Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa2Cu3Oy

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Science Advances  05 Jan 2018:
Vol. 4, no. 1, eaao5235
DOI: 10.1126/sciadv.aao5235


The origin of the pseudogap region below a temperature T* is at the heart of the mysteries of cuprate high-temperature superconductors. Unusual properties of the pseudogap phase, such as broken time-reversal and inversion symmetry are observed in several symmetry-sensitive experiments: polarized neutron diffraction, optical birefringence, dichroic angle-resolved photoemission spectroscopy, second harmonic generation, and polar Kerr effect. These properties suggest that the pseudogap region is a genuine thermodynamic phase and are predicted by theories invoking ordered loop currents or other forms of intra-unit-cell (IUC) magnetic order. However, muon spin rotation (μSR) and nuclear magnetic resonance (NMR) experiments do not see the static local fields expected for magnetic order, leaving room for skepticism. The magnetic resonance probes have much longer time scales, however, over which local fields could be averaged by fluctuations. The observable effect of the fluctuations in magnetic resonance is then dynamic relaxation. We have measured dynamic muon spin relaxation rates in single crystals of YBa2Cu3Oy (6.72 < y < 6.95) and have discovered “slow” fluctuating magnetic fields with magnitudes and fluctuation rates of the expected orders of magnitude that set in consistently at temperatures TmagT*. The absence of any static field (to which μSR would be linearly sensitive) is consistent with the finite correlation length from neutron diffraction. Equally important, these fluctuations exhibit the critical slowing down at Tmag expected near a time-reversal symmetry breaking transition. Our results explain the absence of static magnetism and provide support for the existence of IUC magnetic order in the pseudogap phase.


The mysterious pseudogap phase in high-temperature superconducting cuprates has been the subject of an enormous amount of research (14) but with little consensus on either its origin or its role in high-temperature superconductivity. The name arises from the loss of low-lying electronic excitations below a temperature T*, which depends strongly on the hole concentration on CuO2 planes. This loss results in considerable modification of properties connected with these excitations, but their nature has been difficult to determine. Even whether or not T* is in fact a phase transition temperature has been controversial.

Recent experiments have shown that, consistent with a phase transition, time-reversal symmetry (5, 6) and spatial rotation and inversion symmetries (79) are broken below T* in a number of cuprate superconductors. However, lattice translational symmetry is preserved, indicating that the antiferromagnetism that lies near the pseudogap phase in the temperature-doping phase diagram is not involved. The unbroken translational symmetry has focused attention on the phenomena within the crystalline unit cell. In particular, the broken time-reversal symmetry is predicted by theories that invoke states with intra-unit-cell (IUC) magnetic order (1012). However, doubt has been cast on the existence of IUC magnetic order: Local static magnetic fields of the expected magnitude were not observed in muon spin rotation (μSR) (1318) or nuclear magnetic resonance (NMR) (19, 20) experiments. μSR and NMR are magnetic resonance techniques in which spin probes (implanted muons or nuclei) are sensitive to magnetic behavior on the local (atomic) scale. μSR in particular is capable of detecting nearby static magnetic moments in the range of 0.001 to 0.01 μB and did not do so in cuprate systems where neutron diffraction measurements observed moments of the order of 0.1 μB.

IUC magnetic order would be retained, and the absent static field would be accounted for, if the ordered moments fluctuate among alternate orientations (15, 18, 19). This would average the local field Bloc(t) at spin-probe sites to zero for time scales longer than a characteristic correlation time τc. These fluctuations could be due to finite-size domains of an ordered phase with different field orientations (21), as seen in tunneling spectroscopy (22). For NMR and μSR, the experimental time scale for this averaging is considerably longer (≳10−5 s) than that for the other techniques (≲10−10 s). Thus, all experiments would be consistent if the fluctuations were “slow,” with τc between these limits.

