Strain-tuning of nematicity and superconductivity in single crystals of FeSe
Michele Ghini, Matthew Bristow, Joseph C. A. Prentice, Samuel Sutherland, Samuele Sanna, Amir A. Haghighirad, Amalia I. Coldea
SStrain-tuning of nematicity and superconductivity in single crystals of FeSe
Michele Ghini,
1, 2, ∗ Matthew Bristow, Joseph C. A. Prentice, SamuelSutherland, Samuele Sanna, A. A. Haghighirad,
1, 4 and A. I. Coldea † Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK Department of Physics and Astronomy ”A. Righi”, University of Bologna, via Berti Pichat 6-2, I-40127 Bologna, Italy Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom Institute for Quantum Materials and Technologies (IQMT),Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany (Dated: February 25, 2021)Strain is a powerful experimental tool to explore new electronic states and understand unconventional super-conductivity. Here, we investigate the effect of uniaxial strain on the nematic and superconducting phase ofsingle crystal FeSe using magnetotransport measurements. We find that the resistivity response to the strain isstrongly temperature dependent and it correlates with the sign change in the Hall coefficient being driven byscattering, coupling with the lattice and multiband phenomena. Band structure calculations suggest that understrain the electron pockets develop a large in-plane anisotropy as compared with the hole pocket. Magneto-transport studies at low temperatures indicate that the mobility of the dominant carriers increases with tensilestrain. Close to the critical temperature, all resistivity curves at constant strain cross in a single point, indicatinga universal critical exponent linked to a strain-induced phase transition. Our results indicate that the supercon-ducting state is enhanced under compressive strain and suppressed under tensile strain, in agreement with thetrends observed in FeSe thin films and overdoped pnictides, whereas the nematic phase seems to be affected inthe opposite way by the uniaxial strain. By comparing the enhanced superconductivity under strain of differentsystems, our results suggest that strain on its own cannot account for the enhanced high T c superconductivity ofFeSe systems. I. INTRODUCTION
Uniaxial strain can considerably alter unconventionalsuperconductivity or nematic and magnetic phases in iron-based superconductors , demonstrating that it is a power-ful tool to induce phase transitions and explore the inter-play of different competing phases with superconductivity.A nematic phase is an electronic state of matter in whichthe electronic structure develops strong in-plane anisotropy intransport properties, breaking the rotational symmetry of thetetragonal lattice. Strain is also used as a small perturbation toidentify nematic electronic phases via diverging nematic sus-ceptibility to in-plane anisotropic strain in various iron-basedsuperconductors . Strain-induced phase transitions can beidentified via resistivity scaling nematic critical points andsuperconductor-insulator quantum phase transitions .FeSe is a unique iron-based superconductor, which despiteits simple structure hosts a nematic electronic phase, in the ab-sence of a long-range magnetic order. This unusual electronicphase is driven by orbitally-dependent effects and correla-tions that are responsible for unusual momentum dependentband shifts . The superconductivity emerging from thisnematic phase has a two-fold symmetric superconducting gap,orbitally-selective pairing and a spin-orbital-intertwinednematic state . The nematic electronic phase of FeSe ishighly sensitive to external parameters, being strongly sup-pressed by the isoelectronic substitution with sulphur andexternal hydrostatic pressure, but for higher applied pressureof ∼ GPa a robust superconducting phase with a T c ∼ Kis stabilized, which competes with a spin-density wave .A monolayer of FeSe, on a suitable substrate, can sustainsuperconductivity in excess of 65 K, driven by a strong inter-facial electron-phonon coupling, the charge transfer through the interface, and strain effects . This remarkable super-conducting state is drastically reduced as the number of lay-ers increases and it is highly dependent on the annealing pro-cesses being reduced to close to 20 K for 50 unit cells .Superconductivity is enhanced by anisotropic compression inthin films of FeSe on CaF , but is suppressed and resistivityincreases for films thinner than 100 nm , similar to exfoliatedflakes of FeSe in the absence of a substrate . Thus, uniaxialstrain studies can help to isolate and decouple the differentessential components to enhance superconductivity and pro-vide important insight in understanding its interplay with theelectronic nematic phase of FeSe.In this work, we explore how the electronic behaviour ofbulk single crystals FeSe is affected by uniaxial strain, usingmagneto-transport measurements outside and inside the ne-matic phase. We find that uniaxial strain induces significantchanges in resistivity and its gradient is highly temperaturedependent being closely correlated with the Hall coefficientinside the nematic phase. The resistivity curves cross in a sin-gle point in the vicinity of the normal to the superconductingtransition and we determine a strain-dependent scaling andits critical exponent. Our results under strain in single crys-tals of FeSe are consistent with those found for epitaxiallygrown FeSe films on different substrates. These results in-dicate that the superconducting state is enhanced under com-pressive strain and suppressed under tensile strain. II. EXPERIMENTAL DETAILS
