Suppression of superconductivity by charge density wave order in YBa_2Cu_3O_{6.67}
Mark E. Barber, Hun-ho Kim, Toshinao Loew, Matthieu Le Tacon, Matteo Minola, Marcin Konczykowski, Bernhard Keimer, Andrew P. Mackenzie, Clifford W. Hicks
SSuppression of superconductivity by charge density wave order in YBa Cu O Mark E. Barber, ∗ Hun-ho Kim, Toshinao Loew, Matthieu Le Tacon,
2, 3
Matteo Minola, Marcin Konczykowski, Bernhard Keimer, Andrew P. Mackenzie,
1, 5 and Clifford W. Hicks
1, 6, † Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Straße 40, 01187 Dresden, Germany Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany Karlsruhe Institute of Technology, Institute for Quantum Materials and Technologies,Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany Laboratoire des Solides Irradi´es, CEA/DRF/lRAMIS, Ecole Polytechnique,CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau, France Scottish Universities Physics Alliance, School of Physics and Astronomy,University of St. Andrews, St. Andrews KY16 9SS, U.K. School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, U.K. (Dated: January 11, 2021)Hole-doped cuprate superconductors show a ubiquitous tendency towards charge order. Althoughonset of superconductivity is known to suppress charge order, there has not so far been a decisivedemonstration of the reverse process, namely, the effect of charge order on superconductivity. Togain such information, we report here the dependence of the critical temperature T c of YBa Cu O on in-plane uniaxial stress up to 2 GPa. At a compression of about 1 GPa along the a axis, 3D-correlated charge density wave (3D CDW) order appears. We find that T c decreases steeply as theapplied stress crosses 1 GPa, showing that the appearance of 3D CDW order strongly suppressessuperconductivity. Through the elastocaloric effect we resolve the heat capacity anomaly at T c , andfind that it does not change drastically as the 3D CDW onsets, which shows that the condensationenergy of the 3D CDW is considerably less than that of the superconductivity. In the doping-temperature phase diagram of hole-doped cuprates there is always an antiferromagneticphase at low doping and a dome of superconductivityat higher doping. The proximity of superconductivityto antiferromagnetic order led early on to a hypothesisthat antiferromagnetic fluctuations drive superconduc-tivity, an idea that remains a bedrock of discussion abouthigh-temperature superconductivity. However, dopingis only one possible axis for tuning. Many hole-dopedcuprates show short-range, quasi-two-dimensional CDWorder when the doped hole density is around 1/8 per Cusite [1, 2]. In YBa Cu O (which has p ≈ /
8) a quasi-long-range, 3D-correlated charge order can be induced bymagnetic field [3–5] or uniaxial stress [6]. These forms ofcharge order are referred to as the 2D and 3D CDWs,respectively. If hole-doped cuprate superconductivity isfound to exist generally in proximity to both antiferro-magnetism and charge order, along distinct tuning axes,the question then arises of whether fluctuations of bothforms of order rather than of antiferromagnetism alonedrive the superconductivity [7].To advance discussion, it is essential to understand howCDW order and superconductivity interact. The onsetof superconductivity suppresses the amplitude of the 2DCDW [8–10], and because competition is reciprocal thepresence of the 2D CDW order must also weaken the su-perconductivity. If, however, it has low spectral weight,for example if it condenses only in localised patches wherethe disorder configuration is favourable, then this effectmight be quantitatively negligible, signalling that thesusceptibility to CDW order might not be important inanalysis of the superconductivity. That the