Muon spin rotation investigation of the pressure effect on the magnetic penetration depth in YBa${}_2$Cu${}_3$O${}_x$

The pressure dependence of the magnetic penetration depth $\lambda$ in polycrystalline samples of YBa${}_2$Cu${}_3$O${}_x$ with different oxygen concentrations $x=6.45$, 6.6, 6.8, and 6.98 was studied by muon spin rotation ($\mu$SR). The pressure dependence of the superfluid density ${\rho }_s\propto 1/{\lambda }^2$ as a function of the superconducting transition temperature $T_{\mathrm{c}}$ is found to deviate from the usual Uemura line. The ratio $(\partial T_{\mathrm{c}}/\partial P)/(\partial {\rho }_s/\partial P)$ is smaller by a factor of $\simeq$2 than that of the Uemura relation. In underdoped samples, the zero-temperature superconducting gap ${\Delta }_0$ and the BCS ratio ${\Delta }_0/k_B{T}_{\mathrm{c}}$ both increase with increasing external hydrostatic pressure, implying an increase of the coupling strength with pressure. The relation between the pressure effect and the oxygen isotope effect on $\lambda$ is also discussed. In order to analyze reliably the $\mu$SR spectra of samples with strong magnetic moments in a pressure cell, a special model was developed and applied.

Figure 1

$\mu$SR asymmetry signal $A\left(t\right)$ of YBa${}_2$Cu${}_3$O${}_{6.98}$ measured at $T=4.5$ K and 95 K in an applied field $B_{\mathrm{app}}=0.1$ T (empty and full circles, respectively). The fast relaxation of the $\mu$SR signal (empty circles) is due to the formation of a vortex lattice in the superconducting state. The solid lines are fits of the data to Eq. (1). For better visualization, the spectra and the fits are shown in a rotating reference frame of 0.08 T.

Figure 2

Fourier transform (FT) amplitude as a function of field for the spectra shown in Fig. 1 [panels (a), (b), and (c)]. Panel (b) is the expanded view (along the $y$ axis) of panel (a) showing the signal from the sample. Panel (c) is the expanded view (along the $x$ axis) of panel (a) showing the signal of the pressure cell. Panel (d) shows the FT of the sample with $x=6.6$ below and above $T_{\mathrm{c}}=60$ K. The solid lines are the FTs of the fitted curves shown in Fig. 1. The FT spectra are slightly broadened due to a FT apodization of 4 $\mu$s${}^{-1}$.

Figure 3

(a) Temperature dependence of $\sigma$ of YBa${}_2$Cu${}_3$O${}_x$ measured at $B_{\mathrm{app}}=0.1$ T at zero and applied hydrostatic pressures for $x=6.45$ ($◊$, $P=0$ GPa; $⧫$, $P=1.1$ GPa), $x=6.6$ ($▿$, $P=0$ GPa; $▾$, $P=1.1$ GPa), $x=6.8$ ($▵$, $P=0$ GPa; $▴$, $P=1.1$ GPa), and $x=6.98$ ($\circ$, $P=0$ GPa; $•$, $P=1.4$ GPa). The data were analyzed with Eq. (1). The solid curves are fits to the data with Eq. (2). (b) Diamagnetic shift of the field $\Delta B=B_1-B_{\mathrm{app}}$ in the corresponding samples. $B_1$ is the mean field in the centre of the sample (see text and the appendix).

Figure 4

Temperature dependence of $\sigma$ of YBa${}_2$Cu${}_3$O${}_x$ measured at $B_{\mathrm{app}}=0.5$ T. The meanings of the symbols and the solid lines are the same as in Fig. 3a.

Figure 5

$T_{\mathrm{c}}$ vs $\sigma \left(0\right)$ (Uemura plot) at zero and applied pressure for YBa${}_2$Cu${}_3$O${}_x$ with $x=6.45$, 6.6, 6.8, and 6.98. The solid line is the Uemura line while the dashed line is a guide to the eye. The dotted lines represent the pressure effects on $T_{\mathrm{c}}$ and $\sigma \left(0\right)$.

Figure 6

Relation between ${\Delta }_0$ and $T_{\mathrm{c}}$ for YBa${}_2$Cu${}_3$O${}_x$ with $x=6.6$, 6.8, and 6.98. The solid line corresponds to ${\Delta }_0/k_B{T}_{\mathrm{c}}=3$ (weak-coupling BSC superconductor: ${\Delta }_0/k_B{T}_{\mathrm{c}}=1.76$). Both ${\Delta }_0$ and ${\Delta }_0/k_B{T}_{\mathrm{c}}$ increase with increasing pressure.

Figure 7

The gap ${\Delta }_0$ as a function of $\sigma \left(0\right)$ for the underdoped samples of YBa${}_2$Cu${}_3$O${}_x$ with $x=6.6$ and 6.8. The linear relation between $\sigma \left(0\right)$ and ${\Delta }_0$ is better fulfilled under hydrostatic pressure than the Uemura relation $T_{\mathrm{c}}$ vs $\sigma \left(0\right)$ and ${\Delta }_0/k_B{T}_{\mathrm{c}}$ vs $T_{\mathrm{c}}$ (see Figs. 5 and 6). The line is a guide to the eye.

Figure 8

(a) Schematic sketch of the $\mu$SR pressure instrument GPD at the Paul Scherrer Institute: Cylindrical sample (S), pressure cell (PC), muon stopping distribution (dotted ellipse), and forward and backward positron detectors. (b) Illustration of the surface current on a slice of a homogeneously magnetized cylindrical sample. The magnetic field induced by this slice is equivalent to the magnetic field of the surface currents. (c) Cross section of the cylindrical sample and the surface current distribution in $xz$ plane. (d) Magnetic field map of the surface currents as illustrated in panels (b) and (c).

Figure 9

(a) Contour plot of the field distribution $H_z^{\prime }(y,z)$ in the $yz$ plane for a cylindrical sample with $R=2.5$ mm and $H=15$ mm (the geometry of the sample used in the experiment). (b) Contour plot of the muon stopping distribution in the $yz$ plane. The gray area on the top of the sample corresponds to the empty pressure cell space where no muons stop (this space is filled with a low-density pressure transmission medium). The dashed line indicates the sample space. (c) Magnetic field profile of $H_z^{\prime }$ along the $y$ axis and (d) magnetic field profile of $H_z^{\prime }$ along the $z$ axis.