Moiré-enabled topological superconductivity in twisted bilayer graphene

The two-dimensional nature of van der Waals materials provides a unique playground to engineer emergent states [1–3]. Among others, their 2D nature allows to induce ferromagnetic exchange [4–8], superconductivity and spin–orbit coupling (SOC) [9–13] through proximity to other 2D materials. This versatility increases considerably the range of phenomena that can be explored in van der Waals materials. In particular, the combination of Rashba SOC, Zeeman splitting and superconductivity in van der Waals heterostructures can give rise to topological superconductivity [14–18]. These ingredients, difficult to find in a single material, can be engineered in a van der Waals heterostructure through the appropriate combination of 2D materials. The demonstration of topological superconductivity in CrBr₃/NbSe₂ heterostructures [18] featuring moiré effects [19] opens the possibility to use the interplay of moiré and impurities to probe unconventional quantum states [20–24]. Scanning tunneling microscopy was employed in the experiments [18, 19], showing that despite the presence of a monolayer ferromagnetic insulator atop, bulk and edge modes can be observed by tunneling through the insulator [18, 19].

Moiré patterns in twisted graphene multilayers allow engineering quantum states including superconductivity and correlated insulators [25–31], orbital ferromagnetism [28, 32, 33] and electronic cascades [34–37]. In twisted bilayer graphene (TBG) the moiré pattern opens gaps between low-energy narrow bands and high-energy wide bands, giving rise to a parabolic dispersion close to the Γ and the M points [38–41]. The large unit cell in small twist angle TBG allows gate-doping up to several electron/holes per moiré unit cell, making it possible to probe a doping range spanning the full flat bands and partially the higher energy bands. The TBG bandstructure is highly sensitive to external parameters such as pressure [42, 43], strain [44–46], induced SOC [47] and electric field [48–50]. The intrinsic correlation effects and the superconductivity, which originate in the flatness of the narrow bands, are restricted to angles very close to the magic value $\theta\sim 1.1^\circ$. However, the potential to engineer tunable states from parabolic dispersion at Γ both in low and high energy bands has remained relatively unexplored.

Here we demonstrate that a highly tunable topological superconducting state can be driven in a twisted graphene multilayer with TBG stacked between ferromagnetic and superconducting layers. The mechanism proposed here does not require tuning the twist angle to the flat band value, being generically applicable to twisted bilayers between 1.3 and 3 degrees. The upper limit for twist angle (${\lt}3^\circ$) ensures a large unit cell and the possibility of achieving large doping with a gate electrode. The lower limit (${\gt}1.3^\circ$) ensures that the intrinsic correlated and superconducting states of magic-angle TBG are not present and therefore do not interfere with the mechanism for topological superconductivity proposed here. In this range of twist angles, the band structure used in our work is a good starting point. Rashba SOC, ferromagnetic exchange and an s-wave superconductivity are induced in TBG through the proximity to the other 2D materials. The mini-gaps driven by the moiré pattern enable the appearance of topological superconductivity. Specifically, TBG allows generating topological superconducting states with Chern number up to $|C| = 6$, tunable with doping, strain, and electric field.

We consider a heterostructure with TBG encapsulated between a ferromagnetic insulator and an s-wave superconducting layer, figure 1(a). Breaking of the mirror symmetry at the interfaces between the different materials generates Rashba SOC in TBG. Although graphene has relatively weak SOC [51], SOC can get greatly enhanced via proximity effect to a substrate [9, 11, 52–55], which is the strategy we use here. For instance, transition metal dichalcogenides (TMDs) have been predicted to induce substantial Rashba SOC up to 15 meV in an adjacent graphene layer [10]. Predictions also indicate that ferromagnetic insulators like CrI₃ can similarly amplify SOC effects, along with significant magnetic exchange fields [4, 5]. In our model, the SOC interaction is directly included in the effective model of the twisted bilayer graphene, taking that the degrees of freedom of the ferromagnet and the superconductor are integrated out. In the real device, the SOC interaction will be mainly driven by the SOC of the substrate materials, both the superconductor and ferromagnet. The ferromagnetic layer induces an out-of-plane exchange spin splitting in TBG and the superconducting layer is responsible for an induced s-wave superconducting order parameter. We note that the different ingredients required to realize the Hamiltonian above have been experimentally demonstrated independently in a variety of van der Waals heterostructures [18, 56–63]. We note that an out-of-plane magnetic field will not be induced by the ferromagnetic layer, as such field vanishes for infinite two-dimensional ferromagnets according to classical electromagnetism [64] The resulting Hamiltonian is

$$\begin{equation} \mathcal{H} = \mathcal{H}_{0} + \mathcal{H}_J + \mathcal{H}_{R} + \mathcal{H}_{\text{SC}} \end{equation} \tag{ 1 }$$

