The relationship between transport anisotropy and nematicity in FeSe

Jack Bartlett,1, 2, ∗ Alexander Steppke,1, † Suguru Hosoi,3, 4 Hilary Noad,1 Joonbum Park,1

Carsten Timm,5, 6 Takasada Shibauchi,3 Andrew P. Mackenzie,1, 2 and Clifford W. Hicks1, 7, ‡

1Max Planck Institute for Chemical Physics of Solids, Nothnitzer Str 40, 01187 Dresden, Germany2SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews KY16 9SS, United Kingdom

3Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan4Department of Materials Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

5Institute of Theoretical Physics, Technische Universitat Dresden, 01062 Dresden, Germany6Wurzburg-Dresden Cluster of Excellence ct.qmat,

Technische Universitat Dresden, 01062 Dresden, Germany7School 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 Ts is determined.(2) Resistive anisotropy is measured as nematicity is induced through applied strain at fixed tem-perature above Ts. 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 temperatureTc, 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 BaFe2As2 [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 (ρxx−ρyy)/(ρxx +ρyy) 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 Ts,0, but the elastic compliance of the lattice raisesthe transition temperature to Ts > Ts,0. For T > Ts,applied anisotropic strain ε induces nonzero ψ throughelectron-lattice coupling, with a susceptibility dψ/dε thatdiverges (with divergence temperature Ts,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

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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 ∼ Ts. (2) Strain-tuning is employed to control the twinning as samplesare cooled, allowing measurement of the intrinsic resistiveanisotropy at temperatures below Ts. 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 < Ts, in comparison withthat induced by strain at T > Ts, is shown in Fig. 8.

For orientation, the electronic structure of FeSe aboveand far below Ts 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 〈100〉 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 kx, signatures of the Y pocketdisappear from spectroscopic probes [17, 21, 30–33], andthe hole pocket becomes elongated along the ky 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 μm

500 μm

10 μm

5 μm

FeSe

epoxyTi

sampleplatform

Sample B

Sample C

(b)

X

Y

X

Y(a) T Ts:T > Ts:

a

b yzxzxy

piezoelectric actuators

(c)

(e) (f)

Γ

12 mm

capacitive displacement sensor

platform

stressapparatus

x

zy

epoxy

xy

xy

[100][010]

[100][010]

(d) Sample B

Sample C

FIG. 1: (a) Schematic illustration of the electronic struc-ture above and below the nematic transition temperature Ts.Fermi surfaces are colored by their dominant orbital content.For T < Ts, kx is oriented along the crystalline a axis, wherea > 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.

3

We report data from three samples. Sample A was cutfor application of strain with 〈110〉 principal axes, andSamples B and C with 〈100〉 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 ρc/ρab of FeSeappears not to have been measured, however that ofFeSe0.4Te0.6 is ≈70 at 15 K [35]. The length scale forcurrent injected at the upper surface to spread out overthe full sample thickness is t(ρc/ρab)

1/2, where t ∼ 10 µmis the sample thickness. This length scale is short enoughthat measurements here are not strongly affected by thec-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, ε110. The transverse strainε110 is given by ε110 = −νε110, where ν = 0.32 is thePoisson’s ratio of the platform. The symmetric compo-nent of the strain field is εA1g ≡ 1

2 (ε110 + ε110), whichcomes to 0.34ε110, while the antisymmetric componentis εB2g ≡ 1

2 (ε110 − ε110) = 0.66ε110. These parameters,along with equivalent parameters for Samples B and C,are summarized in Table I. We also label resistivities bythe measurement axis: ρ100, for example, is the resistiv-ity along the [100] direction. For Sample C both ρ100 andρ010 are measured, and so symmetric and antisymmetricresistivities can be defined: ρA1g ≡ 1

2 (ρ100 + ρ010) andρB1g ≡ 1

2 (ρ100 − ρ010).We note that specifying lattice distortions becomes

more 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 〈100〉 directions are Fe-Fe bond directions. SampleA is aligned so that stress is applied along the [110] latticedirection; the strain along this axis, ε110, 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 < 0 denotes compression.

Sample A

SeFe

εA1g ≡ 12(ε110 + ε110) = 1

2(1− ν)ε110 = 0.34ε110

εB2g ≡ 12(ε110 − ε110) = 1

2(1 + ν)ε110 = 0.66ε110

Samples B and C

εA1g ≡ 12(ε100 + ε010) = 1

2(1− ν)ε110 = 0.34ε100

εB1g ≡ 12(ε100 − ε010) = 1

2(1 + ν)ε110 = 0.66ε010

length of the platform. When the sample twins, we use aand 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 ac-axis strain in the sample, ε001 = −2c13εA1g/c33. c-axisstrain preserves the tetragonal symmetry of the T > Ts

lattice, and therefore is in the A1g representation. Whenwe discuss A1g 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/ρA1g)dρB1g/dεB1g at εB1g = 0 is equal to m11 −m12

in that formalism. Most previous elastoresistivity resultshave been reported using the 2-Fe unit cell, in whichm11 −m12 transforms to 2m66.

