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Inverse correlation between quasiparticle mass and Tc in a cuprate high-Tc superconductor

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American Association for the Advancement of Science

high pressure, quantum oscillations, cuprate superconductors

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      Abstract

      Contrary to what is expected near a quantum critical point, pressure decreases the quasiparticle mass of a high-T c superconductor.

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      Incipient charge order observed by NMR in the normal state of YBa2Cu3Oy

      The pseudogap regime of high-temperature cuprates harbours diverse manifestations of electronic ordering whose exact nature and universality remain debated. Here, we show that the short-ranged charge order recently reported in the normal state of YBa2Cu3Oy corresponds to a truly static modulation of the charge density. We also show that this modulation impacts on most electronic properties, that it appears jointly with intra-unit-cell nematic, but not magnetic, order, and that it exhibits differences with the charge density wave observed at lower temperatures in high magnetic fields. These observations prove mostly universal, they place new constraints on the origin of the charge density wave and they reveal that the charge modulation is pinned by native defects. Similarities with results in layered metals such as NbSe2, in which defects nucleate halos of incipient charge density wave at temperatures above the ordering transition, raise the possibility that order–parameter fluctuations, but no static order, would be observed in the normal state of most cuprates if disorder were absent.
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        Direct measurement of the upper critical field in cuprate superconductors