Averaging or “motional narrowing” of Bloc(t) (the term comes from the effect of nuclear motion on static NMR line broadening in an applied field) occurs in the rapid fluctuation limit Embedded Image, where γ is the gyromagnetic ratio of the nucleus or muon and Embedded Image is the root-mean-square (rms) local field. For the muon, γ = γμ = 8.5156 × 108 s−1 T−1. The IUC-ordered moments per triangular plaquette obtained from neutron diffraction [0.05 to 0.1 μB in YBa2Cu3O6.66 (5, 6) and progressively lower at higher y] give dipolar field values Bloc = 1 to 10 mT at candidate muon sites in the unit cell of La2−xSrxCuO4 (15). For these fields, the above inequality is satisfied for τc ≲ 10−6 s, which is in the range of experimental consistency. Thus, Bloc(t) gives rise to dynamic or “spin-lattice” nuclear or muon spin relaxation. Itoh et al. (23) have reported “ultraslow” fluctuations in HgBa2CaCu2O6+δ that may be of this kind.

In μSR experiments, the motionally narrowed dynamic muon spin relaxation rate in zero applied field (ZF) is given by Embedded Image (24, 25). The relaxation rate measured in a longitudinal magnetic field (LF) (that is, the magnetic field parallel to the initial muon polarization) depends on the magnitude HL of the field, an effect of sweeping the muon Zeeman frequency through the fluctuation noise power spectrum (24, 25). For Markovian fluctuations with a single well-defined correlation time τc, the LF relaxation rate λLF in a field HL is given by the so-called Redfield relation (24)Embedded Image(1)

The dependence of λLF on HL, if observed to be of the form of Eq. 1, provides estimates of τc and Embedded Image. More generally, one expects a decrease of λLF for γμHL greater than a characteristic fluctuation rate 1/τc, in which case τc and Embedded Image from fits of Eq. 1 to the data are heuristic estimates of the characteristic time and field scales. The Redfield relation has been widely applied in μSR to characterize dynamic fluctuating magnetic fields in strongly correlated electron systems, for example, in the heavy fermion superconductor PrOs4Sb12 (26).

Here, we report the discovery of slow magnetic fluctuations in single crystals of YBa2Cu3Oy, y = 6.72, 6.77, 6.83, and 6.95 (superconducting transition temperatures Tc = 72, 80, 88, and 91 K, respectively) via the field dependence of the μSR dynamic relaxation. Consistency with Eq. 1 is found, and values of τc and Embedded Image are obtained. We also find maxima at TmagT* in the temperature dependences of the rates in ZF and LF. This is consistent with critical slowing down of magnetic fluctuations near the transition and demonstrates that these fluctuations are associated with the IUC order. There is considerable statistical uncertainty because the measured relaxation rates are near the limit of the technique, but standard statistical analysis techniques (see below) demonstrate the validity of our conclusions.


We performed LF-mSR experiments on samples with y = 6.72, 6.77, and 6.83. The field dependence of the exponential relaxation rate λLF was measured for these samples at temperatures below T* and above Tc for 2 mT ≤ μ0HL ≤ 400 mT. The minimum field was chosen to be much larger than the ~0.1-mT quasi-static field from nuclear dipoles (see Materials and Methods and section S1), so that the dipolar field is “decoupled” (25); that is, the resultant field is nearly parallel to the muon spin, and the dipolar fields do not cause appreciable muon precession. Thus, one observes only dynamic relaxation.

The results are shown in Fig. 1, together with fits of Eq. 1 to the data. The rates are very small, close to the lower limit accessible to the technique, and the statistical uncertainty is large. Control experiments and precautions taken to minimize systematic errors are discussed in section S2.

Fig. 1 Dependence of the LF exponential relaxation rate λLF(HL) on LF HL in YBa2Cu3Oy.

(A) y = 6.72, T = 80 K. (B) y = 6.77, T = 85 K. (C) y = 6.83, T = 93 K. Curves: Fits of Eq. 1 to the data. A fit for y = 6.72 using Eq. 1 with τc fixed at one order larger than the optimal fit result [τc = 5 (2) ns] is plotted in (A) (dashed curve) for comparison of fit quality.

