Growth of nematic susceptibility in the field-induced normal state of an iron-based superconductor revealed by elastoresistivity measurements in a 65 T pulsed magnet

In iron-based superconductors, both nematic and magnetic fluctuations are expected to enhance superconductivity and may originate from a quantum critical point hidden beneath the superconducting dome. The behavior of the nonsuperconducting state can be an important piece of the puzzle, motivating, in this paper, the use of high magnetic fields to suppress superconductivity and measure the nematic susceptibility of the normal state at low temperatures. We describe experimental advances which make it possible to measure a resistive gauge factor (which is a proxy for the nematic susceptibility) in the field-induced normal state in a 65 T pulsed magnet, and report measurements of the gauge factor of a micromachined single crystal of $\mathrm{Ba}{\left({\mathrm{Fe}}_{0.926}{\mathrm{Co}}_{0.074}\right)}_2{\mathrm{As}}_2$ at temperatures down to 1.2 K. The nematic susceptibility increases monotonically in the field-induced normal state as the temperature decreases, consistent with the presence of a quantum critical point nearby in composition.

Figure 1

Schematic diagram of the experiment. Panel (a) shows an optical microscope image of the sample after FIB micromachining. The sample is adhered to the surface of the PZT stack, and the four contacts on the top and bottom of the image are embedded in epoxy. Panel (b) shows the setup for the resistance measurement. The excitation signal is generated by the Red Pitaya STEMlab 125-14 device and the sample voltage is detected by the same device at a digitization rate of 125 MHz. Panel (c) shows the circuit used to drive the piezoelectric device. The wave-form generator is synchronized to produce 18 cycles of a sine wave starting 6 ms after the beginning of the pulse. Panel (d) shows the optical path for the fiber Bragg grating (FBG) strain sensor. A continuous wave, broad-spectrum LED light source illuminates the grating, and the grating only reflects in a narrow band of wavelengths. A spectrometer and line camera records the spectra during the pulse at a frequency of 46.5 kHz.

Figure 2

A representative elastoresistivity measurement of $\mathrm{Ba}{\left({\mathrm{Fe}}_{0.926}{\mathrm{Co}}_{0.074}\right)}_2{\mathrm{As}}_2$ during a magnetic field pulse. This trace was taken at 3 K, well below the zero-field superconducting $T_c$, and superconductivity was fully suppressed with a 65 T pulse. (a) The field (blue line) and amplitude of the sample response (black line) during the magnetic field pulse. The magnet is fired at $t=0$ ms. When the field surpasses $H_{c2}\approx 50T$, the sample becomes resistive. The PZT stack is energized with a 3 kHz drive for 6 ms around peak field (gray shaded region). Inset: Magnification of the sample response at peak field. (b) The sample response (black line) after subtraction of the magnetoresistance background while the PZT stack is driven. The resistance of the sample oscillates in time due to the elastoresistivity response. This is fit with a 3 kHz sine wave (red line). (c) The strain (black line) measured by the FBG grating while the PZT stack is driven. This is also fit by a 3 kHz sine wave (red line).

Figure 3

The extracted resistive gauge factor of a micromachined sample of $\mathrm{Ba}{\left({\mathrm{Fe}}_{0.926}{\mathrm{Co}}_{0.074}\right)}_2{\mathrm{As}}_2$ as a function of temperature and field. Data taken during a pulse is represented by filled symbols and zero-field data taken using the continuous AC elastoresistivity technique [16] is represented by a solid black line. Error bars represent one standard deviation, as calculated in detail in Appendix pp4. Measurements at all field values produce the same result within error bars, and match with the zero-field data at temperatures above $T_c=24.8\mathrm{K}$. Below the zero-field superconducting transition (vertical black bar) the gauge factor (and therefore the nematic susceptibility) continues to increase smoothly with decreasing temperature, indicating an increasing importance of nematic fluctuations. Note that we use a rotated coordinate basis (inset) in which the cartesian $x$ and $y$ axes are rotated by ${45}^{\circ }$ with respect to the in-plane crystal axes.

Figure 4

Linear extrapolation of the zero-field resistivity down to cold temperatures. The deviation between the measured resistance at 65 T and the extrapolated value at the same temperature is largest at the coldest temperatures and highest fields, at 12%. The shaded region is the region over which the linear fit was performed.

Figure 5

Comparison of the extracted gauge factor $G=(-\mathrm{\Delta }R/R_0)/{\varepsilon }_{xx}$ using $R_0=R_0\left(H\right)$ and $R_0=R_0(H=0)$. The two differ only quantitatively and the choice of normalization does not change the conclusions we draw from the data.

Figure 6

$H_{c2}$ as a function of temperature, measured with increasing field before the strain cycles (upward triangles) and decreasing field after the strain cycles (downward triangles). To check for Joule heating within the sample in the resistive phase, several measurements (open symbols) were taken without driving the PZT and with the sample current decreased by a factor of 10. We observe an increase in $H_{c2}$ when the sample is submerged in liquid helium. Inset: Normalized resistance vs field for two 60 T pulses at 7 K. $H_{c2}$ is extracted from the intersection of the two straight line fits. Dotted lines corresponds to the data taken without energizing the PZT and using 10% of the sample excitation current.

Figure 7

Strain per volt measured on a Pst150 2x3x5 PZT stack along both the $x$ and $z$ directions, both perpendicular to the poling axis $y$. The sample is primarily sensitive to the $x$ direction, while the FBG detected strain along the $z$ axis, parallel to the field. The intrinsic response of the PZT stack itself is expected to be identical in both $x$ and $z$ directions, with an approximately 10% decrease in the measured ${\varepsilon }_{zz}$, likely due to the placement of the electrodes along the $yz$ faces of the stack.

Figure 8

Standard deviation of ${\varepsilon }_{xx}$ from sets of pulses taken at each temperature.

Figure 9

Standard deviation of $\mathrm{\Delta }R$ from zero field pulses taken above the superconducting transition. The standard deviation is roughly temperature independent.

Figure 10

Reported values of $m_{B2g}$ and the gauge factor normalized to the value at 50 K (chosen arbitrarily) for $\mathrm{Ba}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ from three different studies: Chu 2012 [11], Kuo 2016 [12], and this paper. Relative to the scale factor used for this paper, data from Kuo and Chu require scaling factors of 1.91 and 0.83, respectively.

Figure 11

The same data presented in Fig. 3, replotted as its inverse for several values of the temperature-independent offset term $G_0$. If a single set of parameters could describe all of our data with a Curie-Weiss fit, the data would appear linear. Even the largest $G_0$ presented here, which is unphysically large for $\mathrm{Ba}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$, is unable to reconcile both the high- and low-temperature behaviors with Curie-Weiss scaling.