Novel topologies in superconducting junctions
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1 Novel topologies in superconducting junctions Yuli V. Nazarov Delft University of Technology The Capri Spring School on Transport in Nanostructures 2018, Anacapri IT, April
2 Overview of 3 lectures ecture #1 Basics of superconducting junctions and structures Old topology Majorana ecture #2 Weyl topology Basics of semiclassical description: circuit theory ecture #3 Semiclassical topology Smiling gaps topology
3 ecture #1 Quantum states in superconductors Scattering approach Andreev reflection Beenakker formula Practical matters Majorana: idea and status Majorana: simple example and consequencies
4 Quantum states for fermions 1 level x 2 spin directions = 4 states
5 Quantum states in superconductors Superconducting condensate Mixtures of
6 BdG Hamiltonian Bogoluybov transform cr/ann operators for the excitations It doubles the basis! Mirror symmetry of eigenvalues Only positive ones matter
7 The most confusing slide :) BdG spectrum Energies of the states
8 Gap in the bulk Old news? Strictly no excitations Enables quantum states in nanostructures Superconducting qubits Andreev bound states 1 2
9 Andreev reflection How to transport? (Given energy conservation) N h e e Ideal nanostructure: no reflection (example: interface) e S E E e < D : e > D : No states in superconductor. Andreev reflection Electron => quasiparticle, still partially Andreev reflected.
10 Andreev reflection Ideal nanostructure e < D N S An electron is reflected back as a hole (the only possibility) et us check conservation laws: Energy: conserved Charge: conserved! Since charge 2e goes into the superconductor as a Cooper pair. (twee halen, een betalen) Momentum: (almost) conserved! (while velocity flips) Spin: conserved. h e E 2e
11 Andreev amplitude y e Exponential decay For N Plane waves y h x cor S E < D electron and hole states decay in the superconductor: Andreev reflection Amplitude of the reflected hole wave: sup. phase y h = -i iarccos / e j - e D y e Energy-dependent phase emarkable universality: does not depend on the scattered wave
12 Scattering approach Nanostructure: can be very complex Can be modelled as: waveguide with transport channels + potential barrier Essence: scattering matrix r b = Incoming amplitudes Outgoing amplitudes r sˆa r éb ê r êë b ù ú úû = érˆ ê ëtˆ a r b r tˆ ' ù ú rˆ' û r éa ê r ëa ù ú û scattering region ideal waveguides reservoir
13 How and why to combine normal scattering and Andreev reflection S a e a e a h a h b e b h b e b h S - j / 2 j / 2 æ b ö æ e a ö ç ç ç be ç ç b ç h ç b ç è h ø è ø e a ˆ 0 e æ s ö = sˆ, ˆ N s N = ç * a 0 sˆ h è ø ah Scattering at the (short) nanostructure sˆ æ r t ö = ç t r ' è ø iq he æ a 0 e ö æ be ö æ e ö ç 0 ç ç iq he ç a 0 e b ç e e = sˆ ç, ˆ A s A = ç a ç i eh h b ç q h e 0 ç ç 0 a ç h b è ø è h ø ç iq eh 0 e è ø
14 Therefore: Beenakker formula r r r y = sˆ y = sˆ sˆ y out N in N A out det( sˆ sˆ - 1) = 0 Satistfied at certain = energy N A
15 Energy dependence of scattering matrix At the scale of inverse dwell time Compared with Δ Short structure can be disregarded Most of these lectures
16 Two-terminal (short) junction Many channels: bound state for each channel E = D -T j p 2 1 p sin / 2 ( ) Phase-dependent part of ground state energy ( ) 2 = - å p = -Då 1- p sin / 2 p p E E T j
17 Majorana: idea Mirror symmetry of BdG spectrum Usual, trivial Topologically distinct = forbidden Majorana pair: realistic Majorana localized entity a placeholder. One quasiparticle requires 2 placeholders Highly degenerate state. Interesting exchange statistics => manipulation
18 Majorana: status: semiconducting nanowires ong, strong spin-orbit, zeeman splitting
19 Majorana: status: magnetic chains
20 Majorana: simple example Minimum Hamiltonian: ong-wave approximation near transition point Zero-energy: polarizations ocalized states: at zeros of a(x)
21 Majorana: scattering approach Combining normal scattering and majorana wire Poles in energy dependence of the scattering matrix at small energy
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