Two-stage magnetic-field-tuned superconductor-insulator transition in underdoped La 2−x Sr x CuO 4
Xiaoyan Shi, Ping V. Lin, T. Sasagawa, V. Dobrosavljević, Dragana Popović
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec Two-stage magnetic-field-tuned superconductor-insulatortransition in underdoped La − x Sr x CuO Xiaoyan Shi † , Ping V. Lin , T. Sasagawa , V. Dobrosavljevi´c & Dragana Popovi´c ∗ December 4, 2014 National High Magnetic Field Laboratory and Department of Physics, Florida State University,Tallahassee, Florida 32310, USA Materials and Structures Laboratory, Tokyo Institute of Technology, Kanagawa 226-8503, Japan † Present address: Sandia National Laboratories, Albuquerque, New Mexico 87185, USA* E-mail: [email protected] 1 n the underdoped pseudogap regime of cuprate superconductors, the normal state is commonlyprobed by applying a magnetic field ( H ). However, the nature of the H -induced resistive statehas been the subject of a long-term debate, and clear evidence for a zero-temperature ( T = 0 ) H -tuned superconductor-insulator transition (SIT) has proved elusive. Here we report mag-netoresistance measurements in underdoped La − x Sr x CuO , providing striking evidence forquantum critical behavior of the resistivity – the signature of a H -driven SIT. The transition isnot direct: it is accompanied by the emergence of an intermediate state, which is a supercon-ductor only at T = 0 . Our finding of a two-stage H -driven SIT goes beyond the conventionalscenario in which a single quantum critical point separates the superconductor and the insu-lator in the presence of a perpendicular H . Similar two-stage H -driven SIT, in which bothdisorder and quantum phase fluctuations play an important role, may also be expected in othercopper-oxide high-temperature superconductors. The SIT is an example of a quantum phase transition (QPT): a continuous phase transitionthat occurs at T = 0, controlled by some parameter of the Hamiltonian of the system, such as dopingor the external magnetic field . A QPT can affect the behavior of the system up to surprisinglyhigh temperatures. In fact, many unusual properties of various strongly correlated materials havebeen attributed to the proximity of quantum critical points (QCPs). An experimental signature ofa QPT at nonzero T is the observation of scaling behavior with relevant parameters in describingthe data. Although the SIT has been studied extensively , even in conventional superconductorsmany questions remain about the perpendicular-field-driven SIT in two-dimensional (2D) or quasi-2D systems . In high- T c cuprates ( T c – transition temperature), which have a quasi-2D nature,early magnetoresistance (MR) experiments showed the suppression of superconductivity with high H , revealing the insulating behavior and hinting at an underlying H -field-tuned SIT . However,2ven though understanding the effects of H is believed to be essential to understanding high- T c cuprate superconductivity and continues to be a subject of intensive research, the evidence for the H -field-driven SIT and the associated QPT scaling in cuprates remains scant and inconclusive.In the conventional picture of type-II superconductors, H penetrates the sample in the formof a solid lattice of interacting vortex lines in the entire mixed state H c ( T ) < H < H c ( T ),where H c is the Meissner field and H c is the upper critical field. This picture, however, neglectsfluctuations which, in high- T c superconductors, are especially important. Indeed, the delicateinterplay of thermal fluctuations, quantum fluctuations and disorder leads to a complex H − T phase diagram of the vortex matter . Thermal fluctuations, for example, cause melting of thevortex solid into a vortex liquid for fields below what is now a crossover line H c ( T ). Quantumfluctuations could result in a vortex liquid persisting down to T = 0. At very low T , the disorderbecomes important and modifies the vortex phase diagram such that there are two distinct vortexsolid phases
9, 10 : a Bragg glass with T c ( H ) > T c = 0at higher fields. The SIT would then correspond to a transition from a VG to an insulator ateven higher H . However, the interplay of this vortex line physics and quantum criticality, the keyquestion in the H -field-tuned SIT, has remained largely unexplored.In this study, we find strong evidence for the H -field-driven SIT in underdoped La − x Sr x CuO (LSCO). The results are consistent with the existence of three phases at T = 0, although thebehavior of the in-plane resistivity ρ suggests the presence of the direct SIT over a surprisinglywide range of T and H . At the lowest T , however, ρ ( T, H ) reveal an intermediate phase with T c = 0 and the true SIT at higher H . We focus on samples with a relatively low T c ( H = 0) toensure that experimentally available H are high enough to fully suppress superconductivity. Unlike3ost other studies, ours includes samples grown using different techniques (Methods), in order toseparate out any effects that may depend on the sample preparation conditions from the moregeneral behavior. In addition to providing evidence for the SIT, the MR data are used to calculatethe contribution of superconducting fluctuations (SCFs) to conductivity, allowing a construction ofthe H − T phase diagram. In-plane resistivity of La − x Sr x CuO One sample was a film with the nominal composition La . Sr . CuO (ref. 11) and a measured T c ( H = 0) = (3 . ± .
1) K. The high-quality single crystal had T c ( H = 0) = (5 . ± .
1) K andthe nominal composition La . Sr . CuO y , with y not precisely known (see Methods for moredetails). Unless otherwise specified, T c is defined throughout as the temperature at which theresistance ( i.e. ρ ) becomes zero for a given H . (The method to determine T c ( H ) is illustrated inSupplementary Fig. S1.)Figure 1a shows the ρ ( T ) curves for different H that were extracted from the MR measure-ments at fixed T for the x = 0 .
07 sample (see Supplementary Fig. S2 for similar data on the x = 0 .
