Odd-frequency pairing in topological superconductors

Yukio Tanaka

It is known that odd-frequency pairing ubiquitously presents in superconductor junctions [1]. Especially, in the presence of zero energy surface Andreev bound state (ZESABS) realized in topological superconductors, the odd-frequency pairing is amplified near the surface or interface. One of the remarkable property generated by odd-frequency pairing is anomalous proximity effect in diffusive normal metal (DN) / superconductor junction where quasiparticle density of states in DN has a zero energy peak (ZEP) of LDOS due to the penetration of odd-frequency spin-triplet s-wave pairing [2,3]. It has been shown that proximity coupled nano-wire junction [4] is an idealistic system to study anomalous proximity effect due to odd-frequency triplet-s wave pairing [5].
We have further clarified the relation between induced odd-frequency pairing and the bulk quantity defined by Green’s function[6]. Odd-frequency Cooper pairs with chiral symmetry emerging at the edges are a useful physical quantity. We have shown that the odd-frequency Cooper pair amplitudes can be expressed by a winding number extended to a nonzero frequency and can be evaluated from the spectral features of the bulk. We have found that the odd-frequency Cooper pair amplitudes are classified into two categories: the amplitudes in the first category have the singular functional form proportional to 1/z (where z is a complex frequency) that reflects the presence of ZESABS, whereas the amplitudes in the second category have the regular form proportional to z.
Recently, we have found that the presence of ZESABS generates new type of thermopower.
We have shown that the thermoelectric effect in ferromagnet / superconductor junctions can be entirely dominated by ingap Andreev reflection processes. Consequently, the electric current from a temperature bias changes sign in the presence of ZESABS and resulting odd-frequency pairing [7].

[1]Y. Tanaka, M. Sato and N. Nagaosa, J. Phys. Soc. Jpn. 81 011013 (2012)
[2]Y. Tanaka and A.A. Golubov, Phys. Rev. Lett. 98 037003 (2007)
[3]Y. Tanaka, A.A. Golubov, S. Kashiwaya, and M. Ueda, Phys. Rev. Lett. 99 037005 (2007); Y. Tanaka, Y. Tanuma, and A. A. Golubov, Phys. Rev. B 96 054552 (2007)
[4]Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70 012507 (2004)
[5]R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010), Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010)
[6]Y. Asano and Y. Tanaka, Phys. Rev. B 87 104513 (2013)
[7]S. Ta mura, S. Hoshino and Y. Tanaka, Phys. Rev. B 99 , 184512 (2019)
[8]T. Savander , S. Tamura, C. Flindt, Y. Tanaka and P. Burset , arXiv: 2008.00849

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