Imaging the Meissner effect and local superfluid stiffness in a graphene superconductor

Abstract

We report the observation of the Meissner effect in a rhombohedral graphene superconductor, realized via direct imaging of the static fringe magnetic field. In our few-micron sample, the onset of superconductivity manifests as a diamagnetic response that screens only $\sim 100$ ppm of the applied magnetic field. Tracking the evolution of the resulting nanotesla-scale fields in real space allows us to observe the entry of superconducting vortices and map the local superfluid stiffness, $ρ_s$. Correlating fringe field signals from both Meissner screening and magnetically ordered states, we show that superconductivity onsets in the midst of a continuous quantum phase transition to a canted spin ferromagnet. Within the superconducting state, we find the temperature dependence of $ρ_s$ to be incompatible with isotropic Bardeen-Cooper-Schrieffer theory and the zero-temperature stiffness $ρ_s^0$ to be linearly proportional to $T_c$, constraining future theoretical models of superconductivity in this system.

Figures (15)

• Figure 1: Meissner effect in R3G/WSe$_2$.a, Measurement and device schematics. The nSOT is mounted at an angle $\theta$ relative to the $z$--axis, which is perpendicular to the sample plane. Transport experiments are realized with four contacts arranged on the right side of the device. The inset illustrates the applied external magnetic fields in the x- and z-directions as well as the fringe field pattern due to the Meissner effect. b, $R_{xx}$ measured as a function of $n_e$ and $D$. The light cyan region corresponds to the superconducting state, characterized by vanishing resistance. The red contour corresponds to $R_{xx}=5\Omega$. c, Fringe magnetic field $\Delta B$ measured at a fixed location above the sample A diamagnetic Meissner signal (rendered in blue) is associated with the superconducting region of panel b. A ferromagnetic region (rendered in red) is also visible at the upper right. d, Spatial image of $\Delta B$ within the ferromagnetic state corresponding to the $n_e, D$ configuration marked by the solid square in panel c. An outline of the device is shown in overlay as a guide to the eye, while the solid square indicates the spatial position where data in panel c were acquired. The scale bar is $1\mu$m throughout this work. e, Spatial image of $\Delta B$ in the superconducting state, corresponding to the $n_e, D$ configuration marked by the solid circle in panel c. f, $R_{xx}$ at fixed $D=0.138V/nm$ as a function of $n_e$ for different values of $T$. g, $\Delta B$, measured at the spatial point marked by the solid star in panel e, for the same parameters as the data in panel f.
• Figure 1: nanoSQUID-on-tip (nSOT) characterization.a, Schematic of nSOT measurement circuit. A voltage bias $V_{\rm{bias}}$ is applied across a bias resistor $R_B=10$ k$\Omega$, and a shunt resistor of $R_s = 5 \Omega$ shunts the nSOT. The current through the nSOT is measured via the voltage $V_{fb}$ across a resistor $R_F = 1$ k$\Omega$ coupled via a series superconducting squid array amplifier with a turn-ratio of 13.5 between the inductors. b, Schematic of the nSOT in the $x-z$ plane of the external magnet. The nSOT is tilted at angle $\theta\approx 12.5^\circ$ relative to the vertical. There is also a small tilt $\alpha \approx 2^\circ$ due to imperfect mounting of the sample, causing an in-plane field $B_x$ to inadvertently apply a finite effective perpendicular field to the sample. c, SEM image of a representative nSOT similar to the one used in this experiment. d, Measured d$V_{fb}$/d$B_z$ of the nSOT as a function of applied bias voltage $V_{\rm{bias}}$ to the nSOT measurement circuit and applied magnetic field in the $z$-direction, $B_z$, at zero in-plane magnetic field, $B_x$. Jumps at high bias are artifacts due to the feedback circuit. e, Measured d$V_{fb}$/d$B_x$ of the nSOT as a function of $V_{\rm{bias}}$ and $B_x$ at $B_z = 0$; the longer period of oscillation is due to the tilt of the nSOT. f, Measured d$V_{fb}$/d$B_z$ of the nSOT as a function of $V_{\rm{bias}}$ and $B_z$ at fixed $B_x = -46$ mT. The black bar denotes the bias and range of magnetic fields for the measurements. g, Representative sensitivity of the nSOT as a function of frequency.
• Figure 2: Spatial imaging of vortices and direct measurement of $B_{c1}$.a, Spatial images of $\Delta B$ at different values of applied $B_z$. Scale bar is 1 $\rm{\mu}$m. $n_e$ and $D$ are chosen at the center of the superconducting region (solid circle in Fig. \ref{['fig:fig1']}c). b, Schematic of magnetic field focusing due to screening currents in the pure Meissner state ($B_z<B_{c1}$, left) and in the presence of a vortex ($B_z>B_{c1}$, right). c, $\Delta B$ as a function of $B_z$ measured at the spatial position indicated by the circle in panel a. Near $B_z = 0$, the behavior is linear consistent with a constant diamagnetic susceptibility. At $B_z = \pm B_{c1} \approx 200$$\rm{\mu} T$, the flux changes sharply due to the appearance of a localized vortex. d, $\Delta B$ measured along the dotted line in panel a, for variable $B_z$. $X$ denotes the coordinate along the dotted line with $X = 0$ in the upper left of the image in panel a. e, Numerical derivative $\rm{d}(\Delta B)/\rm{d} B_z$ for the data in panel d. Orange regions denote constant negative susceptibility, while purple regions are associated with the entry of a new localized vortex to the sample.
• Figure 2: Boxcar measurement.a, Illustration of square-wave gate modulation applied to the bottom gate. b, Illustration of the expected nSOT signal corresponding to the gate modulation considering finite settling time effects.
• Figure 3: Relationship of spin-canted ferromagnetism and superconductivity.a, $\Delta B$ as a function of $n_e$ and $D$ for a spatial position with sensitivity to both in- and out-of-plane magnetic moments (see Methods and Extended Data Fig. \ref{['fig:SuppFigSpinCanting']}), measured at $T=210$ mK and $B_z = 750 \mu$T. b, Schematic phase diagram showing overlap of superconductivity (SC), the spin-canted ferromagnetism, and the valley-imbalanced ferromagnetism (VI). The dashed contour outlining the SC at 70 mK (light cyan region in Fig. \ref{['fig:fig1']}b) is shown as a guide to eye. c, High-resolution linecut of $\Delta B$ along the trajectory marked in panel a, which is symmetrized and anti-symmetrized in $B_z = \pm 100 \mu$T. The antisymmetric contribution chiefly shows the Meissner response in the superconducting pocket, whereas the symmetric contribution highlights the in-plane moments (unaffected by the direction of $B_z$) coming from the ordered, spin-canted phase. d, Schematic of $m_\parallel$ and $m_\perp$ as a function of $n_e$, based off of the data in panel c. Insets show the spin -ordering of the $K$ (orange) and $K'$ (green) valley states, which are canted at higher $|n_e|$ and spin-valley locked at low $|n_e|$.