Majorana Zero Modes in Graphene

A clear demonstration of topological superconductivity (TS) and Majorana zero modes remains one of the major pending goals in the field of topological materials. One common strategy to generate TS is through the coupling of an $s$-wave superconductor to a helical half-metallic system. Numerous proposals for the latter have been put forward in the literature, most of them based on semiconductors or topological insulators with strong spin-orbit coupling. Here, we demonstrate an alternative approach for the creation of TS in graphene-superconductor junctions without the need for spin-orbit coupling. Our prediction stems from the helicity of graphene’s zero-Landau-level edge states in the presence of interactions and from the possibility, experimentally demonstrated, of tuning their magnetic properties with in-plane magnetic fields. We show how canted antiferromagnetic ordering in the graphene bulk close to neutrality induces TS along the junction and gives rise to isolated, topologically protected Majorana bound states at either end. We also discuss possible strategies to detect their presence in graphene Josephson junctions through Fraunhofer pattern anomalies and Andreev spectroscopy. The latter, in particular, exhibits strong unambiguous signatures of the presence of the Majorana states in the form of universal zero-bias anomalies. Remarkable progress has recently been reported in the fabrication of the proposed type of junctions, which offers a promising outlook for Majorana physics in graphene systems.

Popular Summary

Each matter particle in nature has been known, since the time of Paul Dirac, to have an associated antiparticle, with the exception of Majorana particles. Predicted by Ettore Majorana in 1937, Majorana particles are unique in that each is its own antiparticle. We are now tantalizingly close to finding Majorana quantum states, but not in high-energy accelerators as expected. They have recently been predicted to arise in particular solid-state systems where they may acquire a remarkable property, known as non-Abelian anyon statistics, which could revolutionize our ability to exploit quantum entanglement technologically. This property could also finally enable quantum computing. But where are Majoranas hiding—in semiconducting wells, nanowires, or atomic chains? Here, we predict that graphene could be an ideal platform to finally bring Majorana quantum states to light, which would arguably be graphene’s most remarkable application yet.

The setup required is simple and is likely realizable today thanks to recently demonstrated advances in fabrication. We propose that robust Majorana states can be found in graphene as a result of its unique electronic structure under strong magnetic fields, the so-called “zero Landau level with spontaneously broken symmetry.” Remarkably, by bringing this electronic state into close proximity with a conventional superconductor, Majorana states are generated without a stringent requirement of most other Majorana proposals—strong coupling between the electron’s spin and its momentum.

Graphene appears to be an appealing candidate for this specific task, and we anticipate that this material may well turn out to be a key ingredient in the future of quantum computation.

Figure 1

Sketch, band structure, and phase diagram of a 350-nm-wide graphene sample (Fermi velocity $v_F={10}^{6}m/s$) in the quantum Hall regime (out-of-plane field $B_z=1\mathrm{T}$) in various configurations. The chemical potential (dotted line) is tuned within the gap ${\mathrm{\Delta }}_{\mathrm{ZLL}}\approx 20\mathrm{meV}$ of the zero Landau level, which has ferromagnetic (a–c), antiferromagnetic (d–f), or canted antiferromagnetic ordering (g–i) due to electronic interactions. The band structures under each sketch correspond to an infinite graphene ribbon surrounded by vacuum (vac/Gr/vac panels) or coupled to a superconductor of gap ${\mathrm{\Delta }}_{\mathrm{SC}}$ (chosen to be large, around 7 meV, for visibility) along the top edge as in the sketches (vac/Gr/SC panels, Nambu bands). Bands in red correspond to eigenstates localized at the top edge of the ribbon. The zero-energy local density of states is shown in blue in each sketch. (j) Low-energy band structure of states along a graphene-superconductor interface folded onto the $\mathrm{\Gamma }$ point, in the gapless (left panel, zoom of panel f), nontrivially gapped (middle panel, canted order), and trivially gapped (right panel, with strong intervalley scattering) phases. (k) Phase diagram of said interface, computed from its low-energy effective Hamiltonian (see text), as a function of magnetic angle $\theta$ and intervalley coupling $w$. The three possible phases are shown, bounded by threshold values $w_0$, $w_{\mathrm{ins}}$, and ${\theta }_{\mathrm{ins}}$ (see main text).