RESULTS: 〈110〉 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 Ts

by only a few kelvin. Ts 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 dTs/dε110 = 750 K. This slope is dueto the A1g component of the applied strain: under thetetragonal symmetry of FeSe at T > Ts, reversal of thesign of εB2g gives a symmetrically equivalent strain, socoupling to εB2g can give only strain-even components inthe strain dependence of Ts.εA1g = 0.34ε110, so dTs/dε110 = 750 K corresponds to

dTs/dεA1g = 2200 K. In Ref. [38], Ts is found to be sup-

4

FeSe

FIG. 2: The effect of transverse strain. (a) ρ110(T ), the re-sistivity along the [110] direction, for Sample A at variousapplied strains ε110. The principal axes of the nematicity inFeSe are the 〈100〉 axes, so this strain is a transverse field tothe nematicity. The inset is a schematic of the strain axis.(b) Ts versus ε110 for this sample. Ts is identified as the max-imum in d2ρ110/dT

2. The shaded region is a measure of thewidth of the transition; it is where d2ρ/dT 2 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 dTs/dεA1g ≈ 6200 K. (See Appendixsection 3 for details.) The difference between this andour result allows, in principle, separation of the effectof c-axis strain ε001 and that of “pure” in-plane biax-ial strain εA1g, pure that has no associated c-axis strain.Applying again the elastic moduli from Ref. [39], un-der in-plane uniaxial stress ε001 = −0.3 × εA1g, pure,and under hydrostatic stress ε001 = 1.0 × εA1g, pure, so∆Ts ≈ (3200 K)× εA1g, pure + (1000 K)× ε001.

RESULTS: 〈100〉 STRAIN

Stress-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 nonzeroH 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/µ0 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 = Ts 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 B1g distortion,meaning that b contracts by nearly the same amount asa lengthens [40–42]. Therefore, the nematicity-inducedstructural distortion can be described as a spontaneouslocal strain εB1g, local = ±εs(T ), where the quantity εs

is termed the structural strain. The average strain inthe sample must match that of the platform, but when|εB1g| < εs(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 εB1g 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 > Ts: 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 ∼ Ts

Measurements of resistivity confirm this qualitativeform of strain-temperature phase diagram. To facili-tate comparison with measurements of εs, we now plotdata against the antisymmetric strain εB1g. ρ100(εB1g)of Sample B for T ∼ Ts is shown in Fig. 4(a), and thederivative dρ100/dεB1g in panel (b). The neutral strainpoint εB1g = 0 is determined as the strain where the twinboundary density for T < Ts is highest; these data areshown below. Above Ts, the strain dependence of ρ100

is seen to have substantial nonlinearity even over a rela-tively small strain range |εB1g| < 0.1 · 10−2. Its slope is

5

first-ordertransition

crossover

twinned

stress σB1g

tem

pera

ture

0 0–εs(T 0) +εs(T 0)

2nd-ordertransition

strain εB1g

(a) (b)

Ts

2nd-ordertransition

FIG. 3: Schematic phase diagrams. (a) Schematic stress-temperature phase diagram for the nematicity of FeSe, forstress applied with B1g 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, εB1g = ±εs(T ).

largest near, though not precisely at, εB1g = 0.

As T is reduced below Ts, the onset of twinning changesthe form of ρ100(εB1g): a range of strain appears overwhich dρ100/dεB1g 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 andb axes are ρa and ρb, and the equilibrium twin volumeratio is a linear function of applied strain. Therefore,the observed bulk resistivity is an interpolation betweenρb at εB1g = −εs and ρa at εB1g = +εs, that to highprecision is linear under two conditions that are bothsatisfied here. (1) |(ρa − ρb)/(ρa + ρb)| 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 Ts.

Even though the transitions into the twinned regionmust, when εB1g 6= 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 atεB1g = 0, but slightly on the tensile side. This asym-metry is due to the A1g component of the applied strain:as shown with Sample A in Fig. 2, tensile A1g strain in-creases Ts.

ρ versus temperature at a few nonzero εB1g are shownin Fig. 4(d), and Fig. 4(e) shows Ts derived from suchtemperature sweeps as a function of strain. For bothSamples B and C, Ts 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 Ts. (a) ρ100(εB1g), whereρ100 is the resistivity along the [100] direction and εB1g ≡(ε100− ε010)/2, of Sample B at various temperatures near Ts.(b) dρ100/dεB1g for the curves from panel (a). For T < Ts,dρ100/dεB1g becomes nearly constant over the range where thesample twins. This range is indicated for the 86.8 K curve.(c) Schematic of ρ100(εB1g) for T < Ts; the underlying curve isnot accessible for −εs < εB1g < +εs due to the onset of twin-ning, and the observed resistivity instead interpolates overthis range. (d) Temperature ramps at three values of εB1g.(e) Ts versus strain for low strains. The shaded regions indi-cate the transition width, defined by d2ρ100/dT

2 crossing halfits maximum value.

