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Received 13 Aug 2013 | Accepted 19 Jan 2014 | Published 12 Feb 2014

G. Grissonnanche1, O. Cyr-Choinire1, F. Lalibert1, S. Ren de Cotret1, A. Juneau-Fecteau1, S. Dufour-Beausjour1,M.-. Delage1, D. LeBoeuf1,w, J. Chang1,w, B.J. Ramshaw2, D.A. Bonn2,3, W.N. Hardy2,3, R. Liang2,3, S. Adachi4, N.E. Hussey5,w, B. Vignolle6, C. Proust3,6, M. Sutherland7, S. Kramer8, J.-H. Park9, D. Graf9, N. Doiron-Leyraud1 & Louis Taillefer1,3

In the quest to increase the critical temperature Tc of cuprate superconductors, it is essential to identify the factors that limit the strength of superconductivity. The upper critical eld

Hc2 is a fundamental measure of that strength, yet there is no agreement on its magnitude and doping dependence in cuprate superconductors. Here we show that the thermal conductivity can be used to directly detect Hc2 in the cuprates YBa2Cu3Oy, YBa2Cu4O8 and

Tl2Ba2CuO6 d, allowing us to map out Hc2 across the doping phase diagram. It exhibits two peaks, each located at a critical point where the Fermi surface of YBa2Cu3Oy is known to undergo a transformation. Below the higher critical point, the condensation energy, obtained directly from Hc2, suffers a sudden 20-fold collapse. This reveals that phase competition

associated with Fermi-surface reconstruction and charge-density-wave orderis a key limiting factor in the superconductivity of cuprates.

1 Dpartement de physique & RQMP, Universit de Sherbrooke, Sherbrooke, Qubec, Canada J1K 2R1. 2 Department of Physics & Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1. 3 Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8.

4 Superconductivity Research Laboratory, ISTEC, Yokohama, Kanagawa 223-0051, Japan. 5 H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK. 6 Laboratoire National des Champs Magntiques Intenses, Toulouse 31400, France. 7 Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK. 8 Laboratoire National des Champs Magntiques Intenses, Grenoble, France. 9 National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA. w Present address: Laboratoire National des Champs Magntiques Intenses, Grenoble, France (D.L.);cole Polytechnique Fdrale de Lausanne,

CH-1015 Lausanne, Switzerland (J.C.); High Field Magnet Laboratory, Radboud University Nijmegen, The Netherlands (N.E.H.). Correspondence and requests for materials should be addressed to N.D.-L. (email: mailto:[email protected]

Web End [email protected] ) or to L.T. (email:mailto:[email protected]

Web End [email protected] ).

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DOI: 10.1038/ncomms4280 OPEN

Direct measurement of the upper critical eld in cuprate superconductors

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4280

In a type-II superconductor at T 0, the onset of the superconducting state as a function of decreasing magnetic eld H occurs at the upper critical eld Hc2, dictated by the

pairing gap D through the coherence length x0BvF/D, via Hc2

F0/2px02, where vF is the Fermi velocity and F0 is the magnetic ux quantum. Hc2 is the eld 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 eld, Hvs, is equal to Hc2. In cuprate superconductors, the strong 2D character and low superuid density cause a vortex liquid phase to intervene between the vortex-solid phase below Hvs(T) and the normal state above

Hc2(T) (ref. 1). It has been argued that in underdoped cuprates there is a wide vortex-liquid phase even at T 0 (refs 25), so

that Hc2(0)44Hvs(0), implying that D is very large. Whether the gap D is large or small in the underdoped regime is a pivotal issue for understanding what controls the strength of super-conductivity in cuprates. So far, however, no measurement on a

cuprate superconductor has revealed a clear transition at Hc2,

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 elds, but it is difcult to know whether these are due to vortex-like excitations below Hc2 or to uctuations above Hc2 (ref. 7).

