Superconducting spintronic tunnel diode

ARTICLE
Supe conduc ing spin onic unnel diode
E. S ambini 1✉, M. Spies1✉, N. Liga o 1, S. Ilić2, M. Rouco2, Ca men González-O ellana2, Maxim Ilyn2,
Celia Roge o2,3, F. S. Be ge e 2,3, J. S. Moode a4, P. Vi anen5, T. T. Heikkilä 5& F. Giazo o1✉
Diodes a e key elemen s o elec onics, op ics, and de ec ion. Thei e olu ion owa ds low
dissipa ion elec onics has seen he hyb idiza ion wi h supe conduc o s and he ealiza ion o
supe cu en diodes wi h ze o esis ance in only one di ec ion. He e, we p esen he quasi-
pa icle coun e pa , a supe conduc ing unnel diode wi h ze o conduc ance in only one
di ec ion. The di ec ion-selec i e p opaga ion o he cha ge has been ob ained h ough he
b oken elec on-hole symme y induced by he spin selec ion o he e omagne ic unnel
ba ie : a EuS hin film sepa a ing a supe conduc ing Al and a no mal me al Cu laye . The Cu/
EuS/Al unnel junc ion achie es a la ge ec ifica ion (up o ∼40%) al eady o a small ol age
bias (∼200 μV) hanks o he small ene gy scale o he sys em: he Al supe conduc ing gap.
Wi h he help o an analy ical heo e ical model we can link he maximum ec ifica ion o he
spin pola iza ion (P) o he ba ie and desc ibe he quasi-ideal Shockley-diode beha io o
he junc ion. This c yogenic spin onic ec ifie is p omising o he applica ion in highly-
sensi i e adia ion de ec ion o which wo di e en configu a ions a e e alua ed. In addi ion,
he supe conduc ing diode may pa e he way o u u e low-dissipa ion and as supe -
conduc ing elec onics.
h ps://doi.o g/10.1038/s41467-022-29990-2 OPEN
1NEST, Is i u o Nanoscienze-CNR and Scuola No male Supe io e, I-56127 Pisa, I aly. 2Cen o de Física de Ma e iales (CFM-MPC) Cen o Mix o CSIC-UPV/
EHU, E-20018 Donos ia-San Sebas ián, Spain. 3Donos ia In e na ional Physics Cen e (DIPC), E-20018 Donos ia–San Sebas ián, Spain. 4Physics
Depa men and Plasma Science and Fusion Cen e , Massachuse s Ins i u e o Technology, Camb idge, MA 02139, USA. 5Depa men o Physics and
Nanoscience Cen e , Uni e si y o Jy äskylä, P.O. Box 35 (YFL), FI-40014 Jy äskylä, Finland. ✉email: [email p o ec ed];ma [email protected] .i ;
ancesco.giazo [email protected]
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1234567890():,;
Diodes a e non-linea and non- ecip ocal ci cui s in which
a lack o spa ial in e sion symme y p o ides a s ongly
di ec ion-selec i e elec on anspo . In he long and
success ul his o y o diodes, he ma e ial sea ch o his symme y
b eaking has been mainly ocused on semiconduc ing and
me allic junc ions. Howe e , owing o hei la ge ene gy gap,
semiconduc o s cease o wo k a he sub-Kel in empe a u es
ele an o eme ging c yogenic elec onics1and ul asensi i e
de ec ion, especially a sub-THz equencies2. This p oblem could
be pa ially sol ed by using low-dimensional s uc u es like
quan um do s, which do exhibi cu en ec ifica ion3,4. Gi en
ha he elec on-hole symme y in quan um do s can be uned
only wi hin he le el o a single quan um channel, he impedance
o such sys ems ends o be high, and he ec ified cu en s
he eby e y low, limi ing he alue o his app oach. Supe -
conduc o s would be ideal candida es o he ealiza ion o
c yogenic diodes and de ec o s due o hei in insically low
impedance, and he lowe ene gy scales o he supe conduc ing
gap (∼meV) compa ed o semiconduc o s (∼eV). S ill, he
implemen a ion o a supe conduc ing diode u ns ou o be di -
ficul since i equi es b eaking o he elec on-hole symme y,
whe eas he BCS supe conduc ing s a e is, by defini ion, elec on-