effect of the 2D CDW is not negligible is suggested by the facts that T c of YBa Cu O x dips below trend for p ∼ / T c rises while 2D CDWorder is suppressed [12, 13]. However, although this cor-relation is suggestive it remains in principle possible thatthe 2D CDW responds to, without substantially influenc-ing, the superconductivity. For example, there could betwo quantum critical points under the superconductingdome [14], yielding a natural dip in T c between them.Furthermore, until the relationship between the 2Dand 3D CDWs is clarified, their interaction with thesuperconductivity should be regarded as separate linesof inquiry. Unlike the 2D CDW, the strain-induced 3DCDW is suppressed completely when superconductivityonsets [6]. (The field-induced 3D CDW and supercon-ductivity are also mutually exclusive, at higher tempera-tures, but may coexist below ≈
15 K [15]. A similar low-temperature coexistence region in the stress-temperaturephase diagram is not excluded.) However, this observa-tion does not provide information on the effect of the3D CDW on superconductivity. In Ref. [16], under anapplied field the 3D CDW was found to reach maxi-mum amplitude more quickly than the 2D CDW, suggest-ing that it suppresses superconductivity more effectively.But the intrinsic inhomogeneity of the mixed state com-plicates analysis: because the 3D CDW is more rapidlysuppressed by superconductivity, it may be more sharplyconfined to vortex cores.Here, we study the effect of charge order on the su-perconductivity of YBa Cu O through measurementof the uniaxial stress dependence of T c . If the onset of3D CDW order under a -axis stress σ a affects the super- a r X i v : . [ c ond - m a t . s up r- c on ] J a n FIG. 1. (a) Schematic of the piezoelectric-based uniaxial pres-sure cell employed here, including a cutaway showing theintegrated force sensor and the placement of the actuators.The piezoelectric actuators drive motion of moving block A,through which force is applied to the sample. (b) A micro-graph of Sample 2, mounted for measurement. The Hall crosssusceptometer rests on the upper surface of the sample, whilepick-up and burn-through coils are placed beneath the sam-ple. (c) The tip of the microfabricated Hall probe susceptome-ter. The Hall cross itself is defined by proton irradiation, andis not visible. conductivity, an anomaly in T c ( σ a ) is expected. Con-siderable technical development was required to obtainenough precision for meaningful discussion. We em-ploy piezoelectric-based uniaxial stress apparatus [seeFig. 1(a)], with which GPa-level uniaxial stresses areachievable by mounting beam-like samples with epoxy.At these stresses, plastic deformation or fracture in theepoxy that partially relaxes the stress becomes a concern(see Fig. S1), and therefore the apparatus incorporatesa sensor of the applied force: the stress in the sampleis the force divided by its cross-sectional area, regardlessof the state of the epoxy. In addition, to avoid aver-aging over long-length-scale stress and sample inhomo-geneity, T c was measured using a microfabricated sus-ceptometer [17]. The sensor is a GaAs/AlGaAs 2DEGHall cross with an active area of 10 × µ m surroundedby a 70 µ m-diameter excitation coil; see Fig. 1(c). Themeasured quantity is the field at the Hall cross dividedby the excitation current, B/I .Our setup also includes a “burn-through” coil, a large − − − − − −
65 66 67 6810.510.610.7 50 55 60 659.510.010.5 T (K) B / I ( m T / A ) σ a (GPa) B / I ( m T / A ) σ b (GPa) T (K) B / I ( m T / A ) I bt (mA)0.010.020.050.10.20.5125 (a) Sample 3 σ a = 0 GPa (c) Sample 2 σ k a (b) Sample 1 σ k b FIG. 2. (a) The real part of the susceptometer response