06 crystal). Small H clearly lead to a decrease of T c ( H ), such that T c → H ≈ T c > H c ∼
100 Oe is understood to be a consequenceof the pinning of vortices by disorder . As H increases further, a pronounced maximum appearsin ρ ( T ). The temperature at the maximum, T max , shifts to lower T with increasing H , similar toearly observations . At the highest H , ρ ( T ) curves exhibit insulating behavior.The intermediate H regime in which each ρ ( T ) curve exhibits a maximum is specially in-triguing. Here the system shows a tendency towards insulating behavior already at high T &
5T 4T 3T 2T 1T 0.5T 0T 18T 16T 14T 12T 10T 8T 6T ( m c m ) T (K) R /l ay er ( k ) b -3 -2 -1
18T 16T 14T 12T 11T 10T 9T 8T 7T 6T ( m c m ) T (K) c H (T) ( H , T )= ( H ) T ( H ) ( m c m ) Figure Temperature dependence of the in-plane resistivity ρ in different magneticfields H k c for x = 0 .
07 LSCO film. a,
The ρ ( T, H ) data exhibit a change of the sign of dρ/dT as a function of H at high T & R (cid:3) /layer is resistance per square per CuO layer; seeMethods). Except for H = 0, solid lines guide the eye. b, Some of the ρ ( T, H ) in a are shownon a log-log scale to focus on the low- T , intermediate- H regime. Short-dashed lines guide the eye.Solid lines represent power-law fits ρ ( T, H ) = ρ ( H ) T α ( H ) . c, Fitting parameters ρ ( H ) and α ( H ).Error bars indicate 1 standard deviation (s.d.) in the fits for ρ ( H ) and α ( H ). Short-dashed linesguide the eye. 5ut then the sign of dρ/dT changes, suggesting that another mechanism sets in at lower T anddrives ρ ( T →
0) towards zero. The low- T , intermediate- H regime is more evident in Fig. 1b, wherethe same data are presented on a log-log scale. For T < T max , the data are described best withthe phenomenological power-law fits ρ ( T, H ) = ρ ( H ) T α ( H ) , which indicate that ρ ( T = 0) = 0, i.e. that the system is a superconductor only at T = 0. The exponent α depends on H (Fig. 1c), andgoes to zero at H ∼ . T -independent ρ at that field. Thenon-monotonic behavior of ρ ( T ) in the intermediate H regime suggests that high- T ( T > T max )and low- T ( T < T max ) regions should be examined separately and more closely.
Scaling analysis of the high-temperature behavior
Figure 2a shows a more detailed set of MR measurements on the x = 0 .
07 film focused on thebehavior at 5 K . T <
10 K. The MR curves clearly exhibit a well-defined, T -independent crossingpoint at µ H ∗ = 3 .
63 T. In other words, this is the field where dρ/dT changes sign from positiveor metallic at low H , to negative or insulating at H > H ∗ (see also Fig. 1a). We find that, near H ∗ , ρ ( T ) for different H can be collapsed onto a single function by rescaling the temperature,as shown in Fig. 2b. Therefore, ρ ( T, H ) = ρ ∗ f ( T /T ∗ ), where the scaling parameter T ∗ is foundto be the same function of δ = ( H − H ∗ ) /H ∗ on both sides of H ∗ . In particular, T ∗ ∝ | δ | zν with zν ≈ .
73 (Fig. 2c) over a remarkably wide, more than two orders of magnitude range of | δ | . Such a single-parameter scaling of the resistance is precisely what is expected near a T = 0SIT in 2D, where z and ν are the dynamical and correlation length exponents, respectively .Similar scaling on the x = 0 .
06 crystal sample (Supplementary Fig. S4) yields a comparable zν =0 . ± .
08. It is interesting that, although the critical fields H ∗ in the two samples differ byalmost a factor of two ( µ H ∗ = 3 .
63 T for x = 0 .
07 film and µ H ∗ = 6 .
68 T for x = 0 . ρ ∗ are almost the same ( R (cid:3) /layer ≈ (17 . − .
0) kΩ; see also6 .H ( m c m ) H (T) R /l ay er ( k ) b
2T 1.5T 1T 0.5T 0.3T 0.2T 0.1T 3.6T 3.58T 3.5T 3.43T 3.21T 3T 2.5T 3.93T 3.83T 3.74T 3.68T 3.665T 3.65T 3.63T 6T 5.5T 5T 4.5T 4.2T 4T 9T 8.5T 8T 7.5T 7T 6.5T
T/ T * / * c T * ( K ) | | /H H H H H H H zv=0.737 0.006 zv=0.730 0.003
Figure High-temperature ( T & K) behavior of the resistivity ρ in different magneticfields H k c for x = 0 .
07 LSCO film. a,
Isothermal ρ ( H ) curves in the high- T region showthe existence of a T -independent crossing point at µ H ∗ = 3 .
63 T and ρ ∗ = 1 .
15 mΩ · cm (or R (cid:3) /layer ≈ . b, Scaling of the data in a with respect to a single variable T /T ∗ . Thescaling region is shown in more detail in Fig. 4a. c, The scaling parameter T ∗ as a function of | H − H ∗ | /H ∗ on both sides of H ∗ . The lines are fits with slopes zν ≈ .
