Transport across twist angle domains in moiré graphene

Many experiments in twisted bilayer graphene (TBG) differ from each other in terms of the details of their phase diagrams. Few controllable aspects aside, this discrepancy is largely believed to arise from the presence of a varying degree of twist angle inhomogeneity across different samples. Real-space maps indeed reveal TBG devices splitting into several large domains of different twist angles. Motivated by these observations, we study the quantum mechanical tunneling across a domain wall (DW) that separates two such regions. We show that the tunneling of the moiré particles can be understood by the formation of an effective step potential at the DW. The height of this step potential is simply a measure of the difference in twist angles. These computations lead us to identify the global transport signatures for detecting and quantifying the local twist angle variations. In particular, using Landauer-Büttiker formalism, we compute single-channel conductance ($dI/dV$) and Fano factor for shot noise (ratio of noise power and mean current). A reduction in the low-energy conductance and a zero-bias sub-meV gap in the conductance are observed which scale with the twist angle difference. One of the key findings of our work is that transport in presence of twist angle inhomogeneity is “noisy,” though sub-Poissonian. In particular, the differential Fano factor peaks near the van Hove energies corresponding to the domains in the sample. The location and the strength of the peak are simply a measure of the degree of twist angle inhomogeneity.

Figure 1

Schematic of the system. (Left) A TBG sample with two different twist angles (${\theta }_{\mathrm{L}},{\theta }_{\mathrm{R}}$) across a domain wall along the $y$ axis. We analyze tunneling characteristics of incident moiré electrons from the left, with momentum $\vec{k}$ measured from the $M$ point and incident angle ${\varphi }_k$. In TBG, there always exist two evanescent modes, $e_{\mathrm{L},\mathrm{R}}$, near a DW. Moiré electrons can tunnel when there also exist propagating modes, $p_{\mathrm{L},\mathrm{R}}$. In general, the MBZs of sizes $2K_{{\theta }_{\mathrm{L},\mathrm{R}}}$ can tilt by ${\phi }_{\mathrm{L},\mathrm{R}}$ angles with respect to the DW. (Middle) Using the Bistritzer–MacDonald model, the low-energy moiré dispersion ($\theta =1.{18}^{\circ }$) is obtained along the cut (green lines) shown in the left panel. The inset compares this with the effective dispersion obtained in Eq. (4). (Right) For a BZ cut transverse to the zone boundary ($K\text{−}{K}^{\prime }$ line), the dispersion (black curve) is gapless at the Dirac points ($k_y=\pm K_{\theta }$) but is maximally gapped (gray curve) at the $M$ point ($k_y=0$).

Figure 2

Normal tunneling. (Top) Electronic dispersions for $k_y=0$ and ${\phi }_{\mathrm{L},\mathrm{R}}=0$ with ${\theta }_{\mathrm{L}}<{\theta }_{\mathrm{R}}$. An electrons (green dot) can tunnel across the DW only when the Fermi level (${\epsilon }_F$) is greater than the largest among the two energy minima, $\max \left({\epsilon }_{\mathrm{v}}^{\mathrm{L}},{\epsilon }_{\mathrm{v}}^{\mathrm{R}}\right)$. The blue region delineates the classically forbidden ($R=1$) region, though this can contribute to tunneling if another TBG with $\theta <{\theta }_{\mathrm{R}}$ is placed to its right (see inset in the lower panel). The gray region does not allow any propagating modes to exist. The tunneling problem considered here can be understood through formation of an effective step potential of height, ${\mathrm{\Delta }}_{\mathrm{step}}=|{\epsilon }_{\mathrm{v}}^{\mathrm{R}}-{\epsilon }_{\mathrm{v}}^{\mathrm{L}}|$. (Bottom) Normal tunneling across a DW as a function of chemical potential, with ${\theta }_{\mathrm{L}}=1.1^{\circ }<{\theta }_{\mathrm{R}}$ (see legends) and ${\phi }_{\mathrm{L},\mathrm{R}}=0$. For reasons discussed above, $T$ is finite only when $\epsilon >{\epsilon }_{\mathrm{v}}^{1.1^{\circ }}+{\mathrm{\Delta }}_{\mathrm{step}}$ (legends). Clearly, the larger the difference in twist angles, the larger is the tunneling gap, ${\mathrm{\Delta }}_{\mathrm{step}}$. (Inset) We place another DW parallel but far away from the first one. The twist angles from left to right are ${\theta }_1=1.1^{\circ },{\theta }_2=1.2^{\circ },{\theta }_3$ (see legend). When ${\theta }_3>{\theta }_2$ the effective ${\mathrm{\Delta }}_{\mathrm{step}}$ increases otherwise it decreases. For the case ${\theta }_1={\theta }_3$, resonant tunneling occurs since the entire blue region is now allowed to tunnel, see Sec. 4c for details.

