Moiré impurities in twisted bilayer black phosphorus: Effects on the carrier mobility

Moiré patterns on two-dimensional van der Waals heterostructure can give rise to unique electronic and transport properties. In this work we report a theoretical investigation of Moiré patterns on twisted bilayer black phosphorus (tbBP). It is found that the Moiré pattern has extraordinary effects and leads to significant asymmetry with respect to transport direction and carrier type. The high-symmetry local stacking configurations in the Moiré pattern act as impurities with sizes at the Moiré length scale, and these “Moiré impurities” induce flatbands and localized states in tbBPs. Because both the conduction band minimum and valence band maximum are dominated by these localized states, the deformation potential limited carrier mobility is significantly affected: the electron mobility of tbBPs reduces by almost 20-fold when twisting from zero angle $\left(\sim 2560{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}\right)$ to just $1.8^{\circ }$ $\left(\sim 131{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}\right)$. The microscopic physics behind these effects are revealed by the real-space wave functions.

Figure 1

(a) Schematic construction of a tbBP with a small twisted angle $\theta$ which is defined by the orientation between the two constituent BP lattices. The zero angle corresponds to the AA stacking of bilayer BP. (b) Moiré pattern formed on the tbBP with $\theta =2.7^{\circ }$. The colored rectangles indicate the distribution of the high-symmetry local stacking configurations in tbBPs and the black rectangle indicates the simulated 2.7-tbBP supercell containing 3588 atoms. The atomic structures of the four high-symmetry local stacking configurations, named AA, AB, AA', and AB', are shown in the right panel.

Figure 2

(a) Calculated band structure of 2.7-tbBP. (b, c) Spatial distribution of wave functions of the periodic 2.7-tbBP Moiré structure CBM and VBM, respectively. The central black rectangle box in (b) and (c) indicates the simulated supercell of 2.7-tbBP, which contains 3588 atoms. The distribution of four stacking configurations is the same as that in Fig. 1. (d, e) Zoom-in on the supercells indicated by the black rectangles in (b) and (c), respectively. The isosurface value is set at 1.$7\times {10}^{-5}e{\AA }^{-3}$.

Figure 3

The calculated band structures, at the GGA-PBE level, of four tbBPs (except 2.7-tbBP) are plotted. (a)–(d) The band structures of tbBPs with twist angle $\theta =1.8^{\circ },2.2^{\circ },3.6^{\circ }$, and $5.4^{\circ }$, respectively. The 2D Brillouin zone is the same as that in Fig. 2.

Figure 4

Deformation limited carrier mobility along the armchair $\left(y\right)$ direction $(\mathrm{\Gamma }\text{−}Y$ direction in $k$ space) of tbBPs versus the twist angle $\theta$. The dash and solid lines are guides to the eye.

Figure 5

(a)–(e) Spatial distribution of $|{\psi }_c{\left(\mathbf{r}\right)|}^2$ of the tbBP for five twist angles. On the VBM side, the spatial distribution of $|{\psi }_v{\left(\mathbf{r}\right)|}^2$ are plotted in (f)–(g). The twist angles $\theta$ are shown on the top row. The black boxes indicate the supercell of the tbBP, and the distribution of the four stacking configurations is the same as that in Fig. 1. The isosurface value is identical to that in Fig. 2.

Figure 6

High-symmetry local stacking configurations. The detailed geometrical parameters are summarized in Table 3.

Figure 7

The electronic properties of four high-symmetry local stacking configurations. The calculated band structures of (a) AA stacking, (b) AB stacking, (c) AA' stacking, and (d) AB' stacking are plotted. (e) The spatial distribution of partial charge density (wave functions) corresponding to VBM and CBM using an isosurface of 1.$5\times {10}^{-3}e{\AA }^{-3}$.

Figure 8

Contour plot of the interlayer distance between two constituent layers of 2.7-tbBP supercell in Å. The distribution of four stacking configurations in tbBP is the same as those in Fig. 1 of the main text.

Figure 9

Calculated averaged Hartree potential of 2.7-tbBP supercell projected along (a) the zigzag or $x$ direction and (b) armchair or $y$ direction. The corresponding schematics of side views are plotted below the Hartree potential.

Figure 10

Linear regime of the stress-strain relation under uniaxial tension: (a) along zigzag or $x$ direction and (b) armchair or $y$ direction, as obtained by lammps simulations.