Abstract
The phase diagram of underdoped cuprates in a magnetic field (H) is key to understanding the anomalous normal state of these high-temperature superconductors. However, the upper critical field (Hc2), the extent of superconducting (SC) phase with vortices, and the role of charge orders at high H remain controversial. Here we study stripe-ordered La-214, i.e., cuprates in which charge orders are most pronounced and zero-field SC transition temperatures
INTRODUCTION
Cuprates are type II superconductors (1): Above the lower critical field Hc1, an external perpendicular magnetic field penetrates the material in the form of a solid lattice of vortices or quantized magnetic flux lines. Because 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 Tc(H). The disorder also turns the vortex lattice into a Bragg glass (1), 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 (1). In two-dimensional (2D) systems, the VG freezes at Tg = 0, and thus, the VG has zero resistivity only at Tc = 0; signatures of a glassy, viscous VL, however, can be observed at low enough T, i.e., above Tg. In general, the VL persists up to the crossover line Hc2(T), i.e., up to the upper critical field Hc2 ≡ Hc2(T = 0) where the superconducting (SC) gap closes. Hence, the disorder suppresses the quantum melting field of the vortex lattice to be lower than Hc2. Because 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 these 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 < Tc(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 nonzero at all T > 0, but its V − I remains non-ohmic for I ≠ 0. At high enough T, the effects of disorder are no longer important, and the V − I of a (nonglassy) VL becomes ohmic. Although the vortex matter in cuprates has been extensively studied (1), 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 Hc2.
In cuprates, which have a quasi-2D nature, the value of Hc2 continues to be of great interest because the strength of pairing correlations is an essential ingredient in understanding what controls the value of
We focus on La1.8−xEu0.2SrxCuO4 (LESCO) and La1.6−xNd0.4SrxCuO4 (LNSCO), in which charge order coexists with the antiferromagnetic spin density wave order (2) at T < TSO < TCO; here, TSO and TCO are the onsets of static, short-range spin and charge orders or “stripes,” respectively, and
(A and B) T-x phase diagrams of LESCO and LNSCO, respectively, for H = 0;
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
RESULTS
In-plane resistivity of LESCO and LNSCO
Our samples were single crystals with the nominal composition La1.7Eu0.2Sr0.1CuO4 and La1.48Nd0.4Sr0.12CuO4 (Materials and Methods).
Figure 1 (C and D) shows the in-plane magnetoresistance (MR) measurements at fixed T for La1.7Eu0.2Sr0.1CuO4 and La1.48Nd0.4Sr0.12CuO4, respectively. The positive MR, indicative of the suppression of superconductivity, appears below ∼35 to 40 K. However, as T decreases further, a negative MR develops at the highest fields, resulting in a peak in ρab(H) at H = Hpeak(T) for
Figure 1E shows the ρab(T) curves for LESCO extracted from the MR measurements for H ≤ 18 T (i.e., Fig. 1F for LNSCO up to H ≤ 19 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, RQ = h/(2e)2, which occurs for T ≈ 15 K, the ρab(T) curves start to separate from each other: Lower Hρab(T) curves exhibit a metallic-like drop associated with superconductivity, while higher H curves show a tendency toward insulating behavior. This is also remarkably reminiscent of the behavior of 2D films of many materials at T = 0 SIT (19), where the critical resistance is close to RQ. 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 (7, 20, 21), these similarities between bulk systems, LESCO and LNSCO, and 2D films are striking: It took only specially engineered, double-layer transistors to observe (22) the critical resistance of RQ 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 1 (E and F) shows that the increase of H leads to the suppression of Tc(H) and, just like in highly underdoped LSCO (7), to a peak in ρab(T) that shifts to lower T = Tpeak(H). In LSCO, Tpeak(H) was attributed (7) to the onset of the viscous VL regime. The values of Tc(H) (see also fig. S1), Hpeak(T), and Tpeak(H) are shown in Fig. 2 over a wide range of T and H for both materials.