In Fig. 1 (A and B), the data lie above the fits for HL ≲ 5 mT. This might suggest a logarithmic field dependence as an empirical description of the data. However, there is no clear physical mechanism for this in YBa2Cu3Oy. In highly anisotropic tetracyanoquinodimethane (TCNQ) charge transfer salts, nuclear spin relaxation due to diffusion of electronic spin quasi-particles exhibits log H behavior for particular values of the anisotropic hopping rates (27). However, there is no reason to suspect this diffusive conduction in cuprates, and the higher low-field rates in Fig. 1 are more likely to be due to incomplete decoupling for these fields.

Table 1 gives values of τc and Embedded Image from the fits, together with 1 standard deviation (SD) (1 σ) statistical uncertainties. The experimental values of Embedded Image differ from zero by 5 to 9 σ individually and ~10 σ cumulatively; nonzero values are established at this level. Both τc and Embedded Image vary smoothly with y. We carried out dipolar lattice sum calculations for Bloc, assuming candidate muon stopping sites from the literature (28, 29) and approximating the current loops as point dipoles (30). These yield estimates Bloc ≈ 1 to 1.5 mT for 0.1-μB loop-current magnetic moments, of the same order of magnitude as the observed values. The calculated values are not changed significantly for the “criss-cross” bilayer loop-current configurations recently reported by Mangin-Thro et al. (31).

Table 1 Correlation times τc and rms muon local fields Embedded Image from muon spin relaxation rates in YBa2Cu3Oy.
View this table:

It is evident that τc falls in the middle of the range of experimental consistency discussed above. The observed increase of τc with increasing y could be due to the approach to a quantum critical point as Tmag→0. However, fluctuations of the short-range IUC magnetic order may be associated with defects (21), in which case τc could depend on sample preparation and not be an intrinsic property. More work is required to elucidate the nature of the observed fluctuations.

Exponential relaxation is observed in ZF together with the expected Gaussian contribution due to random quasi-static dipolar fields from nuclear moments (compare section S1). Just above Tc, λZF for y = 6.72 (Fig. 2A) is a factor of about 5 higher than λLF above Tc (Fig. 1A). Some of this increase is due to a Lorentzian contribution to the distribution of static fields (13), but some of it is doubtless because of dynamic relaxation; in ZF, the two are hard to disentangle experimentally (section S1).

Fig. 2 Temperature dependence of the dynamic muon relaxation rate λ in YBa2Cu3Oy.

(A) y = 6.72, LF HL = 0. (B) y = 6.72, μ0HL = 4 mT. (C) y = 6.77, μ0HL = 4 mT, (D) y = 6.95, HL = 0. The pseudogap onset temperature T* is shown for each doping.

The temperature dependence of λ was measured in ZF and LF (HL = 4 mT) for various samples. Figure 2 (A and B) shows λZF(T) and λLF(T), respectively, for y = 6.72, Fig. 2C shows λLF(T) for y = 6.77, and Fig. 2D shows λZF(T) for y = 6.95. Maxima at TmagT* and low-temperature increases are observed in all samples and fields, the former with statistical significance levels of 4 to 5 σ individually and ~8 σ cumulatively (see Materials and Methods). A relaxation rate maximum is often observed at second-order magnetic transitions (32), associated with critical slowing down of magnetic fluctuations near Tmag. However, the low-temperature increase is unusual and is discussed below in more detail.

In the phase diagram of Fig. 3, Tmag from our μSR data is plotted versus the hole concentration p (and the oxygen content y) along with Tc and T* from other experiments. Values of Tmag are consistent with results of other experiments: polarized neutron diffraction (5, 6), terahertz birefringence (7), resonant ultrasound (33), and second harmonic generation (8). [T* is also the temperature around which changes in transport properties (34) and thermodynamic properties (35, 36) from those of the strange metal phase begin to be observed.] The observed Tmag also corresponds well to the results of recent torque magnetometry measurements, which support a second-order phase transition at T* (9). The inset of Fig. 3 compares the doping dependence of the square root Embedded Image of the polarized neutron diffraction cross section (37) with that of Embedded Image. These quantities are both proportional to the order parameter for IUC magnetic order, and they follow the same trend.

Fig. 3 Phase diagram of pseudogap and charge-density-wave/charge inhomogeneity onset temperatures in YBa2Cu3Oy.

