Symmetry of spin excitation spectra in the tetragonal paramagnetic and superconducting phases of 122-ferropnictides

We study the symmetry of spin excitation spectra in 122-ferropnictide superconductors by comparing the results of first-principles calculations with inelastic neutron-scattering (INS) measurements on ${\text{BaFe}}_{1.85}{\text{Co}}_{0.15}{\text{As}}_2$ and ${\text{BaFe}}_{1.91}{\text{Ni}}_{0.09}{\text{As}}_2$ samples that exhibit neither static magnetic phases nor structural phase transitions. In both the normal and superconducting (SC) states, the spectrum lacks the three-dimensional $4_2/m$ screw symmetry around the $\left(\frac{1}{2}\frac{1}{2}L\right)$ axis that is implied by the $I4/mmm$ space group. This is manifest both in the in-plane anisotropy of the normal- and SC-state spin dynamics and in the out-of-plane dispersion of the spin-resonance mode. We show that this effect originates from the higher symmetry of the magnetic Fe sublattice with respect to the crystal itself, hence the INS signal inherits the symmetry of the unfolded Brillouin zone (BZ) of the Fe sublattice. The in-plane anisotropy is temperature independent and can be qualitatively reproduced in normal-state density-functional-theory calculations without invoking a symmetry-broken (“nematic”) ground state that was previously proposed as an explanation for this effect. Below the SC transition, the energy of the magnetic resonant mode ${\omega }_{\text{res}}$, as well as its intensity and the SC spin gap inherit the normal-state intensity modulation along the out-of-plane direction $L$ with a period twice larger than expected from the body-centered-tetragonal BZ symmetry. The amplitude of this modulation decreases at higher doping, providing an analogy to the splitting between even and odd resonant modes in bilayer cuprates. Combining our and previous data, we show that at odd $L$ a universal linear relationship $\hslash {\omega }_{\text{res}}\approx 4.3k_{\text{B}}{T}_{\text{c}}$ holds for all the studied Fe-based superconductors, independent of their carrier type. Its validity down to the lowest doping levels is consistent with weaker electron correlations in ferropnictides as compared to the underdoped cuprates.

Figure 1

(Color online) The reciprocal-space structure of the $I4/mmm$ crystal. The BZ polyhedron of ${\text{BaFe}}_2{\text{As}}_2$ is drawn at the left in solid black lines, and two more such polyhedra are drawn to illustrate the 3D stacking of the Brillouin zones. Two $\overrightarrow{\Gamma X}$ vectors are shown by dashed arrows. The SDW vector of the parent compound ${\mathbf{Q}}_{\text{AFM}}=\left(\frac{1}{2}\frac{1}{2}1\right)$ and its in-plane projection ${\mathbf{Q}}_{\parallel }=\left(\frac{1}{2}\frac{1}{2}0\right)$. Symmetry axes are denoted by dashed-dotted lines.

Figure 2

A sketch illustrating the symmetry of spin excitations: (a) as expected from the BZ symmetry in the absence of matrix elements; (b) as actually observed experimentally in doped 122-compounds. The surfaces schematically represent constant-intensity contours of the magnetic INS response. The center of each panel corresponds to the $\Gamma$ point. Note that despite the lower symmetry in (b) due to the absence of the $4_2/m$ screw around $\left(\frac{1}{2}\frac{1}{2}L\right)$, the fourfold $\left(4/m\right)$ rotational symmetry around $\left(00L\right)$ is preserved.

Figure 3

(Color online) Different primitive unit cells in direct space (left) that can be introduced in 122-ferropnictides and their respective Brillouin zones (right): (a) unfolded tetragonal BZ of the Fe sublattice with one Fe atom per unit cell $\left({\text{Fe}}_1\right)$; (b) structural body-centered-tetragonal BZ that corresponds to two iron atoms per primitive unit cell $\left({\text{Fe}}_2\right)$; (c) unfolded magnetic BZ that corresponds to the magnetically ordered Fe sublattice in the SDW state $\left({\text{Fe}}_2\right)$; (d) doubly folded magnetic BZ that results if both the lattice and magnetic structures are taken into account $\left({\text{Fe}}_4\right)$; (e) one of the most commonly used and experimentally convenient reciprocal-space coordinate systems that corresponds to the BZ of a simple-tetragonal direct lattice with the parameters of the real bct crystal—the notation that we also adopt for the present paper (Ref. 63). For the interactive version of this figure, see Ref. 66.

