## Abstract

For some time now, there has been considerable experimental and theoretical effort to understand the role of the normal-state “pseudogap” phase in underdoped high-temperature cuprate superconductors. Recent debate has centered on the question of whether the pseudogap is independent of superconductivity. We provide evidence from zero-field muon spin relaxation measurements in YBa_{2}Cu_{3}O_{6+x} for the presence of small spontaneous static magnetic fields of electronic origin intimately related to the pseudogap transition. Our most significant finding is that, for optimal doping, these weak static magnetic fields appear well below the superconducting transition temperature. The two compositions measured suggest the existence of a quantum critical point somewhat above optimal doping.

Of the high-temperature cuprate superconductors, YBa_{2}Cu_{3}O_{6+x} has been the most widely studied. The parent compound, YBa_{2}Cu_{3}O_{6}, is an antiferromagnetic (AF) insulator that becomes a superconductor with hole (oxygen) doping. Nuclear magnetic resonance (NMR) (1), inelastic neutron scattering (2), and muon spin relaxation (μSR) (3–6) studies clearly show that short-range AF correlations of the Cu spins in the CuO_{2}planes persist for hole concentrations well beyond the three-dimensional AF phase, leading some to suspect a spin-fluctuation pairing mechanism for superconductivity. Recently, the onset of spin fluctuations above the superconducting transition temperature*T*
_{c} has been linked to the formation of the pseudogap (7), first observed (8) as a gap in the spin-excitation spectrum of YBa_{2}Cu_{3}O_{6.7} and later identified as a coinciding gap in the quasiparticle-excitation spectrum (9). Early interpretations of the pseudogap favored a scenario in which phase-incoherent pairing correlations (10), established in the normal state at a crossover temperature *T**, condense into the coherent superconducting state below the transition temperature*T*
_{c}. Experiments showing that the pseudogap above *T*
_{c} evolves continuously into the superconducting gap below *T*
_{c} lent strong support to this idea. However, a careful examination by Tallon and Loram (11) of several key experiments supports an alternative view, whereby the pseudogap is in fact distinct from the superconducting gap, and the temperature *T** defines an energy scale that vanishes (*T** → 0) in the slightly overdoped region at a critical point*p*
_{cr} ≈ 0.19 (where *p* is the fraction of holes per Cu atom in the CuO_{2} plane). A crucial observation is that short-range AF correlations persist at temperatures above 0 K at the critical hole concentration*p*
_{cr} = 0.19.

Consistent with the latter view, several groups have proposed that there is another source of magnetism, which is distinct from the Cu-spin magnetism and more intimately associated with the pseudogap phase. Early on, Varma (12, 13) developed a model that describes *T** as a crossover to a phase with broken time and rotational symmetry, in which a fourfold pattern of circulating current (CC) flows in the basal (*a*,*b*) plane. The CC phase terminates (*T** → 0) at a critical point *p*
_{cr} within the superconducting region of the *T* – *p* phase diagram. More recently, Chakravarty *et al.*(14) have proposed a similar picture whereby the pseudogap phase is characterized by the development of a new order parameter, a so-called*d*
_{x2}
_{–}
_{y2}density wave (DDW) state. At temperatures below *T**, circulating currents associated with the time-reversal breaking DDW phase produce local magnetic fields that point along the*ĉ*-direction at the apical oxygen sites. Like the CC phase, the DDW phase vanishes at a critical doped-hole concentration*p*
_{cr} under the superconducting “dome.” A key property of the DDW transition is that it is strongly suppressed by disorder. Thus, only clean samples are expected to show a true phase transition at *T** corresponding to the onset of DDW order.

We note that physical magnetic fields arising from circulating electronic currents were first introduced as a property of the so-called staggered flux phase of the *t* – *J*model (15), which competes with*d*
_{x2}
_{–}
_{y2}-wave superconductivity. According to Wen and Lee, the fluctuating staggered current loops are stabilized only in a magnetic field that is large enough to suppress the superconducting order (16). However, it has been shown that the addition of off-site Coulomb repulsion to the pure *t* – *J* model leads to ordering of the staggered flux phase in zero external field (17). Of the various models that predict spontaneous magnetic fields associated with a time-reversal breaking phase, only the proposed CC phase preserves the translational symmetry of the underlying lattice.

The zero-field muon spin relaxation (ZF-μSR) technique offers the opportunity to observe the onset of weak spontaneous magnetic fields at the pseudogap crossover in the YBa_{2}Cu_{3}O_{6+x} system. The spin of the positive muon can detect static internal magnetic fields on the order of 0.1 G (18). Moreover, the technique can easily distinguish such static fields from relaxation due to the rapid Cu spin fluctuations observed with neutron scattering in the vicinity of*T**. Earlier ZF-μSR studies of the YBa_{2}Cu_{3}O_{6+x} system (6) show that static magnetic order persisting from the AF phase is absent beyond *x* ≈ 0.5. Measurements on an *x* = 1 polycrystalline sample found no clear evidence for the onset of spontaneous internal magnetic fields at or below *T*
_{c} (19). However, recent neutron scattering experiments on underdoped *x* = 0.5 (20) and *x* = 0.6 (21) samples of YBa_{2}Cu_{3}O_{6+x} have identified the onset of previously undetected magnetism at temperatures well above *T*
_{c}. The observed weak magnetic moments, argued to be distinct from the well-known Cu spin magnetism, are consistent with an orbital current phase that breaks translational symmetry of the lattice. Although this finding appears to be compatible with the DDW and staggered flux phase pictures, angle-resolved photoemission spectroscopy (ARPES) studies have yet to show clear evidence for the existence of Fermi surface hole pockets, which are a feature characteristic of these same theoretical models. Consequently, the origin of the weak moments observed with neutron scattering is currently unknown.

