Valley Jahn-Teller Effect in Twisted Bilayer Graphene

The surprising insulating and superconducting states of narrow-band graphene twisted bilayers have been mostly discussed so far in terms of strong electron correlation, with little or no attention to phonons and electron-phonon effects. We found that, among the 33 492 phonons of a fully relaxed $\theta =1.08°$ twisted bilayer, there are few special, hard, and nearly dispersionless modes that resemble global vibrations of the moiré supercell, as if it were a single, ultralarge molecule. One of them, doubly degenerate at $\mathbf{\Gamma }$ with symmetry $A_1+B_1$, couples very strongly with the valley degrees of freedom, also doubly degenerate, realizing a so-called $E\otimes e$ Jahn-Teller (JT) coupling. The JT coupling lifts very efficiently all degeneracies which arise from the valley symmetry, and may lead, for an average atomic displacement as small as 0.5 m Å, to an insulating state at charge neutrality. This insulator possesses a nontrivial topology testified by the odd winding of the Wilson loop. In addition, freezing the same phonon at a zone boundary point brings about insulating states at most integer occupancies of the four ultraflat electronic bands. Following that line, we further study the properties of the superconducting state that might be stabilized by these modes. Since the JT coupling modulates the hopping between $AB$ and $BA$ stacked regions, pairing occurs in the spin-singlet Cooper channel at the inter-($AB\text{−}BA$) scale, which may condense a superconducting order parameter in the extended $s$-wave and/or $d\pm id$-wave symmetry.

Popular Summary

Two sheets of graphene—a honeycomb lattice of carbon atoms—laid slightly askew to one another induce surprising insulating and superconductive behavior not seen in individual sheets. Up until now, most researchers have focused on how interactions among electrons in this “twisted bilayer graphene” might produce these states, but they have largely ignored the role of phonons—quantized mechanical vibrations in the atomic lattice. Here, we show how phonons could produce the observed insulating states.

When the two graphene layers are slightly rotated by a small angle, a so-called moiré pattern forms, introducing a long-wavelength modulation of the atomic lattice. Using sophisticated calculations, we find that this modulation affects not only the electronic properties of the system but also the collective lattice vibrations, or phonons. In particular, we find a set of phonon modes—dubbed moiré phonons—whose vibration intensity is modulated by this pattern. Among these moiré phonons, we find a special set that strongly couples to the electrons. When the lattice is statically deformed following the vibration of these modes, insulating states at any integer filling (i.e., whenever one electron is added per unit cell) occur, exactly where they have been observed experimentally.

The suggested electron-phonon origin for the insulating states of the twisted graphene bilayer is a game changer. The additional consequence is that superconductivity in the nearby metallic states has the same origin, leading to detailed predictions about pairing symmetries. In our view, these theoretical discoveries push the established field of apparently strongly correlated insulating and superconducting states of magic bilayers out of its present track and into a new direction.

Figure 1

Top: Moiré superlattice formed by two unrelaxed graphene layers (in blue and red) twisted by an angle $\theta$. We indicate the different stacking regions: $AA$ and the two different Bernal regions $AB$ and $BA$. The domain walls (DWs) separate $AB$ from $BA$ regions and connect different $AA$ regions. On the right, an enlargement of the $AA$ stacked region is shown: two overlapping hexagons in distinct graphene layers are rotated in opposite directions around the perpendicular $z$ axis by an angle $\theta /2$. Bottom: The folding procedure in TBLG. The original single-layer Brillouin zones (in red and blue) are folded into the mini Brillouin zone (MBZ). Two inequivalent $\mathbf{K}$ points in different layers (${\mathbf{K}}_{-}$ and ${\mathbf{K}}_{+}^{\prime }$ or ${\mathbf{K}}_{-}^{\prime }$ and ${\mathbf{K}}_{+}$) are folded into the same points, ${\mathbf{K}}_1$ or ${\mathbf{K}}_2$, of the MBZ. The path ${\mathbf{K}}_1\to \mathbf{\Gamma }\to \mathbf{M}\to {\mathbf{K}}_2$ is also shown.

Figure 2

(a) Electronic band structure of twisted bilayer graphene at the angle $\theta =1.08$ after full atomic relaxation. The charge neutrality point is the zero of energy. The irreps at the $\mathbf{\Gamma }$ point are encoded by colored circles, where blue, red, and green stand for the $A_1+B_1$, $A_2+B_2$, and $E_1+E_2$ irreps of the $D_6$ space group, respectively. At the ${\mathbf{K}}_2$ point the $E$ and $A_1+A_2$ irreps of the little group $D_3$ are represented by green and violet triangles, respectively. (b) Enlargement of the FBs region. (c) Wilson loop of the four FBs as function of $k=G_2/\pi$.

Figure 3

(a) The displacement field $\mathit{\eta }(\mathbf{r})$, Eq. (10), restricted on the DWs: the direction of the atomic displacement is depicted by a small arrow, while its magnitude in mÅ is expressed in colors. (b) Displacement along the DW area highlighted by a black dashed line in (a). The overall effect of the distortion is a narrowing of the DW. (c) Low-energy band structure of TBLG at $\theta =1.08$ after the distortion in (a). The twofold degeneracies protected by the valley symmetry are completely lifted.