        In a type-II superconductor at T=0, the onset of the superconducting state as a function of decreasing magnetic field H occurs at the upper critical field H c2, dictated by the pairing gap Δ through the coherence length ξ0~v F/Δ, via H c2=Φ0/2πξ0 2, where v F is the Fermi velocity and Φ0 is the magnetic flux quantum. H c2 is the field below which vortices appear in the sample. Typically, the vortices immediately form a lattice (or solid) and thus cause the electrical resistance to go to zero. So the vortex-solid melting field, H vs, is equal to H c2. In cuprate superconductors, the strong 2D character and low superfluid density cause a vortex liquid phase to intervene between the vortex-solid phase below H vs(T) and the normal state above H c2(T) (ref. 1). It has been argued that in underdoped cuprates there is a wide vortex-liquid phase even at T=0 (refs 2, 3, 4, 5), so that H c2(0)>>H vs(0), implying that Δ is very large. Whether the gap Δ is large or small in the underdoped regime is a pivotal issue for understanding what controls the strength of superconductivity in cuprates. So far, however, no measurement on a cuprate superconductor has revealed a clear transition at H c2, so there are only indirect estimates2 6 7 and these vary widely (see Supplementary Discussion and Supplementary Fig. 1). For example, superconducting signals in the Nernst effect2 and the magnetization4 have been tracked to high fields, but it is difficult to know whether these are due to vortex-like excitations below H c2 or to fluctuations above H c2 (ref. 7). Here we demonstrate that measurements of the thermal conductivity can directly detect H c2, and we show that in the cuprate superconductors YBa2Cu3O y (YBCO) and YBa2Cu4O8 (Y124) there is no vortex liquid at T=0. This fact allows us to then use measurements of the resistive critical field H vs(T) to obtain H c2 in the T=0 limit. By including measurements on the overdoped cuprate Tl2Ba2CuO6+δ (Tl-2201), we establish the full doping dependence of H c2. The magnitude of H c2 is found to undergo a sudden drop as the doping is reduced below p=0.18, revealing the presence of a T=0 critical point below which a competing phase markedly weakens superconductivity. This phase is associated with the onset of Fermi-surface reconstruction and charge-density-wave order, generic properties of hole-doped cuprates. Results Thermal conductivity To detect H c2, we use the fact that electrons are scattered by vortices, and monitor their mobility as they enter the superconducting state by measuring the thermal conductivity κ of a sample as a function of magnetic field H. In Fig. 1, we report our data on YBCO and Y124, as κ vs H up to 45 T, at two temperatures well below T c (see Methods and Supplementary Note 1). All curves exhibit the same rapid drop below a certain critical field. This is precisely the behaviour expected of a clean type-II superconductor (l 0>>ξ0), whereby the long electronic mean free path l 0 in the normal state is suddenly curtailed when vortices appear in the sample and scatter the electrons (see Supplementary Note 2). This effect is observed in any clean type-II superconductor, as illustrated in Fig. 1e and Supplementary Fig. 2. Theoretical calculations8 reproduce well the rapid drop of κ at H c2 (Fig. 1e). To confirm our interpretation that the drop in κ is due to vortex scattering, we measured a single crystal of Tl-2201 for which l 0~ξ0, corresponding to a type-II superconductor in the dirty limit. As seen in Fig. 2a, the suppression of κ upon entering the vortex state is much more gradual than in the ultraclean YBCO. The contrast between Tl-2201 and YBCO mimics the behaviour of the type-II superconductor KFe2As2 as the sample goes from clean (l 0~10 ξ0) (ref. 9) to dirty (l 0~ξ0) (ref. 10) (see Fig. 2b). We conclude that the onset of the sharp drop in κ with decreasing H in YBCO is a direct measurement of the critical field H c2, where vortex scattering begins. Upper critical field H c2 The direct observation of H c2 in a cuprate material is our first main finding. We obtain H c2=22±2 T at T=1.8 K in YBCO (at p=0.11) and H c2=44±2 T at T=1.6 K in Y124 (at p=0.14) (Fig. 1a), giving ξ0=3.9 nm and 2.7 nm, respectively. In Y124, the transport mean free path l 0 was estimated to be roughly 50 nm (ref. 11), so that the clean-limit condition l 0>>ξ0 is indeed satisfied. Note that the specific heat is not sensitive to vortex scattering and so will have a much less pronounced anomaly at H c2. This is consistent with the high-field specific heat of YBCO at p=0.1 (ref. 5). We can verify that our measurement of H c2 in YBCO is consistent with existing thermodynamic and spectroscopic data by computing the condensation energy δE=H c 2/2μ0, where H c 2=H c1 H c2/(ln κGL+0.5), with H c1 the lower critical field and κGL the Ginzburg-Landau parameter (ratio of penetration depth to coherence length). Magnetization data12 on YBCO give H c1=24±2 mT at T c=56 K. Using κGL=50 (ref. 12), our value of H c2=22 T (at T c=61 K) yields δE/T c 2=13±3 J K−2 m−3. For a d-wave superconductor, δE=N F Δ0 2/4, where Δ0=α k B T c is the gap maximum and N F is the density of states at the Fermi energy, related to the electronic specific heat coefficient γN=(2π2/3) N F k B 2, so that δE/T c 2=(3α2/8π2) γN. Specific heat data5 on YBCO at T c=59 K give γN=4.5±0.5 mJ K−2 mol−1 (43±5 J /K−2 m−3) above H c2. We therefore obtain α=2.8±0.5, in good agreement with estimates from spectroscopic measurements on a variety of hole-doped cuprates, which yield 2Δ0/k B T c~5 between p=0.08 and p=0.24 (ref. 13). This shows that the value of H c2 measured by thermal conductivity provides quantitatively coherent estimates of the condensation energy and gap magnitude in YBCO. H—T phase diagram The position of the rapid drop in κ vs H does not shift appreciably with temperature up to T~10 K or so (Fig. 1b,d), showing that H c2(T) is essentially flat at low