Short-Range Nematic Fluctuations in ${\mathrm{Sr}}_{1-x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ Superconductors

Interactions between nematic fluctuations, magnetic order and superconductivity are central to the physics of iron-based superconductors. Here we report on in-plane transverse acoustic phonons in hole-doped ${\mathrm{Sr}}_{1-x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ measured via inelastic x-ray scattering, and extract both the nematic susceptibility and the nematic correlation length. By a self-contained method of analysis, for the underdoped ($x=0.36$) sample, which harbors a magnetically ordered tetragonal phase, we find it hosts a short nematic correlation length $\xi \sim 10\AA$ and a large nematic susceptibility ${\chi }_{\mathrm{nem}}$. The optimal-doped ($x=0.55$) sample exhibits weaker phonon softening effects, indicative of both reduced $\xi$ and ${\chi }_{\mathrm{nem}}$. Our results suggest short-range nematic fluctuations may favor superconductivity, placing emphasis on the nematic correlation length for understanding the iron-based superconductors.

Figure 1

Representative temperature-dependent energy scans for (a) ${\mathrm{Sr}}_{0.64}{\mathrm{Na}}_{0.36}{\mathrm{Fe}}_2{\mathrm{As}}_2$ at wave vector $\mathbf{Q}=(2,k,0)$ with $k=0.075$ from SPring-8 and (b) ${\mathrm{Sr}}_{0.45}{\mathrm{Na}}_{0.55}{\mathrm{Fe}}_2{\mathrm{As}}_2$ with $k=0.05$ from APS. Solid symbols and dashed curves are experimental data and the corresponding fits, respectively. Black arrows mark the extracted phonon energies $E_{\mathrm{ph}}$ for the Stokes and anti-Stokes phonons. The vertical dashed lines are aligned to the phonon energies at the highest measured temperature. Temperature dependence of the fit values of $E_{\mathrm{ph}}$ in panels (a) and (b) are respectively shown in (c) and (d). The open and solid symbols are data from SPring-8 and APS, respectively. Vertical error bars are least-square fit errors of 1 s.d. (e) Schematic phase diagram of ${\mathrm{Sr}}_{1-x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$, showing the antiferromagentic orthorhombic (AFM-O), antiferromagnetic tetragonal (AFM-T), paramagnetic tetragonal (PM-T), unidentified $C_2^{\prime }$, and superconducting (SC) phases (adapted from Ref. [16]). Red dots represent the two compositions studied in this work. Insets show the schematic magnetic structures for the AFM-O and AFM-T phases.

Figure 2

$E_{\mathrm{ph}}/k$ plotted against $k$ (solid symbols) and corresponding fits (solid lines) for (a) ${\mathrm{Sr}}_{0.64}{\mathrm{Na}}_{0.36}{\mathrm{Fe}}_2{\mathrm{As}}_2$ and (b) ${\mathrm{Sr}}_{0.45}{\mathrm{Na}}_{0.55}{\mathrm{Fe}}_2{\mathrm{As}}_2$. A large nematic correlation length leads to more prominent phonon softening and nonlinearity in the plot of $E_{\mathrm{ph}}/k$ versus $k$. Details of the fits are described in the text. All vertical error bars are least-square fit errors of 1 s.d.

Figure 3

Nematic correlation length $\xi$, derived bare nematic susceptibility in units of ${\lambda }^2/C_{66,0}$ and renormalized shear modulus $C_{66}/C_{66,0}$ as a function of temperature for (a)–(c) ${\mathrm{Sr}}_{0.64}{\mathrm{Na}}_{0.36}{\mathrm{Fe}}_2{\mathrm{As}}_2$ and (d)–(f) ${\mathrm{Sr}}_{0.45}{\mathrm{Na}}_{0.55}{\mathrm{Fe}}_2{\mathrm{As}}_2$. Here $\lambda$ is the coupling strength between the nematic order parameter and the lattice. $C_{66,0}$ is the bare shear modulus in the absence of nematic fluctuations. The vertical dashed lines mark phase transitions in the two samples. For comparison, the blue dashed lines are the bare nematic susceptibilities and the Young’s moduli normalized to the unperturbed Young’s modulus, $Y_{[110]}/Y_{[110]}^0$, in ${\mathrm{Ba}}_{1-x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ with $x=0.265$ (UD) and $x=0.401$ (OP), respectively [29]. All vertical error bars are either least-square fit errors of 1 s.d., or obtained through the propagation of fit errors.