Strain manipulation of Majorana fermions in graphene armchair nanoribbons

Graphene nanoribbons with armchair edges are studied for externally enhanced but realistic parameter values: enhanced Rashba spin-orbit coupling due to proximity to a transition-metal dichalcogenide, such as ${\mathrm{WS}}_2$, 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. The topological phase is highly sensitive to the application of uniaxial strain with a transition to the trivial state above a critical strain well below 0.1%. This sensitivity allows for real-space manipulation of Majorana fermions by applying nonuniform strain profiles. Similar manipulation is also possible by applying an inhomogeneous Zeeman field or chemical potential.

Figure 1

(a) Sketch of a graphene armchair ribbon with Majorana zero modes at both ends realized in the topological region of the phase diagram (b)–(d). (b) Phase diagram on the plane of chemical potential $\mu$, (c) Zeeman field $V_z$, (d) and strain direction $\theta$ vs strain $\varepsilon$. The topological phase with Majorana zero modes corresponds to the MZM region in the figures.

Figure 2

Lowest conduction band for an armchair ribbon of width $N_y=81$ 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 $\varepsilon =0,0.01\%,0.03\%$. For $\mathrm{\Delta }\ne 0$ the system develops a gap, which closes at $V_z=\mathrm{\Delta }$ (d)–(f).

Figure 3

Energy eigenvalue closest to zero energy for a finite ribbon of length $N_x={10}^4$ unit cells ($N_y=20$) as a function of (a) $\mathrm{\Delta }$, (b) $V_z$, and (c) strain $\varepsilon$. Berry phase for the infinite ribbon as a function of (d) $\mathrm{\Delta }$, (e) $V_z$, (f) and strain $\varepsilon$.

Figure 4

Effect of a strain profile in the Majorana zero mode wave function for three different profile heights parametrized by ${\varepsilon }_{\max }$ according to Eq. (3).

Figure 5

Effect of (a) nonuniform $\mu$ and (b) $V_z$ in the Majorana zero mode wave function according to Eqs. (4) and (5), respectively. Parameters: $x_0=0.5a{N}_x$ with $N_x={10}^4$ and $\zeta =0.5a$, where $a$ is the lattice spacing in the armchair direction. (c) The device proposed in the present Rapid Communication: graphene armchair nanoribbon sandwiched between high SOC transition-metal dichalcogenide ${\mathrm{WS}}_2$ and a thin layer of ferromagnetic insulator EuS, decorated with alkali-metal atoms Ca or Li.