Flat band surface state superconductivity in thick rhombohedral graphene

Abstract

Rhombohedral multilayer graphene has recently emerged as a rich platform for studying correlation driven magnetic, topological and superconducting states. While most experimental efforts have focused on devices with N$\leq 9$ layers, the electronic structure of thick rhombohedral graphene features flat-band surface states even in the infinite layer limit. Here, we use layer resolved capacitance measurements to directly detect these surface states for $N\approx 13$ layer rhombohedral graphene devices. Using electronic transport and local magnetometry, we find that the surface states host a variety of ferromagnetic phases, including both valley imbalanced quarter metals and broad regimes of density in which the system spontaneously spin polarizes. We observe several superconducting states localized to a single surface state. These superconductors appear on the unpolarized side of the density-tuned spin transitions, and show strong violations of the Pauli limit consistent with a dominant attractive interaction in the spin-triplet, valley-singlet pairing channel. In contrast to previous studies of rhombohedral multilayers, however, we find that superconductivity can persist to zero displacement field where the system is inversion symmetric. Energetic considerations suggest that superconductivity in this regime is described by the existence of two independent surface superconductors coupled via tunneling through the insulating single crystal graphite bulk.

Figures (13)

• Figure 1: Surface states in multilayer rhombohedral graphite.(a) Band structure and (b) density of states for rhombohedral 13-layer graphene calculated within the self-consistent Hartree approximation at charge neutrality for different values of $D$. (c) Layer resolved density of states for $D = 0.73 \,\text{V/nm}$ and $\mu = 30$, $5$, $-28$, and $-60 \,\text{meV}$. (d) Measurement schematic for layer resolved capacitance. An AC voltage is applied to the R13G through an electrical contact and the current on the top ($A_t$) and bottom ($A_b$) are measured with $C_{t(b)}\equiv A_{t(b)}/V_{AC}$. (e) Measured $C_t$ and (f)$C_b$ at T=1.5K in Device 1, normalized by the geometric capacitance $c$ of the two gates. Between the dashed green lines, $C_{t(b)}\gtrapprox c$, indicating the presence of a high compressibility surface state on the top (bottom) layer. (g)$C_{sym}$ and $C_{asym}$ at $D=0$. Lower insets show Hartree band structure and calculated layer density of states for $\mu=0$, $\pm 30 \,\text{meV}$ for $D=0.05$ V/nm. The upper inset shows an equivalent circuit schematic in the dual-surface regime. (h)$C_{sym}$ and $C_{asym}$ at $D=0.72 \,\text{V/nm}$. (i)$C_{sym}$ as a function of $n_e^0$ and $D$. Regions of single-surface, dual-surface, and bulk layer polarization are indicated. (j)$C_{asym}$ as a function of $n_e^0$ and $D$. In a simplified four-plate capacitor model, $C_{asym}\propto dp/dn_e^0$; red and blue regions thus denote where states at the Fermi level are top or bottom layer polarized.
• Figure 2: Spin and valley ferromagnetism and surface state superconductivity.(a) Penetration field capacitance of Device 2 measured at T=100mK. (b) Longitudinal resistance $R_{xx}$ in Device 2 measured at $T=20\,\text{mK}$. Several superconducting states are indicated. (c) Hall measurements at $20\,\text{mK}$ (blue), $6\,\text{K}$ (green) and $8\,\text{K}$ (yellow). The three curves are measured at $(n_e^0,D,T)=(1.77 \times 10^{12}\,\text{cm}^{-2}, 0.89\,\text{V/nm}, 20\,\text{mK})$, $(n_e^0,D,T)=(1.87 \times 10^{12}\,\text{cm}^{-2}, 0.85\,\text{V/nm}, 6\,\text{K})$ and $(n_e^0,D,T)=(2.21 \times 10^{12}\,\text{cm}^{-2}, 0.83\,\text{V/nm}, 8\,\text{K})$, indicated