(A) La1.7Eu0.2Sr0.1CuO4. (B) La1.48Nd0.4Sr0.12CuO4. Tc(H) (black squares) mark the boundary of the pinned vortex lattice, which is a superconductor with ρab = 0 for all T < Tc(H) [region I; Tc(H) > 0]. H*(T) symbols mark the boundary of the viscous VL, in which dV/dI is non-ohmic [for H < H*(T)] and freezes into a VG at T = Tc = 0. Dashed red line guides the eye. Ohmic behavior is found at H > H*(T). H*(T = 0) thus corresponds to the upper critical field Hc2. Hpeak(T) (green dots) represent fields above, which the MR changes from positive to negative. The region H* < H < Hpeak, in which the MR is positive but transport is ohmic, is identified as the VL. Tpeak(H) (open blue diamonds) tracks the positions of the peak in ρab(T).
The quantum melting of the vortex lattice, in which ρab = 0 as the vortices are pinned by disorder (1), occurs when Tc(H) → 0, e.g., for ∼5.5 T in LESCO (Fig. 2A). On general grounds, in type II superconductors, the vortex lattice melts into a VL or glass, i.e., a regime of strong phase fluctuations. At the lowest T in the intermediate H regime, the data are described best with the power-law fits ρab(H, T) = ρ0(H)Tα(H) in both materials (fig. S2), suggesting a true SC state (ρab = 0) only at T = 0 when the vortices are frozen. This finding is similar to that in highly underdoped LSCO (x = 0.06,0.07) and consistent with the expectations for a viscous VL above its glass freezing temperature Tg = 0 [(7) and references therein]. Here, we also use nonlinear transport measurements to directly probe the vortex matter.
Nonlinear transport and SC correlations
The second technique, therefore, involves measurements of V − I characteristics at fixed H and T (Materials and Methods), in addition to the linear resistance Rab discussed above. When T < Tc, dV/dI is zero, as expected in a superconductor (Fig. 3A), because small values of Idc are not able to cause the depinning of the vortex lattice. However, at higher H, where Tc is suppressed to zero, a zero-resistance state is not observed even at Idc = 0 (Fig. 3A), down to the lowest T (Fig. 3B), but the V − I characteristic remains non-ohmic and dV/dI increases with Idc. This type of behavior is expected from the motion of vortices in the presence of disorder, i.e., it is a signature of a glassy, viscous VL (1). At higher H and T, the non-ohmic response vanishes (Fig. 3).
(A) Differential resistance dV/dI as a function of dc current Idc for several H ≤ 18 T at T = 0.067 K. In the bottom trace, for which T < Tc(H = 4.8 T) ≈ 0.08 K, dV/dI is zero as expected in a superconductor. (B) dV/dI versus Idc for several T at H = 9 T. The linear resistance (dV/dI for Idc → 0) has a metallic-like temperature dependence, but at higher Idc > 20 μA, the temperature dependence of dV/dI is insulating-like. Dashed lines guide the eye.
A comprehensive study of nonlinear transport (see, e.g., fig. S3) over the entire range of T and H was performed on La1.7Eu0.2Sr0.1CuO4. The non-ohmic behavior was established for all H < H*(T) and ohmic behavior for H > H*(T) in Fig. 2A; 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 Tpeak(H) obtained from the linear resistivity measurements. Thus, Tpeak(H) can be used to identify the extent of the viscous VL. Moreover, at T < 0.4 K, both H*(T) and Tpeak(H) agree, within the error, with Hpeak(T), suggesting that, for
The extent of SC fluctuations observed in linear transport was determined from the positive MR at high T (Fig. 1, C and D) as the field Hc′(T) above which the MR increases as H2 (fig. S5), as expected in the high-T normal state. As in other studies on cuprates [e.g., (7, 21, 23, 24)], the result can be fitted with Hc′ = H0′[1 − (T/T2)2] (Fig. 2A). As T → 0, both Hc′ and H* extrapolate to ∼20 T in La1.7Eu0.2Sr0.1CuO4, consistent with H*(T = 0) being the depairing field, i.e., Hc2. Region III in Fig. 2 (H > H*, Hc′) then corresponds to the H-induced normal state. Similar, albeit fewer, measurements of Hc′ in La1.48Nd0.4Sr0.12CuO4 (Fig. 2B) are consistent with this picture.
