Twisted Bilayer Bi2Sr2CaCu2O8: Topological Superconductivity

• Twisted bilayer Bi2Sr2CaCu2O8 is a van der Waals heterostructure formed by stacking rotated cuprate superconductors to enable phase-sensitive control of interfacial superconductivity.
• The system exhibits a sharp angular dependence of the Josephson critical current, following a |cos 2θ| behavior that diminishes near 45° due to destructive interference.
• Both experimental and theoretical studies reveal that twist-engineered Bi2Sr2CaCu2O8 hosts chiral d+id superconductivity with emergent Majorana edge modes and vestigial charge-4e phases.

Twisted bilayer Bi₂Sr₂CaCu₂O₈ (BSCCO or Bi-2212) denotes a class of van der Waals heterostructures formed by exfoliating, rotating, and vertically stacking two thin single crystals of the cuprate superconductor Bi₂Sr₂CaCu₂O₈₊ₓ with a well-defined interlayer twist angle θ about the crystallographic c-axis. This architectural motif allows phase-sensitive control of the interfacial superconducting state, providing a platform for manipulating Josephson coupling anisotropy, probing unconventional order parameter symmetry, and accessing emergent chiral topological phases at high critical temperatures. The system combines the nodal d_{x²–y²} superconductivity intrinsic to BSCCO with twist-engineered Josephson tunneling, thus enabling experimental realization and theoretical modeling of phenomena inaccessible in conventional bulk or interface-free geometries.

1. Fabrication and Structural Characterization

Atomically clean twisted BSCCO bilayers are fabricated using two principal protocols, both executed in inert (Ar) atmosphere to prevent surface degradation. In the cryogenic dry-transfer technique, an optimally doped BSCCO flake (thickness 30–60 nm) is exfoliated onto SiO₂/Si, cooled to –90 °C, then cleaved with a PDMS stamp; the substrate is rotated by the target θ and the separated half is reassembled, forming a sharp BiO–BiO interface. The stack is encapsulated with h-BN flakes to preserve superconductivity, followed by deposition of Au/AuCr edge contacts via stencil masks under high vacuum and low temperature, achieving contact areal resistance ≤ 50 kΩ·μm² (Martini et al., 2023, Zhao et al., 2021). Alternative microcleave-and-stack approaches operate similarly, using rapid assembly to avoid contamination and post-stacking high-T annealing in O₂ to ensure atomically sharp tunneling barriers, verified by cross-sectional STEM and EDS (Lee et al., 2021). Throughout, no moiré pattern forms at 45° due to the commensurability of the D₄ lattice.

2. Josephson Coupling: Angle Dependence and Transport

The Josephson critical current density j_c and characteristic voltage IcRₙ are found to exhibit a strong dependence on the interlayer twist angle θ. At θ = 0°, the junctions exhibit j_c ≈ 0.9–1.2 kA/cm² and IcRₙ ≈ 6–23 mV at T = 5–10 K—values commensurate with bulk intrinsic c-axis Josephson junctions (Martini et al., 2023, Lee et al., 2021, Zhao et al., 2021). As θ increases, j_c and IcRₙ follow a |cos 2θ| dependence, falling precipitously by approximately two orders of magnitude as θ approaches 45°, with IcRₙ(θ ≈ 45°) ∼ 0.05–0.07 mV (Zhao et al., 2021). This angular functional form arises from the nodal d_{x²–y²} symmetry of the superconducting order parameter, which induces destructive interference for direct Cooper pair tunneling at 45°, as accounted for in tight-binding and Green's function-based models including incoherent tunneling effects (Lee et al., 2021, Martini et al., 2023). No systematic T_c–θ dependence is seen, with sharp superconducting transitions at T_c ≈ 74–90 K and preserved interface order, indicating robust interfacial superconductivity as a function of twist angle (Martini et al., 2023, Zhao et al., 2021).

3. Theoretical Framework: d-wave, Chiral, and Topological States

At the single-pair level, Josephson coupling between twisted d_{x²–y²} superconductors is described by the current–phase relation

$$I(\varphi, \theta) = \sum_{n=1}^\infty I_n(\theta)\,\sin(n\varphi)$$

with the dominant contribution in coherent tunneling being $I_1(\theta) \propto \cos(2\theta)$ due to the sign change of the d-wave gap under 90° rotation (Martini et al., 2023, Lee et al., 2021). However, near θ = 45°, the first harmonic (n = 1) term vanishes by symmetry, allowing higher-order (n = 2) processes to dominate. This includes second-harmonic “co-tunneling” of two Cooper pairs (charge 4e), resulting in a non-vanishing supercurrent with current–phase relation $I(\varphi) = -I_2\,\sin(2\varphi)$ (Zhao et al., 2021).