Figure 4

(Color online) The Lindhard function ${\chi }_0\left(\mathbf{Q},\omega \right)$, resulting from DFT calculations in the undoped (top), 10% Co-doped (electron-overdoped, middle) and 40% K-doped (optimally hole doped, bottom) ${\text{BaFe}}_2{\text{As}}_2$ compounds. [(a), (d), and (g)] Surface plots of the real (left) and imaginary (right) parts of the Lindhard susceptibility within the $L=0$ and $L=1$ planes. [(b), (e), and (h)] Respective profiles of ${\chi }_0\left(\mathbf{Q},\omega \right)$ along the high-symmetry directions, plotted at $L=0$, 1/2, and 1. [(c), (f), and (i)] $L$ dependence of ${\chi }_0\left(\mathbf{Q},\omega \right)$ along the $\left(\frac{1}{2}\frac{1}{2}L\right)$ symmetry axis and at the incommensurate peak positions (for doped compounds only).

Figure 5

(Color online) Lindhard function (left) and renormalized RPA spin susceptibility (right) in the static limit ($\omega \to 0$), calculated from a 3D tight-binding model (Ref. 117). (a) Lindhard functions ${\chi }_0\left(H,K,0\right)$ and ${\chi }_0\left(H,K,1\right)$ for the parent (undoped) compound. (b) Corresponding renormalized RPA spin susceptibilities ${\chi }_{\text{RPA}}\left(H,K,0\right)$ and ${\chi }_{\text{RPA}}\left(H,K,1\right)$ calculated for $\overline{U}=0.8$ and $\overline{J}=0.25\overline{U}$. [(c) and (d)] The same for 7.5% electron-doped compound within rigid-band approximation.

Figure 6

(Color online) Characterization of the samples used for the present study. (a) Magnetization curves measured in the magnetic field of 10 Oe, applied in plane, after cooling in the field (FC) and in zero field (ZFC). Insets show photos of the samples. (b) Rocking curves measured on the (004) reflection in the $\left(HHL\right)$ scattering plane with a triple-axis spectrometer. (c) Neutron-diffraction pattern of the ${\text{BaFe}}_{1.91}{\text{Ni}}_{0.09}{\text{As}}_2$ sample in the $\left(HHL\right)$ scattering plane. Powder lines coming from the Al sample holder and traces of the (Fe,Co)As flux are marked by the arrows.

Figure 7

(Color online) Several raw $Q$ scans for ${\text{BaFe}}_{1.85}{\text{Co}}_{0.15}{\text{As}}_2$, measured along the longitudinal direction in the SC state (top row, $T=4\text{K}$) and in the normal state (bottom row, $T=60\text{K}$) at three different energies: 3, 9.5, and 16 meV. The solid lines represent Gaussian fits with a linear background. The background is indicated by dashed lines.

Figure 8

(Color online) Imaginary part of the spin susceptibility at odd (top) and even (bottom) $L$ in the normal and SC states. The left column shows data for ${\text{BaFe}}_{1.91}{\text{Ni}}_{0.09}{\text{As}}_2$ at $\mathbf{Q}=\left(\frac{1}{2}\frac{1}{2}1\right)$ and $\left(\frac{1}{2}\frac{1}{2}3\right)$ in (a) and at $\left(\frac{1}{2}\frac{1}{2}2\right)$ in (c). The right column shows corresponding data for ${\text{BaFe}}_{1.85}{\text{Co}}_{0.15}{\text{As}}_2$. The data points were obtained from constant-$\omega$ scans and constant-$\mathbf{Q}$ scans, as described in the text. The solid lines are guides to the eyes. Different symbol shapes represent data obtained in different measurements.

Figure 9

$L$-dependent magnetic intensity of ${\text{BaFe}}_{1.91}{\text{Ni}}_{0.09}{\text{As}}_2$ in the SC state at $\mathbf{Q}=\left(\frac{1}{2}\frac{1}{2}L\right)$ and 8 meV (close to the resonance energy). The dashed line shows the ${\text{Fe}}^{2+}$ spin-only magnetic form factor.

