Quantum Hall superconductivity from moiré Landau levels

It has long been speculated that quasi-two-dimensional superconductivity can reappear above its semiclassical upper critical field due to Landau quantization; yet this reentrant property has never been observed. Here, we argue that twisted bilayer graphene at a magic angle (MATBG) is an ideal system in which to search for this phenomenon because its Landau levels are doubly degenerate and its superconductivity appears already at carrier densities small enough to allow the quantum limit to be reached at relatively modest magnetic fields. We study this problem theoretically by combining a simplified continuum model for the electronic structure of MATBG with a phenomenological attractive pairing interaction and discuss obstacles to the observation of quantum Hall superconductivity presented by disorder, thermal fluctuations, and competing phases.

Figure 1

(a) Schematic of the band model centered around the two moiré Brillouin zone centers in opposite valleys. The dashed red and blue bands qualitatively represent the actual moiré bands, while the solid red and blue bands represent the $\mathbf{k}·\mathbf{p}$ expansion employed in our model. (b) Schematic quantum limit LL spectra for the three cases considered in the main text. For ${\epsilon }_1=0$, the conduction and valence band Landau levels are simply those of doubly degenerate parabolic bands. The nonstandard labeling scheme employed here accounts for the anomalous $N=0,1$ Landau levels that emerge when ${\epsilon }_1\ne 0$. The blue and red levels represent “$+$” and “$-$” valley LLs, respectively. For the massive Dirac cases (${\epsilon }_0=0$), the anomalous $N=0,1$ Landau levels group with opposite bands in opposite valleys when ${\lambda }_{+}={\lambda }_{-}$, but with the same band in both valleys when ${\lambda }_{+}=-{\lambda }_{-}$. In all cases the LLs appear in degenerate pairs, emphasized by gray ovals. The arrows emphasize our assumption of full spin polarization.

Figure 2

(a) Dingle temperature estimate for MATBG. The red dots are extracted from Shubnikov–de Haas data presented in Fig. 5(b) of Ref. [23] and the black curve is a Lifshitz-Kosevich fit. Based on the fit, a Dingle temperature $T_D\sim 2.6$ K is estimated. (b) Dependence of $T_c$ on LL broadening (characterized by the Dingle temperature) in the reentrant regime for the parabolic band (i.e., ${\epsilon }_1=0$) limit of our model.

Figure 3

Magnetic-field-vs-temperature phase diagram at fixed density for the three cases illustrated in Fig. 1: (a) $v=0$, (b) $m\to \infty ,{\lambda }_{+}={\lambda }_{-}$, and (c) $m\to \infty ,{\lambda }_{+}=-{\lambda }_{-}$. The blue lines represent the disorder-free limit, and the red lines take disorder broadening into account using a Dingle temperature $T_D=2.6$ K. The inset in (a) expands the low-field region near the semiclassical $H_{c2}$. The plots show robust reentrant superconductivity when the quantum limit is reached at moderate magnetic fields for case (i) (trivial) and case (iii) (Chern). The most robust domes at $\gtrsim 10$ T fields correspond to pairing in the lowest LLs. The qualitative difference between (b) and (c) is due to the absence in the former case [case (ii)] of degenerate intervalley $N=0,1$ pairs.

Figure 4

Schematic band dispersions for various cases from the $\mathbf{k}·\mathbf{p}$ Hamiltonian in Eq. (1): (a) $v=0$; $\lambda ,1/m\ne 0$; and $\tilde{s}=-$. (b) $1/m=0$; $\lambda ,v\ne 0$; and $\gamma =2$. (c) $v=0$; $\lambda ,1/m\ne 0$; and $\tilde{s}=+$. (d) $\lambda ,v,1/m\ne 0$ and $\tilde{s}=+$. (e) $1/m,\lambda =0$; $v\ne 0$; and $\gamma =2$. (f) $1/m,\lambda =0$; $v\ne 0$; and $\gamma =1$. The black dashed line represents a given position of the chemical potential. The cases discussed explicitly in the main text correspond to (a) and (b), while (c) resembles the Zeeman-split case discussed below corresponding to valley-triplet, spin-singlet pairs and (d) represents a more general case where both bands are present at the Fermi level.

Figure 5

The ladder diagram for the particle-particle scattering function in the LL representation, with the wavy line representing the pairing interaction.

Figure 6

Magnetic-field-vs-temperature phase diagram assuming spin-singlet pairing and a $g$ factor of 2. The field is tilted such that the majority spin LL $N$ and the minority spin LL $N+3$ are degenerate, which corresponds to a tilt angle that satisfies $cos\left(\theta \right)=m^{*}/m_e$.