Magnetic vortex lattice in HgBa${}_2$CuO${}_{4+\delta }$ observed by small-angle neutron scattering

We report the direct observation of the magnetic vortex lattice in the model high-temperature superconductor HgBa${}_2$CuO${}_{4+\delta }$. Using small-angle neutron scattering on high-quality crystals, we observe two equal domains of undistorted triangular vortex lattices well aligned with the tetragonal crystallographic axes. The signal decreases rapidly with increasing magnetic field and vanishes above $0.4$ T, which we attribute to a crossover from a three-dimensional to a two-dimensional vortex system, similar to previous results for the more anisotropic compound Bi${}_{2.15}$Sr${}_{1.95}$CaCu${}_2$O${}_{8+\delta }$. Our result indicates that a triangular vortex lattice (with or without distortion) at low magnetic fields is a generic property of cuprates with critical temperatures above $80$ K.

Figure 1

Left: Neutron diffraction patterns at $B=0.2$ and $0.3$ T and $T=2$ K, averaged over different sample rotation angles. Right: Rocking scans for two Bragg reflections (indicated by the arrows on the left), measured by rotating the sample-magnet assembly with respect to the incident neutron beam.

Figure 2

(a) Azimuthally averaged signal amplitudes as a function of momentum transfer $q$, measured at $T=2$ K in different magnetic fields. Data obtained with two configurations (see text) are rescaled to allow for a common vertical axis. The $B=0.35$ T data are, furthermore, multiplied by a factor of 5 for clarity. Vertical bars indicate the fitted value of the characteristic $q$. (b) VL-structure-dependent coefficient $\sigma$ (see text) at different fields. Filled and open symbols correspond to configurations I and II, respectively. (c) Azimuthal-angle dependence of the signal measured in configuration I at $T=2$ K for $B=0.2$ and 0.25 T. Data in (a) and (c) are vertically offset for clarity.

Figure 3

(a) Intensities measured at $T=2\hspace{0.5em}$K and $B=0.25\hspace{0.5em}$T in sector boxes surrounding the diffraction peaks indicated by corresponding symbols in (b). The data are offset for clarity, and the same offset is used for reflections that are symmetric about the vertical plane. (b) Illustration of a tilt scan (see text). Broken circles are the intersections of the Ewald sphere with the two-dimensional reciprocal plane at the end points of the scan. Potential diffraction spots are marked by crosses. The five diffraction spots marked by symbols have their Bragg conditions satisfied within the range of the scan. (c) VL intensity measured at $T=2\hspace{0.5em}$K as a function of magnetic field. Measurements in different configurations are normalized to the values at common magnetic fields. The dotted line is the expected field dependence in the London limit (Eq. (2)); the dashed line is the expected behavior after vortex-core correction (see text), using $B_{c2}=100\hspace{0.5em}$T (Ref. 24) and the phenomenological parameter $s=7$; the solid line is a smoothing curve describing the data. The inset shows the same data plotted versus $B^{-1/2}$, where the field dependence expected in the London limit is represented by the dashed line.

Figure 4

Temperature dependence of diffraction intensity measured at two different magnetic fields compared with a $\mu$SR result for optimally-doped Hg1201.[30] The value at $T=100\hspace{0.5em}$K is set to zero, and the data are normalized to the values at low temperature. The solid lines are empirical fits as guides to the eye (see text).