6

Sample B, T < Ts

Fig. 5(a) shows ρ100 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 εB1g < −εs(T ), andtemperature ramps in which strain was incremented atT > Ts. The maximum compression reached was εB1g =−0.28 × 10−2, which exceeds the spontaneous T → 0structural distortion of FeSe and fully detwins the sam-ple at all temperatures. It corresponds to a longitudinalstrain of ε100 = −0.42 × 10−2, and was large enough toexceed the elastic limit of the platform. Plastic defor-mation of the platform introduced an anomalous offsetbetween εB1g and εA1g 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 εB1g than εA1g, and sowe continue to plot data against εB1g. 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 εs(T ) can beidentified by a sharp change in slope dρ100/dεB1g, as seenalso in Figs. 4(a–b). To obtain εs at all temperatures, wescale εs(T ) from the X-ray diffraction data of Ref. [40] intemperature to match Ts of this sample, and in strain tomatch the locations of the cusps. This procedure givesεs(T → 0) = 0.22 · 10−2. For comparison, εs(T → 0) =0.27×10−2 and 0.23×10−2 were obtained respectively inRefs. [40] and [43] by X-ray diffraction, 0.24× 10−2 and0.25× 10−2 in Refs. [44] and [45] by neutron scattering,and 0.22×10−2 in Ref. [13] by dilatometry measurements.

Fig. 5(b) shows ρ100(T ) at fixed strain εB1g = −0.25×10−2, where the sample is detwinned at all tempera-tures. ρ100 evolves smoothly from Tc to above Ts, 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 |εB1g| < εs(T ),ρ100 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 0.1B1g (10 2)

20

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100

(cm

)

101.8 K99.8 K97.8 K95.7 K91.7 K

87.8 K83.8 K79.8 K76.3 K72.3 K68.4 K64.4 K60.4 K56.5 K52.5 K48.5 K44.6 K40.9 K36.9 K33.0 K29.0 K25.0 K21.0 K17.1 K14.6 K

s(T )

temperature rampsstrain ramps

Sample BTs = 91.9 K

20 40 60 80 100T (K)

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(cm

) B1g = 0.25%:

fits

underlyingresistivity

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estimatedtwin boundaryresistivity

–0.3 –0.2 –0.1 0.0 0.1

(a)

(c)

(b)

T-ramp dataε-ramp data

FIG. 5: (a) ρ100 of Sample B over a wide temperature andstrain range. Points are data from temperature ramps atconstant εB1g, and lines from strain ramps at constant tem-perature. εs(T ), taken as the data of Ref. [40] scaled in Tand ε to match the data here, is indicated at each tempera-ture. (b) ρ100(T ) for εB1g = −0.25× 10−2, 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 εB1g axis was adjusted to correct for plas-tic deformation of the platform; see Appendix section 4 fordetails.

7

at εB1g = 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 pointεB1g = 0. Evidence for twinning is also directly visiblein the strain-ramp data in Fig. 5(c), there is hysteresisfor |εB1g| < εs that closes when |εB1g| > εs. 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 B1g lattice distortion, the twin boundary density is ex-pected to be symmetric about εB1g = 0. Furthermore,because twin boundaries are oriented along 〈110〉 direc-tions [19], no average change in twin boundary orien-tation is expected for strain with 〈100〉 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 dρ100/dεB1g

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 εB1g = −εs, 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, ρ100 and ρ010, were measured. Results fromstrain ramps are shown in Fig. 6, and from T ramps inAppendix section 6. The neutral strain point εB1g = 0was again taken as the strain where twin boundary den-sity in the T -ramp data was highest. Around εB1g = 0and at temperatures near Ts, ρ100 and ρ010 vary stronglyand oppositely with εB1g, confirming previous reportsthat the low-strain elastoresistivity of FeSe is dominantlyin the B1g channel [18, 19]. Below Ts, the twinning tran-sition at εB1g = −εs(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 εs(T ) for Sample C, we scaleεs(T ) reported in Ref. [40] in temperature to match theobserved Ts of Sample C, but we do not scale it in strain.

The antisymmetric resistivity ρB1g for temperaturesnear Ts is plotted in panel (b). Here it can be seen that al-though |ρB1g| initially grows rapidly with strain-inducednematicity, it eventually reaches a maximum; just aboveTs, this occurs at εB1g ≈ −0.19 · 10−2. The symmetricresistivity ρA1g is plotted in panel (c). For T & Ts, ρA1g

is a minimum near εB1g = 0, and as T is reduced towardsTs this minimum becomes sharper.