Here we demonstrate that measurements of the thermal conductivity can directly detect Hc2, and we show that in the cuprate superconductors YBa2Cu3Oy (YBCO) and YBa2Cu4O8 (Y124) there is no vortex liquid at T 0. This fact allows us

to then use measurements of the resistive critical eld Hvs(T) to

obtain Hc2 in the T 0 limit. By including measurements on the

overdoped cuprate Tl2Ba2CuO6 d (Tl-2201), we establish the full

doping dependence of Hc2. The magnitude of Hc2 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

YBCO p = 0.11

1.8 K

8 K

2.6

0.0 0.5 1.0 1.5 2.0

0.84

0.92

9.4

(W K1 m1 )

Y124

T = 1.6 K

YBCO

T = 1.8 K

2.4

0.88

9.2

9.0

2.2

0.84

0 10 20 30 40 508.8

c d

9.4

2.6

Y124 p = 0.14

1.6 K

9 K

(W K1 m1 )

9.2

Y124

T = 9 K

YBCO

T = 8 K

2.4

9.0

2.2

8.8

e f

1.0

20 30 40 50

Y124

12 K

1.5 K

Hvs

d-wave Clean limit

T 0

KFe2As2

1.0

0.0 0.5 1.0 1.5

(H) / (H c2)

(H) / (55 T)

0.5

0.0

H / Hc2

H (T)

Figure 1 | Thermal conductivity of YBCO and Y124. (ad) Magnetic eld dependence of the thermal conductivity k 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, dening the upper critical eld Hc2 (vertical

dashed line). In Fig. 1a,c, the data are plotted as k vs H/Hc2, with Hc2 22 T for YBCO and Hc2 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 Hc2 conrm the long electronic mean path in this sample. In Fig. 1b,d, the overlap of the two isotherms plotted as k vs H shows that Hc2(T) is independent of temperature in both YBCO and Y124, up to at least 8 K. (e) Thermal conductivity of the type-II superconductor KFe2As2 in the T 0 limit, for a sample in the

clean limit (green circles). Error bars represent the uncertainty in extrapolating k/T to T 0. The data9 are compared with a theoretical calculation

for a d-wave superconductor in the clean limit8. (f) Electrical resistivity of Y124 at T 1.5 K (blue) and T 12 K (red) (ref. 11). The green arrow denes

the eld Hn below which the resistivity deviates from its normal-state behaviour (green dashed line). While Hc2(T) is essentially constant up to 10 K (Fig. 1d), Hvs(T)the onset of the vortex-solid phase of zero resistance (black arrows)moves down rapidly with temperature (see also Fig. 3b).

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phase is associated with the onset of Fermi-surface reconstruction and charge-density-wave order, generic properties of hole-doped cuprates.

ResultsThermal conductivity. To detect Hc2, 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 k of a sample as a function of magnetic eld H. In

Fig. 1, we report our data on YBCO and Y124, as k vs H up to45 T, at two temperatures well below Tc (see Methods and Supplementary Note 1). All curves exhibit the same rapid drop below a certain critical eld. This is precisely the behaviour expected of a clean type-II superconductor (l044x0), whereby the long electronic mean free path l0 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 k at Hc2 (Fig. 1e).

To conrm our interpretation that the drop in k is due to vortex scattering, we measured a single crystal of Tl-2201 for which l0Bx0, corresponding to a type-II superconductor in the dirty limit. As seen in Fig. 2a, the suppression of k 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 (l0B10 x0) (ref. 9) to dirty (l0Bx0) (ref. 10) (see

Fig. 2b). We conclude that the onset of the sharp drop in k with decreasing H in YBCO is a direct measurement of the critical eld Hc2, where vortex scattering begins.

Upper critical eld Hc2. The direct observation of Hc2 in a cuprate material is our rst main nding. We obtain Hc2 222 T at T 1.8 K in YBCO (at p 0.11) and

Hc2 442 T at T 1.6 K in Y124 (at p 0.14) (Fig. 1a), giving

x0 3.9 nm and 2.7 nm, respectively. In Y124, the transport mean

free path l0 was estimated to be roughly 50 nm (ref. 11), so that the clean-limit condition l044x0 is indeed satised. Note that the specic heat is not sensitive to vortex scattering and so will have a much less pronounced anomaly at Hc2. This is consistent with the high-eld specic heat of YBCO at p 0.1 (ref. 5).

