Spin and charge currents driven by the Higgs mode in high-field superconductors
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Spin and cha ge cu en s d i en by he Higgs mode in high- ield supe conduc o s
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Silae , Mikhail A.; Ojajä i, Ris o; Heikkilä, Te o T.
Silae , M. A., Ojajä i, R., & Heikkilä, T. T. (2020). Spin and cha ge cu en s d i en by he Higgs
mode in high- ield supe conduc o s. Physical Re iew Resea ch, 2(3), A icle 033416.
h ps://doi.o g/10.1103/PhysRe Resea ch.2.033416
2020
PHYSICAL REVIEW RESEARCH 2, 033416 (2020)
Spin and cha ge cu en s d i en by he Higgs mode in high-field supe conduc o s
Mikhail A. Silae ,1,2Ris o Ojajä i ,1and Te o T. Heikkilä 1
1Uni e si y o Jy askyla, Depa men o Physics and Nanoscience Cen e , P.O. Box 35 (YFL), FI-40014 Uni e si y o Jy äskylä, Finland
2Moscow Ins i u e o Physics and Technology, Dolgop udny, Moscow Region 141700, Russia
(Recei ed 1 July 2019; e ised 23 Augus 2020; accep ed 25 Augus 2020; published 15 Sep embe 2020)
The Higgs mode in supe conduc ing ma e ials desc ibes slowly decaying oscilla ions o he o de pa ame e
ampli ude. We demons a e ha in supe conduc o s wi h a buil -in spin-spli ing field he Higgs mode is s ongly
coupled o he spin deg ees o eedom, allowing o he gene a ion o ime-dependen spin cu en s. Con e ing
such spin cu en s o elec ic signals by spin-fil e ing elemen s p o ides a ool o he second-ha monic gene a-
ion and he elec ical de ec ion o he Higgs mode gene a ed by he ex e nal i adia ion. The nonadiaba ic spin
o ques gene a ed by hese spin cu en s allow o he magne ic de ec ion o he Higgs mode by measu ing he
p ecession o he magne ic momen in he adjacen e omagne . We discuss also he ecip ocal e ec , which is
he gene a ion o he Higgs mode by he magne ic p ecession. Coupling he collec i e modes in supe conduc o s
o ligh and magne ic dynamics p o ides an oppo uni y o he s udy o supe conduc ing op ospin onics.
DOI: 10.1103/PhysRe Resea ch.2.033416
I. INTRODUCTION
Oscilla ions o he o de pa ame e ampli ude in
condensed-ma e sys ems a e o en called Higgs modes
(HMs) [1–5], in analogy wi h he Higgs boson in pa icle
physics [6]. These collec i e exci a ions a e gene ic o
o de ed s a es such as an i e omagne s, cha ge densi y
wa es [7], supe fluids [8–10], cold a omic gases [11,12] and
supe conduc o s [2,13–27]. In gene al, one can call HMs all
he possible collec i e modes o he o de pa ame e , o he
han he Nambu-Golds one modes [4].
Higgs modes ha e been obse ed by Raman sca e ing in
supe conduc o s wi h cha ge densi y wa e o de [14,15,17,28]
and by he nuclea magne ic esonance in supe fluid 3He
[8–10]. In usual supe conduc o s he HMs a e cha ge neu al
and hus decoupled om cha ge cu en . In such sys ems he
obse a ion o HMs has been acili a ed by he de elopmen
o low- empe a u e e ahe z spec oscopy [18,19,24,29–32].
Wi h his echnique, HMs ha e been obse ed in NbTiN and
NbN compounds [18,19]. Higgs modes ha e been obse ed
indi ec ly as he AC linea conduc ance peak in cu en -
ca ying films o NbN [24] and Al [33].
He e we sugges a di e en mechanism allowing elec ical
de ec ion o HMs due o hei coupling wi h spin and cha ge
deg ees o eedom in high-field supe conduc o / e omagne
junc ions. Unusual anspo p ope ies o such sys ems ha e
a ac ed in ense a en ion [34–37], s imula ing bo h expe i-
men al [38–47] and heo e ical e o s [34,48–57].
The unde lying physical mechanism behind he sugges ed
elec ical measu emen o he HM is oo ed in he s ong cou-
pling be ween he supe conduc ing o de pa ame e dynamics
Published by he Ame ican Physical Socie y unde he e ms o he
C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he
dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)
and he published a icle’s i le, jou nal ci a ion, and DOI.
and elec on spins. The possibili y o ansmi spin signals
by he o de pa ame e exci a ions has been elucida ed using
he example o mobile opological de ec s, i.e., Ab ikoso
o ices [55,56]. He e we demons a e ha ime-dependen
spin cu en s can be gene a ed by he collec i e ampli ude
modes in supe conduc o s.
The s uc u e o his pape is as ollows. In Sec. II we
in oduce he se up and model. Sec ion III shows he e ec o
he HM on AC spin and cha ge cu en s. Thei use in access-
ing he HM ei he in second-ha monic gene a ion o ia he
measu emen o an a oided c ossing be ween e omagne ic
esonance and HMs is discussed in Sec. IV. We conclude in
Sec. Vwi h an ou look o he ange o phenomena a ec ed by
he HM.
II. SETUP AND MODEL
The gene ic se up ha we s udy is shown in Fig. 1(a).I s
basic elemen is a supe conduc ing film placed in con ac wi h
a e omagne ic (FM) ma e ial. An e ec i e spin-spli ing
field hen e ing as he Zeeman e m in he Hamil onian o
he supe conduc o (SC) is induced by an ex e nal in-plane
magne ic field. Al e na i ely, hcould be induced by he p ox-
imi y o a e omagne ic insula o [58–62]. The sys em is
exposed o an ex e nal i adia ion Eei which gene a es a
ime-dependen pe u ba ion o he o de pa ame e ampli-
ude δ( )=2e2i h ough he second-o de nonlinea i y
2∝E2
[63,64].
We model he SC/FM junc ion using he unneling Hamil-
onian app oach [65,66], which has been used ex ensi ely o
s udy bo h AC and DC unnel cu en s [65,67–69],
HT=
kkα
A†
kα(ˆ
ˆ
Bk)α+H.c.,(1)
ˆ
=Tˆτ3+U(m·ˆ
σ).(2)
He e Akα(Bkα) annihila es an elec on wi h momen um k
and spin αin he SC ( e omagne ), he uni ec o m
2643-1564/2020/2(3)/033416(10) 033416-1 Published by he Ame ican Physical Socie y
SILAEV, OJAJÄRVI, AND HEIKKILÄ PHYSICAL REVIEW RESEARCH 2, 033416 (2020)
(a) (b)
FIG. 1. (a) Se up o he supe conduc o / e omagne s uc u e.
