The relationship between transport anisotropy and nematicity in FeSe
Jack Bartlett, Alexander Steppke, Suguru Hosoi, Hilary Noad, Joonbum Park, Carsten Timm, Takasada Shibauchi, Andrew P. Mackenzie, Clifford W. Hicks
TThe relationship between transport anisotropy and nematicity in FeSe
Jack Bartlett,
1, 2, ∗ Alexander Steppke, † Suguru Hosoi,
3, 4
Hilary Noad, Joonbum Park, Carsten Timm,
5, 6
Takasada Shibauchi, Andrew P. Mackenzie,
1, 2 and Clifford W. Hicks
1, 7, ‡ Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Str 40, 01187 Dresden, Germany SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews KY16 9SS, United Kingdom Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan Department of Materials Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan Institute of Theoretical Physics, Technische Universit¨at Dresden, 01062 Dresden, Germany W¨urzburg-Dresden Cluster of Excellence ct.qmat,Technische Universit¨at Dresden, 01062 Dresden, Germany School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, U.K. (Dated: 24 Nov 2020)The mechanism behind the nematicity of FeSe is not known. Through elastoresitivity measure-ments it has been shown to be an electronic instability. However, so far measurements have extendedonly to small strains, where the response is linear. Here, we apply large elastic strains to FeSe, andperform two types of measurements. (1) Using applied strain to control twinning, the nematicresistive 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 tem-perature above T s . In both cases, as nematicity strengthens the resistive anisotropy peaks aboutabout 7%, then decreases. Below ≈
40 K, the nematic resistive anisotropy changes sign. We discusspossible implications of this behaviour for theories of nematicity. We report in addition: (1) Underexperimentally accessible conditions with bulk crystals, stress, rather than strain, is the conjugatefield to the nematicity of FeSe. (2) At low temperatures the twin boundary resistance is ∼
10% of thesample 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. At an electronic-nematic transition, electronic interac-tions drive a spontaneous reduction in rotational sym-metry without introducing translational or time-reversalsymmetry breaking. Electronic nematicity affects all theFermi surfaces of a metal, and therefore its fluctuationscan have powerful effects [1, 2]. It is potentially an in-tegral part of the high-temperature superconductivityof iron-based and cuprate superconductors [3], and themechanisms behind it are therefore a topic of interest.In many iron-based superconductors, nematicity oc-curs in close proximity to a transition into unidirectionalspin density wave order, suggesting that it is a meltedform of the magnetic order [4, 5]. In contrast, the ne-matic transition of FeSe occurs, at 92 K, without a subse-quent magnetic transition. Whereas in other iron-basedsuperconductors magnetic and lattice fluctuations arelinked by a scaling relationship, they are not so linkedin FeSe [6–8]. In spite of these differences, there aresimilarities between FeSe and other iron-based supercon-ductors that suggest that their nematicities are related.For example, unidirectional magnetic order can be in-duced in FeSe [9, 10], and the nematic electronic struc-ture as observed in angle-resolved photoemission quali-tatively matches that of BaFe As [11, 12]. FeSe is avaluable reference material not only because of the ab-sence of magnetic order, but also because of the absenceof intrinsic dopant disorder, and the availability of high-quality, vapor-transport-grown samples [13, 14].Measurements of the strain dependence of resistivity, i.e. the elastoresistivity, have shown that its nematicity,like that of other iron-based superconductors, is an elec-tronic instability. The key observation is that the resis-tive anisotropy ( ρ xx − ρ yy ) / ( ρ xx + ρ yy ) varies with strainat a rate that diverges with cooling [15–19]. The resistiveanisotropy is understood to be proportional to an under-lying electronic anisotropy that can be quantified by a ne-matic order parameter ψ . On a clamped lattice, ψ wouldtransition to a nonzero value at a bare transition temper-ature T s,0 , but the elastic compliance of the lattice raisesthe transition temperature to T s > T s,0 . For T > T s ,applied anisotropic strain ε induces nonzero ψ throughelectron-lattice coupling, with a susceptibility dψ/dε thatdiverges (with divergence temperature T s,0 ) as the sam-ple is cooled. Therefore, because resistive anisotropy isproportional to ψ , its dependence on strain also steepenswith cooling.An assumption of a linear relationship between ψ andresistive anisotropy has become deeply enough embed-ded that resistive anisotropy is often employed as a mea-sure of ψ . Here, we explore elastoresistivity at large | ψ | ,where the relationship becomes strongly nonlinear. FeSeis considered to be a Hund’s metal, meaning that in-terorbital charge fluctuations are suppressed by Hund’scoupling [20]. Strong evidence for the importance of or-bital character is provided by the fact that the magnitudeof the superconducting gap correlates closely with yz or-bital weight [21–23]. Many of the strain effects that weobserve here are also consistent with a prominent role of a r X i v : . [ c ond - m a t . s t r- e l ] F e b the yz orbital in electronic correlations, and we discusshow our data may constitute a test of theories of thenematicity of FeSe.Two types of measurement are presented. (1) Resistiveanisotropy is measured as a function of strain-inducednematicity at constant temperature T ∼ T s . (2) Strain-tuning is employed to control the twinning as samplesare cooled, allowing measurement of the intrinsic resistiveanisotropy at temperatures below T s . Although the T -dependent nematic resistive anisotropy has been reportedpreviously for a few iron-based compounds [19, 24–27],these previous measurements have relied upon assump-tions that twin boundary resistance is negligible, and/orthat a sustained stress applied to detwin samples is weakenough not to substantially alter the electronic struc-ture, even though the iron-based superconductors are ex-tremely sensitive to uniaxial stress [25, 28]. With strain-tuning, samples can be held in a fully or partially de-twinned state without sustained application of externalanisotropic stress.This paper is organized as follows. We first presentour setup and methods, and then define the key param-eters for discussion of elastoresistivity. We then presentresults for application of anisotropic strain with prin-cipal axes rotated by 45 ◦ from the nematic axes, inother words where it constitutes a transverse field to thenematicity [29]. Results are then presented for strainaligned with the nematic axes, where the response ismuch stronger. Our main result, the spontaneous ne-matic resistive anisotropy for T < T s , in comparison withthat induced by strain at T > T s , is shown in Fig. 8.For orientation, the electronic structure of FeSe aboveand far below T s is illustrated schematically in Fig. 1(a).We work with the 1-Fe unit cell, in which the Fe-Fe bonddirections, and the principal axes of the nematicity, arethe (cid:104) (cid:105) directions. In the corresponding Brillouin zone,there is a hole pocket at the Γ point, and two electronpockets, one at the X and the other at the Y point. Inthe nematic state, where the a lattice parameter becomeslarger than b , the pocket at X distorts into a peanut-like shape elongated along k x , signatures of the Y pocketdisappear from spectroscopic probes [17, 21, 30–33], andthe hole pocket becomes elongated along the k y direction. METHODS
To apply large strains to FeSe, we affix samples to plat-forms with a layer of epoxy (Masterbond ® EP29LPSP),and then apply stress to the platform; details of thismethod are presented in Ref. [34]. By preventing sam-ples from buckling under compressive strain the platformallows samples to be very thin. This is helpful for FeSebecause it is a layered compound with a very low elasticlimit for interlayer shear stress, which is minimized whensamples are thin. The epoxy that wicked up the sides of
200 μm500 μm
10 μm5 μm
FeSeepoxyTi sampleplat f ormSample BSample C (b) XY XY (a)
T T s : T > T s : ab yzxzxy piezoelectric actuators(c) (e) (f) Γ
12 mm capacitive displacement sensorplatformstressapparatus xz y epoxy xyxy [100][010][100][010] (d)
Sample BSample C
FIG. 1: (a) Schematic illustration of the electronic struc-ture 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 plat-form. (c)Photograph of Sample B, with contacts attached formeasuring resistivity along the sample’s length. The latticedirections are indicated. Sample A was prepared similarly,though with its crystal axes rotated by 45 ◦ . (d) Scanningelectron (SEM) micrograph of a cut, made with a focused ionbeam, through Sample B and the epoxy layer beneath it. (e)Photograph and (f) SEM micrograph of Sample C, which wasprepared in a Montgomery configuration. the sample may also have served to hinder cleavage. Aschematic of the setup is shown in Fig. 1(b), and imagesof mounted samples are shown in Fig. 1(c–f).Here, the platforms are titanium sheets. The centralportion is cut into a narrow neck within which stress isconcentrated, and samples are attached to this neck. Theplatforms are then mounted onto a piezoelectric-drivenuniaxial stress apparatus. This apparatus incorporates acapacitive sensor of the applied displacement, and there-fore of the longitudinal strain within the neck.We report data from three samples. Sample A was cutfor application of strain with (cid:104) (cid:105) principal axes, andSamples B and C with (cid:104) (cid:105) principal axes. Samples Aand B were prepared as shown in Fig. 1(c): bars withhigh length-to-width ratio, with contacts for measure-ment of resistivity along the sample length. Sample C,shown in Fig. 1(e–f), was prepared in a Montgomery con-figuration for simultaneous measurement of longitudinaland transverse resistivities, as introduced for elastoresis-tivity measurements in Ref. [16]. The conversion frommeasured resistances to longitudinal and transverse re-sistivities in the Montgomery geometry is discussed inAppendix section 1. In Appendix section 2, we considerthe mechanics of strain transmission from the platformto the sample, and show that the lengths and widths ofthe samples here are all long enough that to good preci-sion both the longitudinal and transverse strains can betaken to be locked to those in the platform.Electrical contacts, fabricated from sputtered goldwith no adhesion layer, were deposited on the samples’upper surfaces. The resistivity ratio ρ c /ρ ab of FeSeappears not to have been measured, however that ofFeSe . Te . is ≈
70 at 15 K [35]. The length scale forcurrent injected at the upper surface to spread out overthe full sample thickness is t ( ρ c /ρ ab ) / , where t ∼ µ mis the sample thickness. This length scale is short enoughthat measurements here are not strongly affected by the c -axis resistivity. For Sample C, the contacts also rundown the sides of the sample. KEY PARAMETERS
The applied strain can be resolved into symmetric andantisymmetric components, and throughout this workit will be important to resolve their separate effects.Here, we define quantities for discussion. For Sam-ple A, stress is applied along the [110] lattice direc-tion; the displacement sensor in the stress cell measuresthe strain along this axis, ε . The transverse strain ε is given by ε = − νε , where ν = 0 .