We can verify that our measurement of Hc2 in YBCO is

consistent with existing thermodynamic and spectroscopic data by computing the condensation energy dE Hc2/2m0, where

Hc2 Hc1 Hc2/(ln kGL 0.5), with Hc1 the lower critical eld

and kGL the Ginzburg-Landau parameter (ratio of penetration depth to coherence length). Magnetization data12 on YBCO give

Hc1 242 mT at Tc 56 K. Using kGL 50 (ref. 12), our value

of Hc2 22 T (at Tc 61 K) yields dE/Tc2 133 J K 2 m 3.

For a d-wave superconductor, dE NF D02/4, where D0 a kB Tc

is the gap maximum and NF is the density of states at the Fermi energy, related to the electronic specic heat coefcient gN (2p2/3) NF kB2, so that dE/Tc2 (3a2/8p2) gN. Specic heat

data5 on YBCO at Tc 59 K give gN 4.50.5 mJ K 2 mol 1

(435 J /K 2 m 3) above Hc2. We therefore obtain a 2.80.5,

1.2

Tl-2201 T = 6 K

YBCO T = 8 K

KFe2As2

Dirty

Clean

T 0

9.4

3.6

(W K1 m1 )

0.8

3.2

(H) / (H c2)

9.2

0.6

0.4

2.8

0.2

8.8

2.4

0 0.5 1 1.5

0 0 0.5 1 1.5

H / Hc2

Tl-2201

Hc2

Hvs

(H) / (H c2)

0.9

Tl-2201

2 K

35 K

21 K

17 K

6 K

H(T)

0.8

0.7

0 4 8 12 16 20 24

0 0 5 10 15 20 25 30 35

H (T)

T (K)

Figure 2 | Thermal conductivity and H-T diagram of Tl-2201. (a) Magnetic eld dependence of the thermal conductivity k in Tl-2201, measured at T 6 K on an overdoped sample with Tc 33 K (blue). The data are plotted as k vs H/Hc2, with Hc2 19 T, and compared with data on YBCO at

T 8 K (red; from Fig. 1b), with Hc2 23 T. (b) Corresponding data for KFe2As2, taken on clean9 (red) and dirty10 (blue) samples. (c) Isotherms of

k(H) in Tl-2201, at temperatures as indicated, where k is normalized to unity at Hc2 (arrows). Hc2 is dened as the eld below which k starts to fall with decreasing eld. (d) Temperature dependence of Hc2 (red squares) and Hvs (blue circles) in Tl-2201. Error bars on the Hc2 data represent the uncertainty in locating the onset of the drop in k vs H relative to the constant normal-state behaviour. All lines are a guide to the eye.

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in good agreement with estimates from spectroscopic measurements on a variety of hole-doped cuprates, which yield 2D0/kBTcB5 between p 0.08 and p 0.24 (ref. 13). This

shows that the value of Hc2 measured by thermal conductivity provides quantitatively coherent estimates of the condensation energy and gap magnitude in YBCO.

HT phase diagram. The position of the rapid drop in k vs H does not shift appreciably with temperature up to TB10 K or so (Fig. 1b,d), showing that Hc2(T) is essentially at at low temperature. This is in sharp contrast with the resistive transition at Hvs(T), which moves down rapidly with increasing temperature (Fig. 1f). In Fig. 3, we plot Hc2(T) and Hvs(T) on an H-T diagram, for both YBCO and Y124 (see Methods and Supplementary Methods). In both cases, we see that Hc2 Hvs in the T 0 limit.

This is our second main nding: there is no vortex liquid regime at T 0 (see Supplementary Note 3). With increasing tempera

ture the vortex-liquid phase grows rapidly, causing Hvs(T) to fall

below Hc2(T). The same behaviour is seen in Tl-2201 (Fig. 2d): at low temperature, Hc2(T) determined from k is at, whereas

Hvs(T) from resistivity falls abruptly, and Hc2 Hvs at T-0 (see

also Supplementary Figs 3 and 4, and Supplementary Note 4).

