Vortex phase diagram and the normal state of cuprates with charge and spin orders
Zhenzhong Shi P. G. Baity T. Sasagawa Dragana Popović
cond-mat.supr-con cond-mat.dis-nn cond-mat.str-el





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Vortex phase diagram and the normal state of cuprates with charge and spin orders
    Zhenzhong Shi,^1† P. G. Baity,^1,2†† T. Sasagawa,^3 Dragana Popović^1,2∗



^1National High Magnetic Field Laboratory, Florida State University,

Tallahassee, Florida 32310, USA

^2Department of Physics, Florida State University,

Tallahassee, Florida 32306, USA

^3Materials and Structures Laboratory, Tokyo Institute of Technology,

Kanagawa 226-8503, Japan



^† Present address: Department of Physics, Duke University,

Durham, North Carolina 27708, USA

^†† Present address: James Watt School of Engineering, University of Glasgow,

Glasgow, G12 8QQ, Scotland, United Kingdom

^∗To whom correspondence should be addressed; E-mail: dragana@magnet.fsu.edu

    
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The phase diagram of underdoped cuprates in a magnetic field (H) is the key ingredient in understanding the anomalous normal state of these high-temperature superconductors.  However, the upper critical field (H_c2) or the extent of superconducting phase with vortices, a type of topological excitations, and the role of charge orders that are present at high H, remain under debate.  We address these questions by studying stripe-ordered La-214, i.e. cuprates in which charge orders are most pronounced and zero-field transition temperatures T_c^0 are lowest; the latter opens a much larger energy scale window to explore the vortex phases compared to previous studies.  By combining linear and nonlinear transport techniques sensitive to vortex matter, we determine the T–H phase diagram, directly detect H_c2, and reveal novel properties of the high-field ground state.  Our results  demonstrate that, while the vortex phase diagram of underdoped cuprates is not very sensitive to the details of the charge orders, quantum fluctuations and disorder play a key role as T→ 0.  The presence of stripes, on the other hand, seems to alter the nature of the anomalous normal state, such that the high-field ground state is a metal, as opposed to an insulator.


Introduction


Cuprates are type-II superconductors<cit.>: above the lower critical field H_c1, an external perpendicular magnetic field penetrates the material in the form of a solid lattice of vortices or quantized magnetic flux lines.  Since this vortex lattice is set in motion by the application of a current, the pinning of vortices by disorder ensures the zero-resistivity property of a superconductor below T_c(H).  The disorder also turns the vortex lattice into a Bragg glass<cit.>, i.e. a state that “looks” nearly as ordered as a perfect solid, but with many metastable states and the dynamics of a glass.  This phase melts into a vortex liquid (VL) when the temperature is high enough, or into an amorphous vortex glass (VG) for strong enough disorder.  The latter transition, in particular, can occur at low T as a function of H, which increases the density of vortices and the relative importance of disorder<cit.>.  In two-dimensional (2D) systems, the VG freezes at T_g=0, and thus the VG has zero resistivity only at T_c=0; signatures of a glassy, viscous VL, however, can be observed at low enough T, i.e. above T_g.  In general, the VL persists up to the crossover line H_c2(T), i.e. up to the upper critical field H_c2≡ H_c2(T=0) where the superconducting (SC) gap closes.  Hence, the disorder suppresses the quantum melting field of the vortex lattice to be lower than H_c2.  Since the presence of disorder plays a crucial role in the vortex physics and leads to glassy dynamics, a method of choice for probing the energy landscape of such systems is the response to a small external force.  In particular, the key signatures of a Bragg glass and a glassy VL are nonlinear voltage-current (V–I) characteristics at low excitation currents: for T<T_c(H), a vortex lattice or a Bragg glass has zero resistivity in the I→ 0 limit and a finite critical current; in contrast, the I→ 0 (i.e. linear) resistivity of a glassy VL is non-zero at all T>0, but its V–I remains non-Ohmic.  At high enough T, the effects of disorder are no longer important and the V–I of a (non-glassy) VL becomes Ohmic.  Although the vortex matter in cuprates has been studied extensively<cit.>, the regime of high H and low T, in which disorder and quantum effects are expected to dominate, has remained largely unexplored; in particular, there have been no V–I measurements.  However, it is precisely this regime that is relevant for the determination of H_c2.  

