Nematic spin fluid in the tetragonal phase of BaFe${}_2$As${}_2$

We use inelastic neutron scattering to study spin waves below and above $T_N$ in iron-arsenide BaFe${}_2$As${}_2$. In the low-temperature orthorhombic phase, we find highly anisotropic spin waves with a large damping along the antiferromagnetic $a$-axis direction. On warming the system to the paramagnetic tetragonal phase, the low-energy spin waves evolve into quasi-elastic excitations, while the anisotropic spin excitations near the zone boundary persist. These results strongly suggest the presence of a spin nematic fluid in the tetragonal phase of BaFe${}_2$As${}_2$, which may cause the electronic and orbital anisotropy observed in these materials.

Figure 1

(a) The AF Fe spin ordering in BaFe${}_2$As${}_2$ with the magnetic exchange couplings $J_{1a}$, $J_{1b}$, $J_2$ along different directions. (b) Temperature dependence of the resistivity in detwinned BaFe${}_2$As${}_2$ (from Ref. 14). The inset is a plot of the resistivity for the twinned sample used in our neutron measurements with the blue points corresponding to $T=7$, 125, and 150 K. (c) Color plots describing qualitatively how the spin wave scattering evolves from Q = (1, 0) to (1, 1) as a function of energy using an anisotropic damping Γ. The solid black contours are an overlay of the same model but with no damping. The exchange couplings used in both plots are from best fits of the data. (d) Color plot of the anisotropic damping Γ, which is much stronger along the $H$ direction than along the $K$ direction. (e) Spin wave dispersion along the (1, $K$) direction as determined by energy and Q cuts of the raw data in Fig. 2 below and above $T_N$. The solid line is a Heisenberg model calculation using anisotropic exchange couplings $S{J}_{1a}=59.2$ ± 2.0, $S{J}_{1b}$ = −9.2 ± 1.2, $S{J}_2=13.6$ ± 1.0, $S{J}_c=1.8$ ± 0.3 meV determined by fitting the full cross section. The dotted line is a Heisenberg model calculation assuming isotropic exchange coupling $S{J}_{1a}$ = $S{J}_{1b}=18.3$ ± 1.4, $S{J}_2$ = 28.7 ± 0.5, and $S{J}_c=1.8$ meV. (f) Dispersion along the ($H$, 0) direction; data points beyond $H=1.4$ could not be reliably obtained due to strong damping at higher energies. The red shading stresses how the damping grows as a function of $H$. Error bars are systematic and represent the difference between Q and $E$ cut dispersion points. The statistical error of the Q and $E$ cuts are much smaller.

Figure 2

Wave vector dependence of the spin waves for energy transfers of (a) $E=26$ ± 10 meV [$E_i$ = 450 meV and Q = ($H$, $K$, 1)]; (b) $E=81$ ± 10 meV [$E_i$ = 450 meV and Q = ($H$, $K$, 3)]; (c) $E=113$ ± 10 meV [$E_i$ = 450 meV and Q = ($H$, $K$, 5)]; (d) $E=157$ ± 10 meV [$E_i$ = 600 meV and Q = ($H$, $K$, 5)]; (e) $E=214$ ± 10 meV [$E_i$ = 600 meV and Q = ($H$, $K$, 7)]; (f) The projection of the spin waves on the energy transfer axis and (1, $K$) direction (with integration of $H$ from 0.8 to 1.2 rlu) after subtracting the background integrated from 1.8 < $H$ < 2.2 and from −0.25 < $K$ < 0.25 with $E_i$ = 450 meV. The color bar scales represent the absolute spin wave intensity in units of mbarn·sr${}^{-1}$·meV${}^{-1}$·f.u.${}^{-1}$ and the dashed boxes indicate zone boundaries. The missing low-energy data in (f) is due to imperfect data subtraction. (g)–(l) Model calculation of identical slices as in (a)–(f) using anisotropic exchange couplings from best fits and convolved with the instrumental resolution.

Figure 3

(a)–(c) Spin waves of $E=50$ ± 10 meV; (d)–(f) $E=75$ ± 10 meV; (g)–(i) $E=125$ ± 10 meV; and (j)–(l) $E=150$ ± 10 meV for temperatures of $T=7$, 125, and 150 K. The dashed curves show fixed reciprocal space sizes at different temperatures.