FeSe single crystal were grown by the chemical vapourtransport method . Electrical connections were made us-ing indium soldering in a 5 point-contact configuration for a r X i v : . [ c ond - m a t . s up r- c on ] F e b T s T c (cid:1) xx (m W cm) T ( K ) [ 1 1 0 ] (cid:1) [ 1 1 0 ] > 0 D (cid:3) xx / (cid:3) xx (%) (cid:1) [ 1 1 0 ] ( % ) a b cd e f - 8 0- 6 0- 4 0- 2 002 04 06 0 m66 T ' t e t r a g o n a l n e m a t i c T s d( (cid:3) / (cid:3) ) / d (cid:2) (cid:1) | (cid:2) (cid:1) =0 T ( K ) b e f o r e g l u i n g (cid:3) (m W cm) T ( K ) (cid:1) = - 0 . 2 6 % (cid:1) = - 0 . 1 4 % (cid:1) = - 0 . 0 9 % (cid:1) = - 0 . 0 1 % (cid:1) = + 0 . 0 7 % (cid:1) = + 0 . 1 1 % T s T ' (cid:1) = 0 % ( S 3 ) (cid:1) = - 0 . 2 1 % ( S 1 ) (cid:1) = - 0 . 0 5 % ( S 1 ) (cid:1) = + 0 . 0 9 % ( S 1 ) m H < 1 T RH (cid:1) (10-9 m3/C) T ( K ) FIG. 1. (a)
Transport measurement of the single crystal of FeSe with the current along the [110] tetragonal direction. (b)
Relative changes inresistivity versus applied uniaxial strain at fixed temperatures around the structural transition T s . (c) Resistivity measurements as a functionof temperature around T s at different fixed amount 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 ofnormalized resistivity against uniaxial strain d ( ρ/ρ ε =0 ) /dε for ε → , indicated by the solid blue symbols. The grey diamonds are the valuesof m 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-fieldslope of the ρ xy versus B = µ H below 1 T, measured under different constant uniaxial strain inside the nematic phase. Unstrained bulkmeasurements are indicated by the star symbols, after Ref. 26. magneto-transport measurements. The current flows paral-lel to the direction of the applied stress, which is the [110]direction (Fe-Fe bonds) in the tetragonal symmetry and cor-responds to the B g symmetry channel . Strain experimentswere performed using a CS100 cell from Razorbill . The barshaped single crystal is suspended freely between two mount-ing plates and glued using a two-part epoxy, different fromstudies in which the sample is glued first to a thin titaniumplate, which is itself strained and allows large strain to be ap-plied . As the material is very soft, the glue itself can alsoapply a small tensile strain to the sample ( ε glue ∼ . )(Fig. 1(d)), as found in previous NMR studies . The capaci-tance between the two plates provides a direct estimate of theapplied displacement ( µ m). The amount of nominal stress ap-plied, ε = ∆ L/L was of the order of . - . % and theactual lattice distortion was calculated by finite element sim-ulations, as shown in the Appendix. III. RESULTS AND DISCUSSIONA. Resistivity under strain inside the nematic phase
Fig. 1 shows the effect of the applied strain along the [110]tetragonal direction on the transport behaviour of a singlecrystal of FeSe. In the absence of strain, FeSe enters the ne-matic phase below T s ∼
87 K and it becomes superconductingat T c ∼ RRR ∼ ) (Fig. 1(a)), consistent with previous studies .We performed detailed transport measurements as a functionof uniaxial strain at fixed temperatures, as the strain cell is ca-pable of applying large and tunable uniaxial stress at low tem-peratures. Fig. 1(b) shows the relative variation in resistivity, ∆ ρ xx /ρ xx (0) = [ ρ xx ( ε ) − ρ xx (0)] /ρ xx (0) , at fixed tempera-tures inside and outside the nematic phase and in the vicinityof T c . At high temperatures in the tetragonal phase, tensilestrain (positive strain) increases the resistivity of the systemwhile compressive strain (negative strain) decreases it. Onthe other hand, inside the nematic state, the response to strainchanges significantly and becomes strongly non-linear and theslope changes sign compared with the tetragonal phase.Fig. 1(e) shows the temperature dependence of the slope ofthe normalized resistance as a function of uniaxial strain in thelimit of small strain ( S = d ( ρ/ρ ε =0 ) /dε , ε → ), extractedfrom the data shown in Fig. 1(b). In this low-strain regime, thetemperature dependence of the slope follows closely the diver-gent behaviour of the nematic susceptibility, 2 m , (measuredusing piezostacks) as approaching T s . 