Magnetic excitations in underdoped Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ with $x=0.047$

The magnetic excitations in the paramagnetic-tetragonal phase of underdoped Ba(Fe${}_{0.953}$Co${}_{0.047}$)${}_2$As${}_2$, as measured by inelastic neutron scattering, can be well described by a phenomenological model with purely diffusive spin dynamics. At low energies, the spectrum around the magnetic ordering vector ${\mathbf{Q}}_{\mathrm{AFM}}$ consists of a single peak with elliptical shape in momentum space. At high energies, this inelastic peak is split into two peaks across the direction perpendicular to ${\mathbf{Q}}_{\mathrm{AFM}}$. We use our fittings to argue that such a splitting is not due to incommensurability or propagating spin-wave excitations, but is rather a consequence of the anisotropies in the Landau damping and in the magnetic correlation length, both of which are allowed by the tetragonal symmetry of the system. We also measure the magnetic spectrum deep inside the magnetically ordered phase, and find that it is remarkably similar to the spectrum of the paramagnetic phase, revealing the strongly overdamped character of the magnetic excitations.

Figure 1

Inelastic neutron scattering data measured on Ba(Fe${}_{0.953}$Co${}_{0.047}$)${}_2$As${}_2$ at $T=5$ K using ARCS with $E_{\mathrm{i}}=250$ meV and the crystal aligned with the incident neutron beam along the [0,0,1] direction. (a) Transverse slice of the data along the $[K,-K]$ direction through ${\mathbf{Q}}_{\mathrm{AFM}}=(\frac{1}{2},\frac{1}{2},L)$ after averaging over $H=0.5\pm 0.05$ r.l.u. (b) Longitudinal slice of the data along the $[H,H]$ direction through ${\mathbf{Q}}_{\mathrm{AFM}}=(\frac{1}{2},\frac{1}{2},L)$ after averaging over $K=0\pm 0.1$ r.l.u. In each panel, the color scale represents the intensity of scattered neutrons.

Figure 2

Comparison of inelastic neutron scattering data measured on Ba(Fe${}_{0.953}$Co${}_{0.047}$)${}_2$As${}_2$ at $T=$ 5 K (empty circles) and $T=$ 70 K (filled circles). Transverse cuts of the data at (a${}^{\prime }$) 100 $\pm$ 10 meV, (b${}^{\prime }$) 80 $\pm$ 10 meV, (c${}^{\prime }$) 60 $\pm$ 10 meV, (d${}^{\prime }$) 40 $\pm$ 10 meV. Longitudinal cuts of the data at (a${}^{\prime \prime }$) 100 $\pm$ 10 meV, (b${}^{\prime \prime }$) 80 $\pm$ 10 meV, (c${}^{\prime \prime }$) 60 $\pm$ 10 meV, (d${}^{\prime \prime }$) 40 $\pm$ 10 meV.

Figure 3

Inelastic neutron scattering data measured on Ba(Fe${}_{0.953}$Co${}_{0.047}$)${}_2$As${}_2$ using ARCS with $E_{\mathrm{i}}=250$ meV and the crystal aligned with the incident neutron beam along the [0, 0,1] direction. Data at $T=5$ and 70 K are summed together to improve statistics. (a) Transverse slice of the data along the $[K,-K]$ direction through ${\mathbf{Q}}_{\mathrm{AFM}}=(\frac{1}{2},\frac{1}{2},L)$ after averaging over $H=0.5\pm 0.05$ r.l.u. (b) Longitudinal slice of the data along the $[H,H]$ direction through ${\mathbf{Q}}_{\mathrm{AFM}}=(\frac{1}{2},\frac{1}{2},L)$ after averaging over $K=0\pm 0.1$ r.l.u. Constant energy slices in the $(H,K)$ plane averaged over an energy range of (c) $35\pm 5$ meV, (d) $50\pm 10$ meV, (e) $80\pm 10$ meV, (f) $100\pm 10$ meV, (g) $120\pm 10$ meV, and (h) $150\pm 10$ meV. In each panel, the color scale represents the intensity of scattered neutrons, where the maximum intensity in each panel, in arb. units, is (a) 10, (b) 10, (c) 10, (d) 8, (e) 5, (f) 3, (g) 2, (h) 1; and the minimum intensity in all panels is $-1$ arb. units.