In cuprates, which have a quasi-2D nature, the value of H_c2 continues to be of great interest, because the strength of pairing correlations is an essential ingredient in understanding what controls the value of T_c^0, the nature of the pairing mechanism, and the pseudogap regime<cit.>.  However, the interpretation of experimental data has been controversial and contradictory [<cit.> and refs. therein].  At least a part of the controversy lies in a disagreement on whether the VL regime vanishes<cit.> or persists<cit.> at T=0, although it has been shown that the VL may survive up to surprisingly high fields<cit.> at T≪ T_c^0.  For example, in highly underdoped La_2-xSr_xCuO_4 (LSCO), in which charge order is at best very weak<cit.>, the vortex phase diagram was proposed based on the evidence<cit.> for quantum criticality of the linear resistivity associated with T=0 transitions from the vortex lattice to VG, and then from the VG to insulator, i.e. a 2D superconductor-insulator transition (SIT)<cit.>, at higher H.  Charge orders, however, are ubiquitous in all hole-doped cuprates, in a range of dopings centered near x=1/8, and their interplay with superconductivity at high H is one of the key open questions<cit.>.  To explore the possible effects of charge order on the phases of vortex matter and the nature of the normal state, we study the in-plane resistivity (ρ_ab) of other underdoped cuprates from the La-214 family because, at H=0, they exhibit short-range charge orders with strongest correlations.  Furthermore, they do not have the complication present in YBa_2Cu_3O_6+x (YBCO), in which an additional, longer-range charge order emerges at high H <cit.>.  We also employ nonlinear (V–I) transport as a direct probe of vortex matter.  The measurements extend over an unusually large, more than three-orders-of-magnitude-wide range of T down to T/T_c^0≲ 0.003 (Methods).

We focus on La_1.8-xEu_0.2Sr_xCuO_4 (LESCO) and La_1.6-xNd_0.4Sr_xCuO_4 (LNSCO), in which charge order coexists with the antiferromagnetic spin-density-wave order<cit.> at low enough T<T_SO<T_CO; here T_SO and T_CO are the onsets of static, short-range spin and charge orders or “stripes”, respectively, and T_c^0(x) is suppressed for doping near x=1/8 (Supplementary Fig. 1).  As shown below, the low values of T_c^0 in this regime allow us to reveal and probe deep into the normal state by applying H up to 35 T.  Fields perpendicular to CuO_2 planes, i.e. along the c axis, enhance both spin<cit.> and charge orders<cit.>.  These effects are observed only below T_c^0, not above, and in samples away from x=1/8, such as x∼ 0.10, where the stripe order is weaker.  The emergence and an enhancement of the charge order by H below T_c^0, accompanying the suppression of high-temperature superconductivity, has been observed<cit.> also in YBCO.  While there is no evidence of coincident static spin order in YBCO <cit.>, it has been argued<cit.> that the charge order in YBCO is similar to that in La-based cuprates and, in fact, that charge-density-wave modulations persist in the T=0 field-induced normal state in underdoped YBCO and in Hg- and La-based cuprates<cit.>.  Thus, the stripe order in LESCO and LNSCO is expected to persist even after T_c(H) is suppressed to zero.

We find that the vortex phase diagrams of La-214 underdoped cuprates are qualitatively the same regardless of the presence or strength of the charge orders: as T→ 0, the vortex lattice is separated from the high-field ground state by a wide range of fields where quantum phase fluctuations and disorder (i.e. a viscous VL) dominate.  On general grounds, the same conclusions should apply also to other cuprates below optimum doping.  By establishing the T–H phase diagram over an unprecedented range of parameters and demonstrating that measurements, such as nonlinear transport, are needed down to T≪ T_c^0 to determine H_c2, our work also resolves the lasting controversy in the cuprate research.  Furthermore, we reveal novel properties of the high-field normal (non-SC) state in the presence of stripes: the insulatinglike, ln(1/T) dependence of the in-plane resistivity, the origin of which has been a long-standing puzzle, is suppressed by H, suggesting a metallic high-field ground state.  This is in contrast to the early work<cit.> in the absence of static stripes, where H does not seem to affect the ln(1/T) behavior, and the high-field ground state is an insulator.




Results


In-plane resistivity of La_1.8-xEu_0.2Sr_xCuO_4 and La_1.6-xNd_0.4Sr_xCuO_4.
Our samples were single crystals with the nominal composition   and  (Methods).  T_c^0 was defined as the temperature at which the linear resistance R_ab≡lim_I_dc→ 0 V/I (that is, ρ_ab) becomes zero.  For , T_c^0=(5.7± 0.3) K, T_SO∼ 15 K, T_CO∼ 40 K (Supplementary Fig. 1A), and the pseudogap temperature T_pseudogap∼ 175 K <cit.>.  For , T_c^0=(3.6± 0.4) K, T_SO∼ 50 K, T_CO∼ 70 K (Supplementary Fig. 1B), and T_pseudogap∼ 150 K <cit.>. 

