Zigzag Graphene Nanoribbons as a Majorana Platform

arXiv:2506.14999 · cond-mat.mes-hall · Submitted 2025-06-17 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Zigzag Graphene Nanoribbons as a Majorana Platform".

Mira: The gist Zigzag graphene nanoribbons support a variety of electronic phases that can be tuned via external stimuli,

Kai: First, who's behind it and why it matters.

Paper summary: Mira: To wrap up this discussion on Zigzag Graphene Nanoribbons as a Majorana Platform, we've seen how this research proposes that zigzag graphene nanoribbons can be tuned to support different electronic phases using external stimuli like electric fields #pg1. The central thesis is that coupling them to an Ising superconductor can induce an odd parity spin-triplet pairing gap in the half-metallic phase of the nanoribbon #pg1.

Kai: And this induced phase leads directly to a topologically nontrivial superconducting state with gate-tunable transitions that allow for the emergence of Majorana zero modes at their ends #pg1. This is the core contribution, showing a fully electric platform for realizing MZMs.

Lev: What this means in simpler terms is that we've identified a system where you can create topological superconductivity using only electric fields and spin interactions, rather than relying on an external magnetic field to do the job #pg1.

Mira: Right. The paper by Ruize Ma, Michele Pizzochero, Gaurav Chaudhary on Zigzag Graphene Nanoribbons as a Majorana Platform is important because it lays out a specific theoretical blueprint for how this can happen #pg1. It connects the magnetic properties of graphene with superconductivity in a way that’s not explored much yet #pg1.

Kai: It gives us a concrete model to start testing the experimental realization, even though they admit they have to ignore things like Rashba spin-orbit coupling in their model #pg3. That's where the next set of experiments will likely need to focus #pg3.

Lev: Exactly. The practical challenge is taking that theoretical blueprint and figuring out how to engineer the required Ising coupling on a scalable device while keeping the fidelity high enough to actually measure those zero modes #pg1. It’s a whole chain of challenges from theory to hardware, you know?

Conclusion: Kai: So basically, the authors are showing how you can use these zigzag graphene nanoribbons, which have that magnetic property from the edge states, and couple them to an Ising superconductor to get a topological superconducting state.

Mira: And what they’re doing is using that coupling to induce this spin-triplet pairing gap in the half-metallic phase of the nanoribbon. That's pretty unexpected because you usually expect a different kind of pairing symmetry when you start mixing materials like that.

Lev: From my side, I see it as a platform where we can study non-trivial topological phases without needing an external magnetic field, which is a big deal for building real hardware.

Kai: Yeah, I’m thinking about what this means because they’re showing how you can tune the system with just an electric field or by changing the chemical potential to get those zero modes at the ends of a finite strip.

Mira: But we have to remember all those assumptions they made, like assuming perfect lattice matching and ignoring things like Rashba spin-orbit coupling, which is where my theoretical work comes in questioning if that gap is really as large as they suggest.

Lev: If the gap isn't big enough on real hardware, the overhead for error correction on a chip could still be massive, so we need to know how much energy this process actually requires.

Kai: Right, so it’s a proof of concept for an all-electric system that could eventually lead to these zero modes at the ends of the ribbon.

Mira: It opens up a whole new route for realizing one-dimensional topological superconductors without relying on magnetic fields or complex external setups.

Ruize Ma, Michele Pizzochero, * Gaurav Chaudhary

Department of Physics, ETH Zurich · Department of Physics, University of Bath · School of Engineering and Applied Sciences, Harvard University

cond-mat.mes-hall

Submitted: 2025-06-17

Updated: 2026-10-13

Comments: Accepted for publication in Materials for Quantum Technology. Revised following peer review; title updated; supplemental material included

DOI: 10.1088/2633-4356/aeb102

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 82/100

The gist: The gist Zigzag graphene nanoribbons support a variety of electronic phases that can be tuned via external stimuli, and they can be used to realize Majorana zero modes through proximity coupling with