Figure 4

Phonon density of states $F(\omega )$ for Bernal-stacked bilayer graphene at zero twist angle (red dashed line) and fully relaxed TBLG at $\theta =1.20$, 1.12, 1.08 (blue, green, and black lines).

Figure 5

Enlargement of the optical region of the phonon spectrum of the fully relaxed TBLG at $\theta =1.08$. Among many scattered energy levels of highly dispersive branches (not resolved in this narrow energy window), a set of 10 narrow continuous branches stands out (no line drawn through data points, which just fall next to one another). The degeneracy of these modes is twice that expected by $D_6$ space symmetry, similar in this to electronic bands. The twofold degenerate mode with the highest overlap with the deformation $\mathit{\eta }(\mathbf{r})$, drawn in Fig. 6, is marked by a red arrow, while the mode at $\mathbf{M}$ used in Sec. 6 is marked by a green arrow. The avoided crossing which occurs close to $\mathbf{\Gamma }$ is encircled by a blue dashed line.

Figure 6

(a) Atomic displacements on one of the two layers corresponding to the $A_1$ symmetry moiré mode of the phonon doublet marked by a red arrow in Fig. 5. The direction of displacement is represented by a small arrow centered at each atomic position, while its modulus is encoded in colors. The mean displacement per atom is 0.57 mÅ. (b) Enlargement of the center of an $AA$ region, shown for both layers. (c) Enlargement along one of the domain walls. Note the similarity with the ad hoc displacement in Fig. 3.

Figure 7

The band structure of twisted bilayer graphene at $\theta \approx 1.08°$ in a ribbon geometry with open boundary conditions. The atomic structure inside the unit cell (replicated 7 times along the $x$ direction) has been deformed with the $A_1$-symmetric moiré phonon mode. With a mean deformation amplitude of $\approx 1.14\mathrm{m}\AA$, that mode is so strongly coupled to open a large electronic gap at charge neutrality point of $\approx 25\mathrm{meV}$. The bulk bands have been highlighted in blue to emphasize the presence of edge states both within the phonon-driven FBs gap and the gaps above and below them.

Figure 8

Atomic displacements on one of the two layers corresponding to one of the moiré modes at $\mathbf{M}$ marked by a green arrow in Fig. 5. The direction of displacement is represented by a small arrow centered at each atomic position, while its modulus is encoded in colors. The mean deformation is 1.8 mÅ and leads to the DOS in Fig. 9. The inset shows an enlargement of the $AA$ region close to the origin. The rectangular unit cell, now containing twice the number of atoms, is highlighted by a black dashed line.

Figure 9

Density of states in the flatbands region after the TBLG has been distorted by one of the two degenerate modes at $\mathbf{M}$ marked by a green arrow in Fig. 5. A mesh of $40\times 40$ points in the MBZ has been used. The DOS is expressed as function of energy, the zero corresponding to the charge neutrality point, and corresponding occupancy $\nu$ of the FBs. The band gaps at $\nu =\pm 2,\pm 4$ are highlighted in violet. The mean displacement per atom is 1.8 mÅ.

Figure 10

Atomic displacements on one of the two layers corresponding to multicomponent deformation obtained combining moiré JT phonons at three inequivalent $\mathbf{M}$ points. The mean deformation is 1.7 mÅ and leads to the DOS in Fig. 11. The intensity of displacement is encoded in colors. The inset shows an enlargement of one of the $AA$ regions when both layers are considered. The unit cell, denoted by a black dashed line, is 4 times larger than the original one.

Figure 11

Electronic density of states with a multicomponent lattice distortion obtained by freezing a combination of the modes at the three inequivalent $\mathbf{M}$ points. Band gaps at $\nu =+1,\pm 3,\pm 4$ are highlighted in violet. The mean displacement per atom is 1.7 mÅ.

Figure 12

(a) Sketch of the elementary band representation along $\mathbf{\Gamma }\to {\mathbf{K}}_1$ taking into account the 32 bands closest to the charge neutrality point and without allowing avoided crossing between same-symmetry Bloch states. Blue, red, and green lines refer to Bloch states that at $\mathbf{\Gamma }$ transform like $A_1+B_1$, $A_2+B_2$, and $E_1+E_2$, respectively. Should we allow for avoided crossings, close to charge neutrality we would obtain the four flatbands shown as black lines surrounding the shaded region. (b) The FBs obtained by a model tight-binding Hamiltonian that includes only the solid bands of (a). The details of this Hamiltonian are given in Appendix pp2. (c) Wilson loop corresponding to the four bands in (b) fully occupied.

Figure 13

Phonon-mediated attraction. The two scattering channels corresponds to the two phonons and have the same amplitude $g^2/2\omega$.

Figure 14

Pair amplitudes $\mathrm{\Delta }(k{)}^{†}$ of the leading superconducting channels: $d-id$ wave (a),(b) and extended $s$ wave (c),(d). In each panel we show the hexagonal MBZ and the phase amplitudes restricted to a narrow region close to the Fermi surfaces corresponding to the occupancies $\nu \sim -2$ (a),(c) and $\nu \sim -1$ (b),(d). The phase of the superconducting order parameter is expressed in color. Note that the double lines are due to the fact that two bands cross the chemical potential at different $\mathbf{k}$ points.

Figure 15

Phonon dispersion obtained with our choice of intralayer and interlayer potentials in Bernal-stacked bilayer graphene.