Muon spin relaxation study of superconducting ${\mathrm{Bi}}_2{\mathrm{Sr}}_{2-x}{\mathrm{La}}_x\mathrm{Cu}{\mathrm{O}}_{6+\delta }$

We have performed transverse-field (TF) and zero-field (ZF) $\mu \mathrm{SR}$ measurements of ${\mathrm{Bi}}_2{\mathrm{Sr}}_{2-x}{\mathrm{La}}_x\mathrm{Cu}{\mathrm{O}}_{6+\delta }$ (Bi2201) systems with $x=0.2$, 0.4, 0.6, and 1.0, using ceramic specimens with modest $c$-axis alignment and single-crystal specimens. The absence of static magnetic order has been confirmed in underdoped $(x=0.6)$ and optimally doped $(x=0.4)$ systems at $T=2\mathrm{K}$, while only a very weak signature towards static magnetism has been found at $T=2\mathrm{K}$ in the $x=1.0$ system, which is a lightly hole-doped nonsuperconducting insulator. In the superconducting ($x=0.6$, 0.4, and 0.2) systems, the relaxation rate $\sigma$ in TF-$\mu \mathrm{SR}$, proportional to $n_s∕m^{*}$ (superconducting carrier density and effective mass), followed a general trend found in other cuprate systems in a plot of $T_c$ vs $n_s∕m^{*}(T\to 0)$. Assuming the in-plane effective mass $m^{*}$ for Bi2201 to be comparable to three to four times the bare electron mass $m_e$ as found in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ (LSCO) and ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ (YBCO) systems, we obtain $n_s\sim 0.15–0.2$ per Cu for the $x=0.4$ Bi2201 system. This carrier density is much smaller than the Hall number $n_{\mathit{\text{Hall}}}\sim 10$ per Cu obtained at $T<1.6\mathrm{K}$ in high magnetic fields $(40–60\mathrm{T})$ along the $c$ axis applied to suppress superconductivity. The present results of the superfluid density $(n_s∕m^{*})$ in Bi2201 are compared with those from other cuprate systems, including YBCO systems with very much reduced $T_c<20\mathrm{K}$ studied by microwave, $H_{c1}$, and inductance methods. Additional muon-spin-relaxation $\left(\mu \mathrm{SR}\right)$ measurements have been performed on a single-crystal specimen of Bi2201 $(x=0.4)$ in a high transverse magnetic field of $5\mathrm{T}$ parallel to the $c$ axis, in order to search for the field-induced muon spin relaxation recently found in LSCO and some other high-temperature superconducting cuprate (HTSC) systems well above $T_c$. The nearly temperature-independent and very small relaxation rate observed in Bi2201 above $T_c$ rules out a hypothesis that the field-induced relaxation is directly proportional to the magnitude of the Nernst coefficient, which is a measure of the strength of dynamic superconductivity. We also describe a procedure for angular averaging of $\sigma$ in $\mu \mathrm{SR}$ measurements using ceramic specimens with modest alignment of $c$-axis orientations, together with the neutron-scattering results obtained for determining the orientation distribution of microcrystallites in the present ceramic specimens.

Figure 1

Comparison of phase diagrams, obtained from transport measurements in high magnetic fields up to $60\mathrm{T}$ applied parallel to the $c$ axis, for (a) Bi2201 (Ref. 36) and (b) LSCO (Ref. 33). In the shaded region, the in-plane conductivity (of Bi2201) and in- and out-of-plane conductivity (of LSCO) show an insulating behavior. Bi2201 has much smaller insulating region compared to LSCO.

Figure 2

(Color) Asymmetry $A{G}_z\left(t\right)$ of $\mathrm{ZF}\text{−}\mu \mathrm{SR}$ time spectra observed in (a) a ceramic specimens of ${\mathrm{Bi}}_2{\mathrm{Sr}}_{2-x}{\mathrm{La}}_x\mathrm{Cu}{\mathrm{O}}_{6+\delta }$ (Bi2201) with $x=1$ (nonsuperconducting and nominally undoped parent compound) at $T=100$ and $2.2\mathrm{K}$, and ceramic specimen of ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$; and (b) in ceramic specimens of superconducting Bi2201 with $x=0.6$ (underdoped, $T_c=12\mathrm{K}$) and $x=0.4$ (optimally doped, $T_c=27\mathrm{K}$) at $T=2$ and $100\mathrm{K}$.

