Direct measurement of the upper critical field in cuprate superconductors

Abstract

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 YBa₂Cu₃O_y, YBa₂Cu₄O₈ and Tl₂Ba₂CuO_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 YBa₂Cu₃O_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.

The point at which a magnetic field kills superconductivity in the cuprates has been difficult to measure. Grissonnanche et al. use thermal conductivity measurements to reliably determine this field and find that it drops suddenly below some critical doping, suggesting the onset of a new competing phase.

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=Φ₀/2πξ₀², 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 Tl₂Ba₂CuO_6+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 (l₀>>ξ₀), 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 κ at H_c2 (Fig. 1e).

Figure 1. Thermal conductivity of YBCO and Y124.

(a–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 Fig. 1a,c, 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 Fig. 1b,d, 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 KFe₂As₂ in the T=0 limit, for a sample in the clean limit (green circles). Error bars represent the uncertainty in extrapolating κ/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 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).

To confirm our interpretation that the drop in κ is due to vortex scattering, we measured a single crystal of Tl-2201 for which l₀~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 (l₀~10 ξ₀) (ref. 9) to dirty (l₀~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.

Figure 2. Thermal conductivity and H-T diagram 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 KFe₂As₂, taken on clean9 (red) and dirty10 (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. Error bars on the H_c2 data represent the uncertainty in locating the onset of the drop in κ vs H relative to the constant normal-state behaviour. All lines are a guide to the eye.

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 l₀ was estimated to be roughly 50 nm (ref. 11), so that the clean-limit condition l₀>>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_c²/2μ₀, 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 data12 on YBCO give H_c1=24±2 mT at T_c=56 K. Using κ_GL=50 (ref. 12), our value of H_c2=22 T (at T_c=61 K) yields δE/T_c²=13±3 J K⁻² m⁻³. For a d-wave superconductor, δE=N_F Δ₀²/4, 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 γ_N=(2π²/3) N_F k_B², so that δE/T_c²=(3α²/8π²) γ_N. Specific heat data5 on YBCO at T_c=59 K give γ_N=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_BT_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).

Figure 3. Field-temperature phase diagram of YBCO and Y124.

(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. Error bars on the H_c2 data represent the uncertainty in locating the onset of the downward deviation in κ vs H relative to the normal-state behaviour. The solid lines are a fit of the H_vs(T) data (solid circles) to the theory of vortex-lattice melting1, as in ref. 14. 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 Fig. 3b, we plot the field H_n defined 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 the H_n data represent the uncertainty in locating the onset of the downward deviation in ρ vs H relative to the normal-state behaviour. We see that H_n(T) is consistent with H_c2(T). (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 diffraction24 (open green circles, from Supplementary Fig. 7). Error bars on the T_X data represent the uncertainty in locating the onset of the downward deviation in the x-ray intensity vs T at a given field relative to the data at 17 T (see Supplementary Fig. 7a). We see that T_X(H) follows a curve (red dashed line) that is consistent with H_n(T) (at p=0.14; Fig. 3b) 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 marked increase in H_vs(0) as p goes from 0.12 to 0.18. From these and other data (in Supplementary Fig. 6), we obtain the H_vs(T→0) values that produce the H_c2 vs p curve plotted in Fig. 4a.

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, fields lower than 60 T are sufficient to suppress T_c to zero, and thus directly assess H_vs(T→0), yielding 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 (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 p₁~0.08 and p₂~0.18. (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).

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 1), the onset of the vortex-solid phase at T→0, where H_vs(T) is obtained from high-field 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 p>0.22 (diamonds) are from data on Tl-2201 (Table 1, Fig. 2 and Supplementary Fig. 6). Error bars on the H_c2 data represent the uncertainty in extrapolating the H_vs(T) data to T=0. (b) Critical temperature T_c of YBCO as a function of doping p, for three values of the magnetic field H, as indicated (Table 1). 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 p₁~0.08 and p₂~0.18 (open diamonds). The first peak coincides with the onset of incommensurate spin modulations at p≈0.08, 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-field NMR (ref. 22) and X-ray scattering23,24,25. (c) Condensation energy δE (red circles), given by the product of H_c2 and H_c1 (see Supplementary Note 5 and Supplementary Fig. 8), plotted as δE/T_c² 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.