Hp phase diagram. Having established that Hc2 Hvs at T-0

in YBCO, Y124 and Tl-2201, we can determine how Hc2 varies with doping from measurements of Hvs(T) (see Methods and

Supplementary Methods), as in Supplementary Figs 5 and 6. For po0.15, elds lower than 60 T are sufcient to suppress Tc to

zero, and thus directly assess Hvs(T-0), yielding Hc2 242 T

at p 0.12 (Fig. 3c), for example. For p40.15, however, Tc

cannot be suppressed to zero with our maximal available eld of68 T (Fig. 3d and Supplementary Fig. 5), so an extrapolation procedure must be used to extract Hvs(T-0). Following ref. 14, we obtain Hvs(T-0) from a t to the theory of vortex-lattice melting1, as illustrated in Fig. 3 (and Supplementary Fig. 6). In Fig. 4a, we plot the resulting Hc2 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 nding: the Hp phase

diagram of superconductivity consists of two peaks, located at p1B0.08 and p2B0.18. (A partial plot of Hvs(T-0) vs p was

a b

YBCO

p = 0.12

p = 0.11

Hc2

Hvs

Hc2

H (T)

Hvs

0 0 20 40 60

0 0 25 50 75

T (K)

Y124p = 0.14

c d

100

YBCO

p = 0.180

0.151

H (T)

H vs (T)

Hvs

0.120

0 0 20 40 60

0 0.0 0.5 1.0

T (K)

T / Tc

Figure 3 | Field-temperature phase diagram of YBCO and Y124. (a,b) Temperature dependence of Hc2 (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 Hc2(T) might extrapolate to zero at Tc. Error bars on theHc2 data represent the uncertainty in locating the onset of the downward deviation in k vs H relative to the normal-state behaviour. The solid lines are a t of the Hvs(T) data (solid circles) to the theory of vortex-lattice melting1, as in ref. 14. Note that Hc2(T) and Hvs(T) converge at T 0, in both

materials, so that measurements of Hvs vs T can be used to determine Hc2(0). In Fig. 3b, we plot the eld Hn dened in Fig. 1f (open green squares, from data in ref. 11), which corresponds roughly to the upper boundary of the vortex-liquid phase (see Supplementary Note 3). Error bars on theHn data represent the uncertainty in locating the onset of the downward deviation in r vs H relative to the normal-state behaviour. We see that

Hn(T) is consistent with Hc2(T). (c) Temperature TX below which charge order is suppressed by the onset of superconductivity in YBCO at p 0.12,

as detected by X-ray diffraction24 (open green circles, from Supplementary Fig. 7). Error bars on the TX data represent the uncertainty in locating the onset of the downward deviation in the x-ray intensity vs T at a given eld relative to the data at 17 T (see Supplementary Fig. 7a). We see that TX(H)

follows a curve (red dashed line) that is consistent with Hn(T) (at p 0.14; Fig. 3b) and with the Hc2(T) detected by thermal conductivity at lower

temperature (at p 0.11 and 0.14). (d) Hvs(T) vs T/Tc, showing a marked increase in Hvs(0) as p goes from 0.12 to 0.18. From these and other data

(in Supplementary Fig. 6), we obtain the Hvs(T-0) values that produce the Hc2 vs p curve plotted in Fig. 4a.

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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 Tp plane: the single Tc dome at H 0 transforms into two domes when a magnetic eld

is applied (Fig. 4b).

DiscussionA natural explanation for two peaks in the Hc2 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 Tc domes in the temperature-pressure phase diagram were revealed by adding impurities to weaken superconductivity15: one dome straddles an underlying anti-ferromagnetic transition and the other dome a valence transition16. In YBCO, there is indeed strong evidence of two transitionsone at p1 and another at a critical doping consistent with p2 (ref. 17). In particular, the Fermi surface of YBCO is known to undergo one transformation at p 0.08 and another

near pB0.18 (ref. 18). Hints of two critical points have also been found in Bi2Sr2CaCu2O8 d, as changes in the superconducting

gap detected by ARPES at p1B0.08 and p2B0.19 (ref. 19).