The exchange field his induced by an ex e nal magne ic field B.
The double- equency gap modula ion can be exci ed by he ex e -
nal elec omagne ic i adia ion 2∝E2
and is enhanced due o
he coupling o he HM. (b) Semiconduc o model o he cu en
gene a ion by a slowly a ying pe u ba ion o he o de pa ame e
ampli ude δ( ). Only he spin-down band is shown. The e o-
magne has an equilib ium dis ibu ion n0(ε), while he dis ibu ion
n(ε, ) o he supe conduc o is shi ed wi h espec o he equi-
lib ium dis ibu ion (dashed line) by δ( ) away om he Fe mi
le el. Due o he Zeeman shi h, he pe u ba ion in he numbe o
exci a ions is asymme ic wi h espec o he Fe mi le el εF.This
esul s in ne spin and cha ge cu en s flowing in o he a ached FM
elec ode h ough he unnel ba ie (I).
defines he spin quan iza ion axis o he ba ie , ˆτkand
ˆσka e he Pauli ma ices in Nambu and spin spaces, e-
spec i ely, and Uand Ta e he spin-independen and
spin-dependen ma ix elemen s o he unneling Hamil onian,
espec i ely [70]. The ma ix unneling cu en h ough he
spin-pola ized ba ie s can be exp essed h ough momen um-
a e aged G een’s unc ions (GFs) in he supe conduc ing and
FM elec odes νSˆgS=ˆτ3kTˆ
Ak(τ)ˆ
A†
k(τ)and νFˆgF=
ˆτ3kTˆ
Bk(τ)ˆ
B†
k(τ), espec i ely. He e τand τa e imag-
ina y imes, Tis he ime-o de ing ope a o , and νSand νF
a e he no mal me al densi ies o s a es on he wo sides o he
junc ion. Fo simplici y, we assume momen um-independen
unneling coe ficien s [70,71]. The ime-dependen unneling
cu en o he gene al nonequilib ium s a e in he elec odes
de i ed in Appendix A eads
ˆ
I(τ)=iνSνF
2[ˆgS◦(ˆ
ˆgFˆ
)−(ˆ
ˆgFˆ
)◦ˆgS]τ=τ,(3)
whe e ◦deno es ime con olu ion. The o e all unnel cu en
ampli ude is de e mined by κ=νSνF(T2+U2) and he e -
ec i e spin-fil e ing pola iza ion is P=2TUm/(T2+U2).
T acing he gene al exp ession wi h app op ia e Pauli ma-
ices, we ex ac he cha ge cu en I=eT (ˆτ3ˆ
I) and he spin
cu en Is=T ( ˆ
σˆ
I). The eal- ime esponse is ob ained by he
me hod o analy ic con inua ion, desc ibed in Appendix B.
We assume ha he elec odes a e in he di usi e egime
and can be desc ibed by he ime-dependen Usadel equa ion
o quasiclassical GFs. In he imagina y- ime ep esen a ion i
has he o m (we se ¯h=1 he e and below)
−i{ˆτ3∂τ,ˆg}τ=Dˆ
∂ (ˆg◦ˆ
∂ ˆg)−i[ˆτ3ˆ
H,ˆg]τ,(4)
whe e Dis he di usion cons an , ˆ
H=ˆτ1+h·ˆ
σ, and h
is he exchange field. The quasiclassical GFs also sa is y
he no maliza ion condi ion (ˆg◦ˆg)τ,τ=δ(τ−τ). The ime
de i a i e, con olu ion p oduc , and di e en ial supe ope a o
in Eq. (4)a e
{ˆτ3∂τ,ˆg}τ=ˆτ3∂τ1ˆg(τ1,τ
2)+∂τ2ˆg(τ1,τ
2)ˆτ3,(5)
( ◦g)(τ1,τ
2)=β
0
dτ3 (τ1,τ
3)g(τ3,τ
2),(6)
ˆ
∂ =∂ −ie
c[ˆτ3A(τ),·],(7)
espec i ely, whe e eis he elemen a y cha ge and cis he
speed o ligh .
III. RESULTS
A. Quali a i e desc ip ion
In Fig. 1(b) we adap he usual semiconduc o pic u e o
he unnel cu en in supe conduc o junc ions [72] oshow
how he ime-dependen gap unc ion c ea es a nonequilib-
ium s a e n(ε, ) in he supe conduc ing elec ode. Due o he
Zeeman shi h, his s a e is nonsymme ic wi h espec o he
Fe mi le el εFand he e o e p oduces spin cu en h ough
he unnel ba ie be ween he SC and he adjacen no mal
me al. This quali a i e pic u e is based on he ime-dependen
ene gy spec um Eσ=√ξ2
p+( )2+σh, wi h σ=±1 o
spin-up/down Bogoliubo quasipa icles, espec i ely, whe e
ξpis he kine ic ene gy coun ed om he Fe mi le el εF.
Fo a slow ime-dependen o de pa ame e he spin-
esol ed pe u ba ion o he quasipa icle dis ibu ion unc ion
can be w i en as ( ˙
/)d
dNσ, whe e he numbe o he mally
exci ed s a es in equilib ium is Nσ=dξpn(Eσ(ξp,)), wi h
n(E)= anh(E/2T). The inelas ic sca e ing elaxa ion a e
is gi en by he Dynes pa ame e [73].
The spin-dependen pe u ba ion o he dis ibu ion unc-
ion esul s in he spin cu en
Is( )=κ
˙
d
d(N+−N−),(8)
whe e κis he e ec i e ba ie anspa ency. As shown below,
Eq. (8) is ob ained in he low- equency limi om
he gene al esul (21). The ad an age o Eq. (8) is ha i
allows o he ca oon in e p e a ion in e ms o he semicon-
duc o model in Fig. 1(b). Howe e , o he mos in e es ing
case when he equency o he ( )∝e2i oscilla ion is
compa able o he gap ∼and hence is coupled o he
HM [13,16,74,75], he pic u e becomes mo e complica ed and
equi es calcula ions using Eqs. (3) and (4) as desc ibed in
Sec. IIIC.