73, as shown.7upplementary Information). The exponent product zν ∼ . H -field-tuned transitions also in conventional 2D superconductors ( e.g. in a-Bi and a-NbSi ) and,more recently, in 2D superconducting LaTiO /SrTiO interfaces . The value zν ∼ . XY model and assuming that z = 1 due to the long-range Coulomb interaction between charges
13, 17 . Scaling analysis of the low-temperature behavior
As shown above, the behavior of the system over a range of T above T max appears to be controlledby the QCP corresponding to the transition from a superconductor to an insulator in the absenceof disorder, and driven by quantum phase fluctuations. As T is lowered below T max , however,this transition does not actually take place as some of the insulating curves assume the power-lawdependence ρ ( T, H ) = ρ ( H ) T α ( H ) (Fig. 1), leading to a superconducting state at T = 0. Figure 3ashows a set of MR measurements carried out at very low T < T max , which exhibit a T -independentcrossing point at µ H ∗ = 13 .
45 T, consistent with α ( H = H ∗ ) ≈ H ∗ ,an excellent scaling of ρ with temperature according to ρ ( T, H ) = ρ ∗ f ( T /T ∗ ) is obtained (Fig. 3b),where T ∗ ∝ | δ | zν on both sides of H ∗ (Fig. 3c). Here δ = ( H − H ∗ ) /H ∗ and zν = 1 . ± . z = 1, this type of single-parameter scaling with ν > T = 0 SIT ina 2D disordered system . Superconducting fluctuations
The extent of SCFs can be determined from the transverse ( H k c ) MR by mapping out fields H ′ c ( T ) above which the normal state is fully restored
11, 18–20 . In the normal state at low fields, theMR increases as H (ref. 21), so that the values of H ′ c can be found from the downward deviationsfrom such quadratic dependence that arise from superconductivity when H < H ′ c . The H ′ c ( T ) line8 H (T) ( m c m ) T (mK)121 .H
13 45T R /l ay er ( k ) b
18T 14T 13.2T 10T 17.5T 13.7T 13T 9T 17T 13.6T 12.8T 8T 16.5T 13.55T 12.5T 7T 16T 13.5T 12T 6T 15.5T 13.4T 11.5T 5T 15T 13.35T 11T 4T 14.5T 13.3T 10.5T 3T
T/ T * / * c -2 -1 T * ( K ) zv =1.15 0.05 | | /H H H H H H H
Figure Low-temperature ( T . . K) behavior of the resistivity ρ in different mag-netic fields H k c for x = 0 .
07 LSCO film. a,
Isothermal ρ ( H ) curves in the low- T regionshow the existence of a T -independent crossing point at µ H ∗ = 13 .
45 T and ρ ∗ = 6 .
404 mΩ · cm(or R (cid:3) /layer ≈
97 kΩ). b, Scaling of the data in a with respect to a single variable T /T ∗ . Thescaling region, which is shown in more detail in Fig. 4a, includes the data at the lowest T ≈ .
09 K. c, The scaling parameter T ∗ as a function of | H − H ∗ | /H ∗ on both sides of H ∗ . The line is a fitwith slope zν ≈ .
15. 9etermined using this method (see Supplementary Fig. S5) is shown in Fig. 4a for the x = 0 . (see Supplementary Information and Fig. S6 for the x = 0 .
06 crystal sample). In both cases, H ′ c ( T ) is well fitted by H ′ c ( T ) = H ′ c (0)[1 − ( T /T ) ]. Figure 4a also shows the SCF contributionto the conductivity
18, 19 ∆ σ SCF ( T, H ) = ρ − ( T, H ) − ρ − n ( T, H ), where the normal-state resistivity ρ n ( T, H ) was obtained by extrapolating the region of H magnetoresistance observed at high enough H and T , as illustrated in Supplementary Fig. S5. Phase diagram
In addition to H ′ c ( T ) and ∆ σ SCF ( T, H ), the phase diagram in Fig. 4a includes T c ( H ) and T max ( H ),as well as the critical fields H ∗ and H ∗ . For both samples, T c ( H = H ) → H ≈ H ∗ ( µ H ∗ = 3 .
63 T for x = 0 .
07 film and µ H ∗ = 6 .
68 T for x = 0 .
06 crystal). This is consistent with the T = 0 transition from a pinned vortex solid to anotherphase at higher fields. For the x = 0 .
07 film sample, T max ( H = H ) → µ H = (13 . ± .
7) T, i.e. T max vanishes at the critical field µ H ∗ = 13 .
45 T within the measurement error. In the x = 0 .
06 single crystal, T max ( H ) extrapolates to zero at µ H = (14 . ± .
6) T, i.e. at a fieldsimilar to that in the film sample, even though their H ′ c ( T = 0) are very different. The vanishing ofa characteristic energy scale, such as T max ( H ), in the T = 0 limit is consistent with the existenceof a quantum phase transition at H ∗ .The scaling regions associated with the critical fields H ∗ and H ∗ are also shown in Fig. 4a.It is striking that the “hidden” critical point at H ∗ , corresponding to the SIT in the clean limit,dominates a huge part of the phase diagram. (See also Supplementary Fig. S7 for scaling of ρ ( T, H )in the x = 0 .
07 film over the entire scaling region shown in Fig. 4a.) For
H < H ∗ , this scaling failsat low T precisely where SCFs increase dramatically, leading to a large drop in ρ . For H > H ∗ ,10 H * SCF (S/cm)
T (K) H ( T ) x =0.07 H || c H c ’ T max Zero-resistance T c H * b T * T * T c (H) H * T (K) H ( T ) T maxH * Vortex latticeVortex glassInsulator
Figure Transport H − T phase diagram and scaling regions in underdoped LSCO. The data are shown for x = 0 .