32 is thePoisson’s ratio of the platform. The symmetric compo-nent of the strain field is ε A1g ≡ ( ε + ε ), whichcomes to 0 . ε , while the antisymmetric componentis ε B2g ≡ ( ε − ε ) = 0 . ε . These parameters,along with equivalent parameters for Samples B and C,are summarized in Table I. We also label resistivities bythe measurement axis: ρ , for example, is the resistiv-ity along the [100] direction. For Sample C both ρ and ρ are measured, and so symmetric and antisymmetricresistivities can be defined: ρ A1g ≡ ( ρ + ρ ) and ρ B1g ≡ ( ρ − ρ ).We note that specifying lattice distortions becomesmore complicated when the lattice twins. We adopt herethe convention that [110] for Sample A, and [100] forSamples B and C, always refer to the direction along the TABLE I: Strain parameters. We take the 1-Fe unit cell, inwhich the (cid:104) (cid:105) directions are Fe-Fe bond directions. SampleA is aligned so that stress is applied along the [110] latticedirection; the strain along this axis, ε , is measured by thedisplacement sensor integrated into the stress cell. SamplesB and C are aligned so that stress is applied along the [100]direction. ν = 0 .
32 is the Poisson’s ratio of the platform. Thegraphics illustrate the strain directions. We take the signconvention that ε <
Sample A
SeFe ε A1g ≡ ( ε + ε ) = (1 − ν ) ε = 0 . ε ε B2g ≡ ( ε − ε ) = (1 + ν ) ε = 0 . ε Samples B and C ε A1g ≡ ( ε + ε ) = (1 − ν ) ε = 0 . ε ε B1g ≡ ( ε − ε ) = (1 + ν ) ε = 0 . ε length of the platform. When the sample twins, we use a and b to refer to the directions along which the in-planelattice constant lengthens and shrinks, respectively; inother words, the a and b axes are defined locally, and the[100] and [110] directions globally.For all samples, the applied strain will also generate a c -axis strain in the sample, ε = − c ε A1g /c . c -axisstrain preserves the tetragonal symmetry of the T > T s lattice, and therefore is in the A representation. Whenwe discuss A strain it should be understood that itincludes this associated c -axis strain.Because the aim of this work is to explore the non-linear regime, we do not apply the elastoresistivity ma-trix formalism introduced in Ref. [36]. For compar-ison with previous results we note that the quantity(1 /ρ A1g ) dρ B1g /dε
B1g at ε B1g = 0 is equal to m − m in that formalism. Most previous elastoresistivity resultshave been reported using the 2-Fe unit cell, in which m − m transforms to 2 m . RESULTS: (cid:104) (cid:105)
STRAIN
Although strong transverse strain is predicted to en-hance quantum fluctuations and suppress nematicity [29,37], the range of transverse strain explored here shifts T s by only a few kelvin. T s can be identified from an upturnin the resistivity, and, as shown in Fig. 2, decreases at amodest rate with compression. Within our strain rangeonly a linear component of the strain dependence is re-solved, with slope dT s /dε = 750 K. This slope is dueto the A component of the applied strain: under thetetragonal symmetry of FeSe at T > T s , reversal of thesign of ε B2g gives a symmetrically equivalent strain, socoupling to ε B2g can give only strain-even components inthe strain dependence of T s . ε A1g = 0 . ε , so dT s /dε = 750 K corresponds to dT s /dε A1g = 2200 K. In Ref. [38], T s is found to be sup- FeSe
FIG. 2: The effect of transverse strain. (a) ρ ( T ), the re-sistivity along the [110] direction, for Sample A at variousapplied strains ε . The principal axes of the nematicity inFeSe are the (cid:104) (cid:105) axes, so this strain is a transverse field tothe nematicity. The inset is a schematic of the strain axis.(b) T s versus ε for this sample. T s is identified as the max-imum in d ρ /dT . The shaded region is a measure of thewidth of the transition; it is where d ρ/dT exceeds half itsmaximum value. The line is a fit. pressed by compressive hydrostatic stress with an initialslope of 39 K/GPa. Using the elastic moduli of Ref. [39],this converts to dT s /dε A1g ≈ c -axis strain ε and that of “pure” in-plane biax-ial strain ε A1g, pure that has no associated c -axis strain.Applying again the elastic moduli from Ref. [39], un-der in-plane uniaxial stress ε = − . × ε A1g, pure ,and under hydrostatic stress ε = 1 . × ε A1g, pure , so∆ T s ≈ (3200 K) × ε A1g, pure + (1000 K) × ε . RESULTS: (cid:104) (cid:105)
STRAINStress-temperature versus strain-temperature phasediagram.