B. Second-ha monic gene a ion due o he b oken
pa icle-hole symme y
Spin cu en gene a ed by he HM can be con e ed o
cha ge cu en using spin-fil e ing FM elec odes. In he
se up shown in Fig. 1(a) he spin cu en is e ec i ely
con e ed o he cha ge cu en while passing h ough he
spin-fil e ing ba ie cha ac e ized by he pola iza ion ec-
o P. The ime-dependen cha ge cu en induced in his
way by he o de pa ame e ampli ude oscilla ion is he e o e
quali a i ely gi en by I( )∝P·Is( ), which esul s in he
033416-2
SPIN AND CHARGE CURRENTS DRIVEN BY THE HIGGS … PHYSICAL REVIEW RESEARCH 2, 033416 (2020)
es ima e I( )∝(P·h)∂ . Modula ion o he o de pa am-
e e ampli ude can be induced, o example, by an ex e nal
i adia ion [63,64]( )∝A2( ), whe e A( ) is he ec o
po en ial o he ex e nal field. Hence his cha ge cu en I( )∝
(P·h)∂ A2, being quad a ic in he ec o po en ial, demon-
s a es he second-ha monic gene a ion (SHG) con olled by
he supe conduc ing o de pa ame e .
Despi e he la ge amoun o a en ion o he nonlinea e -
ec s in supe conduc o s, SHG has no been ob ained be o e.1
Hence only hi d-ha monic gene a ion has been s udied in
supe conduc o s [19,32,63,64,76]. We show below ha such
a kind o SHG is no p ohibi ed by he gene ic symme ies o
he p oblem, bu is elimina ed by he app oxima e symme y
o Fe mi su ace sys ems, made exac in he widely used
quasiclassical app oxima ion [77].2This addi ional symme y
o he GF sa is ying he Usadel equa ion (4)is
ˆg(A,h,)=−ˆτ1ˆg(−A,h,
∗)ˆτ1.(9)
The o -diagonal Nambu space Pauli ma ix ˆτ1in e changes
he pa icle and hole blocks in he Hamil onian [77], so he
physical in e p e a ion o Eq. (9) is a pa icle-hole symme y.
Fo he nons a iona y cha ge cu en gene a ed by he ime-
dependen ec o po en ial his symme y yields I(A,)=
−I(−A,
∗). Fu he , in he absence o supe cu en o ex-
e nal o bi al fields we can assume he o de pa ame e o be
eal =∗. Then e en he b oken in e sion symme y nea
su aces does no help o p oduce SHG in supe conduc ing
sys ems in con as o he no mal me al coun e pa o his
e ec . Because o his symme y, he di ec coupling be ween
he HM and he cha ge cu en is p ohibi ed. Howe e , as we
see below, i allows o he spin cu en s d i en by he HM and
ex e nal field e en wi h a non e omagne ic ba ie , ha is, a
P=0.
The pa icle-hole symme y is b oken in supe conduc ing/
FM sys ems leading o la ge he moelec ic [34,78,79] and
anomalous Josephson e ec s [77]. Fo eal he ans o ma-
ion (9) applied o he gene al unnel cu en yields
I(A,h,P)=−I(−A,h,−P).(10)
This ela ion allows o SHG. Because he sign o Pis in e ed
he e is no longe a symme y wi h espec o he me e flipping
o he ec o po en ial, I(A)=−I(−A). Hence, o he AC
ex e nal field Aei ,Eq.(10) allows o he double- equency
cha ge cu en componen I2ei wi h he ampli ude I2∝
||2A2
(P·h) as well as he DC unnel cu en [51]IDC ∝
||2AA−(P·h). The esonan SHG o spin and cha ge
cu en s h ough he exci a ion o HM by elec omagne ic
i adia ion is discussed below in Sec. III C.
C. Calcula ion o spin and cha ge cu en s
We assume ha he supe conduc ing elec ode is d i en
ou o equilib ium by he elec omagne ic field desc ibed
by he ime-dependen ec o po en ial Aei . I p oduces
1He e we exclude he i ial SHG which esul s om he hi d-o de
nonlinea i y when bo h he oscilla ing and cons an fields a e applied.
2The Fe mi su ace symme y p esen in he quasiclassical app ox-
ima ion is b oken he e by he spin pola iza ion o unneling.
he second-ha monic pe u ba ion o he GF and unnel
cu en (3)
ˆgS(τ,τ)=T
ω
ˆgS(ω+,ω
−)eiω+τ−iω−τ,(11)
ˆ
I2=iνSνF
2T
ω
[ˆ
ˆgSˆ
ˆg0(ω−)−ˆg0(ω+)ˆ
ˆgSˆ
],(12)
whe e ω±=ω±a e he e mionic Ma suba a equencies
shi ed by he equency o he ex e nal field. We define
ˆgS=ˆgS(ω+,ω
−) and assume ha he e omagne is in he
equilib ium s a e de e mined by he GF ˆgF(ω)=ˆg0(ω)≡
sgn(ω)ˆτ3.
The e a e wo quali a i ely di e en e ms in he nonequi-
lib ium GF ˆgS=ˆgAA +ˆg. The fi s one is gene a ed by
he di ec coupling o he ex e nal elec omagne ic field. The
second e m is gene a ed by he o de pa ame e oscilla ions,
which can be induced by ei he he elec omagne ic field o
o he sou ces, o example, he spin cu en . Di ec coupling o
he elec omagne ic field is desc ibed by he GF pe u ba ions
o second o de by he ec o po en ial.
F om he Usadel equa ion (4) we find ha he pe u ba ion
ˆgAA(ω+,ω
−) sa isfies he equa ion
s+ˆg0(ω+)ˆgAA −s−ˆgAA ˆg0(ω−)
=DeA
c2
[ˆg0(ω+)ˆτ3ˆg0(ω)ˆτ3−ˆτ3ˆg0(ω)ˆτ3ˆg0(ω−)].
(13)
Expanding he no maliza ion condi ion in pe u ba ion se ies
p o ides an an icommu a ion ule ˆgAA ˆg0(ω−)=−ˆg0(ω+)ˆgAA,
which can be used o sol e Eq. (13),
ˆgAA =DeA
c2ˆτ3ˆg0(ω)ˆτ3−ˆg0(ω+)ˆτ3ˆg0(ω)ˆτ3ˆg0(ω−)
s++s−
,
(14)
whe e s±=ˆs(ω±) and ˆs(ω)=2−(iω−h·ˆ
σ)2. Abo e,
we abuse he no a ion sligh ly by w i ing he ma ix in e se
[s++s−]−1as a scala di ision. No ambigui y is in oduced
as ˆs(ω)’s commu e wi h he e ms in he nume a o . Fu he -
mo e, in he final exp ession o he spin cu en (21)weshi
he ene gy in eg a ion and emo e he exchange field om he
ˆs(ω)’s, making hei spin s uc u e i ial.
Co ec ions o he GF induced by he ime-dependen o -
de pa ame e ampli ude 2e2i can be ound in he o m
ˆg(τ,τ)=Tωei(ω+τ−ω−τ)ˆg(ω+,ω
−). F om he Usadel
equa ion (4) ha ˆg(ω+,ω
−) sa isfies he equa ion
s+ˆg0(ω+)ˆg−s−ˆgˆg0(ω−)=2[ˆτ2ˆg0(ω−)−ˆg0(ω+)ˆτ2].