Here, we report on ZF-μSR measurements of YBa_{2}Cu_{3}O_{6+x} single crystals, using the HELIOS and LAMPF spectrometers on the M20 surface muon beam line at the TRI-University Meson Facility (TRIUMF) (22). Optimally doped *x* = 0.95 crystals with*T*
_{c} = 93.2 ± 0.3 K were prepared by a flux method in yttria-stabilized zirconia crucibles (23). Detwinned ortho-III phase *x* = 0.67 crystals with *T*
_{c} = 67.9 ± 1.2 K were grown by a flux method using BaZrO_{3} crucibles, following the same procedure used to grow ortho-II phase crystals (24). The ortho-III phase is characterized by a unit cell consisting of one completely deoxygenated and two consecutive fully oxygenated CuO chains in the basal plane. Underdoped crystals containing CuO chains that are completely full or empty have optimum oxygen homogeneity and minimal opportunity for phase separation in the oxygen annealing process.

The muons were implanted into the bulk of the sample with their initial spin polarization *P̂*
_{μ}(0) parallel (∥) or perpendicular (⊥) to the *ĉ*-axis of the crystals.Figure 1 shows the measured time evolution of the muon spin polarization in the *x* = 0.95 crystals in zero applied magnetic field for the case*P̂*
_{μ}(0) ⊥ *ĉ*. The parameter *a* is the initial muon-decay asymmetry at time*t* = 0, which depends mainly on the energies of the detected positrons and the solid angles subtended by the detectors. At 115 K and below *T*
_{c} at 55 K, the ZF time spectra are identical, confirming the earlier work (19) in which no evidence was found for the appearance of spontaneous internal magnetic fields upon cooling through *T*
_{c}. The solid curve (Fig. 1A) is a fit to the well-known Kubo-Toyabe function (25) that describes the depolarization (or relaxation) of the muon spin by the randomly orientated static magnetic fields of the nuclear dipoles
(1)where Δ/γ_{μ} is the width of the Gaussian field distribution at the muon site and γ_{μ} = 0.0852 μs^{−1} *G*
^{−1} is the muon gyromagnetic ratio. The first term, which does not evolve in time, corresponds to the 1/3 component of the initial muon-spin polarization (defined to be along the *ẑ*-direction) that is parallel to the local field. This is not evident in the time spectra, because for Δ*t* << 1 (early time and/or small Δ) the Kubo-Toyabe function reduces to a simple Gaussian form*G*
_{z}
^{KT}(*t*) ≈ exp(−Δ^{2}
*t*
^{2}). The fit toEq. 1 yields Δ_{⊥} = 0.1144 ± 0.0006 μs^{−1}. The line width parameter Δ is expected to be independent of temperature below 200 K where the muon is immobile (19).

On cooling below 45 ± 10 K (Fig. 1B), there is a marked increase in the ZF relaxation rate, signifying the occurrence of a small additional magnetic field at the muon site(s). The time evolution of the muon spin polarization below 45 K was better described by the product of Eq. 1 and an exponential function,*G*
_{z}(*t*) =*G*
_{z}
^{KT}(*t*) · exp(−λ*t*), which arises if there is an additional source of magnetic field that convolutes with the field distribution of the nuclear dipoles. The temperature dependence of the exponential relaxation rate λ was determined on separate occasions with the HELIOS and LAMPF spectrometers (Fig. 2A). The error bars represent the statistical uncertainty in the fits, whereas the random scatter of the data is due to small instrumental effects related to variations in the incoming muon rate. No observable change is seen at *T*
_{c}, indicating that any stray field present in the normal state is well below the limit of detection. The sudden increase in the relaxation rate below about 45 K is also observed for the case*P̂*
_{μ}(0) ∥ *ĉ*(Fig. 2B), although with this orientation of the muon spin, λ has a smaller maximum value at the lowest temperature considered. Fits of the ZF time spectra at higher temperatures for this latter orientation yield Δ_{∥} = 0.1065 ± 0.0008 μs^{−1}. We note that the anisotropy ratio Δ_{⊥}/Δ_{∥} ≈ 1.07 is significantly smaller than the ratio λ_{⊥}/λ_{∥} ≈ 1.70 at the lowest temperature.