The effect of strain applied along the principal axesof the nematicity is much more dramatic. Before show-ing results, we discuss the differences between stress- andstrain-temperature phase diagrams for a nematic transi-tion. The distinction between stress and strain is equiv-alent to that between magnetic field H and magnetic induction B . When a ferromagnet is cooled through itsCurie temperature under nonzero H the transition broad-ens into a crossover. Experimentally, controlled H is ap-plied by preparing samples to have a low demagnetizationfactor: thin bars parallel to the applied field. In the op-posite limit, of a thin plate perpendicular to the appliedfield, it is B that is held fixed, and if B/µ is less than thespontaneous magnetization M of the sample then in gen-eral magnetic domains will form such that the sample’saverage magnetization matches the applied B . Domainformation under nonzero applied B requires reversal oflocal magnetization, so it is a first-order transition ratherthan a crossover.For nematic compounds, the difference between stressand strain is illustrated in Fig. 3. In the stress-temperature phase diagram, a first-order transition linecorresponding to reversal of the nematicity runs alongthe zero-stress axis from T = T s to T →
0. In the strain-temperature phase diagram, on the other hand, thereare two lines of first-order transitions. The structuraldistortion in FeSe is to high precision a B distortion,meaning that b contracts by nearly the same amount as a lengthens [40–42]. Therefore, the nematicity-inducedstructural distortion can be described as a spontaneouslocal strain ε B1g, local = ± ε s ( T ), where the quantity ε s is termed the structural strain. The average strain inthe sample must match that of the platform, but when | ε B1g | < ε s ( T ) twin formation is favored, and the ap-plied strain sets the equilibrium twin volume ratio. Likeformation of magnetic domains under nonzero B , forma-tion of twinned domains under nonzero applied ε B1g is afirst-order process, so the twinned region is bounded byfirst-order transitions.In the stress-temperature phase diagram there will beresolvable crossover lines at
T > T s : when the appliedstress is small, there will be a small temperature rangeover which the nematicity-driven strain increases at arapid but non-divergent rate. In this sense, stress acts asa classic conjugate field. We present some evidence belowon whether equivalent crossover lines are discernable inthe strain-temperature phase diagram. Sample B, T ∼ T s Measurements of resistivity confirm this qualitativeform of strain-temperature phase diagram. To facili-tate comparison with measurements of ε s , we now plotdata against the antisymmetric strain ε B1g . ρ ( ε B1g )of Sample B for T ∼ T s is shown in Fig. 4(a), and thederivative dρ /dε B1g in panel (b). The neutral strainpoint ε B1g = 0 is determined as the strain where the twinboundary density for
T < T s is highest; these data areshown below. Above T s , the strain dependence of ρ is seen to have substantial nonlinearity even over a rela-tively small strain range | ε B1g | < . · − . Its slope is first-ordertransitioncrossover twinnedstress σ B1g t e m p e r a t u r e ε s ( T
0) + ε s ( T nd -ordertransition strain ε B1g (a) (b) T s nd -ordertransition FIG. 3: Schematic phase diagrams. (a) Schematic stress-temperature phase diagram for the nematicity of FeSe, forstress applied with B principal axes; the first-order tran-sition is where the direction of the nematicity flips. (b) Thecorresponding strain-temperature phase diagram. In the indi-cated region, the lattice is unstable and breaks up into twinswhere, locally, ε B1g = ± ε s ( T ). largest near, though not precisely at, ε B1g = 0.As T is reduced below T s , the onset of twinning changesthe form of ρ ( ε B1g ): a range of strain appears overwhich dρ /dε B1g becomes nearly constant. This changeis easiest to see in Fig. 4(b), where we have marked thetwinned region for the 86.8 K curve. The origin of thisbehavior is illustrated schematically in Fig. 4(c). Withineach twin domain the resistivities along the local a and b axes are ρ a and ρ b , and the equilibrium twin volumeratio is a linear function of applied strain. Therefore,the observed bulk resistivity is an interpolation between ρ b at ε B1g = − ε s and ρ a at ε B1g = + ε s , that to highprecision is linear under two conditions that are bothsatisfied here. (1) | ( ρ a − ρ b ) / ( ρ a + ρ b ) | is much less than1, so that redistribution of current into lower-resistivitydomains does not substantially alter the observed bulkresistivity. (2) The domain wall resistance is negligible,which we show later to be the case for T near T s .Even though the transitions into the twinned regionmust, when ε B1g (cid:54) = 0, be first-order, no hysteresis is re-solved, indicating that the energy barrier for twin forma-tion is low. Separately, close inspection of Figs. 4(b)reveals that twinning does not initially onset right at ε B1g = 0, but slightly on the tensile side. This asym-metry is due to the A component of the applied strain:as shown with Sample A in Fig. 2, tensile A strain in-creases T s . ρ versus temperature at a few nonzero ε B1g are shownin Fig. 4(d), and Fig. 4(e) shows T s derived from suchtemperature sweeps as a function of strain. For bothSamples B and C, T s follows a downward quadratic form,consistent with the schematic strain-temperature phasediagram illustrated in Fig. 3(b). observed ρ(ε) underlying ρ(ε) FeSe
FIG. 4: Elastoresistivity near T s . (a) ρ ( ε B1g ), where ρ is the resistivity along the [100] direction and ε B1g ≡ ( ε − ε ) /
2, of Sample B at various temperatures near T s .(b) dρ /dε B1g for the curves from panel (a). For
T < T s , dρ /dε B1g becomes nearly constant over the range where thesample twins. This range is indicated for the 86.8 K curve.(c) Schematic of ρ ( ε B1g ) for
T < T s ; the underlying curve isnot accessible for − ε s < ε B1g < + ε s due to the onset of twin-ning, and the observed resistivity instead interpolates overthis range. (d) Temperature ramps at three values of ε B1g .(e) T s versus strain for low strains. The shaded regions indi-cate the transition width, defined by d ρ /dT crossing halfits maximum value. Sample B,
T < T s Fig. 5(a) shows ρ of Sample B over a much widertemperature and strain range. Here, the contributionof twin boundaries to the total sample resistance be-comes apparent. Two data sets are shown: strain rampsin which T was incremented at ε B1g < − ε s ( T ), andtemperature ramps in which strain was incremented at T > T s . The maximum compression reached was ε B1g = − . × − , which exceeds the spontaneous T → ε = − . × − , and was large enough toexceed the elastic limit of the platform. Plastic defor-mation of the platform introduced an anomalous offsetbetween ε B1g and ε A1g at large strains. Data shown inAppendix section 4, where the plastic deformation is de-scribed in more detail, show that the resistivity of FeSedepends much more sensitively on ε B1g than ε A1g , and sowe continue to plot data against ε B1g . Crucially, the sam-ple residual resistivity did not change, showing that itsown deformation remained elastic even as the platformdeformed plastically.For T above ≈
60 K, the structural strain ε s ( T ) can beidentified by a sharp change in slope dρ /dε B1g , as seenalso in Figs. 4(a–b). To obtain ε s at all temperatures, wescale ε s ( T ) from the X-ray diffraction data of Ref. [40] intemperature to match T s of this sample, and in strain tomatch the locations of the cusps. This procedure gives ε s ( T →
0) = 0 . · − . For comparison, ε s ( T →
