Majorana fermions and odd-frequency Cooper pairs in a normal-metal nanowire proximity-coupled to a topological superconductor

Abstract

We discuss a strong relationship between Majorana fermions and odd-frequency Cooper pairs which appear at a disordered normal (N) nanowire attached to a topologically nontrivial superconducting (S) one. The transport properties in superconducting nanowire junctions show universal behaviors irrespective of the degree of disorder: the quantized zero-bias differential conductance at 2e 2 /h in NS junction and the fractional current-phase (J -ϕ) relationship of the Josephson effect in SNS junction J ∝ sin(ϕ/2). Such behaviors are exactly the same as those found in the anomalous proximity effect of odd-parity spin-triplet superconductors. We show that the odd-frequency pairs exist wherever the Majorana fermions stay.

Non-triviality matters

“…−ikr e −r/ξ0 Note, that this expression is consistent with the result found by Tanaka and Asano in a semiconductor/superconductor device [130]. In their case, an additional magnetic field is applied through which only the spin up part survives of Eq.…”

“…However, in this case, we have a full oddfrequency state at zero energy. Therefore the conclusion of Asano and Tanaka [130] holds that in these devices pure odd-frequency Cooper pairs and the Majorana zero-energy mode are one and the same thing.…”

“…The presence of peaks in the conductance at zero-energy has been interpreted as experimental evidence for the existence of Majorana modes in such systems [132][133][134]154]. It is also known from theoretical studies that a zero-bias conductance peak and conductance dips at the characteristic gap energy are typical features of Majorana zero-energy modes [8,34,66,130,[155][156][157] (see also Chap. 4).…”

“…In order to determine the pairing relations for a nanowire with strong SOC we start with the relation [E − H]G = 1 where G is the Green function of the system, 1 the identity matrix and H is in the basis ψ , ψ , ψ † , ψ † [130]. For a system with spin orbit coupling [E − H] is given by…”

Experimentally observable signatures of odd-frequency pairing in multiband superconductors

“…Notably, there are no zero-energy or subgap states, otherwise often associated with oddfrequency pairing. 43,48,49,[56][57][58][59] Recent works have pointed out that zero-energy states do not always accompany odd-frequency pairing, 41,54,60,61 and odd-frequency, oddinterband pairing provides another example when this happens.…”