High-field normal state
The highest field ρab(T) data are shown in Fig. 4. In La1.7Eu0.2Sr0.1CuO4, ρab ∝ 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
(A) La1.7Eu0.2Sr0.1CuO4. (B) La1.48Nd0.4Sr0.12CuO4. Solid lines are fits to
Similar ρab ∝ ln (1/T) behavior is found in La1.48Nd0.4Sr0.12CuO4 (Fig. 4B), in agreement with earlier experiments (26) performed up to 15 T and down to 1.5 K. However, while that work reported (26) that ρab 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 La1.48Nd0.4Sr0.12CuO4 (Fig. 4B). The apparent saturation at low T is attributed to the presence of Tpeak (H), which moves to lower T with increasing H. The onset of metallic behavior is anticipated at ∼70 T (Fig. 4B, 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 La1.7Eu0.2Sr0.1CuO4) cannot be completely ruled out based on these measurements, as Hc′(T) may acquire a “tail” at low T such that the fitted H0′ < Hc′(T = 0) (7). However, magnetotransport studies that also use 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 2 shows that, in H = 0, phase fluctuations become dominant at T ≲ 15 K ∼ (2 to
The existence of a broad phase fluctuation 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 ρab(T) close to
In films with weak and homogeneous disorder, the application of H itself can lead to the emergent granularity of the SC state (42). Generally speaking, because increasing H increases the effective disorder (1), as T → 0, it also leads to the suppression of Tc(H), i.e., the suppression of the ordered vortex lattice phase (region I in Fig. 2) to fields below Hc2. This results in an intermediate regime characterized by large quantum phase fluctuations, i.e., a viscous VL in Fig. 2, which extends up to Hc2. Therefore, consistent with general expectations (1, 34), H*(T → 0) = Hc2 in Fig. 2 corresponds to the upper critical field of the clean system, i.e., in the absence of disorder.
Our results (Fig. 2) indicate that, in analogy to 2D SC films near the H-tuned SIT (19, 34), the MR peak in LESCO and LNSCO allows access to both bosonic (positive MR) and fermionic (negative MR) regimes. While Hpeak(T) tracks H*(T) and Tpeak(H) at low temperatures, there is a clear bifurcation at higher T. This results in the H*(T) < H < Hpeak(T) bosonic regime in which, in contrast to the viscous VL, the transport is ohmic. Therefore, we identify this regime as the (ohmic) VL. This is consistent with the finding, based on other types of probes, that a line similar to Tpeak(H) separates the VL from the viscous VL in the high-temperature (
DISCUSSION
Our key results are summarized in the T-H phase diagrams (Fig. 2) 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
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 (6, 43). 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 Hc2. Therefore, probes sensitive to vortex matter, used over a wide range of T and H, and studying the response of the system to a small external force, are needed to determine Hc2. For transport measurements, for example, Fig. 3 demonstrates that, due to the intrinsically nonlinear nature of the V-I characteristics of the vortex matter, using high excitations Idc would yield much higher measured values of ρab and even change the sign of its temperature dependence (Fig. 3B). As a result, the observation of Tpeak(H) and the identification of the viscous VL regime would not be possible. In YBCO, low-T magnetization measurements have reported (4) 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. 2, but Hc2 was not reached with the accessible fields. Conflicting MR results, i.e., both a positive MR only (44) and a peak (45), have been reported for the same material and doping, leaving the question of Hc2 in YBCO open. A quantitative comparison of the typical current densities (∼10−3 A/cm2) used to measure linear ρab in our study (Materials and Methods) to those used in the studies of Hc2 in YBCO [e.g., (3)] is difficult, because the latter information is missing. However, the related literature (46) suggests current excitations of ∼1 mA and the corresponding current densities of >1 A/cm2 in YBCO. In any case, nonlinear transport studies at low excitations of YBCO with a very low
