Evidence for a Square-Square Vortex Lattice Transition in a High-$T_c$ Cuprate Superconductor

Using sound velocity and attenuation measurements in high magnetic fields, we identify a new transition in the vortex lattice state of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$. The transition, observed in magnetic fields exceeding 35 T and temperatures far below zero field $T_c$, is detected in the compression modulus of the vortex lattice, at a doping level of $x=p=0.17$. Our theoretical analysis based on Eilenberger’s theory of the vortex lattice shows that the transition corresponds to the long-sought 45° rotation of the square vortex lattice, predicted to occur in $d$-wave superconductors near a van Hove singularity.

Figure 1

(a) The change in sound velocity of the in-plane longitudinal mode (called $c_{11}$) of a LSCO $p=0.17$ crystal at different temperatures, as a function of field applied along the $c$ axis. All data are from pulsed field downsweeps, except the lowest temperature curve which is an upsweep in a DC field magnet. This curve shows a series of spikes above 5 T or so, associated with flux jumps. Curves have been offset vertically (though not by a constant amount). The arrows mark the field of the anomaly, $B_{\mathrm{anom}}$, which is visible but more subtle up to 1.90 K in the sound velocity, and the depinning field $B_{\mathrm{dp}}$. Measurements were taken at frequencies of 155 MHz in pulsed field and 127 MHz in DC field. (b) Sound velocity of the $c_{11}$ mode at 1.5 K. The upsweep (downsweep) is shown in red (blue). For increasing fields, the VL transition is found at 41 T while it appears at 37 T or so for decreasing fields, as indicated by the black arrows. The fact that above the irreversibility field the two curves overlap and that the depinning field is identical for both upsweeps and downsweeps shows isothermal conditions during the magnetic field pulse. (c) Temperature-magnetic field phase diagram, showing the area covered by ultrasound measurements in different magnet systems and the location of the anomaly during downsweeps (blue) and vortex melting (red). Dashed lines are guides to the eye.

Figure 2

A comparison of the field dependence of three different ultrasound modes in LSCO $p=0.17$, taken at 1.5 K, from downsweeps in pulsed field measurements. The table shows the propagation $\mathbf{k}$ and polarization $\mathbf{u}$ directions for different modes. The different elastic constants of the square and hexagonal VL probed for different propagation and polarization directions used here are listed in Table SII in the Supplemental Material [24]. We have also probed the $L110$ mode of LSCO $p=0.17$, but the signal in this configuration is strongly attenuated below the high temperature tetragonal-low temperature orthorhombic (HTT-LTO) structural transition, resulting in a poor signal-to-noise ratio and preventing us from detecting an anomaly (Fig. S2 in the Supplemental Material [24]).

Figure 3

(a) Left: the $\mathrm{Cu}─\mathrm{O}$ plane [37] of LSCO, with the $c$ axis out of the page. Right: illustrations of the hexagonal, $v_F$-centered square and superconducting gap-centered square vortex lattices, respectively, relative to the crystal lattice, with vortices portrayed as black circles. Dashed lines mark the $\mathrm{Cu}─\mathrm{O}─\mathrm{Cu}$ bond directions, also the [100] axes of the high temperature tetragonal phase. (b) A phase diagram for $p=0.17$ showing the calculated VL configurations as a function of $T$ and $B$. The hexagonal phase is stable in the green-colored region at high $T$, low $B$. In regions $A$ and $D$ (orange and blue), one of the two square lattices is stable. The brown line ($B\text{−}C$ boundary) marks the transition between the two square VL arrangements in slowly varying magnetic field. Metastable regions $B$ and $C$ emerging in pulsed fields are shown with slashed lines. The vortex solid is unobservable with ultrasound for $B>B_{\mathrm{dp}}$. Solid circles are results of the calculation, and lines connecting them are guides to the eye. The depinning transition was not taken into account for the calculations.