hole symme ic. Recen ly, supe cu en diodes ha e been engi-
nee ed wi h me allic supe la ices in s ong magne ic fields,
o e ing he equi ed lack o spa ial in e sion5,6, wi h hin films
pa e ned wi h nanoholes7, wi h uncon en ional Josephson
junc ions8–11 and wi h supe conduc o s wi h la ge spin-o bi
in e ac ions12–14. Al e na i e app oaches o ealizing a supe -
conduc ing diode a e also possible wi h he quasi-pa icle coun-
e pa in spin-selec i e unnel junc ions. When bo h spin
fil e ing and spli ing a e p esen i is possible o b eak he
elec on-hole symme y o he sys em and gene a e di ec ion-
selec i e elec on anspo 15. Fe omagne ic insula o s (FI) like
Eu-based chalcogenides combined wi h supe conduc o s (S) o e
b igh pe spec i es o he ealiza ion o his amily o supe -
conduc ing spin onic echnologies16. De ices based on hin films
o FI/S bilaye s showing ideal spin fil e ing and spin spli ing17,18
ha e been al eady demons a ed in a numbe o seminal expe i-
men s pe o med on EuS/Al-based unnel junc ions17,19,20. He e,
we show a supe conduc ing diode based on a spin-selec i e Al/
EuS/Cu unnel junc ion. The obse ed di ec ion-selec i e elec on
anspo is a he basis o cha ge ec ifica ion and
he moelec ici y21–23 and makes he design o he p esen
supe conduc ing spin onic de ice a p omising app oach o he
implemen a ion o biasless ul asensi i e THz de ec o s24.
The wo king p inciple and de ice cha ac e is ics o he no mal
me al- e omagne ic insula o -supe conduc o (N/FI/S) unnel
junc ion, cen al o his pape , a e shown in Fig. 1. The schema ic o
he de ice s uc u e and measu emen configu a ions o he unnel
spec oscopy can be seen in panel (a). A N s ip o Cu and a S s ip
o Al a e o ien ed pe pendicula o one ano he o ming a c oss-
ba , and a e sepa a ed by a FI ba ie o EuS (see Me hods o
ab ica ion de ails). The EuS laye induces a spin spli ing by an
ene gy wi h magni ude (h) in S h ough in e ace exchange
in e ac ion19,25,26, and i s FI na u e causes a spin fil e ing (P)o he
elec on unneling ac oss he junc ion. The o me esul s in an
opposi e ene gy shi o he BCS densi y o s a es (DoS) o he wo
spin species, as ske ched in Fig. 1(c), while he la e o ms a un-
neling ba ie wi h di e en heigh s o he wo spin species. This
wo old e ec can be p obed expe imen ally by measu ing he
di e en ial conduc ance o he unnel junc ion and leads o qua-
li a i e changes in he sys em’s anspo cha ac e is ics19,27,28.An
example o a unneling conduc ance measu emen as a unc ion o
bias ol age ac oss he N/FI/S junc ion is shown in Fig. 1(d). A
small ol ages (∣V∣≲200 μV) he conduc ance is s ongly sup-
p essed due o he lack o s a es wi hin he supe conduc ing ene gy
gap. A highe bias ol ages, ou dis inc peaks can be obse ed in
o al, co esponding o he ou peaks o he wo BCS DoS a
e∣V∣=Δ±h. The di e en ampli udes o he conduc ance peaks
s em om he spin fil e ing P, p omo ing one spin channel wi h
espec o he o he . All hese pa ame e s can be ex ac ed by fi ing
he conduc ance wi h a nume ical model (see Eq. (3)and(14)in
he “Me hods” o de ails on he model) ha akes in o accoun he
spin spli ing, spin elaxa ion, and o bi al depai ing due o he
magne ic field28, as shown by he ed cu e in Fig. 1(d). Addi-
ionally, he applica ion o an ex e nal magne ic field can
s eng hen he pola iza ion o he EuS laye and enhance bo h h
and P, as shown in Fig. 1(e) and ( ). No ably, hanks o he e o-
magne ism o he EuS, bo h a sizable spli ing and pola iza ion a e
obse ed e en a ze o field (h
0
≃0.025Δ,P
0
≃0.2, and Δ=370
μeV). These anish a he EuS coe ci e field (≃10 mT).
