Topological Floquet engineering of twisted bilayer graphene

We investigate the topological properties of Floquet-engineered twisted bilayer graphene above the so-called magic angle driven by circularly polarized laser pulses. Employing a full Moiré-unit-cell tight-binding Hamiltonian based on first-principles electronic structure, we show that the band topology in the bilayer, at twisting angles above $1.{05}^{\circ }$, essentially corresponds to the one of single-layer graphene. However, the ability to open topologically trivial gaps in this system by a bias voltage between the layers enables the full topological phase diagram to be explored, which is not possible in single-layer graphene. Circularly polarized light induces a transition to a topologically nontrivial Floquet band structure with the Berry curvature analogous to a Chern insulator. Importantly, the twisting allows for tuning electronic energy scales, which implies that the electronic bandwidth can be tailored to match realistic driving frequencies in the ultraviolet or midinfrared photon-energy regimes. This implies that Moiré superlattices are an ideal playground for combining twistronics, Floquet engineering, and strongly interacting regimes out of thermal equilibrium.

Figure 1

Atomic structure and tight-binding model bands of equilibrium twisted bilayer graphene. (a) Real-space atomic lattice. Red (blue) points denote the carbon atoms of the bottom (top) layer of one supercell. The black dotted line indicates the mirror-symmetry line which maps the in-plane atomic coordinates of both layers onto each other. The origin is chosen on an $AA$-stacked carbon site. The twist angle is $\mathrm{\Theta }=7.{34}^{\circ }$, which corresponds to a supercell with 244 atoms. The full (dashed) circle indicates a region of $A-A$ ($A-B$) stacking. (b) Equilibrium band structure along $\mathrm{\Gamma }-K_1$-M, where $|\mathrm{\Gamma }-K_1|=0.22{\AA }^{-1}$. The dashed lines show the single-layer band structure without interlayer coupling but folded back into the reduced zone. The bandwidth at $\mathrm{\Gamma }$ (M) is indicated by a red (blue) arrow. (c) Equilibrium bandwidth of the valence and conduction bands at $\mathrm{\Gamma }$ (red line) and $M$ (blue line) for different twist angles, $\mathrm{\Theta }$ ($\mathrm{\Theta }=7.{34}^{\circ }$ values colored black). The inset shows small angles, whose associated bandwidths lie within the midinfrared photon energy range.

Figure 2

Equilibrium band structure and corresponding valence-band Berry curvatures in close proximity to the Dirac points of the mBZ along $\mathrm{\Gamma }-K_1-M-K_2-\mathrm{\Gamma }$ (interval of length 0.022 ${\AA }^{-1}$ around $K_1$ and $K_2$) for different onsite potentials $V_{AB}$ and backgate voltages $V_{BG}$. [(a), (c), (e)] Antisymmetric $\pm /\mp$ sublattice potential within the layers. [(g), (i), (l)] Symmetric $\pm /\pm$ sublattice potentials. [(b), (f), (j), (l)] Local and valley band curvature shows opposite sign, respectively. [(d), (h)] Nonvanishing valley Berry curvature but zero integrated Berry curvature due to sign flip between valleys. We use the following values for the potentials: [(a), (b)] $V_{AB}^{asym}=0.0001\mathrm{eV}, V_{BG}=0.0001\mathrm{eV}$, [(c), (d)] $V_{AB}^{sym}=0.0001\mathrm{eV}, V_{BG}=0.0001\mathrm{eV}$, [(e), (f)] $V_{AB}^{asym}=0.01\mathrm{eV}, V_{BG}=0.001\mathrm{eV}$, [(g), (h)] $V_{AB}^{sym}=0.01\mathrm{eV}, V_{BG}=0.001\mathrm{eV}$, [(i), (j)] $V_{AB}^{asym}=0.01\mathrm{eV}, V_{BG}=0.05\mathrm{eV}$, and [(k), (l)] $V_{AB}^{sym}=0.01\mathrm{eV}, V_{BG}=0.05\mathrm{eV}$.

Figure 3

Floquet band structures for a strong circularly polarized driving field with $E_{\max }=13.7$ MV/cm ($A_{max}=0.15a_0^{-1}$ at $\mathrm{\Omega }=2.23$ eV) for different onsite potentials and back-gate voltages: (a) $V_{AB}^{sym}=0.0001\mathrm{eV}, V_{BG}=0.0001\mathrm{eV}$, and (b) $V_{AB}=0, V_{BG}=0.05\mathrm{eV}$. The gray dashed lines indicate the equilibrium bands. The overlap between the Floquet bands and the equilibrium bare bands is denoted by the red color as indicated in the color bars. In both cases, the breaking of time-reversal symmetry adds mass to the Dirac bands, which results in a finite energy gap at the Dirac points. Additionally, gaps open away from the Dirac points, due to resonant coupling between Floquet bands. The inset in panel (b) shows a closeup of the region near $K_2$.