07 film. a , The color map (on a log scale) shows the SCF contributionto conductivity ∆ σ SCF vs. T and H k c . The dashed red line is a fit with µ H ′ c [T]= 15[1 − ( T [K] / ]. The error bars indicate the uncertainty in H ′ c that corresponds to 1 s.d. in the slopesof linear fits in Supplementary Fig. S5. The horizontal dashed black lines mark the values of the T = 0 critical fields H ∗ and H ∗ for scaling. The pink lines show the high- T scaling region: thehashed symbols mark areas beyond which scaling fails, and the dots indicate areas beyond whichthe data are not available. The green lines show the low- T scaling region. b , Simplified phasediagram showing the crossover temperatures T ∗ and T ∗ corresponding to H ∗ and H ∗ , respectively,and three phases at T = 0. The error bars for T max indicate 3 s.d. from fitting ρ ( T ) in Fig. 1a.11he scaling works, of course, only for T > T max . The tendency towards insulating behavior thatis observed already at low H & H ∗ and T & H ≈ H ′ c at low T . It is apparent from Fig. 2 (alsoSupplementary Fig. S7) that the insulating ρ ( T ) does not follow the ln T dependence . The dataare instead consistent (not shown) with the variable-range hopping (VRH), although the range ofavailable fields is too small here to observe an orders-of-magnitude change in ρ that is characteristicof VRH. A larger change was observed in some early studies
5, 22 of the H -field induced localizationin underdoped LSCO with a similar T c ( H = 0), and attributed to a 2D or 3D Mott VRH.The scaling region corresponding to the critical point at H ∗ , on the other hand, is remarkablysmall (Fig. 4a), but it dominates the behavior at the lowest T . On the insulating side, i.e. for H > H ∗ , there is evidence for the presence of SCFs, as the applied H . H ′ c . The lowest- T insulatingbehavior (Fig. 3b) may also be fitted to a VRH form, but the change of ρ is too small again todetermine the VRH exponent with high certainty.A simplified version of the same phase diagram is shown in Fig. 4b, which includes crossovertemperatures T ∗ and T ∗ corresponding to the two QCPs H ∗ and H ∗ , respectively. The logarithmic T -scale also makes it more apparent that the vortex solid phase with T c ( H ) > H ∗ , i.e. that the QCP H ∗ is associated with the quantum melting of the pinned vortex solid.The other phases shown in Fig. 4b are discussed below. Discussion
To account for our observation of three distinct phases as T → H : one thatdescribes superconductivity and another one that describes vortex matter (see, e.g. , ref. 23).At low fields below the T c ( H ) line, ρ ( T ) = 0, which is attributed to the pinning of the vortexsolid . T c ( H ) is known
24, 25 to correspond to the so-called irreversibility temperature T irr ( H )below which the magnetization becomes hysteretic, indicating a transition of the vortex systembetween a low- T , low- H pinned regime and an unpinned one . This low- H vortex solid phase hasbeen identified experimentally as a Bragg glass in LSCO (ref. 26), as well as in other cuprates
9, 10 and some conventional superconductors ( e.g. H -NbSe (ref. 27)). The Bragg glass forms whenthe disorder is weak
28, 29 : it retains the topological order of the Abrikosov vortex lattice (see sketchof a Bragg glass in Fig. 5) but yields broadened diffraction peaks. Since such a distorted Abrikosovlattice has many metastable states and barriers to motion, it is, strictly speaking, a glass.At higher H , where the density of vortices is larger
28, 29 , a topologically disordered, amorphousvortex glass is expected (see sketch of a VG in Fig. 5). A transition from a Bragg glass to a VGwith increasing H has been observed in LSCO (ref. 26) and other cuprates
9, 10 , consistent with ourconclusion about two distinct superconducting phases at T = 0: a superconductor with T c ( H ) > H < H ∗ and a superconductor with T c = 0 for H ∗ < H < H ∗ . The power-law ρ ( T ) observedfor T < T max and
H > H ∗ is indeed reminiscent of the behavior expected in a vortex liquid abovethe glass transition occurring at T g = 0, consistent with theoretical considerations that foundthe VG phase to be unstable at non-zero temperature
33, 34 even in 3D.In general, the theoretical description of the T = 0 melting of the vortex lattice (Bragg glassin Fig. 5), associated with the QCP at H ∗ (see Fig. 4b), remains an open problem. However, it isknown that, at finite T , melting of the vortex lattice by proliferation of dislocations corresponds13 raggglass Vortexglass ρ = 0 ρ → ∞ ρ = 0 H * H * HT QCR1 QCR2 T c ( H ) T m ( H ) dρ/dT > 0 dρ/dT < T max ( H ) dρ/dT > Figure Sketch of the interplay of vortex physics and quantum critical behavior in the H − T phase diagram. Two critical fields, H ∗ and H ∗ , are reported, separating three distinctphases at T = 0: i) a superconductor with ρ = 0 (dark blue) for all T < T c ( H ) [ T c ( H ) > H < H ∗ , ii) a superconductor with ρ = 0 only at T = 0 ( i.e. T c = 0) for H ∗ < H < H ∗ ,and iii) an insulator (red), where ρ ( T = 0) → ∞ , for H ∗ < H . The difference between the twosuperconducting ground states is in the ordering of the vortex matter: a pinned vortex solid (Braggglass) for H < H ∗ and a vortex glass for H > H ∗ , as shown schematically. The quantum criticalregions (QCRs) corresponding to H ∗ and H ∗ are also shown schematically (dashed lines). TheQC scaling associated with H ∗ does not extend to the lowest T (see dotted lines); apparently, thisQPT is “hidden” at low T by thermal fluctuations that cause the melting of the pinned vortexsolid (Bragg glass) at T m ( H ) for H < H ∗ , and by the effects of disorder for H > H ∗ . T max is thetemperature at which dρ/dT changes sign. 14o a phase transition in the 2D XY model
