Absence of $\mu \mathrm{SR}$ evidence for magnetic order in the pseudogap phase of ${\mathrm{Bi}}_{2+x}{\mathrm{Sr}}_{2-x}{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$

We present an extended zero-field muon spin relaxation (ZF-$\mu \mathrm{SR}$) study of overdoped ${\mathrm{Bi}}_{2+x}{\mathrm{Sr}}_{2-x}{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (Bi2212) single crystals, intended to elucidate the origin of weak quasistatic magnetism previously detected by $\mu \mathrm{SR}$ in the superconducting and normal states of optimally doped and overdoped samples. New results on heavily overdoped single crystals show a similar monotonically decreasing ZF-$\mu \mathrm{SR}$ relaxation rate with increasing temperature that persists above the pseudogap (PG) temperature $T^{*}$ and does not evolve with hole doping ($p$). Additional measurements using an ultralow-background apparatus confirm that this behavior is an intrinsic property of Bi2212, which cannot be due to magnetic order associated with the PG phase. Instead we show that the temperature-dependent relaxation rate is most likely caused by structural changes that modify the contribution of the nuclear dipole fields to the ZF-$\mu \mathrm{SR}$ signal. Our results for Bi2212 emphasize the importance of not assuming that the nuclear-dipole field contribution is independent of temperature in ZF-$\mu \mathrm{SR}$ studies of high-temperature (high-$T_c$) cuprate superconductors, and do not support a recent $\mu \mathrm{SR}$ study of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ that claims to detect magnetic order in the PG phase.

Figure 1

Temperature dependence of the bulk magnetic susceptibility for the OD55 (blue circles) and OD70 (red circles) samples measured under (a) zero-field-cooled (ZFC) and (b) field-cooled (FC) conditions. The slight decrease of the OD55 data with decreasing temperature between the two downward pointing arrows in (b) indicates the onset of superconductivity in some small volume fraction of the OD55 sample prior to the main superconducting transition at 55 K.

Figure 2

Schematic of the arrangement of the external muon, external positron, internal muon, and internal veto counters, where external and internal refer to outside and inside the helium-gas flow cryostat, respectively. Muons passing through the internal ${\mu }^{+}$ counter that trigger the veto counter are rejected. Also shown are internal Ag and plastic scintillator counter annuli with equivalent 3-mm-diameter holes. These serve as a second type of veto (“active collimator”) in the “ultralow background” (ULB) setup. Muons stopping in the Ag annulus do not pass through the internal muon counter and are rejected. The Ag annulus protects the scintillator counter annulus from muons, so that it can be used to detect and reject positrons from decay muons in the Ag annulus.

Figure 3

Normalized ZF-$\mu \mathrm{SR}$ asymmetry spectra for (a) OD70 at $T=150$ K (red circles) and $T=4$ K (black circles), and (b) OD55 at $T=150$ K (blue circles) and $T=4$ K (black circles). Normalized ZF-$\mu \mathrm{SR}$ asymmetry spectra for OD70 (red circles) and OD55 (blue circles) at (c) $T=150$ K and (d) $T=4$ K. The solid curves are fits to $A\left(t\right)=a_0{G}_z\left(t\right)$, where $G_z\left(t\right)$ is given by Eq. (3).

Figure 4

Temperature dependence of (a) the ZF relaxation rate $\lambda$ and (b) the stretching exponent $\beta$ resulting from fits of the ZF-$\mu \mathrm{SR}$ asymmetry spectra assuming Eq. (3). The two data sets for the OD80 sample in (a) correspond to data collected during two separate experiments, demonstrating the reproducibility of the results. The range of $T^{*}$ values shown at the bottom of panel (a) are compiled values from angle-resolved photoemission spectroscopy and tunneling experiments [34]. (c) Temperature dependence of the ZF relaxation rate in high-purity Ag measured in the same experimental setup. Note that $\lambda$ in this case comes from fits to Eq. (3) with $\beta =1$.

Figure 5

The time at which $A\left(t\right)=a_0/2$ as a function of hole doping for several temperatures. The multiple data points for the OD80 sample at 50 K, 100 K, and 150 K correspond to independent measurements.

Figure 6

(a) Normalized ZF-$\mu \mathrm{SR}$ asymmetry spectra from ULB measurements of the OD80 Bi2212 sample at $T=150$ K (blue circles) and $T=5$ K (red squares). Solid curves are fits to Eq. (3). Temperature dependence of (b) the ZF relaxation rate and (c) $\beta$ for the OD80 sample from measurements using the ULB cyrostat insert (gray triangles) and two sets of measurements in different muon beam periods using the standard muon veto cryostat insert (orange circles). (d) Temperature dependence of the ZF relaxation rate in high-purity Ag from experiments using the ULB (blue circles) and the standard (black squares) muon veto cryostat inserts, and previous results [31] (pink triangles) using the standard muon veto cryostat insert. The values of $\lambda$ for Ag are from fits assuming $\beta =1$.

Figure 7

(a) The orthorhombic crystal structure of Bi2212 (only half of the crystal structure is illustrated for clarity). Contour plots of $\mathrm{\Delta }$ in the SrO layer (shaded plane) are calculated for (b) $T=150$ K and (c) $T=5$ K. (d) The difference between the $\mathrm{\Delta }$ values in (b) and (c) at $100\times$ magnification. (e) The difference between the $\mathrm{\Delta }$ values within the SrO layer for $T=150$ K and within a parallel plane 0.15 Å toward the nearest BiO layer for $T=5$ K. Note that the large values for $\mathrm{\Delta }$ near the apical oxygen atoms are due to the Cu and Bi nuclei located directly below or above in the adjacent ${\mathrm{CuO}}_2$ and BiO layers.