B/I (field in the Hall cross divided by current in the excitationcoil) for different currents in the burn-through coil I bt . Thefield from the excitation coil of the susceptometer at the sam-ple surface was ∼ µ T at 211 Hz, while the burn-throughfield was applied at 20 kHz. 5 mA in the burn-through coilcorresponds to a field at the sample of ∼ µ T. (b)
B/I versus T for Sample 1 at various b -axis pressures σ b . (c) B/I versus T for Sample 2 at various σ a . coil placed below the sample whose applied ac field aidsflux motion. As shown in Fig. 2(a), the width of the su-perconducting transition, measured with the susceptome-ter, decreases from ∼ ∼ T c in-teracting patches of slightly higher T c can form a Joseph-son network that hinders flux motion, giving the appear-ance of bulk superconductivity when the probing field isweak; this effect has been seen in Sr RuO with Ru inclu-sions [19]. Finally, there is a ∼ µ m-diameter pick-upcoil beneath the sample, which in combination with thefield from the burn-through coil can be used for a longer-length-scale measurement of T c . This will be useful forcomparison with elastocaloric effect measurements.Uniaxial stress applied at room temperature can alterthe oxygen ordering, and T c [20, 21]. Here, large stress is σ k a Sample 2Sample 3 σ k b Sample 1
Y Ba Cu O b c a
CuOchainsCuO planes T c ( K ) − − σ a (GPa) − − σ b (GPa)01020 d T c / d ( σ a − σ b ) ( K / G P a ) − − σ a (GPa) − − σ b (GPa) (a)(b) FIG. 3. (a) Dependence of T c on uniaxial stress, appliedalong the b axis for Sample 1, and the a axis for Samples2 and 3. The stars mark the stresses, for Samples 2 and 3,where | d T c / d σ a | is maximum, which we take as the onsetstress σ CDW for 3D CDW order. The error bars in the σ axesindicate the error on the zero-force calibration. The insetshows the crystal structure of YBa Cu O . (b) Derivativesd T c / d( σ a − σ b ), obtained from quintic smoothing splines ofthe data in the upper panel. applied only at temperatures well below 100 K. We alsoperiodically release the pressure and remeasure T c to seeif it was altered after the application of high pressures,but observe no such effects. An example of the transitionbefore and after the application of high pressure is shownin Fig. S2.Three samples were cut from the same single, de-twinned crystal, Sample 1 for application of stress alongthe b axis, and Samples 2 and 3 for stress along the a axis. Fig. 2(b–c) shows susceptibility data from Samples1 and 2 at various stresses. The superconducting transi-tions remain narrow up to the highest stresses reached,even when T c changes rapidly, showing that the probedregions have highly homogeneous T c . Susceptibility datafor Sample 3 are shown in Fig. S3. T c data from all three samples, taking T c as the pointof maximum slope d( B/I ) / d T , are shown together inFig. 3(a). d T c / d σ a and d T c / d σ b have opposite signs,which shows that the in-plane orthorhombicity b − a af- σ CDW SC3DCDW σ a T . . . − D C D W S C A F M D C D W S C σ a ( G P a ) p T ( K ) FIG. 4. Doping - uniaxial stress - temperature phase dia-gram of YBa Cu O x , as resolved in experiments so far.The antiferromagnetic, 2D CDW, and superconducting zeropressure phase boundaries are reproduced from [25], [26],and [11], respectively. The superconducting boundary in thestress-temperature plane is an average of Samples 2 and 3from Fig. 3(a). The onset temperature of the 3D CDW, for | σ a | > | σ CDW | , has not yet been measured. Inset: schematicstress-temperature phase diagram for superconductivity sup-pressed by onset of uniaxial CDW order. fects T c more strongly than the unit cell area ab or c -axislattice constant (which also vary under uniaxial stress).We therefore plot T c ( σ a ) and T c ( σ b ) together, with the σ a and σ b axes mirrored. b/a = 1 .
