Edge Properties and Majorana Fermions in the Proposed Chiral $d$-Wave Superconducting State of Doped Graphene

We investigate the effect of edges on the intrinsic $d$-wave superconducting state in graphene doped close to the van Hove singularity. While the bulk is in a chiral $d_{x^2-y^2}+i{d}_{xy}$ state, the order parameter at any edge is enhanced and has $d_{x^2-y^2}$-symmetry, with a decay length strongly increasing with weakening superconductivity. No graphene edge is pair breaking for the $d_{x^2-y^2}$ state, and there are no localized zero-energy edge states. We find two chiral edge modes which carry a spontaneous, but not quantized, quasiparticle current related to the zero-energy momentum. Moreover, for realistic values of the Rashba spin-orbit coupling, a Majorana fermion appears at the edge when tuning a Zeeman field.

Figure 1

(a) Graphene with NN bonds ${\delta }_{\alpha }$, NNN bonds ${\gamma }_{\alpha }$, and zigzag and armchair edges indicated. (b) Different $d$-wave superconducting order parameters for NN and NNN pairing with negative (blue) and positive (red) sign.

Figure 2

(a) Order parameter profile for the zigzag edge for NN $J=0.75t$ at the VHS with real (black lines) and imaginary (red lines) parts for ${\Delta }_1$ (thick line), ${\Delta }_2$ (thin line), and ${\Delta }_3$ (dashed line) [the black dashed line is hidden behind the black solid line since $\mathrm{Re}({\Delta }_2)\approxeq \mathrm{Re}({\Delta }_3)$]. (b) Character of the order parameter in (a): $d_1$ (thick black line), $d_2$ (thick red line), and $s$ (thin black line). The dotted line marks the bulk value. (c) Decay length $\xi$ of the $d_1$ character as a function of $\Delta F$ for different doping levels, edges, and superconducting pairing: NN pairing, zigzag edge, and $\mu =t$ (black cross), $\mu =0.8t$ (red circle), $\mu =1.2t$ (green diamond), or armchair edge and $\mu =t$ (blue cross), NNN pairing, zigzag edge, and $\mu =t$ (black asterisk), $\mu =0.8t$ (red square), $\mu =1.2t$ (green upward triangle) [blue, cross symbols are often completely overlaying black, cross symbols since no notable difference is found between zigzag and armchair edges]. The inset shows $\Delta F$ as a function of the pairing potential for NN pairing (black lines) and NNN pairing (red lines) for $\mu =t$ (thick line) and $\mu =1.2t$ (thin line). $\mu <t$ has a $\Delta F$ curve similar to $\mu >t$.

Figure 3

(a) Band structure for a zigzag edge slab with NN $J=0.75t$, $\mu =t$, and self-consistent $\Delta$ (thick black line), constant $d_1+i{d}_2$ state corresponding to the bulk state (thin black line), and constant $d_1$ state corresponding in amplitude to the $d_1$ state at the surface. (b) LDOS across the ribbon for the self-consistent solution in (a) interpolating between 0.2 (black) to 0 (white) states/(eV unit cell), showing a bulk gap of 0.18 eV and gapless edge states. (c) Quasiparticle edge current in units of $e/h$ as a function of superconducting bulk order parameter $\Delta (1,e^{2\pi i/3},e^{4\pi i/3}$) for the zigzag edge with $\mu =t$ (black cross) and $\mu =0.8t$ (red circle) and the armchair edge with $\mu =t$ (green upward triangle).

Figure 4

(a) Eigenvalue spectrum for a zigzag slab with NN $J=1.2t$, $\mu =t$, ${\lambda }_R=0.2t$, and $h_z=0.4$, 0.535, and 0.6 eV (left to right), with $h_c=0.535\mathrm{eV}$. Small gaps in the surface states are due to limited $k$-point sampling. (b) Self-consistent $\Delta$ as a function of $h_z$ for $J=1.2t$ (black line) and $0.9t$ (red line) for ${\lambda }_R=0.05t$ (thick line), $0.2t$ (thin line), and $0.3t$ (dashed line). The dotted line marks the $h_c=2\Delta$ one-band model phase transition. Crosses mark the numerical phase transition. (c) Eigenvalue amplitude squared for the Majorana mode in (a) for $h_z=0.54\mathrm{eV}$ (black line), 0.56 (dashed line), and 0.6 (red line).