(15)
Again using he no maliza ion condi ion, he solu ion o his
equa ion is gi en by
ˆg=2
ˆg0(ω+)ˆτ2ˆg0(ω−)−ˆτ2
s++s−
.(16)
The ampli ude 2can be ound om he sel -consis ency
equa ion
2=−λT
ω
T [ˆτ2(ˆgAA +ˆg)],(17)
033416-3
SILAEV, OJAJÄRVI, AND HEIKKILÄ PHYSICAL REVIEW RESEARCH 2, 033416 (2020)
whe e we in oduce he dimensionless pai ing cons an λand
he Pauli ma ix ˆτ2co esponds o he supe conduc ing ampli-
ude e ex.
The pa which is di ec ly p oduced by he i adia ion p o-
ides a sou ce o he Higgs mode
F(2)=−λT
ω
T [ˆτ2ˆgAA].(18)
The o he pa de e mines he sel -induced co ec ions o he
o de pa ame e ˜
2=(2)2desc ibed by he pola iza-
ion ope a o
(2)=1+πλT
ω
T 2+2
s−s+(s−+s+),(19)
whe e he ace is aken o e he spin deg ee o eedom. Col-
lec ing all he con ibu ions o he sel -consis ency equa ion
(17), we ge
2=F/[1 −(2)].(20)
This exp ession desc ibes he HM exci a ion in he supe -
conduc o d i en ou o equilib ium by a con inuous-wa e
i adia ion as shown schema ically in Fig. 1(a). The esonance
condi ion co esponding o he HM is sa isfied o =∗
when 1 −(2∗)=0+o(√). Hence he maximal ampli-
ude o he o de pa ame e oscilla ions is de e mined by
he b oadening pa ame e , leading o a sha p peak in
2(, T) o ≈∗(T). In he absence o spin elaxa ion
p ocesses ∗=(T).
Using he ound GF co ec ions (14) and (16), we calcu-
la e he spin and cha ge componen s o he unneling cu en
(12). The HM con ibu ion is de e mined by he e m ˆg.
Using he p ocedu e o analy ical con inua ion desc ibed in
Appendix B, we ob ain he ampli ude o eal- equency spin
cu en Is()e2i d i en by he HM,
Is()=iκh2
σ
σ
hdε
4π
ε[n(ε+)−n(ε−)]
sR
+sA
−(sR
++sA
−),(21)
whe e n(ε) is he equilib ium dis ibu ion unc ion. He e he
spin spli ing has been shi ed om he spec al unc ions o
he dis ibu ion unc ions, so ε±=ε±+σhand sR,A=
−i(ε±i)2−2. In he low- equency limi we
ob ain Eq. (8) when he spin cu en is d i en by he adiaba ic
ime dependence o in acco dance wi h he quali a i e pic-
u e shown schema ically in Fig. 1(b).
In he p esence o he HM, which is he slowly decaying os-
cilla ions o he o de pa ame e ( )[13,16], he spin cu en
is gi en by he sum o he co esponding Fou ie componen s
wi h he ampli udes gi en by (21). As a esul o Eq. (21)we
ge slowly decaying oscilla ions o he spin cu en Is( ) which
can be measu ed using elec ical p obes a e he supe con-
duc o is ini ially d i en in o a nonequilib ium s a e by a field
pulse.
Taking in o accoun he ela ion (20), we ob ain he SHG
spin and cha ge cu en s induced by he ex e nal i adia ion in
acco dance wi h he quali a i e discussion in Sec. III B.The
esonan beha io o he double- equency spin cu en Is()
esul ing om he HM mode exci a ion is shown in Fig. 2.
FIG. 2. Ampli ude o double- equency spin cu en Is()e2i
d i en h ough he SC/ e omagne unnel junc ion by an ex e -
nal field Aei . The cu en is no malized o I0=κD(eA/c)2;
Tcis he c i ical empe a u e. The sha p maximum a equencies
(2)≈1 co esponds o he esonan exci a ion o he Higgs
mode. (a) Is(, T)and(b)Is(T) a di e en equencies /0=
0.4,0.8,0.95,1.2. The exchange field is h=0.20and he Dynes
pa ame e =0.0050. The peaks a e a empe a u es de e mined
by (2)≈1.
IV. DISCUSSION
A. Elec ical de ec ion o he Higgs mode
The sugges ed e ec o SHG cha ge cu en coupled o he
HM can be measu ed, o example, in hin films o Al supe -
conduc o placed in a unnel con ac wi h FM i on elec odes
simila o he se ups used in he measu emen o he non-
local spin signals [38–40,43–47]. Wi h Tc=1.6 K and gap
0=2×10−4eV, he spin-spli ing field h=0.20can be
ob ained wi h an ex e nal in-plane magne ic field B≈0.5T,
and he pola iza ion o his ype o FM con ac [44]isP=0.2.
Wi h la ge enough a ea, he no mal-s a e unnel conduc ance
can be eκ=10−2S. The elec omagne ic pa o he se up can
be simila o he expe imen s on s imula ed supe conduc i i y
[80,81]. The elec omagne ic powe is cha ac e ized by he pa-
ame e α=D(eA/c)2, which can be made as la ge as α=
0.1Tcwi hou des oying he supe conduc i i y [82]. Fo Al i
yields α≈10−5eV. Wi h such pa ame e s he cha ge cu en
ampli ude co esponding o Fig. 2is ePI0=20 nA, which is
wo o de s o magni ude la ge han he nonlocal he moelec-
ic cu en measu ed ecen ly in a simila se up [47]. The
maximal HM esonance equency in Al is 2=100 GHz,
which is wi hin he capabili y o mode n spec um analyse s.
A 10-nA cu en ac oss 50 co esponds o he signal ampli-
ude −113 dBm, which means ha he signal- o-noise le el
exceeding uni y can be ob ained wi hin a 1-s measu emen
ime wi h s a e-o - he-a high- equency mic owa e spec um
analyse s wi h a noise floo abou −120 dBm/Hz.
B. Spin o ques gene a ed be he Higgs mode
I he exchange field hin he SC is noncollinea wi h he
magne iza ion min he e omagne , he HM gene a es a spin
o que ac ing on m. The gene ic sys em which can ealize his
configu a ion is shown in Fig. 3(a). He e he exchange field
hm0is c ea ed by he e omagne ic insula o laye wi h a
fixed magne ic momen m0[34].