015 in unstressed YBa Cu O [22], andfor all three samples T c initially increases when b − a is decreased. This is in agreement with thermal ex-pansion data showing a reduction in b − a below T c inunstressed samples [23]. We observe d T c / d σ b | σ b =0 = − . T c / d σ a | σ a =0 = +1 . . Cu O from Ref. [24]: d T c / d σ b | σ b =0 = − . T c / d σ a | σ a =0 = +1 . T c for σ a < − T c / d σ a , which is seen to be approximately constantfor σ a . − . T c , we introducea Ginzburg-Landau free energy functional for competingorders. F = α d × ( T − T c , ) | ∆ d | + β d | ∆ d | + g d ( ε o − ε o , ) | ∆ d | + α C × ( T − T CDW , ) | ∆ C | + β C | ∆ C | + g C ( ε o − ε o , ) | ∆ C | + λ | ∆ C | | ∆ d | . ∆ d and ∆ C are the amplitudes of the superconduct-ing and 3D CDW order parameters, respectively. ε o = ε xx − ε yy is the orthorhombic strain. (We neglect termsfor coupling to c -axis strain, ε zz , and changes in unit cellarea, ε xx + ε yy .) The term g d ( ε o − ε o , ) | ∆ d | representsdirect coupling between the lattice and superconductiv-ity, through the effect of lattice distortion on the pairinginteraction. While the lowest-order coupling between d -wave superconductivity and ε o is quadratic in ε o , the 3DCDW is uniaxial [5], and therefore couples linearly in ε o .For both the superconductivity and CDW, the presenceof the CuO chains introduces an offset ε o , = 0 (which inreality need not be the same for the CDW and supercon-ductivity). Conceptually, this is the strain at which theelectronic interactions become tetragonally symmetric.Because the 3D CDW is not seen in unstressedYBa Cu O , we must have T c , > T CDW , . Becausethe superconductivity and 3D CDW compete, we mustalso have λ >
0. In addition, the 3D CDW will choosethe orientation that gives g C ( ε o − ε o , ) <
0. In the insetof Fig. 4 we illustrate a schematic σ - T phase diagramwhere there is no direct coupling of the superconductiv-ity to the lattice (that is, g d = 0): T c is unchanged until3D CDW order onsets at σ CDW . Competition then sup-presses T c , and, due to the linear coupling between the3D CDW and ε o , the suppression is stress-linear. T c ( σ a ) has this qualitative form. For | σ a | . . T c has a weak quadratic dependence on σ a , that is po-tentially explainable through a small direct-interactioncoefficient g d >
0. The crossover to a linear dependencefor σ a . − . σ CDW in YBa Cu O as the maximumin | d T c / d σ a | , we obtain σ CDW = − .
97 GPa for Sam-ple 2 and − . a -axis strain between − . − . · − , correspond-ing to σ CDW between − . − . − . · − ( i.e. σ CDW = − . σ CDW from Ref. [6] agrees well with that found here. Thedoping - uniaxial stress - temperature phase diagram ofYBa Cu O x , as resolved so far, is shown in Fig. 4.In principle, strain-induced changes in the 2D CDWcould also drive changes in T c . However, recent x-raydata [28] show that, in contrast to T c , the 2D CDWresponds in a qualitatively symmetric way to uniaxialstress: a -axis compression strengthens the componentwith q k b , and b -axis compression that with q k a .On the other hand, for stresses up to ≈ a -axis compression. Thereforethe effects of the 3D CDW appear to dominate changesin T c .To further investigate the relationship between 3DCDW order and superconductivity, we measure the heatcapacity anomaly at T c through the elastocaloric effect(ECE), the change in temperature in response to appliedstress. Competition means that the presence of the 3DCDW reduces the condensation energy of the supercon-ductivity, and so for σ a < σ CDW a reduction in the heatcapacity jump at T c , ∆ C , is expected. The field-induced3D CDW has nearly the same onset temperature as zero-field superconductivity [29, 30], suggesting a close rela-tionship between these phases. If they were nearly degen-erate, with similar condensation energies and entropy-temperature relations S ( T ), the reduction in ∆ C wouldbe substantial. This is possible: in one theoretical model,superconductivity and pair-density-wave order are foundto be nearly degenerate [31], and in experiments thestripe order of La − x Ba x CuO is seen to replace super-conductivity with small changes in doping [32] or uniaxialstress [33], suggesting near-degeneracy.The elastocaloric effect arises due to stress-drivenchanges in entropy. We employ the model of Ref. [34],in which the total heat capacity C = C (nc) + C (c) , where C (c) is the heat capacity of critical electronic fluctuationsassociated with the onset of superconductivity and C (nc) is the non-critical background. C (c) is taken to have theform C (c) ( T, σ ) = C (c) ( T − T c ( σ )). In this model, and inthe adiabatic limit,d T d σ = C (c) C d T c d σ + d T (nc) d σ , where d T (nc) / d σ is the contribution from non-criticalchanges in entropy [34]. Measurements were performedon Sample 3, through the addition of a thermocoupleto the sample setup. The measurement frequency was23 Hz, at which the thermal diffusion length is ∼ µ min our temperature range (Ref. [35], and Fig. S4), mean-ing that this measurement averages over essentially theentire exposed portion of the sample.Raw data at various σ a are shown in Fig. 5(a). InFig. 5(b), an approximate normalisation is obtained bysubtracting a common slope from all the curves, as anestimate for d T (nc) / d σ , then dividing each curve byd T c / d σ a . (Curves are also divided by a correction factorof ≈ σ a = − .