Figure <ref>A shows the in-plane magnetoresistance (MR) measurements at fixed T for   (see Supplementary Fig. 2A for similar data on ).  The positive MR, indicative of the suppression of superconductivity, appears below ∼ 35 K.  However, as T decreases further, a negative MR develops at the highest fields, resulting in a peak in ρ_ab(H) at H=H_peak(T) for T< T_c^0.  In cuprates, only positive MR has been discussed so far, with the exception of a broad MR maximum<cit.> in underdoped LSCO.   
Its similarities to that in 2D films of conventional superconductors near a field-tuned SIT have led to the suggestion<cit.> that, in both cases, H_peak represents the field scale above which the SC gap, a measure of the pairing strength between electrons that form Cooper pairs, vanishes.  

Figure <ref>B (Supplementary Fig. 2B for LNSCO) shows the ρ_ab(T) curves extracted from the MR measurements for H≤ 18 T; the MR data at higher H are discussed further below.  When the normal state sheet resistance, R_□/layer, is close to the quantum resistance for Cooper pairs, R_Q=h/(2e)^2, which occurs for T≈ 15 K, the ρ_ab(T) curves start to separate from each other: lower-H ρ_ab(T) curves exhibit a  metalliclike drop associated with superconductivity, while higher-H curves show a tendency towards insulating behavior.  This is also remarkably reminiscent of the behavior of 2D films of many materials at a T=0  SIT <cit.>, where the critical resistance is close to R_Q.  Such a transition is generally attributed to quantum fluctuations of the SC phase and, hence, loss of phase coherence; Cooper pairs that form the SC condensate thus survive the transition to the insulator.  Although underdoped cuprates are highly anisotropic, layered materials, and thus behave effectively as 2D systems<cit.>, these similarities between bulk systems, LESCO and LNSCO, and 2D films are striking: it took only specially engineered, double-layer transistors to observe<cit.> the critical resistance of R_Q for the H=0, electric-field-driven SIT in LSCO.  These similarities thus suggest the importance of phase fluctuations in LESCO and LNSCO at low T.    
Figure <ref>B shows that the increase of H clearly leads to the suppression of T_c(H) and, just like in highly underdoped LSCO <cit.>, to a peak in ρ_ab(T) that shifts to lower T=T_peak(H).  In LSCO, T_peak(H) was attributed<cit.> to the onset of the viscous VL regime.  The values of T_c(H) (see also Supplementary Fig. 3), H_peak(T), and T_peak(H) are shown in Fig. <ref> over a wide range of T and H for both materials.

The quantum melting of the vortex lattice, in which ρ_ab=0 as the vortices are pinned by disorder<cit.>,  occurs when T_c(H)→ 0, e.g. for ∼ 5.5 T in LESCO (Fig. <ref>A).  On general grounds, in type-II superconductors the vortex lattice melts into a vortex liquid or glass, i.e. a regime of strong phase fluctuations.  Indeed, at the lowest T in the intermediate H regime, the data are described best with the power-law fits ρ_ab(H,T) = ρ_0(H)T^α(H) in both materials (Supplementary Fig. 4), suggesting a true SC state (ρ_ab=0) only at T=0 when the vortices are frozen, and consistent with the expectations for a viscous VL above its glass freezing temperature<cit.> T_g=0.  This finding is similar to that<cit.> in highly underdoped LSCO (x=0.06, 0.07).  Here we employ also nonlinear transport measurements to probe the vortex matter directly.




Nonlinear transport and superconducting correlations.
The second technique, therefore, involves measurements of V–I characteristics at fixed H and T (Methods), in addition to the linear resistance R_ab discussed above.  When T<T_c, dV/dI is zero as expected in a superconductor (Fig. <ref>A), since small values of I_dc are not able to cause the depinning of the vortex lattice.  However, at higher H, where T_c is suppressed to zero, a zero-resistance state is not observed even at I_dc=0 (Fig. <ref>A), down to the lowest T (Fig. <ref>B), but the V–I characteristic remains non-Ohmic and dV/dI increases with I_dc.  This type of behavior is indeed expected from the motion of vortices in the presence of disorder, i.e. it is a signature of a glassy, viscous VL<cit.>.  At higher H and T, the non-Ohmic response vanishes (Fig. <ref>).  