0) =0 . × − and 0 . × − were obtained respectively inRefs. [40] and [43] by X-ray diffraction, 0 . × − and0 . × − in Refs. [44] and [45] by neutron scattering,and 0 . × − in Ref. [13] by dilatometry measurements.Fig. 5(b) shows ρ ( T ) at fixed strain ε B1g = − . × − , where the sample is detwinned at all tempera-tures. ρ evolves smoothly from T c to above T s , withno feature apparent that could be identified as a nematiccrossover. In other words, it does not appear to be use-ful to consider strain as a conjugate field to nematicityin FeSe, because even under a strain that is only barelylarge enough to detwin the sample any nematic crossoverappears to be so broad as to be indistinguishable fromthe background.We now discuss twin boundaries. For | ε B1g | < ε s ( T ), ρ from the temperature ramps systematically exceedsthat from the strain ramps. Panel (c) shows a closeup ofdata at 36.9 and 14.6 K: the T -ramp data have a peakedform that the strain-ramp data do not. The magnitudeof this peak is very similar at the two temperatures, eventhough the intrinsic resistivity at 36.9 K is more thandouble that at 14.6 K, which shows that its origin is ex-trinsic. It is due to twin boundaries. The elastic mis-match between the sample, which distorts orthorhombi-cally, and the platform, which does not, will be strongest –0.4 –0.3 –0.2 –0.1 0.0 0.1 B1g (10 )20406080100120140 ( c m ) s ( T )temperature rampsstrain rampsSample B T s = 91.9 K20 40 60 80 100 T (K)050100150 ( c m ) B1g = 0.25%: fitsunderlyingresistivity -2 estimatedtwin boundaryresistivity –0.3 –0.2 –0.1 0.0 0.1 (a)(c)(b) T -ramp data ε -ramp data FIG. 5: (a) ρ of Sample B over a wide temperature andstrain range. Points are data from temperature ramps atconstant ε B1g , and lines from strain ramps at constant tem-perature. ε s ( T ), taken as the data of Ref. [40] scaled in T and ε to match the data here, is indicated at each tempera-ture. (b) ρ ( T ) for ε B1g = − . × − , where the sampleis fully detwinned at all T . (c) Close-up of the data in panel(a) at 14.6 and 36.9 K. The squares mark points where theposition along the ε B1g axis was adjusted to correct for plas-tic deformation of the platform; see Appendix section 4 fordetails. at ε B1g = 0, leading to a peak in the equilibrium twinboundary density. This peak is resolvable for tempera-tures up to ∼
70 K, at a temperature-independent strain,which we therefore identify as the neutral strain point ε B1g = 0. Evidence for twinning is also directly visiblein the strain-ramp data in Fig. 5(c), there is hysteresisfor | ε B1g | < ε s that closes when | ε B1g | > ε s . In Appendixsection 5 we show that ramping the strain back and forthcan partially anneal twin boundaries out of the sample.A method to estimate the twin boundary contributionto the measured resistivity is illustrated in Fig. 5(c). Fora B lattice distortion, the twin boundary density is ex-pected to be symmetric about ε B1g = 0. Furthermore,because twin boundaries are oriented along (cid:104) (cid:105) direc-tions [19], no average change in twin boundary orien-tation is expected for strain with (cid:104) (cid:105) principal axes.We therefore fit lines to the temperature-ramp data oneither side of the cusp and average their slopes to ob-tain an underlying slope, meaning the slope dρ /dε B1g that would be observed if the twin boundary resistancewere zero. The line labelled “underlying resistivity” inFig. 5(c) is a line of this slope placed to intersect the dataat ε B1g = − ε s , where the sample is de-twinned. In thisway, we find that at 14.6 K the twin boundary contribu-tion to the sample resistance is as high as 15%, for thissample geometry. Twin boundary density may be lowerfor thicker and/or free-standing samples. Sample C
In Sample C both the longitudinal and transverse re-sistivities, ρ and ρ , were measured. Results fromstrain ramps are shown in Fig. 6, and from T ramps inAppendix section 6. The neutral strain point ε B1g = 0was again taken as the strain where twin boundary den-sity in the T -ramp data was highest. Around ε B1g = 0and at temperatures near T s , ρ and ρ vary stronglyand oppositely with ε B1g , confirming previous reportsthat the low-strain elastoresistivity of FeSe is dominantlyin the B channel [18, 19]. Below T s , the twinning tran-sition at ε B1g = − ε s ( T ) is broader than for Sample B. Al-though this could indicate lower sample quality, we alsonote that strain inhomogeneity will generally be worsein a square sample geometry than in the linear geometryof Sample B. To estimate ε s ( T ) for Sample C, we scale ε s ( T ) reported in Ref. [40] in temperature to match theobserved T s of Sample C, but we do not scale it in strain.The antisymmetric resistivity ρ B1g for temperaturesnear T s is plotted in panel (b). Here it can be seen that al-though | ρ B1g | initially grows rapidly with strain-inducednematicity, it eventually reaches a maximum; just above T s , this occurs at ε B1g ≈ − . · − . The symmetricresistivity ρ A1g is plotted in panel (c). For T (cid:38) T s , ρ A1g is a minimum near ε B1g = 0, and as T is reduced towards T s this minimum becomes sharper. (c)(b) ρ B1g ( ρ - ρ ) : ρ A1g ( ρ + ρ ) : T (K):10090304050607080804060100180160140120 ρ : ρ :Sample C ( T s = 89.5 K): (a) T ≈ T s FIG. 6: Data from Sample C, the Montgomery-configurationsample. (a) ρ (left) and ρ (right), from strain ramps atvarious fixed temperatures. The hysteresis is shown for twotemperatures. The vertical ticks mark − ε s ( T ), taken fromRef. [40] and scaled in temperature to match the T s observedhere. (b) ρ B1g ≡ ( ρ − ρ ) /
2, derived from the data inpanel (a), at temperatures above T s . Note that by symmetry ρ B1g ( T > T s ) = 0 at ε B1g = 0, but measurement error gives asmall deviation from this. (c) ρ A1g ≡ ( ρ + ρ ) / There are indications that other iron-based su-perconductors will have similar behavior. InBa(Fe . Co . ) As (for which the nematicityalso aligns with the (cid:104) (cid:105) directions) ρ and ρ bothhave upward curvature against ε B1g , that grows sharperas T is reduced to T ≈ T s [46], suggesting that in thismaterial too ρ A1g is a minimum for ε B1g ≈
0. ForBaFe As , ρ near T s has been observed to have anS-shaped dependence on ε , with the steepest slope ap-pearing near ε = 0 [47], matching the qualitative form(though with opposite sign) of ρ ( ε B1g ) observed here.Similar behavior is seen in Sr − x Ba x Fe . Ni . As [48]. EFFECT OF BIAXIAL STRAIN.
Data from Sample C allow effects of the A and B strain components to be separated. The A elastoresis-tivity dρ A1g /dε
A1g can be obtained by noting that withinthe twinned region B strain does not couple locallyto the sample, because the local B strain is fixed at ± ε s ( T ), but A strain does couple locally. We take ρ A1g within the twinned region as ρ A1g = ( ρ a + ρ b ) /
2, and nowdetermine dρ A1g /dε
A1g at ε B1g = ε A1g = 0.Under the approximation of linear interpolation be-tween ρ a and ρ b and neglecting twin boundary resistance, ρ and ρ in the twinned region are given by ρ = f ρ a + (1 − f ) ρ b , (1) ρ = f ρ b + (1 − f ) ρ a , (2)where f = ( ε s + ε B1g ) / ε s is the volume fraction of thesample with the nematic a axis oriented along the longaxis of the platform. Differentiating with respect to ε B1g gives: dρ dε B1g = ρ a − ρ b ε s + f dρ a dε B1g + (1 − f ) dρ b dε B1g , (3) dρ dε B1g = ρ b − ρ a ε s + f dρ b dε B1g + (1 − f ) dρ a dε B1g . (4)Under the experimental conditions here, d/dε B1g =( dε A1g /dε