31 GPaare compared with the susceptibility as measured withthe combination of pick-up and burn-through coils. Thetransition widths match, confirming that the feature seenin the ECE is the superconducting transition. ∆ C at T c is found to be ∼ − − − − (cid:9) σ CDW − − d T / d σ a ( m K / G P a ) σ a (GPa) − − σ a (GPa) T c ( K ) ∆ CC C ( c ) / C susc.measuredwith:coilsHall probe T (K) χ ( a r b . un i t s ) C ( c ) / C (a)(b)(c) FIG. 5. (a) Raw elastocaloric effect data from Sample 3. The y -axis is the change in temperature with stress under quasi-adiabatic conditions. The inset shows the locations of thetemperature ramps on the T c ( σ a ) curve. (b) A normalisationof the data in panel (a), described in the text. C (c) /C isthe critical part of the heat capacity, associated with the su-perconducting transition, divided by the total heat capacity.(c) Comparison of data at σ a = − through direct measurement on unstressed samples, 0.3–0.5% [36]. This demonstrates the capability of ECE mea-surements to resolve tiny heat capacity anomalies, evenwith inhomogeneity broadening.We find that any change in ∆ C across σ CDW is small,meaning that although the presence of 3D CDW orderstrongly suppresses T c , its condensation energy is notclose to that of superconductivity. We do not resolveelastocaloric anomalies that could indicate onset of 3DCDW order, possibly because our data do not extend tohigh enough temperature, or because the stress depen-dence of the transition temperature is too small.Identifying the phase boundary where the 3D CDWonsets is an important future task to better understandthe relevance of charge order for superconductivity. 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Zhang, E. M. Levenson-Falk, B. J. Ramshaw, D. A. Bonn, R. Liang, W. N. Hardy, S. A. Hartnoll, andA. Kapitulnik, “Anomalous thermal diffusivity in un-derdoped YBa Cu O x ,” Proc. Natl. Acad. Sci. U.S.A. , 5378–5383 (2017).[36] J. W. Loram, K. A. Mirza, J. R. Cooper, and W. Y.Liang, “Electronic specific heat of YBa Cu O x from1.8 to 300 K,” Phys. Rev. Lett. , 1740–1743 (1993). upplemental Materials for:Suppression of superconductivity by charge density wave order in YBa Cu O Mark E. Barber, ∗ Hun-ho Kim, Toshinao Loew, Matthieu Le Tacon,
2, 3
Matteo Minola, Marcin Konczykowski, Bernhard Keimer, Andrew P. Mackenzie,
1, 5 and Clifford W. Hicks
1, 6, † Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Straße 40, 01187 Dresden, Germany Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany Karlsruhe Institute of Technology, Institute for Quantum Materials and Technologies,Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany Laboratoire des Solides Irradi´es, CEA/DRF/lRAMIS, Ecole Polytechnique,CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau, France Scottish Universities Physics Alliance, School of Physics and Astronomy,University of St. Andrews, St. Andrews KY16 9SS, U.K. School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, U.K.
