Heterostrain Determines Flat Bands in Magic-Angle Twisted Graphene Layers

The moiré of twisted graphene bilayers can generate flat bands in which charge carriers do not possess enough kinetic energy to escape Coulomb interactions with each other, leading to the formation of novel strongly correlated electronic states. This exceptionally rich physics relies on the precise arrangement between the layers. Here, we survey published scanning tunneling microscope measurements to prove that near the magic-angle, native heterostrain, the relative deformations between the layers, dominates twist in determining the flat bands as opposed to the common belief. This is demonstrated at full filling where electronic correlations have a weak effect and where we also show that tip-induced strain can have a strong influence. In the opposite situation of zero doping, we find that electronic correlation further renormalizes the flat bands in a way that strongly depends on experimental details.

Figure 1

Heterostrained twisted graphene layers. (a)–(c) STM images of twisted graphene layers near the magic angle adapted from (a) Refs. [8], (b) [7], and (c) [9], respectively. The scale bar is 10 nm in each image. Topographies were taken at 0.5 V and 30 pA, 0.4 V and 50 pA, and 0.5 V and 50 pA, respectively. Insets present the parameters describing the relative arrangement of the layers in good agreement with estimates of the original studies [7, 8, 9]. (d) Local density of states measured in the $AA$ regions for each of the samples of (a)–(c). (e) Corresponding tight-binding calculation of the LDOS including heterostrain. (f) Tight-binding prediction for the local density of states in $AA$ regions as function of energy and for increasing uniaxial heterostrain. The variations are plotted for three different angles ${\theta }_s$ of application of heterostrain. (g) Local density of states calculated for 0.4% of heterostrain applying along varying ${\theta }_s$. The white dotted lines indicate the values of angles used in (f). (h) Sketch of uniaxial heterostrain configuration. One layer is deformed by a uniaxial heterostrain applied along the direction defined by ${\theta }_s$ and then rotated by an angle $\theta$ with respect to the undeformed layer.

Figure 2

Comparison between theory and experiments. (a) Representative doping dependence of the spacing between van Hove singularities. The horizontal line corresponds to prediction by tight-binding calculations with heterostrain. (b) Experimental spacing of van Hove singularities at large doping ($\mathrm{\Delta }{E}_{\exp }^0$), zero doping ($\mathrm{\Delta }{E}_{\exp }^{*}$), theoretical tight binding ($\mathrm{\Delta }{E}_{\mathrm{TB}}$), low-interaction Hartree-Fock ($\mathrm{\Delta }{E}_{\mathrm{HF}}^{\epsilon =\infty }$), and large interaction ($\mathrm{\Delta }{E}_{\mathrm{HF}}^{\epsilon =5}$). Such presentation of all the data on a single graph is justified by (i) the weak dependence of $\mathrm{\Delta }{E}_{\exp }$ on the twist angle near the magic angle (see Ref. [18] and Supplemental Material [20], Fig. S5) and (ii) the weak dependence of $\mathrm{\Delta }E$ on ${\theta }_s$ if it is estimated as the spacing between the outermost singularities [Fig. 1]. This is also why the Hartree-Fock model was calculated for a twist angle of 1.1° and ${\theta }_s=30°$. On the contrary, $\mathrm{\Delta }{E}_{\mathrm{TB}}$ were calculated with the full relative arrangement of the experimental data and show a better agreement with experiments. The twist angle has still a small influence, which explains the small deviations from a purely linear strain dependence. The measurements showing a cascade of transitions are indicated by a double border. (c) Relative deviation of $\mathrm{\Delta }{E}_{\exp }^0$ to $\mathrm{\Delta }{E}_{\mathrm{TB}}$ as a function of the tunneling resistance $R_t=V_b/i_t$. The figure includes new unpublished data for the sample of Fig. 1 (red crosses).

Figure 3

Tip-induced strain. (a)–(c) STM images measured at decreasing tunnel resistance. The tunneling set point is $i_t=100\mathrm{pA}$ and the bias is (a) $V_b=175$, (b) $V_b=55$, and (c) $V_b=15\mathrm{mV}$. The images are $25\times 25{\mathrm{nm}}^2$. (d)–(f) Local density of states measured by STM for decreasing tunneling conductance in (d) $AA$, (e) $AB$, and (f) intermediate regions. The vertical dotted lines are guides for the eyes to track the position of the LDOS peaks. The tip-sample interaction was defined by the tunneling set point prior to switching off the feedback loop and subsequent sweeping of the bias voltage between $-200$ and 200 mV. In each panel from top to bottom, the tunneling set point is $i_t=100\mathrm{pA}$ and the bias is, respectively, $V_b=-800$, $V_b=-200$, $V_b=-125$, and $V_b=-50\mathrm{mV}$. The $dI/dV$ (V) signal was measured using phase sensitive detection with a 2 mV oscillation at 263 Hz added to the tunnel bias. The $dI/dV$ curves were normalized to 1 at $-200\mathrm{mV}$.