Normal-state spin dynamics in the iron-pnictide superconductors BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$ and Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ probed with NMR measurements

The NMR results in iron pnictides BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$ and Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ are analyzed based on the self-consistent renormalization (SCR) spin-fluctuation theory. The temperature dependence of the NMR relaxation rate $T_1^{-1}$ as well as the electrical resistivity is well reproduced by a SCR model where two-dimensional antiferromagnetic (AF) spin fluctuations are dominant. The successful description of the crossover feature from non-Fermi-liquid to Fermi-liquid behavior strongly suggests that low-lying spin fluctuations in BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$ and Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ possess an itinerant AF nature, and that chemical substitution in the two compounds tunes the distance of these systems to an AF quantum critical point. The close relationship between spin fluctuations and superconductivity is discussed compared with the other unconventional superconductors, cuprate and heavy fermion superconductors. In addition, it is suggested that magnetism and lattice instability in these pnictides are strongly linked via orbital degrees of freedom.

Figure 1

The Knight shift of the ${}^{31}$P nucleus for different P concentrations of BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$.

Figure 2

The ${}^{31}$P nuclear spin-lattice relaxation rate divided by temperature ${\left(T_{1}T\right)}^{-1}$ for BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$. The data for $x=0$ (BaFe${}_2$As${}_2$) were cited from Ref. 21.

Figure 3

The $T$ dependence of the NMR relaxation rate arising from the $\mathbf{q}\sim {\mathbf{Q}}_{\mathrm{AF}}$ mode of spin fluctuations ${\left(T_1\right)}_{\mathrm{AF}}^{-1}$ of Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ for $H\parallel ab$, cited from Ref. 6. The solid lines represent simulations with the SCR parameters listed in Table .

Figure 4

The experimental (Ref. 32) (solid line) and calculated (circles) damping constant $\Gamma \left(T\right)$ for the dynamical susceptibility ${\chi }^{\prime \prime }(Q,\omega )=\frac{{\chi }_T\Gamma \left(T\right)\omega }{{\omega }^2+{\Gamma }^2\left(T\right)(1+{\xi }_T^2|\mathit{Q}-{\mathit{Q}}_{\mathrm{AFM}}\left|\right)}$ observed in Ba(Fe${}_{0.925}$Co${}_{0.075}$)${}_2$As${}_2$. The circles represent a calculation with $y_0=0.025$, $y_1=2.5$, and $T_0=450$ K.

Figure 5

The $T$ dependence of the electrical resistivity ${\rho }_{ab}$ of Ba(Fe${}_{0.92}$Co${}_{0.08}$)${}_2$As${}_2$, cited from Ref. 27. The solid line represents a calculation using $y_0=0.025$, $y_1=2.5$, and $T_0=450$ K, and residual resistivity ${\rho }_0^{ab}=0.1$ m$\Omega$ cm.

Figure 6

The $T_1^{-1}$ of ${}^{31}$P and ${}^{75}$As in BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$ with $x=0.33$ is plotted. We estimated the hyperfine coupling constant of ${}^{31}$P nucleus from the slope as ${}^{31}{A}_{\mathrm{hf}}=0.674$ T/${\mu }_B$ by using ${}^{75}{A}_{\mathrm{hf}}=1.72$ T/${\mu }_B$ (Ref. 27).

Figure 7

The experimental and calculated NMR relaxation rate arising from the $\mathit{q}\sim {\mathit{Q}}_{\mathrm{AF}}$ mode of spin fluctuations in BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$. The data points represent the experimental data, while the solid lines indicate the calculated data using the SCR parameters listed in Table .

Figure 8

The experimental (Ref. 5) and calculated electrical resistivity of BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$. The solid lines represent the experimental data, while the dotted lines indicate the calculated data using the SCR parameters listed in Table .

Figure 9

Phase diagrams of BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$ and Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$, where $T_{\mathrm{N}}$ and $T_c$ denote an AF transition temperature and SC transition temperature, respectively. In both materials, the concentration where $T_c$ peaks [$x\sim 0.3$ for BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$, and $x\sim 0.06$ for Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$] exists near the region where $y_0=0$. Since $y_0=0$ corresponds to an AF QCP, these phase diagrams suggest a close link between superconductivity and AF quantum criticality.

Figure 10

The SC transition temperature $T_c$ vs the spin-fluctuation temperature (a) $T_0$ and (b) $T_A$ for various superconductors. The data for $T_c$ and $T_0$ were cited from Refs. 35 and 37. The dotted lines represent linear curve fittings. $T_c$ is linear with $T_0$, as suggested in Ref. 35, and the iron-pnictide superconductors lie on the same line. This may be the signature that Fe pnictides, heavy-fermion, and cuprate superconductors are mediated by AF spin fluctuations.

Figure 11

The $T$ dependence of ${\left(T_{1}T\right)}^{-1}$ normalized by ${\left(T_{1}T\right)}^{-1}$ at 1.25$T_c$ in various unconventional superconductors (Refs. 44, 45, 46, 47) are plotted against $T/T_c$. ${\left(T_{1}T\right)}^{-1}$ at 1.25$T_c$ is adopted in order to avoid the suppression by the pseudogap effect. The characteristic energy of the spin fluctuations in these compounds seems to be scaled to $T_c$, since the normalized ${\left(T_{1}T\right)}^{-1}$ data are approximately on the same curve.

Figure 12

(a) Low-temperature magnetic structure at the Fe layer. The Ba sites are also shown. The dotted lines indicate the tetragonal unit cell above $T_{\mathrm{S}}$, and are deformed below $T_{\mathrm{S}}$. The stripe magnetic structure is shown by the arrows. (b) As $4p_{x,y}$ and Fe $3d_{yz}$ and $3d_{zx}$ orbitals are shown. The difference in the electronic population is shown by the contrasting density. Magnetic interactions between the nearest neighbor and next nearest neighbor are denoted by arrows.

Figure 13

The $T$ dependence of the elastic constant $C_{66}$ in Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$, measured by Yoshizawa et al. (Ref. 50), is plotted against ${\left(T_{1}T\right)}_{\mathrm{AF}}^{-1}$ of Fig. 6. An apparent proportionality holds between $C_{66}$ and ${\left(T_{1}T\right)}_{\mathrm{AF}}^{-1}$. The dotted lines are guides to the eyes.

Figure 14

The nematic transition temperature $T^{*}$ and ${\left(T_{1}T\right)}^{-1}$ are compared. The magnitude of ${\left(T_{1}T\right)}^{-1}$ is shown as a contour plot. ${\left(T_{1}T\right)}^{-1}$ seems to increase below $T^{*}$, particularly in the low P concentration, suggestive of a close relationship between $T^{*}$ and AF spin fluctuations.