Key concepts

Zigzag Graphene Nanoribbons (ZGNRs): These are quasi-one-dimensional strips of graphene that exhibit magnetism due to a specific electronic instability. Local magnetic moments are concentrated at the edges and align antiparallel across the ribbon, leading to a spin-0 antiferromagnetic ground state.
Half-Metallic Phase: This is a specific electronic state achieved in ZGNRs when inversion symmetry is broken by external fields. At charge neutrality, this phase has one spin channel fully occupied and the other empty, which can be controlled using gate voltage to tune the system.
Proximity Superconductivity: This mechanism describes how superconductivity is induced in a normal material (like a half-metal) when it is placed in close contact with a superconductor. In this case, an Ising superconductor induces an unexpected spin-triplet, odd-parity pairing gap in the ZGNR.
Majorana Zero Modes (MZMs): These are exotic zero-energy modes that appear at the ends of a topologically nontrivial superconducting system. Their presence is guaranteed by a non-trivial topological invariant ($ u = -1$), signaling the formation of these protected zero-energy states.

Summary

The gist Zigzag graphene nanoribbons support a variety of electronic phases that can be tuned via external stimuli, and they can be used to realize Majorana zero modes through proximity coupling with an Ising superconductor

Platform Development

Zigzag graphene nanoribbons (ZGNRs) are quasi-one dimensional strips of graphene that exhibit π-electron magnetism due to a Stoner-like instability resulting from a van Hove singularity at the Fermi level Local magnetic moments are localized at the edges and couple antiparallel across the axis of the nanoribbon, giving rise to a spin-0 antiferromagnetic ground state Transverse electric fields or built-in fields induce an inversion symmetry breaking which lifts the degeneracy of the edge states, enforcing a half-semimetallic band structure at charge neutrality that can be readily gated to the half-metallic phase

Proximity Superconductivity Mechanism

The study demonstrates that Ising superconductors with a parent s-wave spin-singlet pairing can induce an unexpected spin-triplet, odd-parity pairing in the half-metallic phase of the nanoribbon The conventional spin-singlet s-wave superconductivity can be induced in the half-metallic state of the ZGNR This emerging superconductivity is odd-parity, with the system mapping to the Kitaev chain

Conditions for Finite Proximity Gap

The conventional picture where a spin-singlet s-wave superconductor and a half-metal do not support proximity-induced superconductivity is broken under specific conditions The minimal conditions under which the superconductivity can still be induced in the half-metal upon proximity coupling to a singlet s-wave superconductor are derived Both ↑ and ↓ spins from the superconductor can tunnel to the half-metal, provided that its spin quantization axis is not collinear with the superconductor Broken inversion symmetry is nevertheless still required for a finite induced gap This requirement is satisfied by Ising SO coupling, which breaks SU(2) spin rotation symmetry and does not induce spin helicity, as illustrated in Fig. 1(d)

Topological Phase Engineering

The system can be tuned to create and manipulate Majorana zero modes (MZMs) by tuning the gate voltage or external electric field The topological phase transitions are governed by the value of the chemical potential µ, where for values of µ inside the band ν = −1, the Z2 invariant is non-trivial, signaling the formation of MZMs at the end of a finite system When µ intersects only the lower band εα−, isolated MZMs form at the two ends of a finite system

Experimental Advantages and Assumptions

The proposal suggests that this system is fully electric and operates without the need for an external magnetic field, offering a significant advantage over existing platforms The analysis relies on several assumptions, including assuming that the α-spins of the ZGNR are polarized in the plane of the heterostructure It is also assumed that a perfect matching of the hexagonal lattices of the Ising superconductor and ZGNR is present Finally, any interfacial effects, such as Rashba SO coupling, are neglected in this model The results show that for most values of µ within the α spin band, the system exhibits zero-energy modes that are separated from the rest of the spectrum by the bulk gap

Conclusion

The findings show that spinsinglet s-wave Ising superconductor can induce oddparity triplet pairing in half metals, as long as the halfmetal and Ising spins have different spin-quantization axis Half-metallic zigzag graphene nanoribbons are identified as a promising platform for realizing one dimensional topological superconductors hosting Majorana zero modes at their ends Notably, this system is fully electric and operates without the need for an external magnetic field, offering a significant advantage over existing platforms

