Critical divergence of the symmetric ($A_{1g}$) nonlinear elastoresistance near the nematic transition in an iron-based superconductor

We report the observation of a nonlinear elastoresistivity response for the prototypical underdoped iron pnictide $\mathrm{Ba}{\left({\mathrm{Fe}}_{0.975}{\mathrm{Co}}_{0.025}\right)}_2{\mathrm{As}}_2$. Our measurements reveal a large quadratic term in the isotropic ($A_{1g}$) electronic response that was produced by a purely shear ($B_{2g}$) strain. The divergence of this quantity upon cooling towards the structural phase transition reflects the temperature dependence of the nematic susceptibility. This observation shows that nematic fluctuations play a significant role in determining even the isotropic properties of this family of compounds.

Figure 1

Representative data showing the resistivity response to strain of $\mathrm{Ba}{\left({\mathrm{Fe}}_{0.975}{\mathrm{Co}}_{0.025}\right)}_2{\mathrm{As}}_2$ at 116 K. The left-hand column (a) shows data for a crystal oriented with the crystal axes parallel to the normal strain frame [represented by the schematic pink-colored crystal in the inset to (a)(i)], such that the crystal experiences an admixture of $A_{1g}$ and $B_{1g}$ symmetry strain. The right-hand column (b) shows data for a crystal with the axes oriented at 45 deg to the normal strain frame [shown schematically by the blue crystal in the inset to (b)(i)], such that the crystal experiences an admixture of $A_{1g}$ and $B_{2g}$ symmetry strain. The top graph (i) in each column shows the resistive response of the sample along the $x^{\prime }$ and $y^{\prime }$ axes due to the strain, where the $x^{\prime }$ and $y^{\prime }$ axes are defined by the normal strain frame (inset). The zero antisymmetric strain condition is marked by a vertical line in (b). The middle graph (ii) shows the antisymmetric response, given by the difference $\frac{1}{2}[{(\mathrm{\Delta }\rho /{\rho }_0)}_{x^{\prime }{x}^{\prime }}-{(\mathrm{\Delta }\rho /{\rho }_0)}_{y^{\prime }{y}^{\prime }}]={\left(\frac{\mathrm{\Delta }\rho }{{\rho }_0}\right)}_{B_{1g}/B_{2g}}$. For both crystal orientations, the antisymmetric response is linear (black lines show linear fits). The bottom graph (iii) shows the symmetric ($A_{1g}$) response, given by the sum $\frac{1}{2}[{(\mathrm{\Delta }\rho /{\rho }_0)}_{x^{\prime }{x}^{\prime }}+{(\mathrm{\Delta }\rho /{\rho }_0)}_{y^{\prime }{y}^{\prime }}]={\left(\frac{\mathrm{\Delta }\rho }{{\rho }_0}\right)}_{A_{1g}}$. This response is found to be always linear for samples that experience $A_{1g}+B_{1g}$ symmetry strain (black line shows linear fit), while that of the samples that experience $A_{1g}+B_{2g}$ symmetry strain is clearly nonlinear and is fit by a second order polynomial (black line).

Figure 2

Temperature dependence of (a) the antisymmetric ($B_{2g}$) elastoresistivity response, and (b) the isotropic ($A_{1g}$) elastoresistivity response, of a single crystal of $\mathrm{Ba}{\left({\mathrm{Fe}}_{0.975}{\mathrm{Co}}_{0.025}\right)}_2{\mathrm{As}}_2$ oriented with the crystal axes at 45 deg to the normal strain frame (blue schematic insets). The anisotropic response is always linear, whereas the isotropic response shows a large quadratic component with a minimum close to the $B_{2g}$ neutral strain point. Both responses exhibit a strong temperature dependence. Note that the accessible strain range shifts with temperature, due in part to differences in the thermal expansion of the PZT and sample, and in part to the temperature dependence of the dynamic range of the PZT stack. For clarity, each fixed temperature strain sweep for the $A_{1g}$ response are offset by $-7.5\times {10}^{-4}$ per trace from the 100 K sweep. The data showing the $B_{2g}$ response are not offset.

Figure 3

Temperature dependence of the elastoresistance coefficients of $\mathrm{Ba}{\left({\mathrm{Fe}}_{0.975}{\mathrm{Co}}_{0.025}\right)}_2{\mathrm{As}}_2$ for all symmetry channels measured. Blue circles show the response for a sample that experiences an admixture of $A_{1g}+B_{2g}$ symmetry strain (blue schematic insets), while pink triangles show the response for a sample that experiences an admixture of $A_{1g}+B_{1g}$ symmetry strain (pink schematic insets). (a) The linear response to anisotropic strain, $m_{B_{2g}}^{B_{2g}}$ (left axis) and $m_{B_{1g}}^{B_{1g}}$ (right axis). $m_{B_{2g}}^{B_{2g}}$ can be well fit by a Curie-Weiss functional form (black line; see main text). (b) The linear response to isotropic strain $m_{A_{1g}}^{A_{1g}}$. For crystals that experience $A_{1g}+B_{1g}$ symmetry strain (pink), $m_{A_{1g}}^{A_{1g}}$ is extracted from a linear fit; for crystals that experience $A_{1g}+B_{2g}$ symmetry strain (blue), the data are extracted from the linear term of a second order polynomial fit. (c) The weighted quadratic coefficients $[{\left(\frac{1-{\nu }_P}{1+{\nu }_P}\right)}^2{m}_{A_{1g}}^{A_{1g},A_{1g}}+m_{A_{1g}}^{B_{2g},B_{2g}}]$ (blue data, left axis) and $[{\left(\frac{1-{\nu }_P}{1+{\nu }_P}\right)}^2{m}_{A_{1g}}^{A_{1g},A_{1g}}+m_{A_{1g}}^{B_{1g},B_{1g}}]$ (pink data, right axis) describing the isotropic response to ($A_{1g}+B_{2g}$) and ($A_{1g}+B_{1g}$) symmetry strains, respectively, extracted from the second order polynomial fit of the isotropic response as a function of anisotropic strain. The only measurably nonzero nonlinear coefficient is $m_{A_{1g}}^{B_{2g},B_{2g}}$, the isotropic response to $B_{2g}$ symmetry anisotropic strain. The temperature-dependence of this coefficient can be well fit by $\frac{a}{{(T-\mathrm{\Theta })}^2}+\frac{b}{T-\mathrm{\Theta }}+c$ (black line; see main text), with $\mathrm{\Theta }$ taken from the Curie-Weiss fit to $m_{B_{2g}}^{B_{2g}}$. Error bars represent 95% confidence intervals from statistical fits. If an error bar is not shown, the uncertainty of the fit is contained within the size of the data point.

