Strain tuning of nematicity and superconductivity in single crystals of FeSe

Strain is a powerful experimental tool to explore new electronic states and understand unconventional superconductivity. Here, we investigate the effect of uniaxial strain on the nematic and superconducting phase of single crystal FeSe using magnetotransport measurements. We find that the resistivity response to the strain is strongly temperature dependent and it correlates with the sign change in the Hall coefficient being driven by scattering, coupling with the lattice and multiband phenomena. Band-structure calculations suggest that under strain the electron pockets develop a large in-plane anisotropy as compared with the hole pocket. Magnetotransport studies at low temperatures indicate that the mobility of the dominant carriers increases with tensile strain. Close to the critical temperature, all resistivity curves at constant strain cross in a single point, indicating a universal critical exponent linked to a strain-induced phase transition. Our results indicate that the superconducting state is enhanced under compressive strain and suppressed under tensile strain, in agreement with the trends observed in FeSe thin films and overdoped pnictides, whereas the nematic phase seems to be affected in the opposite way by the uniaxial strain. By comparing the enhanced superconductivity under strain of different systems, our results suggest that strain on its own cannot account for the enhanced high $T_{\mathrm{c}}$ superconductivity of FeSe systems.

Figure 1

(a) Transport measurement of the single crystal of FeSe with the current along the [110] tetragonal direction. (b) Relative changes in resistivity versus applied uniaxial strain at fixed temperatures around the structural transition $T_s$. (c) Resistivity measurements as a function of temperature around $T_s$ at different fixed amounts of uniaxial strain. The dotted line is measured before gluing the sample to the strain cell. (d) Single crystal of FeSe glued with epoxy and suspended between the two titanium plates of the strain cell. (e) Slope of the linear fit of normalized resistivity against uniaxial strain $d(\rho /{\rho }_{\varepsilon =0})/d\varepsilon$ for $\varepsilon \to 0$, indicated by the solid blue symbols. The grey diamonds are the values of $m_{66}$ normalized at 200 K and the dashed line is a fit to the Curie-Weiss law, after Ref. [6]. (f) Hall coefficient $R_H$, estimated as the low-field slope of the ${\rho }_{xy}$ versus $B$ below 1 T, under different constant uniaxial strain inside the nematic phase. Unstrained bulk measurements are indicated by the star symbols, after Ref. [26].

Figure 2

(a) Temperature dependence of the resistivity near the superconducting to normal transition under uniaxial strain. The superconducting state is clearly enhanced under compressive uniaxial strain. Opposite trends are found for tensile strain, which can be detected from the constant temperature strain loops in Fig. 7 in the Appendix. As the sample is glued to the strain cell, it will be exposed to an additional small tensile strain, ${\varepsilon }_{\mathrm{glue}}\sim 0.02\%$. (b) Scaling analysis of superconducting to normal resistivity near the crossing point, $T^{*}=9.8\mathrm{K}$, where ${\rho }^{*}$ is the resistivity at ${\rho }^{*}=\rho \left(T^{*}\right)$. The ratio between $\rho /{\rho }^{*}$ versus ${\left|\varepsilon \right|}^{1/\nu z}·|T-T^{*}|/T^{*}$ describes the universality of the transition with a critical exponent of $\nu z\approx 7$. (c) Variation of $T_{\text{c}}$, defined as the temperature with zero resistance, and of the width of the transition, $\mathrm{\Delta }{T}_{\text{c}}$, under applied strain. Solid lines are guides to the eye.

Figure 3

Superconducting phase diagrams versus strain of (a) bulk FeSe based on this study and (b) thin films of FeSe epitaxially growth on different substrates, adapted after Ref. [37]. Panel (b) contains the single crystal data (open symbols) from (a), which are adjusted to include the additional small tensile strain caused by the glue ${\varepsilon }_{\mathrm{glue}}\sim 0.02\%$. Compressive strain ($\varepsilon <0$) enhances superconductivity and seems to suppress the nematic state. Black data points in (b) refer to $T_s$ and $T_{\text{c}}$ of bulk FeSe. (c) The sensitivity of $T_{\text{c}}$ to strain, $\mathrm{\Delta }{T}_{\text{c}}/({\varepsilon }_{\text{max}}·T_c^{\varepsilon =0})$, for a variety of unconventional superconductors, adapted after Ref. [2] and compared with bulk, thin films of FeSe and monolayers of FeSe on ${\mathrm{SrTiO}}_3$.

Figure 4

(a) Freestanding single crystals of FeSe cut along the [110] tetragonal direction measured before being mounted on the strain cell in (b). (c) The sample model used to estimate the strain transmission in the sample inside the cell strain secured at both ends by epoxy.