Figure 4

Calculations of the inelastic neutron scattering spectrum as obtained from a model of itinerant diffusive spin fluctuations, as described by Eq. (6). The parameters of the model were obtained by fits to the neutron scattering data for Ba(Fe${}_{0.953}$Co${}_{0.047}$)${}_2$As${}_2$ as shown in Figs. 6 and 7. (a) Transverse slice of the data along the $[K,-K]$ direction through ${\mathbf{Q}}_{\mathrm{AFM}}=(\frac{1}{2},\frac{1}{2},L)$ after averaging over $H=0.5\pm 0.05$ r.l.u. (b) Longitudinal slice of the data along the $[H,H]$ direction through ${\mathbf{Q}}_{\mathrm{AFM}}=(\frac{1}{2},\frac{1}{2},L)$ after averaging over $K=0\pm 0.1$ r.l.u. Constant energy slices in the $(H,K)$ plane averaged over an energy range of (c) $35\pm 5$ meV, (d) $50\pm 10$ meV, (e) $80\pm 10$ meV, (f) $100\pm 10$ meV, (g) $120\pm 10$ meV, and (h) $150\pm 10$ meV. In each panel, the color scale represents the intensity of scattered neutrons, where the maximum intensity in each panel, in arb. units, is (a) 10, (b) 10, (c) 10, (d) 8, (e) 5, (f) 3, (g) 2, (h) 1; and the minimum intensity in all panels is 0.

Figure 5

Calculations of the inelastic neutron scattering spectrum as obtained from a ballistic model of overdamped spin waves, as described by Eq. (11). The parameters of the model were obtained by fits to the neutron scattering data for Ba(Fe${}_{0.953}$Co${}_{0.047}$)${}_2$As${}_2$ as shown in Figs. 6 and 7. (a) Transverse slice of the data along the $[K,-K]$ direction through ${\mathbf{Q}}_{\mathrm{AFM}}=(\frac{1}{2},\frac{1}{2},L)$ after averaging over $H=0.5\pm 0.05$ r.l.u. (b) Longitudinal slice of the data along the $[H,H]$ direction through ${\mathbf{Q}}_{\mathrm{AFM}}=(\frac{1}{2},\frac{1}{2},L)$ after averaging over $K=0\pm 0.1$ r.l.u. Constant energy slices in the $(H,K)$ plane averaged over an energy range of (c) $35\pm 5$ meV, (d) $50\pm 10$ meV, (e) $80\pm 10$ meV, (f) $100\pm 10$ meV, (g) $120\pm 10$ meV, and (h) $150\pm 10$ meV. In each panel, the color scale represents the intensity of scattered neutrons, where the maximum intensity in each panel, in arb. units, is (a) 10, (b) 10, (c) 10, (d) 8, (e) 5, (f) 3, (g) 2, (h) 1; and the minimum intensity in all panels is 0.

Figure 6

Fitting results for the diffusive (red lines) and ballistic (blue lines) models to the summed $T=5$ and 70 K data (open symbols). Transverse cuts (${}^{\prime }$) and longitudinal cuts (${}^{\prime \prime }$) through $\mathbf{Q}=(\frac{1}{2},\frac{1}{2},L)$ at (a) $E=120\pm 10$ meV, (b) $E=100\pm 10$ meV, (c) $E=80\pm 10$ meV, (d) $E=55\pm 5$ meV, (e) $E=45\pm 5$ meV, and (f) $E=35\pm 5$ meV. Shaded regions represent 95$\%$ confidence intervals for the fitting results.

Figure 7

Fitting results for the diffusive (red lines) and ballistic (blue lines) models to the summed $T=5$ and 70 K data (open symbols). Spectra cuts centered at (a) $(0.5,0.5,L)$, (b) $(0.55,0.45,L)$, (c) $(0.6,0.4,L)$, (d) $(0.65,0.35,L)$, and (e) $(0.7,0.3,L)$. Shaded regions represent 95$\%$ confidence intervals for the fitting results.