SUPERCONDUCTIVITY: INTERTWINING OF CORRELATION AND TOPOLOGY

Superconductivity in twisted bilayer graphene was observed at the magic angle of approximately θ ≈ 1.1°[6], with the critical temperature up to 2 K at anomalously low carrier density close to the half filling (Fig. 5a). In this section, we discuss how the correlation and band topology reshape the unconventional superconducting behaviors in MATBG.

_Figure 5.: Signatures of unconventional superconductivity in MATBG: (a) Superconducting phase diagram in a carrier density-temperature mapping plot [6]. (b) Logarithmic plot of critical temperature TC versus Fermi temperature TF for various superconductors, showing the superconductivity in MATBG is in the strong-coupling limit [6]. (c) Power-law fit exponent n of the temperature-dependent shift in resonant frequency due to varying superfluid across the entire superconducting dome in both electron and hole-doped regime in MATBG, showing anisotropic superconducting gap in the fermi liquid framework [91]. (d) Tunneling spectrum on MATBG in the superconducting state, showing V-shaped spectra that can be fitted using the model quasiparticle density of states (DOS) for a nodal superconductor [93]. (e) Superfluid weight as a function of superfluid density [87]. The black dash line shows calculated \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}_s}$\end{document} using the conventional band dispersion relation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}_s}( T ) = {{{\mathrm{e}}}^2}{{n}_s}( T )/{{m}^*}$\end{document}. The red line and green dashed line denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}s}$\end{document} extracted with the measured critical supercurrent density and a mean-field theory that considers the interaction-driven quantum geometry.

Non-BCS superconductivity

In a conventional weak-coupling BCS superconductor, electrons pair with a coherence length larger than the interparticle distance and at an energy window close to the Fermi level. The phonon-mediated superconducting pairing energy in BCS theory that determines the superconducting transition temperature T_C is much smaller than the Fermi energy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{E}{\mathrm{F}}}$\end{document}, yielding the result of _T_C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\ \ll$\end{document} T_F (here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{T}{\mathrm{F}}}$\end{document} is the Fermi temperature). Given the extremely low superfluid density (due to the reset state after the half filling) and low Fermi energy (due to the flat band dispersion), superconductivity in MATBG is expected to be strongly coupled. The coherence length (ξ ≈ 50 nm at optimal doping) and averaged interparticle distance are found to be at the same order, revealing the superconducting state in MATBG appears in the strong-coupling regime of the BCS to Bose–Einstein condensate (BEC) crossover. In the Uemura plot which compares the superconducting transition temperature (_T_C) to the estimated Fermi temperature (_T_F), conventional weak-coupling BCS superconductors exhibit ratio of _T_C/_T_F ≪ 1, as shown in Fig. 5b whereas most unconventional superconductors, such as cuprates, heavy fermion systems, and organic materials fall in the range of 0.01–0.05 [86]. MATBG shows a _T_C/_T_F ratio of about 0.08 at optimal doping, suggesting that its superconductivity likely arises from electron correlation rather than merely from conventional phonon-mediated BCS pairing [6,87].

Other unconventional features further distinguish the superconducting state of MATBG from that of conventional isotropic BCS superconductors. One prominent example is the emergence of electronic nematicity in both the normal and superconducting state, arising from the spontaneous breaking of the underlying lattice symmetry due to electronic correlation [88]. In alternating magic angle twisted trilayer graphene (MATTG), the preferred direction of superconducting transport aligns with the principal axis of the metallic phase that has the highest resistivity, while the strange metal behavior is oriented along the principal axis with the lowest resistivity [89]. In addition, MATTG also exhibits superconductivity that violates the Pauli limit by a factor of 2–3, indicating possible spin-triplet superconductivity [90].

Further evidence for possible unconventional superconductivity in MATBG and MATTG was obtained through microwave circuit quantum electrodynamics, where superfluid stiffness \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\rho }_s}$\end{document} can be measured via the probed kinetic inductance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{L}_K}$\end{document} [91,92]. In superconductors, the superfluid stiffness \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\rho }_s}$\end{document} is intimately connected to the superconducting pairing symmetry and gap structure due to quasiparticle spectrum at a finite temperature. The observed power-law temperature dependence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\rho }_s}$\end{document} and nonlinear Meissner effect in the current-bias dependence contradicts with the exponential dependence in isotropic s-wave superconductors, as shown in Fig. 5c, indicating an anisotropic superconducting gap in MATBG [91]. Both signatures indicate nodal structures in the superconducting order parameter, in contrast to conventional BCS superconductors where the strength of Cooper pairing is characterized by the isotropic superconducting gap in momentum space.