The spin- ans e o que (STT) gene a ed by he HM is
shown schema ically in Fig. 3(a). The pola iza ion o he
033416-4
SPIN AND CHARGE CURRENTS DRIVEN BY THE HIGGS … PHYSICAL REVIEW RESEARCH 2, 033416 (2020)
FIG. 3. Se ups o s udying ans e se spin cu en s coupled o
he HM in he supe conduc o (S). The exchange field in S is gene -
a ed by he adjacen e omagne ic insula o (FI) wi h magne iza ion
m0. (a) The spin o que τis gene a ed in an adjacen e omagne
(F) wi h noncollinea magne iza ion m∦m0. (b) Magne iza ion p e-
cession m( ) induces he spin cu en Isand he spin ba e y e ec
leading o a pe u ba ion o he o de pa ame e ampli ude δ.
nonequilib ium spin cu en Isis de e mined by he di ec-
ion o he exchange field h. Assuming ha he ans e se
componen o he spin cu en is abso bed in he e omag-
ne [83–87], we ob ain he STT τ=Ish⊥/h, whe e h⊥=
h−m(m·h) is he pe pendicula componen o he exchange
field.
The ecip ocal e ec shown in Fig. 3(b) is he pe u ba ion
o he gap δ by he magne ic p ecession. The pumped spin
cu en [83]Is∝m×˙
mhas a longi udinal componen Ish
which gene a es a ime-dependen spin accumula ion μsin he
SC. In combina ion wi h he spin-spli ing field h, his esul s
in [51,88]
δ =λ
1−μs∂(N+−N−),(22)
whe e 1 −∝λis he low- equency asymp o ic o he
pola iza ion ope a o . This exp ession demons a es he pos-
sibili y o couple he o de pa ame e ampli ude wi h he
magne iza ion dynamics. Thus he highe - equency magne-
iza ion p ecession wi h ∼gene a es he HM in he
supe conduc o wi h a spin-spli ing field.
This e ec can be iewed as he HM-media ed ans e o
he spin angula momen um om he e omagne ic insula o
o he me allic e omagne shown in Fig. 3(a). Oscilla ing
STT gene a ed by he o de pa ame e ampli ude mode can
exci e he e omagne ic esonance (FMR) in he a ached
e omagne . Hyb idiza ion o he FMR and Higgs esonance
should show up as he a oided c ossing o he peaks in
he second-ha monic esponse o he sys ems. Such an ex-
pe imen will di ec ly demons a e he dynamical coupling
o he magne ic and supe conduc ing o de s. Modifica ion
o he FMR linewid h by supe conduc ing co ela ions in
e omagne /SC s uc u es has been obse ed [89–91]. In
pe malloy films he FMR has been measu ed in fields up o
0.3 T co esponding o a equency o 20 GHz. Fo such a
equency he Higgs esonance in Al is expec ed o occu
a T≈0.92Tc. Thus a ying he field, one can measu e he
empe a u e-con olled hyb idiza ion o he HM and FMR
mode in Al/pe malloy s uc u es wi hin he cu en ly acces-
sible ange o pa ame e s.
V. CONCLUSION
We ha e demons a ed ha spin and cha ge cu en s can
be e ec i ely gene a ed by he collec i e ampli ude modes o
he supe conduc ing o de pa ame e . Owing o he ac ha
he HM can be gene a ed by ex e nal i adia ion [27,92], ou
esul pa es he way o a concep ually di e en di ec ion o
supe conduc ing op ospin onics: he s udy o spin cu en s
and spin o ques gene a ed by ligh in e ac ing wi h supe con-
duc ing ma e ials.
We ha e sugges ed a de ec ion scheme o he HM based on
measu ing esonan elec ic signals, ei he he cha ge cu en
o ol age gene a ed ac oss he spin-pola ized unnel junc ion
by he ex e nal field. Because hese signals appea a he
doubled equency o he ex e nal field, ou se up in oduces
a sys em ea u ing second-ha monic gene a ion con olled by
supe conduc i i y. The sugges ed SHG can be s udied using
op ical o mic owa e de ec o s [93] and he unneling cu en
I2can be de ec ed using elec ical p obes. This ea u e o he
SHG as compa ed o he p e iously known nonlinea esponse
echniques allows o an elec ical de ec ion o he HM in
supe conduc o s.
Finally, a quali a i ely simila e ec should occu p o ided
he e omagne is eplaced by ano he spin-fil e ing elemen
such as a semiconduc o nanowi e in p oposed Majo ana-
based qubi s [94–97]. The cha ge noise which is impo an in
such de ices [97] can cause he o de pa ame e oscilla ions
coupled o he spli ing o Majo ana ze o modes. This cou-
pling opens possibili ies o many in e es ing e ec s o s udy.
ACKNOWLEDGMENTS
This wo k was suppo ed by he Academy o Finland
(P ojec s No. 297439 and No. 317118), Jenny and An i
Wihu i Founda ion, Russian Science Founda ion (G an No.
19-19-00594), and he Eu opean Union’s Ho izon 2020 e-
sea ch and inno a ion p og am unde G an Ag eemen No.
800923 (SUPERTED).
APPENDIX A: TUNNEL CURRENT
We model he spin-dependen unneling h ough he
SC/ e omagne in e ace by he unneling Hamil onian (1).
We calcula e he unneling cu en as a unc ion o he ime
on he con ou unning along he imagina y axis om 0 o
β=1/T.
The ma ix unneling cu en in e ms o he imagina y ime
unc ions eads
ˆ
I(τ)=i
2
k
[∂τˆ
GS(τ,τ,k,k)+∂τˆ
GS(τ,τ,k,k)]τ=τ.