temperature. This is in sharp contrast with the resistive transition at H vs(T), which moves down rapidly with increasing temperature (Fig. 1f). In Fig. 3, we plot H c2(T) and H vs(T) on an H-T diagram, for both YBCO and Y124 (see Methods and Supplementary Methods). In both cases, we see that H c2=H vs in the T=0 limit. This is our second main finding: there is no vortex liquid regime at T=0 (see Supplementary Note 3). With increasing temperature the vortex-liquid phase grows rapidly, causing H vs(T) to fall below H c2(T). The same behaviour is seen in Tl-2201 (Fig. 2d): at low temperature, H c2(T) determined from κ is flat, whereas H vs(T) from resistivity falls abruptly, and H c2=H vs at T→0 (see also Supplementary Figs 3 and 4, and Supplementary Note 4). H—p phase diagram Having established that H c2=H vs at T→0 in YBCO, Y124 and Tl-2201, we can determine how H c2 varies with doping from measurements of H vs(T) (see Methods and Supplementary Methods), as in Supplementary Figs 5 and 6. For p 0.15, however, T c cannot be suppressed to zero with our maximal available field of 68 T (Fig. 3d and Supplementary Fig. 5), so an extrapolation procedure must be used to extract H vs(T→0). Following ref. 14, we obtain H vs(T→0) from a fit to the theory of vortex-lattice melting1, as illustrated in Fig. 3 (and Supplementary Fig. 6). In Fig. 4a, we plot the resulting H c2 values as a function of doping, listed in Table 1, over a wide doping range from p=0.05 to p=0.26. This brings us to our third main finding: the H—p phase diagram of superconductivity consists of two peaks, located at p 1~0.08 and p 2~0.18. (A partial plot of H vs(T→0) vs p was reported earlier on the basis of c-axis resistivity measurements14, in excellent agreement with our own results.) The two-peak structure is also apparent in the usual T—p plane: the single T c dome at H=0 transforms into two domes when a magnetic field is applied (Fig. 4b). Discussion A natural explanation for two peaks in the H c2 vs p curve is that each peak is associated with a distinct critical point where some phase transition occurs. An example of this is the heavy-fermion metal CeCu2Si2, where two T c domes in the temperature-pressure phase diagram were revealed by adding impurities to weaken superconductivity15: one dome straddles an underlying antiferromagnetic transition and the other dome a valence transition16. In YBCO, there is indeed strong evidence of two transitions—one at p 1 and another at a critical doping consistent with p 2 (ref. 17). In particular, the Fermi surface of YBCO is known to undergo one transformation at p=0.08 and another near p~0.18 (ref. 18). Hints of two critical points have also been found in Bi2Sr2CaCu2O8+δ, as changes in the superconducting gap detected by ARPES at p 1~0.08 and p 2~0.19 (ref. 19). The transformation at p 2 is a reconstruction of the large hole-like cylinder at high doping that produces a small electron pocket18 20 21. We associate the fall of T c and the collapse of H c2 below p 2 to that Fermi-surface reconstruction. Recent studies indicate that charge-density wave order plays a role in the reconstruction22 23 24 25. Indeed, the charge modulation seen with X-rays23 24 25 and the Fermi-surface reconstruction seen in the Hall coefficient18 26 emerge in parallel with decreasing temperature (see Fig. 5). Moreover, the charge modulation amplitude drops suddenly below T c, showing that superconductivity and charge order compete23 24 25 (Supplementary Fig. 7a). As a function of field24, the onset of this competition defines a line in the H—T plane (Supplementary Fig. 7b) that is consistent with our H c2(T) line (Fig. 3). The flip side of this phase competition is that superconductivity must in turn be suppressed by charge order, consistent with our interpretation of the T c fall and H c2 collapse below p 2. We can quantify the impact of phase competition by computing the condensation energy δE at p=p 2, using H c1=110±5 mT at T c=93 K (ref. 27) and H c2=140±20 T (Table 1), and comparing with δE at p=0.11 (see above): δE decreases by a factor 20 and δE/T c 2 by a factor 8 (see Supplementary Note 5). In Fig. 4c, we plot the doping dependence of δE/T c 2 (in qualitative agreement with earlier estimates based on specific heat data28—see Supplementary Fig. 8). We attribute the tremendous weakening of superconductivity below p 2 to a major drop in the density of states as the large hole-like Fermi surface reconstructs into small pockets. This process is likely to involve both the pseudogap formation and the charge ordering. Upon crossing below p=0.08, the Fermi surface of YBCO undergoes a second transformation, where the small electron pocket disappears, signalled by pronounced changes in transport properties18 21 and in the effective mass m* (ref. 29). This is strong evidence that the peak in H c2 at p 1~0.08 (Fig. 4a) coincides with an underlying critical point. This critical point is presumably associated with the onset of incommensurate spin modulations detected below p~0.08 by neutron scattering30 and muon spectroscopy31. Note that the increase in m* (ref. 29) may in part explain the increase in H c2 going from p=0.11 (local minimum) to p=0.08, since H c2~1/ξ0 2~1/v F 2~m *2. Our findings shed light on the H-T-p phase diagram of cuprate superconductors, in three different ways. In the H-p plane, they establish the boundary of the superconducting phase and reveal a two-peak structure, the likely fingerprint of two underlying critical points. In the H-T plane, they delineate the separate boundaries of vortex solid and vortex liquid phases, showing that the latter phase vanishes as T→0. In the T-p plane, they elucidate the origin of the dome-like T c curve as being primarily due to phase competition, rather than fluctuations in the phase of the superconducting order parameter32, and they quantify the impact of that competition on the condensation energy. Our finding of