I. MORE INFORMATION ON THE EXPERIMENT SETUP
Samples were grown using a flux method [37] producing large single crystals whose oxygen content was later adjustedto 6.67 ( T c ≈
65 K) by annealing under well-defined oxygen partial pressure. The large samples were mechanicallydetwinned by heating under slight uniaxial stress, 50–60 MPa, to 400 ◦ C, before being cut into smaller pieces andmechanically polished to the required dimensions for the pressure cell. Samples were mounted into the uniaxial stresscell with Stycast 2850FT epoxy, cured for 4 hours at 65 ◦ C. The epoxy layers on both the top and bottom faces ofthe sample ends were ≈ µ m thick.The uniaxial pressure cell uses three piezoelectric actuators, arranged to cancel their own differential thermalexpansion, to apply pressure to the sample, which is shaped into a long, thin bar. Making reference to Fig. 1(a),one end of the sample is attached to moving block A and the other to moving block B. These blocks are joined withflexures to the outer frame of the cell. The actuators drive motion of moving block A, which is constrained to movealong a straight line by four flexures. Block B is held by four thicker flexures, and moves slightly in response to theforce transmitted through the sample. A parallel-plate capacitive sensor measures the displacement of block B, andcombining with the known spring constant of the thick flexures the force on the sample is determined. A completedescription of the apparatus and its calibration is given in Ref. [38].There is in addition a capacitive displacement sensor between moving blocks A and B, which measures the displace-ment applied to the sample and the epoxy that holds it. Data from this sensor provides information on the mechanicalstate of the sample and epoxy. Force-displacement data from Sample 3, as the applied stress approached − T = 68 K, are shown in Fig. S1(a–b). Panel (a) shows F ( d ), where F is force and d displacement, over the entirecourse of measurements, including the point where the sample fractured under compression and F dropped abruptly.Panel (b) shows a close-up of F ( d ) at high stresses, approaching the point of fracture. Here, it is seen that F ( d )followed a gentler slope as | σ a | was increased than on the return strokes. This difference is visible in, for example,the gentler slope between the points numbered 1 and 9 in the figure, where | σ a | was increased monotonically, and thesteeper slope between points 9 and 10, where | σ a | was decreased. This behaviour most likely shows that the epoxyrelaxes plastically as high stresses are approached. That this possible plastic flow is in the epoxy, not the sample, isshown by the data in Fig. S2, where the transition observed in Sample 3 at − .
79 GPa is the same before and afterramping to − . ∼ F = 0 reading could be determined withhigh precision. Samples 1 and 2, on the other hand, were fractured in tension, and the fractures occurred inside theepoxy, such that there remained a frictional connection between the two sample ends. F ( d ) traces leading up to andfollowing the fractures for Samples 1 and 2 are shown in Figs. S1(c) and (d), respectively. For both samples, afterfracture there is a portion of the force-displacement curve where the slope is strongly reduced, which we interpret asthe crossover from tensile to compressive stress in the sample, broadened by frictional and plastic effects in the epoxy. a r X i v : . [ c ond - m a t . s up r- c on ] J a n − − − . − . − . − . − . . . . − − − − − − fracture d ( µ m) F ( N ) σ a ( G P a ) − − − − − − − d ( µ m) F ( N ) σ a ( G P a ) − − − fracture d ( µ m) F ( N ) σ b ( G P a ) − − − fracture d ( µ m) F ( N ) σ a ( G P a ) Sample 1 T = 70 KSample 2 T = 70 KSample 3 T = 68 KSample 3 T = 68 K (b) (c)(d)(a) FIG. S1. (a) Force-displacement data for a series of T c measurements for Sample 3. The markers indicate points where thetemperature was ramped from 68 K to below T c and then back to 68 K at fixed strain. Between these points the displacement d was ramped at 1.5 nm/s. (b) Close-up of the data in panel (a), showing the probable plastic relaxation of the epoxy holdingthe sample as | σ a | became large. The numbers on each marker indicate the order in which temperature ramps were performed.(c) Force-displacement data for Sample 1, before and after it was fractured under tension. As in panel (a), the markers indicatestrains where temperature ramps were performed to measure T c , and the lines are the strain ramps between these points. Thedisplacement was ramped at 1.5 nm/s. (d) Same as (c), for Sample 2.
