Spin density wave induced disordering of the vortex lattice in superconducting La${}_{2-x}$Sr${}_x$CuO${}_4$

We use small-angle neutron scattering to study the superconducting vortex lattice in La${}_{2-x}$Sr${}_x$CuO${}_4$ as a function of doping and magnetic field. We show that near optimally doping the vortex lattice coordination and the superconducting coherence length $\xi$ are controlled by a Van Hove singularity crossing the Fermi level near the Brillouin zone boundary. The vortex lattice properties change dramatically as a spin-density-wave instability is approached upon underdoping. The Bragg glass paradigm provides a good description of this regime and suggests that spin-density-wave order acts as a source of disorder on the vortex lattice.

Figure 1

Phase diagram of (a) the ($T<3$ K) vortex lattice structure and (b) the magnetism in La${}_{2-x}$Sr${}_x$CuO${}_4$, both revealed by neutron diffraction. Data points in (a) for $x=0.12$, 0.145, 0.16, and 0.22 are from this work whereas $x=0.105$, 0.17, and 0.20 are from Refs. 22, 23, 33. Circular points are defined by the onset of a square VL coordination. The field-doping plane in (b), adapted from Ref. 26, shows schematically the ordered SDW moment normalized to that at the 1/8 doping. Field-induced order was reported by Khaykovich et al.[31] and later confirmed in Ref. 26. The temperature-doping plane shows the superconducting dome together with the onset of static incommensurate SDW order $T_{\mathrm{SDW}}$ as seen by neutron diffraction.[26, 34] We remark that a similar phase diagram was proposed for YBa${}_2$Cu${}_3$O${}_y$.[35] The dashed lines indicate the samples studied in this paper.

Figure 2

A schematic diagram illustrating the experimental geometry chosen for our experiments. A square VL in real space forms a two-dimensional reciprocal VL, the properties of which are recorded by SANS using a position-sensitive detector. The Bragg spots in reciprocal space exhibit finite widths $w_{\ell }$, $w_{\mathrm{q}}$, and $w_{\perp }$ that are dependent on both the instrumental resolution and properties of the VL. These three lengths are estimated at the detector by recording the angular widths ${\tau }_{\mathrm{r}}$ and ${\tau }_{\mathrm{A}}$ within the detector plane, and ${\tau }_{\omega }$ perpendicular to the detector plane. ${\tau }_{\omega }$ is determined experimentally by recording the rocking curve, and corresponds to the rocking curve width.

Figure 3

(a), (b) Vortex lattice diffraction patterns, recorded using a position-sensitive detector (see Fig. 2), of LSCO $x=0.145$ under applied magnetic fields of 0.2 T and 1.3 T, respectively. Notice that the Cu-O axes are along the diagonal. (c), (d) Azimuthally averaged momentum $\left|q\right|$ dependence of the scattered intensity, summed over rocking angles, for dopings $x=0.145$ and 0.12, and applied magnetic fields as indicated. For visibility, the 0.2 T data in (c) have been divided by a factor of ten. Inset of (c) is a zoom on the 0.2 T data. The vertical bars above the diffraction peaks indicate the expected positions for square and regular hexagonal VL coordinations. The solid black lines in (c) and (d) are Lorentzian fits to the data.

Figure 4

(a) Dimensionless constant $\sigma$, defined in the text, as a function of magnetic field for LSCO $x=0.145$, 0.16 $0.17$ (Ref. 22), 0.20 (Ref. 23), and 0.23. $\sigma =\sqrt{3}/2$ is expected for a hexagonal lattice and a square vortex lattice has $\sigma =1$. The change form $\sigma =\sqrt{3}/2$ to 1 therefore reveals the a hexagonal-to-square transition of the vortex lattice structure (see also Fig. 1). (b) SANS intensity $I$ versus applied magnetic field for NCCO $x=0.15$ (Ref. 53) and La${}_{2-x}$Sr${}_x$CuO${}_4$ with $x=0.105$ (Ref. 54), and $0.145\text{--}0.23$, (this work). For clarity, the intensities for each of the compositions have been given an arbitrary vertical offset. Solid lines are fits to the Clem model form factor where the superconducting coherence length $\xi$ is the only parameter (see Fig. 5). Dashed lines indicate power-law dependencies; $I\sim H^{-0.5}$ and $I\sim H^{-2}$. Notice that the Bragg glass paradigm for vortices in the presence of disorder is consistent with a crossover from $I\sim H^{-0.5}$ to $I\sim H^{-2}$ (see text).

Figure 5

Superconducting coherence length $\xi$ in LSCO extracted from magnetoresistance (solid blue points),[52] specific heat (open black points),[51] and our SANS measurements (solid red points) and plotted as a function of the hole concentration $x$. The high-field magnetoresistance study measures the upper critical field $H_{c2}$ at low temperatures and we used $H_{c2}={\Phi }_0/\left(2\pi {\xi }^2\right)$ to estimate the coherence length $\xi$. On the other hand, the specific heat and the SANS experiments were carried out at fields smaller or comparable to $H_{c2}$. To extract the superconducting coherence length from the SANS data we used the Clem model for the VL form factor. Notice that in presence of vortex lattice disorder, the Clem model will overestimate the coherence length (see text). This may explain why the coherence length extracted from the SANS data lies systematically above the specific heat and magnetoresistance measurements. All lines are guides to the eye.