Here H₀ is the unperturbed tight-binding Hamiltonian for TBG, $H_\textbf{J}$ and $H_\textbf{SC}$ account for the exchange fields and superconducting order parameter induced by proximity. The term $H_\textbf{R}$ accounts for the Rashba SOC that results from the proximity-induced SOC due to mirror symmetry breaking. We use an atomistic tight-binding Hamiltonian for TBG as $\mathcal{H}_{0} = \sum_{\langle i,j\rangle,s}t c^\dagger_{i,s}c_{j,s}+\sum_{{i,j},{s}}t^{\perp}_{i,j}c^\dagger _{i,s} c_{j,s}+ \sum_{i,s}\mu_0c^\dagger_{i,s}c_{i,s}$, where $s = \uparrow,\downarrow$ denotes the spin index, i labels sites at position $r_{i} = (x_i,y_i,z_i)$ located at $z_i = \pm d/2$ and µ₀ is an onsite potential controlling the chemical potential. The inter-layer hopping depends on the distance as $t^{\perp}_{i,j} = t_\perp[(z_i-z_j)^2/|\textbf{r}_i-\textbf{r}_j|^2]e^{-(|\textbf{r}_i-\textbf{r}_j|-d)/\lambda}$. The typical energy scales in twisted bilayer graphene are t = 3 eV and $t_\perp = 350$ meV [65] and we focus on a twist angle $\alpha = 1.6 ^\circ$ TBG [66].

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The ferromagnet and the superconductor induce respectively an out-of-plane exchange field in the top layer of TBG $\mathcal{H}_{J} = J \sum_{i\in T,s,s^{^{\prime}}} \sigma_z^{s,s^{^{\prime}}} c^{\dagger}_{i,s}c_{i,s^{^{\prime}}}$ and a pairing term in the bottom layer of TBG $\mathcal{H}_\textbf{SC} = \Delta \sum_{i \in B} c^{\dagger}_{i,\uparrow}c^{\dagger}_{i,\downarrow}+h.c.$ . Here $\sigma_z^{s,s^{^{\prime}}}$ is the spin Pauli matrix, and $i\in T$ and $i\in B$ respectively run over sites of the top and bottom layers. The mirror symmetry breaking produced by Rashba SOC is incorporated within both TBG layers and it is written as $\mathcal{H}_{R} = i\lambda_R \sum_{\langle i j \rangle, s s^{^{^{\prime}}}} \mathbf{d}_{ij} \cdot \mathbf{\sigma} ^{s,s^{^{^{\prime}}}} c^{\dagger}_{i, s}c_{j,s^{^{^{\prime}}}}$, where $\mathbf{d}_{ij} = (\mathbf{r}_i - \mathbf{r}_j)\times \hat z.$ We take $J = 0.018 t_\perp$, $\lambda_R = J$ and leave Δ and µ₀ as tuning parameters.

The bandstructure of the unperturbed $\alpha \approx 1.6 ^\circ$ TBG is shown in figure 1(b). Narrow bands are separated from the high energy bands by a gap at Γ, created by the moiré pattern. Parabolic bands are found at the Γ and M points of the narrow bands and at the Γ point of the high energy bands. The large moiré unit cell permits tuning experimentally the chemical potential to be around any of these points, doping the TBG with a gate voltage. In our setup, similar to the configuration in [26], a gate is a fundamental component to control the chemical potential of the twisted bilayer. Controlling both the interlayer bias and chemical potential of twisted multilayers is standard in these experiments, and this is the strategy we use in our model. The addition of the exchange field, induced by the ferromagnetic layer, and the Rashba SOC breaks the spin degeneracy, splitting the bands in figure 1(c). Rashba SOC creates a momentum-dependent splitting, while the exchange field opens up a gap at the time-reversal invariant points, creating a pair of helical states. Except for a rigid energy shift, in the absence of superconductivity, the bandstructure of this model does not depend on doping. The exchange proximity and superconductivity stem from the different top and bottom substrate materials. While both order parameters do compete, the twisted bilayer acting as a spacer prevents that the ferromagnet quenches the superconducting order. The superconducting order Δ and exchange J assumed in the model are the ones resulting after such competition is taken into account.