Figure 3

Time spectra $A{G}_x\left(t\right)\cos \left(\omega t\right)$ observed in transverse-field $\mu \mathrm{SR}$ measurements in a ceramic specimen of Bi2201 $(x=0.4)$ with $B_{\mathit{ext}}=200\mathrm{G}$ at (a) $T=50\mathrm{K}$ and (b) $T=2\mathrm{K}$. (b) Taken after field cooling in $200\mathrm{G}$.

Figure 4

(Color) Temperature dependence of the muon-spin-relaxation rate ${\sigma }_{sc}$ due to superconductivity, observed in transverse-field $\mu \mathrm{SR}$ in $B_{\mathit{ext}}=200\mathrm{G}$ in ceramic specimens of Bi2201 with $x=0.6$ (underdoped), 0.4 (optimally doped), and 0.2 (overdoped). The background relaxation rate due to nuclear dipolar fields ${\sigma }_{\mathit{nd}}$, estimated by averaging the observed relaxation rate $\sigma$ above $T_c$, is subtracted quadratically following Eq. (2). Negative values of ${\sigma }_{sc}$ indicate cases with $\sigma ⩽{\sigma }_{\mathit{nd}}$.

Figure 5

(Color) Field dependence of the muon-spin-relaxation rate $\sigma$ observed in transverse magnetic field in single-crystal and modestly oriented ceramic specimens of optimally doped Bi2201 $(x=0.4)$ at $T=2\mathrm{K}$. The contribution from the nuclear dipolar field is not subtracted. The flux vortex has 3D structure in lower fields, and the 2D pancake structure in higher fields, as illustrated.

Figure 6

(Color) Line shapes of Fourier-transformed spectra of transverse-field $\mu \mathrm{SR}$ in (a) a single-crystal specimen and (b) a mildly oriented ceramic specimen of optimally doped Bi2201 $(x=0.4)$ observed at $T=2\mathrm{K}$ in a few different values of external field. Asymmetric line shapes due to the internal-field profile of the 3D Abrikosov flux vortex lattice are seen in the results for the single-crystal sample in a low-field region.

Figure 7

(Color) Temperature dependence of the muon-spin-relaxation rate observed in transverse fields of $200\mathrm{G}$ and $5\mathrm{T}$ in a single-crystal specimen of optimally doped Bi2201 $(x=0.4)$. The contribution from nuclear dipolar fields is not subtracted. The open-square results in $5\mathrm{T}$ include contributions from spatial inhomogeneity and drift of the applied field. The solid square symbols show the results after the correction for these applied-field effects.

Figure 8

Intensity profile, as a function of the angle $\theta$ between the line bisecting the scattering angle and the line normal to the sample disc surface, observed in a modestly oriented ceramic specimen of Bi2201 $(x=0.4)$ by elastic neutron scattering at the setting of the scattering angle at that of the [006] Bragg reflection. The broken line shows the background contribution from nuclear incoherent scattering determined by setting the scattering angle off from the Bragg peak position. The inset figure shows the scanning electron microscope (SEM) image of grain size at the surface of the ceramic specimen of Bi2201 $(x=0.4)$ used in the $\mu \mathrm{SR}$ and neutron measurements.

Figure 9

(Color) Simulation results of the muon-spin-relaxation rate $\sigma$ expected in oriented ceramic specimen with a Gaussian distribution of crystallite orientations [Eq. (5)] with the width ${\Delta }_{\theta }$ calculated assuming a factor $C=1.43$ for the contribution of shift broadening in Eq. (8). The values of $\sigma$ are normalized to ${\sigma }_{xtal}$ expected for a single-crystal specimen. The results corresponding to the cases for Bi2201 specimens with $x=0.2$, 0.4, and 0.6 are shown, respectively, at their values of ${\Delta }_{\theta }$ determined by neutron-scattering intensity (shown in Fig. 8 for the $x=0.4$ specimen).

