Recently, gate-defined Josephson junctions in magic angle twisted bilayer graphene (MATBG) were studied experimentally, and highly unconventional Fraunhofer patterns were observed. In this work, we show that an interaction-driven valley-polarized state connecting two superconducting regions of MATBG would give rise to a long-sought-after purely electric controlled ${\phi }_0$-junction in which the two superconductors acquire a finite phase difference ${\phi }_0$ in the ground state. We point out that the emergence of the ${\phi }_0$-junction stems from the valley-polarized state which breaks time-reversal symmetry, and trigonal warping effects which break intravalley inversion symmetry. Importantly, a spatially nonuniform valley-polarization order parameter at the junction can explain the key features of the observed unconventional Fraunhofer patterns. Our work explores the novel transport properties of the valley-polarized state, and we suggest that gate-defined MATBG Josephson junctions could realize the first purely electric controlled ${\phi }_0$-junctions.

• Received 16 February 2022
• Accepted 10 March 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.023029

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society