Resul s and discussion
Measu emen configu a ions. In he ollowing, wo measu e-
men configu a ions (ske ch in Fig. 1a) ha e been adop ed o
quan i y he diode cha ac e is ics. In configu a ion (i) he cu en
flows om he S o he N laye , he eby e ec i ely c ossing he
junc ion. A con en ional ec ifica ion is obse ed in his case. In
configu a ion (ii) he cu en flows along he N s ip, and a
ans e se ec ifica ion is obse ed. In bo h cases he ol age d op
is measu ed om he S o he N laye ac oss he unnel junc ion.
No ably, in configu a ion (ii) he cu en and ol age pa hs a e
decoupled, p o iding mo e flexibili y in p ojec ing he impedance
o he de ice, wi h ad an ages o he in eg a ion o an an enna
and he implemen a ion o a de ec o . Measu emen s o he wo
configu a ions a e compa ed and discussed. A ypical cu en s.
ol age (I(V)) cha ac e is ic o he unnel junc ion shows a con-
en ional ec ifica ion, as can be seen in Fig. 2. I co esponds o
measu emen configu a ion (i) in ol age bias. The cu en bias
configu a ion is conside ed in he Supplemen a y In o ma ion
(sec ion I) oge he wi h an al e na i e choice o ma e ial laye s,
namely EuS/Al/AlO
x
/Co (sec ion II).
Rec ifica ion. The p esence o he supe conduc ing gap can be
clea ly ecognized in he I(V) cha ac e is ic displayed in Fig. 2(b):
he absence o cu en flow a low bias, and an Ohmic beha io
o ela i ely la ge ol age (eV ≫Δ). In an in e media e ol age
ange, non-linea i ies and non- ecip oci y appea , which can be
isualized in he symme ic and an isymme ic pa s o he I(V)
cha ac e is ic. They a e defined as ISym ¼IðVÞþIðVÞ
2and
IAn isym ¼IðVÞIðVÞ
2, and a e shown in Fig. 2(c). The sizable
I
Sym
(V) al eady sugges s an e ficien cha ge ec ifica ion i.e. he
capabili y o con e an AC inpu in o a DC ou pu signal. Rec-
ifica ion (R) o a ci cui can be defined as he a io be ween he
di e ence o he o wa d and backwa d cu en di ided by he
sum o he wo, RðVÞ¼IðVÞþIðVÞ
IðVÞIðVÞ¼ISym=IAn isym, and is shown
in Fig. 2(d). Fo ideal ec ifie s R=1, while o R=0 no ec ifi-
ca ion is p esen . In he junc ion a Rup o ∼40% can be achie ed
in he in e media e ol age ange (eV∼Δ). This uppe limi ,
equi alen o he pola iza ion Po he EuS junc ion can be
unde s ood using a simple analy ical model o he N/FI/S unnel
junc ions, which neglec s spin-dependen sca e ing and o bi al
depai ing. Wi hin hese app oxima ions he I(V) unneling cu -
en can be simplified o he ins uc i e exp ession:
IðVÞ¼ISeeV=ðkBTÞ1

þIScosh eV
kBT

1

ðP1Þ:ð1Þ
The cu en scale I
S
depends on he physical cha ac e is ics o he
de ice, such as he no mal-s a e esis ance, supe conduc ing
ene gy gap and he exchange field, as de ailed in he Me hods
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sec ion, Eq. (4). The exp ession is alid a low empe a u es
(k
B
T≪h) and ol ages (e∣V∣≪Δ−h). No e ha subgap s a es
due o inelas ic sca e ing can p o ide an addi ional con ibu ion
o he cu en δI, which also sa isfies δI(V)≠−δI(−V), and
becomes pa icula ly impo an a e y low empe a u es (see
Me hods o mo e de ails). Equa ion (1) is composed by wo
elemen s. The fi s one ep esen s he Shockley ideal diode
equa ion29 and domina es when Pis close o uni y. I desc ibes
he asymme ic I(V) cu es cha ac e is ic o diodes. The second
con ibu ion is he fi s co ec ion o an ideal diode due o he
non-ideal spin pola iza ion. This yields he simple esul o he
ec ifica ion, R¼P anh½eV=ð2kBTÞ. The maximum ec ifica ion
a ∣eV∣≳2k
B
Tis hence dic a ed by he spin fil e ing e ficiency P.