Red diamonds: Temperatures Tmag of maxima in μSR exponential relaxation rates (Fig. 2). Open green squares: Pseudogap temperatures T* from polarized neutron diffraction (5, 6). Open blue triangles: T* from THz birefringence (7). Pink pentagons: T* from resonant ultrasound (33). Orange circles: T* from second harmonic generation (8). Green stars: T* from magnetic torque (9). Magenta left triangles: Charge-density-wave (CDW) onset temperatures TCDW from NMR (20). Filled blue triangle: TCDW from nuclear quadrupole resonance (NQR) (43). Black squares: TCDW from Hall effect (53). Blue circle: TCDW from high-energy x-ray diffraction (54). Black circles: Onset of “charge inhomogeneity” (CDW or lattice change) from μSR experiments (14). Gray points: Superconducting transition temperatures. Inset: Doping dependences of the square root Embedded Image of the polarized neutron diffraction cross section (37) and the rms magnitude Embedded Image of the fluctuating local field (Table 1).


The observed increase of motionally narrowed muon spin relaxation with decreasing temperature below T*, shown in Fig. 2, cannot occur in a transition to a uniform ordered state. It is, however, consistent with low-frequency fluctuations in domains of IUC magnetic order, of increasing magnitude with the increasing order parameter (21). Scanning tunneling spectroscopy experiments (22) have found these domains in the pseudogap phase associated with defects, with linear dimensions of ~20 unit cells; these provide a mechanism for the pseudogap in the one-particle spectra (21). Other experiments (38, 39) have observed mysterious anomalous low-frequency fluctuations in the pseudogap phase ascribed to extended defects.

Phenomena other than IUC magnetism can affect μSR experiments. Sonier and co-workers (14, 16) observed features in ZF data from YBa2Cu3Oy and attributed them to charge and structural inhomogeneities from lattice changes and CDW formation. Fits of an exponentially damped Gaussian Kubo-Toyabe (KT) to data from a nearly fully doped sample (y = 6.985) (14) yielded correlations between Δ(T) and λ(T), leading to the conclusion that the exponential rate could not be determined unambiguously (40). Structure in ZF λ(T) with Δ(T) held fixed was related to the onset of short-range CDW order (41) for that doping.

For slightly lower doping (y = 6.95), however, fits with Δ(T) a free parameter (42) yield a clear peak in λ(T) at Tmag = 80 K with no corresponding structure in Δ(T) at that temperature (compare section S3). NQR measurements (43) on a sample of similar oxygen content showed that CDW order sets in at a considerably lower temperature (~35 K). For y < 6.95, ZF CDW transition temperatures from NMR/NQR and other measurements are significantly lower than Tmag from our experiments. For all y, the onset of long-range CDW order is at still lower temperatures and is only observed in high-applied magnetic fields (20). Recent torque magnetometry measurements also provide strong evidence that CDW order and the pseudogap phase are characterized by distinct broken symmetries at different temperatures (9). The detailed temperature-doping phase diagram (Fig. 3) shows that lattice change/CDW and pseudogap phases set in at distinct temperatures and are distinct phenomena. We conclude that there is no evidence for associating the maxima in λ(T) and its increase at low temperatures with either CDW or charge inhomogeneity.

It has been claimed (40) that the maxima for y = 6.72 and 6.77 at ~210 and ~165 K, respectively (Fig. 2, A and C), could be due to the onset of thermally activated muon hopping (diffusion) (14); this causes unwanted dynamic relaxation by nuclear dipolar fields. However, in the usual trapping-detrapping scenario (44), the maximum is due to a temperature sequence in which the muons are trapped and static at low temperatures, begin hopping with increasing temperature, find deep impurity traps and become static again at an intermediate temperature, and finally detrap definitively at high temperatures. It has been argued (42) that this is highly unlikely in the present case because independent μSR determinations (14, 42, 45) of the muon hopping rate using dynamic Gaussian KT fits (25) show that hopping is too slow to produce the observed decrease in λ(T) below the maximum (compare section S4). Alternatively, the maximum might be purely dynamic because with increasing temperature, the muon hopping rate passes through the muon Zeeman frequency. In that case, however, the application of HL = 4 mT field would move the maximum to temperatures where the hopping rate is ~γμHL = 3.4 μs−1. According to the dynamic KT fits, this is well above 300 K, whereas in our data (Fig. 2B), the position of the maximum is unchanged. We conclude that muon hopping is not the origin of the maximum for y = 6.72. Note that, for y = 6.67, no maximum in λZF(T) was seen near T* ~200 K (13), which is probably due to significant muon hopping at this temperature.

