Colloquium Spin-orbit effects in superconducting hybrid structures Morten Amundsen Nordita KTH Royal Institute of Technology and Stockholm University

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Colloquium: Spin-orbit eﬀects in superconducting hybrid structures

Morten Amundsen

Nordita, KTH Royal Institute of Technology and Stockholm University,

Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

Center for Quantum Spintronics, Department of Physics, Norwegian University of Science & Technology,

NO-7491 Trondheim, Norway ∗

Jacob Linder

Center for Quantum Spintronics, Department of Physics, Norwegian University of Science & Technology,

NO-7491 Trondheim, Norway†

Jason W. A. Robinson

Department of Materials Science & Metallurgy, University of Cambridge, 27 Charles Babbage Road,

Cambridge CB3 0FS, United Kingdom‡

Igor Žutić

Department of Physics, University at Buﬀalo, State University of New York, Buﬀalo, New York 14260, USA§

Niladri Banerjee

Department of Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom¶

Spin-orbit coupling (SOC) relates to the interaction between an electron’s motion and its spin, and is

ubiquitous in solid-state systems. Although the effect of SOC in normal-state phenomena has been

extensively studied, its role in superconducting hybrid structures and devices opens many unexplored

questions. In conjunction with broken symmetries and material inhomogeneities within supercon-

ducting hybrid structures, SOC may have additional contributions, beyond its effects in homogenous

materials. Remarkably, even with well-established magnetic or nonmagnetic materials and conven-

tional s-wave spin-singlet superconductors, SOC leads to emergent phenomena including equal-spin

triplet pairing and topological superconductivity (hosting Majorana states), a modiﬁed current-phase

relationship in Josephson junctions, and nonreciprocal transport. SOC is also responsible for trans-

forming quasiparticles in superconducting structures which enhances the spin Hall effect and changes

spin dynamics. Taken together, SOC in superconducting hybrid structures and the potential for elec-

tric tuning of the SOC strength, creates fascinating possibilities to advance superconducting spintronic

devices for energy-efﬁcient computing, and enable topological fault-tolerant quantum computing. By

providing a description of experimental techniques and theoretical methods to study SOC, this Collo-

quium describes the current understanding of resulting phenomena in superconducting structures and

offers a framework to select and design a growing class of materials systems where SOC plays an

important role.

CONTENTS

I. Introduction 2

II. Background 3

A. Spin–orbit coupling 3

B. Validity of Rashba and Dresselhaus models 4

C. Triplet superconductivity 5

D. Theoretical frameworks 6

1. General considerations of spin-dependent ﬁelds 6

2. Superconducting proximity effect 7

3. The Ginzburg-Landau formalism 8

4. Bogoliubov-de Gennes method 9

5. Quasiclassical theory 9

∗Corresponding author:morten.amundsen@ntnu.no

†Corresponding author:jacob.linder@ntnu.no

‡Corresponding author:jjr33@cam.ac.uk

§Corresponding author:zigor@buffalo.edu

¶Corresponding author:n.banerjee@imperial.ac.uk

III. Experimental techniques 11

A. Transition temperature measurements 11

B. Josephson junctions 11

C. Magnetization dynamics 11

D. Spectroscopic techniques 12

E. Low-energy muon spin rotation technique 12

IV. Recent developments 12

A. Majorana zero modes 13

B. Superconducting critical temperature 15

C. Modiﬁcation of magnetic anisotropy 15

D. Interfacial magnetoanisotropy 16

E. Josephson junctions 17

F. Supercurrent diodes 20

G. Spin-pumping 21

H. Spin-Hall phenomena with superconductors 21

V. Open questions and future directions 23

Acknowledgments 24

References 26

arXiv:2210.03549v2 [cond-mat.supr-con] 1 Jul 2024

I. INTRODUCTION

As a relativistic effect, the motion of an electron in an elec-

tric ﬁeld creates a magnetic ﬁeld in its rest frame (Jackson,

1998). The resulting spin-orbit coupling (SOC) in solid-state

systems can have different contributions. In addition to the

coupling of electron spin with the average electric ﬁeld from

the periodic crystal potential, other SOC terms arise due to

an applied or built-in electric ﬁeld, for example due to broken

inversion symmetry. One can also distinguish intrinsic, ex-

trinsic, and synthetic SOC, due to electronic structure, impuri-

ties, and magnetic textures, respectively. With SOC, at a given

wave vector, k, the twofold spin degeneracy is removed result-

ing in a k-dependent Zeeman energy and an effective magnetic

ﬁeld (Winkler, 2003; Žuti´

cet al., 2004). In superconducting

heterostructures, the role of SOC can be even more striking

by transforming the orbital and spin symmetry of the Cooper

pairs, through which exotic states may emerge—even from

simple s-wave spin-singlet superconductors.