Due o he s ong asymme y induced by he spin fil e ing o his
specific junc ion, Ris maximized a ound 225–280 μV whe e i
ob ains alues as high as ∼40%, in good ag eemen wi h he
pola iza ion alue ex ac ed om he dI/dV fi s (see Fig. 1( )).
T ans e se ec ifica ion. No ably, he geome y o he de ice
oge he wi h he small esis ance o he unnel junc ion allows
o he implemen a ion o a “ h ee- e minal”diode in which he
pa h o he ec ified signal (in his case he ol age) is decoupled
wi h espec o he exci a ion cu en (I
H
) pa h. This co esponds
o measu emen configu a ion (ii) and is ske ched in Fig. 3(a) and
(b). The de ice is ope a ed wi h a cu en bias I
H
applied along
he Cu bo om lead, while he ol age d op is measu ed o ho-
gonal o i . A he junc ion, I
H
can pa ially flow in he S lead and
gene a e a ol age due o he non-symme ic esponse o he
junc ion o he flowing cu en (see he ske ch displayed in
Fig. 3(a)). The esul ing measu ed ol age V
sym
, symme ized o
disca d he i ial Ohmic componen o igina ed in he Cu lead, is
shown in Fig. 3(c) o di e en magne ic fields. A mono onic
inc ease o V
sym
(I
H
) is isible and mo e p onounced a la ge fields
due o he la ge hand Po he junc ion. No ably, a sizable
ans e se ec ifica ion is p esen also a ze o field hanks o he
e omagne ic na u e o he FI laye . This cha ac e is ic is espe-
cially ele an o applica ions since no addi ional lines o gen-
e a e an ex e nal magne ic field need o be in eg a ed while
ope a ing he de ice. On he o he hand, a he EuS coe ci e field
(≃14 mT a base empe a u e) he ec ified signal is no isible,
confi ming he spin onic na u e o his e ec .
F om he expe imen al I(V) cha ac e is ics o he diode shown
in Fig. 2(b) i is possible o model he ans e se esponse o he
diode (see Me hods o calcula ion de ails). Ou heo e ical
calcula ions compa ed wi h he da a shown in he inse o
Fig. 3(c) a e in ag eemen wi h he expe imen bu a e gene ally
la ge han he measu ed da a by abou ∼30%. This di e ence
likely s ems om he he moelec ic e ec ha , simila o
ec ifica ion, is also p esen in he junc ion wi h lack o
elec on-hole symme y21.
The modynamic conside a ions. F om a he mal model ha
conside s he Joule hea ing induced by I
H
, we can es ima e he
esul ing he mo ol age and find ha i is smalle and o he
opposi e sign wi h espec o he ec ifica ion ol age (see sec ion
IV o he Supplemen a y In o ma ion o mo e de ails), he e o e
confi ming he p esence o he moelec ici y in he junc ion.
No ably, he ela i e ampli ude o he wo e ec s depends on he
leng h o he unnel junc ion, wi h ans e se ec ifica ion dom-
ina ing o junc ions longe han ∼100 μm ( his leng h scale
depends on a ious sample specific pa ame e s as desc ibed by
Eq. (S18) in he Supplemen a y In o ma ion). A simila analysis
can be ca ied ou on he ec ified cu en o configu a ion (i).
Also in his case, he exci a ion ol age can o e hea he S me al
and gene a e a he moelec ic cu en opposi e o he ec ified
signal.