Figure 10

Comparison of momentum profiles at even and odd $L$ at fixed energies that are below ${\omega }_{\text{sg}}$ for even $L$, but above it for odd $L$. (a) ${\text{BaFe}}_{1.91}{\text{Ni}}_{0.09}{\text{As}}_2$, $T=3\text{K}$ and $\omega =3\text{meV}$. (b) ${\text{BaFe}}_{1.85}{\text{Co}}_{0.15}{\text{As}}_2$, $T=4\text{K}$ and $\omega =4\text{meV}$.

Figure 11

(Color online) (a) Experimental intensity distributions for ${\text{BaFe}}_{1.85}{\text{Co}}_{0.15}{\text{As}}_2$ near ${\mathbf{Q}}_{\parallel }$ (left) and ${\mathbf{Q}}_{\text{AFM}}$ (right), measured in the $\left(HK\left[H+K\right]\right)$ scattering plane in the SC state $\left(T=4\text{K}\right)$ at the resonance energy (9.5 meV). The small black ellipse around $\left(\frac{1}{2}\frac{1}{2}1\right)$ is a 9.5 meV cross section of the spin-wave dispersion for the ${\text{CaFe}}_2{\text{As}}_2$ parent compound (Refs. 61, 62), shown for comparison. The white dotted lines are BZ boundaries. (b) Comparison of the LO and TR cross sections of the data from panel (a) around $L=0$ (left) and $L=1$ (right). (c) The Lindhard function $\text{Im}{\chi }_{\text{DFT}}\left(\omega \right)/\omega$ at 7.5% Co doping, calculated by DFT in the same reciprocal space regions. (d) The same for the RPA-renormalized low-energy spin susceptibility $\text{Im}{\chi }_{\text{RPA}}\left(\omega \right)/\omega$ [same as in Fig. 5d], calculated from a TB model in the rigid-band approximation.

Figure 12

(Color online) (a) LO and TR widths of the commensurate peaks around $Q_{\text{AFM}}$ $\left(L=1\right)$ and $Q_{\parallel }$ $\left(L=0\right)$ at 9.5 meV vs temperature. (b) The same widths vs energy transfer at low temperatures. Solid lines are results of a global fit to the data in both panels (see text) using Eq. (6). Resolution-corrected dependencies are shown by dashed lines. (c) The resolution-corrected anisotropy ratio $A=\left(w_{\text{TR}}-w_{\text{LO}}\right)/\left(w_{\text{TR}}+w_{\text{LO}}\right)$ as a function of temperature, compared to the respective values for the magnetically ordered (Refs. 61, 62) and paramagnetic (Ref. 32) states of ${\text{CaFe}}_2{\text{As}}_2$. (d) The same ratio as a function of energy transfer. The dashed line gives the anisotropy from the global fit. The dotted line is derived from the high-energy dispersion reported for ${\text{BaFe}}_{1.87}{\text{Co}}_{0.13}{\text{As}}_2$ in Ref. 30.

Figure 13

(Color online) (a) Doping dependence of ${\omega }_{\text{res}}$ at odd and even $L$ in BFNA and BFCA studied here (full symbols) and in previous works (empty symbols), as referred to in (a) and (c). The lower dotted line follows the average $T_{\text{c}}$, rescaled to 4.3 at its optimum (Refs. 127, 128, 129, 130). The upper dotted line is a guide to the eyes. (b) Linear extrapolation of the resonance intensities to the energy axis, as compared to the onset of particle-hole continuum. The hatched region covers the range of directly measured $2\Delta$ values for the larger gap in nearly optimally doped BFCA, estimated by various experimental techniques (Refs. 133, 134, 135, 136, 137, 138, 139). (c) Resonance energy vs $T_{\text{c},\text{opt}}$ for different Fe-based superconductors. For the compounds with dispersing resonance, only ${\omega }_{\text{res}}$ at odd $L$ is shown.

Figure 14

(Color online) Illustration of the evolution from an $\infty$-layer system like BFCA to a bilayer system like YBCO in terms of interbilayer and intrabilayer distances and effective interactions. For the equidistant limit (left), a single dispersing resonant mode is observed, whose intensity modulation (shown here by the brightness of the curve) is mainly governed by the closeness to the particle-hole continuum with an onset at $2\Delta$. The dashed line depicts the replica that gains intensity only after the equivalency of the layers is broken (middle panel). For alternating interlayer coupling, the resonance splits into odd and even modes, which become nondispersive for the case of ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+y}$ with nearly independent bilayers (right).