(c)

(b) ρB1g (ρ100 - ρ010) :12

ρA1g (ρ100 + ρ010) :12

T (K):

100

90

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120

ρ100: ρ010:

Sample C (Ts = 89.5 K):(a)

T ≈ Ts

FIG. 6: Data from Sample C, the Montgomery-configurationsample. (a) ρ100 (left) and ρ010 (right), from strain ramps atvarious fixed temperatures. The hysteresis is shown for twotemperatures. The vertical ticks mark −εs(T ), taken fromRef. [40] and scaled in temperature to match the Ts observedhere. (b) ρB1g ≡ (ρ100 − ρ010)/2, derived from the data inpanel (a), at temperatures above Ts. Note that by symmetryρB1g(T > Ts) = 0 at εB1g = 0, but measurement error gives asmall deviation from this. (c) ρA1g ≡ (ρ100 + ρ010)/2.

8

There are indications that other iron-based su-perconductors will have similar behavior. InBa(Fe0.975Co0.025)2As2 (for which the nematicityalso aligns with the 〈100〉 directions) ρ100 and ρ010 bothhave upward curvature against εB1g, that grows sharperas T is reduced to T ≈ Ts [46], suggesting that in thismaterial too ρA1g is a minimum for εB1g ≈ 0. ForBaFe2As2, ρ100 near Ts has been observed to have anS-shaped dependence on ε100, with the steepest slope ap-pearing near ε100 = 0 [47], matching the qualitative form(though with opposite sign) of ρ100(εB1g) observed here.Similar behavior is seen in Sr1−xBaxFe1.97Ni0.03As2 [48].

EFFECT OF BIAXIAL STRAIN.

Data from Sample C allow effects of the A1g and B1g

strain components to be separated. The A1g elastoresis-tivity dρA1g/dεA1g can be obtained by noting that withinthe twinned region B1g strain does not couple locallyto the sample, because the local B1g strain is fixed at±εs(T ), but A1g strain does couple locally. We take ρA1g

within the twinned region as ρA1g = (ρa+ρb)/2, and nowdetermine dρA1g/dεA1g at εB1g = εA1g = 0.

Under the approximation of linear interpolation be-tween ρa and ρb and neglecting twin boundary resistance,ρ100 and ρ010 in the twinned region are given by

ρ100 = fρa + (1− f)ρb, (1)

ρ010 = fρb + (1− f)ρa, (2)

where f = (εs + εB1g)/2εs is the volume fraction of thesample with the nematic a axis oriented along the longaxis of the platform. Differentiating with respect to εB1g

gives:

dρ100

dεB1g=ρa − ρb

2εs+ f

dρadεB1g

+ (1− f)dρbdεB1g

, (3)

dρ010

dεB1g=ρb − ρa

2εs+ f

dρbdεB1g

+ (1− f)dρadεB1g

. (4)

Under the experimental conditions here, d/dεB1g =(dεA1g/dεB1g)d/dεA1g = [(1− ν)/(1 + ν)]d/dεA1g. Sum-ming Eqs. (3) and (4) yields the A1g elastoresistivity:

dρA1g

dεA1g=

1 + ν

2(1− ν)

(dρ100

dεB1g+dρ010

dεB1g

)(5)

To obtain underlying slopes dρ100/dεB1g anddρ010/dεB1g, that is, that exclude the effect of twinboundaries, we average the observed slopes on eitherside of εB1g = 0, as shown in Fig. 5(c).

The A1g elastoresistivity is shown in Fig. 7(a). It isnormalized by ρA1g at εB1g = 0 with an estimate ofthe twin boundary resistance subtracted (see Appendix

section 6 for details). For temperatures below ≈60 K,dρA1g/dεA1g < 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 Tc— again, when thesample is twinned only the A1g component of the straincouples locally. Both the increase in Tc and ρA1g are op-posite to the generic expectation that compression shouldincrease bandwidths. A similar correlation between re-sistivity and Tc is also seen in strained Sr2RuO4 [49].

At large |εB1g|, the plastic deformation of the platformcauses a gradual relaxation of the applied A1g strain, andso for εB1g . −0.15×10−2 the Tc curve bends downwardsubtly. For εB1g < −εs, the sample detwins, and theB1g component of the applied strain couples locally tothe sample. Tc turns downward more sharply. Depend-ing on the resistivity level selected as the criterion forTc, 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 Ts and that induced by strain at T ∼ Ts.We obtain ρa − ρb at T < Ts by analyzing temperature-ramp data at small strains. At εB1g = 0, f in Eqs. (1)and (2) is 0.5, yielding

ρa − ρb = εs

(dρ100

dεB1g− dρ010

dεB1g

). (6)

The underlying slopes dρ100/dεB1g and dρ010/dεB1g areobtained, as before, by averaging the observed slopesfrom εB1g > 0 and < 0.