A comprehensive study of nonlinear transport (see, e.g., Supplementary Fig. 5) over the entire range of T and H was performed on .  The non-Ohmic behavior was established for all H<H^∗(T) and Ohmic behavior for H>H^∗(T) in Fig. <ref>A; as T→ 0, H^∗  extrapolates to ∼ 20 T.   We note the quantitative agreement between H^∗(T), the boundary of the viscous VL obtained from nonlinear transport, with the values of T_peak(H) obtained from the linear resistivity measurements.  Thus T_peak(H) can be indeed used to identify the extent of the viscous VL.  Moreover, at T<0.4 K, both H^∗(T) and T_peak(H) agree, within the error, with H_peak(T), suggesting that, for T≪ T_c^0, H_peak may be indeed interpreted as the field scale corresponding to the closing of the SC gap<cit.>.  In , T_peak(H) and H_peak(T) are also found to be the same, within the error, at low enough T<0.7 K (Fig. <ref>B); V–I measurements at low T (Supplementary Fig. 6), confirm that transport remains non-Ohmic, for example, up to H^∗=(22± 3) T at T∼ 0.05 K, comparable to H_peak(T∼ 0.05 K)∼ 27 T, and well beyond the quantum melting field of the vortex lattice (∼ 4 T). 

The extent of SC fluctuations observed in linear transport was determined from the positive MR at high T (Fig. <ref>A) as the field H_c'(T) above which the MR increases as H^2 (Supplementary Fig. 7), as expected in the high-T normal state.  As in other studies on cuprates<cit.>, the result can be fitted with H_c' = H_0'[1-(T/T_2)^2] (Fig. <ref>A).  Remarkably, as T→ 0, both H_c' and H^∗ extrapolate to ∼ 20 T in , consistent with H^∗(T=0) being the depairing field, i.e. H_c2.  Region III in Fig. <ref> (H>H^∗, H_c') then corresponds to the H-induced normal state.  Similar, albeit fewer measurements of H_c' in   (Fig. <ref>B) are consistent with this picture.




High-field normal state.
The highest field ρ_ab(T) data are shown in Fig. <ref>.  In , ρ_ab∝ln(1/T) is observed in this regime over a temperature range of one-and-a-half decades, without any sign of saturation down to at least ∼ 0.07 K or T/T_c^0∼ 10^-2 (Fig. <ref>A).  The ln(1/T) behavior was first reported<cit.> in underdoped La_2-xSr_xCuO_4;  it was not dependent on H, and its origin is still not well understood<cit.>.  Nevertheless, experiments suggest that the  high-H normal-state in underdoped La_2-xSr_xCuO_4 is an insulator<cit.>.  In contrast, a clear weakening of the insulatinglike, ln(1/T) behavior with H is observed in   (Fig. <ref>A), strongly suggesting that ρ_ab becomes independent of T, i.e. metallic, at much higher fields ≳ 55 T (Fig. <ref>A inset).   

Similar ρ_ab∝ln(1/T) behavior is found in   (Fig. <ref>B), in agreement with earlier experiments<cit.> performed up to 15 T and down to 1.5 K.  However, while that work reported<cit.> that ρ_ab became independent of the field strength for H⩾ 11 T, lower T and higher H have allowed us to reveal the weakening of the ln(1/T) behavior by H also in   (Fig. <ref>B).  The apparent saturation at low T is attributed to the presence of T_peak(H), which moves to lower T with increasing H.  The onset of metallic behavior is anticipated at ∼ 70 T (Fig. <ref>B inset).  The results thus strongly suggest that the high-field ground state of striped cuprates is a metal.

We note that the possibility that SC fluctuations persist in the H>H^∗ regime (e.g. beyond H^∗∼ 20 T as T→ 0 in ) cannot be completely ruled out based on these measurements, as H_c'(T) may acquire a “tail” at low T such that the fitted H_0'<H_c'(T=0) <cit.>.   However, magnetotransport studies that employ also parallel magnetic fields do confirm the absence of any observable remnants of superconductivity for H>H^∗(T→ 0); those results will be presented elsewhere.  In any case, it would be interesting to perform additional studies of this peculiar high-field normal state that is characterized by the ln(1/T) temperature dependence and the negative MR, as well as to extend the measurements to H>55 T to probe the properties of the anticipated high-field metal phase. 