A comparison of spectroscopic data on a variety of hole-doped cuprates, in which the La-214 family was not considered, has established (47) that the SC gap
In La1.7Eu0.2Sr0.1CuO4 and La1.48Nd0.4Sr0.12CuO4, which have different strengths of stripe correlations, we have qualitatively established the same vortex phase diagrams as in highly underdoped LSCO (7), in which there is no clear evidence of charge order (8). The LSCO study (7) 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 Hc2 than those argued previously. Even in conventional superconductors, the interplay of vortex matter physics, disorder, and quantum fluctuations leads to the enhancement of Hc2 as T → 0, a long-standing puzzle in the field (51). It should thus come as no surprise that the precise values of Hc2 may also be affected by the presence of stripes. In particular, stronger stripe correlations in La1.48Nd0.4Sr0.12CuO4 than in La1.7Eu0.2Sr0.1CuO4 seem to enhance the VL regime as T → 0, but this issue is beyond the scope of this work. On the other hand, because 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 qualitatively modify the vortex phase diagrams. 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 VL regime extends out to overdoped regions of the cuprate phase diagram where the (normal state) pseudogap closes remains an open question for future study.
MATERIALS AND METHODS
Samples
Several single crystal samples of LESCO (Fig. 1A) with a nominal x = 0.10 and LNSCO (Fig. 1B) with a nominal x = 0.12 were grown by the traveling-solvent floating-zone technique (52). 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 SD less than 0.003 for a ∼1 mm2 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 (53) on a bar-shaped La1.48Nd0.4Sr0.12CuO4 sample with dimensions of 0.24 mm × 0.41 mm × 1.46 mm (a × b × c) and
The values of
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 and ∼2 × 10−3 A cm−2 for LESCO and LNSCO, respectively). The Lakeshore 372 AC resistance bridge, which minimizes power dissipation at low temperatures, was used for linear resistivity measurements. dV/dI measurements were performed by applying a dc current bias (density) down to 2 μA (∼1 × 10−3 and ∼4 × 10−4 A cm−2 for LESCO and LNSCO, respectively) and a small ac current excitation Iac ≈ 1 μA (∼13 Hz) through the sample while measuring the ac voltage across the sample. For each value of Idc, the ac voltage was monitored for 300 s, and the average value was 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 Idc > 100 μA at the lowest T; at higher T, such as 1.7 K, that current was Idc ≳ 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 SC magnets (H up to 18 T) using 0.1 to 0.2 T/min sweep rates; a portable dilution refrigerator (0.02 K ⩽ T ⩽ 0.7 K) in a 35-T resistive magnet using a 1 T/min sweep rate; and a 3He system (0.3 K ⩽ T ⩽ 20 K) in a 31-T resistive magnet using 1 to 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 millikelvin range. This results in a slight weakening of the ρab(T) curves below ∼0.06 K for all fields. We note that this does not make any qualitative difference to the phase diagram (Fig. 2). The fields, applied perpendicular to the CuO2 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 CuO2 layer R□/layer = ρab/l, where l = 6.6 Å is the thickness of each layer.
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/7/eaay8946/DC1
Fig. S1. Determination of the zero-resistance Tc(H).
Fig. S2. The dependence of the in-plane resistivity on T at intermediate fields.
Fig. S3. Nonlinear in-plane transport in La1.7Eu0.2Sr0.1CuO4.
Fig. S4. Nonlinear in-plane transport in La1.48Nd0.4Sr0.12CuO4.
Fig. S5. In-plane MR of La1.7Eu0.2Sr0.1CuO4 versus H2 for several
Fig. S6. Comparison of studies of upper critical field in various underdoped cuprates at different hole concentrations (p).
Fig. S7. Temperature dependence of the magnetic susceptibility and in-plane resistivity.
Fig. S8. Comparison of the irreversibility fields Hirr(T) with the resistive Tc(H) in La1.7Eu0.2Sr0.1CuO4.
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REFERENCES AND NOTES
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