One-dimensional topological superconductivity at the edges of twisted bilayer graphene nanoribbons

Zhen-Hua Wang, Fuming Xu, Lin Li, Rong Lü, Bin Wang, and Wei-Qiang Chen

Phys. Rev. B 100, 094531 – Published 27 September 2019

Abstract

Twisted bilayer graphene is one of the simplest van der Waals structures, and its inhomogeneous interlayer coupling can induce rich electronic properties. In twisted bilayer graphene nanoribbons (tBLGNRs), the interlayer coupling strengths are different for two ribbon edges due to the inhomogeneous bonding, which splits the edge states into two individuals in energy. The lower-energy state, localizing at the ribbon edge with the stronger interlayer coupling, is a good candidate to generate one-dimensional (1D) topological superconductivity in the presence of Rashba spin-orbit coupling, Zeeman field, and $s$ -wave superconductivity. Majorana zero modes (MZMs) are found to be localized at both ends of this edge. The topological invariants of the system are explored by evaluating the Berry phase for infinite-length ribbons and Majorana polarization for quasi-1D ribbons, giving the same topological phase diagram. More importantly, by adjusting interlayer dislocation and uniaxial strain of tBLGNRs across the critical values, the lower-energy edge changes and 1D topological superconductivity can “jump” from one ribbon edge to the other one. Finally, by applying a gate voltage bias between bilayers or changing the interlayer distance, a MZM can transfer along the ribbon edge. The tBLGNRs provide an alternative platform to study 1D topological superconductivity and MZMs.

Received 27 March 2019
Revised 14 August 2019

DOI:https://doi.org/10.1103/PhysRevB.100.094531

Physics Subject Headings (PhySH)

Majorana bound states Topological superconductors

Graphene

Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Zhen-Hua Wang^1,2,3, Fuming Xu¹, Lin Li^4,2, Rong Lü^5,6, Bin Wang^1,7,*, and Wei-Qiang Chen^2,7,†

¹Shenzhen Key Laboratory of Advanced Thin Films and Applications, College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China
²Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
³Beijing Computational Science Research Center, Beijing 100193, China
⁴College of Physics and Electronic Engineering and Center for Computational Sciences, Sichuan Normal University, Chengdu 610068, China
⁵Department of Physics, Tsinghua University, Beijing 100084, China
⁶Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
⁷Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518055, China

^*binwang@szu.edu.cn
^†chenwq@sustech.edu.cn

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Issue

Vol. 100, Iss. 9 — 1 September 2019

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Images

Figure 1
(a) Lattice structure of AB-stacked bilayer graphene. Solid and dashed lines represent lattices of top and bottom layers, respectively. $δ$ are the vectors connecting the nearest-neighbor atoms, $a_{1}$ and $a_{2}$ are the primitive vectors, and $θ$ is the direction of applied tension $T$ . (b) Lattice structure of (1,2) tBLGNR with $N_{y} = 4$ and rotation angle $φ = 21.8^{\circ}$ based on $a_{1}^{(l)}$ and $a_{2}^{(l)}$ . The blue parallelogram in the lower right corner defined by $L_{1}$ and $L_{2}$ indicates a unit cell of (1,2) tBLG. Inset of (b): The atomic structure and the interlayer coupling distribution in the unit cell. The interlayer coupling is stronger in edge 1. (c) Configuration of experimental proposals: tBLGNR sandwiched between ${WS}_{2}$ and ferromagnetic insulator EuS, decorated with alkali-metal atoms Ca or Li.
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Figure 2
The lowest energy band for (1,2) tBLGNRs with $N_{y} = 6$ . (a) Split edge states due to inhomogeneous interlayer coupling. State 1 localizes at edge 1 with the stronger interlayer coupling, and state 2 localizes at edge 2 with the weaker interlayer coupling. (b) Rashba SOC induced spin split of state 1 with $λ = 0.54$ meV and (c) spin-resolved band shift due to Zeeman coupling with spin gap equal to $2 V_{Z} (V_{Z} = 1.08$ meV). Solid curves, bands with interlayer distance $d_{0} = 0.335$ nm; dashed curves, bands with a little smaller interlayer distance. (d–f) Topological trivial to nontrivial evolution of superconducting gap versus $Δ$ and $V_{Z}$ .
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Figure 3
Majorana zero modes and topological invariants. (a) Energy eigenvalues (blue curve, solid circles) closest to zero and MP $C$ (green curve, open circles) as a function of $Δ / V_{Z}$ for a quasi-1D (1,2) tBLGNR with $N_{x} = 10 000$ and $N_{y} = 6$ . (b) Berry phase $γ$ versus $Δ / V_{Z}$ for a periodic (1,2) tBLGNR with $N_{y} = 6$ . (c, d) Real-space distribution of the MZMs in top layer and bottom layer of quasi-1D (1,2) tBLGNRs. The quasi-1D topological superconductivity is achieved at edge 1, and the MZMs localize at both ends of edge 1. (e, f) Real-space distribution of the MZMs in a finite AB-stacked armchair GNR with $N_{x} = 10 000$ and $N_{y} = 21$ eight-atom unit cells. The arrows in (c–f) represent local MP at partial carbon atoms for the zero-energy states.
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Figure 4
The 1D topological superconductivity and Majorana end states “jump” from one edge of (1,2) tBLGNRs to the opposite edge, induced by the (a, b) interlayer dislocation and (c, d) uniaxial strain. In (a) and (c), dislocation and strain are below the critical values, and in (b) and (d) they are above. With the increase of $d_{s}$ and/or $ɛ$ , the lower-energy edge state changes and the 1D topological superconductivity appears at opposite edges. The “ferromagnetic” MP structures in the ellipses confirm the existence of Majorana states in a quasi-1D system. The used parameters are the same as in Fig. 2; the finite tBLGNR size is $N_{y} = 6$ and $N_{x} = 1.5 \times 10^{4}$ .
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Figure 5
(a, b) The evolution of the two edge states as function of interlayer dislocation $d_{s}$ and uniaxial strain $ɛ$ . At the critical values, $d_{s} = 0.143$ and $ɛ / | t_{0} | = 0.023 %$ , the two edge states cross at $k = π$ and the lower-energy edge changes beyond the critical values. The chemical potential $μ$ (black dashed line) is fixed in the topological region of the lower-energy edge state. (c, d) The jump behaviors between edge 1 and edge 2 in a wider ribbon. The local MP shows that the two “quasi-Majorana” end states overlap with each other in the middle of the edge.
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Figure 6
The transfer of a MZM along the stronger interlayer coupling edge of (1,2) tBLGNR as a function of (a–c) gate voltage bias $Δ μ$ and (d–f) interlayer distance $d_{0}$ . (a) Three gates are used to control the chemical potential of related regions. The chemical potential of the bottom layer is fixed by gate $V_{0}$ , and the chemical potentials of the two ends of the top layer are controlled by gates $V_{1}$ and $V_{2}$ , respectively. (b) Berry phase (green lines) for the infinite ribbon and the MP $C$ (blue dots) for the fully open ribbon are given as a function of $Δ μ$ . The same phase diagram is obtained. Above the critical value, the related region enters into a trivial phase and the MZMs are transferred, as shown in (c). (d–f) The interlayer distance can be decreased by pressure and, therefore, transfer a MZM. The used parameters are the same as in Fig. 2.
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