Direct measurement of the upper critical field in a cuprate superconductor
G. Grissonnanche, O. Cyr-Choiniere, F. Laliberte, S. Rene de Cotret, A. Juneau-Fecteau, S. Dufour-Beausejour, M.-E. Delage, D. LeBoeuf, J. Chang, B. J. Ramshaw, D. A. Bonn, W. N. Hardy, R. Liang, S. Adachi, N. E. Hussey, B. Vignolle, C. Proust, M. Sutherland, S. Kramer, J.-H. Park, D. Graf, N. Doiron-Leyraud, Louis Taillefer
11 Direct measurement of the upper critical field in a cuprate superconductor
G. Grissonnanche , O. Cyr-Choinière , F. Laliberté , S. René de Cotret , A. Juneau-Fecteau , S. Dufour-Beauséjour , M.-È. Delage , D. LeBoeuf , J. Chang , B. J. Ramshaw , D. A. Bonn , W. N. Hardy , R. Liang , S. Adachi , N. E. Hussey , B. Vignolle , C. Proust , M. Sutherland , S. Krämer , J.-H. Park , D. Graf , N. Doiron-Leyraud & Louis Taillefer † Present address : Laboratoire National des Champs Magnétiques Intenses, Grenoble, France. ‡ Present address : École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland.
The upper critical field H c2 is a fundamental measure of the pairing strength, yet there is no agreement on its magnitude and doping dependence in cuprate superconductors . We have used thermal conductivity as a direct probe of H c2 in the cuprates YBa Cu O y and YBa Cu O to show that there is no vortex liquid at T = 0, allowing us to use high-field resistivity measurements to map out the doping dependence of H c2 across the phase diagram. H c2 ( p ) exhibits two peaks, each located at a critical point where the Fermi surface undergoes a transformation . The condensation energy obtained directly from H c2 , and previous H c1 data , undergoes a 20-fold collapse below the higher critical point. These data provide quantitative information on the impact of competing phases in suppressing superconductivity in cuprates. 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 ξ ~ v F / Δ , via H c2 = Φ / 2 πξ , where v F is the Fermi velocity and Φ is the 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. 8). It has been argued that in underdoped cuprates there is a wide vortex-liquid phase even at T = 0 (refs. 2,9,10,11), so that H c2 (0) >> H vs (0), implying that Δ is very large. So far, however, no measurement on a cuprate superconductor has revealed a clear transition at H c2 , so there are only indirect estimates (refs. 1,2,3) and these vary widely (see Fig. S1 and associated discussion). For example, superconducting signals in the Nernst effect and the magnetization 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. 3). To resolve this question, 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 two cuprate superconductors, YBa Cu O y (YBCO) and YBa Cu O (Y124), as κ vs H up to 45 T, at two temperatures well below T c . 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 >> ξ ), whereby the long electronic mean free path l in the normal state is suddenly curtailed when vortices appear in the sample and scatter the electrons (see Fig. S2, and associated discussion). This effect is observed in any clean type-II superconductor, as illustrated in Fig. 1e and Fig. S2. Theoretical calculations 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 have measured a single crystal of the cuprate Tl Ba CuO δ (Tl-2201) with a much shorter mean free path, such that l ~ ξ . 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 behavior of the type-II superconductor KFe As as the sample goes from clean ( l ~ 10 ξ ) (ref. 13) to dirty ( l ~ ξ ) (ref. 14) (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. 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 ξ = 3.9 nm and 2.7 nm, respectively. In Y124, the transport mean free path l was estimated to be roughly 50 nm (ref. 15), so that the clean-limit condition l >> ξ is indeed satisfied. Note that the specific heat is not sensitive to vortex scattering and so should not have a marked anomaly at H c2 , as indeed found in YBCO at p = 0.1 (ref. 10). We can verify that our measurement of H c2 in YBCO is consistent with existing thermodynamic and spectroscopic data by computing the condensation energy δ E = H c2 / 2µ , where H c2 = H c1 H c2 / (ln κ GL + 0.5), with H c1 the lower critical field and κ GL the Ginzburg-Landau parameter (ratio of penetration depth to coherence length). Magnetization data on YBCO give H c1 = 24 ± 2 mT at T c = 56 K. Using κ GL = 50 (ref. 6), our value of H c2 = 22 T (at T c = 61 K) yields δ E / T c2 = 13 ± 3 J / K m . For a d -wave superconductor, δ E = N F Δ / 4, where Δ = 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 γ N = (2 π /3) N F k B2 , so that δ E / T c2 = (3 α / 8 π ) γ N . Specific heat data on YBCO at T c = 59 K give γ N = 4.5 ± 0.5 mJ / K mol (43 ± 5 J / K m ) 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 2 Δ / k B T c ~ 5 between p = 0.08 and p = 0.24 (ref. 16). 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. The position of the rapid drop in κ vs H does not shift appreciably with temperature up to T ~ 10 K or so (Figs. 1b and 1d), 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. 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. Of course, 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 → Having established that H c2 = H vs at T → H c2 varies with doping in YBCO from measurements of H vs ( T ) (as in Figs. S5 and S6). For p < 0.15, fields lower than 60 T are sufficient to suppress T c to zero, and thus directly access H vs ( T → H c2 = 24 ± 2 T at p = 0.12 (Fig. 3c), for example. For p > 0.15, however, T c cannot be suppressed to zero with our maximal available field of 68 T (Figs. 3d and S5), so an extrapolation procedure must be used to extract H vs ( T → H vs ( T →