Previous transverse-field (TF) μSR experiments in the pseudogap phase (46) observed exponential relaxation and ascribed it to static spatial inhomogeneity of superconducting fluctuations. Our observed LF-μSR rates (Fig. 1) are an order of magnitude slower than the TF-μSR rates. This is consistent with the assumption that the latter are static and precludes detecting the dynamic relaxation in TF-μSR.

The observed critical slowing down at TmagT* (Fig. 2) indicates that this temperature marks the onset of broken time-reversal symmetry, as is also found in each of the four hole-doped cuprate families amenable so far to polarized neutron diffraction experiments. The observed magnitude of the order parameter of about 0.1μB staggered moment per unit cell has a condensation energy of ~50 J/mol, similar to the maximum superconducting energy in cuprates (47). These properties are all consistent with our results. Note that a recent LF-μSR study of Bi2+xSr2−xCaCu2O8+δ (45) also reported a quasi-static internal magnetic field in the pseudogap phase.

Our discovery of fluctuating magnetic fields provides an understanding of the absence of static magnetic fields due to IUC magnetic order in YBa2Cu3Oy. The expected fields are present but fluctuating. Although μSR is a point probe in real space and, thus, is not directly sensitive to the spatial symmetry, our results are strong evidence for IUC order and its excitations and establish them as important for understanding the unusual behavior of cuprates.


Sample growth and characterization

High-quality single crystals of YBa2Cu3Oy were grown by the top-seeded solution growth polythermal method using 3BaO-5CuO solvent (48). A YBa2Cu3Oy single crystal with an ab plane area of 10 mm × 10 mm and c axis length of 8 mm was synthesized with a cooling rate of 0.5 K per hour in air. The crystal was then cut into small pellets with thicknesses of 0.55 mm and lateral dimensions of 2 mm × 2 mm. Single crystals with optimal Tc = 91 K were achieved by annealing at 400°C for 180 hours in flowing oxygen. A range of oxygen concentrations of YBa2Cu3Oy was achieved by post-annealing in flowing oxygen at different temperatures as described in the study by Gao et al. (49), resulting in superconducting transition temperatures between 72 and 88 K.

Figure 4 shows the temperature dependences of the magnetization and the resistivity in our samples. The data indicate that the superconducting transitions are sharp. The values of T* from the departure of the resistivity from linearity are in agreement with previous results (50).

Fig. 4 Characterization data from YBa2Cu3Oy single crystals.

(A) Magnetization of YBa2Cu3Oy, y = 6.72, 6.77, 6.83, and 6.95, showing sharp superconducting transitions. (B) Temperature dependence of electrical resistivity ρab for y = 6.77 showing a deviation from linearity. (C) Temperature dependence of the deviation Δρab from linear resistivity for y = 6.77. The deviation sets in below T* = 156 K, which is consistent with reported results (50).

Muon spin relaxation experiments

In the time differential μSR technique (51), spin-polarized muons are implanted into the sample. In the muon decay to a positron and two neutrinos, the positron momentum is preferentially oriented along the direction of the muon spin at the time of decay. The time evolution of the ensemble muon polarization can therefore be monitored via the asymmetry in positron emission count rates.

Our μSR experiments were performed using the Los Alamos Meson Physics Facility spectrometer at the M20 beamline of TRIUMF (Vancouver, Canada) and the MUSR and EMU spectrometers at ISIS, Rutherford Appleton Laboratory (Chilton, United Kingdom). At both facilities, muons were implanted into the sample with their initial spin polarization Pμ perpendicular to the ab plane.