2 m provides a di-rect measure of the electronic nematic order parameter and itstemperature dependence has a Curie-Weiss behaviour, as re-ported previously in Ref. 6 (see diamond symbols in Fig. 1e)and Refs. 9 and 14. Inside the nematic phase, the effect ofapplied strain is unusual and the slope S changes sign as afunction of temperature. We identify a characteristic temper-ature, T (cid:48) ∼ K, as the temperature at which the strain hasthe weakest effect on resistivity and the slope changes sign, ascompared with the high temperature regime. This behaviour isconsistent with the sign change of m below 65 K as wellas the change in resistivity anisotropy induced by the strain ofa PEEK substrate . Remarkably, the change in anisotropyat T (cid:48) coincides also with the temperature at which a largeanisotropy develops in the local spin susceptibility, as detectedfrom the line splitting of the Knight shift . Thus, the changesin the anisotropy of the local magnetism are likely to affectthe scattering and the coherent coupling between local spinsand itinerant electrons.To further address this, we look at magneto-transport mea-surements inside the nematic phase (raw data shown inFig. S3. To describe the non-linear effects in magnetic fieldof ρ xx and ρ xy , a three-band model was employed to ac-count for a small electron-like pocket, besides almost compen-sated hole and electron pockets . In the presence of the ap-plied strain, the overall magnetotransport behaviour does notchange significantly and still requires a multi-band model toexplain these features. These findings are in contrast to spec-troscopic surface-sensitive studies under strain that involve thepresence of a single (uncompensated) electron peanut shapepocket . The mobility spectra for bulk FeSe suggest thatunder tensile strain the mobility of the dominant carriers in-crease, as shown in Fig. S3(c) and (d), in agreement withmobilities of thin films of FeSe under strain . Furthermore,the Hall coefficient of bulk FeSe, R H = ρ xy /B ( B < T)shown in Fig. 1(f), becomes negative below 65 K under differ-ent uniaxial strain, similar to the unstrained case , suggestingsignificant changes in scattering below T (cid:48) . However, in thinfilms that have a higher degree of disorder, the Hall coeffi-cient is always positive as disorder hinders and averages outthe effects responsible for the negative Hall coefficient .An extremum of the Hall coefficient has been assigned to amaximum in the scattering anisotropy by spin fluctuations or driven by the currents renormalized by vertex correctionsdominated by the majority carriers . Coupling with the lat-tice of these fluctuations may have additional consequencessuch as the suppression of superconductivity and of the ne-matic critical fluctuations .To further understand the changes of the electronic struc-ture under strain, we have calculated the Fermi surface ofFeSe. Fig. S5 shows the evolution of the Fermi surfacewith strain and for the renormalized and shifted band struc-ture, which was brought in agreement with high temperatureARPES data . Interestingly, the DFT calculations suggestthat the in-plane anisotropy is larger for the electron bandswhereas out-of-plane anisotropy changes for all pockets withincreasing strain (Fig. S5). The sizes and the number of chargecarriers for all three pockets shrink with increasing strain sug-gesting a potential increase in the Hall coefficient dominatedby the most mobile carriers, as illustrated in Fig. 1(f) andRef. 45. ARPES studies in thin films of FeSe indicate thatthe tensile strain promotes significant shifts of the electronbands that can lead to the formation of highly-mobile Diraccarriers . Shifts of 2-3 meV under tensile strain can reducethe size of both electron and hole pockets ; the inner electronband could eventually disappear with increasing the strength - 5 - 4 - 3 - 2 - 1 a bc (cid:1) = - 0 . 1 1 % (cid:1) = - 0 . 0 9 % (cid:1) = - 0 . 0 6 % (cid:1) = - 0 . 0 5 % (cid:1) = - 0 . 0 3 % (cid:1) = - 0 . 0 2 % (cid:1) = + 0 . 0 0 % (cid:1) = + 0 . 0 3 % (cid:2) ( mW m) T ( K ) (cid:2) z = 7 T * = (cid:3) (cid:1)(cid:2) (cid:1) = (cid:1) (cid:3) ( T * ) (cid:2) / (cid:2) (cid:1) | (cid:2) (cid:1) | (cid:1) z (cid:215) | T - T * | /
T * (cid:1) Tc (K) Tc (K) (cid:1) ( % ) FIG. 2. (a)
Temperature dependence of the resistivity near the super-conducting to normal transition under uniaxial strain. The supercon-ducting state is clearly enhanced under compressive uniaxial strain.Opposite trends are found for tensile strain which can be detectedfrom the constant temperature strain loops in Fig. S4d. As the sam-ple is glued to the strain cell, it will be exposed to an additional smalltensile strain ε glue ∼ . . (b) Scaling analysis of superconduct-ing to normal resistivity near the crossing point, T ∗ = 9 . K, where ρ ∗ is the resistivity at ρ ∗ = ρ ( T ∗ ) . The ratio between ρ/ρ ∗ versus | ε | /νz ·| T − T ∗ | /T ∗ describes the universality of the transition witha critical exponent of νz ≈ . (c) Variation of T c , defined as the tem-perature with zero resistance, and of the width of the transition, ∆ T c ,under applied strain. Solid lines are guides to the eye. of the orbital order .Next, we focus on the effect of strain on the nematic tran-sition at T s shown in Fig. 1(c). In the absence of strain, thereis a well-defined anomaly in resistivity at T s , as shown inFig. S2(e). As the nematic order parameter and associated lat-tice distortion have a B g symmetry that breaks the fourfoldrotational symmetry, the applied