In order to confirm that the increase in the exponential relaxation rate below about 45 K is due to small static internal local fields, rather than to slowing down of the fast Cu spin fluctuations, we carried out measurements above and below 45 K with a longitudinal field (LF) of 0.5 kOe applied along the initial direction of the muon spin (for the case*P̂ _{μ}
*(0) ∥

*ĉ*). A field was chosen that was large enough so that magnetic flux could penetrate the interior of the sample in the form of a vortex lattice. The LF completely decouples the effect of the static nuclear dipoles [

*G*

_{z}

^{KT}(

*t*) → 1] and strongly reduces λ

_{∥}below 45 K (Fig. 2B), indicating that the muon spin is fully decoupled from the local fields. The observed increase of λ

_{∥}in ZF is therefore attributed to a static or quasi-static (i.e., field fluctuation frequency <1 MHz) local field distribution at the muon site. Although the data have been fit to an exponential relaxation function, the increased relaxation is too small to clearly distinguish between an exponential function characteristic of a dilute concentration of magnetic moments and forms of the relaxation function consistent with a dense system of weak moments. In a dense static spin system, the size of the local magnetic field experienced by the muon may depend somewhat on its position in the crystallographic lattice. Although it is well established that the muon binds to an oxygen atom in YBa

_{2}Cu

_{3}O

_{6+x}with a muon-oxygen bond length of the order of 1 Å, there is no clear consensus on which of the oxygen sites the muon prefers (26–29).

An important distinction between ZF measurements with*P̂*
_{μ}(0) ∥ *ĉ* and*P̂*
_{μ}(0) ⊥ *ĉ* is that only in the latter configuration is the muon spin polarization relaxed by static internal fields along the*ĉ*-direction of the crystals. Below 45 K, we find that λ_{⊥} > λ_{∥}, which implies that the anomalous magnetic field at the muon site is mainly in the*ĉ*-direction, with smaller components in the*â*-*b̂* plane. From the λ values, we estimate the characteristic field at the muon site to have components of ∼0.3 G and ∼0.19 G perpendicular and parallel to the basal plane, respectively. It is important to note that this anisotropy refers to the field at the muon site and does not necessarily imply a similar anisotropy of the spontaneous moments. Thus, one should not necessarily conclude that the field originates from moments preferentially oriented in the *ĉ*-direction.

Figure 3 shows the temperature dependence of the ZF exponential relaxation rate below 200 K in the underdoped *x* = 0.67 sample for the case*P̂*_{μ}(0) ⊥ *ĉ*. Over this temperature range, the relaxation rate due to nuclear dipolar broadening is Δ_{⊥} = 0.1065 ± 0.0027 μs^{−1}. This value is comparable to the value at optimal doping, which suggests that the muon-stopping sites are the same in the underdoped sample. Above 200 K, the line width parameter Δ_{⊥} decreases with increasing temperature, as observed in earlier ZF studies (19). This behavior is attributed to the onset of muon diffusion, resulting in motional averaging of the interaction between the muon and the nuclear dipoles. Below 160 ± 15 K, there is a significant increase in λ_{⊥}. Near this temperature, the Cu spin fluctuation rate is within the neutron time window (10^{10} to 10^{12} s^{−1}) [for example, (7)] and far too fast to be detected with ZF-μSR. Thus, we do not attribute the increased exponential relaxation rate to a growth of the AF Cu spin correlations. At the lowest temperature investigated, the value of λ_{⊥} is substantially smaller than that in the *x* = 0.95 sample, which implies that the spontaneous magnetic fields are weaker. Although this may seem surprising, the trend is consistent with the staggered-flux phase model (15), whereby the magnitude of the orbital currents decreases as the hole concentration decreases. Nevertheless, additional measurements on similar high-quality samples of intermediate oxygen content are certainly needed to clearly establish the dependence of the field strength on the value of*x*.

We define a temperature *T**_{ZF} at which the onset of spontaneous magnetic fields is observed, and this is plotted versus the oxygen content *x* along with*T*
_{c}(*x*) in Fig. 4. At *x* = 0.67,*T**_{ZF} is between the pseudogap transition *T** ≈ 140 K determined from the peak in the ^{63}Cu NMR spin-lattice relaxation rate 1/*T*
_{1}
*T* (30) and the departure of the resistivity ρ(*T*) from linearity (31), and *T** ≈ 200 K estimated from the downturn in the ^{89}Y NMR Knight shift (32). The hole concentration *p* can be estimated from the following empirical equation (33)(2)The inset of Fig. 4 is a plot of*T*
_{c} and*T**_{ZF} versus the hole concentration*p*. If one assumes a linear extrapolation through the data points of*T**_{ZF}(*p*) corresponding to *x* = 0.67 and *x* = 0.95,*T**_{ZF} falls to zero at a critical doped-hole concentration of *p*
_{cr} = 0.182 ± 0.009.

Our investigation of highly pure and homogeneous crystals of YBa_{2}Cu_{3}O_{6+x} reveals the onset of spontaneous static magnetic fields at a temperature dependent on the oxygen content *x*. In the underdoped sample, the onset is near the pseudogap crossover temperature *T** deduced from other methods, whereas the onset occurs well below*T*
_{c} at optimal doping. Although the occurrence of magnetic moments below *T** is consistent with some recent theories of the pseudogap phase, the increased ZF relaxation rate is too small to clearly determine whether the static fields arise from a dilute or dense concentration of magnetic moments.