0) from a fit to the theory of vortex-lattice melting , as illustrated in Fig. 3 (and Fig. S6). In Fig. 4a, we plot the resulting H c2 values as a function of doping, listed in Table S1, over a wide doping range from p = 0.05 to p = 0.205. This brings us to our third main finding: the H - p phase diagram of superconductivity consists of two peaks, located at p ~ 0.08 and p ~ 0.18. (A plot of H vs ( T →
0) vs p was reported earlier on the basis of c -axis resistivity measurements , in excellent agreement with our own results, but the two peaks where not observed because the data were limited to 0.078 < p < 0.162.) 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). 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 CeCu Si , where two T c domes in the temperature-pressure phase diagram were revealed by adding impurities to weaken superconductivity : one dome straddles an underlying antiferromagnetic transition and the other dome a valence transition. In YBCO, there is indeed strong evidence of two transitions – one at p and another at a critical doping consistent with p (ref. 20). In particular, the Fermi surface of YBCO is known to undergo one transformation at p = 0.08 and another near p ~ 0.18 (ref. 4). Hints of two critical points have also been found in Bi Sr CaCu O δ , as changes in the superconducting gap detected by ARPES at p ~ 0.08 and p ~ 0.19 (ref. The transformation at p is a reconstruction of the large hole-like cylinder at high doping that produces a small electron pocket . We associate the fall of T c and the collapse of H c2 below p to that Fermi-surface reconstruction. Recent studies indicate that charge-density wave order is most likely the cause of the reconstruction . Indeed, the charge modulation seen with X-rays and the Fermi-surface reconstruction seen in the Hall coefficient emerge in parallel with decreasing temperature (see Fig. S7). Moreover, the charge modulation amplitude drops suddenly below T c , showing that superconductivity and charge order compete (Fig. S8a). As a function of field , the onset of this competition defines a line in the H - T plane (Fig. S8B) 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 . We can quantify the impact of phase competition by computing the condensation energy δ E at p = p , using H c1 = 110 ± 5 mT at T c = 93 K (ref. 7) and H c2 = 150 ± 20 T, and comparing with δ E at p = 0.11 (see above): δ E decreases by a factor 20, and δ E / T c2 by a factor 8. In Fig. 4c, we plot the doping dependence of δ E / T c2 and find good qualitative agreement with earlier estimates based on specific heat data (see Fig. S9 and associated discussion). The tremendous weakening of superconductivity below p is attributable to a drop in the density of states as the large hole-like Fermi surface reconstructs into small pockets. This process may well involve both the pseudogap formation and the charge ordering. Upon crossing below p , the Fermi surface of YBCO undergoes a second transformation, signalled by pronounced changes in transport properties and in the effective mass m é (ref. 27), where the small electron pocket disappears. This is strong evidence that the peak in H c2 at p ~ 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 scattering and muon spectroscopy . Note that the increase in m é naturally explains the increase in H c2 going from p = 0.11 (local minimum) to p = 0.08, since H c2 ~ 1 / ξ ~ 1 / v F2 ~ m é . Our findings shed light on the H - T - p phase diagram of YBCO, 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 →
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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. Figure 1 | Field dependence of thermal conductivity. a), b), c), d)