9, 10 . It is thus plausible that the (2+1)D XY model inthe clean limit could describe the T = 0 melting of the pinned vortex solid, consistent with ourfindings. The QC scaling associated with this QPT does not extend all the way down to the lowest T . For H > H ∗ , the data suggest that the effects of disorder become important at low T , causingthe freezing of the vortex liquid into a VG phase with T c = 0. T max ( H ) may represent a crossoverenergy scale associated with the glassy freezing of vortices, although there is some evidence in othercuprates that a similar line is a continuous (second-order) glass transition
9, 10 . For
H < H ∗ , thescaling no longer works at low T where ρ exhibits a large drop (Fig. 4a), i.e. for T lower than afield-dependent temperature scale resembling the T m ( H ) line sketched in Fig. 5. It is known that asharp drop in ρ is associated with a jump in the reversible magnetization , which is indicative ofthe thermal melting of the low-field vortex solid phase
9, 10, 25 . In general, the thermal melting takesplace at T m ( H ) > T irr ( H ). Therefore, the failure of scaling for H < H ∗ on the low- T side (Fig. 4a)may be attributed to the melting of the pinned state. Indeed, our observation of QC scaling abovethe non-zero melting temperature is consistent with general expectations .At even higher H , the transition from the superconducting VG phase to an insulator withlocalized Cooper pairs occurs at H ∗ (where H ∗ < H ∗ . H ′ c ), consistent with boson-dominatedmodels of the SIT (ref. 13) and earlier arguments . The QC behavior associated with this QPT isobserved down to the lowest measured T .The scaling behavior observed near critical fields H ∗ and H ∗ is consistent with a model wheresuperconductivity is destroyed by quantum phase fluctuations in a 2D superconductor. We notethat such scaling is independent of the nature of the insulator
1, 13 . In particular, the same scaling,except for the value of zν , is expected when the insulating phase is due to disorder and when it is15aused by inhomogeneous charge ordering, which is known to be present in low-doped LSCO (refs.35–39), including cluster charge glass
11, 40, 41 and charge density wave .We expect a similar two-stage H -field-tuned SIT to occur for other doping levels in the entireunderdoped superconducting regime. This is supported by our observation of the same behavior insamples that were prepared in two very different ways and in which the number of doped holes wasprobably not exactly the same. Also, the possibility of two critical points was suggested recently for LSCO films with x = 0 . , .
09 and 0.10 (Supplementary Information). Our results provideimportant insight into the interplay of vortex line physics and quantum criticality. Apparently,the high-temperature, clean-limit SIT quantum critical fluid falls into the grip of disorder uponlowering T and splits into two transitions: first, a T = 0 vortex lattice to vortex glass transition,followed by a genuine superconductor-insulator QPT at higher H . Methods
Samples.
The LSCO film with a nominal x = 0 .
07 was described in detail in ref. 11. The LSCOsingle crystal with a nominal x = 0 .
06 was grown by the traveling-solvent floating-zone technique .Measurements were carried out on a sample that was cut out along the main crystallographic axesand polished into a bar with dimensions 3 . × . × .
34 mm suitable for direct measurementsof the in-plane resistance. Electrical contacts were made by evaporating gold on polished crystalsurfaces, followed by annealing in air at 700 ◦ C. For current contacts, the whole area of two opposingside faces was covered with gold to ensure a uniform current flow through the sample. In turn, thevoltage contacts were made narrow ( ∼ µ m) in order to minimize the uncertainty in the absolutevalues of the resistance. The distance between the voltage contacts is 1.41 mm. Gold leads were16ttached to the sample using the Dupont 6838 silver paste. This was followed by the heat treatmentat 450 ◦ C in the flow of oxygen for 6 minutes. The resulting contact resistances were less than 1 Ωat room T . As a result of annealing, the sample composition is probably La . Sr . CuO y , with y not precisely known. Measurements.
The in-plane sample resistance and magnetoresistance were measured with astandard four-probe ac method ( ∼
11 Hz) in the Ohmic regime at current densities as low as(3 × − − × − ) A/cm for the x = 0 .
07 film and 7 × − A/cm for the x = 0 .
06 crystal.To cover a wide range of T and H , several different cryostats were used: a He system at T downto 0.3 K and with magnetic fields up to 9 T, using 0 . − .
05 T/min sweep rates; a dilutionrefrigerator with T down to 0.02 K and a He system (0.3 K ≤ T .
30 K) in superconductingmagnets with fields up to 18 T at the National High Magnetic Field Laboratory (NHMFL), using0 . − . ≤ T .
30 K), using 1 T/min sweep rates. Therefore, the MR measurements span morethan two orders of magnitude in T , down to 0.09 K, i.e. much lower than temperatures commonlyemployed in studies of underdoped cuprates. For that reason, we use excitations ( i.e . currentdensities) that are orders of magnitude lower than in similar studies. It was not possible to coolthe samples below 0.09 K. The fields, applied perpendicular to the CuO planes, were swept atconstant temperatures. The sweep rates were always low enough to avoid the heating of the sampledue to eddy currents.The resistance per square R (cid:3) = ρ/d = ρ/ ( nl ) ( d – sample thickness, n – number of CuO layers, l = 6 . layer R (cid:3) /layer = nR (cid:3) = ρ/l . 17 eferences
1. Sachdev, S. Quantum Phase Transitions (Cambridge University Press, Cambridge, 2011).2. Gantmakher, V.F. & Dolgopolov, V.T. Superconductor-insulator quantum phase transition.