uniaxial strain in FeSe alongthe [110] tetragonal direction induces a finite order parame-ter at all temperatures. Therefore, it turns the phase transitioninto a crossover, smearing all the related features at T s andthe resistivity increases under tensile strain, as the scatteringfrom nematic domain boundaries becomes significant. Inter-estingly, this effect is in the opposite direction to what happensin the vicinity of the superconducting transition where resis-tivity in the normal state is reduced due to tensile strain, asshown in Fig. 2(a). In our bulk FeSe, the sharp transition at T s is suppressed from 87 K towards 83 K and replaced by abroad crossover with uniaxial stress ( in addition to the effectof the glue that applies ε glue ∼ . (Fig. S2(e,f)). Afterapplying compressive strain, the resistivity is reduced and thecurves seem to recover the signature of the unstrained sam-ple, as shown in Fig. 1(c). This behaviour is similar to thatfound for underdoped Co-doped BaFe As , where the featurecorresponding to the nematic phase transition is quickly sup-pressed (under small strain of × − ) and replaced by abroad crossover . In contrast, the ε B g strain is a continuoustuning parameter and has a quadratic variation as a functionof strain for underdoped Co-doped BaFe As . B. Resistivity scaling under strain at low temperatures
Fig. 2(a) shows the resistivity ρ ( T, ε ) of bulk FeSe as afunction of the temperature, under various amounts of uniax-ial applied strain in the vicinity of the superconducting tran-sition. In the normal state the resistivity decreases under ten-sile strain, as the mobility of the dominant carriers increases(Fig. S3d). We find that all resistivity curves, independentfrom the amount of uniaxial strain applied, cross through asingle point in the vicinity of the superconducting transitionaround T ∗ ∼ . K. Using finite-size scaling analysis, all datacan collapse onto a single curve by re-scaling the resistivitywith the value at the crossing point ρ ∗ = ρ ( T ∗ ) and plot-ting it against the functional relation | ε | /νz · | T − T ∗ | /T ∗ , asshown in Fig. 2 (b). Scaling allows the determination of thecritical exponents and the universality class of the transition .The best results for bulk FeSe under strain are for a value of νz ≈ , as shown in Fig. S4, larger than νz ≈ / found fordirty thin films of FeSe, which describes the universality classof quantum percolation transitions .Scaling relations to describe superconductor-insulatorquantum phase transitions and superconductor-metaltransitions report values of νz ranging from . to .Large values of νz can occur when the dynamical criti-cal exponent shows a divergence by approaching the zero-temperature quantum critical point, as in the case of a Grif-fiths singularity . A Griffiths phase can be found in thevicinity of a critical point in the presence of disorder for atwo-dimensional superconductor-metal quantum phase tran-sition; in this case, rare superconducting regions could formin the normal matrix both as a function of temperature andstrain. Signatures of Griffiths phases have also been detectedin the vicinity of a nematic critical point in FeSe . S . tuned by magnetic field and applied hydrostatic pressure .Our findings suggest that strain leads to a superconducting-to-insulating phase transition in bulk FeSe with the possibleformation of inhomogeneous superconducting regions insidethe normal matrix. This important role played by strain andthe consequent coupling with the lattice is likely to have aneffect on the critical fluctuations of iron-chalcogenides . Ts (K) - 0 . 3 - 0 . 2 - 0 . 1 0 . 0 + 0 . 19 . 09 . 51 0 . 01 0 . 5 (cid:1) [ 1 1 0 ] ( % ) F e S e b u l k n e m a t i c T c T s S C Tc (K)
02 04 06 08 01 0 01 2 0 Ts (K) - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 + 0 . 5 + 1 . 0 + 1 . 50481 21 62 02 42 8 T o n s e t a b c n e m a t i c T c F e S e t h i n f i l m s T s S C (cid:1) ( % ) Tc (K) G d B a C u O d H g B a C u O d F e S e ( t h i n f i l m )F e S e ( b u l k )B a ( F e
C o ) A s B a ( F e
C o ) A s | (cid:1) Tc / Tc0 (cid:2) max | T c ( K ) F e S e ( 1 M L / S r T i O )S r R u O FIG. 3. Superconducting phase diagrams versus strain of (a) bulk FeSe based on this study and (b) thin films of FeSe epitaxi-ally growth on different substrates, adapted after Ref. 37. Panel (b) contains the single crystals data (open symbols) from (a) , which areadjusted to include the additional small tensile strain caused by theglue ε glue ∼ . . Compressive strain ( ε < ) enhances super-conductivity and seems to suppress the nematic state. Black datapoints in (b) refers to T s and T c of bulk FeSe. (c) The sensitivityof T c to strain, ∆ T c /( ε max · T ε =0 c ), for a variety of unconventionalsuperconductors, adapted after Ref. and compared with bulk, thinfilms of FeSe and monolayer of FeSe on SrTiO . C. Phase diagram of FeSe under uniaxial strain