The transformation at p2 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 Tc and the collapse of Hc2 below p2 to that Fermi-surface reconstruction. Recent studies indicate that charge-density wave order plays a role in the reconstruction2225. Indeed, the charge modulation seen with X-rays2325 and the Fermi-surface reconstruction seen in the Hall coefcient18,26 emerge in parallel with decreasing temperature (see Fig. 5). Moreover, the charge modulation amplitude drops suddenly below Tc, showing that superconductivity and charge order compete2325 (Supplementary Fig. 7a). As a function of eld24, the onset of this competition denes a line in the HT plane (Supplementary Fig. 7b) that is consistent with our Hc2(T)

line (Fig. 3). The ip side of this phase competition is that superconductivity must in turn be suppressed by charge order, consistent with our interpretation of the Tc fall and Hc2 collapse below p2.

We can quantify the impact of phase competition by computing the condensation energy dE at p p2, using

Hc1 1105 mT at Tc 93 K (ref. 27) and Hc2 14020 T

(Table 1), and comparing with dE at p 0.11 (see above): dE

decreases by a factor 20 and dE/Tc2 by a factor 8 (see Supplementary Note 5). In Fig. 4c, we plot the doping dependence of dE/Tc2 (in qualitative agreement with earlier estimates based on specic heat data28see Supplementary Fig. 8). We attribute the tremendous weakening of super-conductivity below p2 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.

YBCO

p1 p2

150

100

H c2( T )

0 0.1 0.2 0.3

Hole doping, p

100

YBCO

H = 0

15 T30 T

T c ( K )

Figure 4 | Doping dependence of Hc2, Tc and the condensation energy. (a) Upper critical eld Hc2 of the cuprate superconductor YBCO as a function of hole concentration (doping) p. Hc2 is dened as Hvs(T-0)

(Table 1), the onset of the vortex-solid phase at T-0, where Hvs(T) is

obtained from high-eld resistivity data (Fig. 3, and Supplementary Figs 5 and 6). The point at p 0.14 (square) is from data on Y124 (Fig. 3b). The

points at p40.22 (diamonds) are from data on Tl-2201 (Table 1, Fig. 2 and Supplementary Fig. 6). Error bars on the Hc2 data represent the uncertainty in extrapolating the Hvs(T) data to T 0. (b) Critical temperature Tc of

YBCO as a function of doping p, for three values of the magnetic eld H, as indicated (Table 1). Tc is dened as the point of zero resistance. All lines are a guide to the eye. Two peaks are observed in Hc2(p) and in Tc(p; H40), located at p1B0.08 and p2B0.18 (open diamonds). The rst peak coincides with the onset of incommensurate spin modulations at pE0.08, detected by neutron scattering30 and muon spin spectroscopy31. The second peak coincides with the approximate onset of Fermi-surface reconstruction18,21,

attributed to charge modulations detected by high-eld NMR (ref. 22) and X-ray scattering2325. (c) Condensation energy dE (red circles), given by the product of Hc2 and Hc1 (see Supplementary Note 5 and Supplementary Fig. 8), plotted as dE/Tc2 vs p. Note the eightfold drop below p2 (vertical dashed line), attributed predominantly to a corresponding drop in the density of states. All lines are a guide to the eye.

0 0.1 0.2 0.3

Hole doping, p

200

YBCO

2 (J K2 m3 )

150

100

E / T c

0 0.1 0.2 0.3

Hole doping, p

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Table 1 | Samples used in resistance measurements.