Vortex phase diagram.
Figure <ref> shows that, in H=0, phase fluctuations become dominant at T≲ 15 K ∼ (2-3)T_c^0 in ,  and at T≲ 20 K ∼ 6 T_c^0 in , i.e. about an order of magnitude below T_pseudogap.  Gaussian fluctuations of amplitude and phase become observable below ∼ 35 K in , i.e. ∼ (33-48) K in   as in previous work<cit.>.  Therefore, we find no evidence for pairing at temperatures comparable to T_pseudogap, similar to the conclusions of other, recent studies<cit.>.  Measurements at relatively high T<T_c suggest low values of H^∗ needed to close the gap (Fig. <ref>), consistent with some reports<cit.> of low values of H_c2.  For example, it was reported<cit.> that H_c2∼8 T on   with a similar T_c^0, but those experiments were performed at fairly high T (see Supplementary Fig. 8).  Figure <ref>A shows that H^∗∼ 8 T at a comparably high T∼ 1 K.  By extrapolating to T→ 0 in different cuprate families, it was then argued<cit.> that there is no VL at T=0.  However, as our study demonstrates, it is only by systematically tracking the evolution of SC correlations with H and T down to T/T_c^0≪ 1, that reliable T→ 0 extrapolations can be made.  Indeed, Fig. <ref> shows clearly that the regime of viscous VL broadens dramatically as T is reduced: the VL thus survives up to fields H_c2=H^∗∼ 20 T in   (∼ 25 T in ) as T→ 0, i.e. well beyond the quantum melting field of the vortex lattice (H_m∼ 5.5 T and H_m∼ 4 T, respectively, for   and ).  The relatively high values of H_c2 are consistent with the H=0 opening of the SC gap at T several times higher than T_c^0 (albeit ≪ T_pseudogap).

The existence of a broad phase fluctuations regime in H=0 can be attributed to several factors.  The first one is the effective 2D nature of the materials, which leads to the Berezinskii-Kosterlitz-Thouless transition and broadening of ρ_ab(T) close to T_c^0 <cit.>.  The second one is the presence of stripes, i.e. the intrinsic nanoscale phase separation or intrinsic electronic inhomogeneity, which can result from competing interactions even in the absence of disorder<cit.>: the presence of stripes in cuprates has been suggested to lead to an intrinsically granular SC state<cit.>.  Indeed, in contrast to highly underdoped LSCO <cit.>, the ρ_ab(T) curves of striped LESCO and LNSCO exhibit a two-step development of superconductivity (Supplementary Figs. 2-4, 9B) similar to novel hybrid structures with engineered nanoscale phase separation<cit.>, granular films of conventional superconductors<cit.>, relatively clean heavy-fermion<cit.> and iron-based superconductors<cit.> in the presence of competing orders, and some novel 2D materials<cit.>.  Finally, even homogeneous disorder may lead to the inhomogeneity of the pairing amplitude<cit.> on the scale of the coherence length, forming SC islands weakly coupled by the Josephson effect; although inhomogeneous, the pairing amplitude is finite throughout the system, and the SC transition occurs at T_c^0 because of the loss of phase coherence.  Spatially inhomogeneous pairing gaps persisting above T_c^0 have been observed in cuprates, such as Bi_2Sr_2CaCu_2O_8+δ [e.g. <cit.>], and in films of conventional superconductors<cit.>, including on both sides of the SIT<cit.>.

Interestingly, in films with weak and homogeneous disorder, the application of H itself can lead to the emergent granularity of the SC state<cit.>.  Generally speaking, since increasing H increases the effective disorder<cit.>, as T→ 0 it also leads to the suppression of T_c(H), i.e. the suppression of the ordered, vortex lattice phase (region I in Fig. <ref>) to fields below H_c2.  This results in an intermediate regime characterized by large quantum phase fluctuations, i.e. a viscous VL in Fig. <ref>, that extends up to H_c2.  Therefore, consistent with general expectations<cit.>, H^∗(T→ 0)=H_c2 in Fig. <ref> corresponds to the upper critical field of the clean system, i.e. in the absence of disorder.  

