Relationship between Transport Anisotropy and Nematicity in FeSe

The mechanism behind the nematicity of FeSe is not known. Through elastoresitivity measurements it has been shown to be an electronic instability. However, thus far measurements have extended only to small strains, where the response is linear. Here, we apply large elastic strains to FeSe and perform two types of measurement. (1) Using applied strain to control twinning, the nematic resistive anisotropy at temperatures below the nematic transition temperature $T_s$ is determined. (2) Resistive anisotropy is measured as nematicity is induced through applied strain at fixed temperature above $T_s$. In both cases, as nematicity strengthens, the resistive anisotropy peaks at about 7%, then decreases. Below $\approx 40\mathrm{K}$, the nematic resistive anisotropy changes sign. We discuss possible implications of this behavior for theories of nematicity. In addition, we report the following. (1) Under experimentally accessible conditions with bulk crystals, stress, rather than strain, is the conjugate field to the nematicity of FeSe. (2) At low temperatures the twin boundary resistance is $\sim 10\%$ of the sample resistance, and must be properly subtracted to extract intrinsic resistivities. (3) Biaxial in-plane compression increases both in-plane resistivity and the superconducting critical temperature $T_c$, consistent with a strong role of the $yz$ orbital in the electronic correlations.

Popular Summary

At room temperature, the electrical resistivity in iron selenide (FeSe) is exactly the same in the $x$ and $y$ directions. But at 90 K, there is a transition: Electrons conduct better along one direction than the other. This change is known as electronic nematicity. It is usually, but does not have to be, accompanied by a lattice distortion, and it may play a key role in high-temperature superconductivity and related phenomena. However, the physical processes that cause and accompany this transition remain unclear. Here, we shed light on those processes as we document changes in a sample of FeSe upon inducing electronic nematicity through the application of large strains.

We find that resistive anisotropy grows rapidly upon the onset of nematicity, but then its behavior gets more complicated. As nematicity grows—as measured by, for example, the amount of lattice distortion—resistive anisotropy first increases rapidly, but then goes into reverse and actually shrinks.

A strong component of the resistivity is magnetic fluctuations: In a microscopic patch of the crystal, a little bit of magnetism appears for a tiny fraction of a second, and this is enough to scatter electrons and cause resistance. A prominent class of theories predicts that these magnetic fluctuations also drive the nematicity. Our work shows that this is unlikely. Since the resistive anisotropy grows and shrinks as nematicity appears, the anisotropy of the magnetic fluctuations probably also grows and shrinks, while the nematicity continues to grow.

Our observations therefore provide key information on how the electronic structure of FeSe changes as nematicity develops.

Figure 1

(a) Schematic illustration of the electronic structure above and below the nematic transition temperature $T_s$. Fermi surfaces are colored by their dominant orbital content. For $T<T_s$, $k_x$ is oriented along the crystalline $a$ axis, where $a>b$. (b) Piezoelectric uniaxial stress apparatus with a platform. (c) Photograph of sample B, with contacts attached for measuring resistivity along the sample’s length. The lattice directions are indicated. Sample A was prepared similarly, though with its crystal axes rotated by 45°. (d) Scanning electron (SEM) micrograph of a cut, made with a focused ion beam, through sample B and the epoxy layer beneath it. (e) Photograph and (f) SEM micrograph of sample C, which was prepared in a Montgomery configuration.

Figure 2

The effect of transverse strain. (a) ${\rho }_{110}(T)$, the resistivity along the [110] direction, for sample A at various applied strains ${ϵ}_{110}$. The principal axes of the nematicity in FeSe are the $⟨100⟩$ axes, so this strain is a transverse field to the nematicity. Inset: schematic of the strain axis. (b) $T_s$ versus ${ϵ}_{110}$ for this sample. $T_s$ is identified as the maximum in $d^2{\rho }_{110}/d{T}^2$. The shaded region is a measure of the width of the transition: it is where $d^2\rho /d{T}^2$ exceeds half its maximum value. The line is a fit.

Figure 3

Schematic phase diagrams. (a) Schematic stress-temperature phase diagram for the nematicity of FeSe, for stress applied with $B_{1g}$ principal axes; the first-order transition is where the direction of the nematicity flips. (b) The corresponding strain-temperature phase diagram. In the indicated region, the lattice is unstable and breaks up into twins where, locally, ${ϵ}_{B1g}=\pm {ϵ}_s(T)$.

Figure 4

Elastoresistivity near $T_s$. (a) ${\rho }_{100}({ϵ}_{B1g})$, where ${\rho }_{100}$ is the resistivity along the [100] direction and ${ϵ}_{B1g}\equiv ({ϵ}_{100}-{ϵ}_{010})/2$, of sample B at various temperatures near $T_s$. (b) $d{\rho }_{100}/d{ϵ}_{B1g}$ for the curves from (a). For $T<T_s$, $d{\rho }_{100}/d{ϵ}_{B1g}$ becomes nearly constant over the range where the sample twins. This range is indicated for the 86.8 K curve. (c) Schematic of ${\rho }_{100}({ϵ}_{B1g})$ for $T<T_s$; the underlying curve is not accessible for $-{ϵ}_s<{ϵ}_{B1g}<+{ϵ}_s$ due to the onset of twinning, and the observed resistivity instead interpolates over this range. (d) Temperature ramps at three values of ${ϵ}_{B1g}$. (e) $T_s$ versus strain for low strains. The shaded regions indicate the transition width, defined by $d^2{\rho }_{100}/d{T}^2$ crossing half its maximum value.