by the colored dots in panel b. (d) Temperature dependent $R_{xx}$ in SC1, measured at $D = 0.47\,\text{V/nm}$ and $B_\parallel=0.9T$. (e) Fringe magnetic field arising from in-plane magnetization measured by SQUID-on-tip microscopy at T=350mK. As shown in the inset, the out-of-plane fringe field $B$ is measured over a sample edge, once with applied external field $(B_\parallel,B_\perp)=(20 \,\text{mT},22\,\text{mT})$ and once with $(B_\parallel,B_\perp)=(-20 \,\text{mT},20\,\text{mT})$; the plotted data is the difference between the two measurements. Overlays (gray dashed lines) indicate single-surface, dual-surface, and bulk regions extracted from Fig. \ref{['fig:fig1']}e and f. (f) Top half: schematic phase diagram of symmetry breaking and superconductivity. Spin unpolarized phases are indicated in purple; spin polarized phases in pink, and valley polarized phases in tan. Superconductors are rendered in cyan and labeled. The cyan region at low $n_e^0$ and intermediate $D$ has high resistivity and large negative compressibility, and may be a Wigner crystal like state. Bottom half shows the spin imbalance density calculated within a simplified two-flavor Hartree-Fock model described in the methods.
• Figure 3: Field induced superconductivity and Pauli limit violation in SC1.(a)$R_{xx}$ at B=0 and $B_\parallel=0.75 T$. (b) Differential resistance $dV/dI$ at $n_e^0=0.05 \times 10^{12}\,\text{cm}^{-2}$ and $D=0.47 \,\text{V/nm}$ as a function of applied current at different temperatures for $B_\parallel=0.9T$. $T_C\approx 90mK$ as defined by the onset of the resistivity drop. (c)$dV/dI$ at $n_e^0=0.30 \times 10^{12}\,\text{cm}^{-2}$ and $D=0.51 \,\text{V/nm}$ as a function of $B_\perp$ at fixed $B_\parallel=0.9 \,\text{T}$. (d)$R_{xx}$ at small bias as a function of $n_e^0$ and $B_\parallel$ for $D=0.47V/nm$. Superconductivity at this displacement field is induced by a magnetic field. $T_C$ never exceeds 100mK, implying a Pauli limiting field of $B_p\approx 140mT$ which is exceeded by at least a factor of 7 at the maximum field available. (e) Penetration field capacitance as a function of $n_e^0$ and $B_\parallel$ in the same range as panel d. The bright line is associated with a first order transition to the spin polarized state detected in the magnetization measurements of Fig. 2e. The domain of superconductivity is overlaid and occurs on the spin unpolarized side of the transition.
• Figure 4: Dual-surface superconductivity.(a) Detail of $R_{xx}$ near SC2 and SC3 at $B_\parallel=0 \,\text{mT}$. (b)$R_{xx}$ as a function of temperature at $D=0\,\text{V/nm}$ and $D=0.02\,\text{V/nm}$. (c) Magnetometry at $B_{||} = 100$ mT, $B_\perp = 7$ mT, $T =$ 350mK in Device 1, measured over the edge of the device where the fringe field arising from in-plane magnetic moments dominates that arising from out of plane moments. (d) Derivative of measured magnetic field with respect to electron density, at $D=0$ and $0.04 \,\text{V/nm}$. (e) Schematic phase diagram showing regions of single- and dual-surface spin ferromagnetism as well as single- and dual-surface superconductivity.
• Figure S1: Detailed circuit schematic for capacitance measurements. DC voltages are denoted $\text{V}_i$ while AC voltages are denoted $v_i$. $\text{V}_{1-4}$ are used to tune the impedance of the HEMTs (model FHX35X). $\text{V}_{t/b/s}$ are applied in combination to control $n_e^0$ and $D$ within the device. AC excitations $v_{t/b/s}$ are applied to measure the relevant capacitance components. The AC signals are first nulled by balancing them with $v_{1/2}$ through the reference capacitors $c_{ref1/2}$. The resulting output signal from sweeping $n_e$ and $D$ is read out at room temperature as a voltage ($v_{\text{out}}$).