For decades, SOC effects have been identiﬁed as crucial

for many normal-state phenomena, such as spin-photon and

spin-charge conversion (Meier and Zakharchenya, 1984), var-

ious topological states (Armitage et al., 2018; Shen, 2012),

the family of spin Hall effects (D’yakonov and Perel’, 1971a;

Maekawa et al., 2012), magnetocrystalline anisotropy and

noncolinear spin textures (including skyrmions and chiral do-

main walls) (Tsymbal and Žuti´

c, 2019). They also formed the

basis for early spintronic applications, which can be traced

back to the discovery of anisotropic magnetoresistance in

1857 (Thomson, 1857; Žuti´

cet al., 2004). In contrast, the

relevance of SOC in superconducting structures was largely

absent, or limited to speciﬁc aspects without fully recogniz-

ing many connections (Bergeret et al., 2005; Buzdin, 2005;

Golubov et al., 2004; Meservey and Tedrow, 1994; Tedrow

and Meservey, 1971). Motivated by recent advances in stud-

ies of hybrid superconducting structures where SOC plays a

prominent role, this review aims to provide an experimental

and theoretical framework to highlight many such connec-

tions between different phenomena and emerging applications

in these structures.

The quest to realize topological superconductivity and elu-

sive Majorana states for fault-tolerant topological quantum

computing in structures with strong SOC relies on equal-spin-

triplet superconductivity (Elliot and Franz, 2015; Nayak et al.,

2008). This triplet superconductivity is also sought in super-

conducting spintronics (Eschrig, 2015; Linder and Robinson,

2015; Ohnishi et al., 2020; Yang et al., 2021) as it supports

dissipationless spin currents and allows for the coexistence of

superconductivity and ferromagnetism. Josephson junctions

(JJs) with tunable SOC, which enable spin-triplet supercon-

ductivity, are important building blocks for topological super-

conductivity and superconducting spintronics (Dartiailh et al.,

2021; Mayer et al., 2020). These JJs also reveal the supercon-

ducting diode effect (Baumgartner et al., 2022; Dartiailh et al.,

2021) an example of a nonreciprocal phenomenon (Nadeem

et al., 2023). While nonreciprocal effects are technologi-

cally important (Marder, 2010; Shockley, 1952) and known

since the nineteenth century in the normal state (Faraday,

1846; Kerr, 1877), experimental demonstrations of supercon-

ducting counterparts were largely absent, until a few years

ago (Nadeem et al., 2023). Analogous to multiferroic ma-

terials which allow electrical control of magnetic properties

and, conversely, magnetic control of electrical properties, we

can view SOC in the superconducting state as enabling vari-

ous magnetoelectric effects (Tkachov, 2017) and facilitating

the coupling between different order parameters. Since SOC

changes the properties of quasiparticles in superconductors, it

has also been shown to produce strongly enhanced spin Hall

phenomena in superconducting structures.

With controllable SOC, the previous efforts to integrate

superconductors and ferromagnets can be radically simpli-

ﬁed. Instead of engineering complex noncollinear mag-

netic structures at the superconductor/ferromagnet (S/F) inter-

face (Banerjee et al., 2014; Keizer et al., 2006; Khaire et al.,

2010; Robinson et al., 2012, 2010; Usman et al., 2011), a sin-

gle common F with SOC in a superconducting heterostruc-

ture with broken inversion symmetry is sufﬁcient to support

spin-triplet superconductivity and large magnetoresistive ef-

fects (Banerjee et al., 2018; Cai et al., 2021; González-Ruano

et al., 2021, 2020; Jeon et al., 2018, 2019a, 2020b; Martínez

et al., 2020). Theoretically, the observed role of SOC in

singlet-to-triplet pair conversion has been studied for both

ballistic and diffusive transport (Bergeret and Tokatly, 2013,

2014; Feng et al., 2008; Högl et al., 2015; Jacobsen et al.,

2015; Yokoyama et al., 2006) and preceded by the related

effect of spin-active interfaces (Eschrig et al., 2003; Hal-

terman and Valls, 2009; Linder et al., 2009; Žuti´

c and Das

Sarma, 1999) and SOC generated k-anisotropic triplet con-

densates (Edelstein, 2003; Gor’kov and Rashba, 2001).

