Crossover from spin waves to diffusive spin excitations in underdoped $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x{)}_2{\mathrm{As}}_2$

Using inelastic neutron scattering, we show that the onset of superconductivity in underdoped $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x{)}_2{\mathrm{As}}_2$ coincides with a crossover from well-defined spin waves to overdamped and diffusive spin excitations. This crossover occurs despite the presence of long-range stripe antiferromagnetic order for samples in a compositional range from $x=0.04$ to 0.055, and is a consequence of the shrinking spin-density wave gap and a corresponding increase in the particle-hole (Landau) damping. The latter effect is captured by a simple itinerant model relating Co doping to changes in the hot spots of the Fermi surface. We argue that the overdamped spin fluctuations provide a pairing mechanism for superconductivity in these materials.

Figure 1

Temperature dependence of INS data for Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ with $x=0.015$ [(a), (c), (e)] and $x=0.033$ [(b), (d), (f)] plus fits to the spin-wave model. (a), (b) Energy scans at ${\mathbf{Q}}_{\mathrm{AFM}}=(0.5,0.5,1)$ performed at the indicated temperatures are offset vertically. (c), (d) Reduced temperature dependence of spin-wave model parameters $\alpha$ (open symbols) and $\Delta$ (solid symbols). (e), (f) Reduced temperature dependence of the ordered magnetic moment $\mu$ normalized to its low-temperature value. Light gray symbols in (a) and (b) represent measured intensity which was excluded from fitting due to concerns with the validity of background estimates at those points.

Figure 2

Background subtracted INS intensity of Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ corrected for the Bose thermal population factor and the Fe${}^{2+}$ single-ion magnetic form factor plus best fit lines to the diffusive (light green lines) and the damped spin-wave (black lines) models. (a)–(e) Constant-${\mathbf{Q}}_{}$ energy scans at ${\mathbf{Q}}_{\mathrm{AFM}}=(0.5,0.5,1)$ for five compositions. (f)–(h) Constant-$E$ scans in the $[h,h,0]$ direction across ${\mathbf{Q}}_{\mathrm{AFM}}=(0.5,0.5,1)$ at $E=7$ meV. (i)–(j) Constant-$E$ scans in the $[h,h,0]$ direction across ${\mathbf{Q}}_{\mathrm{AFM}}=(0.5,0.5,3)$ at $E=10$ meV. (k)–(m) Constant-$E$ scans in the $[0,0,l]$ direction, perpendicular to the Fe layer, across ${\mathbf{Q}}_{\mathrm{AFM}}=(0.5,0.5,1)$ at $E=7$ meV. (n)–(o) Constant-$E$ scans in the $[0,0,l]$ direction across ${\mathbf{Q}}_{\mathrm{AFM}}=(0.5,0.5,1)$ at $E=10$ meV. Light gray symbols represent measured intensity which was excluded from fitting due to concerns with the validity of background estimates at those points.

Figure 3

(a) Phase diagram of Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ showing regions of AFM, SC, and their coexistence; colored symbols show the locations in phase space of the measurements performed in this study. (b)–(f) Select model parameters as a function of composition for the diffusive (green diamonds) and spin-wave (black circles) models. All data points shown in (b)–(h) were determined at the lowest temperature indicated in (a), and the lightly shaded background indicates compositions which exhibit SC at low temperature. Spin-wave model: (c) spin gap $\Delta$ (solid), damping $\alpha$ (open), and ${\Gamma }_{\mathrm{s}}={\Delta }^2/\alpha$ (diamonds); (e) interplane spin-wave velocity. Diffusive model: (b) Landau damping $\gamma$ and the corresponding theoretical prediction; (d) spin relaxation characteristic energy $\Gamma$ (solid) and the effective magnetic energy $E_{\mathrm{SF}}=1/\gamma$ (open); (f) correlation length $\xi$. In (c) and (d) an estimate for the SDW gap—derived from $\mu \left(x\right)$ from Ref. [6] and ${\Delta }_{\mathrm{SDW}}(x=0)$ from Ref. [17]—is also shown (tan line). (g) Residual ${\chi }^2$ for each fit model. (h) Spectral weight of the $(0.5,0.5,1)$ excitation. The spectral weight is the ${\mathbf{Q}}_{}$-averaged energy integration of the trace of the imaginary component of the magnetic susceptibility tensor. The averaging range in ${\mathbf{Q}}_{}$ here is $0<H<1$, $0<K<1$, $0<L<2$; the energy integration is over the range $0<E<35$ meV. All error bars represent the combined errors for all function parameters. The solid green and black lines in (c)–(h) are guides to the eye.