Figure 5

(a) ${\rho }_{100}$ of sample B over a wide temperature and strain range. Points are data from temperature ramps at constant ${ϵ}_{B1g}$. The positions of points at ${\epsilon }_{B1g}<-0.11·10^{-2}$ have been adjusted along the ${\epsilon }_{B1g}$ axis to correct for plastic deformation of the platform; see Appendix pp1-s4 for details. The lines are strain ramps at constant temperature. ${ϵ}_s(T)$, taken as the data of Ref. [40] scaled in $T$ and $ϵ$ to match the data here, is indicated at each temperature. (b) ${\rho }_{100}(T)$ for ${ϵ}_{B1g}=-0.25\times {10}^{-2}$, where the sample is fully detwinned at all $T$. (c) Close-up of the data in (a) at 14.6 and 36.9 K. The squares mark points whose position along the ${ϵ}_{B1g}$ axis was corrected.

Figure 6

Data from sample C, the Montgomery configuration sample. (a) ${\rho }_{100}$ (left) and ${\rho }_{010}$ (right), from strain ramps at various fixed temperatures. The hysteresis is shown for two temperatures. The vertical ticks mark $-{ϵ}_s(T)$, taken from Ref. [40] and scaled in temperature to match the $T_s$ observed here. (b) ${\rho }_{B1g}\equiv ({\rho }_{100}-{\rho }_{010})/2$, derived from the data in (a), at temperatures above $T_s$. Note that by symmetry ${\rho }_{B1g}(T>T_s)=0$ at ${ϵ}_{B1g}=0$, but measurement error gives a small deviation from this. (c) ${\rho }_{A1g}\equiv ({\rho }_{100}+{\rho }_{010})/2$.

Figure 7

Effect of biaxial strain. (a) $A_{1g}$ elastoresistivity $(1/{\rho }_{A1g})d{\rho }_{A1g}/d{ϵ}_{A1g}$ versus $T$ of sample C, determined as explained in the text. (b) $T_c$ versus strain, determined as the temperature where the resistivity crosses specific values, as shown in the inset. Note that within the twinned region, ${ϵ}_{B1g}$ does not couple locally to the sample, and instead the effect on $T_c$ is through the applied $A_{1g}$ component of the strain. When the platform deformation is elastic, this is ${ϵ}_{A1g}=0.52{ϵ}_{B1g}$. The observed slope therefore corresponds to $d{T}_c/d{ϵ}_{A1g}=-450\mathrm{K}$.

Figure 8

Nematic resistive anisotropy. (a) The spontaneous resistive anisotropy below $T_s$, obtained from $T$-ramp data as described in the text. The inset shows ${\rho }_a$ and ${\rho }_b$ near $T_s$. (b) $B_{1g}$ elastoresistivity, $(1/{\rho }_{A1g})d{\rho }_{B1g}/d{ϵ}_{B1g}$, obtained from both long strain ramps and from a small oscillating strain. For sample B, ${\rho }_{010}$ was not measured, so the [100] elastoresistivity $(1/{\rho }_{100})\times d{\rho }_{100}/d{ϵ}_{100}$ is plotted instead. Fits are to a Curie-Weiss form; see the text. (c) Resistivity anisotropy of sample C against strain at $T\approx T_s$. The curve has been shifted vertically to set ${\rho }_{B1g}=0$ at ${ϵ}_{B1g}=0$, canceling a small geometrical error in the measurement. Also shown is an estimate of the strain-induced nematicity $\psi$, taken as the $xz\text{−}yz$ energy splitting at the $X$ point, obtained from evaluation of Ginzburg-Landau parameters.

Figure 9

Plastic deformation of the platform. (a) ${\rho }_{100}$ of sample B versus displacement $D$ applied to the platform. The datasets were taken in the following order. (1) Strain ramps at fixed temperature. (2) Temperature ramps at fixed strain. (3) Strain ramps at higher compression. The offset between datasets (1) and (3) is due to plastic deformation of the platform that occurred over the course of the temperature ramps. (b) When data from set 3 are offset along the $D$ axis, the match with dataset 1 is excellent. (c) Schematic illustration of the process of plastic deformation. (d) Low-temperature resistivity measured before and after the platform plastic deformation. To compare datasets where $T_c$ was the same, the before data are taken at $D=-1.6\mu \mathrm{m}$ and the after data at $D=0.8\mu \mathrm{m}$.

Figure 10

Annealing twin boundaries out of the sample by ramping the applied strain.

Figure 11

(a),(b) Temperature-ramp data from sample C. Panel (a) shows ${\rho }_{100}$ and (b) ${\rho }_{010}$. (c) Change in slope $d\rho /d{ϵ}_{B1g}$ across ${ϵ}_{B1g}=0$. This quantity is proportional to the twin boundary contribution to sample resistivity at ${ϵ}_{B1g}=0$.

Figure 12

${\rho }_{110}$ versus $T$ of sample A over a wide temperature range.

Figure 13

Nematic resistivity anisotropy $({\rho }_a-{\rho }_b)/({\rho }_a+{\rho }_b)$ of sample C, derived from the strain-ramp data shown in Fig. 6. This determination is based on extraction of equilibrium slopes $d\rho /d{ϵ}_{B1g}$ at ${ϵ}_{B1g}=0$, obtained by averaging the observed increasing-$ϵ$ and decreasing-$ϵ$ slopes at ${ϵ}_{B1g}=0$, as shown in the inset.