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Double criticality in the magnetic field-driven transition of a high- T c super-conductor. Preprint at http://arxiv.org/abs/1306.4583 (2013).22 cknowledgements We thank P. Baity for experimental assistance, and A.T. Bollinger and I. Boˇzovi´c for the filmsample. The work by X.S., P.V.L. and D.P. was supported by NSF/DMR-0905843, NSF/DMR-1307075, and the NHMFL, which was supported by NSF/DMR-0654118, NSF/DMR-1157490 andthe State of Florida. The work of T.S. was supported by MSL Collaborative Research Project.V.D. was supported by NSF/DMR-1005751. V.D. and D.P. thank the Aspen Center for Physics,where part of this work was done, for hospitality and support under NSF/PHYS-1066293.
Author contributions
X.S. and D.P. conceived the project; the single crystal was grown by T.S.; X.S. and P.V.L per-formed the measurements and analysed the data; V.D. and D.P. contributed to the data analysisand interpretation; X.S., P.V.L and D.P. wrote the manuscript; D.P. planned and supervised theinvestigation. All authors commented on the manuscript.
Additional information
The authors declare that they have no competing financial interests. Supplementary informa-tion accompanies this paper. Correspondence and requests for materials should be addressed toD.P. (email: [email protected]). 23 upplementary Information
Characteristic temperatures T c ( H ) and T max ( H ) noise floor=0.0003 H (T) ( m c m ) Figure S1 : Determination of the zero-resistance T c in x = 0 .
07 LSCO film.
In a mag-nified plot of ρ ( H ) curves at fixed temperatures, the field where the resistivity is smaller than theexperimental noise floor (here 3 × − mΩ · cm) is defined as the “zero-resistance field” at a giventemperature. That temperature, in turn, is identified as the zero-resistance T c ( H ) in this field.The zero-resistance T c ( H ) was determined as illustrated in Fig. S1. T max ( H ) is the temper-ature at the maximum of the ρ ( T ) curves in the intermediate-field regime (see Figs. 1a and S2).Figure S3 shows T c ( H ) and T max ( H ) with the phenomenological fits H ( T ) = H exp( − T /T ) (or,equivalently, T c ( H ) = T ln( H /H ) for H > T c ( H ): µ H = (4 . ± .
5) T and T = (1 . ± .
2) K for x = 0 .
07 film (Fig. S3a), µ H = (6 . ± .
3) Tand T = (1 . ± .
1) K for x = 0 .
06 single crystal (Fig. S3b). For T max ( H ): µ H = (13 . ± .
7) Tand T = (3 . ± .
3) K for x = 0 .
07 film, µ H = (14 . ± .
6) T and T = (3 . ± .
1) K for x = 0 . H || c T (K) ( m c m ) b T (K) ( m c m ) Figure S2 : Temperature dependence of the in-plane resistivity ρ in different magneticfields H k c for x = 0 .
06 LSCO single crystal. a, ρ ( T, H ) data are shown for 1.2 K < T <
35 Kand H up to 34 T. b, ρ ( T, H ) in a shown magnified in the intermediate field regime, where amaximum in ρ ( T ) emerges at low temperatures.single crystal. A similar exponential decay was found in the irreversibility field H irr ( T ) in magne-tometry measurements in LSCO [S1], with T ∼ x = 0 .
06 single crystal. The correspondingirreversibility temperature in a given field, T irr ( H ), is identified from the resistivity measurementsas the zero-resistance T c ( H ) [S2, S3].We note that, in general, T irr ( H ) ( i.e. T c ( H )) is distinct from the thermal melting transitionof the vortex solid to a vortex liquid at T m ( H ) > T irr ( H ) (see sketch in Fig. 5) [S4]. Nevertheless,the zero-resistance T c ( H ) data are sometimes fitted to the melting curve given by [S5, S6] p b m ( t )1 − b m ( t ) t √ − t " (cid:0) √ − (cid:1)p − b m ( t ) + 1 = 2 πc L √ Gi , (1)where t ≡ T /T c , b m ( t ) ≡ B m /µ H c ( t ) = B m / [ µ H c (0)(1 − t )], c L is the Lindemann number, and Gi is the Ginzburg number. The Lindemann number c L represents the ratio of the root-mean-25quare amplitude of vortex lattice thermal fluctuations over the lattice constant. The Ginzburgnumber Gi is given by Gi = 12 k B T c γ πµ ( µ H c ( T = 0)) ξ ! (2) ≈ (cid:0) . × (cid:2) Wb − K − (cid:3) × µ H c (0) T c λ ab λ c (cid:1) , (3)where γ is the anisotropy ratio γ ≡ λ c /λ ab , and the definition H c (0) ≡ πµ λ ab [ µ H c ( T = 0)] / Φ has been used ( λ ab and λ c are the penetration depths parallel and perpendicular to the ab -plane atzero temperature; Φ is the flux quantum, and the mean-field Ginzburg-Landau coherence length ξ can be represented by H c (0) from µ H c ( T = 0) = Φ / (2 πξ )). The right hand side of Eq. 1can be thus rewritten as 2 πc L √ Gi ≈ Kµ H c (0) T c , (4)where K ≡ πc L / (cid:0) . × (cid:2) Wb − K − (cid:3) × λ ab λ c (cid:1) is a fitting parameter in addition to H c (0)and T c . Figure S3 insets show the melting line fits to our T c ( H ) data with the following fittingparameters: µ H c (0) = (5 ±
1) T, T c = (4 ±
1) K and K = (14 ±
5) T · K for x = 0 .