Figure 10

(Color) Superconducting transition temperature $T_c$ of HTSC and other type-II superconductors plotted against the muon-spin-relaxation rate $\sigma \propto n_s∕m^{*}$ at $T\to 0$. (a) compares the present Bi2201 results with those of YBCO (open circle), LSCO (open triangle) and other cuprates (Refs. 2, 3), Zn-doped cuprates (closed triangle) (Ref. 5), overdoped Tl2201 (closed circle) (Ref. 8), $A_3{\mathrm{C}}_{60}$ (Ref. 69), and LCO, LSCO (Ref. 6), and LESCO (Ref. 7) systems with partial volume having static stripe magnetic order (blue striped square). The blue “dot in open circle” points represent the $\mu \mathrm{SR}$ results of $\lambda$ for single-crystal specimens of YBCO (Refs. 70, 71, 72), converted following the procedures in the text. An expanded plot in (b) compares the present $\mu \mathrm{SR}$ results on Bi2201 with the superfluid density $n_s∕m^{*}$ estimated in YBCO by microwave (Ref. 39) and ${\mathrm{H}}_{c1}$ (Ref. 38) measurements on single crystals and by inductance measurements (Ref. 40) on thin-film specimens.

Figure 11

(Color) (a) Transition temperature $T_c$ of various HTSC systems plotted against the product of the muon-spin-relaxation rate $\sigma (T\to 0)$ and the average distance $c_{\mathit{int}}$ of adjacent $\mathrm{Cu}{\mathrm{O}}_2$ planes. The horizontal axis represents the 2D superfluid density $n_{s2d}∕m^{*}$. The $T_{\mathit{KT}}$ line shows system-independent superfluid density expected in the Kosterlitz-Thouless theory for purely 2D superfluid systems just below the transition temperature (Ref. 79). The inset shows the relationship between $T_c$ and $c_{\mathit{int}}$ obtained for MBE-made thin films of alternating layers of optimally doped YBCO and insulating PBCO $\left(\mathrm{Pr}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7\right)$ stacked along the $c$-axis direction (Ref. 73). (b) A plot of $T_c$ vs carrier density per Cu for Bi2201 obtained in the present work for the values of $m^{*}$ estimated for YBCO and LSCO systems (Ref. 74), and the Hall number $n_{\mathit{\text{Hall}}}$ obtained in the high-field Hall measurements (Ref. 34).

Figure 12

(Color) Plot of $T_c$ vs the energy of the roton minimum of bulk superfluid ${}^4\mathrm{He}$ (values for both axes multiplied by a factor 60) measured under applied pressures (Ref. 83), compared with the relationship seen in HTSC systems for the energies of the neutron resonance mode at $(\pi ,\pi )$ (Ref. 80), the Raman $A{1}_g$ mode (Ref. 81), and the ARPES SCP (Ref. 82). Also included are the neutron energy transfers of spectral weight maximum near the $(\pi \pm \delta ,\pi )$ point in YBCO (Ref. 80) and LSCO (Ref. 84) (closed squares). Taken from Uemura (Ref. 1).

Figure 13

(Color) Comparison of (a) the $\mu \mathrm{SR}$ and (b) the Nernst-effect results for optimally doped Bi2201 $(x=0.4)$ and ${\mathrm{La}}_{1.88}{\mathrm{Sr}}_{0.12}\mathrm{Cu}{\mathrm{O}}_4$ (LSCO:0.12). (a) shows the present results for Bi2201 and our recent results (Ref. 45) for LSCO:0.12 in TF-$\mu \mathrm{SR}$ with $B_{\mathit{ext}}=5\mathrm{T}$. (b) shows the Nernst coefficient (see text) in Bi2201 (Ref. 48) and LSCO:0.12 (Ref. 85).