The resistivity studies of bulk FeSe under strain have re-vealed that the compressive strain enhances superconductivityand narrows the width of the superconducting transition, asshown in Fig. 2(a) and (c). Under tensile strain, FeSe displaysa superconductor-to-metal quantum phase transition (see alsoFig. S2d), similar to underdoped iron-pnictides . Fig. 3 showsthe phase diagrams for bulk FeSe under strain and thin filmsof FeSe epitaxially strained by different substrates. For thinfilms, the strain is mainly induced by the mismatch in the lat-tice parameters between the film and the substrate . In-terestingly, the compressive uniaxial strain enhances super-conductivity but it seems to suppress the nematic phase forboth bulk and epitaxial films, suggesting that the two phasesare competing with each other, similar to what is found undersmall applied hydrostatic and chemical pressure How-ever, these opposite trends of the superconductivity versus ne-maticity under strain are in contrast to the effect of growthconditions and disorder . In thin flakes of FeSe, T c and T s decrease at the same time suggesting that disorder as wellas scattering of twin boundary formation inside the nematicphase may have an important role on the transport behaviour.The response of the superconductivity of FeSe to straincan be compared with that of Co-doped BaFe As singlecrystals . For underdoped and near-optimally doped composi-tions, the superconducting critical temperature has a quadraticdependence on applied strain being rapidly suppressed byboth compressive and tensile stress ε B g . In the overdopedregime, the response of T c to strain is smaller in magni-tude and no longer symmetric for tensile and compressivestress, resembling the behaviour found for FeSe, which hasa nematic phase but no long-range order, similar to over-doped pnictides (Fig. 3(c)). The sensitivity to strain of T c in bulk FeSe agrees with uniaxial high-resolution thermal-expansion measurements which suggest that superconductiv-ity couples strongly to the in-plane area , being anisotropicfor the two in-plane directions ( dT c /dp a =2.2(5) K/GPa and dT c /dp b =3.1(1.1) K/GPa ) whereas under uniaxial strain dT c /da is ∼
54 K/ ˚A (Fig. 3(c)). Thus, to stabilize a 90 Ksuperconductor, the applied uniaxial strain needs to be ex-tremely large, much higher than the strain generated by aSrTiO substrate. This suggests that strain on its own is in-sufficient to enhance superconductivity and electron dopingplays an essential role. IV. CONCLUSIONS.
To summarize, we have investigated the electronic responseto external uniaxial strain of bulk single crystals of FeSe. Wehave identified a direct correlation between the transport re-sponse to strain, which is is strongly temperature dependent,and the temperature changes in the Hall coefficient. The nor-mal resistivity increases under tensile strain in the vicinityof the nematic transition but it decreases in the proximity ofthe superconducting transition. This suggest that there is astrong coupling between different scattering processes whichare strongly temperature dependent involving the formation ofnematic domains, multi-band effects and the coupling with thelattice. Band structure calculation suggest that under strain theelectron bands would develop the largest in-plane anisotropy.Superconductivity of FeSe is enhanced under compressiveuniaxial strain, similar to overdoped 122 iron-based super-conductors, whereas the the nematic electronic state respondsto strain in a opposite way. Furthermore, close to the criti-cal temperature, we identify a universal crossing point for all resistivity curves measured under constant strain. The scal-ing behaviour of resistivity versus temperature in the vicinityof the superconducting transition indicates the existence of astrain-induced phase transition, that would be consistent withthe development of rare superconducting regions inside of anormal metal matrix as a function of temperature and strain.This study establishes that uniaxial strain, on its own, is notsufficient to stabilize a high- T c FeSe-based superconductor.
Note after review:
After completing this work we becomeaware of another study of FeSe under strain , in which thesingle crystals are glued to titanium sheets and these plat-forms are strained to higher values that would be possible fora stand-alone crystal (as in our study). The reported findingsare in broad agreement with the results presented in our work. V. ACKNOWLEDGMENTS
We are very grateful to Dragana Popovic for helpfulcommunication related to the scaling analysis. We thank S. J.Singh and P. Reiss for technical support, Oliver Humphriesfor the development of the mobility analysis spectrum andA. Morfoot for useful comments on the manuscript. Thiswork was mainly supported by EPSRC (EP/I004475/1)and the Oxford Centre for Applied Superconductivity. M.Ghini acknowledges the financial support of the Univer-sity of Bologna, Scuola di Scienze. JCAP acknowledgesthe support of St Edmund Hall, University of Oxford,through the Cooksey Early Career Teaching and ResearchFellowship. The DFT calculations were performed on theUniversity of Oxford Advanced Research Computing Service(https://doi.org/10.5281/zenodo.22558). AIC acknowledgesan EPSRC Career Acceleration Fellowship (EP/I004475/1).