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	30±5
0.078	6.45	45.0	11.5	4.9	50±5
0.102	6.51	59.0	12.4	0	30±2
0.109	6.54	61.3	7.8	0	24±2
0.120	6.67	66.0	10.5	0	24±2
0.135	6.80	78.5	28.7	−	35±5
0.140	Y124	80.0	30.8	11.8	45±2
0.151	6.86	91.0	52.1	31.9	70±10
0.161	6.92	93.5	67.9	51.0	115±15
0.173	6.99	93.0	69.1	56.0	140±20
0.180	6.998	90.5	69.3	57.6	150±20
0.190	Ca-1.4%	87.0	65.2	53.3	140±20
0.205	Ca-5%	77.0	49.2	36.4	100±20
0.225	Tl-2201	59	—	—	63±5
0.248	Tl-2201	33	—	—	20±2
0.257	Tl-2201	20	—	—	14±2
0.260	Tl-2201	15	—	—	10±3

List of all samples whose resistivity data are used in this paper. Doping p; oxygen content y; zero-resistance T_c at H=0, H=15 T and H=30 T; upper critical field H_c2=H_vs(T=0). The H_c2 values are plotted vs p in Fig. 4a. The T_c values at H=0, 15 T and 30 T are plotted vs p in Fig. 4b. The value of H_c2=H_vs(T=0) in Tl-2201 at T_c=15 K is obtained from analysis of published data (see Supplementary Figs 3 and 4).

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 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 p~0.18 (ref. 18). Hints of two critical points have also been found in Bi₂Sr₂CaCu₂O_8+δ, as changes in the superconducting gap detected by ARPES at p₁~0.08 and p₂~0.19 (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 T_c and the collapse of H_c2 below p₂ 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 p₂.

Figure 5. Fermi-surface reconstruction and charge order.

Hall coefficient R_H(T) of YBCO as a function of temperature at a doping p=0.12 (T_c=66 K), for a field H=15 T (red line, from ref. 26). T_max is the temperature at which the Hall coefficient R_H(T) peaks, before it falls to reach negative values—a signature of Fermi-surface reconstruction18,21. T_H is the inflexion point where the downturn in R_H(T) begins. The evolution of R_H(T) is compared with the growth of charge-density-wave modulations in YBCO detected by X-ray diffraction, at the same doping and field24. As seen, the onset of the modulations, at T_CO~130 K, coincides with T_H. This suggests a causal connection between charge order and Fermi-surface reconstruction.

We can quantify the impact of phase competition by computing the condensation energy δE at p=p₂, 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_c² by a factor 8 (see Supplementary Note 5). In Fig. 4c, we plot the doping dependence of δE/T_c² (in qualitative agreement with earlier estimates based on specific heat data28—see Supplementary Fig. 8). We attribute the tremendous weakening of superconductivity 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.

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 p₁~0.08 (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/ξ₀²~1/v_F²~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 La_1.8-xEu_0.2Sr_xCuO₄ (ref. 20) and HgBa₂CuO_4+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).

Table 2. Samples used in thermal conductivity measurements.

Sample	Hc2 (T)	Figs	Hvs (T)	Figs
YBCO 6.54 no 4	22±2	1	Yes	3a, S5
YBCO 6.54 no 5	22±2		Yes
YBCO 6.54 b	23±2		No
YBCO 6.56	23±2		No
Y124 D	44±2	1	No
Y124 YTA2	43±2		Yes	1f, 3b, S5
Tl-2201 Tc =33 K	20±2	2	Yes	2d
Tl-2201 Tc =20 K	14±2		Yes

List of all samples whose thermal conductivity was measured in this study. In the second column, we give the value of H_c2 obtained from κ vs H at T~2 K. In the third column, we state the figure(s) where κ data are displayed. We also state whether the resistance of a sample was used to determine H_vs(T), and refer to the corresponding figure(s).

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