Majorana modes in graphene strips: polarization, wavefunctions, disorder, and Andreev states

Abstract

Topologically protected Majorana zero modes (MZMs) have attracted intense interest due to their potential application in fault-tolerant quantum computation (TQC). Graphene nanoribbons, with tunable edge terminations and compatibility with planar device architectures, offer a promising alternative to semiconductor nanowires. Here we present a comprehensive theoretical study of finite graphene strips with armchair, zigzag, and nearly square geometries, proximitized by an s-wave superconductor and subject to Rashba spin-orbit coupling, Zeeman fields, and disorder. Using exact diagonalization of the Bogoliubov-de Gennes tight-binding Hamiltonian, we analyze Majorana polarization, low-energy spectra, and real-space wavefunctions to identify the non-trivial topological phases supporting MZMs and distinguish them from from partially separated Andreev bound states (psABS) or the quasi-Majoranas. We systematically chart the robustness of these modes across geometries and disorder regimes, finding that armchair strips with short zigzag edges provide the most stable platform. Our results unify polarization diagnostics with spatial wavefunction analysis and disorder effects, yielding concrete design guidelines for graphene-based topological superconductors.

Figures (12)

• Figure 1: A finite-size graphene strip of various geometries. (a) Armchair type strip (A-strip) with N = 5 and L = 15 (zigzag short edges). (b) Zigzag type strip (Z-type) with N = 21 and L = 3 (armchair short edges). (c) Nearly square type strip (S-type) with N = 12 and L = 5. N represents the number of layers, and L represents the number of unit cells in a system. Red and black dots denote the sublattices.
• Figure 2: Spatial profiles of disorder potentials used in our calculations for different ribbon geometries: (a) armchair strip, (b) zigzag strip, and (c) square strip. Panels (i), (ii), and (iii) correspond to increasing disorder strengths: weak disorder with $\mathcal{Z} = 20$ impurities and $V_0 = 1.0$ (top row), moderate disorder with $\mathcal{Z} = 50$ impurities and $V_0 = 1.0$ (middle row), and strong disorder with $\mathcal{Z} = 150$ impurities and $V_0 = 1.5$ (bottom row). The color scale represents the disorder potential values, ranging from $-2.2$ to $+2.2$.
• Figure 3: Majorana polarization in a finite-size armchair graphene strip. Each column (i-iv) corresponds to a different disorder strength: (i) clean system (no disorder), (ii) weak disorder ($\mathcal{Z} = 20$), 20 impurities, $V_0 = 1.0$, (iii) moderate disorder ($\mathcal{Z} = 50$), 50 impurities, $V_0 = 1.0$, (iv) strong disorder ($\mathcal{Z} = 150$), 150 impurities, $V_0 = 1.5$. Panels: (a) Heatmap of the absolute value of Majorana polarization in the lower half of the strip, $|\mathcal{P}^d|$, as a function of Zeeman energy ($h_x/t$) and chemical potential ($\mu/t$). (b) Energy spectrum as a function of $h_x/t$ at fixed chemical potential $\mu = 1.5$. (c) Absolute value of Majorana polarization $|\mathcal{P}^\nu|$ with $\nu = u, d, l, r$ (upper, lower, left, right halves) as a function of $h_x/t$, also at $\mu = 1.5$.
• Figure 4: Real-space probability distribution of the low-energy Majorana modes on an armchair graphene nanoribbon under varying magnetic field and disorder strength. Panels (d)-(g) correspond to increasing values of Zeeman field $h_x/t = 0.40,\ 0.45,\ 0.70,$ and $2.0$, respectively, at fixed chemical potential $\mu/t = 1.5$. Each row (i-iv) shows the effect of increasing disorder strength: (i) Clean (no disorder), (ii) Weak disorder ($\mathcal{Z} = 20,\ V_0 = 1.0$), (iii) Moderate disorder ($\mathcal{Z} = 50,\ V_0 = 1.0$), and (iv) Strong disorder ($\mathcal{Z} = 150,\ V_0 = 1.5$). The adjacent table presents the absolute value of the Majorana polarization $|\mathcal{P}|$ for the lower ($\mathcal{P}^d$), upper ($\mathcal{P}^u$), left ($\mathcal{P}^l$), and right ($\mathcal{P}^r$) halves of the strip for each case.
• Figure 5: Majorana polarization in a finite-size zigzag graphene strip. Each column (i-iv) corresponds to a different disorder strength: (i) clean system (no disorder), (ii) weak disorder ($\mathcal{Z} = 20$), 20 impurities, $V_0 = 1.0$, (iii) moderate disorder ($\mathcal{Z} = 50$), 50 impurities, $V_0 = 1.0$, (iv) strong disorder ($\mathcal{Z} = 150$), 150 impurities, $V_0 = 1.5$. Panels: (a) Heatmap of the absolute value of Majorana polarization in the lower half of the strip, $|\mathcal{P}^d|$, as a function of Zeeman energy ($h_y/t$) and chemical potential ($\mu/t$). (b) Energy spectrum as a function of $h_y/t$ at fixed chemical potential $\mu = 2.0$. (c) Absolute value of Majorana polarization $|\mathcal{P}^\nu|$ with $\nu = u, d, l, r$ (upper, lower, left, right halves) as a function of $h_y/t$, also at $\mu = 2.0$.