Designing flat bands by strain

We study the effects of heterostrain on moiré bands in twisted bilayer graphene and bilayer transition metal dichalcogenide (TMD) systems. For bilayer graphene with a twist angle near $1^{\circ }$, we show that heterostrain significantly increases the energy separation between conduction and valence bands as well as the Dirac velocity at charge neutrality, which resolves several puzzles in scanning tunneling spectroscopy and quantum oscillation experiments at once. For bilayer TMD, we show that applying small heterostrain generally leads to flat moiré bands that are highly tunable.

Figure 1

(a)The moiré pattern of bilayer graphene with twist angle $\theta =1.{05}^{\circ }$ and uniaxial strain $\epsilon =0.7\%,\phi =0^{\circ }$. Notice that the moiré superlattice is not a regular triangular lattice. The moiré dots are elliptical which is commonly observed in STM experiments [32, 33]. (b) The moiré Brillouin zone with the same geometrical parameters. In the plot, for clarity, we only label the rescaled ${\mathbf{K}}_{-}$ points from the two layers. Correspondingly, the red and black dots are the position of the shifted Dirac points from ${\mathbf{K}}_{-}$ valley. The other valley can be obtained by time reversal operation. The red dashed line is the momentum path for the band structure plots.

Figure 2

In (a), we show the band structure (left) and the density of states (right) for the ${\mathbf{K}}_{-}$ valley of twisted bilayer graphene at twist angle $\theta \cong 1.{05}^{\circ }$ with (solid line) and without (dashed line) uniaxial heterostrain $\epsilon =0.6\%,\phi ={30}^{\circ }$. The heterostrain increases the separation between the valence and conduction bands, while keeping them rather flat in most areas of the Brillouin zone. In (b), we show that the two bands are still connected to each other through two Dirac points. With heterostrain, the positions of the two Dirac crossings are no longer at the corner of the moiré Brillouin zone, as shown in the insets of (b). The hexagons represent the stretched Brillouin zone and the blue/red dots are the positions of two Dirac crossings. We plot the band dispersions along the red/blue dashed lines in the Brillouin zone. The conduction and valence bands are rather flat in most parts of the momentum space except near the Dirac crossings. The energy shift between the two Dirac points is $\sim 9\mathrm{meV}$. As seen from (b), the Dirac velocity is anisotropic and reaching $0.15v_F$, which is much larger than the unstrained case (about $0.018v_F$ with the current parameters). This velocity is comparable to the one observed in transport and capacitance experiments [1]. Since the Dirac points are shifted in energy, a finite electron/hole fermi surfaces appear at charge neutrality. Due to the enhanced Dirac velocity, the size of the fermi pocket is found to be much smaller than the size of the Brillouin zone. The estimated electron/hole density (including the valley and spin degeneracy) is $\sim 3\times {10}^{10}{\mathrm{cm}}^{-2}$ for current parameters.

Figure 3

Normalized density of state plot with fixed $\theta =1.{05}^{\circ },\epsilon =0.3\%$, and varying $\phi$. (The curves are relatively shifted to make the plot clear.) We observe many subtle transitions in the structure of the van Hove singularities as the direction of the strain is changes. However, the bandwidth of the middle two bands stays approximately constant. (The density of states has only one prominent peak in each band for $\phi$ near ${20}^{\circ }\sim {30}^{\circ }$.)

Figure 4

The splitting of the van Hove singularities, $\mathrm{\Delta }$, as a function of the twist angle $\theta$. We fixed the strain direction $\phi ={25}^{\circ }$ and different curves represent different strain magnitudes. The splitting of the van Hove singularities stay approximately constant for nearly magic twist angles.

Figure 5

The blue line shows the saturation values of the van Hove singularity splitting, labeled by $\mathrm{\Delta }$, as a function of the magnitude of the uniaxial strain. The red line shows the maximal energy shift of the two Dirac crossings within a valley $\delta$ as a function of strain magnitude. The fitting shows that both quantities are linearly proportional to the magnitude of uniaxial strain.