Figure 4

Left: Photograph of a representative $\mathrm{Ba}{\left({\mathrm{Fe}}_{0.975}{\mathrm{Co}}_{0.025}\right)}_2{\mathrm{As}}_2$ sample prepared for an elastoresistance measurement using the modified Montgomery method. The sample is cut into a square, affixed with electrical contacts, and glued onto the PZT stack. This sample is aligned with its crystallographic axes rotated 45 deg with respect to the normal strain frame. Right: Schematic diagram showing the PZT stack prepared for an elastoresistance measurement, seen from the side. The sample is glued to the top face of the PZT. Strain is measured via a strain gauge glued to the back of the PZT stack.

Figure 5

Strain transmission through a large ($3140\mu \text{m}\times 3330\mu \text{m}\times 50\mu \text{m}$) ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ crystal under $A_{1g}+B_{2g}$ symmetry strains. A strain gauge is glued on top of the sample ($\mathrm{\Delta }{\epsilon }_{y^{\prime }{y}^{\prime }}^{\text{Samp}}$) and a second strain gauge is affixed directly to the back of the PZT stack ($\mathrm{\Delta }{\epsilon }_{y^{\prime }{y}^{\prime }}^{\text{PZT}}$). For this particular test, $\mathrm{\Delta }{\epsilon }_{x^{\prime }{x}^{\prime }}$ is estimated based on the measured Poisson ratio of the PZT, allowing for the estimation of the antisymmetric strains $\mathrm{\Delta }{\epsilon }_{B_{2g}}^{\text{PZT}}$ and $\mathrm{\Delta }{\epsilon }_{B_{2g}}^{\text{Samp}}$; $\frac{1}{2}(\mathrm{\Delta }{\epsilon }_{x^{\prime }{x}^{\prime }}^{\text{PZT}}-\mathrm{\Delta }{\epsilon }_{y^{\prime }{y}^{\prime }}^{\text{PZT}})$ and $\frac{1}{2}(\mathrm{\Delta }{\epsilon }_{x^{\prime }{x}^{\prime }}^{\text{Samp}}-\mathrm{\Delta }{\epsilon }_{y^{\prime }{y}^{\prime }}^{\text{Samp}})$, respectively. (a) The temperature dependence of the ratio of the range of antisymmetric strain experienced by the two strain gauges during fixed temperature voltage sweeps. Below 250 K the strain transmission through the samples is $\ge 80$% and only has a weak temperature dependence. (b) The extracted $m_{B_{2g}}^{B_{2g}}$ elastoresistivity response calculated from both strain gauges. The two traces are in good agreement below 250 K, indicating that the temperature dependence of the response is dominated by the intrinsic temperature dependence of the electronic sample properties over the temperature dependence of the strain transmission.

Figure 6

Comparison of sample size on the linear anisotropic elastoresistivity response $m_{B_{2g}}^{B_{2g}}$ and strain transmission in ${\mathrm{BaFe}}_2{\mathrm{As}}_2$. Three sample sizes are studied: small ($280\mu \text{m}\times 300\mu \text{m}\times 20\mu \text{m}$), medium ($3760\mu \text{m}\times 750\mu \text{m}\times 40\mu \text{m}$), and large ($3140\mu \text{m}\times 3330\mu \text{m}\times 50\mu \text{m}$). The strain at the surface of the large sample has been measured to be $\ge 80$% below 250 K (Fig. 5). For the data shown here the strain is measured by a strain gauge glued directly to the PZT stack and strain is assumed to be fully transmitted from the PZT to the sample. The top plot (a) shows $m_{B_{2g}}^{B_{2g}}$ data for all three samples. The elastoresistivity responses of the medium and large samples have the same magnitude and temperature dependence suggesting they are in a regime of similar strain transmission ($\ge 80$%). The small sample has a significantly smaller response. This is attributed to imperfect strain transmission in the smallest sample, resulting in the overestimation of the strain experienced by the sample. The bottom plot (b) shows the normalized elastoresistivity response for the small and large sample. The two curves exhibit the same temperature dependence, indicating that imperfect strain transmission results in a temperature independent scaling of the response.