a collapse in condensation energy due to phase competition is likely to be a generic property of hole-doped cuprates, since Fermi-surface reconstruction—the inferred cause—has been observed in materials such as La1.8-x Eu0.2Sr x CuO4 (ref. 20) and HgBa2CuO4+δ (refs 33, 34), two cuprates whose structure is significantly different from that of YBCO and Y124. This shows that phase competition is one of the key factors that limit the strength of superconductivity in high-T c cuprates. Methods Samples Single crystals of YBa2Cu3O y (YBCO) were obtained by flux growth at UBC (ref. 35). The superconducting transition temperature T c was determined as the temperature below which the zero-field resistance R=0. The hole doping p is obtained from T c (ref. 36). To access dopings above p=0.18, Ca substitution was used, at the level of 1.4% (giving p=0.19) and 5% (giving p=0.205). At oxygen content y=6.54, a high degree of ortho-II oxygen order has been achieved, yielding large quantum oscillations37 38, proof of a long electronic mean free path. We used such crystals for our thermal conductivity measurements. Single crystals of YBa2Cu4O8 (Y124) were grown by a flux method in Y2O3 crucibles and an Ar/O2 mixture at 2,000 bar, with a partial oxygen pressure of 400 bar (ref. 39). Y124 is a stoichiometric underdoped cuprate material, with T c=80 K. The doping is estimated from the value of T c, using the same relation as for YBCO (ref. 36). Because of its high intrinsic level of oxygen order, quantum oscillations have also been observed in the highest quality crystals of Y124 (ref. 40). We used such crystals for our thermal conductivity measurements. Single crystals of Tl2Ba2CuO6 (Tl-2201) were obtained by flux growth at UBC. Compared with YBCO and Y124, crystals of Tl-2201 are in the dirty limit (see Supplementary Note 4). We used such crystals to compare thermal conductivity data in the clean and dirty limits. The thermal conductivity (and resistivity) was measured on two strongly overdoped samples of Tl-2201 with T c=33 K and 20 K, corresponding to a hole doping p=0.248 and 0.257, respectively. The doping value for Tl-2201 samples was obtained from their T c, via the standard formula T c/T c max=1–82.6 (p–0.16)2, with T c max=90 K. Resistivity measurements The in-plane electrical resistivity of YBCO was measured in magnetic fields up to 45 T at the NHMFL in Tallahassee and up to 68 T at the LNCMI in Toulouse. A subset of those data is displayed in Supplementary Fig. 5. From such data, H vs(T) is determined and extrapolated to T=0 to get H vs(0), as illustrated in Fig. 3 and Supplementary Fig. 6. The H c2=H vs(0) values thus obtained are listed in Table 1 and plotted in Fig. 4a. Corresponding data on Y124 were taken from ref. 11 (see Supplementary Fig. 5). The resistance of a Tl-2201 sample with T c=59 K (p=0.225) was also measured, at the LNCMI in Toulouse up to 68 T (see Supplementary Figs 5 and 6). In all measurements, the magnetic field was applied along the c axis, normal to the CuO2 planes. (See also Supplementary Methods.) Thermal conductivity measurements The thermal conductivity κ of four ortho-II oxygen-ordered samples of YBCO, with p=0.11, was measured at the LNCMI in Grenoble up to 34 T and/or at the NHMFL in Tallahassee up to 45 T, in the temperature range from 1.8 K to 14 K. Data from the four samples were in excellent agreement (see Table 2). The thermal conductivity κ of two single crystals of stoichiometric Y124 (p=0.14) was measured at the NHMFL in Tallahassee up to 45 T, in the temperature range from 1.6 K to 9 K. Data from the two samples were in excellent agreement (see Table 2). A constant heat current Q was sent in the basal plane of the single crystal, generating a thermal gradient dT across the sample. The thermal conductivity is defined as κ=(Q/dT) (L/w t), where L, w and t are the length (across which dT is measured), width and thickness (along the c axis) of the sample, respectively. The thermal gradient dT=T hot−T cold was measured with two Cernox thermometers, sensing the temperature at the hot (T hot) and cold (T cold) ends of the sample, respectively. The Cernox thermometers were calibrated by performing field sweeps at different closely spaced temperatures between 2 K and 15 K. Representative data are shown in Fig. 1. (See also Supplementary Note 1.) Author contributions G.G., S.R.d.C. and N.D.-L. performed the thermal conductivity measurements at Sherbrooke. G.G., O.C.-C., S.D.-B., S.K. and N.D.-L. performed the thermal conductivity measurements at the LNCMI in Grenoble. G.G., O.C.-C., A.J.-F., D.G. and N.D.-L. performed the thermal conductivity measurements at the NHMFL in Tallahassee. N.D.-L., D.L., M.S., B.V. and C.P. performed the resistivity measurements at the LNCMI in Toulouse. S.R.d.C., J.C., J.-H.P. and N.D.-L. performed the resistivity measurements at the NHMFL in Tallahassee. M.-È.D., O.C.-C., G.G., F.L., D.L. and N.D.-L. performed the resistivity measurements at Sherbrooke. B.J.R., R.L., D.A.B. and W.N.H. prepared the YBCO and Tl-2201 single crystals at UBC (crystal growth, annealing, de-twinning, contacts). S.A. and N.E.H. prepared the Y124 single crystals. G.G., O.C.-C., F.L., N.D.-L. and L.T. wrote the manuscript. L.T. supervised the project. Additional information How to cite this article: Grissonnanche, G. et al. Direct measurement of the upper critical field in cuprate superconductors. Nat. Commun. 5:3280 doi: 10.1038/ncomms4280 (2014). Supplementary Material Supplementary Information Supplementary Figures 1-8, Supplementary Notes 1-5, Supplementary Discussion, Supplementary Methods and Supplementary References
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          Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor

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            10.1126/sciadv.1501657
            27034989
            4803492

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