Other than conventional electrical transport measurements, STM measurements in MATBG and MATTG systems have also provided compelling evidence for anisotropic nodal superconductivity, pointing toward an unconventional superconducting pairing mechanism [93,94]. In MATBG, tunneling spectra near half-filling reveal a distinct V-shaped gap structure in the differential conductance, shown in Fig. 5d, indicative of nodes in the superconducting gap function, where quasiparticle excitation remains gapless along specific momentum directions. In MATTG, STM uncovers an intriguing evolution from a U-shaped to a V-shaped gap as the filling is tuned away from ν ≈ −2 toward ν ≈ −2.3. This transformation signals a change from a fully gapped state to gapless paired states, or a transition from BEC and BCS phases with a single nodal order parameter [94]. Moreover, both systems exhibit a pseudogap regime above the superconducting transition temperature as well as particle-hole asymmetry, further support the role of strong electronic interactions and spontaneous symmetry breaking. Recently, combined tunneling spectroscopy and transport measurements on MATTG reveal a V-shaped tunneling gap that directly links to the superconducting state observed in transport. The tunneling spectra show a linear gap-filling with increasing temperature and magnetic field, consistent with a nodal superconducting order parameter [95]. Crucially, these phenomena are highly sensitive to the relative alignment between the graphene layers and hBN, consistent with observations in transport measurements [9,10]. Both the pseudogap and superconductivity are absent when MATBG is commensurately aligned with the hBN substrate, suggesting that the structural characteristics and/or the C2zT symmetry of unaligned MATBG are required for stabilizing these ground states. On the other hand, misalignment appears to preserve or even enhance nodal superconducting features, likely by preserving underlying symmetries or enhancing valley coherence. These findings point to a rich interplay between twist angle, band topology, substrate alignment, and electron interactions in shaping the symmetry and superconducting order parameter in moiré graphene systems.

Quantum-geometry enabled superconductivity

The superfluid weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}_s}$\end{document}, which characterizes the superfluid phase stiffness in a superconductor, is conventionally believed to be associated with electronic kinetic energy, i.e. the band dispersion. In a parabolic band dispersion, the superfluid weight contributed by electronic kinetic energy is approximated to be:

(5) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} {{D}_s} = {{D}_{s,\mathrm{disp}}}( T) = {{{\mathrm{e}}}^2}{{n}_s}( T)/{{m}^*}, \end{eqnarray*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{n}_s}( T )$\end{document} is the superfluid density and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{m}^}$\end{document} is the effective mass of carriers. Since Fermi velocity is vanishing and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{m}^}$\end{document} is extremely large in flat bands, the superfluid weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}_s}$\end{document} of MATBG is expected to be small. Small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}s}$\end{document} yields a low upper bound of Berezinskii–Kosterlitz–Thouless (BKT) transition temperature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{T}{{\mathrm{BKT}}}}$\end{document} according to the Nelson–Kosterlitz criterion:

(6) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*}\frac{{{{\hbar }^2}{{D}_s}\left( {T = 0} \right)}}{{8{{{\mathrm{e}}}^2}{{k}_{\mathrm{B}}}}} \ge \frac{{{{T}_{{\mathrm{BKT}}}}}}{\pi }, \end{eqnarray*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{k}{\mathrm{B}}}$\end{document} is the Boltzmann constant [96]. Through Schwinger-limited nonlinear transport analysis from which the effective mass \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{m}^*}$\end{document} and thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}{s,\mathrm{disp}}}$\end{document} can be obtained, the estimated upper bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{T}{{\mathrm{BKT}}}}$\end{document} is much smaller than experimental results [87]. The association between superconductivity and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}s}$\end{document} can be also reflected through the behavior of critical 2D supercurrent density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{J}{\mathrm{c}}}$\end{document} and kinetic inductance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{L}{\mathrm{K}}}$\end{document} according to the following relations:

(7) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} {{D}_s}\left( {T = 0} \right) = \frac{{2\pi {{J}_{\mathrm{c}}}\xi }}{{{{{\mathrm{\Phi }}}_0}}}, \end{eqnarray*}\end{document}

(8) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*} {{L}_{\mathrm{K}}} = 1/{{D}_s}, \end{eqnarray*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{{\mathrm{\Phi }}}0} = h/2{\mathrm{e}}$\end{document} is the flux quantum. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}s}$\end{document} is experimentally obtained in superconducting twisted graphene systems via measurements of critical supercurrent density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{J}{\mathrm{c}}}$\end{document} and kinetic inductance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{L}{\mathrm{K}}}$\end{document} [87,91,92,97]. However, both probes show much larger values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}_s}$\end{document} than those derived only from the flat band dispersion (Fig. 5e).

This discrepancy is now understood in terms of the superfluid weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}s}$\end{document}, which depends on both the band dispersion and the quantum geometry of the electronic states [98–100]. The total superfluid weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}s}$\end{document} is expressed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}s} = \ {{D}{s,\mathrm{disp}}} + {{D}{s,\mathrm{geom}\ }}$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}{s,\mathrm{geom}\ }}$\end{document}arises from nontrivial quantum geometry and is proportional to the quantum metric in a multiband system. This quantum metric manifests as the overlapping of Wannier functions in neighboring lattices and hence enables the transport of quasiparticles like the Cooper pairs. In MATBG, flat bands exhibit highly nontrivial band topology with nonzero Chern number and quantum metric, creating extra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}_{s,\mathrm{geom}\ }}$\end{document} contribution and pronounced \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}s}$\end{document} even when its conventional dispersion part \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{D}{s,\mathrm{disp}}}$\end{document} is strongly suppressed. Superconductivity in MATBG is therefore proposed to be enhanced by nontrivial quantum geometry of flat bands, even though their flatness would otherwise favor Cooper-pair localization. The contribution of quantum geometry to _T_C can account for the large _T_C/_T_F ratio that violates the noninteracting BCS theory.