VI. APPENDIXA. Strain calibration and simulations
Hysteresis strain loops were performed at constant temper-ature to investigate the sample response to uniaxial tensile andcompressive stress. The hysteresis loops were performed atleast twice for each set temperature. Multiples measurementswere conducted on the sample over several weeks to verifyreproducibility between different measurements (Fig. S2 (c) ).FeSe crystals, with typical dimensions of ≈ × × µ m , were cut into a rectangular shape along the [110] tetrag-onal direction, which corresponds applying strain along the B g symmetry channel , and mounted on the CS100 uni-axial strain cell. The amount of nominal stress applied, ε external = ∆ L/L , is defined as the displacement ∆ L = L ( T )- L ( T ) divided by the unstrained length L of the sam-ple. An ultra-precise capacitance sensor (Andeen-HagerlingAH270 Capacitance bridge) was used to measure the positionof the two titanium plates, in order to evaluate the amount ofstrain applied with the uniaxial strain cell. Capacitance as a [110] ab c FIG. S1. (a)
Free-standing single crystals of FeSe cut along the[110] tetragonal direction measured before being mounted on thestrain cell in (b) . (c) The sample model used to estimate the straintransmission in the sample inside the cell strain secured at both endsby epoxy. function of the distance between two plates follows the equa-tion: C ( d ) = ε external A/ [ d + d cell ] , with d cell = 36 . µ m, A = 6 . mm , ε external = ε × k = 8 . pF/m, and where d is the displacement applied from the cell.In order to asses the strain transmission through the sam-ple and quantify the internal strain compared to the appliedstress, we have performed finite element analysis simulationsin COMSOL using the Linear Elastic Material material modelfrom the structural mechanics module, testing our approachagainst previous reports . The geometry was constructed andparameterized to match the geometry of the sample as shownin Fig. S1. ε external was averaged over the the volume be-tween the voltage contacts to get a value for the fraction ofstrain applied that is transmitted to the sample, and averagedover the top and bottom surfaces between the contacts to get avalue for the strain inhomogeneity across the sample. For theparameters used, ε [110] = . ε external . The strain gener-ated by the epoxy was also simulated and contributes a tensilestrain of about ε glue ∼ . to the sample.To extract the value of nominal strain from the measuredcapacitance, we account for its temperature dependence (thethermal contractions of titanium were negligible). Fig. S2 (b) shows the behaviour of the capacitance against the displace-ment of the two plates C ( d ) at room temperature and the de-pendence of the capacitance sensor versus temperature. Vari-ous measurements were carried out over the temperature range ∆ T = 3 − K without any applied strain (zero voltageapplied to the piezostacks) in order to establish the trend of C ( T , d ). To account for this effect, the corrected capacitanceis obtained by subtracting from the measured capacitance at a given temperature the variation in capacitance between thattemperature and room temperature.As bulk FeSe is a soft layered material, large amounts ofstrain tend to break the sample. Bending, cracking and exfo-liation were observed in various samples with this suspendedconfiguration for higher values of stress. Fig. S2 (e) showsthe resistivity of the sample without applied strain in differentstages of the experiment: before and after gluing the sampleto the cell (run0 and run1), after all the measurements withoutmagnetic field (run2) and after the last measurement (run3).The behaviour of the system is consistent between all the dif-ferent runs (measured over several weeks) but the first one,which corresponds to the free-standing case which was notglued to the cell. Fig. S2 (f) illustrates the derivative of re-sistivity versus temperature in the proximity of the nematictransition, showing the impact of glue on transport properties.It is possible to observe the evolution of T s under the effect ofthe applied strain by assessing the shifts of the high tempera-ture minimum as well as the position of the middle peak.For a limited temperature range in which the capacitance ofthe cell remains constant, resistivity measurements at constantstrain were performed as a function of temperature. Precau-tions were taken to avoid unwanted effects of thermal drift andfluctuation. All transport measurements as a function of tem-perature reported in this work were collected during warm-ing ramps and with slow warming rates, in order to limit ef-fects of thermal load at low temperatures. Fig. S2 (a) showsthe thermal drift of a cooling ramp compared to the warm-ing measurement. Fig. S2 (d) shows the transition betweenthe superconducting to normal state induced by the uniaxialcompressive strain at a fixed temperature ( T = 9 . K).
B. Magnetotransport studies
