Interface currents and magnetization in singlet-triplet superconducting heterostructures: Role of chiral and helical domains

Alfonso Romano, Canio Noce, Ilya Vekhter, and Mario Cuoco

Phys. Rev. B 96, 054512 – Published 14 August 2017

Abstract

Chiral and helical domain walls are generic defects of topological spin-triplet superconductors. We study theoretically the magnetic and transport properties of superconducting singlet-triplet-singlet heterostructure as a function of the phase difference between the singlet leads in the presence of chiral and helical domains inside the spin-triplet region. The local inversion symmetry breaking at the singlet-triplet interface allows the emergence of a static phase-controlled magnetization and generally yields both spin and charge currents flowing along the edges. The parity of the domain wall number affects the relative orientation of the interface moments and currents, while in some cases the domain walls themselves contribute to spin and charge transport. We demonstrate that singlet-triplet heterostructures are a generic prototype to generate and control nondissipative spin and charge effects, putting them in a broader class of systems exhibiting spin-Hall, anomalous Hall effects and similar phenomena. Features of the electron transport and magnetic effects at the interfaces can be employed to assess the presence of domains in chiral/helical superconductors.

Received 12 June 2017

DOI:https://doi.org/10.1103/PhysRevB.96.054512

Physics Subject Headings (PhySH)

Superconductivity Topological superconductors

Unconventional superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Alfonso Romano¹, Canio Noce¹, Ilya Vekhter², and Mario Cuoco¹

¹CNR-SPIN, I-84084 Fisciano (Salerno), Italy and Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno, I-84084 Fisciano (Salerno), Italy
²Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA

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Issue

Vol. 96, Iss. 5 — 1 August 2017

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Figure 1
Schematic representation of the singlet-triplet-singlet (S-T-S) heterostructure in the presence of a spin-triplet superconductor with single (a) or double chiral domain (c) or single (b) or double (d) helical domain. In (c) and (d) the domain wall is indicated with a black dashed line to separate the two regions with opposite winding of the superconducting order parameter for each spin direction. The arrows (green and red for up and down spin) along the domain wall and at the singlet-triplet interface indicate the spin dependent charge currents due to the presence of Andreev bound states.
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Figure 2
(a) Schematic description of the singlet-triplet-singlet (S-T-S) heterostructure with a single chiral domain in the spin-triplet region having $p_{x} + i p_{y}$ orbital symmetry and zero spin projection along the $z$ direction. The encircling arrows for the electron pair are used to sketch the coherent spin-triplet state with zero projection along the direction perpendicular to the spin plane. The donutlike shape schematically indicates the electron spatial probability associated to the $p_{x} + i p_{y}$ orbital state. Range 1 (2) indicates half (entire) extension of the S-T-S system. Spatial profile of the real [(b) and (d)] and imaginary [(c) and (e)] parts of spin-singlet and spin-triplet order parameters, respectively, is shown at different values of the phase difference $ϕ$ between the spin-singlet sides of the heterostructure. Panels (f), (h), and (j) indicate the spatial evolution of the $z$ -projected magnetization (i.e., $2 S_{Z}$ ), $z$ component of the spin current ( $J_{S}^{Z}$ ), and charge current ( $J_{C}$ ), respectively. (g), (i), and (k) describe the phase dependent behavior of the integrated quantities over the range 1 (blue squares) and 2 (black circles) for $S_{Z}$ , $J_{S}^{Z}$ , and $J_{C}$ , respectively.
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Figure 3
(a) Schematic description of the singlet-triplet-singlet (S-T-S) heterostructure with two chiral domains with opposite orbital winding in the spin-triplet region for a spin configuration with zero spin projection along the $z$ direction (i.e., $d_{z}$ order parameter). Range 1 (2) indicates half (entire) extension of the S-T-S system. Spatial profile of the real and imaginary parts of the spin-singlet [(b) and (c)] and spin-triplet [(d) and (e)] order parameters is reported at different values of the phase difference $ϕ$ between the spin-singlet sides of the heterostructure. (f), (h), and (j) indicate the spatial evolution along the $x$ direction of the $z$ -projected magnetization, spin current, and charge current, respectively. (g), (i), and (k) describe the phase dependent behavior of the integrated quantities over the range 1 (blue squares) and 2 (black circles) for the $z$ -projected magnetization, spin current, and charge current, respectively.
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Figure 4
(a) Schematic description of the singlet-triplet-singlet (S-T-S) heterostructure with a single helical domain in the spin-triplet region having $p_{x} + i p_{y} (p_{x} - i p_{y})$ orbital symmetry for the spin up (down) electron pairs. Range 1 (2) indicates half (entire) extension of the S-T-S system. Spatial profile of the real and imaginary parts of the spin-singlet [(b) and (c)] and spin-triplet [(d) and (e) for up spin and (f) and (g) for down spin polarization] order parameters is shown at different values of the phase difference $ϕ$ between the spin-singlet sides of the heterostructure. (h), (j), and (l) indicate the spatial evolution along the $x$ direction of the $z$ -projected magnetization, $x$ - and $z$ -projected spin currents, respectively. (i), (k), and (m) describe the phase dependent behavior of the integrated quantities over the range 1 (blue squares) and 2 (black circles) for the $z$ -projected magnetization, $x$ - and $z$ -projected spin currents, respectively.
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Figure 5
(a) schematic description of the singlet-triplet-singlet (S-T-S) heterostructure with two helical domains. Range 1 (2) indicates half (entire) extension of the S-T-S system. Spatial profile of the real and imaginary parts of the spin-singlet [(b) and (c)] and spin-triplet [(d) and (e) for up spin and (f) and (g) for down spin] order parameters is shown at different values of the phase difference $ϕ$ between the spin-singlet sides of the heterostructure. (h), (j), and (l) indicate the spatial evolution along the $x$ direction of the $z$ -projected magnetization, $x$ - and $z$ -projected spin currents, respectively. (i), (k), and (m) describe the phase dependent behavior of the integrated quantities over the range 1 (blue squares) and 2 (black circles) for the $z$ -projected magnetization, $x$ - and $z$ -projected spin currents, respectively.
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