-0.2 -0.1 0.0 0.1 0.2
-0.4
-0.2
0
0.2
0.4
Magne ic ield (T)
Exchange in e ac ion
Pola iza ion
-(i)
e
abc
d
Al
Cu
500 µm
Al
EuS
Cu
(ii)
-
+
+
EuSAl Cu
-0.8 -0.4 0.0 0.4 0.8
0
0.1
0.2
0.3
Measu emen
Fi
Fi Spin Up
Fi Spin Down
)S( ecna cudnoC lai ne e iD
V
Al
- V
Cu
(mV)
-0.5 0.0 0.5
0.2
0.1
0.0
-0.1
-0.2
Bias Vol age (mV)
)T( dlei ci engaM
0.0
0.1
0.2
0.3
)S( ecna cudnoC lai ne e iD
Fig. 1 Wo king p inciple and cha ac e is ics o he supe conduc ing unnel diode. a Schema ic o he de ice s uc u e: a Cu s ip (o ange) is co e ed by a
EuS laye (g een) and a pe pendicula Al s ip (g ay). Measu emen se ups: The elec ic cu en is applied (i) om he Al o he Cu s ip o (ii) along he Cu
s ip. The ol age d op is measu ed be ween he Al and he Cu s ip on he emaining wo wi es o he ou -wi e se -up. bVisible ligh mic oscopy image o
he de ice. cSchema ic o he DoS along he e ical axis o he s uc u e (Al/EuS/Cu om op o bo om). The dashed line indica es he Fe mi le el. No e
ha he EuS laye induces spin spli ing in he supe conduc ing DoS, and spin fil e ing hanks o he di e en heigh s o he unnel ba ie o he wo spin
species. The ed (blue) line co esponds o he spin up (down) DoS in he Al laye . dExempla y di e en ial conduc ance (black) measu ed as a unc ion o
ol age ac oss he junc ion a an applied ex e nal magne ic field Bo 0.1 T a ≃100 mK. By employing a nume ical model (de ailed in he Me hods sec ion,
Eqs. (3) and (14)), he fi o he di e en ial conduc ance ( ed) and he con ibu ions o he spin up (ligh blue) and spin down (ligh ed) elec ons we e
calcula ed wi h hese fi ing pa ame e s: Δ
0
=0.33meV, h=0.32Δ
0
,P=0.48, Γ=0.01Δ
0
,T=300 mK. eColo map o he di e en ial conduc ance
dI/dV(V) measu ed o B anging om −0.2 T o 0.2 T. The sweep di ec ion is indica ed by he a ow. The da a in panel dco esponds o he dash-do ed
line (B =0.1 T). The coe ci e field a he empe a u e o his measu emen (100 mK) co esponds o −9 mT, indica ed by a dashed line. Exchange field (h)
induced in he supe conduc ing Al s ip (blue) and pola iza ion (P) o he EuS unnel ba ie as a unc ion o he ex e nal magne ic field B. Bo h quan i ies
a e ex ac ed om he bes fi ing esul s o he da a as shown in panel d. The sweep di ec ion is again indica ed by an a ow.
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0 0.5 1 1.5
10
20
30
40
)
%
(
o
caF n
o
i a
c
i
i
c
e
R
V
Al
-V
Cu
(mV)
measu emen
Fi wi h nume ical model
Fi wi h simpli ied
analy ical model
-1.5 -1 -0.5 0 0.5 1 1.5
-0.2
-0.1
0.0
0.1
0.2
)Am( ne uC
V
Al
-V
Cu
(mV)
-0.4 -0.2 0.0 0.2 0.4
-0.02
0.00
0.02
0
ba
cd
L
I
Al
EuS
Cu
0 0.5 1 1.5
0
2
4
6
8
10
12
14
ne uC ci emmyS
Vol age (mV)
0.0
0.1
0.2
0.3
An isymme ic Cu en (mA)
Fig. 2 Rec ifica ion o he supe conduc ing unnel diode. a Schema ic o he N/FI/S unnel junc ion. The pa h o he unneling cu en is indica ed by he
black line and i s a ows. In e ms o elec onic ci cui elemen s his junc ion beha es like he indica ed diode: he cu en flows p e e en ially om he Al
laye o he Cu laye while he e e se flow is inhibi ed. bCu en - o-Vol age (I(V)) cha ac e is ics o he junc ion measu ed a T≃50 mK, B=0.1 T in he
ou -wi e configu a ion (i). cSymme ic and an isymme ic pa s o he I(V) cha ac e is ic o panel cshowing a sizable symme ic componen o he
cu en . dRec ifica ion coe ficien R(V)=I
Sym
(V)/I
An isym
(V) e alua ed om e(black line) along wi h he compa ison wi h he ec ifica ion ex ac ed om
he app oxima ed analy ical model R¼P anh½eV=ð2KBTÞ (blue line) and he ull nume ical ones ( ed line). De ails o he nume ical model can be ound in
he Me hods sec ion, specifically in Eqs. (3) and (14). No ice he good quali a i e ag eemen wi h he simplified model p edic ing he sa u a ion a
R≃P∼40% a 225–280 μV. The model ceases o wo k when eV ≳Δ−h∼250 μeV. The disc epancy be ween he analy ical model and he expe imen
mos ly comes om weak inelas ic sca e ing, and o a lesse ex en om spin elaxa ion and o bi al depai ing.