Another example where SOC fundamentally modiﬁes the

underlying physics is within superconducting random-access

memories using ferromagnetic JJs. Here, nonvolatile control

of the zero and πground state phase encoding binary informa-

tion (Birge and Houzet, 2019; Dayton et al., 2018), needs to

be revisited in the presence of SOC where, in addition to the

spin-singlet and the spin-triplet states, their admixture is also

possible. The resulting anomalous Josephson effect (Buzdin,

2008; Reynoso et al., 2008) supports an arbitrary phase shift

other than just zero and π, leading to novel challenges and

opportunities for non-binary information processing and stor-

age. Just as magnetic JJs are the building blocks for super-

conducting memories, their nonmagnetic counterparts are the

key elements for low-power and high-speed superconducting

logic (Holmes et al., 2013; Tafuri, 2019) and superconduct-

ing quantum computing (Krantz et al., 2019; Wendin, 2017).

This means that SOC may not only modify such devices, but

also provide entirely new functionalities in their operation, as

current-phase relation, Josephson energy, critical temperature

and critical current, can all strongly change with SOC. As

in the normal state, SOC is the major source of spin relax-

ation and decoherence, as well as the underlying mechanism

for spin dynamics, in the superconducting state (Žuti´

cet al.,

2004). Since both long and short spin relaxation times (Lin-

demann et al., 2019; Nishikawa et al., 1995) can be desirable

in the normal state, their SOC-controlled tunability in the su-

perconducting state would be similarly advantageous. Taken

together, the presence of SOC and its tunability in hybrid su-

perconducting structures offers an intriguing prospect to both

identify novel phenomena as well as advance various quantum

technologies, from storing, transferring, and processing infor-

mation, to improving quantum sensing. While some of the

resulting efforts simply extend the current concepts and appli-

cations of superconductivity, others, like topological quantum

computing, would radically change the paths towards realiz-

ing computational architectures (Cai et al., 2023). Even if the

most ambitious proposals remain aspirational, the advances in

our understanding of SOC have already transformed the way

we view various superconducting phenomena.

By focusing on more common materials, where their super-

conductivity is well established, simpliﬁes the understanding

of the role of SOC. For example, a large part of this review

focuses on hybrids with conventional elemental or nitride s-

wave superconductors, including Nb, Al, V, and NbN. How-

ever, we note that other superconducting systems to investi-

gate SOC effects are possible; for over two decades supercon-

ducting spintronics has been studied with high-temperature d-

wave superconductors (Chen et al., 2001; Vas’ko et al., 1997;

Wei et al., 1999), where even their normal-state properties re-

main debated. In brieﬂy mentioning other oxide supercon-

ductors, two-dimensional superconductors, such as NbSe2,

and proximity-induced superconductivity in III-V and group

IV semiconductor nanostructures, we complement our Collo-

quium by providing relevant reviews on these topics.

While we recognize that signiﬁcant development is re-

quired before SOC-driven superconducting phenomena can

be applied in the ﬁeld of spintronics or quantum technolo-

gies, sufﬁcient progress has already been made both theoreti-

cally and experimentally where one can start to think about

potential areas of application. Using dissipationless super-

currents offers alternatives for energy-efﬁcient information-

communication and quantum technologies. Just data cen-

ters alone are predicted to require 8% of globally generated

electrical power by 2030 (Jones, 2018). A potential solu-

tion may combine superconducting electronics with recent ad-

vances in spintronics (Hirohata et al., 2020; Tsymbal and Žu-

ti´

c, 2019) to seamlessly integrate logic and memory (Birge

and Houzet, 2019) and thereby overcome the von Neumann

bottleneck (Dery et al., 2012). More importantly, through

this review we hope to drive future innovations beneﬁting

ﬁelds like superconducting spintronics and Majorana physics

which will assist in solving material challenges that are key to

progress in these areas.

We start with a brief introduction to relevant theoretical and

experimental background of superconductivity in the presence

of SOC in hybrid structures, followed by recent developments

and, ﬁnally, concluding with open questions and highlight

promising research directions.

II. BACKGROUND

This section reviews basic concepts which build a basis for

the results outlined in the following sections. We start by de-

scribing SOC, speciﬁcally the Rashba and Dresselhaus mod-

els, in bulk materials and structures with inversion asymmetry.

We then introduce the physics emerging from SOC supercon-

ductivity in proximity structures and include a discussion on

theoretical methods that can be used to study such systems.

