetic spin direction. Taking the quantum nature of the spins into account, we calculate the magnetic excitation spectra in the presence of both, superexchange and double exchange. A dramatic increase in the spin-wave energies at the competing Néel ordering wave vector is found, in agreement with recent neutron-scattering data. The spectra are fitted to a spin-only model with a strong spatial anisotropy and additional longer-ranged couplings along the ferromagnetic chains. Over a realistic parameter range, the effective couplings along the chains are negative corresponding to unfrustrated stripe antiferromagnetism.

Figure 1

(Color online) Illustration of the degenerate double-exchange model for the pnictides. (a) The local moments are coupled by nearest- and next-nearest-neighbor exchanges $J_1$ and $J_2$, respectively, and interact via a ferromagnetic Hund coupling $J_{\text{H}}$ with the itinerant electrons of the degenerate $d_{xz}$, $d_{yz}$ orbitals. (b) Illustration of the hopping parameters in a two-band model of these orbitals (shown as projections in the plane). (c) Resulting ferro-orbital order which stabilizes $\left(\pi ,0\right)$ antiferromagnetism by directing the kinetic energy of the electrons along the ferromagnetic spin direction.

Figure 2

(Color online) Classical phase diagrams of the degenerate double-exchange model as a function of $J_{\text{H}}/t_1$ and $J_2/J_1$ for different filling levels $n=0.05$, $n=0.10$, and $n=0.15$. The tight-binding hopping parameters are $t_2=-0.1t_1$, $t_3=0.2t_1$, and $t_4=0.05t_1$, the nearest-neighbor superexchange $J_1=0.04t_1$.

Figure 3

(Color online) (a) and (b) The dispersions of the itinerant-electron bands for $J_{\text{H}}=0.1t_1$ (a) and $J_{\text{H}}=1.0t_1$ (b), along the path $\left(0,0\right)\text{−}\left(\pi ,0\right)\text{−}\left(\pi ,\pi \right)\text{−}\left(0,0\right)$ in the Brillouin zone containing one Fe atom per unit cell. The tight-binding parameters are chosen to be $t_2=-0.1t_1$, $t_3=0.2t_1$, and $t_4=0.05t_1$ throughout the paper. The bands are colored according to the weight of the $d_{yz}$ orbital. (c) and (d) Total orbital polarization $n_o$ (c) and spin polarization $n_s$ (d) as a function of the filling $n$ for various Hund’s couplings $J_{\text{H}}$.

Figure 4

(Color online) Spin-wave dispersion $\omega \left(q\right)$ in the presence of both superexchange and double exchange along the path $\left(0,0\right)\text{−}\left(\pi ,0\right)\text{−}\left(\pi ,\pi \right)\text{−}\left(0,0\right)$ for (a) different filling levels $n$, $J_{\text{H}}=2.0t_1$, and (b) different Hund’s couplings $J_{\text{H}}$, $n=0.1$. In both cases, $J_1=0.04t_1$, $J_2=0.6J_1$, and $S=1/2$.

Figure 5

(Color online) (a) Best fits of the spin-wave spectrum of the DDEX for $n=0.1$ and $J_{\text{H}}=3.0t_1$ obtained from a spin-only model with effective couplings illustrated as inset. (b) and (c) Effective exchange constants as functions of Hund’s coupling $J_{\text{H}}$ for bare exchange constants $J_1=0.04t_1$ (b), $J_1=0.08t_1$ (c), and $J_2/J_1=0.6$.

Figure 6

(Color online) Spin-wave dispersion $\omega \left(q\right)$ in the presence of both, superexchange and double exchange along the path $\left(0,0\right)\text{−}\left(\pi ,0\right)\text{−}\left(\pi ,\pi \right)\text{−}\left(0,0\right)$ for different Hund’s couplings $J_{\text{H}}$. Parameters are the same as in Fig. 4b, except for $J_2/J_1=0.4$. In this regime, $\left(\pi ,0\right)$ order is classically stable for $J_{\text{H}}\gtrsim 1.6t_1$. (Inset: the fitted, effective exchange constants ${\tilde{J}}_{1x}$, ${\tilde{J}}_{1y}$, ${\tilde{J}}_2$, and ${\tilde{J}}_{3y}$ as functions of $J_{\text{H}}$.)