Measuring local moiré lattice heterogeneity of twisted bilayer graphene

We introduce a new method to continuously map inhomogeneities of a moiré lattice and apply it to large-area topographic images we measure on open-device twisted bilayer graphene (TBG). We show that the variation in the twist angle of a TBG device, which is frequently conjectured to be the reason for differences between devices with a supposed similar twist angle, is about ${0.08}^{\circ }$ around the average of ${2.02}^{\circ }$ over areas of several hundred nanometers, comparable to devices encapsulated between hexagonal boron nitride slabs. We distinguish between an effective twist angle and local anisotropy and relate the latter to heterostrain. Our results imply that for our devices, twist angle heterogeneity has an effect on the electronic structure roughly equal to that of local strain. The method introduced here is applicable to results from different imaging techniques and on different moiré materials.

Figure 1

(a) Moiré pattern created by stacking two hexagonal lattices with a twist angle of $5^{\circ }$. (b) Schematic representation of our spatial lock-in algorithm to map the local twist angle and anisotropy. The measured lattice can be thought of as the result of a series of transformations applied to an ideal lattice. The scaling transformation $D$ holds information about the local twist angle ${\theta }^{*}\left(\mathbf{r}\right)$ and the intrinsic local strain present in the device, $\kappa \left(\mathbf{r}\right)$. V gives the direction of this local strain, $\psi \left(\mathbf{r}\right)$. Finally, $W$ indicates the relative angle between the bilayer and the underlying hBN substrate.

Figure 2

(a) STM topography of a device with an average twist angle of $\theta =2.{38}^{\circ }$ (setup conditions: V = 250 mV, I = 100 pA). The topography shows both the atomic and moiré lattice. (b) Fourier transform of (a), with zoom-ins of the moiré peaks (green inset) and the bottom left atomic peak (blue inset). Satellite peaks of the moiré lattice are visible around the atomic peak as well. (c) Large-scale topography measured on a different device with an average twist angle of $2.{02}^{\circ }$ (setup conditions: V = 250 mV, I = 20 pA).

Figure 3

Lock-in in 1D. The panels in the left column show, from top to bottom, the signal (an almost periodic sinusoid), the real part of the reference, the real part of the product of the signal and reference, and the wavelength calculated by taking the gradient of the phase of the product signal. The right column displays the Fourier transform of the (complex) signals in the left column. Finally, in the bottom right curve, the orange dashed line represents the Gaussian filter used for the lock-in procedure.

Figure 4

(a) STM topography of a device with an average twist angle of $2.{02}^{\circ }$ [V = 250 mV, I = 20 pA, same data as in Fig. 2]. The blue circle at the bottom right indicates the size of the filter used by the algorithm (see text). (b) Fourier transform of (a), showing the Bragg peaks of the moiré lattice visible in the image. The Bragg peaks are labeled $q_1$–$q_3$. (c) Effective twist angle map extracted from (a), by the algorithm discussed. The red square indicates the area over which the average twist angle and standard deviation are calculated. (d) Local moiré anisotropy map $\kappa \left(\mathbf{r}\right)$ extracted by the algorithm from (a). (e) Local moiré anisotropy direction $\psi \left(\mathbf{r}\right)$ extracted by the algorithm from (a). (f) Heterostrain map extracted as described in the text.

Figure 5

Phase maps of the data shown in Fig. 4 [and Fig. 8]. (a)–(c) correspond to the phase maps of the Bragg peak labeled ${\mathbf{q}}_1$, ${\mathbf{q}}_2$, and ${\mathbf{q}}_3$, respectively [see Fig. 4]. Because the map corresponding to ${\mathbf{q}}_2$ shows some phase singularities, we use ${\varphi }_1$ and ${\varphi }_2$ for determining the displacement field.

Figure 6

Schematic overview of the devices studied in this paper. The contact labeled GND can be used to bias or ground the bilayer.

Figure 7

(a) Lawler-Fujita corrected STM topography of Fig. 4 [and Fig. 8]. (b) Extracted effective twist angle map of (a). (c) Extracted residual local anisotropy map of (a). (d) Local anisotropy angle of (c). (e) Heterostrain map of (a). (f) Local moiré rotation of (a). This angle corresponds to the angle in the $W$ matrix [Eq. (A2)].

Figure 8

Spatial lock-in output for sequentially measured topographies in the same field of view. (f) was measured at 65 nm/s, whereas (a)–(e) were measured with a scan speed of 54 nm/s. The setup condition was kept constant between measurements: V = 250 mV, I = 20 pA.

Figure 9

(a) STM topography of a TBG device with an average twist angle of $2.{16}^{\circ }$ (setup conditions: V = 170 mV, I = 20 pA). (b) Extracted effective twist angle map of (a). (c) Extracted local anisotropy map of (a). (d) Local anisotropy angle of (c). (e) Heterostrain map of (a). (f) Local moiré rotation of (a). This angle corresponds to the angle in the $W$ matrix [Eq. (A2)].

Figure 10

(a) STM topography of a TBG device with an average twist angle of $2.{01}^{\circ }$ (setup conditions: V = 350 mV, I = 100 pA). (b) Extracted effective twist angle map of (a). (c) Extracted local anisotropy map of (a). (d) Local anisotropy angle of (c). (e) Heterostrain map of (a). (f) Local moiré rotation of (a). This angle corresponds to the angle in the $W$ matrix [Eq. (A2)].

Figure 11

(a) STM topography of a device with an average twist angle of $2.{02}^{\circ }$ [V = 250 mV, I = 20 pA, same data as in Fig. 4]. (b) Effective twist angle map extracted from (a), by the algorithm discussed. The red square indicates the area over which the average twist angle and standard deviation are calculated. (c) Effective twist angle map corresponding to the area marked by the red square in (b).

Figure 12

(a) Effective twist angle map extracted from the data shown in Fig. 4. (b) Local moiré anisotropy map $\kappa \left(\mathbf{r}\right)$ extracted from the data shown in Fig. 4. (c) Heterostrain map extracted from the data shown in Fig. 4. (d)–(f) Histograms of the maps displayed above. The light blue histograms count the full data as shown, whereas the dark blue histograms exclude a border of two times the filter radius used in the lock-in procedure [pixels outside of the dark blue border in (a)]. Finally, the red histograms count the data inside the area marked by the red square in (a).

Figure 13

(a) STM topography of a device with an average twist angle of $2.{02}^{\circ }$ [V = 250 mV, I = 20 pA, same data as in Fig. 4]. (b) Effective twist angle map extracted from (a), by the algorithm discussed. (c) Twist angle map of (b) after smearing with a Gaussian filter with a sigma of 15 nm.

Figure 14

(a) STM topography of a device with an average twist angle of $2.{02}^{\circ }$ [V = 250 mV, I = 20 pA, same data as in Fig. 4]. (b) Zoom-in of (a). (c) Same as in (b), but plotted with interpolative shading. The red marker indicates the position of the Gaussian fitted to that moiré site, and the green cross represents the extent of the 95% confidence interval.