Our results (Fig. <ref>) indicate that, in analogy to 2D SC films near the H-tuned SIT<cit.>, the MR peak in LESCO and LNSCO allows access to both bosonic (positive MR) and fermionic (negative MR) regimes.  While H_peak(T) tracks H^∗(T) and T_peak(H) at low temperatures, there is a clear bifurcation at higher T.  This results in the H^∗(T)<H<H_peak(T) bosonic regime in which, in contrast to the viscous VL, the transport is Ohmic.  Therefore, we identify this regime as the (Ohmic) vortex liquid.  This is consistent with the finding, based on other types of probes, that a line similar to T_peak(H) separates the VL from the viscous VL in the high-temperature (T<T_c^0), low-H regime of several cuprate families<cit.>.  It is still an open question whether that line represents a true second-order glass transition or just a crossover energy scale.  The nonmonotonic behavior of H_peak(T) observed at low T, however, is a new and unexpected finding.  It suggests that, for certain H, the system can pass from a regime dominated by SC phase fluctuations (i.e. a VL) to a normal, albeit very unusual, state, and then again to a phase-fluctuations-dominated regime (i.e. a viscous VL) as T is lowered.  The understanding of this intriguing behavior, however, is beyond the scope of this study.





Discussion

Our key results are summarized in the T-H phase diagrams (Fig. <ref>) of striped LESCO and LNSCO, which have been mapped out over more than three orders of magnitude of T and deep into the H-induced normal state using two different, complementary techniques that are sensitive to global phase coherence.  In particular, by relying only on transport techniques, we have been able to obtain a self-consistent set of data points and achieve quantitative agreement of our results, allowing us to compare different energy scales.  Furthermore, by using cuprates with a very low T_c^0, we have been able to access a much larger energy scale window to explore the vortex phases compared to previous studies.  Indeed, Supplementary Fig. 8 shows that our measurements extend to effectively much lower T and higher H compared to previous studies of H_c2 on these and other underdoped cuprates.  This has allowed us to probe the previously inaccessible regimes, including deep into the normal state.  We have revealed that the unusual, insulatinglike behavior [ρ_ab∝ln(1/T)] of the normal state is suppressed by even higher H, leading to a metallic high-field ground state in striped cuprates.

Our findings have important implications for other cuprates, especially YBCO, which is considered the cleanest cuprate, but just like any other real material, it does contain some disorder<cit.>.  Even if the disorder is weak, its effects will be amplified at low T with increasing H, leading to the quantum melting of the vortex lattice into a VL below H_c2.  Therefore, probes sensitive to vortex matter, employed over a wide range of T and H, and studying the response of the system to a small external force, are needed to determine H_c2.  Indeed, for transport measurements, for example, Fig. <ref> demonstrates that, due to the intrinsically nonlinear nature of the V-I characteristics of the vortex matter, using high excitations I_dc would yield much higher measured values of ρ_ab and even change the sign of its temperature dependence (Fig. <ref>B).  As a result, the observation of T_peak(H) and the identification of the viscous VL regime, would not be possible.  In YBCO, low-T magnetization measurements have reported<cit.> the melting of the vortex lattice into a “second vortex solid” with a much weaker shear modulus, somewhat reminiscent of the viscous VL regime in Fig. <ref>, but H_c2 was not reached with the accessible fields.  Indeed, conflicting MR results, i.e. both a positive MR only<cit.> and a peak<cit.>, have been reported for the same material and doping, leaving the question of H_c2 in YBCO open.  Nonlinear transport studies at low excitations of YBCO with a very low T_c^0, comparable to that in our striped LESCO and LNSCO crystals or in highly underdoped LSCO <cit.>, are thus needed to identify the vortex phases, determine H_c2, and probe the normal state.  

A comparison of spectroscopic data on a variety of hole-doped cuprates, in which the La-214 family was not considered, has established<cit.> that the SC gap ∼ 5k_BT_c^0 with the value of ∼40 meV at optimal doping.  This indeed suggests inaccessibly high H_c2 (∼ 100 T), although this same relationship yields an order of magnitude lower gap ∼2–3 meV for T_c^0≃ 4–6 K that are comparable to those in our LESCO and LNSCO crystals.  Similarly, in the La-214 family at optimal doping, where T_c^0∼ 40 K, it was found<cit.> that the SC energy scale ∼20 meV.  It is thus reasonable to expect a SC gap ∼2 meV when T_c^0≃ 4–6 K, i.e. of the same order as H_c2 found in our study.  