63 6410.410.510.610.7 T (K) B / I ( m T / A ) Sample 3, σ a = − FIG. S2. Two measurements of Sample 3 at σ a = − σ a was ramped to − Because there is no single sharp feature that can be identified as zero stress, we estimate an error of ± .
05 GPa onthe zero-stress determination for these two samples.The burn-through field for all measurements with the Hall cross susceptometer shown in this paper, apart fromFig. 2(a), was ∼ µ T at 20 kHz. The probing field from the susceptometer was ∼ µ T, applied at 211 Hz, for allmeasurements.
II. ADDITIONAL DATA
Susceptibility data across T c for Sample 3 are shown in Fig. S3. (Susceptibility data from Samples 1 and 2 areshown in Fig. 2.) For this set of measurements, the signal from the Hall cross drifted over time, so data are scaled bydividing each sweep by the starting value at 68 K. − − − −
40 50 600.980.991.00 T (K) B I (cid:14) B I ( K ) σ a (GPa)Sample 3 FIG. S3. Real part of the susceptometer response
B/I for Sample 3 at various σ a . Due to drift, data are scaled by the readingsat 68 K. III. FURTHER DISCUSSION OF σ CDW
As noted in the main text, in Ref. [6] the strain at which 3D CDW order onsets in YBa Cu O was reportedto be between − . − . · − , corresponding to σ CDW between − . − . σ CDW found here for Samples 2 and 3, respectively − .
97 and − . σ CDW = − . σ CDW . IV. FURTHER INFORMATION ON THE ELASTOCALORIC EFFECT MEASUREMENTS
Elastocaloric effect (ECE) measurements are ideally performed in the adiabatic limit, meaning that the measurementfrequency should be high enough that temperature oscillations do not dissipate into the bath through the ends ofthe sample, yet low enough that the thermocouple thermalises to the sample. Here, this frequency window was notbroad. In analysis below, we show that at our measurement frequency of 23.11 Hz the observed signal was about 80%of its adiabatic limit. This factor is included in the normalised ECE data of Fig. 5, and the good agreement with theheat capacity jump at T c , ∆ C , measured directly in Ref. [36] shows that it is at least approximately correct. Thenarrowness of the frequency window prevents us from commenting on small apparent changes in ∆ C , but our dataare sufficient to conclude that ∆ C does not change drastically across σ CDW .We now provide more details on the measurement. A photograph of Sample 3, including the attached thermocouple,is shown in Fig. S4(a). The thermocouple is chromel-AuFe , and was secured to the sample with Stycast 2850FT.It was made from wire from the same spools as in Ref. [39], and we therefore apply the calibration reported there.The stress apparatus contains two sets of actuators, labelled compression and tension actuators, whose actions on thesample have opposite sign. For the ECE measurements, the compression actuators were used to apply the dc stress,and the tension actuators the ac stress, with a temperature-independent applied voltage of 1.4 V rms . The capacitance . . .