Magneto-transport measurements were conducted as afunction of the magnetic field up to . T inside the nematicphase (between T = 10 K and T = 60 K), under three dif-ferent amounts of uniaxial strain between ε = − . % to +0 . %. Magneto-transport measurement were conductedmeasuring both the longitudinal resistivity, ρ xx , (with the cur-rent along the [110] direction), and the Hall component, ρ xy ,as a function of the applied magnetic field applied along the[0 0 1] direction. These values have been symmetrized andantisymmetrized, with respect to the applied magnetic field(as reported in Fig. S3 (a,b) ). Fig. S3 (c) reports the mobilityspectrum generated from the magneto-transport data for dif-ferent amount of strain at T = 30 K. The mobilities of thedominant charge carriers are consistently enhanced under ten-sile strain and suppressed under compressive strain, inside thenematic phase (Fig. S3 (d) ). The low field Hall coefficient, R H (in B < T), is surprisingly insensitive to compressivestrain but changes under tensile strain, as shown in Fig. S3 (e) . a b c d e f - 1 0 - 5 0 + 5 + 1 01 . 21 . 41 . 61 . 82 . 0 - 0 . 0 5 0 . 0 0 + 0 . 0 50 . 9 81 . 0 01 . 0 21 . 0 40 1 0 0 2 0 0 3 0 00 . 00 . 40 . 8 6 0 7 0 8 0 9 0 1 0 00 . 0 10 . 0 20 . 0 3 - 0 . 0 6 - 0 . 0 3 0 . 0 0 + 0 . 0 302468 (cid:3) ( mW m) T ( K ) w a r m i n g c o o l i n g C (pF) d ( m m ) T ( K ) T s (cid:1) ( % ) (cid:1) ( % ) r u n 0 1 r u n 5 0 r u n 7 5 T = 2 0 K T ( K ) r u n 0 r u n 1 r u n 2 r u n 3 (cid:3) ( T ) (cid:1) (cid:3) (cid:1) (
297 K ) d (cid:2) (cid:1) d T ( mW m/K) T ( K ) (cid:1) = - 0 . 2 6 % (cid:1) = - 0 . 1 4 % (cid:1) = - 0 . 1 1 % (cid:1) = - 0 . 0 7 % (cid:1) = - 0 . 0 1 % (cid:1) = + 0 . 1 1 % (cid:1) = + 0 . 1 0 % (cid:1) = - 0 . 1 2 % T ( K ) d (cid:2) (cid:1) d T ( mW m/K) (cid:3) ( (cid:2) ) / (cid:3) ( (cid:2) = T = 9 . 0 K (cid:3) ( (cid:2) ) / (cid:3) ( (cid:2) = FIG. S2. (a)
Transport measurements performed while cooling and warming, showing the small temperature variation are due to the thermalload of the strain cell. (b)
Capacitance as a function of the displacement d ( µ 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 thereproducibility and consistency between different measurements acquired at the same temperature. (d)
Direct evidence that compressivestrain enhances the superconducting state showing the superconducting to normal transition induced by uniaxial compressive strain at fixedtemperature T = 9 . K. (e) Transport measurements without applied strain at different stages of the experiment. All the values are normalizedat room temperature ρ ( T ) / ρ (RT). Inset shows the derivative of the resistivity against temperature of the same runs, indicating the smearingof the transition after gluing the sample to the strain cell as compared with the free standing sample (run0). (f) Derivative of resistivity againsttemperature around the nematic transition at different applied strain. The structure of the transition developed an additional feature after thegluing of the sample to the strain cell. Here, T s is defined as the position of the Gaussian fit of the middle peak. C. Scaling analysis
Fig. S4 reports scaling of the reduced resistivity ρ/ρ ∗ around the strain-independent crossing point T ∗ as a func-tion of | ε | /νz · | T − T ∗ | /T ∗ (Fig. S4 (b) ). This functionalrelation leads to a collapse of the experimental data with thesame exponent νz ≈ . Fig. S4 (c) reports the standard de-viation of the dispersion at ρ/ρ ∗ = 0 . versus different val-ues of the critical exponent νz as a way to estimate the op-timal exponent. Optimal results were obtained with a valueclose to the minimum νz = 7 . Fig. S4 (d) shows transportmeasurements at the superconductive transition collected byvarying the uniaxial strain at fixed temperatures, instead ofvarying the temperature with fixed strain. As before, we ob-serve that compressive strain increases the resistivity abovethe transition and enhances the superconductive state (whiletensile strain induces opposite effects) and the presence of thetemperature T ∗ at the middle of the transition. D. Band structure calculations
Fig. S5 (a-c) reports Fermi surfaces and slices through thecentre of the Brillouin zone ( Γ -M plane) for bulk FeSe fordifferent applied strain along [110] of ε = ± . . The samerenormalizations and band shifts were chosen to agree withARPES data in the unstrained tetragonal case and these shiftwere applied at all calculations under strain (Fig. S5 (d-f)).The outer hole band and the two electron bands were renor-malized by a factor of 3 and 4 respectively, as suggested byARPES data [6], and then shifted by -45meV and 45 meV re-spectively, to ensure charge compensation. Additionally, theinner and middle hole bands are shifted away from the Fermilevel entirely, as compared with the unshifted case (Fig. S5(a-c)). We find that both the outer hole band and the elec-tron bands shrink with increasing tensile strain in both the un-shifted and shifted cases. This reflects clearly in the numberof charge carriers that decreases with increasing tensile strainfor all three Fermi surface pockets, as shown in Fig. S5(i).The understand the effect of the strain on the in-plane andout-of-plane anisotropy of the Fermi surface pockets b, wehave also computed the penetration depth, λ , along the threelattice vectors, and taking suitable ratios. Fig. S5(g) showsthe in-plane anisotropy ( λ a /λ b ) and Fig. S5 (h) shows the B ( T ) T = 1 0 K T = 2 0 K T = 3 0 K T = 4 0 K T = 5 0 K T = 6 0 K (cid:1) = - 0 . 