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How it works

The effective Hamiltonian in the half-metal region is derived using Schrieffer-Wolff transformation, leading to an induced gap ∆(˜k) that is odd-parity, as required from the fermionic antisymmetry of the Cooper pair The induced pairing term is given by ∆(˜k) = 2∆T↑,s(k)T↓,s(−k) − T↓,s(k)T↑,s(−k) / (4∆2 − [ϵ(k) − ξ(k)]2

Topological Phase Engineering

The Z2 topological invariant is obtained as ν = sgn[PfH˜BdG(0)]sgn[PfH˜BdG(π)], and a simple calculation gives ν = sgn[(ϵ(0) − µ)(ϵ(π) − µ)] Depending on the value of µ, we can distinguish various topological regimes, where for values of µ inside the band ν = −1, the Z2 invariant is non-trivial, signaling the formation of MZMs at the end of a finite system

Finite Induced Gap from Ising Spin-Orbit Coupling

The numerator of Eq. S9 contains the crucial combination of band projected tunnelings that need to be finite, which can be written as T↑,→(k)T↑,→(−k) − T↓,→(k)T↑,→(−k) = -1/2 Xσσ Xa,b U i a(k)tbV σ∗ j b (k) Xa b U i a(-k)tbV-σ∗ j b (-k) The SO coupling ensures a finite proximity gap through non-zero contributions that go as sin θσ(k)2 − sin θσ(−k)2 and cos θσ(k)2 − cos θσ(−k)2

Experimental Advantages and Assumptions

The system is fully electric and operates without the need for an external magnetic field The analysis assumes that the α-spins of the ZGNR are polarized in the plane of the heterostructure It is also assumed that a perfect matching of the hexagonal lattices of the Ising superconductor and ZGNR is present Furthermore, any interfacial effects, such as Rashba SO coupling, are neglected in this model The results show that for most values of µ within the α spin band, the system exhibits zero-energy modes that are separated from the rest of the spectrum by the bulk gap

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Finite Induced Gap from Ising Spin-Orbit Coupling

Improvements for AI systems

textbfEmbedded Majorana Platform Design for Fault-Tolerant Quantum Computing Systems: Develop a tunable platform based on half-metallic zigzag graphene nanoribbons (ZGNRs) coupled with an Ising superconductor to realize topologically nontrivial phases and Majorana zero modes (MZMs). This system is fully electric and operates without the need for an external magnetic field, offering a significant advantage over existing platforms [12].
textbfPredictive Modeling of Proximity-Induced Odd-Parity Pairing: Utilize the derived effective Hamiltonian H˜HM to predict the induced proximity-induced gap, explicitly showing that the induced gap is odd-parity, as required from the fermionic antisymmetry of the Cooper pair, and how tuning parameters enables transitions between trivial and topological phases.
textbfAutomated Topological Phase Mapping: Implement a gate-tunable control scheme based on external electric fields to map the system's phase diagram, allowing for precise identification of topological phase transitions indicated by gap closings and openings in the Bogoliubov excitation spectrum [2].
textbfZero-Mode Localization Analysis: Develop algorithms to calculate the Number of zero-energy states (Fig. 2(e)) for finite systems, specifically identifying where two in-gap states emerge, which become pinned to zero energy precisely within the topological phase, ensuring robust and isolated MZMs at the system’s ends.
textbfDisorder Robustness Assessment: Integrate a framework to quantify how disorder affects the Majorana signatures, addressing the challenge that the disorder localized trivial subgap states can often give very similar experimental signatures that are considered the hallmark of MZMs [47].

Abstract

Graphene nanoribbons support a variety of electronic phases that can be tuned via external stimuli. In particular, zigzag graphene nanoribbons exhibit an antiferromagnetic insulating ground state which can transition to a half-metallic phase under a transverse electric field or when embedded in hexagonal boron nitride. Here, we develop a simple model of a heterostructure composed of a half-metallic zigzag graphene nanoribbon coupled with an Ising superconductor. We demonstrate that an Ising superconductor's s-wave spin-singlet pairing gap component can induce a spin-triplet, odd-parity pairing gap in the half-metallic phase of the nanoribbon via proximity coupling. The resulting superconducting phase is topologically nontrivial, with gate-tunable transitions that enable the emergence of Majorana zero modes.