Twist-angle tunable Josephson junctions in three-dimensional superconductors

TL;DR

The paper addresses twist-induced Josephson coupling in 3D superconductors without a conventional barrier, exploring how twist-angle and Fermi-surface geometry determine transport. It develops a self-consistent Bogoliubov-de Gennes framework within an effective moiré description to compute perpendicular supercurrents and current-phase relations. Key findings include a phase jump at the twisted interface forming a twist Josephson junction, a finite perpendicular current persisting even when Fermi surfaces are momentum-disconnected, and a tunable Jc/Jd with twist angle, with Gamma-pocket presence further enhancing the ratio. The work suggests twist-angle control as a route to tunable Josephson devices and provides momentum-resolved insights relevant to NbSe2-like multiband superconductors, with potential extensions to barrier and ferromagnetic junctions.

Abstract

We theoretically investigate the superconducting phase and perpendicular Josephson supercurrent in twisted three-dimensional (3D) superconductors, where two layered 3D materials are stacked with a relative twist. We formulate the Bogoliubov-de Gennes Hamiltonian and develop a self-consistent method to calculate the superconducting order parameter and the resulting supercurrent. Applying this framework to a toy model with Fermi surfaces located near the Brillouin zone corners, we demonstrate a phase discontinuity at the twisted interface, indicating that a Josephson junction is formed purely by the twist. Our calculations reveal that the interface supports a finite critical current even when the Fermi surfaces of the two superconductors are completely separated, unlike in the case of a twisted normal-metal interface. We further show that the critical current can be effectively controlled by the twist angle, transitioning from a high-transparency regime at small angles to a low-transparency regime at larger angles.

Figures (11)

• Figure 1: Schematic illustration of a twisted three-dimensional superconductor with a twist angle $\theta$. The black arrow indicates a possible Josephson supercurrent flowing in the perpendicular direction.
• Figure 2: (a) Three-dimensional Fermi surface consisting of a cylindrical region with caps at $k_z = \pm \pi/c$. The Fermi energy is $E_\mathrm{F} = 0.5 \, \mathrm{eV}$. (b) Fermi surface projected onto the $k_x k_y$ plane, appearing as a filled circle. The dashed lines represent the edge of the Brillouin zone. (c) Schematic of the 3D Brillouin zone, depicted as a hexagonal prism. High-symmetry points such as $\Gamma$ and $K_\pm$ are indicated.
• Figure 3: Side view of a twisted three-dimensional superconductor, where the lower (upper) slabs are shown by the orange (green) layers. The periodic boundary condition is employed for the calculation of the perpendicular supercurrent. The system contains $2N$ bilayer units, which are labeled by an integer $n$ (see the text). The twisted interfaces are located between $n = N$ and $N+1$ with the angle $\theta$, and between $n = 1$ and $n = 2N$ with the angle $-\theta$.
• Figure 4: (a) Brillouin zones of the lower (yellow) and upper (green) honeycomb lattices in the extended-zone scheme. The coupling wavevectors $\vb*{q}_1$, $\vb*{q}_2$, and $\vb*{q}_3$ are indicated by red arrows (see the text). A moiré Brillouin zone is defined by a small black hexagon near $\vb*{q}_1$, which is magnified in (b). (b) Misaligned Fermi surfaces of the lower and upper 3D metals around the $K_+$ point are illustrated as red and blue circles, respectively. With the moiré Brillouin zone (black hexagon), the moiré reciprocal vectors $\vb*{G}^\mathrm{M}_1$, $\vb*{G}^\mathrm{M}_2$ and high-symmetry points are shown as well.
• Figure 5: (a) Variation of the superconducting phase (on the non-dimer sites) along the stacking direction. The phase twist $\delta\varphi$ is varied, while the twist angle is fixed to $\theta=10^\circ$. $\delta\varphi_\mathrm{int}$ denotes the phase jump at the twisted interface (between layers $20n$ and $20n+1$). (b) Similar plots for various twist angles $\theta$, with fixing $\delta\varphi = \pi$. (c) Misaligned Fermi surfaces projected onto the $k_x k_y$ plane for the three twist angles. The red (blue) circle represents the Fermi surface of the lower (upper) part of the system.