Nematic Susceptibility of Hole-Doped and Electron-Doped ${\mathrm{Ba}\mathrm{F}\mathrm{e}}_2{\mathrm{As}}_2$ Iron-Based Superconductors from Shear Modulus Measurements

The nematic susceptibility, ${\chi }_{\phi }$, of hole-doped ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ and electron-doped $\mathrm{Ba}{({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x)}_2{\mathrm{As}}_2$ iron-based superconductors is obtained from measurements of the elastic shear modulus using a three-point bending setup in a capacitance dilatometer. Nematic fluctuations, although weakened by doping, extend over the whole superconducting dome in both systems, suggesting their close tie to superconductivity. Evidence for quantum critical behavior of ${\chi }_{\phi }$ is, surprisingly, only found for $\mathrm{Ba}{({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x)}_2{\mathrm{As}}_2$ and not for ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$—the system with the higher maximal $T_c$ value.

Figure 1

(a) Schematic drawing of our three-point-bending setup in the capacitance dilatometer. The sample (gray) is supported by three wires (red) and the capacitor gap is indicated by small arrows. (b) Definition of sample dimensions and orientation relative to the wire supports. Young’s modulus $Y_{[110]}$ of (c) Co-Ba122 and (d) K-Ba122, normalized at room temperature. For comparison, the temperature dependence of the $C_{66}$ mode of pure ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ from Ref. [21] (gray circles) is shown in (c). The inset in (c) shows a fit of the data of the 33% Co-Ba122 sample to the Varshni formula used as the phonon background.

Figure 2

Nematic susceptibility, ${\chi }_{\phi }$ in units of ${\lambda }^2/C_{66,0}$, obtained from the critical part of the Young’s modulus of (a) Co-Ba122 and (b) K-Ba122. Dashed lines correspond to a Curie-Weiss fit using Eq. (3). Insets show the parameter $c_0/Y_{[110]}(293\mathrm{K})$ used for separating the critical part of $Y_{[110]}$ from the background (see text for details). Triangles mark the inflection point of ${\chi }_{\phi }(T)$, $T^{*}$, which is a lower temperature limit for its Curie-Weiss-like divergence. (c), (d) Temperature-dependent Curie-Weiss temperature $T_0(T)=1-1/A{\chi }_{\phi }$ of the nematic susceptibility for (c) Co-Ba122 and (d) K-Ba122. Linear extrapolations beyond the structural or superconducting transition are shown as thin black lines.

Figure 3

(a) Temperature and doping dependence of the magnitude of the normalized nematic susceptibility ${\lambda }^2{\chi }_{\phi }/C_{66,0}$ for Co-Ba122 and K-Ba122 from Fig. 2. (b) $T_c/T_{c,\max }$ as a function of the maximum value of $({\lambda }^2{\chi }_{\phi }/C_{66,0})(T)$ for all overdoped samples. (c) Phase diagram showing the doping dependence of $T^{*}$, $T_s^{\mathrm{CW}}$ and $T_c$. The red dashed area corresponds to the “critical” region above $T^{*}$ where ${\chi }_{\phi }(T)$ diverges. (d) Doping dependence of the electron-lattice coupling energy $T_s^{\mathrm{CW}}-T_0={\lambda }^2/A{C}_{66,0}$. (e) Curie-Weiss temperature $T_0$ at various temperatures obtained from Fig. 2 2. An abrupt change from Curie-Weiss to non-Curie-Weiss behavior occurs in the shaded region. Lines are a guide to the eye.