Figure 2

Normalized critical current $I_c/I_c^0$ as a function of magnetic flux through a Josephson junction. Curves from bottom to top (shifted for better visibility) correspond to the noninteracting case (black), the gapless AF phase (purple), a trivially gapped AF phase (light gray), and a ferromagnetic phase with a topologically nontrivial gap (orange). Only the latter shows nondecaying nonzero minima in $I_c$, a consequence of an Andreev spectrum [inset (a)] with an odd number (one) of edge-resolved zero-energy crossings (bottom-right panel). States colored in red and blue are located at the top and bottom edges, respectively [inset (b)], while states in gray are spread across the width of the junction [inset (c)].

Figure 3

(a) In blue, density of the lowest Andreev level formed by the hybridization of four Majorana bound states in a square $500\text{−}\mathrm{nm}\times 500\text{−}\mathrm{nm}$ Josephson junction at different values of canting angle $\theta$. The rest of the parameters are as in Fig. 1. In A and C ($\theta <{\theta }_L$ and $\theta >{\theta }_W$, respectively), the hybridization is strong, while in B, it is exponentially suppressed, and the level remains a fourfold degenerate zero mode. (b–e) Transport spectroscopy $dI/dV$ from a normal point contact $N$ (transparent, single spinful channel) [see (a)]. (b, d) The $dI/dV$ (at $\theta =0$ and $\theta =0.1\pi$, respectively) as a function of the junction phase difference $\varphi$. Both exhibit an odd number (one) of zero-energy crossings of edge-resolved $dI/dV$ resonances, signaling a topologically nontrivial system. (c) $dI/dV$ as a function of canting angle $\theta$ at $\varphi =0$. It exhibits a zero-bias anomaly in the window ${\theta }_L<\theta <{\theta }_W$ (see dashed lines). (e) The $dI/dV$ dependence with out-of-plane magnetic field $B_z$ through the sample ($\varphi =0$, $\theta =0.1\pi$), showing the same zero crossings as with $\varphi$ each time the total flux $\mathrm{\Phi }=B_{z}LW$ is increased by a flux quantum ${\mathrm{\Phi }}_0$.

Figure 4

(a) $350\mathrm{nm}\times 350\mathrm{nm}$ sample with a single superconducting contact, otherwise identical to system of Fig. 3, main text. (b) and (c) show the differential conductance as a function of canting angle $\theta$ and out of plane magnetic field $B_z$. Both show the same phenomenology as panels (b) and (e) of the two contact setup in Fig. 3, albeit with a simpler two-Majorana geometry, and with the total length of the vacuum edge $L^{*}=2L+W$ playing the role of $L$.

Figure 5

Transport spectroscopy $dI/dV$ obtained in various arrangements of normal, trivial superconducting (SC), and topological superconducting (TS) nanowires, to be compared to Fig. 3. The wires are modeled after Refs. [16, 17]. (a) A trivial SC/normal/SC Josephson junction is probed by a third normal contact at bias $V$, as a function of the superconducting phase difference $\varphi$. No protected zero-bias $dI/dV$ anomalies arise. (b) Same as (a) for a TS/normal/TS junction. A universal $4e^2/h$ zero-bias anomaly (white) is obtained at a certain $\varphi$, here $\varphi =\pi$, for which the two Majoranas in the junction (blue circles in the inset) decouple. (c) A normal/SC junction that transitions into normal/TS with two Majoranas as the longitudinal Zeeman field $V_Z$ exceeds a critical value $V_Z^c$. Apart from a small oscillatory splitting that decays exponentially with TS length, the $dI/dV$ in the latter case shows a universal $2e^2/h$ zero-bias anomaly (orange).

Figure 6

(a) Sketch of a hybrid AF-SC ribbon, with region colors as those used in the band structure. (b) Band structure of a hybrid ribbon with doped SC and armchair interface. Band structure of a hybrid ribbon with undoped (c) and doped (d) SC with zigzag interface, which shows the different interface states in each valley.

Figure 7

(a) Scheme of the effect of chemical doping in the Dirac spectrum of proximized graphene. (b) Schematic phase diagram of the interface electronic spectrum between the SC and the QH ferromagnet, as a function of the in-plane field and the chemical doping of the SC. Quasiparticle energies for a gapless (c) and gapped (d) interface. Phase boundary obtained by numerical calculation at low dopings (e) and at large doping (f). The green and red colored states of the band structures correspond to the states localized at the interface, where the topological superconducting gap is calculated between the red states shown in (c,d). The blue gapless states correspond to the chiral states between the QH and vacuum.

Figure 8

Band structure of the interface between a square superconductor and an antiferromagnet (a) or canted antiferromagnet (c) honeycomb lattice. Panels (b) and (d) show the local density of states in a finite system, which correspond to gapless interface channels (without canting) (b) and Majorana states (with canting) (d).