The purpose of this section is thus to provide the reader with

a set of theoretical concepts necessary to discuss the interest-

ing spintronics phenomena that arise due to the combination

of superconductivity and SOC in heterostructures.

A. Spin–orbit coupling

Coupling between the motion of an electron and its spin

stems from the fact that in the reference frame of the electron,

it is the positively charged lattice that moves. Moving charges

create a magnetic ﬁeld, which may couple to the electron spin.

In a Lorentz-invariant formulation, a SOC term emerges, as

shown in the Dirac equation (Dirac, 1928)

mc2+V(r)−i¯

hcσ·∇

−i¯

hcσ·∇−mc2+V(r)ψe

ψh=ε+mc2ψe

ψh,

(1)

and taking the nonrelativistic limit, ε,V≪mc2, where ε

is the particle energy without its rest mass. Here, V(r)is

the lattice potential, mis the free electron mass, and σis

a vector of Pauli matrices. The resulting Hamiltonian is

H=p2/2m+V(r) + ¯

h/(4m2c2)σ·(∇V×p),for the elec-

tron wavefunction ψe, where irrelevant terms are discarded.

The last term represents SOC, large near a lattice site (Fabian

et al., 2007). Within second quantization, in the basis of the

Bloch functions, it takes the form (Samokhin, 2009),

HSO =∑

∑

nn′

∑

ss′

Qnn′(k)·σss′c†

knsckn′s′,(2)

where Qnn′is a phenomenological model function which ex-

presses the coupling between momentum and spin, with nand

n′band indices, and kthe crystal momentum. For a cen-

trosymmetric material, the terms Qnn vanish, SOC can only

be described by models containing at least two bands. How-

ever, in a noncentrosymmetric material a one-band model is

possible, ∑nn′Qnn′→Qin Eq. (2), while Q(k) = −Q(−k).

One can distinguish bulk and structure inversion asymmetry

(BIA, SIA), which lead to a spin splitting and SOC

HSO(k) = ¯

hσ·Ω(k)/2,(3)

where Ω(k)is the Larmor frequency for the electron spin

precession in the conduction band (Žuti´

cet al., 2004) or,

equivalently, SOC ﬁeld. Here momentum scattering, Ω(k)

is responsible for spin dephasing. Related SOC manifesta-

tions in semiconductors usually focuses on effective models

SIA

[111]

BIA

[001]

BIA

BIA [110]

110

001

100

010

FIG. 1: Vector ﬁelds Ω(k)on a circular Fermi surface for

structure (SIA) and bulk (BIA) inversion asymmetry. Since

Ω(k)is the spin quantization axis, the vector pattern is also

the pattern of the spin on the Fermi surface. As opposite

spins have different energies, the Fermi circle splits into two

concentric circles with opposite signs of spin; shown here

only for the SIA case, but the analogy extends to all

examples. The ﬁeld for BIA [110] perpendicular to the plane,

with the magnitude varying along the Fermi surface. All

other cases have constant ﬁelds lying in the plane.

From Žuti´

cet al., 2004.

which capture the low-energy properties of the conduction

and valence bands. An example of BIA is the Dresselhaus

SOC (Dresselhaus, 1955), given by ΩD= (2γ/¯

h)[kx(k2

y−

z),ky(k2

z−k2

x),kz(k2

x−k2

y)],where γis the SOC strength.

In two-dimensional (2D) systems with quantum conﬁnement

along the unit vector ˆn,ΩDcan be linearized in k,

Ω2D

D∼k2

n[2nx(nyky−nzkz) + kx(n2

y−n2

z)]ˆ

x+c.p., (4)

where k2

nis the expectation value of the square of the wave

number operator normal to the plane in the lowest subband

state while ˆn= (nx,ny,nz)is the conﬁnement unit vector of

the quantum well, and c.p. denotes the cyclic index permuta-

tion. For a rectangular well of width a,k2

n= (π/a)2, while for

a triangular well k2

nis given in de Sousa and Das Sarma, 2003.

With a strong conﬁnement, k2

∥≪k2

n, where k∥is the in-plane

(IP) wave vector (⊥ˆn), cubic terms in kin ΩDfrom Eq. (4)

can be neglected.