Doping y Tc(0) (K) Tc(15T) (K) Tc(30T) (K) Hc2 (T)0.063 6.35 19.5 2.0 0 3050.078 6.45 45.0 11.5 4.9 5050.102 6.51 59.0 12.4 0 3020.109 6.54 61.3 7.8 0 2420.120 6.67 66.0 10.5 0 2420.135 6.80 78.5 28.7 355

0.140 Y124 80.0 30.8 11.8 4520.151 6.86 91.0 52.1 31.9 70100.161 6.92 93.5 67.9 51.0 115150.173 6.99 93.0 69.1 56.0 140200.180 6.998 90.5 69.3 57.6 150200.190 Ca-1.4% 87.0 65.2 53.3 140200.205 Ca-5% 77.0 49.2 36.4 100200.225 Tl-2201 59 6350.248 Tl-2201 33 2020.257 Tl-2201 20 1420.260 Tl-2201 15 103

List of all samples whose resistivity data are used in this paper. Doping p; oxygen content y; zero-resistance T at H 0, H 15 T and H 30 T; upper critical eld H H (T 0). The H values are

plotted vs p in Fig. 4a. The T values at H 0, 15 Tand 30 Tare plotted vs p in Fig. 4b. The value of H H (T 0) in Tl-2201 at T 15 K is obtained from analysis of published data (see Supplementary

Figs 3 and 4).

YBCO H = 15 T

p = 0.12

Tmax

200

X-ray intensity (counts s1 )

RH (mm

3 C 1)

100

0 50 100 150 2005

T (K)

Figure 5 | Fermi-surface reconstruction and charge order. Hall coefcient RH(T) of YBCO as a function of temperature at a doping p 0.12

(Tc 66 K), for a eld H 15 T (red line, from ref. 26). Tmax is the

temperature at which the Hall coefcient RH(T) peaks, before it falls to reach negative valuesa signature of Fermi-surface reconstruction18,21. TH is the inexion point where the downturn in RH(T) begins. The evolution of RH(T) is compared with the growth of charge-density-wave modulations in

YBCO detected by X-ray diffraction, at the same doping and eld24. As seen, the onset of the modulations, at TCOB130 K, coincides with TH. This suggests a causal connection between charge order and Fermi-surface reconstruction.

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 Hc2 at p1B0.08 (Fig. 4a)

coincides with an underlying critical point. This critical point is presumably associated with the onset of incommensurate spin modulations detected below pB0.08 by neutron scattering30 and muon spectroscopy31. Note that the increase in m* (ref. 29) may in part explain the increase in Hc2 going from p 0.11 (local

minimum) to p 0.08, since Hc2B1/x02B1/vF2Bm*2.

Our ndings 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 ngerprint 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 Tc curve as being primarily due to phase competition, rather than uctuations in the phase of the superconducting order parameter32, and they quantify the impact of that competition on the condensation energy.

Our nding of a collapse in condensation energy due to phase competition is likely to be a generic property of hole-doped cuprates, since Fermi-surface reconstructionthe inferred cause has been observed in materials such as La1.8-xEu0.2SrxCuO4

(ref. 20) and HgBa2CuO4 d (refs 33,34), two cuprates whose

structure is signicantly 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-Tc cuprates.

Methods

Samples. Single crystals of YBa2Cu3Oy (YBCO) were obtained by ux growth at UBC (ref. 35). The superconducting transition temperature Tc was determined as the temperature below which the zero-eld resistance R 0. The hole doping p is

obtained from Tc (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 ux 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, withTc 80 K. The doping is estimated from the value of Tc, 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 ux 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 Tc 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 Tc, via the standard formula Tc/Tcmax 182.6 (p0.16)2, with Tcmax 90 K.

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4280 ARTICLE

Table 2 | Samples used in thermal conductivity measurements.

Sample Hc2 (T) Figs Hvs (T) Figs YBCO 6.54 no 4 222 1 Yes 3a, S5

YBCO 6.54 no 5 222 YesYBCO 6.54 b 232 NoYBCO 6.56 232 NoY124 D 442 1 NoY124 YTA2 432 Yes 1f, 3b, S5 Tl-2201 Tc 33 K 202 2 Yes 2d

Tl-2201 Tc 20 K 142 Yes

List of all samples whose thermal conductivity was measured in this study. In the second column, we give the value of H obtained from k vs H at TB2 K. In the third column, we state the gure(s) where k data are displayed. We also state whether the resistance of a sample was used to determine H (T), and refer to the corresponding gure(s).