In Fig. 8(a) we show the nematic resistive anisotropyat T < Ts, normalized by ρA1g (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 ≈7%, 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 −1.5%.This is surprisingly small: in ARPES data, the length-to-width ratios of the Fermi surfaces at 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-

9

08 9 10

T (K)

5

15

10ρ 1

00 (μ

Ω c

m)

0.130.05-0.03

εB1g (10-2):-0.11-0.25-0.32

11

6 μΩ cm4 μΩ cm2 μΩ cm

9.8

9.6

8.4

8.6

8.8

9.0

9.2

9.4

(b)

-0.3 -0.2 0.10-0.1

-20

0

20

0 80604020 100T (K)

(a)

criteria:

FIG. 7: Effect of biaxial strain. (a) A1g elastoresistivity(1/ρA1g)dρA1g/dεA1g versus T of Sample C, determined asexplained in the text. (b) Tc versus strain, determined asthe temperature where the resistivity crosses specific values,as shown in the inset. Note that within the twinned region,εB1g does not couple locally to the sample, and instead theeffect on Tc is through the applied A1g component of thestrain. When the platform deformation is elastic, this isεA1g = 0.52εB1g. The observed slope therefore correspondsto dTc/dεA1g = −450 K.

netic order is much larger, for example on the order of100% in underdoped Ba(Fe,Co)2As2 [24].

In Ref. [19], ρa and ρb were obtained by comparingthe resistivities of stress-detwinned and unstressed sam-ples, taking the resistivity of the latter to be (ρa +ρb)/2.(ρa−ρb)/(ρa +ρb) was found to be ≈3%, with weak tem-perature dependence, in qualitative disagreement withthe results here. However, this analysis method treatsthe twin boundary resistance as negligible, which we haveshown not to be a good approximation at lower temper-atures.

This is, however, a valid approach near Ts, where twinboundary resistance is low compared with the total sam-ple resistance. We show in the inset of Fig. 8(a) ρa andρb of Sample C, derived by taking the T -ramp resistiv-ity at εB1g = 0 as (ρa + ρb)/2 and then applying theanisotropy plotted in the main panel to obtain ρa andρb. Upon cooling into the nematic phase, ρb is seen todecrease and ρa to increases.

In Fig. 8(b) we compare the B1g resistivity derivedfrom the long strain ramps shown in Fig. 6(a) to that

0 1208040T (K)

ρ a -

ρ bρ a

+ ρ

b(%

)ρ B

1g

ρ A1g

(%)

0

20

40

60

-20

0 80604020T (K)

100

(a)

(b)

(c)

dρ10

0

dε10

0

1 ρ 100

×

FeSe

2

0

-6

-4

-2

0.01

0

-0.03

-0.02

-0.01

ρ 100

- ρ 0

10

ρ 100

+ ρ

010

=

long strain rampsoscillating strainfit; Ts,0 = 60.7 K

Sample C

long strain rampsfit; Ts,0 = 54.8 K

Sample B

B1g

ela

stor

esis

tivity

0

20

40

60

-20

εB1g

8

6

4

2

0

-2

ρ (μ

Ω-c

m)

120

140

160

80 90 100T (K)

Sample C

ρa

ρb

FIG. 8: Nematic resistive anisotropy. (a) The spontaneousresistive anisotropy below Ts, obtained as described in thetext. The inset shows ρa and ρb near Ts. (b) B1g elas-toresistivity, (1/ρA1g)dρB1g/dεB1g, obtained from both longstrain ramps and from a small oscillating strain. For Sam-ple B, ρ010 was not measured, so the [100] elastoresistivity(1/ρ100)× dρ100/dε100 is plotted instead. Fits are to a Curie-Weiss form; see the text. (c) Resistivity anisotropy of Sam-ple C against strain at T ≈ Ts. The curve has been shiftedvertically to set ρB1g = 0 at εB1g = 0, cancelling a smallgeometrical error in the measurement. Also shown is an esti-mate of the strain-induced nematicity ψ, taken as the xz-yzenergy splitting at the X point, obtained from evaluation ofGinzburg-Landau parameters.

10

from a classic elastoresistivity measurement, also per-formed on Sample C, in which the strain was oscillatedby a small amplitude (here, a peak-to-peak amplitudeof 3.4× 10−5 at 0.0167 Hz) and the resulting oscillationamplitude of the resistivity was measured. For the longstrain ramps, the nematic resistive anisotropy at T < Ts

was determined by methods similar to those describedabove (See Appendix section 8 for details), and the B1g

elastoresistivity is taken as (1/εs)× (ρa − ρb)/(ρa + ρb).For T > Ts and for the small-amplitude strain oscillationdata, the B1g elastoresistivity is (1/ρA1g)× dρB1g/dεB1g;these two definitions are equivalent at T = Ts. Perhapssurprisingly, the small-amplitude elastoresistivity tracksthe long-strain-ramp data to well below Ts, which showsthat even with a very small strain oscillation amplitudetwin boundaries shift with the applied strain.