B1g ) d/dε A1g = [(1 − ν ) / (1 + ν )] d/dε A1g . Sum-ming Eqs. (3) and (4) yields the A elastoresistivity: dρ A1g dε A1g = 1 + ν − ν ) (cid:18) dρ dε B1g + dρ dε B1g (cid:19) (5)To obtain underlying slopes dρ /dε B1g and dρ /dε B1g , that is, that exclude the effect of twinboundaries, we average the observed slopes on eitherside of ε B1g = 0, as shown in Fig. 5(c).The A elastoresistivity is shown in Fig. 7(a). It isnormalized by ρ A1g at ε B1g = 0 with an estimate ofthe twin boundary resistance subtracted (see Appendix section 6 for details). For temperatures below ≈
60 K, dρ A1g /dε
A1g <
0, meaning that biaxial compression in-creases the average in-plane resistivity of FeSe. A similartemperature dependence is seen in the elastoresistivity ofSample A; see Appendix section 7.We show in panel (b), with data from Sample B, thatbiaxial compression also increases T c — again, when thesample is twinned only the A component of the straincouples locally. Both the increase in T c and ρ A1g are op-posite to the generic expectation that compression shouldincrease bandwidths. A similar correlation between re-sistivity and T c is also seen in strained Sr RuO [49].At large | ε B1g | , the plastic deformation of the platformcauses a gradual relaxation of the applied A strain, andso for ε B1g (cid:46) − . × − the T c curve bends downwardsubtly. For ε B1g < − ε s , the sample detwins, and theB component of the applied strain couples locally tothe sample. T c turns downward more sharply. Depend-ing on the resistivity level selected as the criterion for T c , it may even decrease. This behavior suggests thatincreasing the lattice orthorhombicity is detrimental tosuperconductivity. THE NEMATIC RESISTIVE ANISOTROPY
We now report the central result of this paper,the nematic resistive anisotropy, both the spontaneousanisotropy below T s and that induced by strain at T ∼ T s .We obtain ρ a − ρ b at T < T s by analyzing temperature-ramp data at small strains. At ε B1g = 0, f in Eqs. (1)and (2) is 0.5, yielding ρ a − ρ b = ε s (cid:18) dρ dε B1g − dρ dε B1g (cid:19) . (6)The underlying slopes dρ /dε B1g and dρ /dε B1g areobtained, as before, by averaging the observed slopesfrom ε B1g > < T < T s , normalized by ρ A1g (with, again, an esti-mate for the twin boundary resistivity subtracted; seeAppendix section 6). Separate derivations from strain-ramp data from Sample C, and from data from SampleB, are shown in Appendix section 8; the agreement isexcellent, which confirms that the twin boundary resis-tance has been properly cancelled. The nematic resis-tive anisotropy peaks at ≈ T ≈
80 K, but thendecreases as T is reduced further, eventually changingsign at ≈
40 K. The low-temperature resistive anisotropy,where the nematicity is fully developed, is about − . X and Γ is 2–3 [22]. Any anisotropy in conduction from these Fermisurfaces individually appears to cancel almost perfectly.In contrast, resistive anisotropy in materials with mag- T (K)51510 ρ ( μ Ω c m ) B1g (10 -2 ):-0.11-0.25-0.3211 (b) -0.3 -0.2 0.10-0.1-200200 80604020 100 T (K) (a) criteria: FIG. 7: Effect of biaxial strain. (a) A elastoresistivity(1 /ρ A1g ) dρ A1g /dε
A1g versus T of Sample C, determined asexplained in the text. (b) T c versus strain, determined asthe 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 theeffect on T c is through the applied A component of thestrain. When the platform deformation is elastic, this is ε A1g = 0 . ε B1g . The observed slope therefore correspondsto dT c /dε A1g = −
450 K. netic order is much larger, for example on the order of100% in underdoped Ba(Fe,Co) As [24].In Ref. [19], ρ a and ρ b were obtained by comparingthe resistivities of stress-detwinned and unstressed sam-ples, taking the resistivity of the latter to be ( ρ a + ρ b ) / ρ a − ρ b ) / ( ρ a + ρ b ) was found to be ≈ T s , where twinboundary resistance is low compared with the total sam-ple resistance. We show in the inset of Fig. 8(a) ρ a and ρ b of Sample C, derived by taking the T -ramp resistiv-ity at ε B1g = 0 as ( ρ a + ρ b ) / ρ a and ρ b . Upon cooling into the nematic phase, ρ b is seen todecrease and ρ a to increases.In Fig. 8(b) we compare the B resistivity derivedfrom the long strain ramps shown in Fig. 6(a) to that T (K) ρ a - ρ b ρ a + ρ b ( % ) ρ B ρ A ( % ) T (K) 100 (a)(b)(c) d ρ d ε ρ × FeSe ρ - ρ ρ + ρ = long strain rampsoscillating strainfit; T s,0 = 60.7 KSample C long strain rampsfit; T s,0 = 54.8 KSample B B e l a s t o r e s i s t i v i t y ε B1g ρ ( μ Ω - c m ) T (K) Sample C ρ a ρ b FIG. 8: Nematic resistive anisotropy. (a) The spontaneousresistive anisotropy below T s , obtained as described in thetext. The inset shows ρ a and ρ b near T s . (b) B elas-toresistivity, (1 /ρ A1g ) dρ B1g /dε
B1g , obtained from both longstrain ramps and from a small oscillating strain. For Sam-ple B, ρ was not measured, so the [100] elastoresistivity(1 /ρ ) × dρ /dε is plotted instead. Fits are to a Curie-Weiss form; see the text. (c) Resistivity anisotropy of Sam-ple C against strain at T ≈ T s . The curve has been shiftedvertically to set ρ B1g = 0 at ε B1g = 0, cancelling a smallgeometrical error in the measurement. Also shown is an esti-mate of the strain-induced nematicity ψ , taken as the xz - yz energy splitting at the X point, obtained from evaluation ofGinzburg-Landau parameters. . × − at 0.0167 Hz) and the resulting oscillationamplitude of the resistivity was measured. For the longstrain ramps, the nematic resistive anisotropy at T < T s was determined by methods similar to those describedabove (See Appendix section 8 for details), and the B elastoresistivity is taken as (1 /ε s ) × ( ρ a − ρ b ) / ( ρ a + ρ b ).For T > T s and for the small-amplitude strain oscillationdata, the B elastoresistivity is (1 /ρ A1g ) × dρ B1g /dε
B1g ;these two definitions are equivalent at T = T s . Perhapssurprisingly, the small-amplitude elastoresistivity tracksthe long-strain-ramp data to well below T s , which showsthat even with a very small strain oscillation amplitudetwin boundaries shift with the applied strain.The B elastoresistivity of Sample C peaks at 62. Pre-viously reported values, from conventional measurementsin which samples are affixed directly to piezoelectric ac-tuators, are 61 [19], 38 [18], and 300 [17]. We fit thesmall-amplitude data at T > T s to a Curie-Weiss form,1 ρ A1g dρ B1g dε B1g = aT − T s,0 , which yields T s,0 = 60 . ρ was not measured, weanalyse the quantity (1 /ρ ) dρ /dε ] yields T s,0 =54 . T → ∞ limitcompression would cause resistivity to increase, which isnot expected.)In Fig. 8(c) we show the normalized resistiveanisotropy of Sample C as a function of strain at T ≈ T s .This peaks at ≈ ε B1g = − . × − , then shrinksas ε B1g becomes more negative. In order to estimate themagnitude of the strain-induced nematicity at this strain,we evaluate parameters in a Ginzburg-Landau free en-ergy, F = α × ( T − T s,0 )2 ψ + b ψ + c ε − λε B1g ψ. (7)We take ψ to be the splitting between the xz and yz or-bitals at the X point, which grows in an order-parameter-like fashion with cooling below T s and reaches 0.05 eV as T → ψ can thenbe obtained by solving dF/dψ = 0 under conditions offixed strain. Doing so and evaluating at 90 K gives theresult shown in Fig. 8(b). The maximum in the resis-tive anisotropy is found to occur when ψ ≈ .
025 eV, inother words when ψ is approximately half of its T → ψ reaches half its T → ≈
80 K [50], and so we can conclude that resistiveanisotropy is a maximum for ψ/ψ ( T → ≈ . ψ is induced through applied strain or by allowing thesample to cool. DISCUSSION
We first summarize our findings.(1) The resistive anisotropy ( ρ a − ρ b ) / ( ρ a + ρ b ) evolvesnonmonotonically as nematicity ψ grows, peaking at ≈
7% and then decreasing [Fig. 8(a)]. Both when ψ grows spontaneously with cooling and when it is inducedthrough strain at T ≈ T s , resistive anisotropy is maxi-mum when | ψ | is about half its spontaneous T → T ∼