We first investigate the existence of topological superconductivity in close proximity to the Γ point of the lower narrow band, at a chemical potential µ₁ marked with a black dashed line in figure 1(c). By introducing a superconducting order parameter Δ, both band inversion and an energy gap E_g , smaller than Δ, emerge in the Bogoliubov-de Gennes electronic structure, see figure 1(d). We evaluate the Chern number C by integrating the Berry curvature over the entire Brillouin zone and obtain $C = -2$, showing the emergence of topological superconductivity. This Chern number stems from each of the two valleys, originating from the graphene K and K' points, contributing with $C = -1$.

A topological superconductor is expected to show C chiral gapless edge states. To confirm their presence we build a ribbon, finite along the y-direction and infinite in the x-direction. The lower energy states of a wide ribbon, discrete due to the finite width, are shown in figure 1(e), and the color gives the projection of the states in the upper edge, lower edge, and bulk of the ribbon. Bulk states, in black, are spread along the width of the ribbon. The topologically non-trivial nature of the electronic structure is confirmed by the emergence of gapless edge states within the gap. two co-propagating chiral states emerge in the upper edge (magenta) and two states along the lower one (cyan), as expected from the $C = -2$ Chern invariant.

Topological transitions can be detected by a gap closing. In figure 1(f) the gap E_g is shown as a function of µ₀ and Δ. Regions with $C = -2$ and C = 0 are separated by a zone with zero gap. Experimentally the system can be tuned through the topological transition via a gate voltage, which modifies the chemical potential. Alternatively, Δ can be changed with temperature or switching the superconducting layer. We now move the chemical potential to µ₂ close to the energy minima at the M point, blue dashed line in figure 1(c). For small Δ, we observe a Chern number C = 6 until a topological transition to a trivial state with C = 0 takes place at larger Δ in figure 2(a). The Chern number $C = -6$ is consistent with the chiral gapless edge states found in a ribbon, see figure 2(k). The different Chern numbers at µ₂ and µ₁ result from their different Fermi surfaces, figure 2(f) and inset in figure 1(c). At µ₂, C = 6 originates in the four pockets at K and K' two in each valley, and the two pockets at Γ, one in each valley. With µ₃, in figure 1(c), in the high energy bands, and $\Delta = 0.4 J$ we have found C = 6 originating from a complex Fermi surface centered at Γ and being again consistent with the ribbon edge states (not shown).

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Other topological superconducting states can appear by the application of strain in the sample. The strain is directly included in hopping terms of the Hamiltonian, such that t_ij is renormalized as $t_{ij}(1+s_{ij})$ where $s_{ij} = s_0 (\cos{(\theta_{ij}-\phi}))^2$, θ_ij is the angle between sites i and j, whereas φ is the polar angle measured with respect to the X direction parametrizing the direction of the applied strain. We assume that the same strain is applied in both graphene layers for concreteness. We focus on the influence of strain at µ₂, close to the energy minima at the M point, originally showing $|C| = 6$. The band structure subjected to a strain magnitude of $s_0 = 0.025$ applied along $\phi = \pi/4$, plotted in figure 2(b) in the absence of Δ, shows important differences with respect to the one in figure 1(c). Topological phase transitions between different topological phases appear in figure 2(d) as a function of strain magnitude. The origin of those transitions stems from an avalanche of Lifshitz transitions driven by the strain [46], as observed in the Fermi surfaces in figures 2(f)–(j). The phases with different Chern numbers show associated chiral modes, see figures 2(k)–(o). For a fixed magnitude of the strain, the band structure depends on the direction in which it is applied, see bands in the superconducting state in figure 2(c) with the strain of magnitude of $s_0 = 0.015$ applied under three distinct directions. Consequently, the sequence of topological transitions depends on the strain direction, which can be seen comparing figure 2(e) for $\phi = \pi/2$ with figure 2(d) for $\phi = \pi/4$. The tunability with strain allows to engineering of a plethora of topological states in a single twisted multilayer.