Magnetic field dependence of the thermal conductivity κ in YBCO ( p = 0.11) and Y124 ( p = 0.14), for temperatures as indicated. The end of the rapid rise marks the end of the vortex state, defining the upper critical field H c2 (vertical dashed line). In Figs. 1a and 1c, the data are plotted as κ vs H / H c2 , with H c2 = 22 T for YBCO and H c2 = 44 T for Y124. The remarkable similarity of the normalized curves demonstrates the good reproducibility across dopings. The large quantum oscillations seen in the YBCO data above H c2 confirm the long electronic mean path in this sample. In Figs. 1b and 1d, the overlap of the two isotherms plotted as κ vs H shows that H c2 ( T ) is independent of temperature in both YBCO and Y124, up to at least 8 K. e) Thermal conductivity of the type-II superconductor KFe As in the T = 0 limit, for a sample in the clean limit (green circles). The data are compared to a theoretical calculation for a d -wave superconductor in the clean limit . f) Electrical resistivity of Y124 at T = 1.5 K (blue) and T = 12 K (red) (ref. 15). The green arrow defines the field H n below which the resistivity deviates from its normal-state behaviour (green dashed line). While H c2 ( T ) is essentially constant up to 10 K (Fig. 1d), H vs ( T ) – the onset of the vortex-solid phase of zero resistance (black arrows) – moves down rapidly with temperature (see also Fig. 3b). Figure 2 | Thermal conductivity of Tl-2201. a)
Magnetic field dependence of the thermal conductivity κ in Tl-2201, measured at T = 6 K on an overdoped sample with T c = 33 K (blue). The data are plotted as κ vs H / H c2 , with H c2 = 19 T, and compared with data on YBCO at T = 8 K (red; from Fig. 1b), with H c2 = 23 T. b) Corresponding data for KFe As , taken on clean (red) and dirty (blue) samples. c) Isotherms of κ ( H ) in Tl-2201, at temperatures as indicated, where κ is normalized to unity at H c2 (arrows). H c2 is defined as the field below which κ starts to fall with decreasing field. d) Temperature dependence of H c2 (red squares) and H vs (blue circles) in Tl-2201. The error bars reflect the uncertainty in locating the drop in κ vs H . All lines are a guide to the eye. κ ( H ) / κ ( H c ) H ( T ) Tl-22012 K6 K17 K21 K35 K c H ( T ) T ( K ) Tl-2201 d κ ( W / K m ) H / H c2 Tl-2201T = 6 K
YBCOT = 8 K a κ ( H ) / κ ( H c ) H / H c2 KFe As CleanDirty T 0 b H vs H c2 Figure 3 | Field-temperature phase diagrams. a), b)
Temperature dependence of H c2 (red squares, from data as in Fig. 1) for YBCO and Y124, respectively. The red dashed line is a guide to the eye, showing how H c2 ( T ) might extrapolate to zero at T c . The solid lines are a fit of the H vs ( T ) data (solid circles) to the theory of vortex-lattice melting , as in ref. 17. Note that H c2 ( T ) and H vs ( T ) converge at T = 0, in both materials, so that measurements of H vs vs T can be used to determine H c2 (0) in YBCO. In Fig. 3b, we plot the field H n defined in Fig. 1f (open green squares, from data in ref. 15), which corresponds roughly to the upper boundary of the vortex-liquid phase (see Supplementary Material). We see that H n ( T ) is consistent with H c2 ( T ). p = 0.11 H vs H c2 H ( T ) T ( K )
YBCO p = 0.12 H vs T X H ( T ) T ( K )
YBCO a bc d H n H ( T ) T ( K )
Y124 p = 0.14 H c2 H vs H vs ( T ) T / T c YBCO c) Temperature T X below which charge order is suppressed by the onset of superconductivity in YBCO at p = 0.12, as detected by X-ray diffraction (open green circles, from Fig. S8). We see that T X ( H ) follows a curve (red dashed line) that is consistent with H n ( T ) (at p = 0.14; Fig. 1f) and with the H c2 ( T ) detected by thermal conductivity at lower temperature (at p = 0.11 and 0.14). d) H vs ( T ) vs T / T c , showing a dramatic increase in H vs (0) as p goes from 0.12 to 0.18. From these and other data (in Fig. S6), we obtain the H vs ( T →
0) values that produce the H c2 vs p curve plotted in Fig. 4a. H c ( T ) Hole doping, p YBCO p1 p2 a T c ( K ) Hole doping, p YBCO H = 015 T30 T bc δ E / T c ( J / K m ) Hole doping, p Figure 4 | Doping dependence of H c2 , T c and the condensation energy. a) Upper critical field H c2 of the cuprate superconductor YBCO as a function of hole concentration (doping) p . H c2 is defined as H vs ( T →
0) (Table S1), the onset of the vortex-solid phase at T →
0, where H vs ( T ) is obtained from high-field resistivity data (Figs. 3, S5 and S6). The point at p = 0.14 (square) is from data on Y124 (Fig. 3b). b) Critical temperature T c of YBCO as a function of doping p , for three values of the magnetic field H , as indicated (Table S1). T c is defined as the point of zero resistance. All lines are a guide to the eye. Two peaks are observed in H c2 ( p ) and in T c ( p ; H > 0), located at p ~ 0.08 and p ~ 0.18 (open diamonds). The first peak coincides with the onset of incommensurate spin modulations at p ≈ and muon spin spectroscopy . The second peak coincides with the approximate onset of Fermi-surface reconstruction , attributed to charge modulations detected by high-field NMR (ref. 23) and X-ray scattering . c) Condensation energy δ E (full red circles), given by the product of H c2 and H c1 (see text and Fig. S9), plotted as δ E / T c2 . Note the 8-fold drop below p (vertical dashed line), attributed predominantly to a corresponding drop in the density of states.(vertical dashed line), attributed predominantly to a corresponding drop in the density of states.