Time-delay matrix, midgap spectral peak, and thermopower of an Andreev billiard

Abstract

We derive the statistics of the time-delay matrix (energy derivative of the scattering matrix) in an ensemble of superconducting quantum dots with chaotic scattering (Andreev billiards), coupled ballistically to $M$ conducting modes (electron-hole modes in a normal metal or Majorana edge modes in a superconductor). As a first application we calculate the density of states ${\rho }_0$ at the Fermi level. The ensemble average $⟨{\rho }_0⟩={\delta }_0^{-1}M[\max (0,M+2\alpha /\beta ){]}^{-1}$ deviates from the bulk value $1/{\delta }_0$ by an amount depending on the Altland-Zirnbauer symmetry indices $\alpha ,\beta$. The divergent average for $M=1,2$ in symmetry class D ($\alpha =-1$, $\beta =1$) originates from the mid-gap spectral peak of a closed quantum dot, but now no longer depends on the presence or absence of a Majorana zero-mode. As a second application we calculate the probability distribution of the thermopower, contrasting the difference for paired and unpaired Majorana edge modes.

arXiv comment13 pages, 6 figures

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• Random-matrix theory of Majorana fermions and topological superconductors
• Wigner time delay and related concepts -- Application to transport in coherent conductors
• Generic quantized zero-bias conductance peaks in superconductor-semiconductor hybrid structures
• Separation of heat and charge currents for boosted thermoelectric conversion
• Efficient semiclassical approach for time delays
• X-shaped and Y-shaped Andreev resonance profiles in a superconducting quantum dot
• Correlators for the Wigner-Smith time-delay matrix of chaotic cavities
• Random matrix theory of quantum transport in chaotic cavities with non-ideal leads
• Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble
• Effect of chiral symmetry on chaotic scattering from Majorana zero modes
• Joint moments of proper delay times
• Random matrix theory for the robustness, quantization, and end-to-end correlation of zero-bias conductance peaks in a class D ensemble
• Thermoelectric processes of quantum normal-superconductor interfaces
• Effect of a tunnel barrier on the scattering from a Majorana bound state in an Andreev billiard
• Delay times in chaotic quantum systems