For commonly considered quantum well conﬁnements, one

obtains for [001]: Ω2D

D∼k2

n(−kx,ky,0), for [111]: Ω2D

D∼

n(k×n), and for [110]: Ω2D

D∼k2

nkx(−1,1,0), as shown in

Fig. 1. Several features can be readily seen, BIA [100] dis-

plays a “breathing" pattern, while BIA [110] Ω(k)is perpen-

dicular to the plane such that, within the linear in kapproxi-

mation, the perpendicular spins do not dephase.

An extensively studied SIA example is given by Bychkov-

Rashba (or just Rashba) SOC (Bychkov E. I. and Rashba,

1984), which arises in asymmetric quantum wells or in de-

formed bulk systems, expressed by

ΩR=2α(k×n),(5)

where αparametrizes its strength and the inversion symmetry

is broken along the n-direction. We see in Fig. 1 that its func-

tional form, ΩR, coincides with BIA Ω2D

Din [111] quantum

wells. A desirable property of Rashba SOC is that αis tunable

by an applied electric ﬁeld. While these linearized forms of

the Rashba and Dresselhaus SOC are the most common mod-

els, there is an increasing interest in the study of phenomena

which go beyond their range of validity.

For example, in Rashba SOC there is a growing class of

materials where cubic terms in kcan play an important role,

or even be dominant (Alidoust et al., 2021). Finally, while

we have here focused on intrinsic SOC, we note that extrin-

sic SOC caused by impurities plays a key role in spintronics

both with and without superconductors, giving rise to impor-

tant contributions to spin Hall effects and spin relaxation (Žu-

ti´

cet al., 2004).

B. Validity of Rashba and Dresselhaus models

The validity of effective low-energy SIA and BIA SOC

models can be examined from electronic structure calcula-

tions, using ﬁrst-principles, k·pmethod, or a tight-biding

model. Another contribution to Rashba-like spin splitting,

HL=−p·E, arises in systems with localized orbital mo-

mentum, L(Park et al., 2012, 2011), where Eis the electric

ﬁeld and the electric dipole moment, p∝L×k, is produced

by the asymmetric charge distribution. Rashba SOC strength

is renormalized by the orbital contribution, while local sym-

metry breaking can induce local orbital Rashba spin splitting

even in centrosymmetric systems (Lee and Kwon, 2020).

Going beyond Rashba and Dresselhaus model might be

necessary for interfacial SOC in junctions with interface-

induced symmetry reduction in the individual bulk con-

stituents and for multiorbital conﬁgurations (Mercaldo et al.,

2020). This is exempliﬁed in an Fe/GaAs junction, where the

cubic and Tdsymmetries of bulk Fe and GaAs, respectively,

are reduced to C2v(Fabian et al., 2007; Žuti´

cet al., 2019).

Since the interfacial SOC is present only near the interface,

its effects can be controlled electrically via a gate voltage or

an applied external bias capable of pushing the carrier wave

function into or away from the interface. Interfacial SOC can

also be controlled magnetically, as it strongly depends on the

the orientation of Min the Fe layer, as illustrated from the

ﬁrst-principles calculation in Figs. 2(b) and (c) (Gmitra et al.,

2013). The bias-dependence of the SOC can be inferred from

the transport anisotropy in Figs. 2(d) and (e).

While the resulting interfacial SOC for Fe/GaAs junction

corresponds neither to Rashba, nor Dresselhaus models, its

existence can be probed through tunneling anisotropic mag-

netoresistance (TAMR), which gives the dependence of the

tunneling current in a junction with only one magnetic elec-

trode on the orientation of M(Gould et al., 2004). For an IP

FIG. 2: (a) Schematic of a Fe/GaAs slab. For in-plane

TAMR, M, is rotated in the plane of Fe. (b) Angular k-space

dependence of the amplitude of the interfacial SOC ﬁeld for

Malong the GaAs [1¯

10]direction (green arrow). (c) Same as

in (b) but for Malong the [110] direction (Gmitra et al.,

2013). The tunneling resistance R(φ)is normalized to its

φ=0 value, R[110]. Measurements for bias ±90 meV are

shown in (d) and (e), respectively (Moser et al., 2007). (f)

Angular dependence of the TAMR in the out-of-plane (OOP)

conﬁguration. Left (right) panels correspond to CoPt/AlOx/Pt

(Co/AlOx/Pt) tunnel junctions. An extra Pt layer with strong

SOC yields in CoPt/AlOx/Pt two orders of magnitude larger

TAMR than in Co/AlOx/Pt. The insets show M

measurements in OOP magnetic ﬁelds (Park et al., 2008).