Figure 4

The blue diamonds in (a)–(d) are constant-Q cuts at Q = (1, 0.05), (1, 0.2), (1, 0.35), and (1, 0.5), respectively, at $T=7$ K. The green squares and red circles in (a)–(d) are identical constant-Q cuts at $T=125$ and 150 K, respectively. The dashed lines are guides to the eye for the observed paramagnetic scattering. (e) and (f) Q dependence of the spin wave excitations below and above $T_N$ obtained through constant-$E$ cuts at $E=19$ ± 5 and 128 ± 5 meV. The solid lines in (a)–(f) are fits to the anisotropic spin-wave model discussed in the text, and the horizontal bars represent the instrumental energy ($E$)/wave vector (Q) resolution. (g) Energy dependence of the dynamic spin-spin correlation lengths below and above $T_N$ obtained by Fourier transform of constant-$E$ cuts similar to (e) and (f).

Figure 5

Heisenberg model calculations with $J_{1a}\ne J_{1b}$ demonstrating the effect of twinning on spin waves in BaFe${}_2$As${}_2$. All calculations are performed with $L=1$ and with an equal amount of twinning in both directions. Units of intensity are arbitrary. The primary effect of twinning is to introduce a second scattering center rotated ${90}^o$ away with intensity split equally between the domains. As a result, magnetic anisotropy can still be clearly observed in twin samples since the scattering at these centers are distinct, with overlap only occurring at the zone boundary.

Figure 6

(a)-(d) Heisenberg model calculations with $J_{1a}\ne J_{1b}$ demonstrating the effect of twinning on spin waves in BaFe${}_2$As${}_2$. All calculations assume equal amounts of twinning in both directions. Units of intensity are arbitrary. (e), (f) Cartoons depicting the cut directions of the dispersions plotted in (a)-(d). The primary effect of twinning is to introduce weak twin bands that only contribute to the observed dispersion near the zone boundary.

Figure 7

Constant energy cuts of the spin wave excitations at 7 K and our model fits to the data using an anisotropic Heisenberg Hamiltonian convolved with the instrumental resolution. The solid lines are the output from the Toby fit program${}^{34}$ using fitting parameters as discussed in the text of the paper.

Figure 8

Heisenberg model calculations with $J_{1a}=J_{1b}$ demonstrating the effect of twinning on spin waves in BaFe${}_2$As${}_2$. All calculations are performed with $L=1$ and with an equal amount of twinning in both directions. Units of intensity are arbitrary. In general, isotropic in plane exchange couplings result in a loss in intensity along the K-direction as the energy is increased. This is exactly the opposite of what we observe thereby ruling out an isotropic Heisenberg model.

Figure 9

Heisenberg model calculations with $J_{1a}=J_{1b}$ demonstrating the effect of twinning on spin waves in BaFe${}_2$As${}_2$. All calculations assume equal amounts of twinning in both directions. Units of intensity are arbitrary. Note that whereas the dispersion along the K-direction peaks at $K=1$ when $J_{1a}\ne J_{1b}$, when these exchange couplings are made equal, the dispersion softens with $K=1$ now corresponding to a minima in the dispersion. This is exactly the opposite of what we observe both below and above $T_N$. This suggests that the effective in plane exchange coupling remain anisotropic in both the orthorhombic and tetragonal state.

Figure 10

Constant energy cuts of the spin wave excitations at 7 K and the $J_{1a}$ = $J_{1b}$ model fits to the data using an anisotropic Heisenberg Hamiltonian convolved with the instrumental resolution. The solid lines are the output from the Toby fit program${}^{34}$ using fitting parameters as discussed in the supplementary material. While this model fits the low-energy spin wave data reasonably well, it completely fails to describe the data for spin wave energies above 100 meV.

Figure 11

Constant energy cuts of the spin wave excitations at 7 K for BaFe${}_2$As${}_2$ and CaFe${}_2$As${}_2$ in absolute units within the first Brillouin zone. The data for CaFe${}_2$As${}_2$ are from Ref. 7.

Figure 12

Comparison of a normalized RPA calculation from Ref. 27 and our data. Given the normalization correction, RPA appears to fit the data.