(A1)
To find he pe u ba ion we conside he con ou -o de ed GF
ˆ
GS(τ1,τ
2,k,k)=Tˆ
Sˆ
Ak(τ1)ˆ
A†
k(τ2),(A2)
whe e Tis he con ou -o de ing ope a o and
ˆ
S≈1−β
0
dτ3HT(τ3).(A3)
033416-5
SILAEV, OJAJÄRVI, AND HEIKKILÄ PHYSICAL REVIEW RESEARCH 2, 033416 (2020)
In he in e ac ion ep esen a ion wi h espec o he unneling Hamil onian he equa ion o mo ion is
∂τˆ
Ak=[ˆ
Ak,HT]=
k
(ˆ
kkˆ
Bk).(A4)
Using he equa ion o mo ion we ge
−∂τ1ˆ
GS(τ1,τ
2,k,k)=−
qTˆ
S[ˆ
kq ˆ
Bq(τ1)] ˆ
A†
k(τ2)
≈
qTβ
0
dτ3HT(τ3)ˆ
kq ˆ
Bq(τ1)ˆ
A†
k(τ2)
=
k1,k
1,qTβ
0
dτ3ˆ
B†
k
1(τ3)ˆ
†
k1k
1
ˆ
Ak1(τ3)ˆ
kq ˆ
Bq(τ1)ˆ
A†
k(τ2)
=
k1,k
1,qβ
0
dτ3ˆ
kqTˆ
Bq(τ1)ˆ
B†
k
1(τ3)ˆ
k
1k1Tˆ
Ak1(τ3)ˆ
A†
k(τ2)
=
k1,k
1,qβ
0
dτ3ˆ
kq ˆ
GF(τ1,τ
3,q,k
1)ˆ
k
1k1ˆ
GS(τ3,τ
2,k1,k)(A5)
and
−∂τ2ˆ
GS(τ1,τ
2,k,k)=−
qTˆ
Sˆ
Ak(τ)[ˆ
kq ˆ
Bq( )]†
≈
qTβ
0
dτ3HT(τ3)ˆ
Ak(τ1)[ˆ
kq ˆ
Bq(τ2)]†
=
k1,k
1,qTβ
0
dτ3ˆ
k1k
1ˆ
Bk
1(τ3)ˆ
A†
k1(τ3)ˆ
Ak(τ1)ˆ
B†
q(τ2)ˆ
†
qk
=−
k1,k
1,qβ
0
dτ3Tˆ
Ak(τ1)ˆ
A†
k1(τ3)ˆ
k1k
1Tˆ
Bk
1(τ3)ˆ
B†
q(τ2)ˆ
qk
=−
k1,k
1,qβ
0
dτ3ˆ
GS(τ1,τ
3,k,k1)ˆ
k1k
1ˆ
GF(τ3,τ
2,k
1,q)ˆ
qk.(A6)
Hence he ma ix cu en is gi en by
ˆ
I(τ)=i
2
k,k1,k
1,qβ
0
dτˆ
GS(τ,τ,k,k1)ˆ
k1k
1ˆ
GF(τ,τ,k
1,q)ˆ
qk
−ˆ
kq ˆ
GF(τ,τ,q,k1)ˆ
k1k
1ˆ
GS(τ,τ,k
1,k).(A7)
We assume ha GFs a e spa ially homogeneous, so ˆ
GF(τ,τ,q,k1)=δq,k1ˆ
GF(τ,τ,q) and he ma ix elemen is momen um
independen ˆ
kk=ˆ
. Then we can in oduce he quasiclassical unc ions qˆ
GF,S(τ,τ,q)=νF,Sˆτ3ˆgF,S(τ,τ) o w i e he
cu en as
ˆ
I(τ)=iνSνF
2[ˆgS◦(ˆτ3ˆ
ˆgFˆ
ˆτ3)−(ˆτ3ˆ
ˆgFˆ
ˆτ3)◦ˆgS]τ=τ.(A8)
Taking in o accoun ha he no mal me al GF ˆgFcommu es
wi h ˆτ3,Eq.(A8) can be educed o Eq. (3).
APPENDIX B: ANALYTICAL CONTINUATION
In o de o find he eal- equency esponse we need
o implemen he analy ic con inua ion o Eq. (12). These
second-o de esponses a e ob ained by he summa ion o
exp essions which depend on he mul iple shi ed e mionic
equencies such as g(ω1,ω
2,ω
3). The analy ic con inua ion
o he sum by Ma suba a equencies is de e mined acco ding
o he gene al ule [98]
T
ω
g(ω1,ω
2,ω
3)→
3
l=1dε
4πin0(εl)g...,−iεR
l,...
−g...,−iεA
l,...,(B1)
033416-6
SPIN AND CHARGE CURRENTS DRIVEN BY THE HIGGS … PHYSICAL REVIEW RESEARCH 2, 033416 (2020)
whe e n0(ε)= anh(ε/2T) is he equilib ium dis ibu ion
unc ion. On he igh -hand side o (B1) we subs i u e in
each e m ωk<l=−iεR
kand ωk>l=−iεA
k o k=1,2,3, and
we define εk=ε+(2 −k),εR=ε+i, and εA=ε−i.
He e he e m wi h >0 is added o shi he in eg a ion
con ou in o he co esponding hal plane. A he same ime,
can be used as he Dynes pa ame e [73] o desc ibe he e ec
o di e en depai ing mechanisms on spec al unc ions in he
supe conduc o .
We implemen he analy ical con inua ion in such a way
ha s(−iεR,A)=−i(εR,A)2−2, assuming ha he b anch
cu s un om (, ∞) and (−∞,−). In he p esence o he
spin-spli ing field he ene gy in Eq. (B1) should be shi ed o
ε+σh, whe e σ=±1 is he spin subband index.
The equilib ium GF in he imagina y equency domain is
gi en by ˆg0(ω)=(ˆτ3ω−ˆτ2)/s(ω). The eal- equency con-
inua ion eads ˆgR,A
0(ε)=(ˆτ3εR,A−iˆτ2)/(εR,A)2−2.
Example. To demons a e he analy ical con inua ion in
p ac ice we calcula e he spin cu en d i en by he Higgs
mode. Fo eal equencies he spin cu en ob ained om
(A8) can be w i en in e ms o he Keldysh componen
Is=κ
8π
σ
σdεT [ˆgF(ε+)ˆgS(ε)−ˆgS(ε)ˆgF(ε−)]K
=κ
8π
σ
σdε[n(ε+)−n(ε−)]T ˆτ3ˆga
S,(B2)
whe e ε±=ε+σh±ω. In de i ing (B2) we used he ac
ha ˆgR(A)
F=±1 do no depend on ene gy. The anomalous pa
o he nonequilib ium GF in he supe conduc o is
ˆga
S=2
ˆgR(ε+)ˆτ2ˆgA(ε−)−ˆτ2
sR
++sA
−
,(B3)
whe e we define sR,A
±=sR,A(ε±). Subs i u ing he solu ion
(B3) and using T [τ3gR
+τ2gA
−]=2i0ε/sR
+sA
−, we ge
Is=iκ02
σ
σdε
4π
(ε+σh)[n(ε+)−n(ε−)]
sR
+sA
−(sR
++sA
−)
=iκ02
(ω+i)dε
16π
σ
σ[n(ε+)−n(ε−)]1
sR
+−1
sA
−,
(B4)
whe eweuse(sR
+)2−(sA
−)2=4(ε+σh)(ω+i). In he
low- equency limi we can subs i u e n(ε+)−n(ε−)=
2ω∂εnand sR
+=−sA
+=−i√ε2−2. Then he spin cu en
can be w i en in he simple o m
Is=κ
σ
σd
d dξpn(Eσ(ξp,( )))
=κ
˙
d
d(N+−N−),(B5)
whe e Eσ(ξp,( ))=√ξ2
p+2( )+σhis he spec um o
Bogoliubo quasipa icles shi ed by he spin-spli ing field h.