Adapted with permission from Žuti´

cet al., 2019.

rotation of Min Fig. 2(a), we can deﬁne TAMR as the nor-

malized resistance difference,

TAMR = (R(φ)−R[110])/R[110],(6)

where R(φ=0)≡R[110]is the resistance along the [110] crys-

tallographic axis. The out-of-plane (OOP) TAMR is deﬁned

analogously. TAMR appears because the electronic struc-

ture depends on the Morientation, due to SOC. The surface

or an interface electronic structure can strongly deviate from

its bulk counterparts and host pure or resonant bands. With

SOC, the dispersion of these states depends on the Morien-

tation (Chantis et al., 2007). As a result, the tunneling con-

ductance, which, in a crystalline junction, is very sensitive to

the transverse wave vector, develops both OOP and IP MR,

shown in Figs. 2(d)-(f), whose angular dependence reﬂects

the crystallographic symmetry of the interface. For example,

the TAMR inherits the C4vsymmetry for the Fe (001) sur-

face (Chantis et al., 2007) and the reduced C2vsymmetry for

the Fe(001)/GaAs interface (Moser et al., 2007).

Our prior discussion of SOC and its manifestations in the

normal-state properties have important superconducting coun-

terparts as well as enable entirely new phenomena, absent

in the normal state. Even when the SOC results in only a

small normal-state transport anisotropy, as shown in Figs. 2(d)

and (e), the superconducting analogs of such phenomena can

lead to much greater effects (Cai et al., 2021; Martínez et al.,

2020).

C. Triplet superconductivity

Conventional s-wave superconductors are well-described

by the Bardeen-Cooper-Schrieffer (BCS) microscopic theory.

The superconducting correlations consist of Cooper pairs in a

spin-singlet state and carry no net spin, unlike the proximity-

induced spin-triplet superconducting correlations. There are

materials believed to exhibit intrinsic triplet superconductivity

such as Bechgaard salts (Sengupta et al., 2001), UPt3(Joynt

and Taillefer, 2002), as well as the ferromagnetic supercon-

ductors (Aoki et al., 2011). A direct interaction between su-

perconductivity and SOC is also found in noncentrosymmetric

superconductors, where electron pairing is a mixure of spin-

singlet and spin-triplet (Smidman et al., 2017).

Through proximity effects, triplet superconducting correla-

tions can be generated using only conventional materials. In

S/F bilayers, the spin splitting in the latter leads to oscillations

in the pair correlation between the singlet and triplet spin con-

ﬁgurations due to a process akin to the Fulde-Ferrell-Larkin-

Ovchinnikov (FFLO) oscillations (Buzdin, 2005; Fulde and

Ferrell, 1964; Larkin and Ovchinnikov, 1965). Nevertheless,

such a coupling between S and homogeneous F is rapidly sup-

pressed as one moves away from the interface region, leading

to a short-range proximity effect. The situation is different in

F with an inhomogeneous Mdirection, where the spin of the

short-ranged triplet correlations is orthogonal to M. Here, the

short-ranged triplets decay over the coherence length of the

Cooper pairs in the F layer. If the orientation of Mchanges,

the triplet spin will obtain a component parallel to M. This

component, referred to as a long-ranged triplet component,

is not inﬂuenced by the spin splitting to the same degree as

their short-ranged counterparts. Remarkably, it may persist

for long distances as in nonmagnetic metals (Giroud et al.,

1998; Lawrence and Giordano, 1999; Petrashov et al., 1994,

1999), of the order pD/2πTin the diffusive limit, where D

is the diffusion coefﬁcient of the F region and Tis the temper-

ature (Bergeret et al., 2001, 2005; Kadigrobov et al., 2001).

Engineering superconducting hybrid structures for generat-

ing spin-polarized triplets have been extensively studied. Us-

ing magnets with rare earth materials (holmium) with intrinsi-

cally inhomogeneous Min JJs, provides evidence of triplet

pair creation (Robinson et al., 2010; Sosnin et al., 2006).

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Colloquium:Spin-orbiteffectsinsuperconductinghybridstructuresMortenAmundsenNordita,KTHRoyalInstituteofTechnologyandStockholmUniversity,HannesAlfvénsväg12,SE-10691Stockholm,SwedenCenterforQuantumSpintronics,DepartmentofPhysics,NorwegianUniversityofScience&Technology,NO-7491Trondheim,Norway∗JacobLinde...

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Colloquium Spin-orbit effects in superconducting hybrid structures Morten Amundsen Nordita KTH Royal Institute of Technology and Stockholm University

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