Defect-induced magnetism and Yu-Shiba-Rusinov states in twisted bilayer graphene

Atomic defects have a significant impact on the low-energy properties of graphene systems. By means of first-principles calculations and tight-binding models we provide evidence that chemical impurities modify both the normal and the superconducting states of twisted bilayer graphene. A single hydrogen atom attached to the bilayer surface yields a triple-point crossing, whereas self-doping and threefold symmetry breaking are created by a vacant site. Both types of defects lead to time-reversal symmetry breaking and the creation of local magnetic moments. Hydrogen-induced magnetism is found to exist also at the doping levels where superconductivity appears in magic-angle graphene superlattices. As a result, the coexistence of superconducting order and defect-induced magnetism yields in-gap Yu-Shiba-Rusinov excitations in magic-angle twisted bilayer graphene.

Figure 1

Sketch of the two defects considered in tBLG: (a) a carbon vacancy and (b) a chemisorbed hydrogen atom. Blue represents the upper layer; red, the lower layer; and green, the hydrogen atom. Both defects create impurity bands close to the charge neutrality point that give rise to local magnetic moments, substantially impacting the low-energy properties of twisted bilayers.

Figure 2

Real-space redistribution of the charge density of two different defects in twisted bilayer graphene at $\alpha =5.{09}^{\circ }$. Top view of the charge distribution of (a) a single vacancy and (b) a single H atom. A single H atom creates a triangular distribution around the defect, while the vacant site enhances the net charge localization at the $\sigma$ dangling bonds. Isosurfaces correspond to charge densities of ${10}^{-3}{e}^{-}{\AA }^{-3}$ (a, b) and ${10}^{-2}{e}^{-}{\AA }^{-3}$ (c).

Figure 3

(a) First-principles paramagnetic electronic band structure of a bilayer graphene with a twist angle of $\alpha =5.{09}^{\circ }$ and a C atom vacancy. Spin-polarized calculations yield a spin splitting of the defect-induced band. (b) A vacancy in the AB stacking zone yields a magnetic moment of $1.5{\mu }_B$, which is lowered to $0.9{\mu }_B$ when one electron is removed. A divacancy exhibits a paramagnetic configuration and an empty nearly flat state at the Fermi level. In all four cases the impurity state exhibits at the $\mathrm{\Gamma }$ point an energy in between those of the two Dirac cones.

Figure 4

(a) First-principles paramagnetic band structure of monohydrogenated $\alpha \approx 5^{\circ }$ tBLG. When an H atom is attached to a C atom in the AB (b) and AA (c) regions, a local magnetic moment arises in the system, lifting the original spin degeneracy. (b, c) Unbalanced occupation of the nearly flat band in the up/down channels, yielding the local magnetic moment mentioned above. In this large-angle regime, $\alpha \approx 5^{\circ }$, removing one electron from the system leads to a nonmagnetic and gapped configuration, with an empty electronic flat state in the vicinity of the Fermi level.

Figure 5

(a) Band structure of monohydrogenated tBLG with a twist angle $\alpha \approx 3^{\circ }$, showing the emergence of a flat band with a triple-point crossing. (b) The self-consistent band structure exhibits the emergence of an exchange splitting as a consequence of interactions. (c) Real-space representation of the local density of states at the Fermi level in (a). (d) Magnetism arising from the impurity state obtained after including interactions.

Figure 6

(a, c, e) Local density of states and (b, d, f) band structure for tBLG with an H atom on the surface calculated with the low-energy tight-binding model for $\alpha \approx 1.3^{\circ }$. In (a) and (b), the H atom is in the AA stacking region; in (c) and (d), in the AB region; and in (e) and (f), in the AB/BA interface. The color code of the band structures (b, d, f) reflects the localization of the state in the supercell. A triple-point crossing at the Fermi level is observed in all three cases.

Figure 7

(a) Noninteracting band structure of a single vacancy in the AB region for a twist angle of $\alpha \approx 1^{\circ }$ and a doping of two holes with respect to charge neutrality. (b) Exchange splitting arises when electronic interactions are included. (c) The noninteracting band structure opens a gap when a pairing $\mathrm{\Delta }$ is included. (d) The combined effect of the exchange interactions and superconductivity is the emergence of a band with an energy lower than $\mathrm{\Delta }$, signaling the emergence of Shiba-Rusinov states in the tBLG. (e) The self-consistent magnetization in (b) and (f) shows the local density of states of the in-gap bands in (d), highlighting that the in-gap Yu-Shiba-Rusinov state inherits the spatial distribution of the impurity-induced magnetic moment. We took $U=2t$ and $\mathrm{\Delta }=0.02t$.