Strain Manipulation of Majorana Fermions in Graphene Armchair Nanoribbons
Update: 2019-10-22 09:18:45      Author:

Majorana fermions—particles which are their own antiparticles [1]—have recently been the subject of intense research due to the real prospect of realizing such exotic particles in condensed-matter platforms [2,3]. Graphene with its highly tunable properties is a tempting platform when it has a vanishing spin-orbit coupling (SOC). Recently, Zhen-Hua Wang, Eduardo V. Castro, and Hai-Qing Lin have studied the Majorana zero mode in Graphene nanoribbons with armchair edges: enhanced Rashba spin-orbit coupling due to proximity to a transition-metal dichalcogenide, such as WS2, and enhanced Zeeman field due to exchange coupling with a magnetic insulator, such as EuS under an applied magnetic field. The presence of s-wave superconductivity induced either by proximity or by decoration with alkali-metal atoms, such as Ca or Li, leads to a topological superconducting phase with Majorana end modes (see Fig. 2(c)).

For an insulating armchair ribbon, Rashba SOC lifts spin degeneracy by displacing the parabolic bands horizontally in opposite directions. The spectrum can be seen in Fig. 1(a) for a conservative SOC value of λ ≈ 0.6 meV. The Zeeman coupling lifts the remaining spin degeneracy at k = 0, opening up a gap of value 2Vz as shown in Fig. 1(b). Strain has a strong impact on the system and this is shown in Fig. 1(c) for strains ε = 0, 0.01%, 0.03%. In the presence of a finite s-wave superconducting pairing Δ the system becomes gapped as shown in Fig. 1(d) for Vz < Δ. The system goes through a gapless transition point at Vz = Δ [Fig. 1(e)] and becomes gapped again for Vz > Δ [Fig. 1(f)]. The presence of a finite gap with a gap closing point separates the topological and trivial phases.

The advantage of graphene compared to other platforms is that not only strain, but also other parameters—such as the chemical potential μ and the Zeeman coupling Vz— can be used to manipulate Majorana zero modes. Fig. 2(d) shows the Majorana zero mode wave-function amplitude along the ribbon for different strain profile heights εmax. If εmax < εc, the two Majorana zero modes appear localized at the two opposite ends of the ribbon (top panel). At εmax = εc the right side of the ribbon (x>x0) goes through a topological phase transition, and the gap closes (middle panel). For εmax > εc the region x>x0 becomes trivial, and one of the Majorana zero modes localizes at x0 (bottom panel). The Majorana zero mode has thus been transferred to x0 through strain manipulation. Applying a nonuniform chemical potential and Zeeman field, the topological region can be confined to an acquired region (Fig. 2(a)-(b)). Graphene armchair nanoribbon appears to be good candidates to realize topological superconductivity and provides new platforms to design topological quantum computation [4].


Fig. 1: Lowest conduction band for an armchair ribbon of width Ny = 81 unit cells. The effect of Rashba SOC is shown in (a), and the effect of Zeeman coupling is shown in (b), whereas in (c) we show the evolution of the lowest band with strain ε = 0, 0.01%, 0.03%. For Δ ≠ 0 the system develops a gap, which closes at Vz = Δ (d)–(f).


Fig. 2: Effect of (a) nonuniform μ and (b) Vz in the Majorana zero mode wave function. (c) The device proposed: graphene armchair nanoribbon sandwiched between high SOC transition-metal dichalcogenide WS2 and a thin layer of ferromagnetic insulator EuS, decorated with alkalimetal atoms Ca or Li. (d) Effect of a strain profile in the Majorana zero mode wave function for three different profile heights parametrized by εmax.


[1]     F. Wilczek, Nat. Phys. 5, 614 (2009).

[2]     J. Alicea, Rep. Prog. Phys. 75, 076501 (2012).

[3]     C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys. 4, 113 (2013).

       [4]     Zhen-Hua Wang, Eduardo V. Castro, Hai-Qing Lin, Phys. Rev. B, 97, 041414(R) (2018).

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