Online Supplementary Material: Low temperature vortex liquid in $\mathrm{L}{\mathrm{a}}_{2-\mathrm{x}}\mathrm{S}{\mathrm{r}}_{\mathrm{x}}\mathrm{Cu}{\mathrm{O}}_4$

Abstract

We report here the supplementary material for Ref. [1]. Using torque magnetometry, we measured the magnetization anisotropy of 7 crystals of $\mathrm{L}{\mathrm{a}}_{2-\mathrm{x}}\mathrm{S}{\mathrm{r}}_{\mathrm{x}}\mathrm{Cu}{\mathrm{O}}_4$ (LSCO). In optimally-doped cuprates, the bulk susceptibility is dominated by the paramagnetic van-Vleck orbital term ${\chi }_{orb}$ which has a significant anisotropy (${\chi }_c^{orb}>{\chi }_{ab}^{orb}$) that changes weakly with $T$ (subscripts $c$ and $ab$ identify quantities measured with $H||c$ and $\mathbf{H}\perp \mathbf{c}$, respectively). Moreover, in the lightly-doped regime, the paramagnetic spin susceptibilities ${\chi }_c^s$ and ${\chi }_{ab}^s$ become significantly large below the interval 40-60 K. However, the spin susceptibility is very nearly isotropic (except below 10 K where its anisotropy becomes measurable). Against the large orbital and spin terms, the weak diamagnetic signal is very difficult to resolve using standard bulk magnetometry in lightly-doped cuprates. By contrast, torque magnetometry selectively detects the orbital diamagnetism generated by supercurrents confined to the CuO${}_2$ layers while ignoring the large spin contribution when it is isotropic. The orbital van-Vleck contribution is also detected, but as a ``background'' that is $H$-linear to intense fields and only mildly $T$ dependent.

11 pages, 9 figures, Supplement to cond-mat/0702121, Nature Physics, in press