40 K, and at low temperature, where the nematicityis fully developed, it is only ≈ − .
5% [Fig. 8(a)].(3) At T ≈ T s , ρ A1g ≡ ( ρ a + ρ b ) is a minimum whenthe sample is tetragonal [Fig. 6(c)].(4) Below ≈
60 K biaxial compression increases both ρ a + ρ b [Fig. 7(a)] and T c [Fig. 7(b)], in opposition tothe general expectation that compression increases band-widths and weaken correlations.This data set places previous low-strain measure-ments [17–19] in context of the response over a widerstrain range, over which elastoresistivity is a nontrivialfunction of nematicity ψ . It allows definitive determi-nation of the spontaneous nematic resistive anisotropy.These results are described above, so we focus the re-maining discussion on possible microscopic origins.We first consider whether the observed elastoresistiv-ity is a property of the mean-field nematic state. Thenematic transition point at ε B1g = 0 and T = T s is acritical point of the twinning transition [see Fig. 3(b)],and the fact that elastoresistivity is particularly large inits vicinity, but shrinks quickly upon moving away fromit in either temperature or strain, raises the possibilitythat strong elastoresistivity is a consequence of criticalnematic fluctuations rather than a property of the mean-field nematic state. However, two observations argueagainst this possibility. One is that for T ≈ T s , ρ A1g is a minimum near ε B1g = 0 [Fig. 6(c)], whereas if crit-ical fluctuations contributed strongly to resistivity onewould expect it to be maximum. The other is that theelastoresistivity is much stronger for strain aligned withthan transverse to the principal axes of the nematicity(that is, | dρ B1g /dε
B1g | (cid:29) | dρ B2g /dε
B2g | ), as expected formean-field nematic susceptibility. We therefore interpretthe resistivities observed here as those of the mean-fieldnematic state.The effects of biaxial strain at low temperature, likethe observation that the superconducting gap magnitudecorrelates with yz orbital weight [21, 22], point to an im-portant role for the yz orbital in electronic correlations.The yz orbital is the only one with weight both on the1Γ and X pockets, and so is thought to be the dominantcontributor to ( π,
0) spin fluctuations [53]. Inelastic neu-tron scattering measurements have shown that the onsetof nematicity correlates with stronger ( π,
0) spin fluctua-tions; Refs. [45, 54] show that there is transfer of weight,at energies ∼ k B T s relevant for transport at T ∼ T s ,from ( π, π ) to ( π,
0) and/or (0 , π ), while in Ref. [55] itis shown that the transfer is to ( π,
0) rather than (0 , π ).At low temperatures the maximum yz weight on the Γpocket is only 20% [22]. Biaxial compression, by weaken-ing nematicity and increasing bandwidths, will increasethis value, potentially strengthening the channel for ( π, T c .We focus the rest of our discussion on the nonmono-tonic dependence of the resistive anisotropy on both tem-perature and strain. We first point out that the signchange in ρ a − ρ b occurs within the inelastic compo-nent of the resistivity. A possible explanation for asign change in resistive anisotropy is that the inelasticand elastic components of the resistivity contribute op-positely, but balance at some temperature. At 40 K,however, the resistivity is about four times the residualresistivity (based on reasonable extrapolation of the re-sistivity to T → ∼ ∼
28% at verylow temperatures, in disagreement with observation thatit reaches only 1–2%.The observed temperature dependence of the resistiveanisotropy does not track thermodynamic measures ofnematicity. The orthorhombicity of the unstressed lat-tice [13, 40, 43–45], the anisotropy of the magnetic sus-ceptibility [56], and the energy splitting between the xz and yz bands [51, 52] all increase in a monotonic, order-parameter-like fashion below T s . Several factors couldcause temperature-dependent changes in resistivity. Forexample, in Ref. [57] it is found that shifting the relativeimportance of impurity versus spin fluctuation scatteringcan change the sign of the resistive anisotropy in iron-based superconductors. It is therefore important thatthis nonmonotonicity is also observed when nematicityis induced at fixed temperature, showing that it is not atemperature effect alone but intrinsic to the developmentof nematicity.The importance of this observation rests on the rela-tionship between resistive anisotropy and spin fluctua-tions. Spin fluctuations are found in theoretical work todominate the resistivity at higher temperatures [53, 57–61], and in optical conductivity measurements the DC re-sistive anisotropy is indeed found to track the scatteringrate rather than the Drude weight [62]. In Ref. [53], ( π, yz orbital weight were foundto give ρ a > ρ b , as observed, because on the hole pocketstronger scattering of quasiparticles with yz weight sup- presses conduction in the x direction. At lower tem-peratures, when spin fluctuations are weak, the preciselocations of nesting-driven hot spots on the Fermi sur-face may be decisive in determining the sign of resistiveanisotropy [27, 63], making it sensitive to details, but astemperature is raised the precise nesting conditions be-come less important [57].A further intuitive reason to expect ( π,
0) spin fluc-tuations to play a strong role in transport is that theyconnect the Γ and X Fermi surface pockets, providinga channel for umklapp scattering and momentum relax-ation along the k x direction. In a clean lattice, momen-tum is ultimately transferred to the lattice through umk-lapp scattering. In systems with closed Fermi surfaces,small-angle electron-phonon scattering can transfer mo-mentum between the electrons and phonons, but does notrelax the momentum of the combined system, and so doesnot contribute to dc resistivity. This is seen in weaklycorrelated metals (where the electron-phonon term isreadily observable) as a modification of the usual T de-pendence for electron-phonon resistivity to exponentiallyactivated, with the activation energy corresponding to aphonon that connects Fermi surfaces [64, 65]. The factthat ρ a increases when nematicity onsets [see the inset ofFig. 8(a)], while ρ b decreases, is qualitatively consistentwith the ( π,
0) spin fluctuations providing a mechanismfor preferential relaxation of transport currents along k x .We propose a specific mechanism for the non-monotonic dependence of resistive anisotropy, consistentwith data so far. ( π,
0) spin fluctuations, and the asso-ciated resistive anisotropy, strengthen as nematicity ini-tially onsets and the Fermi velocity on the yz sections ofFermi surface is reduced. These fluctuations then weakenas the nematicity grows further and suppresses the yz orbital weight on the hole pocket, cutting off this fluctu-ation channel. This is a proposal and a point for furtherinvestigation; the relative contributions of spin fluctu-ation strength and nematicity-driven changes in Fermisurface shape to resistive anisotropy need to be deter-mined. However, direct measurement of spin fluctuationsunder tunable lattice strain, through inelastic neutronscattering, would be a very challenging experiment. Itis nevertheless an important route to attempt becauseit could provide a direct test of a major class of theo-ries of the nematicity of FeSe, in which it is proposed tobe driven by the increase in phase space that it allowsfor spin fluctuations [66–68]. The potential challenge tothese theories, if the nonmonotonic resistive anisotropyobserved here indeed correlates with nonmonotonic spinfluctuation strength, is to explain why the nematicitygrows well past the point where it maximises spin fluc-tuation strength.Regardless of how that path of inquiry develops, we an-ticipate that the strain-tuning capabilities demonstratedhere will allow resolution of the separate orbital contri-butions to the electronic properties of FeSe, and theories2of the nematicity of FeSe to be tested. ACKNOWLEDGEMENTS
We thank Hiroshi Kontani, Andreas Kreisel, KazuhikoKuroki, Seiichiro Onari, Sahana R¨oßler, J¨org Schmalian,Roser Valent´ı, Matthew Watson, and Steffen Wirth foruseful discussions. S.H. and T.S. thank S. Kasahara, Y.Matsuda, K. Matsuura, and Y. Mizukami for early-stagecollaboration on sample growth. We thank the MaxPlanck Society for financial support. C.W.H., A.P.M.,and C.T. acknowledge support by the DFG (DE) throughthe Collaborative Research Centre SFB 1143 (projectsC09 and A04). C.T. acknowledges support by the DFG(DE) through the Cluster of Excellence on Complexityand Topology in Quantum Matter ct.qmat (EXC 2147).Work in Japan was supported by Grants-in-Aid for Sci-entific Research (KAKENHI) (Nos. JP19H00649 andJP18H05227), and Grant-in-Aid for Scientific Researchon innovative areas “Quantum Liquid Crystals” (Nos.JP19H05824 and JP20H05162) from Japan Society forthe Promotion of Science (JSPS).
