Project MASSS – Marie-Sklodowska Curie Action by Leandro Tosi
Building Andreev States: Intro
An introduction in 19 pictures: 1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 10 – 11 – 12 – 13 – 14 – 15 – 16 – 17 – 18 – 19
2-Can be described in terms of conduction channels. The simplest case with a single spin-degenerate channel
3-The Supercunductor is charaterized by a gap in the density of states and by a phase of the order parameter
4-Electrons close to the Fermi level, occupy the conduction channel and suffer Andreev Reflection at the interface
5-The reflected hole arrives to the opposite interface and is in turn Andreev reflected as an electron
6-The same for an electron travelling to the left
7- The phase accumulated during the propagation of the electron and the hole can be writen in terms of addimensional parameters: an effective length lambda, which depends on the velocity at the Fermi level and the energy in units of the gap
9-The imperfect transmission of the conduction channel can be taken into account by considering a delta-like barrier which couples left and right movers, thus opening gaps in the spectrum
10-ABS for different lambda with not perfect transmission
11-Since the conduction channel is spin-degenerate, electrons with spin or down lead to the formation of Andreev States with the same energy and therefore the Andreev spectrum is also degerate
12-Rashba spin-orbit coupling is a key ingredient to lift the spin degeneracy. In a one-dimensional wire, the dispersion relations for spin up and down are split
13-This spliting, however, doesn't lift the degeneracy of ABS since electrons with spin up and down have the same velocity at the Fermi level and therefore the same effective length
14-In the case of a wire with many channels (which is more generally the case), the Rashba spin-orbit coupling also introduces an hibridization of different transverse channels having opposite spin.
15-Leading to effective conduction channels with an energy-dependent spin (i.e. a spin texture)
16-When these effective modes are used to build Andreev States, electrons with different spin have different velocities, and therefore give Andreev states at different energies. If you are more interested in this idea, check the references: A. A. Reynoso et al., “Spin-orbit-induced chirality of Andreev states in Josephson junctions”, Phys. Rev. B 86, 214519 (2012) and Sunghun Park and A. Levy Yeyati, “Andreev spin qubits in multichannel Rashba nanowires”, Phys. Rev. B 96, 125416 (2017).
17-As condisdered before, the effect of scattering is to couple left and right movers, lifting the degeracy and opening gaps.
18-Comparison between two long weak-links with strong spin-orbit coupling in the single channel and the multichannel case
19-Given the Exitation Spectrum of Andreev States, we consider the possible transitions that can be observed by shining microwave photons
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