Formation of topological domain walls and quantum transport properties of zero-line modes in commensurate bilayer graphene systems

doi:10.1007/s11467-022-1185-y

Front. Phys.

2022, Vol. 17

Issue (6) : 63503 https://doi.org/10.1007/s11467-022-1185-y

RESEARCH ARTICLE

Formation of topological domain walls and quantum transport properties of zero-line modes in commensurate bilayer graphene systems

Junjie Zeng¹, Rui Xue¹, Tao Hou¹, Yulei Han^1,², Zhenhua Qiao¹(

)

¹. ICQD, Hefei National Laboratory for Physical Sciences at Microscale, CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, Department of Physics, University of Science and Technology of China, Hefei 230026, China
². Department of Physics, Fuzhou University, Fuzhou 350108, China

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Abstract

We study theoretically the construction of topological conducting domain walls with a finite width between AB/BA stacking regions via finite element method in bilayer graphene systems with tunable commensurate twisting angles. We find that the smaller is the twisting angle, the more significant the lattice reconstruction would be, so that sharper domain boundaries declare their existence. We subsequently study the quantum transport properties of topological zero-line modes which can exist because of the said domain boundaries via Green’s function method and Landauer−Büttiker formalism, and find that in scattering regions with tri-intersectional conducting channels, topological zero-line modes both exhibit robust behavior exemplified as the saturated total transmissionG_tot ≈ 2e²/h and obey a specific pseudospin-conserving current partition law among the branch transport channels. The former property is unaffected by Aharonov−Bohm effect due to a weak perpendicular magnetic field, but the latter is not. Results from our genuine bilayer hexagonal system suggest a twisting angle aroundθ ≈ 0.1° for those properties to be expected, consistent with the existing experimental reports.

Keywords twistronics lattice reconstruction topological domain wall zero-line mode quantum transport

Corresponding Author(s): Zhenhua Qiao

Issue Date: 28 July 2022

Cite this article:

Junjie Zeng,Rui Xue,Tao Hou, et al. Formation of topological domain walls and quantum transport properties of zero-line modes in commensurate bilayer graphene systems[J]. Front. Phys. , 2022, 17(6): 63503.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-022-1185-y
https://academic.hep.com.cn/fop/EN/Y2022/V17/I6/63503

Fig.1 Boundary condition (BC II) in use. (a) Profiles of the envelope function Eq. (3) with

w / π = 1, 1 / 20

, which vanish identically at the two Bernal (AB/BA) stacking centers as well as the saddle point (SP). (b) The light-blue regular hexagonal region represents the domain

D

(a Moiré primitive cell for

θ = 0.2745 °

) over which the coupled equations Eq. (1) is to be solved; the periphery regular hexagon in red is the boundary

? D

of the domain

D

. The black arrows starting from

? D

schematize disproportionately the actual vectorially-valued boundary condition, which forms clockwise rotation pattern around each corners of

D

. Besides, those arrows enjoy the invariance of Morié lattice translation.

Fig.2 Boundary value problem facilitation. (a) Radial profile of the initial trial function with an in-plane full rotation symmetry, the monotonously increasing window

ρ ∈ [0, ρ 0]

mainly corresponds to the AA stcaking region, which is the magnitude of a counterclockwise vector field given in Eq. (5). Beyond this range we expect it to decrease gradually as the distance from the center increases. (b) Strategic domain discretization of a same domain

D

shown in Fig.1(b), special attention is paid to regions as AA stcaking center, six AA-SP straight lines, and the domain boundary

? D

are discretized into smaller cells compared to other regions, in the anticipation of achieving a balance between calculation accuracy and resource requirement.

Fig.3 In-plane vector relaxation field. Relaxation field magnitude and binding energy density from the vanishing boundary condition BC I Eq. (2) (a) and our devised boundary condition BC II Eq. (4) (b). The major difference between them lies at around the saddle point, where BC I leaves a higher binding energy profile than that from BC II. (c) The complete vector relaxation field from BC II Eq. (4), with the arrows indicating the correct flow directions both at the center and the corners, in addition to color-encoded magnitude.

Fig.4 Commensurate Moiré bandstructures after atomic reconstruction. (a?c) Bandstructures for three cases with different twisting angles subjected to vanishing interlayer potential bias. As twisting angle decreases, flat bands and subgaps (indicated by the light blue rectangles) appear. (d?f) Bandstructures for a single twisting angle with three different interlayer potential biases. The increasing of interlayer potential bias closes the subgaps in this

1.05 °

case. Charge neutrality point has been subtracted, and energy is measured in eV for comparison with results in Ref. [31].

Fig.5 Setup for investigating quantum transport properties of topological zero-line modes. (a) Central scattering region is constructed from a relaxed Moiré primitive cell (light green hexagonal background), where pink and black dots represent atoms from upper and lower layer, respectively. The narrow domain walls are unambiguously observable. Red and blue dots are two bilayer primitive cells of each semi-infinite armchair-edged leads, whose width is chosen almost as wide as the side length of the Moiré primitive cell. (b) Bandstructures of corresponding armchair nanoribbons constructed by the leads primitive cells in (a) with a full one-dimensional lattice translational symmetry, where in-gap (nearly gapless) zero-line modes are easily identifiable; the similar with (c) for a scattering region with a different twisting angle. Energy is measured in

V pp π

Fig.6 Transmissions and local density of states (LDOS) for three different twisting angles [corresponding to angles around

0.1 °

, c.f. Eq. (12) for rationale] without magnetic field of a six-terminal device. Panels (a1), (b1), and (c1) are branch transmissions

G i 1

, where

i = 2, 3, ?, 6

is the lead number assigned in Fig.5(a), and the total transmission

G tot = ∑ i = 2 6 G i 1

. Panels (a2), (b2), and (c2) are the corresponding local density of states near the charge neutrality point. The distribution of local density of states are approximately symmetric with respect to left and right. Energy is measured in

V pp π

and charge neutrality point is at

E cn ≈ 0.15

Fig.7 Transmissions and LDOS with magnetic field (AB phase is

2 π ? ~ B = 2 π / 10000

). Magnetic field introduces further fluctuations for the transmission curves, and makes it left-right asymmetric in the local density of states pattern. Energy is measured in

V pp π

and charge neutrality point is at

E cn ≈ 0.15

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