Figure 5

(a) Transport measurements performed while cooling and warming, showing the small temperature variations are due to the thermal load of the strain cell. (b) Capacitance as a function of the displacement $d$ ($\mu \mathrm{m}$) of the two plates, calculated at room temperature (in black). Temperature dependence of the capacitance without any external strain applied (in red). (c) Resistance versus strain at 20 K, indicating the reproducibility and consistency between different measurements acquired at the same temperature. (d) Direct evidence that compressive strain enhances the superconducting state, showing the superconducting to normal transition induced by uniaxial compressive strain at fixed temperature $T=9.0\mathrm{K}$. (e) Transport measurements without applied strain at different stages of the experiment. All the values are normalized at room temperature $\rho$ ($T$) / $\rho$(RT). Inset shows the derivative of the resistivity against temperature of the same runs, indicating the smearing of the transition after gluing the sample to the strain cell as compared with the freestanding sample (run 0). (f) Derivative of resistivity against temperature around the nematic transition at different applied strains. The structure of the transition developed an additional feature after the gluing of the sample to the strain cell. Here, $T_s$ is defined as the position of the Gaussian fit of the middle peak.

Figure 6

(a) The longitudinal resistivity, ${\rho }_{xx}$, and (b) Hall component, ${\rho }_{xy}$, measured under different constant uniaxial strain applied along [110] axis for FeSe at constant temperatures inside the nematic phase. (c) Mobility spectrum at 30 K extracted from (a) and (b) for different values of uniaxial strain. (d) The position of the main peaks from the mobility spectrum for the main electron and hole in the range 20K-50K. For both carriers, tensile strain increases the mobility (in terms of absolute values) while compressive strain tends to decrease it. (e) Direct comparison between the Hall component ${\rho }_{xy}$for different strain at 30 K.

Figure 7

(a) Temperature dependence of the resistivity near the superconductive transition, measurements performed by varying the temperature while keeping constant the applied strain. Negative uniaxial strain (compressive) increases the resistivity while positive uniaxial strain (tensile) has an opposite effect. The superconductive state is slightly enhanced under compressive strain and suppressed under tensile strain. (b) Scaling analysis of (a) near the crossing point at $T^{*}=9.8\mathrm{K}$, where ${\rho }^{*}$ is the resistivity at the critical point ${\rho }^{*}=\rho (T=T^{*})$. The ratio between $\rho /{\rho }^{*}$ versus the functional relation ${\left|\varepsilon \right|}^{1/\nu z}|T-T^{*}|/T^{*}$ is able to describe the universality of the transition with a critical exponent of $\nu z\approx 7$. (c) Width of the dispersion of the scaling analysis at $\rho /{\rho }^{*}=0.5$ as a function of the critical exponent of $\nu z$, for measurements conducted at fixed applied strain [from (b)]. (d) Resistivity near the superconductive transition, with different amounts of applied uniaxial strain. The measurements conducted at constant temperature to confirm findings from previous measurements performed at constant strain. (e) Scaling analysis of (d) near the crossing point at $T^{*}=9.63\mathrm{K}$, from the data collected at constant temperature with an exponent of $\nu z\approx 7$, similar to that illustrated in (b). (f) The first derivative of the data in (a), showing the evolution of width of the superconducting transition with uniaxial strain.

Figure 8

Calculated Fermi surface plots and slices through the center of the Brillouin zone ($\mathrm{\Gamma }\text{−}\mathrm{M}$ plane) for FeSe for different applied strain $\varepsilon$ along [110] of $\varepsilon =-1.5\%$ in (a), (d), $\varepsilon =0\%$ in (b), (e), and $\varepsilon =1.5\%$ in (c), (f). 3D Fermi surface plots are colored by the magnitude of the Fermi velocity. Color bars for each row are displayed on the far right. Panels (a)–(c) show the Fermi surface with increasing strain as calculated in Wien2K using the GGA approximation (PBE functional) [44, 54], including the spin-orbit coupling. We use the experimental orthorhombic parameters ($Cmma$ symmetry and $a=5.308\AA$, $b=5.334\AA$, $c=5.486\AA$ and the volume cell of the unit cell is conserved under applied uniaxial strain. Panels (d)–(f) show the Fermi surface after band renormalization and shifting are applied to match the unstrained experimental ARPES data at high temperatures above $T_s$ [6]. Band renormalization of 3 and 4 are applied to the outer hole band at $\mathrm{\Gamma }$ and both electron bands at M, respectively, and a band shift of 45 meV is applied to the electron bands and $-45$ meV is applied to the hole band to ensure charge compensation, as suggested by previous experimental ARPES data [6]. (g) In-plane and (h) out-of-plane anisotropy of the penetration depth. (i) The number of charge carriers per unit cell for each quasi-two-dimensional cylinder of each shifted Fermi surface pocket as a function of strain. The solid lines are guides to the eye.