In   and , which have different strengths of stripe correlations, we have established qualitatively the same vortex phase diagrams as in highly underdoped LSCO<cit.>, in which there is no clear evidence of charge order<cit.>.   The LSCO study<cit.> was performed using a different method, as described above, and on samples grown by two different techniques.  Therefore, the qualitative agreement between the vortex phase diagrams obtained on three different materials confirms that our findings are robust, and that the presence of different phases of vortex matter in underdoped La-214 cuprates is not very sensitive to the details of the competing charge orders.  Our data also highlight the key role of disorder in understanding the T→ 0 behavior of underdoped cuprates, and demonstrate that the SC phase with vortices persists up to much higher fields H_c2 than those argued previously.  Even in conventional superconductors, the interplay of vortex matter physics, disorder, and quantum fluctuations leads to the enhancement of H_c2 as T→ 0, a long-standing puzzle in the field<cit.>.  It should thus come as no suprise that the precise values of H_c2 may be also affected by the presence of stripes.  In particular, stronger stripe correlations in   than in   seem to enhance the VL regime as T→ 0, but this issue is beyond the scope of this work.  On the other hand, since strong stripe correlations have not altered the vortex phase diagrams, there is no reason to expect that much weaker, at least in H=0, charge orders in other cuprates will modify the vortex phase diagrams qualitatively.  Our results thus strongly suggest that our conclusions should apply to all underdoped cuprates, as supported by the agreement with the spectroscopic data on other cuprates, including the La-214 family. 
However, whether the vortex liquid regime extends out to overdoped regions of the cuprate phase diagram where the (normal state) pseudogap closes remains an open question for future study.



Samples.  Several single crystal samples of La_1.8-xEu_0.2Sr_xCuO_4 (Supplementary Fig. 1A) with a nominal x=0.10 and La_1.6-xNd_0.4Sr_xCuO_4 (Supplementary Fig. 1B) with a nominal x=0.12 were grown by the traveling-solvent floating-zone technique<cit.>.  From the X-ray fluorescence analysis using an X-ray analytical and imaging microscope (HORIBA XGT-5100), it was confirmed that the chemical compositions were close to the nominal values within the experimental error and spatially homogeneous with the standard deviation less than 0.003 for a ∼ 1 mm^2 area mapping with ∼ 10 μm resolution.  This is supported by the fact that the structural phase transition from the low-temperature orthorhombic to low-temperature tetragonal phase, which reflects the global chemical composition, was very sharp as observed in the temperature dependence of the c-axis resistivity, measured<cit.> on a bar-shaped   sample with dimensions 0.24× 0.41× 1.46 mm^3 (a× b× c) and T_c^0=(3.51± 0.06) K.  SQUID and torque magnetometry measurements were performed on a bar-shaped   crystal with dimensions 1.19× 0.24× 0.78 mm^3 (a× b× c).  Detailed transport measurements were performed on two crystals, shaped as rectangular bars suitable for direct measurements of the in-plane resistance, with dimensions 3.06× 0.37× 0.53 mm^3 and 3.82× 1.19× 0.49 mm^3 (a× b× c) for   and , respectively.  Gold contacts were evaporated on polished crystal surfaces, and annealed in air at 700 ^∘C. The current contacts were made by covering the whole area of the two opposing sides with gold to ensure uniform current flow, and the voltage contacts were made narrow  to minimize the uncertainty in the absolute values of the resistance.  Multiple voltage contacts on opposite sides of the crystals were prepared.  The distance between the voltage contacts for which the data are shown is 1.53 mm for LESCO and 2.00 mm for LNSCO.  Gold leads (≈ 25 μm thick) were attached to the samples using the Dupont 6838 silver paste, followed by the heat treatment at 450 ^∘C in the flow of oxygen for 15 minutes. The resulting contact resistances were less than 0.1 Ω for LESCO (0.5 Ω for LNSCO) at room temperature. 