Figure 4

Floquet-engineered Berry curvature near the two Dirac points of the mBZ along $\mathrm{\Gamma }-K_1-M-K_2-\mathrm{\Gamma }$ for different onsite potentials $V_{AB}$, back-gate voltages $V_{BG}$, and increasing circularly polarized field amplitudes $E_{\max }$. The size of the points indicate the overlap with the original equilibrium bands. Throughout, ${\mathrm{\Delta }}_{\mathrm{\Gamma }}=2.23$ eV is chosen as Floquet driving frequency. [(a), (e), (i), (m)] Equilibrium case. In panel (e), the Berry curvature is not defined at $K_1$ ($K_2$), due to band degeneracy, and zero otherwise. (b) For infinitesimally small potentials, a finite driving amplitude immediately opens a gap and induces a nontrivial Floquet-band topology. [(f), (i), (j), (m), (n)] Topologically trivial phase, characterized by zero integrated Berry curvature of the valence (conduction) bands. [(g), (k), (o)] Topological transition indicated by gap closing. (g) Trivial shift of bands while the sign of associated Berry curvature is preserved. (k) Gap closing and flow of Berry curvature between lowermost conduction band and uppermost valence band. (o) Gap closing and flow of Berry curvature both between bottom conduction band and valence band and between top conduction band and valence band, respectively. [(b), (c), (d), (h), (i), (p)] Topologically nontrivial phase, characterized by reopened gaps and a finite integrated Berry curvature in the valence (conduction) bands. We choose the following parameter sets: [(a)–(d)] $V_{AB}^{sym}=V_{BG}=0.0001\mathrm{eV}, E_{\max }=0.0,0.9,1.8,2.7\mathrm{MV}/\mathrm{cm}$, [(e)–(h)] $V_{AB}=0, V_{BG}=0.05\mathrm{eV}, E_{\max }=0.0,2.7,7.7,8.2\mathrm{MV}/\mathrm{cm}$, [(i)–(l)] $V_{AB}^{asym}=0.01\mathrm{eV}, V_{BG}=0.001\mathrm{eV}, E_{\max }=0.0,2.7,3.6,4.6\mathrm{MV}/\mathrm{cm}$, and [(m)–(p)] $V_{AB}^{sym}=0.01\mathrm{eV}, V_{BG}=0.001\mathrm{eV}, E_{\max }=0.0,2.7,3.6,4.6\mathrm{MV}/\mathrm{cm}$.

Figure 5

Floquet topological gaps and phase diagrams. (a) Energy gap at $K$ as a function of the driving field amplitude with vanishing local potentials. The red line shows the band gap extracted from the Floquet band structure. The blue (black) line shows the quadratic dependence of the gap on the driving amplitude, $A_{\max }$, and the equilibrium TBG (monolayer graphene) Fermi velocity, $v_F$ ($v_F^0$), in the low-amplitude, high-frequency limit. The vertical black dotted line indicates the field strength chosen in panel (b). (b) Energy gap as a function of twist angle for a fixed field strength. The value used throughout the paper, $\mathrm{\Theta }=7.{34}^{\circ }$, is indicated by red color. The black dashed line indicates the corresponding single-layer Floquet band gap. The blue dashed line shows the gap values which would be naively expected from the renormalized Fermi velocity. [(c), (d)] The pink (light blue) area indicates the topologically nontrivial (trivial) phase. (c) With symmetric sublattice potential and a small backgate voltage, $V_{BG}=0.001$ eV. Because of the additional back-gate voltage, the topologically nontrivial phase always requires nonzero driving amplitude. (d) With back-gate voltage only. Throughout, a photon frequency, $\mathrm{\Omega }=2.23\mathrm{eV}$, is assumed.

Figure 6

Tight-binding hopping elements. Only $p_z$ orbitals are taken into account. (a) Intralayer hopping is restricted to $pp\pi$ terms. There are two channels for interlayer hopping: Vertical hopping is dominated by the $pp\sigma$ term (b). For nonvertical transitions, also $pp\pi$ terms (c) contribute.

Figure 7

Onsite potentials that break inversion symmetry. The back-gate voltage, $V_{BG}$, is always chosen symmetric between both layers. (a) Symmetric $\pm /\pm$ sublattice potential, $V_{AB}^{sym}$. (b) Asymmetric $\pm /\mp$ sublattice potential, $V_{AB}^{asym}$.

Figure 8

Calculation of the Berry flux. The Berry flux of the $n\mathrm{th}$ band, $F^{\left(n\right)}$, for a quasimomentum $\mathit{k}=(k_x,k_y)$ is calculated by a closed loop along the eigenstates at position 1 to 4 in momentum space. Throughout, we use a grid spacing of $\mathrm{\Delta }k={10}^{-5}{\AA }^{-1}$.

Figure 9

Full energy gap at the Dirac point as a function of the driving amplitude for different twist angles, $\mathrm{\Theta }$ (colored dots). In all three cases, the Floquet driving frequency is $\mathrm{\Omega }=2.23$ eV. The black dashed line indicates the monolayer dependence of the gap on the driving amplitude and the Fermi velocity, $2{\left(v_F^0{A}_{\text{max}}\right)}^2/\mathrm{\Omega }$. The colored dashed lines indicate the corresponding relation for the twisted case.