Appropriate functional forms, discussed in section S1, were least-squares fit to the asymmetry data using the MUSRFIT μSR analysis program (52). The LF rates were very low and were near the resolution limit of the technique. The difference between the rates was small but was resolved. Note that λLF in Bi2+xSr2−xCaCu2O8+δ is also quite small for HL > 1 mT (45). To our knowledge, no LF-μSR data for other cuprate superconductors in comparable fields have been reported. We nevertheless speculate that the scatter in Fig. 1 was typical for the hole-doped superconducting cuprate family.

Statistical analysis

Random error in μSR experiments arises from the Poisson distribution of positron count rates. The MUSRFIT analysis software computes the propagation of this error to that of the parameters of the fit function and reports their probable SD σ. All parameter uncertainties quoted in this article are 1 σ.

The inverse relative SDs (IRSDs) (the N in “Nσ”) for Embedded Image (Table 1) and for the maxima in the temperature dependences at Tmag (Fig. 3) are shown in Table 2. The IRSD for Embedded Image is simply its value divided by its SD. For the maxima at Tmag, baseline points were chosen above and below each maximum, and baseline values at intermediate points were estimated by linear interpolation. The IRSD of each point is its amplitude relative to the baseline divided by its SD, and the IRSD of the maximum is the square root of the sum of squares of the IRSDs of the points. The sign of the amplitude was included in this sum to account for negative contributions. For both Embedded Image and the maxima, the cumulative IRSD is the square root of the sum of squares of the individual sample IRSDs. It can be seen that some individual IRSDs are marginal, but the cumulative values are quite satisfactory.

Table 2 IRSDs of Embedded Image and relaxation rate maxima near Tmag from muon spin relaxation rates in YBa2Cu3Oy.
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Supplementary material for this article is available at

section S1. Muon relaxation functions

section S2. Control experiments

section S3. Temperature dependence of static Gaussian KT relaxation rate ΔZF(T)

section S4. Muon hopping, superconductivity

section S5. Superconducting fluctuations

section S6. High-temperature relaxation

fig. S1. ZF relaxation of the muon asymmetry Aμ(t) in YBa2Cu3O6.72.

fig. S2. Time evolution of the positron count rate asymmetry Aμ at various temperatures and fields in single-crystal YBa2Cu3Oy.

fig. S3. LF muon spin relaxation rates in silver samples.

fig. S4. Fits of representative ZF-μSR spectra from YBa2Cu3O6.95 for ΔZF fixed and free.

fig. S5. Temperature dependence of ZF exponential damping rate λZF and static Gaussian KT rate ΔZF for YBa2Cu3O6.95.

fig. S6. Damped dynamic Gaussian KT fit of ZF data from YBa2Cu3O6.72.

References (5558)

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Acknowledgments: We are grateful to C. M. Varma for proposing these experiments and for numerous discussions. We thank the support teams at TRIUMF and ISIS for help during the experiments and J. H. Brewer for discussions and correspondence. Funding: The research performed in this study was partially supported by the National Key Research and Development Program of China (nos. 2016YFA0300503, 2017YFA0303104, and 2016YFA0300403), the National Natural Science Foundation of China (nos. 11774061 and 11474060), and the Science and Technology Commission of Shanghai Municipality of China (grant no. 15XD1500200). Research at University of California (UC), Riverside was supported by the UC Riverside Academic Senate. Work at California State University (CSU), Los Angeles was funded by NSF/DMR (Division of Materials Research)/Partnerships for Research and Education in Materials–1523588. Research at CSU-Fresno was supported by NSF DMR-1506677. Author contributions: L.S. and D.E.M. conceived the experiment and wrote the beam-time proposals. J.Z., Z.D., C.T., K.H., O.O.B., P.-C.H., G.D.M., A.D.H., P.K.B., S.P.C., D.E.M., and L.S. performed the μSR measurements, with D.E.M. and L.S. overseeing the experimental work. J.Z., H.X., and X.Y. grew the single-crystal samples. J.Z., D.E.M., and L.S. performed the data analysis; these authors wrote the manuscript. All authors discussed the data and contributed to the analysis. 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.
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