[Paper Review] Scheme for Majorana Manipulation Using Magnetic Force Microscopy

Benjamin H. November, Jay D. Sau+2 more|arXiv (Cornell University)|May 23, 2019

Topological Materials and Phenomena22 references9 citations

TL;DR

This paper proposes a scheme to manipulate Majorana zero modes in Fe(Se,Te) topological superconductors using magnetic force microscopy (MFM), enabling controlled dragging of vortices to braid Majoranas and measure their parity via magnetic force. The method enables robust, non-invasive readout of Majorana pair parity through excess supercurrent detection, offering a scalable path to topological quantum computing with intrinsic topological superconductors.

ABSTRACT

We propose a scheme for the use of magnetic force microscopy to manipulate Majorana zero modes emergent in vortex cores of topological superconductors in the Fe(Se,Te) family. We calculate the pinning forces necessary to drag two vortices together and the resulting change in current and charge density of the composite fermion. A possible algorithm for measuring and altering Majorana pair parity is demonstrated.

Motivation & Objective

• To develop a practical method for braiding Majorana zero modes in intrinsic topological superconductors like Fe(Se,Te), which avoids the challenges of semiconductor nanowire platforms.
• To overcome the limitations of high magnetic fields and chemical potential tuning in nanowire systems by leveraging the intrinsic topological superconductivity of Fe(Se,Te).
• To enable non-invasive, robust measurement of Majorana pair parity using magnetic force microscopy (MFM) without quasiparticle poisoning.
• To demonstrate a feasible experimental protocol for detecting and manipulating Majorana states via vortex manipulation and magnetic force readout.
• To provide a scalable and stable platform for topological quantum computation using native topological superconductors with minimal interface complexity.

Proposed method

• Utilizes a cantilever-based MFM tip to detect and manipulate Majorana zero modes (MZMs) in vortex cores of Fe(Se,Te) without direct contact.
• Employs large tip-sample separation during vortex dragging to prevent quasiparticle poisoning, ensuring coherence during manipulation.
• Applies magnetic force to drag two vortices together, forming a double vortex, and measures the resulting change in supercurrent and charge density.
• Uses the magnetic force generated by the excess supercurrent in a fused vortex pair (parity 1) as a readout signal for Majorana parity.
• Solves the radial Bogoliubov-de Gennes equation (Eq. 4) in the chiral limit (μ=0, m=1/2) to analytically determine the zero-energy MZM wavefunction as Ψ(r) = Nr sech(r) Ψ₀.
• Matches small-r and large-r boundary conditions via matrix determinant condition M(E) = 0 to find bound state energies and construct the full wavefunction.

Experimental results

Research questions

• RQ1 Can magnetic force microscopy (MFM) be used to non-invasively detect and manipulate Majorana zero modes in Fe(Se,Te) vortex cores?
• RQ2 What are the pinning forces required to drag two vortices together in a topological superconductor, and how do they affect the system's current and charge density?
• RQ3 How can the parity of a Majorana pair (0 or 1) be measured via magnetic force in a double vortex configuration?
• RQ4 What is the feasibility of using MFM for repeated parity measurements to assess Majorana state lifetime?
• RQ5 Can the MFM-based scheme enable controlled braiding of Majoranas for logical operations in a scalable topological quantum computing architecture?

Key findings

• The MFM-based scheme enables non-invasive manipulation of Majorana zero modes in Fe(Se,Te) by dragging vortices together using magnetic force, avoiding quasiparticle poisoning.
• The excess supercurrent in a fused double vortex (parity 1) generates a measurable magnetic force, providing a direct readout signal for Majorana pair parity.
• In the chiral limit (μ=0, m=1/2), the zero-energy Majorana wavefunction is analytically solvable and takes the form Ψ(r) = Nr sech(r) Ψ₀, with Ψ₀ satisfying σ_z τ_z Ψ₀ = -Ψ₀.
• The bound state energy E is determined by solving the matrix determinant condition M(E) = 0, ensuring consistency between small- and large-r boundary conditions.
• The current density j(r) vanishes at leading order (j(r) = 0), indicating that higher-order μ corrections are required to observe measurable current, which is consistent with the topological protection of the zero mode.
• The method allows for repeated parity measurements, enabling the experimental assessment of Majorana parity lifetime and paving the way for logical operations in topological quantum computation.