07 film (Fig. S3ainset), µ H c (0) = (9 ±
2) T, T c = (6 . ± .
5) K and K = (27 ±
5) T · K for x = 0 .
06 single crystal(Fig. S3b inset). The values of K obtained from the fits are consistent with λ ab ∼ µ m and γ ∼ c L ∼ . − . T c ( H )data reasonably well. In fact, even the T max ( H ) values can be fitted well using Eq. 1 with thefollowing fitting parameters: µ H c (0) = (15 ±
2) T, T c = (25 ±
5) K and K = (160 ±
20) T · K for x = 0 .
07 film (Fig. S3a inset), µ H c (0) = (17 ±
3) T, T c = (14 ±
2) K and K = (160 ±
20) T · Kfor x = 0 .
06 single crystal (Fig. S3b inset), where the obtained values of K are still reasonable.However, there is no known reason to associate T max ( H ) with the thermal melting of the vortex26 T (K) H ( T ) H ( T ) T (K) x=0.07 H || c T max
Zero-resistance T c b T (K) H ( T ) T (K) H ( T ) T max Zero-resistance T c x=0.06 H || c Figure S3 : T c ( H ) and T max ( H ) for two different samples. The dashed lines show phe-nomenological exponential fits H ( T ) = H exp( − T /T ) for x = 0 .
07 LSCO film in a , and x = 0 . b . Solid lines in the insets show fits to the same data using the expression for thethermal melting of the vortex lattice (Eq. 1). 27olid. Likewise, as noted above, T c ( H ) is, in general, distinct from the thermal melting transition[S4]. In addition, since the melting line fits have much larger error bars than the phenomenologicalexponential fits in all cases, the H values obtained from exponential fits are used in the discussionsin the main text. The precise values of H , however, do not affect any of our conclusions. Scaling near the SIT
Scaling arguments [S9] imply that the critical resistance ( i.e. R (cid:3) /layer at the SIT) for a giventype of system must be universal. We do find that the critical resistances at H ∗ are almost thesame in both samples and estimate that they would be of the same order of magnitude at H ∗ .The precise value of the critical resistance for the SIT, however, may depend on the nature ofinteractions and disorder, and its understanding remains an open question [S10, S11].The possibility of two critical points was suggested recently [S12] for LSCO films with x = 0 . , .
09 and 0.10. However, unlike our study that was performed down to 0 .
09 K, thosemeasurements extended down to only 1.5 K making it impossible to identify the form of ρ ( T ) for T < T max . In addition, T c ( H ) and T max ( H ) were not discussed and thus no connection was madeto H ∗ and H ∗ , respectively. The scaling regions associated with the two critical points were alsonot identified, H ′ c and ∆ σ SCF were not determined, and the phase diagram was not constructed.
Superconducting fluctuations
For both samples, the field H ′ c ( T ), determined as shown in Fig. S5, is fitted by H ′ c ( T ) = H ′ c (0)[1 − ( T /T ) ] (see Fig. S6 for the x = 0 .
06 single crystal). Although the values of T c ( H = 0)in these samples are very similar, their H = 0 onset temperatures for SCFs, T , are very different.In particular, T = 29 K in the film is consistent with the results from terahertz spectroscopy [S13]obtained on similar films. This value is lower than T = 44 K found in the single crystal, but the28 R /l ay er ( k ) ( m c m ) H (T) . b T/ T * / * c H H zv=0.59 0.08 | | /H H H T * ( K ) H H
Figure S4 : High-temperature ( T & K) behavior of the resistivity ρ in different mag-netic fields H k c for x = 0 .
06 LSCO single crystal. a,
Isothermal ρ ( H ) curves in the high- T region show the existence of a T -independent crossing point at µ H ∗ = 6 .