Figure 6

The energy shift between two Dirac points in a valley, labeled by $\delta$, as a function of the direction of the strain. In this plot, we fix $\theta =1.{05}^{\circ }$ and choose four different strain magnitudes $\epsilon =0.1\%$(blue), $0.3\%$ (red), $0.5\%$ (green), $0.7\%$ (yellow). We notice that the energy shift of the Dirac fermions is very sensitive to the direction of the strain. The maximal value of the shift is linear proportional to the strain magnitude.

Figure 7

(a)/(b) The equal energy contour plots for the valence/conduction bands with parameters $\theta =1.{05}^{\circ },\epsilon =0.3\%$, and $\phi ={25}^{\circ }$. We can identity multiple ordinary (dot) and higher order (star) van Hove singularities. Since these van Hove singularities happen to be nearby in energy, they contribute to a single enhanced density of state peak in each band as shown in (c).

Figure 8

(a)/(b) The equal energy contour plots for the valence/conduction bands with parameters $\theta =1.{05}^{\circ },\epsilon =0.53\%$, and $\phi ={50}^{\circ }$. We observe ordinary and higher order van Hove singularities in both valence and conduction bands. They contribute to the sharp peaks in the density of states in (c). In (d) we zoom in near the density of state peak at $E_{vH}\cong 10.4\mathrm{meV}$ and make a log-log plot for $\rho \left(E\right)$ vs $|E-E_{vH}|$ for $E-E_{vH}>0$. The linear fitting indicates that the density of states has a power law divergence near the van Hove singularity, namely $\rho \left(E\right)\sim |E-E_{vH}{|}^{-\nu }$ with $\nu \cong 0.255$. This is the key feature for the higher order van Hove singularities.

Figure 9

We focus on the band structures for ${\mathbf{K}}_{-}$ valley of bilayer ${\mathrm{WSe}}_2$ near the top of the valence band. (a)/(b) show band structures with volume preserving heterostrain at $\epsilon =1.5\%$ and $\phi =0^{\circ }/{30}^{\circ }$, respectively. (We take ${\beta }_{\mathrm{TMD}}\cong 2.3$.) We notice that the first three bands are quite flat. The top two bands are close in energy and well separated from other bands with gap $\mathrm{\Delta }\sim 8\mathrm{meV}$. We observe that for $\phi =0$, the first two valence bands are topologically trivial, while for $\phi ={30}^{\circ }$ they carry $\pm 1$ Chern number.

Figure 10

(a) shows the band structure of ${\mathbf{K}}_{-}$ valley with parameters $\beta =2.10,\epsilon =2\%$, and $\phi =0$. The system is at a topological phase transition. A Dirac crossing is found at one of the $\mathbf{M}$ points in the Brillouin zone. The three $\mathbf{M}$ points are not equivalent because of the lack of $C_3$ rotation symmetry. (b) shows the contour plot of the gap between the first two bands with the same parameters. (c) Phase diagram of the top two valence bands as a function of $\beta$ and $\phi$. There is an interesting pattern of topological and trivial states (topological state refers to $\pm 1$ Chern number for the top two bands and trivial state refers to 0 Chern number) as we tune the angle $\phi$ from 0 to $\pi$. $\phi =0$ and $\pi$ are physically equivalent by an exchange of the two layers. The dashed line is the estimated value of $\beta \cong 2.30$ for ${\mathrm{WSe}}_2$. The phase diagram is obtained with $\epsilon =2\%$. However, the phase diagram is insensitive to the strain magnitude for small strain $<3\%$. (d) The band structure for $\epsilon =2\%$ and $\phi =0$ with an interlayer bias $\mathrm{\Delta }V=10\mathrm{meV}$. The displacement field separates a nearly flat band on the top of the spectrum.

Figure 11

We consider twisted bilayer ${\mathrm{WSe}}_2$ at $\theta =1^{\circ }$ with pressure. As the parameter $p$ is tuned from 1 to 1.5, there are two successive topological phase transitions. (a) At $p\cong 1.195$, the second band and third band touches with quadratic dispersion at the $\mathrm{\Gamma }$ point, which changes the Chern number by 2. (b) At $p\cong 1.410$, the first and second band close gap with a Dirac dispersion, which changes the Chern number by 1.