[1] C. M. Va ma, Higgs boson in supe conduc o s, J. Low Temp.
Phys. 126, 901 (2002).
[2] D. Podolsky, A. Aue bach, and D. P. A o as, Visibili y o he
ampli ude (Higgs) mode in condensed ma e , Phys. Re . B 84,
174522 (2011).
[3] A. Pashkin and A. Lei ens o e , Pa icle physics in a supe con-
duc o , Science 345, 1121 (2014).
[4] G. E. Volo ik and M. A. Zubko , Higgs bosons in pa icle
physics and in condensed ma e , J. Low Temp. Phys. 175, 486
(2014).
[5] D. Pekke and C. M. Va ma, Ampli ude/Higgs modes in con-
densed ma e physics, Annu. Re . Condens. Ma e Phys. 6,
269 (2015).
[6] P. W. Higgs, B oken Symme ies and he Masses o Gauge
Bosons, Phys.Re .Le .13, 508 (1964).
[7] G. G üne , The dynamics o cha ge-densi y wa es, Re . Mod.
Phys. 60, 1129 (1988).
[8] D. N. Paulson, R. T. Johnson, and J. C. Whea ley, P opaga ion
o Collisionless Sound in No mal and Ex ao dina y Phases o
Liquid 3He below 3 mK, Phys. Re . Le . 30, 829 (1973).
[9] D. T. Lawson, W. J. Gully, S. Golds ein, R. C. Richa dson,
and D. M. Lee, A enua ion o Ze o Sound and he Low-
Tempe a u e T ansi ions in Liquid 3He, Phys. Re . Le . 30, 541
(1973).
[10] V. V. Za jalo , S. Au i, V. B. El so , P. J. Heikkinen, and
G. E. Volo ik, Ligh Higgs channel o he esonan decay
o magnon condensa e in supe fluid 3He-B, Na . Commun. 7,
10294 (2016).
[11] U. Bissbo , S. Gö ze, Y. Li, J. Heinze, J. S. K ause , M.
Weinbe g, C. Becke , K. Sengs ock, and W. Ho s e e , De ec -
ing he Ampli ude Mode o S ongly In e ac ing La ice Bosons
by B agg Sca e ing, Phys. Re . Le . 106, 205303 (2011).
[12] M. End es, T. Fukuha a, D. Pekke , M. Cheneau, P. Schauß, C.
G oss, E. Demle , S. Kuh , and I. Bloch, The ‘Higgs’ ampli ude
mode a he wo-dimensional supe fluid/Mo insula o ansi-
ion, Na u e (London) 487, 454 (2012).
[13] A. F. Volko and S. M. Kogan, Collisionless elaxa ion o he
ene gy gap in supe conduc o s, Zh. Eksp. Teo . Fiz. 65, 2038
(1973) [So . Phys.—JETP 38, 1018 (1974)].
[14] R. Soo yakuma and M. V. Klein, Raman Sca e ing by
Supe conduc ing-Gap Exci a ions and hei Coupling o
Cha ge-Densi y Wa es, Phys.Re .Le .45, 660 (1980).
[15] P. B. Li lewood and C. M. Va ma, Ampli ude collec i e modes
in supe conduc o s and hei coupling o cha ge-densi y wa es,
Phys. Re . B 26, 4883 (1982).
[16] R. A. Ba anko , L. S. Le i o , and B. Z. Spi ak, Collec i e Rabi
Oscilla ions and Soli ons in a Time-Dependen BCS Pai ing
P oblem, Phys.Re .Le .93, 160401 (2004).
[17] R. G asse , T. Cea, Y. Gallais, M. Cazayous, A. Sacu o, L.
Ca io, L. Ben a o, and M.-A. Méasson, Higgs-mode adiance
and cha ge-densi y-wa e o de in 2H-NbSe2,Phys.Re .B97,
094502 (2018).
[18] R. Ma sunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H. Te ai,
Z. Wang, and R. Shimano, Higgs Ampli ude Mode in he BCS
Supe conduc o s Nb1−xTixN Induced by Te ahe z Pulse Exci-
a ion, Phys.Re .Le .111, 057002 (2013).
033416-7
SILAEV, OJAJÄRVI, AND HEIKKILÄ PHYSICAL REVIEW RESEARCH 2, 033416 (2020)
[19] R. Ma sunaga, N. Tsuji, H. Fuji a, A. Sugioka, K. Makise, Y.
Uzawa, H. Te ai, Z. Wang, H. Aoki, and R. Shimano, Ligh -
induced collec i e pseudospin p ecession esona ing wi h Higgs
mode in a supe conduc o , Science 345, 1145 (2014).
[20] D. She man, U. S. P ach , B. Go shuno , S. Po an, J. Jesudasan,
M. Chand, P. Raychaudhu i, M. Swanson, N. T i edi, A.
Aue bach, M. Sche fle , A. F ydman, and M. D essel, The
Higgs mode in diso de ed supe conduc o s close o a quan um
phase ansi ion, Na . Phys. 11, 188 (2015).
[21] N. Tsuji and H. Aoki, Theo y o Ande son pseudospin eso-
nance wi h Higgs mode in supe conduc o s, Phys.Re .B92,
064508 (2015).
[22]R.Ma sunaga,N.Tsuji,K.Makise,H.Te ai,H.Aoki,andR.
Shimano, Pola iza ion- esol ed e ahe z hi d-ha monic gene -
a ion in a single-c ys al supe conduc o NbN: Dominance o he
Higgs mode beyond he BCS app oxima ion, Phys. Re . B 96,
020505(R) (2017).
[23] K. Ka sumi, N. Tsuji, Y. I. Hamada, R. Ma sunaga, J.
Schneeloch, R. D. Zhong, G. D. Gu, H. Aoki, Y. Gallais,
and R. Shimano, Higgs Mode in he d-Wa e Supe conduc o
Bi2S 2CaCu2O8+xD i en by an In ense Te ahe z Pulse, Phys.
Re . Le . 120, 117001 (2018).
[24] S. Nakamu a, Y. Iida, Y. Mu o ani, R. Ma sunaga, H. Te ai,
and R. Shimano, In a ed Ac i a ion o he Higgs Mode by
Supe cu en Injec ion in Supe conduc ing NbN, Phys. Re .
Le . 122, 257001 (2019).
[25] H. Uema su, T. Mizushima, A. Tsu u a, S. Fujimo o, and J. A.