Figure 3

Oblique Incidence for $({\theta }_{\mathrm{L}},{\theta }_{\mathrm{R}})=(1.1^{\circ },1.2^{\circ })$. The left panel shows a polar plot of tunneling amplitude as a function of quasiparticles incidence angle. The curves are obtained by numerically solving Eq. (13) for various energies (measured in the units of ${\epsilon }_{\mathrm{v}}^{\mathrm{L}}$). All four quadrants in the polar plot are symmetric. The polar spread of the curves decreases with decreasing energy. This is explained through the top right panel. The allowed value of incidence angle or $k_y$ momentum for propagating states decreases as one approaches the Dirac point. Note, for a given value of $\varphi$, there can be more that two real propagating modes, especially for $\epsilon \lesssim {\epsilon }_{\mathrm{v}}$. (Bottom right) plotting the tunneling as a function of energy demonstrates the $k_y$ dependence of the ${\mathrm{\Delta }}_{\mathrm{step}}$.

Figure 4

Corrections to normal tunneling: (a) for finite ${\phi }_j$, $T$ receives a perturbatively corrected. This result is obtained in Eq. (27). The solid (blue) curve is for a longitudinal DW and the dashed (gray) curve is for a tilted DW. (b) When the DW is considered to be of small but finite width, tunneling receives a decremental correction, see Sec. 4d. The widths ($w$) of the DWs are indicated in units of inverse Dirac momentum. The inset, drawn for different energy slices of the $({\theta }_{\mathrm{L}},{\theta }_{\mathrm{R}})=(1.1^{\circ },1.3^{\circ })$ curve, shows that tunneling probability decreases rapidly with increasing width of the DW.

Figure 5

Quantum transport using Landauer-Büttiker formalism: (a) irrespective of the twist disorder, since there are no tunneling states for zero energy, conductance vanishes at zero bias. In presence of twist disorder, a transport gap appears (magnified in the inset) that scales with ${\mathrm{\Delta }}_{\mathrm{step}}$. The location of the inflection point (gray dashed line), at $\epsilon ={\epsilon }_{\mathrm{v}}^{\mathrm{L}}+{\mathrm{\Delta }}_{\mathrm{step}}\equiv {\epsilon }_{*}$, characterizes the strength of the twist disorder. (b) Due to the absence of tunneling, Fano factor at $V=0$ assumes a much larger constant, though $\le 1$; we do not obtain this value here due to reduced numerical stability. As tunneling increases, $F$ reduces and ultimately vanishes for high bias potential (since for high energies, $T=1$). A peak appears at ${\epsilon }_{*}$, the height of which scales with ${\mathrm{\Delta }}_{\mathrm{step}}$.

Figure 6

(a) Cartoon rendition of Snell's law for Dirac electrons. When ${\theta }_{\mathrm{R}}>{\theta }_{\mathrm{L}}$, for particles incident at an angle larger than a critical angle, ${\varphi }_{\mathrm{L}}>{\varphi }_{\mathrm{L}}^c$, there is perfect reflection (gray line). The orange (dashed) line in (b) demonstrates this for $({\theta }_{\mathrm{L}},{\theta }_{\mathrm{R}})=(1.1^{\circ },1.2^{\circ })$. For ${\theta }_{\mathrm{R}}<{\theta }_{\mathrm{L}}$ tunneling persists for all incidence angle, see the blue (solid) curve in (b) for $({\theta }_{\mathrm{L}},{\theta }_{\mathrm{R}})=(1.1^{\circ },0.7^{\circ })$. Both curves are obtained for $\epsilon =0.1{\epsilon }_{\mathrm{v}}^{1.1^{\circ }}$, however, as evident from Eq. (C8), the dependence of $T\left({\varphi }_{\mathrm{L}}\right)$ on energy (when finite) is extremely weak. For normal incidence ($q_y=0$), irrespective of the tilt angle, $T=1$ for the gapless ($\epsilon =0$) particles. For low-energy gapped particles, $T=T_0^{\perp }$ is a constant in energy. The inset in (b) shows how $T_0^{\perp }$ decreases from 1 with increasing difference in twist angles.