Theoretical Ginzburg–Landau and Bogoliubov–de Gennes models predict that for twist angles θ near 45°, time-reversal symmetry can be spontaneously broken at the interface, stabilizing a fully gapped chiral d_{x²–y²} + id_{xy} superconducting state with nonzero Chern number (C = ±4 for spin-degenerate bilayers), protected chiral Majorana edge modes, and quantized thermal Hall conductance (Can et al., 2020, Haenel et al., 2022, Tummuru et al., 2022). The stability of this phase persists for a finite range $|\theta-45°| \lesssim 3-6°$ (experimentally and in models including disorder-mediated incoherent tunneling), with a minimum induced gap Δ_{min} ∼ 5–15 meV and T_c set by the bulk value (~90 K) at θ=45° (Haenel et al., 2022). This regime is experimentally accessible in Bi-2212 due to its high native gap and large interlayer coupling (Tummuru et al., 2022).

4. Topological, Chiral, and Exotic Vestigial Phases

Microscopically, the second-harmonic Josephson coupling at θ = 45° produces a 4e condensate, permitting vestigial charge-4e superconductivity above the chiral d+id transition temperature. Renormalization-group analyses show that as temperature increases, the fully chiral phase yields to either a charge-4e superconductor (with only the total phase quasi-ordered) or a “chiral metal” (with only the relative phase ordered), depending on the relative phase stiffness. Both vestigial phases lie above the mean-field d+id T_c but below the bare paring scale, and may be detected via half-integer Shapiro steps (indicative of 4e pairing), anomalous Fraunhofer interference, or Kerr rotation in transport and magneto-optical probes (Liu et al., 2023). The absence of a moiré superlattice at 45° yields a quasi-crystal with eightfold rotational symmetry and no translational invariance, further distinguishing this geometry at the group-theoretical level (Liu et al., 2023).

5. Experimental Signatures and Phase-sensitive Probes

Twisted bilayer Bi-2212 devices at or near θ=45° display several unambiguous signatures of unconventional interfacial states:

• Suppression of first harmonic critical current: Near full cancellation of Ic for θ = 45°, with finite residual current attributed to higher harmonics/co-tunneling (Zhao et al., 2021, Lee et al., 2021, Martini et al., 2023).
• Half-integer Shapiro steps and doubled periodicity Fraunhofer patterns: These are observed under microwave irradiation (half-quantum steps) and in perpendicular fields (corresponding to charge 4e transport and chiral order), confirming dominance of the sin(2φ) Josephson term (Zhao et al., 2021).
• Spontaneous time-reversal symmetry breaking and edge modes: Predicted by both mean-field and lattice models, the chiral d+id regime is characterized by a full spectral gap, quantized edge transport, polar Kerr rotation, and possible zero-bias conductance peaks in tunneling (Can et al., 2020, Haenel et al., 2022, Tummuru et al., 2022).
• Loop supercurrents and local magnetism: Variational cluster approximation (VCA) studies demonstrate that spontaneous interlayer loop supercurrents occur in regions of density imbalance, yielding chiral domains with circulating currents detectable via scanning SQUID and μSR (Bélanger et al., 2023).

6. Effects of Disorder, Doping Asymmetry, and Hubbbard Interactions

Experimental and theoretical work finds that disorder-induced incoherent tunneling (resulting from oxygen vacancies) narrows yet does not eliminate the topological chiral d+id wedge in the θ–Γ (incoherence) phase diagram. Typical experimentally relevant ranges for Bi-2212 are 42° ≲ θ ≲ 48° for the full chiral phase at low disorder (Haenel et al., 2022). An interlayer bias leading to doping asymmetry between the two layers further deforms the chiral phase region in density space, leading to a “crescent” region in the (n₁,n₂) plane where time-reversal symmetry breaking persists, tracked exactly by the spontaneous loop current operator (Bélanger et al., 2023). The presence of substantial Hubbard U enhances competition among chiral, charge-4e, and nodal phases as confirmed in variational cluster studies.

7. Outlook and Implications for High-Tc Twistronics

Twisted bilayer Bi-2212 provides an experimentally robust, tunable, and atomically precise platform for exploring twist-engineered Josephson phenomena, high-temperature topological superconductivity, and vestigial charge 4e or chiral metallicity (Lee et al., 2021, Martini et al., 2023, Zhao et al., 2021). The demonstrated angular control over Ic spanning two orders of magnitude, together with phase-sensitive edge, optical, and transport probes, allows direct tests of d-wave symmetry and time-reversal symmetry breaking. Integration into van der Waals heterostructures with other correlated oxides or pnictides will facilitate studies of proximity-induced topological phases, and the potential observation of quantized thermal Hall response and Majorana edge states at elevated temperatures (Can et al., 2020, Tummuru et al., 2022). Future directions include optimization of interlayer tunneling via strain/dielectric engineering, deterministic induction of higher-harmonic Josephson terms, and exploration of multilayer or superlattice extensions for designer quantum matter (Tummuru et al., 2022, Liu et al., 2023).