Sauls, Chi al Higgs Mode in Nema ic Supe conduc o s, Phys.
Re . Le . 123, 237001 (2019).
[26] A. Moo , A. F. Volko , and K. B. E e o , Ampli ude Higgs
Mode and Admi ance in Supe conduc o s wi h a Mo ing Con-
densa e, Phys. Re . Le . 118, 047001 (2017).
[27] M. Silae , Nonlinea elec omagne ic esponse and Higgs-mode
exci a ion in BCS supe conduc o s wi h impu i ies, Phys. Re .
B99, 224511 (2019).
[28] R. G asse , Y. Gallais, A. Sacu o, M. Cazayous, S. Mañas-
Vale o, E. Co onado, and M.-A. Méasson, P essu e-Induced
Collapse o he Cha ge Densi y Wa e and Higgs Mode Visi-
bili y in 2H-TaS2,Phys. Re . Le . 122, 127001 (2019).
[29] M. Beck, M. Klamme , S. Lang, P. Leide e , V. V. Kabano ,
G. N. Gol’ sman, and J. Demsa , Ene gy-Gap Dynamics
o Supe conduc ing NbN Thin Films S udied by Time-
Resol ed Te ahe z Spec oscopy, Phys. Re . Le . 107, 177007
(2011).
[30] M. Beck, I. Rousseau, M. Klamme , P. Leide e , M.
Mi endo , S. Winne l, M. Helm, G. N. Gol’ sman, and J.
Demsa , T ansien Inc ease o he Ene gy Gap o Supe con-
duc ing NbN Thin Films Exci ed by Resonan Na ow-Band
Te ahe z Pulses, Phys.Re .Le .110, 267003 (2013).
[31] R. Ma sunaga and R. Shimano, Nonequilib ium BCS S a e
Dynamics Induced by In ense Te ahe z Pulses in a Supe con-
duc ing NbN Film, Phys. Re . Le . 109, 187002 (2012).
[32] F. Gio gianni, T. Cea, C. Vica io, C. P. Hau i, W. K. Wi hanage,
X. Xi, and L. Ben a o, Legge mode con olled by ligh pulses,
Na . Phys. 15, 341 (2019).
[33] W. V. Budzinski, M. P. Ga unkel, and R. W. Ma kley, Magne ic
field dependence o he su ace esis ance o pu e and impu e
supe conduc ing aluminum a pho on ene gies nea he ene gy
gap, Phys. Re . B 7, 1001 (1973).
[34] F. S. Be ge e , M. Silae , P. Vi anen, and T. T. Heikkilä,
Colloquium: Nonequilib ium e ec s in supe conduc o s wi h a
spin-spli ing field, Re . Mod. Phys. 90, 041001 (2018).
[35] T. T. Heikkilä, M. Silae , P. Vi anen, and F. S.
Be ge e , The mal, elec ic and spin anspo in
supe conduc o / e omagne ic-insula o s uc u es, P og.
Su . Sci. 94, 100540 (2019).
[36] D. Beckmann, Spin manipula ion in nanoscale supe conduc o s,
J. Phys.: Condens. Ma e 28, 163001 (2016).
[37] C. H. L. Quay and M. Ap ili, Ou -o -equilib ium spin anspo
in mesoscopic supe conduc o s, Philos. T ans. R. Soc. A 376,
20150342 (2018).
[38] F. Hüble , M. J. Wol , D. Beckmann, and H. . Löhneysen,
Long-Range Spin-Pola ized Quasipa icle T anspo in Meso-
scopic Al Supe conduc o s wi h a Zeeman Spli ing, Phys. Re .
Le . 109, 207001 (2012).
[39] M. J. Wol , C. Sü ge s, G. Fische , and D. Beckmann,
Spin-pola ized quasipa icle anspo in exchange-spli supe -
conduc ing aluminum on eu opium sulfide, Phys. Re . B 90,
144509 (2014).
[40] M. Rouco, S. Chak abo y, F. Aikebaie , V. N. Golo ach, E.
S ambini, J. S. Moode a, F. Giazo o, T. T. Heikkilä, and F. S.
Be ge e , Cha ge anspo h ough spin-pola ized unnel junc-
ion be ween wo spin-spli supe conduc o s, Phys. Re . B 100,
184501 (2019).
[41] G. De Simoni, E. S ambini, J. S. Moode a, F. S. Be ge e , and
F. Giazo o, Towa d he absolu e spin- al e e ec in supe con-
duc ing unnel junc ions, Nano Le . 18, 6369 (2018).
[42] E. S ambini, V. N. Golo ach, G. De Simoni, J. S. Moode a,
F. S. Be ge e , and F. Giazo o, Re ealing he magne ic p oxim-
i y e ec in EuS/Al bilaye s h ough supe conduc ing unneling
spec oscopy, Phys. Re . Ma e . 1, 054402 (2017).
[43] S. Kolenda, M. J. Wol , and D. Beckmann, Obse a ion
o The moelec ic Cu en s in High-Field Supe conduc o -
Fe omagne Tunnel Junc ions, Phys. Re . Le . 116, 097001
(2016).
[44] M. J. Wol , F. Hüble , S. Kolenda, H. . Löhneysen, and D.
Beckmann, Spin injec ion om a no mal me al in o a meso-
scopic supe conduc o , Phys.Re .B87, 024517 (2013).
[45] C. H. L. Quay, D. Che allie , C. Bena, and M. Ap ili, Spin
imbalance and spin-cha ge sepa a ion in a mesoscopic supe -
conduc o , Na . Phys. 9, 84 (2013).
[46] C. H. L. Quay, C. Du eix, D. Che allie , C. Bena, and M.
Ap ili, F equency-domain measu emen o he spin-imbalance
li e ime in supe conduc o s, Phys.Re .B93, 220501(R)
(2016).
[47] J. Heid ich and D. Beckmann, Nonlocal he moelec ic e ec s
in high-field supe conduc o - e omagne hyb id s uc u es,
Phys. Re . B 100, 134501 (2019).
[48] M. Silae , P. Vi anen, F. S. Be ge e , and T. T. Heikkilä, Long-
Range Spin Accumula ion om Hea Injec ion in Mesoscopic
Supe conduc o s wi h Zeeman Spli ing, Phys. Re . Le . 114,
167002 (2015).
[49] I. V. Bobko a and A. M. Bobko , Long- ange spin imbalance
in mesoscopic supe conduc o s unde Zeeman spli ing, JETP
Le . 101, 118 (2015).
[50] T. K ish op, M. Houze , and J. S. Meye , Nonequilib ium spin
anspo in Zeeman-spli supe conduc o s, Phys. Re . B 91,
121407(R) (2015).
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