Using top and bottom gates it is possible to tune selectively the doping and an out-of-plane electric field. We now address the tunability of the topological phases with an out-of-plane electric field. Specifically, we focus on the high energy bands obtained at doping of $\pm (4 + \epsilon)$ electron per unit cell. An interlayer bias adds a term to the Hamilonian of the form $\mathcal{H}_V = V \sum_{i,s} c^\dagger_{i,s} c_{i,s} \tau_{i}$, with $\tau_i = \pm 1$ for upper/lower layer. Due to the broken inversion symmetry of structure, present already in the absence of an external electric bias, there can be an imbalance between the layers, that can be further changed with a displacement field. This allows controlling the electronic structure with an interlayer bias, as shown in figure 3(a) in the absence of superconductivity. The normal state band structure tunability suggests that topological phase transitions driven by a bias can emerge. We now include superconducting proximity and show in figure 3(b) the dependence of the superconducting gap as a function of the inter-layer electric bias V. The gap closings driven by the bias V suggest the appearance of a topological phase transition, confirmed by the calculation of the Chern numbers. Taking the specific case of $V = 0.1 t_\perp, 0.16 t_\perp$, we observe that the gap closing leads to a topological transition between a topological and a trivial state with different edge channels as shown in figures 3(c) and (d). Very much like in the case of strain and doping, the topological phase transitions are associated with Lifshitz transitions in the Fermi surface driven by the electric bias, as shown in figures 3(e) and (f).

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We finally note that the superconducting and exchange parameters in our model depend not only on the superconductivity and magnetism and top and bottom substrates but also on the coupling to the twisted bilayer. This coupling will be substrate-dependent, and therefore different encapsulations may correspond to different regimes in our topological phase diagram. In our model we have not included the super-moiré pattern which originates in the interplay between the moiré length scale of the proximity order and the original moiré of the twisted bilayer. In the real system the superconducting and exchange order parameters will have a spatial modulation, dependent on the rotation angle between the twisted bilayer and the substrates. The competition of two moiré length scales in our system, the twisted bilayer and the exchange moiré, would provide a potential platform to realize topological superconductors enabled by super-moiré physics. While this regime is not included in the analysis of our manuscript, our results provide a starting point for such a super-moiré superconductor.

To summarize, here we demonstrated the appearance of a family of topological superconducting states in an encapsulated twisted bilayer graphene between a ferromagnet and superconductor. We demonstrated that topological states with tunable Chern numbers appear for multilayers with chemical potential in the bottom, middle of the flat bands, and even in high energy bands. Our results show that high Chern number topological phases appear with the chemical potential close to 2 electrons per moiré unit cell, leading to topological phases highly tunable via in-plane strain. Furthermore, we showed that topological states can also be tuned with an out-of-plane electric field, specifically in the higher energy bands. For concreteness, we have considered specific configurations of strain, Rashba SOC, proximity-induced exchange, and superconducting order parameters, and that our mechanism applies to other configurations with heterostrain, or Rashba SOC at a single interface. Importantly, stabilizing these topological states does not rely on fine-tuning the twist angle, but solely on the emergence of mini-bands in the electronic structure of the twisted multilayer. In summary, our results establish twisted bilayer graphene as a promising platform to realize a wide variety of topological superconducting states hosting Majorana bound states, and ultimately, establishing a potential platform for a van der Waals-based topological quantum computer.

We acknowledge the computational resources provided by the Aalto Science-IT project, and the financial support from the Academy of Finland Project Nos. 331342, 336243, and 358088, the Jane and Aatos Erkko Foundation, the Jubilee Fund of the Foundation for Aalto University Science and Technology, and from PID2021-125343NB-100 (MCIN/AEI/FEDER,EU). M K thanks the hospitality of the Instituto de Ciencia de Materiales de Madrid. We thank P Liljeroth for useful discussions.

All data that support the findings of this study are included within the article (and any supplementary files).