APPENDIX1. Montgomery conversion
To measure the resistivity ρ xx parallel to the direc-tion of applied strain in FeSe we used a four-point setupwith bar-shaped samples. By applying compressive andtensile strain the resistive anisotropy in the nematic statecan be extracted. To decompose the elastoresistance intoits irreducible representations, and access to the nematicsusceptibility requires the knowledge of both ρ xx and ρ yy under applied strain.This can be achieved, for example, by measuring twosamples in perpendicular orientations, special arrange-ment of contact geometry with respect to sample ori-entation and strain direction, or by a Montgomery-typesetup which allows the simultaneous measurement of ρ xx and ρ yy in a single sample. These different approacheshave been recently reviewed in Ref. [69]. In our platform-based measurements, the center region of the platform issmall, the strain transmission length requires the sampledimension along the longitudinal axis of the platform toexceed ≈ µ m. Therefore a Montgomery configura-tion is more suitable.The Montgomery method [70, 71] allows us to converta sample with anisotropic resistivities ρ i but rectangularshape into an isotropic sample, with a single ρ , and dif-ferent effective dimensions. With a rectangular shapedsample of dimensions L , L , and thickness L , the func-tion H determines the relation between resistivity andmeasured resistance R . Following the derivations from dos Santos et al. . [72], the resistivity of an isotropic sam-ple ρ and a rectangular sample with dimensions L , L ,thickness L and measured resistances R , R can beexpressed as ρ = H t eff R (A1)where H is only a function of the geometry of the sample,i.e. H = H ( L , L ), H = H ( L , L ) and the effectivethickness t eff = t eff ( L ).Now we can compare the ratios H /H = R /R (A2)which can be used to calculate L /L in several waysEither with the definition of1 /H = 4 /π ∞ (cid:88) n =0 / { (2 n + 1) sinh[ π (2 n + 1) L /L )] } (A3)from [73], or using the approximation L L ≈ π ln R R + (cid:115)(cid:20) π ln R R (cid:21) + 4 (A4)derived by dos Santos et al. . [72].The infinite series converges rapidly, we therefore com-pute the first few terms and use a bisection algorithm tosolve eq. (A2).Furthermore we require two relations from Wasscher’stransformation [71]: L i = L (cid:48) i (cid:114) ρ i ρ (A5)and ρ = ρ ρ ρ , (A6)which connects the length of an isotropic sample L i with the corresponding dimensions and resistivity of theanisotropic sample L (cid:48) i and ρ i . With the definition of effec-tive thickness t (cid:48) eff = t eff ( L (cid:48) /L ) and in the limit of thinsamples, i.e. L / ( L L ) / < . t eff /L ≈ t (cid:48) eff ≈ L (cid:48) .This allows us to derive( ρ ρ ) / = H t (cid:48) eff R (A7)which yields a relationship between ρ and ρ : ρ = L (cid:48) L (cid:48) L L ρ . (A8) ρ is derived from the measured resistances and sampledimensions: ρ = H t (cid:48) eff R L (cid:48) L (cid:48) L L . (A9)3 Applied strain contribution.
When applying uniaxialstrain to a sample the apparent elastoresistance consistsof a purely geometric contribution from the change of itsdimensions and the strained material exhibits a differentresistivity. We take the geometric contribution into ac-count by calculating the strained sample dimensions inthe limit of small strains.For strains within the plane we assume that the plat-form is coupled rigidly enough to the sample that itsdimensions follow the applied strain from the platform: L (cid:48) , strained = L (cid:48) (1 + (cid:15) xx ) (A10) L (cid:48) , strained = L (cid:48) (1 + (cid:15) yy ) (A11)The c -axis of the sample is not constrained in the ex-periment. If we assume almost rigid coupling within theplane, the corresponding response of the sample alongthe c -axis can become significant. To include this effectwe can define a renormalized Poisson’s ratio ν ∗⊥ for theout-of-plane component: ν ∗⊥ = βν ⊥ (A12)Here β depends on the elastic moduli and the Poisson’sratios as follows: β = E (cid:107) E ⊥ − ν eff − ν (cid:107) (A13)
2. Strain transmission
When the epoxy and sample layers are both thin andthe epoxy elastic moduli are low, strain transfer to thesample can be characterized to good accuracy by a straintransmission length λ , given by λ = ( ctd/G ) / , where c is the relevant elastic modulus of the sample, t the samplethickness, d the epoxy thickness, and G the epoxy shearmodulus [74]. Under the conditions that the c -axis strainin the sample is unconstrained while the transverse strainis fixed, c = c − c /c [34]. Even though the Young’smodulus of FeSe becomes nearly zero for T ≈ T s [7], c remains substantial, at ≈
40 GPa based on the elasticmoduli reported in Refs. [39, 75, 76]. Physically, thismeans that the lattice remains stiff against biaxial com-pression, even as it becomes soft against orthorhombicdistortion. To determine d , a focused ion beam was usedto slice through some of the samples at a few points; anexample of a cross section through Sample B is shown inFig. 1(d). d was found to be 5–10 µ m. To estimate G we take the Young’s modulus of Stycast 1266, reportedin Ref. [77], and assume a Poisson’s ratio of 0.3, whichgives G = 1 . λ . For samples much narrower than λ , the transverse strain isthe longitudinal strain multiplied by the sample’s Pois-son’s ratio, while for samples much wider than λ , it is thelongitudinal strain multiplied by the platform’s Poisson’sratio. For FeSe this is an important distinction becauseits Poisson’s ratio for T ∼ T s is close to 1, while that oftitanium is 0.32. We find that all of the samples have awidth larger than ≈ λ , ensuring good locking of bothlongitudinal and transverse strains to the platform. Inparticular, Sample A is 31 µ m thick, yielding λ ≈ µ m,while its width is 280 µ m. Sample B is 10 µ m thick,yielding λ ≈ µ m, and 230 µ m wide. Complete sampledimensions are shown in Table II.
3. Elastic moduli
Ref. [39] gives elastic moduli of FeSe at T ≈ T s : c ≈ c ≈
50 GPa, c ≈
40 GPa, and c ≈
20 GPa. Un-der conditions of hydrostatic pressure, σ/ε xx = ( c c + c c − c ) / ( c − c ), where σ is the applied stress,and ε zz /ε xx = ( c + c − c ) / ( c − c ). Underconditions of in-plane biaxial stress, where σ xx = σ yy and σ zz = 0, σ xx /ε xx = c + c − c /c , and ε zz /ε xx = − c /c .
4. Plastic deformation of the platform
Sample B was driven to high compressions, and theplatform deformed plastically when the displacement D applied to it exceeded ≈ µ m, causing the strain in theneck to exceed the elastic limit of the platform material, ≈ × − . Data from Sample B were taken in the fol-lowing order: (1) Strain ramps were performed at T ≈ T s up to modest strains. (2) Temperature ramps were per-formed at constant strain, incrementing the strain at103.7 K, and moving gradually to high compressions.(3) Further strain ramps were performed at high com-pression. Data from these three sets are plotted against D in Fig. 9(a–b). There is low hysteresis within eachstrain ramp data set, and the two strain ramp data setsmatch closely except for an offset along the D axis. Thetemperature ramp data bridge this offset smoothly. Weconclude that the platform deformation was essentiallyelastic within each strain ramp data set, and that theoffset between them is due to plastic deformation causedby the large change in applied strain over the course ofthe temperature ramps.Fig. 9(c) shows a schematic illustration of the expectedform of the plastic deformation. Initially, when the plat-form deformation is elastic, ε A1g and ε B1g are linear in D : ε B1g = 0 . D/l eff and ε A1g = 0 . D/l eff (where l eff is theeffective length of the platform). Beyond its elastic limit,the platform material resists further volume compressionby flowing plastically outward: ε B1g starts to vary more4
TABLE II: Sample parameters: length, width, thickness, sep-aration l contact of the voltage contacts, and the residual resis-tivity ratio ρ (300 K) /ρ (12 K). Note that at 12 K there is stillstrong inelastic scattering.Sample l ( µ m) w ( µ m) t ( µ m) l contact ( µ m) RRRA 2370 280 31 970 26B 1150 230 10 630 22C 434 425 ≈ ρ of sam-ple B versus displacement D applied to the platform. Thedata sets were taken in the following order: (1) Strain rampsat fixed temperature. (2) Temperature ramps at fixed strain.(3) Strain ramps at higher compression. The offset betweendata sets (1) and (3) is due to plastic deformation of the plat-form that occurred over the course of the temperature ramps.(b) When data from set 3 are offset along the D axis, thematch with data set 1 is excellent. (c) Schematic illustrationof the process of plastic deformation. (d) Low-temperatureresistivity measured before and after the platform plastic de-formation. To compare data sets where T c was the same, thebefore data are taken at D = 0 . µ m and the after data at D = 1 . µ m. steeply with D , and ε A1g less steeply. When the direc-tion of the applied displacement is reversed, the platformdeformation is again elastic over some range, but for agiven
D ε
B1g is larger and ε A1g smaller than before.That the sample deformation remained elastic even asthe platform deformed plastically is shown in Fig. 9(d),in which low-temperature data from before and afterthe plastic deformation, taken at strains where T c is thesame, are plotted together. The residual resistivity isunchanged.The sign of the offset between the pre- and post-plastic-deformation data shows that ρ is controlled domi-nantly by ε B1g , rather than ε A1g . The fact that a hori-zontal displacement works so well to match the pre- andpost-plastic deformation data shows that the effect of ε A1g on ρ is small; if it were strong then it would haveto be finely balanced, over a wide temperature range,with that of ε B1g for the net effect to be so neatly a hor-izontal shift of the ρ ( D ) curves. Furthermore, the dataof Fig. 2 show directly that the dependence of ρ on ε A1g is weak.In Fig. 5(a) and (c), to account for this plastic plat-form deformation data from the high-strain strain rampsare offset by ∆ ε B1g = − . × − . Because this de-formation occurred gradually over the course of the tem-perature ramps, for ε B1g < − . × − each individualtemperature ramp is offset along the ε B1g axis to matchthe resistivity at 103.7 K with that from the strain ramps.