68 T and ρ ∗ = 1 .
19 mΩ · cm(or R (cid:3) /layer ≈ . b, Scaling of the data in a with respect to a single variable T /T ∗ . Thescaling region is shown in more detail in Fig. S6. The upper branch of the scaling curve is made ofthe H > H ∗ data, while the lower branch corresponds to H < H ∗ . c, The scaling parameter T ∗ asa function of | H − H ∗ | /H ∗ on both sides of H ∗ . The dashed line is a fit with slope zν ≈ . µ H ′ c ( T = 0) = 31 T in the crystal is a factor of two largerthan µ H ′ c ( T = 0) = 15 T in the film. These observations suggest that the origin of the discrepancybetween onset temperatures for SCFs and H ′ c ( T = 0) obtained from different experiments may be,at least partly, attributed to the differences in the sample preparation conditions.We remark that H ′ c ( T ) values determined from the MR measurements have been shown todecrease with underdoping [S16], including also non-superconducting samples [S17]. Supplementary Information References [S1] Li, L., Checkelsky, J.G., Komiya, S., Ando, Y. & Ong, N.P. Low-temperature vortex liquid inLa − x Sr x CuO . Nature Phys. , 311–314 (2007).[S2] Ando, Y. et al. Resistive upper critical fields and irreversibility lines of optimally doped high- T c cuprates. Phys. Rev. B , 12475–12479 (1999).[S3] Sasagawa, T. et al. Magnetization and resistivity measurements of the first-order vortex phasetransition in (La − x Sr x ) CuO . Phys. Rev. B , 1610–1617 (2000).[S4] Le Doussal, P. Novel phases of vortices in superconductors. Int. J. Mod. Phys. B , 3855–3914 (2010).[S5] Houghton, A., Pelcovits, R.A. & Sudbø, A. Flux lattice melting in high- T c superconductors. Phys. Rev. B. , 6763–6770 (1989).[S6] Blatter, G., Feigel’man, M.V., Geshkenbein, V.B., Larkin, A.I. & Vinokur, V.M. Vortices inhigh-temperature superconductors. Rev. Mod. Phys. , 1125–1388 (1994).[S7] Lemberger, T.R., Hetel, I., Tsukada, A., Naito, M. & Randeria, M. Superconductor-to-metalquantum phase transition in overdoped La − x Sr x CuO . Phys. Rev. B , 140507(R) (2011).30S8] Rosenstein, B. & Li, D. Ginzburg-Landau theory of type II superconductors in magnetic field. Rev. Mod. Phys. , 109–168 (2010).[S9] Fisher, M.P.A., Grinstein, G. & Girvin, S.M. Presence of quantum diffusion in two dimensions:Universal resistance at the superconductor-insulator transition. Phys. Rev. Lett. , 587–590(1990).[S10] Gantmakher, V.F. & Dolgopolov, V.T. Superconductor-insulator quantum phase transition. Physics-Uspekhi , 1–49 (2010).[S11] Dobrosavljevi´c, V., Trivedi, N. & Valles, J.M. Conductor-Insulator Quantum Phase Transi-tions (Oxford University Press, Oxford, 2012).[S12] Leridon, B. et al. Double criticality in the magnetic field-driven transition of a high- T c su-perconductor. Preprint at http://arxiv.org/abs/1306.4583 (2013).[S13] Bilbro, L. S. et al. Temporal correlations of superconductivity above the transition tempera-ture in La − x Sr x CuO probed by terahertz spectroscopy. Nature Phys. , 298–302 (2011).[S14] Li, L. et al. Diamagnetism and Cooper pairing above T c in cuprates. Phys. Rev. B ,054510 (2010).[S15] Wang, Y. et al. Onset of the vortexlike Nernst signal above T c in La − x Sr x CuO andBi Sr − y La y CuO . Phys. Rev. B , 224519 (2001).[S16] Rullier-Albenque, F., Alloul, H. & Rikken, G. High-field studies of superconducting fluctua-tions in high- T c cuprates: Evidence for a small gap distinct from the large pseudogap. Phys. Rev.B , 014522 (2011).[S17] Shi, X. et al. Emergence of superconductivity from the dynamically heterogeneous insulatingstate in La − x Sr x CuO . Nature Mater. , 47–51 (2013).31
300 600 900 120001020304050 x =0.06 H || c ( H) (T ) M R ( % ) Figure S5 : Transverse ( H k c ) in-plane magnetoresistance vs. H for the x = 0 . Red lines are fits representing the contributions from normal state transport, i.e.they correspond to [ R ( H ) − R (0)] /R (0) = [ R n (0) − R (0)] /R (0) + αH . The intercept of the red lineshows the relative difference between the fitted normal state resistance and the measured resistanceat zero field. The difference between the red lines and the measured magnetoresistance is due to thesuperconducting contribution. Arrows show H ′ c , the field above which superconducting fluctuationsare fully suppressed and the normal state is restored.32 H c ’ T max Zero-resistance T c x =0.06 H || c SCF (S/cm) H ( T ) T (K) H * Figure S6 : Transport H − T phase diagram and scaling region for the x = 0 .
06 singlecrystal.
The color map (on a log scale) shows the SCF contribution to conductivity ∆ σ SCF asa function of T and H k c . The dashed red line is a fit with µ H ′ c [T]= 31[1 − ( T [K] / ]. Thedashed black line indicates the value of the T = 0 critical field H ∗ for scaling. The pink lines showthe high- T scaling region: the hashed symbols mark areas beyond which scaling fails, and the dotsindicate areas beyond which the data are not available. Even though this sample was measuredonly down to 1.2 K and thus the low-T scaling and H ∗ were not observed, the shape of its high- T scaling region closely resembles that of the x = 0 .
07 film sample (Fig. 4a). In particular, it isconsistent with the presence of another transition near 14 T, where T max ( H ) extrapolates to zero.33
18T 17.5T 17T 16.5T 16T 15.5T 15T 14.5T 14T 13.5T 13T 12.5T 12T 11.5T 11T 10.5T 10T 9.5T 9T 8.5T 3.43T 3.21T 3T 2.5T 2T 1.5T 1T 0.5T 0.3T 0.2T 0.1T 3.93T 3.83T 3.74T 3.68T 3.665T 3.65T 3.63T 3.6T 3.58T 3.5T 8T 7.5T 7T 6.5T 6T 5.5T 5T 4.5T 4.2T 4T
T/ T * / * b T * ( K ) | | /H H H H H H H zv=0.737 0.006 zv=0.730 0.003
Figure S7 : Scaling of ρ ( T , H ) near µ H ∗ = 3 .
63 T for x = 0 .
07 LSCO film. a,
Scaling of ρ ( T, H ) with respect to a single variable
T /T ∗ over the entire scaling region shown in Fig. 4a. b, The scaling parameter T ∗ as a function of | H − H ∗ | /H ∗ on both sides of H ∗ . The lines are fitswith slopes zν ≈ ..