The B1g 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 > Ts to a Curie-Weiss form,

1

ρA1g

dρB1g

dεB1g=

a

T − Ts,0,

which yields Ts,0 = 60.7 K. A similar fit to data fromSample B [where, because ρ010 was not measured, weanalyse the quantity (1/ρ100)dρ100/dε100] yields Ts,0 =54.8 K. (We note that we do not include a high-temperature offset term in these fits, because doing so re-turns negative values, implying that in the 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 ≈ Ts.This peaks at ≈6%, at εB1g = −0.18×10−2, then shrinksas εB1g 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 − Ts,0)

2ψ2 +

b

4ψ4 +

c

2ε2

B1g − λεB1gψ. (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 Ts and reaches 0.05 eV asT → 0 [50–52]. Numerical values for each parameter aredetermined from experimental data, as explained in Ap-pendix section 9. The strain dependence of 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 ≈ 0.025 eV, inother words when ψ is approximately half of its T → 0value. This conclusion is robust against reasonable vari-ation of the Ginzburg-Landau parameters. When an un-stressed sample is cooled, ψ reaches half its T → 0 value

at ≈80 K [50], and so we can conclude that resistiveanisotropy is a maximum for ψ/ψ(T → 0) ≈ 0.5 whetherψ is induced through applied strain or by allowing thesample to cool.

DISCUSSION

We first summarize our findings.(1) The resistive anisotropy (ρa−ρb)/(ρa +ρb) evolves

nonmonotonically as nematicity ψ grows, peaking at≈7% and then decreasing [Fig. 8(a)]. Both when ψgrows spontaneously with cooling and when it is inducedthrough strain at T ≈ Ts, resistive anisotropy is maxi-mum when |ψ| is about half its spontaneous T → 0 value.

(2) The nematic resistive anisotropy changes sign atT ∼ 40 K, and at low temperature, where the nematicityis fully developed, it is only ≈ −1.5% [Fig. 8(a)].

(3) At T ≈ Ts, ρA1g ≡ 12 (ρa + ρb) is a minimum when

the sample is tetragonal [Fig. 6(c)].(4) Below ≈60 K biaxial compression increases both

ρa + ρb [Fig. 7(a)] and Tc [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 εB1g = 0 and T = Ts 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 ≈ Ts, ρA1g

is a minimum near εB1g = 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, |dρB1g/dεB1g| |dρB2g/dεB2g|), 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 the

11

Γ 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 ∼ kBTs relevant for transport at T ∼ Ts,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, 0)spin fluctuations and causing the increase in both resi-tivity and Tc.

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 ρa − ρb 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 → 0), so for this explanation to apply theelastic resistive anisotropy would need to be about fourtimes the inelastic resistive anisotropy, or ∼ 28%. Theresistive anisotropy would then grow to ∼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 xzand yz bands [51, 52] all increase in a monotonic, order-parameter-like fashion below Ts. 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, 0)fluctuations relying on the yz orbital weight were foundto give ρa > ρb, 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 kx 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 5 de-pendence for electron-phonon resistivity to exponentiallyactivated, with the activation energy corresponding to aphonon that connects Fermi surfaces [64, 65]. The factthat ρa increases when nematicity onsets [see the inset ofFig. 8(a)], while ρb decreases, is qualitatively consistentwith the (π, 0) spin fluctuations providing a mechanismfor preferential relaxation of transport currents along kx.

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 yzorbital 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 theories

12

of the nematicity of FeSe to be tested.

ACKNOWLEDGEMENTS

We thank Hiroshi Kontani, Andreas Kreisel, KazuhikoKuroki, Seiichiro Onari, Sahana Roßler, Jorg Schmalian,Roser Valentı, 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).

APPENDIX

1. Montgomery conversion

To measure the resistivity ρxx 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 ρxx and ρyyunder 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 ρxxand ρyy 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 ≈ 200 µm. Therefore a Montgomery configura-tion is more suitable.

The Montgomery method [70, 71] allows us to converta sample with anisotropic resistivities ρi but rectangularshape into an isotropic sample, with a single ρ, and dif-ferent effective dimensions. With a rectangular shapedsample of dimensions L1, L2, and thickness L3, 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 L1, L2,thickness L3 and measured resistances R1, R2 can beexpressed as

ρ = H1teffR1 (A1)

where H is only a function of the geometry of the sample,i.e. H1 = H(L1, L2), H2 = H(L2, L1) and the effectivethickness teff = teff(L3).