5. Annealing twin boundaries
In Fig. 10 we show results of a twin boundary anneal-ing experiment. Sample B was cooled from above T s to14.69 K at a fixed strain. The strain was then rampedback and forth. Over the first few cycles of strain ramp-ing, the sample resistance falls, but then settles at a lowervalue. When the strain ramp amplitude is then increased,the decrease in resistance resumes, and then the resis-tance settles at a yet lower value. This behavior showsthat twin boundaries can be partially annealed out ofthe sample through strain ramps, and confirms that thepeaked form of the resistance in T -ramp data, shown inFig. 5, is due to twin boundaries. T -ramp data from Sample C, and twin boundaryresistivity Temperature-ramp data from Sample C are shown inFig. 11. At low temperatures, the cusp in ρ ( ε B1g ) due tothe maximum in domain wall density is visible in both ρ and ρ . Its location differs slightly in the two mea-surements, possibly because in the Montgomery configu-ration measurements of ρ and ρ do not probe pre-cisely the same area of the sample. We take ε B1g = 0 as5
FIG. 10: Annealing twin boundaries out of the sample byramping the applied strain. See the Appendix text for details. the average of the cusp locations in ρ and ρ .In Fig. 11(c), we show the change in slope dρ/dε B1g across the cusp at ε B1g = 0 versus temperature. Thisquantity is proportional to the twin boundary contribu-tion to sample resistivity at ε B1g = 0. The twin bound-ary resistivity is seen to be nearly T -independent up to ∼
30 K, and then to decrease. Note that this is thetwin boundary resistivity when the sample is cooled fromabove T s at ε B1g = 0; when it is brought to ε B1g = 0 byramping strain at constant temperature, the twin bound-ary density is lower.In Figs. 7(a) and 8(a), elastoresistivities normalized by ρ a + ρ b are shown. For this normalization we subtractedoff an estimated twin boundary resistivity, ρ TB ( T ); forexample, in Fig. 8(a) the quantity that is plotted is ρ a − ρ b , determined by the underlying slopes methoddescribed in the text, divided by ρ ( ε B1g = 0) + ρ ( ε B1g = 0) − ρ TB ( T ). Based on the illustration inFig. 5(c), we estimate ρ TB ( T →
0) = 3 µ Ω-cm. We take ρ TB = ρ TB ( T → × [1 − ( T /T s ) ]. This form overesti-mates somewhat the true twin boundary resistance as T approaces T s , however the effect is tiny.
7. Elastoresistivity of Sample A
Fig. 12 shows the elastoresistivity of Sample A overa wide temperature range. The behavior qualitativelymatches the A elastoresistivity determined from Sam-ple C, and plotted in Fig. 7(a): at higher tempera-tures, compression causes a decrease in resistivity, and atlower temperatures an increase. The sign of the responsechanges at T ≈
45 K, against 60 K for the A elastore-sistivity of Sample C. The measured resistivity of SampleA will also be affected by the B elastoresistivity; how-ever, because this is transverse to the nematic axes it isnot expected to be large, and the qualitative agreementwith the A elastoresistivity suggests that it is indeedmuch smaller than the A elastoresistivity. Note alsothat T c increases with compression, as observed in Sam- ρ ( μ Ω - c m ) temperature (K) ε B1g (10 -2 )(a) ρ : (b) ρ : T = T s temperature (K) c hange i n s l ope a c r o ss ε B = ( μ Ω - c m ) black: ρ red: ρ (c) FIG. 11: (a–b) Temperature ramp data from Sample C; panel(a) shows ρ and panel (b) ρ . (c) Change in slope dρ/dε B1g across ε B1g = 0. This quantity is proportional to thetwin boundary contribution to sample resistivity at ε B1g = 0. ple B [Fig. 7(b)].
8. Additional derivations of the nematic resistiveanisotropy
Above, we presented a determination of the nematicresistive anisotropy for
T < T s based on Sample Ctemperature-ramp data, in which the twin distributioncan be assumed to be in near equilibrium with the appliedstrain. Here, we analyze strain-ramp data. As described6 FIG. 12: ρ versus T of Sample A over a wide temperaturerange. above, the determination of nematic resistive anisotropydepends on extraction of the slopes dρ /dε B1g and dρ /dε B1g at ε B1g = 0 and under the condition thatthe twin boundary configuration does not change. In thestrain ramps, the density and location of twin bound-aries lags the applied strain, and we therefore obtainthese slopes by averaging the observed slopes from theincreasing-strain and decreasing-strain ramps, as illus-trated in the inset of Fig. 13. Applying Eq. (6) yields thenematic resistive anisotropy plotted in Fig. 13. The closeagreement with T -ramp data shows that the twin bound-ary resistance has been properly excluded. Note that, be-cause the twin boundary density is lower in strain-rampthan temperature-ramp data, we do not subtract off atwin boundary contribution.Also shown in Fig. 13 is the resistivity anisotropy de-termined from Sample B. For Sample B, only ρ wasmeasured. Evaluating Eq. (1) at f = 0 . ρ a − ρ b = 2 ε s (cid:18) dρ dε B1g − − ν ν dρ A1g dε A1g (cid:19) . (A14) dρ A1g /dε
A1g must be taken from data from Sample C[see Fig. 7(a)]; the data plotted in Fig. 13 includes thiscorrection. For the normalization our estimate for twinboundary resistivity is subtracted (see Appendix section6).
9. Ginzburg-Landau parameters
In the Ginzburg-Landau free energy [Eq. (7)], thestrain is the B strain, for which the elastic constant c is c − c . This elastic constant must be evaluatedwithout the influence of nematic susceptibility. Ref. [39]finds c ≈
80 GPa at T ≈
250 K, and electronic struc-ture calculations give c = 95 GPa [78]. We take theestimate c = c − c = 60 GPa. The structural strain dρ / dε B1g
FIG. 13: Nematic resistivity anisotropy ( ρ a − ρ b ) / ( ρ a + ρ b )of Sample C, derived from the strain-ramp data shown inFig. 6(a). This determination is based on extraction of equi-librium slopes dρ/dε B1g at ε B1g = 0, obtained by averagingthe observed increasing- ε and decreasing- ε slopes at ε B1g = 0,as shown in the inset. is obtained by noting that dF/dε = 0 at ε = ε s , whichgives ε s = ( λ/c ) ψ . Although the Ginzburg-Landau for-malism only applies, strictly, very near to T s , we eval-uate parameters at considerably lower temperature inorder to obtain approximate evaluations of the coeffi-cients. ε s → . × − as T → λ ≈ . T s,0 = 60 . T s,0 = 60 K. T s is defined by the relationship λ c − α × ( T s − T s,0 ) = 0 . (A15)Taking T s = 90 K yields α = 0 . -K. Finally,we evaluate b from the observation that ψ reaches halfits T →
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