840 50 60 7050556065 T (K) d l / d V ( n m / V ) µ m thermocouplechromel-AuFe
45 K 55 K50 K 60 K68 K F (Hz) θ ( d e g . ) E / E
30 40 50 60 70 80 904006008001 , T (K) p D / ω ( µ m ) Sample 3, σ a = − (a) (c)(d)(b) FIG. S4. (a) Micrograph of Sample 3 showing the chromel-AuFe thermocouple. (b) Temperature dependence of thepiezoelectric actuator response to applied voltage and a linear fit to the data. (c) Temperature dependence of the characteristicthermal length of YBa Cu O at 23.1 Hz [35]. (d) Frequency dependence of the elastocaloric effect signal, amplitude andphase, at various temperatures at σ a = − bridge monitoring the force sensor did not have enough bandwidth for measurements at 23 Hz, and we therefore obtaina dc calibration of the tension actuator response, and, following Refs. [34] and [40], make the approximation that thiscalibration applies at 23 Hz. For Sample 3, at σ a ∼ T = 68 K the stress-voltage response was 4.9 MPa/V.After the sample was fractured, we then tested the temperature response of the tension actuator by ramping theapplied voltage between −
10 and +10 V, yielding the data plotted in Fig. S4(b). Between 69 and 45 K, the actuatorresponse decreases linearly with temperature by about 16%. We therefore take our applied ac stress to be 4.9 MPa/V × rms = 6.9 MPa rms at 68 K, and for lower temperatures scale this by the response curve shown in Fig. S4(b).ECE measurements yield an elastocaloric coefficient E ≡ d T / d σ . We now discuss analysis to determine E/E ,where E is the elastocaloric coefficient in the adiabatic limit. Ikeda et al . [34] showed that the frequency dependenceof the elastocaloric response can be well captured by a discretized thermal model. In this model, EE = (cid:0) a + b (cid:1) − / ,θ = arctan( ab ) , where θ is the phase of the signal relative to the stress oscillation, and a = 1 ωτ i − ωτ θ ,b = 1 + C θ C s + τ θ τ i .τ i is the time constant for thermalisation of the sample to the bath (approximated as being frequency-independent), τ θ is that for thermalisation of the temperature sensor to the sample, C θ is the heat capacity of the sensor, and C s isthe heat capacity of the sample. We estimate C θ /C s , and leave τ i and τ θ as fitting parameters. We estimate C θ asthe heat capacity of 700 µ m of 25 µ m-diameter chromel wire and 1 mm of 25 µ m-diameter AuFe wire, where thelengths are determined from the thermal diffusivities of these materials at 23 Hz and 65 K, in addition to a volume150 × × µ m of Stycast 2850FT. At T = (45 , , , ,
68) K, we find C θ ∼ (0 . , . , . , . , . µ J/K, andemploy these values for the fitting. C s is much larger than C θ , and we therefore take its frequency dependence intoaccount. Following Ref. [34], we take an approximate model in which at low frequencies C s is the heat capacity ofthe entire exposed portion of the sample, and at high frequencies it is the heat capacity of a portion of the sample oflength twice the thermal diffusion length ξ : C s = ( CA s l exp , ω < ω , CA s ξ , ω > ω , where C is the specific heat capacity of YBa Cu O , which we take from Ref. [36], A s is the cross-sectional area ofthe sample, and l exp = 1 mm is the exposed length of the sample. ξ = (2 D/ω ) / , where D is the thermal diffusivity,which we take from Ref. [35]. ω is the angular frequency at which 2 ξ crosses l exp . In Fig. S4(c) we show thetemperature dependence of ξ . At our measurement frequency of 23.11 Hz, we find C θ /C s ∼ . τ i and τ θ are obtained from simultaneous fitting of the observed frequency dependence of E and θ at σ a = − . E is scaled by E , which is obtained from independent fitting ateach temperature. The measurement frequency, 23.11 Hz, was selected as the frequency where E/E is close to itsmaximum over the entire temperature range studied here. Based on a linear fit of E/E with temperature, we find E/E to vary between 0.796 at 45 K and 0.810 at 68 K. We apply this T -dependent correction to the data in Fig. 5 inthe main text. Because C θ is small, a large change in our estimate for C θ has minimal effect. For example, doubling C θ would reduce the maximum E/E in the fitting to ∼ ∗ Present address: Department of Applied Physics and Geballe Laboratory for Advanced Materials, Stanford University,Stanford, CA 94305; [email protected] † [email protected][37] C. T. Lin, W. Zhou, W. Y. Liang, E. Sch¨onherr, and H. Bender, “Growth of large and untwinned single crystals ofYBCO,” Physica C , 291–300 (1992).[38] M. E. Barber, A. Steppke, A. P. Mackenzie, and C. W. Hicks, “Piezoelectric-based uniaxial pressure cell with integratedforce and displacement sensors,” Rev. Sci. Instrum. , 023904 (2019).[39] U. Stockert and N. Oeschler, “Thermopower of chromel–AuFe . thermocouples in magnetic fields,” Cryogenics51