Now we can compare the ratios

H1/H2 = R1/R2 (A2)

which can be used to calculate L1/L2 in several waysEither with the definition of

1/H1 = 4/π

∞∑n=0

2/(2n+ 1) sinh[π(2n+ 1)L1/L2)]

(A3)from [73], or using the approximation

L2

L1≈ 1

2

1

πlnR2

R1+

√[1

πlnR2

R1

]2

+ 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]:

Li = L′i

√ρiρ

(A5)

and

ρ3 = ρ1ρ2ρ3, (A6)

which connects the length of an isotropic sample Li

with the corresponding dimensions and resistivity of theanisotropic sample L′i and ρi. With the definition of effec-tive thickness t′eff = teff (L′3/L3) and in the limit of thinsamples, i.e. L3/(L1L2)1/2 < 0.5 the ratio teff/L3 ≈ 1,and therefore also t′eff ≈ L′3.

This allows us to derive

(ρ1ρ2)1/2

= H1t′effR1 (A7)

which yields a relationship between ρ1 and ρ2:

ρ1 =L′22L′21

L21

L22

ρ2. (A8)

ρ2 is derived from the measured resistances and sampledimensions:

ρ2 = H1t′effR1

L′1L′2

L2

L1. (A9)

13

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′1,strained = L′1 (1 + εxx) (A10)

L′2,strained = L′2 (1 + εyy) (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‖

E⊥

1− νeff

1− ν‖(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)1/2, where cis 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 = c11 − c213/c33 [34]. Even though the Young’smodulus of FeSe becomes nearly zero for T ≈ Ts [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 µm. To estimate Gwe take the Young’s modulus of Stycast 1266, reportedin Ref. [77], and assume a Poisson’s ratio of 0.3, whichgives G = 1.6 GPa at low temperature.

Samples A and B are both long, ensuring good cou-pling of longitudinal strain to the platform, and so thekey question is of their width in comparison with 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 ∼ Ts is close to 1, while that oftitanium is 0.32. We find that all of the samples have awidth larger than ≈ 4λ, ensuring good locking of bothlongitudinal and transverse strains to the platform. Inparticular, Sample A is 31 µm thick, yielding λ ≈ 60 µm,while its width is 280 µm. Sample B is 10 µm thick,yielding λ ≈ 40 µm, and 230 µm wide. Complete sampledimensions are shown in Table II.

3. Elastic moduli

Ref. [39] gives elastic moduli of FeSe at T ≈ Ts: c11 ≈c12 ≈ 50 GPa, c33 ≈ 40 GPa, and c13 ≈ 20 GPa. Un-der conditions of hydrostatic pressure, σ/εxx = (c11c33 +c12c33 − 2c213)/(c33 − c13), where σ is the applied stress,and εzz/εxx = (c11 + c12 − 2c13)/(c33 − c13). Underconditions of in-plane biaxial stress, where σxx = σyyand σzz = 0, σxx/εxx = c11 + c12 − 2c213/c33, andεzz/εxx = −2c13/c33.

4. Plastic deformation of the platform

Sample B was driven to high compressions, and theplatform deformed plastically when the displacement Dapplied to it exceeded ≈ 7 µm, causing the strain in theneck to exceed the elastic limit of the platform material,≈ 2 × 10−3. Data from Sample B were taken in the fol-lowing order: (1) Strain ramps were performed at T ≈ Ts

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 againstD 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, εA1g and εB1g are linear in D:εB1g = 0.66D/leff and εA1g = 0.34D/leff (where leff is theeffective length of the platform). Beyond its elastic limit,the platform material resists further volume compressionby flowing plastically outward: εB1g starts to vary more

14

TABLE II: Sample parameters: length, width, thickness, sep-aration lcontact 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 (µm) w (µm) t (µm) lcontact (µm) RRRA 2370 280 31 970 26B 1150 230 10 630 22C 434 425 ≈10

FIG. 9: Plastic deformation of the platform. (a) ρ100 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 Tc was the same, thebefore data are taken at D = 0.8 µm and the after data atD = 1.6 µm.

steeply with D, and εA1g less steeply. When the direc-tion of the applied displacement is reversed, the platformdeformation is again elastic over some range, but for agiven D εB1g is larger and εA1g 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 Tc is thesame, are plotted together. The residual resistivity isunchanged.

The sign of the offset between the pre- and post-plastic-deformation data shows that ρ100 is controlled domi-nantly by εB1g, rather than εA1g. The fact that a hori-zontal displacement works so well to match the pre- andpost-plastic deformation data shows that the effect ofεA1g on ρ100 is small; if it were strong then it would haveto be finely balanced, over a wide temperature range,with that of εB1g 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 εA1g

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 ∆εB1g = −0.072 × 10−2. Because this de-formation occurred gradually over the course of the tem-perature ramps, for εB1g < −0.11× 10−2 each individualtemperature ramp is offset along the εB1g 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 Ts 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.

6. 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(εB1g) due tothe maximum in domain wall density is visible in bothρ100 and ρ010. Its location differs slightly in the two mea-surements, possibly because in the Montgomery configu-ration measurements of ρ100 and ρ010 do not probe pre-cisely the same area of the sample. We take εB1g = 0 as

15

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 ρ100 and ρ010.