Resistivity measurements. The in-plane electrical resistivity of YBCO was measured in magnetic elds up to 45 T at the NHMFL in Tallahassee and up to68 T at the LNCMI in Toulouse. A subset of those data is displayed in Supplementary Fig. 5. From such data, Hvs(T) is determined and extrapolatedto T 0 to get Hvs(0), as illustrated in Fig. 3 and Supplementary Fig. 6. The

Hc2 Hvs(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 Tc 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 eld was applied along the c axis, normal to the CuO2 planes. (See also Supplementary Methods.)

Thermal conductivity measurements. The thermal conductivity k 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 k 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 dened as k (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 Thot Tcold was measured with two Cernox thermometers,

sensing the temperature at the hot (Thot) and cold (Tcold) ends of the sample,

respectively. The Cernox thermometers were calibrated by performing eld sweeps at different closely spaced temperatures between 2 K and 15 K. Representative data are shown in Fig. 1. (See also Supplementary Note 1.)

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4280

Acknowledgements

We thank Y. Ando, A. Carrington, S.A. Kivelson, A.J. Millis, S. Sachdev and A.-M. Tremblay for fruitful discussions. We thank C. Marcenat for his assistance with the experiments at the LNCMI in Grenoble, and J. Corbin, S. Fortier, and F. Francoeur for their assistance with the experiments at Sherbrooke. R.L., D.A.B. and W.N.H. acknowledge support from NSERC. L.T. acknowledges support from the Canadian Institute for Advanced Research and funding from NSERC, FQRNT, the Canada Foundation for Innovation, and a Canada Research Chair. The work in Toulousewas supported by the French ANR SUPERFIELD, Euromagnet II, and theLABEX NEXT.

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.-.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.

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How to cite this article: Grissonnanche, G. et al. Direct measurement of the upper critical eld in cuprate superconductors. Nat. Commun. 5:3280 doi: 10.1038/ ncomms4280 (2014).

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In the quest to increase the critical temperature T_c of cuprate superconductors, it is essential to identify the factors that limit the strength of superconductivity. The upper critical field H_c2 is a fundamental measure of that strength, yet there is no agreement on its magnitude and doping dependence in cuprate superconductors. Here we show that the thermal conductivity can be used to directly detect H_c2 in the cuprates YBa₂ Cu₃ O_y , YBa₂ Cu₄ O₈ and Tl₂ Ba₂ CuO_6+δ , allowing us to map out H_c2 across the doping phase diagram. It exhibits two peaks, each located at a critical point where the Fermi surface of YBa₂ Cu₃ O_y is known to undergo a transformation. Below the higher critical point, the condensation energy, obtained directly from H_c2 , suffers a sudden 20-fold collapse. This reveals that phase competition--associated with Fermi-surface reconstruction and charge-density-wave order--is a key limiting factor in the superconductivity of cuprates.

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Title

Direct measurement of the upper critical field in cuprate superconductors

Author

Grissonnanche, G; Cyr-choinière, O; Laliberté, F; René De Cotret, S; Juneau-fecteau, A; Dufour-beauséjour, S; Delage, M -È; Leboeuf, D; Chang, J; Ramshaw, B J; Bonn, D A; Hardy, W N; Liang, R; Adachi, S; Hussey, N E; Vignolle, B; Proust, C; Sutherland, M; Krämer, S; Park, J -h; Graf, D; Doiron-leyraud, N; Taillefer, Louis

Pages

3280

Publication year

2014

Publication date

Feb 2014

Publisher

Nature Publishing Group

e-ISSN

20411723

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms4280

ProQuest document ID

1497197631

Direct measurement of the upper critical field in cuprate superconductors

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