The values of T_c^0 and the behavior of the samples did not depend on which voltage contacts were used in the measurements, reflecting the absence of extrinsic (e.g. compositional) inhomogeneity in these crystals.  Furthermore, both T_c^0 and the sample behavior remained astonishingly stable with time (see, e.g., Supplementary Fig. 3), without which it would have not been possible to conduct such an extensive and systematic study that requires matching data obtained using different cryostats and magnets (see below) over the period of 2-3 years during which this study was performed, thus further attesting to the high quality of the crystals.  Even the value of T_c^0=(3.6± 0.4) K in   indicates high sample homogeneity: first, it agrees with the value  (∼ 4 K) obtained<cit.> from magnetic susceptibility, a bulk probe, and second, it is lower than the values of T_c^0 away from this doping at which the superconductivity is most suppressed [<cit.> and refs. therein; also Supplementary Fig. 1B].  In contrast, some of the other transport studies of   have found T_c^0∼ 6 K <cit.> and T_c^0∼ 7 K <cit.>, pointing to either a different effective doping or inhomogeneity in their samples.  Likewise, T_c^0≈ 5.7 K of our   crystal agrees with the bulk susceptibility measurements (Supplementary Fig. 9A), while its high-T resistivity (Supplementary Fig. 9B) is about five times, that is, significantly lower than in some other studies<cit.> of the same material.  Similarly, the high-T resistivity of our   crystal is at least ∼ 30% lower than previously published values<cit.>, indicating that our samples are at least as homogeneous. The agreement between the resistive T_c(H) and the irreversibility field H_irr(T) obtained from torque magnetometry in LESCO (Supplementary Fig. 10) further confirms the absence of extrinsic inhomogeneity and high crystal quality.  In fact, it is probably because of the relatively low level of disorder that our striped LESCO and LNSCO crystals exhibit a two-step ρ_ab(T) (Supplementary Figs. 2-4), known to reflect the onset of local and global superconductivity in various 2D systems<cit.>, as discussed above.  A similar two-step ρ_ab(T) is observed<cit.> also in other La-214 cuprates near x=1/8.





Measurements.   The standard four-probe ac method (∼ 13 Hz) was used for measurements of the sample resistance, with the excitation current (density) of 10 μA (∼ 5 × 10^-3 A cm^-2 and ∼ 2 × 10^-3 A cm^-2 for LESCO and LNSCO, respectively).  dV/dI measurements were performed by applying a dc current bias (density) down to 2 μA (∼ 1 × 10^-3 A cm^-2 and ∼ 4 × 10^-4 A cm^-2 for LESCO and LNSCO, respectively) and a small ac current excitation I_ac≈ 1 μA (∼ 13 Hz) through the sample, while measuring the ac voltage across the sample.  For each value of I_dc, the ac voltage was monitored for 300 s and the average value recorded.  The data that were affected by Joule heating at large dc bias were not considered.  In the high-field normal state, for example, the dc current bias where Joule heating becomes relevant, identified as the current above which the V-I characteristic changes from Ohmic to non-Ohmic, was I_dc>100 μA at the lowest T; at higher T, such as 1.7 K, that current was I_dc≳10 mA.  In all measurements, a π filter was connected at the room temperature end of the cryostat to provide a 5 dB (60 dB) noise reduction at 10 MHz (1 GHz). 




The experiments were conducted in several different magnets at the National High Magnetic Field Laboratory: a dilution refrigerator (0.016 K ⩽ T ⩽ 0.7 K), a ^3He system (0.3 K ⩽ T ⩽ 35 K), and a variable-temperature insert (1.7 K ⩽ T ⩽ 200 K) in superconducting magnets (H up to 18 T), using 0.1 – 0.2 T/min sweep rates; a portable dilution refrigerator (0.02 K ⩽ T ⩽ 0.7 K) in a 35 T resistive magnet, using 1 T/min sweep rate; and a ^3He system (0.3 K ⩽ T ⩽ 20 K) in a 31 T resistive magnet, using 1 – 2 T/min sweep rates.  Below ∼0.06 K, it was not possible to achieve sufficient cooling of the electronic degrees of freedom to the bath temperature, a common difficulty with electrical measurements in the mK range.  This results in a slight weakening of the ρ_ab(T) curves below ∼0.06 K for all fields.  We note that this does not make any qualitative difference to the phase diagram (Fig. <ref>).  The fields, applied perpendicular to the CuO_2 planes, were swept at constant temperatures. The sweep rates were low enough to avoid eddy current heating of the samples.  




The resistance per square per CuO_2 layer R_□/layer=ρ_ab/l, where l=6.6 Å  is the thickness of each layer.








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Acknowledgements

We acknowledge helpful discussions with L. Benfatto, V. Dobrosavljević, and N. Trivedi.  Funding: This work was supported by NSF Grants Nos. DMR-1307075 and DMR-1707785, and the National High Magnetic Field Laboratory (NHMFL) through the NSF Cooperative Agreement No. DMR-1157490 and the State of Florida.  Author contributions: Single crystals were grown and prepared by T.S.; Z.S. and P.G.B. performed the measurements and analysed the data; D.P. contributed to the data analysis and interpretation; Z.S., P.G.B. and D.P. wrote the manuscript; D.P. planned and supervised the investigation.  All authors commented on the manuscript.  Competing interests: The authors declare that they have no competing interests.  Data and materials availability:  All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.  Additional data related to this paper may be requested from the authors.































Supplementary Materials









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