In Fig. 11(c), we show the change in slope dρ/dεB1g

across the cusp at εB1g = 0 versus temperature. Thisquantity is proportional to the twin boundary contribu-tion to sample resistivity at εB1g = 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 Ts at εB1g = 0; when it is brought to εB1g = 0 byramping strain at constant temperature, the twin bound-ary density is lower.

In Figs. 7(a) and 8(a), elastoresistivities normalized byρa + ρb are shown. For this normalization we subtractedoff an estimated twin boundary resistivity, ρTB(T ); forexample, in Fig. 8(a) the quantity that is plotted isρa − ρb, determined by the underlying slopes methoddescribed in the text, divided by ρ100(εB1g = 0) +ρ010(εB1g = 0) − 2ρTB(T ). Based on the illustration inFig. 5(c), we estimate ρTB(T → 0) = 3 µΩ-cm. We takeρTB = ρTB(T → 0) × [1 − (T/Ts)

2]. This form overesti-mates somewhat the true twin boundary resistance as Tapproaces Ts, 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 A1g 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 A1g elastore-sistivity of Sample C. The measured resistivity of SampleA will also be affected by the B2g elastoresistivity; how-ever, because this is transverse to the nematic axes it isnot expected to be large, and the qualitative agreementwith the A1g elastoresistivity suggests that it is indeedmuch smaller than the A1g elastoresistivity. Note alsothat Tc increases with compression, as observed in Sam-

20

40

60

80

100

120

140

160

180

15.018.421.825.228.631.935.338.742.145.548.952.355.759.062.465.869.272.676.079.482.886.189.592.996.3

-0.3

ρ (μ

Ω-c

m)

temperature (K)

0-0.1-0.20.10-0.1-0.2 0.1

εB1g (10-2)

(a) ρ100: (b) ρ010:

0 80604020T=Ts

100

temperature (K)

chan

ge in

slo

pe

acro

ss ε

B1g

=0(1

000

μΩ-c

m)

0

-5

-4

-3

-2

-1

black: ρ100

red: ρ010

(c)

FIG. 11: (a–b) Temperature ramp data from Sample C; panel(a) shows ρ100 and panel (b) ρ010. (c) Change in slopedρ/dεB1g across εB1g = 0. This quantity is proportional to thetwin boundary contribution to sample resistivity at εB1g = 0.

ple B [Fig. 7(b)].

8. Additional derivations of the nematic resistiveanisotropy

Above, we presented a determination of the nematicresistive anisotropy for T < Ts 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 described

16

50

00 20

FIG. 12: ρ110 versus T of Sample A over a wide temperaturerange.

above, the determination of nematic resistive anisotropydepends on extraction of the slopes dρ100/dεB1g anddρ010/dεB1g at εB1g = 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 ρ100 wasmeasured. Evaluating Eq. (1) at f = 0.5 yields

ρa − ρb = 2εs

(dρ100

dεB1g− 1− ν

1 + ν

dρA1g

dεA1g

). (A14)

dρA1g/dεA1g 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 B1g strain, for which the elastic constantc is c11 − c12. This elastic constant must be evaluatedwithout the influence of nematic susceptibility. Ref. [39]finds c11 ≈ 80 GPa at T ≈ 250 K, and electronic struc-ture calculations give c11 = 95 GPa [78]. We take theestimate c = c11 − c12 = 60 GPa. The structural strain

dρ100/dεB1g

FIG. 13: Nematic resistivity anisotropy (ρa − ρb)/(ρa + ρb)of Sample C, derived from the strain-ramp data shown inFig. 6(a). This determination is based on extraction of equi-librium slopes dρ/dεB1g at εB1g = 0, obtained by averagingthe observed increasing-ε and decreasing-ε slopes at εB1g = 0,as shown in the inset.

is obtained by noting that dF/dε = 0 at ε = εs, whichgives εs = (λ/c)ψ. Although the Ginzburg-Landau for-malism only applies, strictly, very near to Ts, we eval-uate parameters at considerably lower temperature inorder to obtain approximate evaluations of the coeffi-cients. εs → 0.27× 10−2 as T → 0 [40], yielding a valuefor the coupling constant: λ ≈ 3.2 GPa/eV. As shownin fig. 8(b), a fit to elastoresistivity data yields a barenematic transition temperature Ts,0 = 60.7 K; we takeTs,0 = 60 K. Ts is defined by the relationship

λ2

c− α× (Ts − Ts,0) = 0. (A15)

Taking Ts = 90 K yields α = 0.0057 GPa/eV2-K. Finally,we evaluate b from the observation that ψ reaches halfits T → 0, or 0.025 eV, value at T ≈ 0.9Ts [50], whichgives b = 82 GPa/eV4.

∗ These authors contributed equally.† These authors contributed equally.; Electronic address:

steppke@cpfs.mpg.de‡ Electronic address: hicks@cpfs.mpg.de

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