2 1 % b c d e B ( T ) (cid:1) = - 0 . 0 5 % B ( T ) (cid:1) = + 0 . 0 9 % T = 1 0 K T = 2 0 K T = 3 0 K T = 4 0 K T = 5 0 K T = 6 0 K (cid:1) xy ( mW m) (cid:1) xy ( mW m) B ( T ) (cid:1) = - 0 . 2 1 % B ( T ) (cid:1) = - 0 . 0 5 % B ( T ) (cid:1) = + 0 . 0 9 % s ( (cid:1) ) (cid:1) ( m / V s ) T = 3 0 K (cid:1) = - 0 . 2 1 % (cid:1) = - 0 . 0 5 % (cid:1) = + 0 . 0 9 % (cid:1) h o l e (cid:1) e l e c t r o n peak position (m2/Vs) T ( K ) a (cid:1) = - 0 . 2 1 % (cid:1) = - 0 . 0 5 % (cid:1) = + 0 . 0 9 % (cid:2) xy ( mW m) B ( T ) (cid:1) = - 0 . 2 1 % (cid:1) = - 0 . 0 5 % (cid:1) = + 0 . 0 9 % T = 3 0 K FIG. S3. (a)
The longitudinal resistivity, ρ xx , and (b) Hall component, ρ 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 differentvalues 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)
Directcomparison between the Hall component ρ xy for different strain at 30 K. out-of-plane anisotropy ( λ c /( λ a + λ b )), calculated from therenormalized and shifted Fermi surface pockets for severalvalues of strain. The inner electron pocket is much smallerthan the others, so numerical errors affect the results muchmore strongly, as seen. We find that the hole pocket is essen-tially isotropic in-plane, whilst the electron pockets have some in-plane anisotropy. However, all three pockets exhibit out-of-plane anisotropy, with the hole band exhibiting the largestanisotropy. The anisotropy in the overall penetration depth,calculated by combining the contributions from each pocket,remains fairly constant with strain, both in-plane and out-of-plane. ∗ corresponding author:[email protected] † corresponding author:[email protected] Alexander Steppke, Lishan Zhao, Mark E. Barber, Thomas Scaf-fidi, Fabian Jerzembeck, Helge Rosner, Alexandra S. Gibbs,Yoshiteru Maeno, Steven H. Simon, Andrew P. Mackenzie, and Clifford W. Hicks, “Strong peak in T c of Sr RuO under uniaxialpressure,” Science (2017), 10.1126/science.aaf9398. Paul Malinowski, Qianni Jiang, Joshua Sanchez, Zhaoyu Liu,Joshua Mutch, Preston Went, Jian Liu, Philip Ryan, Jong-WooKim, and Jiun-Haw Chu, “Drastic suppression of superconduct- - 7 - 5 - 3 - 1 - 5 - 4 - 3 - 2 - 1 (cid:1) = - 0 . 1 1 % (cid:1) = - 0 . 0 9 % (cid:1) = - 0 . 0 6 % (cid:1) = - 0 . 0 5 % (cid:1) = - 0 . 0 3 % (cid:1) = - 0 . 0 2 % (cid:1) = + 0 . 0 0 % (cid:1) = + 0 . 0 3 % (cid:2) ( mW m) T ( K ) n z = 7 std. dev. n z a t (cid:2) (cid:1) / (cid:2) * = 0 . 5 x a x i s i n l o g s c a l e a b c d e f (cid:1) = - 0 . 0 5 % (cid:1) = - 0 . 0 4 % (cid:1) = - 0 . 0 2 % (cid:1) = - 0 . 0 1 % (cid:1) = + 0 . 0 1 % (cid:1) = + 0 . 0 2 % (cid:2) ( mW m) T ( K ) (cid:2) z = 7 T * = (cid:3) * = (cid:1)(cid:3) ( T * ) (cid:1) | (cid:2) (cid:1) | / (cid:1) z | T - T * | /
T * (cid:2) / (cid:2) (cid:1) | (cid:2) (cid:1) | / (cid:1) z | T - T * | /
T * (cid:2) z = 7 T * = (cid:3) * = (cid:1)(cid:3) ( T * ) (cid:1) (cid:2) / (cid:2) (cid:1) d (cid:2) /d T ( mW m/K) T ( K ) + 0 . 0 3 % + 0 . 0 0 % - 0 . 0 2 % - 0 . 0 3 % - 0 . 0 5 % - 0 . 0 6 % - 0 . 0 9 % - 0 . 1 1 % FIG. S4. (a)
Temperature dependence of the resistivity near the superconductive transition, measurements performed by varying the temper-ature 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 . K, where ρ ∗ is the resistivity at the critical point ρ ∗ = ρ ( T = T ∗ ) . The ratiobetween ρ/ρ ∗ vs the functional relation | ε | /νz · | T − T ∗ | /T ∗ is able to describe the universality of the transition with a critical exponentof νz ≈ . (c) Width of the dispersion of the scaling analysis at ρ/ρ ∗ = 0 . as a function of the critical exponent of νz , for measurementsconducted at fixed applied strain (from b ). (d) Resistivity near the superconductive transition, with different amount of applied uniaxial strain.The measurements conducted at constant temperature to confirm findings from previous measurements performed at constant strain. (e)
Scal-ing analysis of d near the crossing point at T ∗ = 9 . K, from the data collected at constant temperature with an exponent of νz ≈ , similarto that illustrated in b . (f) The first derivative of the data in (a) showing the evolution of width of the superconducting transition with uniaxialstrain.ing T c by anisotropic strain near a nematic quantum critical point,”arXiv:1911.03390 (2019). A. E. B¨ohmer, A. Sapkota, A. Kreyssig, S. L. Bud’ko,G. Drachuck, S. M. Saunders, A. I. Goldman, and P. C.Canfield, “Effect of Biaxial Strain on the Phase Transitions of
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