Josephson vortex state across the phase diagram of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$: A magneto-optics study

We present a detailed doping dependent study of the Josephson vortex state in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ using infrared spectroscopy. A magnetic field as high as $17\text{tesla}$, applied along the $\mathrm{Cu}{\mathrm{O}}_2$ planes, is found to suppress the Josephson plasmon in all measured samples. We find the strongest suppression in samples with dopings close to $x=1∕8$ and attribute this effect to the spontaneous formation of in-plane charge inhomogenities at this doping level. Several theoretical models of the Josephson vortex state are applied to explain the observed effects.

Figure 1

(a) Josephson vortex sandwiched between two $\mathrm{Cu}{\mathrm{O}}_2$ planes. (b) Josephson vortex between two $\mathrm{Cu}{\mathrm{O}}_2$ planes with in-plane charge inhomogeneities. Note that the figure is not drawn to scale as the size of the patches ${\xi }_s$ is typically 100–1000 times smaller than the vortex size ${\lambda }_c$ (Table ).

Figure 2

Temperature dependence of infrared reflectance $R(\omega ,T,H=0\mathrm{T})$ for the $x=0.125$ LSCO sample. Note that the frequency scale changes from linear to logarithmic above $100{\mathrm{cm}}^{-1}$.

Figure 3

Temperature dependence of infrared reflectance $R(\omega ,T,H=0\mathrm{T})$ for all LSCO samples. Only spectra taken at temperatures below $T_c$ are shown for clarity. Both absolute $\omega$ and relative frequency units $\omega ∕{\omega }_0$ are shown; ${\omega }_0$ is the frequency of the plasma minimum at $6–7\mathrm{K}$ (Table ).

Figure 4

Temperature dependence of the superfluid plasma frequency in zero field ${\omega }_s\left(T\right)∕{\omega }_s\left(6\mathrm{K}\right)$. ${\omega }_s\left(6\mathrm{K}\right)$ values are reported in Table .

Figure 5

Magnetic field dependence of infrared reflectance $R(\omega ,T=6\mathrm{K},H)$. For each sample only a few in-field spectra are shown for clarity.

Figure 6

Magnetic field dependence of the superfluid plasma frequency at $6\mathrm{K}$ ${\omega }_s\left(H\right)∕{\omega }_s\left(0T\right)$.

Figure 7

The model of Bulaevskii et al.[19] applied to our ${\omega }_s\left(H\right)∕{\omega }_s\left(6\mathrm{K}\right)$ data. Apart from $x\sim 1∕8$ doping the model provides a satisfactory account of the data below $H∕H_0<0.3$; the model breaks down at higher fields.

Figure 8

The models of Coffey and Clam[21, 22, 23] (CC) and Tachiki, Koyama, and Takahashi[8] (TKT) applied to our ${\omega }_s\left(H\right)∕{\omega }_s\left(6\mathrm{K}\right)$ data. These models reproduce the data for all samples, except for $x\sim 1∕8$ doping. Moreover, for dopings different from $1∕8$ the model yields physically realistic values of the parameters, such as the viscosity coefficient $\eta$ and the pinning constant ${\kappa }_p$ (Table ).

Figure 9

Field-induced pair-breaking effects in $s$-wave and $d$-wave superconductors. Left panel: for $s$-wave gap (shaded region), small magnetic field (dashed line) is ineffective in pair breaking until its magnitude exceeds the gap value. Middle and right panels: for superconductors with $d$-wave symmetry of the order parameter, the gap value goes to zero at the so-called nodal points and any finite field can cause pair breaking. However, when the maximum gap (at $[0,\pi ]$, $[\pi ,0]$, and symmetry related points) is small, a larger portion of the Fermi surface is influenced by the field (circled region) and the impact of the field is enhanced. The latter situation is relevant to LSCO samples with dopings close to $x=1∕8$.

Figure 10

Won, Jang, and Maki’s model[24, 25] combined with the TKT and applied to 0.125 data, as described in the text. Also shown are individual components that are set to produce equal contribution to a net suppression of superfluid plasma frequency. Gray line illustrates the Zeeman effect alone [Eq. (8)].

Figure 11

Doping dependence of the superfluid plasma frequency ${\omega }_s$ [panel (a), left axis], DC conductivity at $T_c$ ${\sigma }_{\mathit{dc}}$ [panel (a), right axis], the ratio ${\omega }_s^2∕{\sigma }_{\mathit{dc}}$ [panel (b)], and of the suppression of the superfluid plasma frequency at $17\mathrm{T}$ ${\omega }_s\left(17\mathrm{T}\right)∕{\omega }_s\left(0\mathrm{T}\right)$ [panel (c)]. Anomalous behavior near $x=1∕8$ is attributed to inhomogeneous superconducting state, as described in the text.