Strain-controlled crossover between Majorana and Andreev bound states in disordered superconductor-semiconductor heterostructures

Technical Summary: Strain-controlled crossover between Majorana and Andreev bound states in disordered superconductor-semiconductor heterostructures

Problem Statement
The unambiguous identification of topological Majorana-bound states (MBSs) in superconducting hybrid systems is currently hindered by the prevalence of trivial low-energy excitations. Specifically, partially separated Andreev bound states (psABSs), often arising from disorder, smooth confinement, or spatial inhomogeneity, mimic the zero-energy signatures and Majorana-like characteristics of true MBSs while lacking topological protection. This "quasi-Majorana" problem poses a fundamental obstacle to distinguishing topological phases from trivial ones in realistic, disordered environments. Furthermore, while one-dimensional (1D) nanowires offer a clean single-channel platform, graphene nanoribbons present a richer but more complex multiband setting where edge states and band hybridization further obscure the identification of MBSs. A critical open question is whether experimentally accessible parameters can systematically control the crossover between trivial states, psABSs, and topological MBSs.

Methodology
The authors employ tight-binding Bogoliubov–de Gennes (BdG) simulations to investigate two distinct systems: (i) one-dimensional semiconductor nanowires and (ii) graphene nanoribbons. Both systems are modeled with proximity-induced $s$-wave superconductivity, Rashba spin–orbit coupling, Zeeman fields, and spatially varying disorder.

• Disorder Modeling: Disorder is introduced as a superposition of randomly distributed impurities with finite spatial extent, generating a smooth electrostatic potential that mimics charge puddles and gate fluctuations.
• Strain Modeling: Spatially nonuniform strain is incorporated phenomenologically by renormalizing position-dependent tight-binding parameters.
  • In 1D nanowires, strain modulates the nearest-neighbor hopping amplitude ($t$) and Rashba coupling ($\alpha_R$) based on a local strain profile $\epsilon_i$. Both symmetric (centered) and asymmetric (gradient) strain configurations are tested.
  • In graphene nanoribbons, strain is modeled via a continuum elasticity framework where a displacement field modifies bond lengths. This leads to anisotropic hopping renormalization and position-dependent Rashba coupling. The study focuses on diagonal strain profiles to isolate hopping renormalization effects without invoking strong pseudomagnetic fields.
• Diagnostics: The nature of low-energy states is characterized using:
  • Majorana Polarization (MP): A wave-function-based diagnostic measuring the local particle-hole structure. In chiral-symmetric 1D systems, a simplified scalar definition is used; in quasi-1D and graphene systems (Symmetry Class D), a generalized complex particle-hole formulation is employed.
  • Nonlocal Correlations: The product of polarizations on opposite halves of the system ($P_{left} \times P_{right}$) is used to quantify spatial separation.
  • Spectral Properties: Energy splitting ($\delta E$) of low-energy modes and the bulk excitation gap ($\xi$) are analyzed.
  • Criteria for MBS: Identification relies on a combined set of criteria: near-zero energy within a finite bulk gap, high Majorana polarization ($|P| \approx 1$), and spatial separation of wavefunction components.

Key Contributions and Results

1. Strain as a Control Parameter for Crossovers:
   The paper demonstrates that spatially nonuniform strain acts as a versatile tuning parameter capable of driving continuous and reversible transitions between trivial states, psABSs, and topological MBSs.

   • In 1D Nanowires: Strain primarily modifies the spatial overlap of Majorana components and shifts the effective topological phase boundary.
     • Clean Systems: Increasing symmetric strain drives a crossover from well-separated MBSs to psABSs by pushing the wavefunctions from the edges toward the interior, increasing their overlap and reducing topological protection.
     • Disordered Systems: Strain can convert disorder-induced psABSs into well-separated, robust MBSs. By shifting the topological phase boundary and enhancing nonlocality, strain effectively "cleans" the trivial states, restoring the spatial separation required for topological protection.
     • Re-entrant Behavior: The system can exhibit re-entrant behavior, transitioning from trivial states to MBSs and then back to psABSs as strain strength increases, highlighting the dual role of strain in stabilizing and destabilizing topological phases.

2. Multiband Effects in Graphene Nanoribbons:
   In graphene, the presence of multiple dispersive subbands leads to strong hybridization and a dense low-energy spectrum.

   • Clean Systems: Uniaxial strain drives a crossover from localized MBSs to psABSs by moving bulk states toward zero energy and causing mode coalescence.
   • Disordered Systems: Unlike in nanowires where strain primarily shifts boundaries, in disordered graphene, asymmetric strain acts to suppress inter-subband hybridization and lift accidental degeneracies. This reorganizes the spectrum, opening a finite band gap and stabilizing boundary-localized modes with finite Majorana polarization. Thus, strain serves to isolate low-energy states from the crowded bulk spectrum.

3. Analytical Framework:
   The authors develop an analytical theory based on a position-dependent topological mass, $M(x) = h^2 - \Delta^2 - \mu_{eff}^2(x)$.

   • Disorder and strain enter the local topological criterion through an effective potential $V_{eff}(x)$.
   • The motion of Majorana components is described as domain-wall dynamics where $M(x)=0$. The theory derives a real-space criterion for the crossover based on the nonlocal action $S_{12}$, which encodes the separation and decay length of the two Majorana components.
   • A strain-induced crossover from psABS to MBS occurs when strain increases the nonlocal action ($dS_{12}/d\epsilon_0 > 0$), effectively pushing components apart or reducing their decay length. Conversely, a reduction in nonlocal action drives the MBS-to-psABS transition.

Significance and Claims
The paper claims that strain is not merely a perturbation but a critical control parameter for manipulating the nature of subgap excitations in realistic, disordered systems. The primary significance lies in providing a mechanism to distinguish and stabilize topological MBSs from trivial psABSs. By systematically altering the spatial structure and spectral properties of low-energy states, strain allows for the conversion of trivial states into topological ones (and vice versa), offering a pathway to better control and identification of Majorana modes.

The authors emphasize that the role of strain is dictated by dimensionality and band structure: in single-channel systems, it acts as a direct tuning knob for the topological phase boundary, while in multiband systems, it functions as a mechanism for spectral reorganization to suppress hybridization. These findings suggest that strain engineering could be a practical tool for experimentalists working with hybrid nanowire and graphene-based superconducting systems to probe the interplay between topology and nonlocality, ultimately aiding the realization of topological quantum computation in complex environments.