Temperature dependence of the paramagnetic spin excitations in BaFe${}_2$As${}_2$

We use inelastic neutron scattering to study temperature dependence of the paramagnetic spin excitations in iron pnictide BaFe${}_2$As${}_2$ throughout the Brillouin zone. In contrast to a conventional local moment Heisenberg system, where paramagnetic spin excitations are expected to have a Lorentzian function centered at zero energy transfer, the high-energy ($\hslash \omega >100$ meV) paramagnetic spin excitations in BaFe${}_2$As${}_2$ exhibit spin-wave-like features up to at least 290 K ($T=2.1T_N$). Furthermore, we find that the sizes of the fluctuating magnetic moments $\langle m^2\rangle \approx 3.6{\mu }_B^2$ per Fe are essentially temperature independent from the antiferromagnetic ordered state at $0.05T_N$ to $2.1T_N$, which differs considerably from the temperature dependent fluctuating moment observed in the iron chalcogenide Fe${}_{1.1}$Te [Zaliznyak et al., Phys. Rev. Lett. 107, 216403 (2011)]. These results suggest unconventional magnetism and strong electron correlation effects in BaFe${}_2$As${}_2$.

Figure 1

(a)–(c) compare the $E=50$ meV magnetic scattering deep inside the ordered state (7 K) to scattering in the paramagnetic phase for temperatures (225 and 290 K) well away from the $T_N=138$ K phase transition. (d)–(f) and (g)–(i) are a similar comparison for energy transfers of 100 and 150 meV, respectively. The dotted ellipses and boxes are guides to the eye to more easily facilitate comparison. Data in (a)–(f) and (g)–(i) were collected using $E_i=250$ and 450 meV, respectively. All data were background subtracted using the average intensity from the region $1.8<H<2.2$, $-0.2<K<0.2$ r.l.u. as the background point. Data in the region $H<0$ were folded into the equivalent $H>0$ positions in order to improve statistics.

Figure 2

(a)–(d) compare the energy vs $K$ intensity slices for 7, 125, 225, and 290 K. All data are background subtracted and folded in an identical manner as described in the caption of Fig. 1. The solid line is the Heisenberg dispersion obtained using anisotropic exchange couplings $S{J}_{1a}=59.2\pm 2.0$, $S{J}_{1b}=-9.2\pm 1.2$, $S{J}_2=13.6\pm 1.0$, $S{J}_c=1.8\pm 0.3$ meV, determined by fitting the full cross section to the 7 K data (Ref. 19).

Figure 3

(a)–(f) Temperature overplot of the evolution of the spin excitations as a function of increasing energy. The green diamonds, yellow squares, red circles, cyan upward facing triangles, and blue downward facing triangles are for the 7, 125, 150, 225, and 290 K data, respectively. The data have been artificially offset for clarity and empirically fit using Gaussian functions. The insets are the fits without offset.

Figure 4

(a) Dispersion along the $[1,K]$ direction as determined by energy and $Q$ cuts of the raw data. The solid line is the anisotropic Heisenberg dispersion (Ref. 19). (b) Dispersion along the $[H,0]$ direction is built using the same method. The light blue upward facing triangular points in (c)–(f) are constant-$Q$ cuts at $Q=(1,0.05)$, $(1,0.2)$, $(1,0.35)$, and $(1,0.5)$, respectively, at 225 K. The dark blue downward facing triangular points in (c)–(f) are identical constant-$Q$ cuts at 290 K. The solid green and red lines are guides to the eye describing the observed 7 and 150 K scattering, respectively. These constant-$Q$ cuts correspond to cuts across the dispersion as depicted in the inset of (c). The horizontal bars in (d) and (f) are instrumental energy resolution.

Figure 5

The local susceptibility plots in (a)–(e) represent the total $Q$-integrated intensity across the magnetic zone of size $0<H\le 2$, $-1<K\le 1$ r.l.u. In practice, it is not possible to use the actual full zone size in $H$ and $K$ because of gaps in the detector array and consequent limited accessibility of certain reciprocal space regions. Thus, for each $E_i$ a smaller region that either contains all of the scattering and/or has the requisite symmetry is chosen instead. These regions are then all normalized to the entire zone area as required by ${\chi }^{\prime \prime }\left(E\right)=\frac{\int {\chi }^{\prime \prime }(Q,E)dQ}{\int dQ}$. The solid black lines in (a)–(e) are empirical fits of the local susceptibility. (f) An overplot of these fits to aid in a cross comparison of the temperature dependence. (g) The dynamic moment as determined by integrating the fits from the previous panels. The static moment is reproduced from Ref. 18.