Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle

The ability in experiments to control the relative twist angle between successive layers in two-dimensional (2D) materials offers an approach to manipulating their electronic properties; we refer to this approach as “twistronics.” A major challenge to theory is that, for arbitrary twist angles, the resulting structure involves incommensurate (aperiodic) 2D lattices. Here, we present a general method for the calculation of the electronic density of states of aperiodic 2D layered materials, using parameter-free Hamiltonians derived from ab initio density-functional theory. We use graphene, a semimetal, and ${\mathrm{MoS}}_2$, a representative of the transition-metal dichalcogenide family of 2D semiconductors, to illustrate the application of our method, which enables fast and efficient simulation of multilayered stacks in the presence of local disorder and external fields. We comment on the interesting features of their density of states as a function of twist angle and local configuration and on how these features can be experimentally observed.

Figure 1

Simulated local electronic density of states (LEDoS) at four different angles of twisted bilayer graphene. Each line corresponds to a different real-space configuration along the line connecting AA to AB stacking. The insets show a real-space image of the density of states in the bilayer system at the energy value identified by a dashed line. For the $0.{88}^{\circ }$ angle (a) an orange line showing the AA to AB configuration path is shown in the real-space image. The figure was constructed to facilitate comparison with experiment [32], which shows excellent agreement in the positions and relative heights of the VHS. For these calculations we use a disk cutoff radius $R=500\AA$ which contains $591344$ atoms.

Figure 2

Local electronic density of states as a function of shift distance across the unit-cell diagonal. (a) Scan of a single orbital's LEDoS with the coloring corresponding to distance across the diagonal. The insets show the real-space configurations $\omega$ for three types of stacking with the atoms in each layer represented by different color circles (red and blue for top and bottom). The unit cell of the bottom layer is outlined in blue and the shifted orbital is highlighted as a filled red dot. (b) LEDoS at the selected VHS peak as a function of shift for one orbital (triangles) and for the average of both orbitals (circles), which is properly symmetric. The peak varies smoothly with shift and has critical points at the three special stacking configurations. For the calculations in (a,b) we use a disk cutoff radius $R=500\AA$ which contains $591344$ atoms. (c) Ground-state energy (GSE) of $\theta =0^{\circ }$ twist angle with and without Li-ion doping (see text for details).

Figure 3

(a) Average EDoS as a function of twist angle for tBLG. (b) Average EDoS for monolayer graphene in the presence of varying magnetic field. (c) LEDoS for AA stacked tBLG with $3.1^{\circ }$ (solid line) and $1.1^{\circ }$ (dashed line) twist angle at different values of the magnetic field and Hall conductivity ${\sigma }_{xy}$ in units of $e^2/h$, with the horizontal red dashed lines at $\pm 2e^2/h$. For these calculations we use a disk cutoff radius $R=750\AA$ ($1330550$ atoms) and averages are over 100 configurations across the unit cell ($10\times 10$ grid).

Figure 4

(a) EDoS for twisted bilayer ${\mathrm{MoS}}_2$ from $0^{\circ }$ (blue) to $28.{65}^{\circ }$ (red) twist angle. (b) and (c) EDoS near the valence- and conduction-band extrema, with the logarithmic scales showing the changes in greater detail. These calculations use a disk cutoff radius $R=300\AA$ ($193700$ atoms) and are averaged over 100 configurations across the unit cell ($10\times 10$ grid).