VS
0 50 100 150 200 250
5
10
15
20
ega loV ci emmyS
Applied Cu en
B(T)
-0.1
-0.014
0
0.1
0.2
0.4
-0.1
0.1
a B = 0.2T
calcula ed
measu ed
he mo-
elec ic
e ec
ac
V
b
L
IH
Al
EuS
Cu
+
-
-+
0x
Fig. 3 T ans e se ec ifica ion o he supe conduc ing unnel diode. a Schema ic o he N/FI/S unnel junc ion and cu en pa h. A biasing cu en I
H
is
applied om one end o he Cu s ip o he o he , while he ol age d op ac oss he junc ion is measu ed om he Al con ac o he Cu one. The pa h o
unneling cu en is indica ed by he black line and i s a ows. bElec onic ci cui diag am o he se up. No e ha he EuS laye e ec i ely ac s as a wo old
ec ifie o he dis ibu ed incoming and ou going cu en s unneling h ough he FI ba ie . cT ans e se ol age d op V
sym
(I
Cu
) measu ed ac oss he
ba ie as a unc ion o he applied cu en I
H
a di e en Band a 50 mK. No e ha e en a ze o applied magne ic field (o ange cu e) a ol age d op
occu s, while a he coe ci e field (B≃−14 mT) he signal is ze o due o he non-pola iza ion o he EuS laye . The V(I) was symme ized in o de o
disca d he Ohmic (linea ) componen o igina ing om he N lead. In he inse , he V
sym
measu ed a 0.2 T is compa ed wi h he calcula ed da a poin s
ob ained h ough a heo e ical model o he ci cui and using he ec ifica ion alue ob ained om he expe imen al da a.
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Figu e 4shows he empe a u e dependence o he di e en ial
conduc ance and he ans e se ec ifica ion ol age o he
discussed unnel junc ion. No ably, despi e he e iden he mal
b oadening o he dI/dV(V) (see Fig. 4(a)), he ans e se
ec ifica ion is only ma ginally a ec ed below 1 K (see Fig. 4(b)),
making he e ec e y obus e en a a empe a u e up o nea ly
hal o he Al c i ical empe a u e (T
C
≃2.3 K). Howe e , o
empe a u es la ge han T
C
/2, a clea damping o he signal is
isible wi h measu able e ec s up o ∼1.9 K. This high
empe a u e ange o ope a ion makes ou supe conduc ing
unnel diode appealing o supe conduc ing elec onics schemes
whe e obus ness agains empe a u e fluc ua ions is desi able.
Mo eo e , his beha io is expec ed o hold o o he supe -
conduc ing ma e ials. The e a e se e al FI/S bilaye sys ems
whose T
C
/2 lies abo e 4 K ( o ins ance, GdN/NbN bilaye s30).
These ma e ials ha e he ad an age ha hey can be ope a ed a
s anda d 4He c yogenic empe a u es and deposi ed wi h la ge-
scale spu e ing sys ems.
In conclusion, we ha e shown he capabili ies o a N/FI/S
unnel junc ion o unc ion bo h as a con en ional diode (i) and
as a ans e se ec ifie (ii). The ans e se ec ifie benefi s om
a lowe impedance and a di ec decoupling be ween he AC
exci a ion line ( he an enna) and he DC sensing line. This
ad an age allows o mo e flexibili y in he design o he de ice
impedance when compa ed o configu a ion (i) o op imizing
he impedance ma ching be ween he ec ifie and he pho on
abso be , owa ds op imal quan um e ficiency. Bo h supe -
conduc ing ec ifie s can be ope a ed in ze o applied magne ic
field showing simila de ec ion sensi i i ies and noise equi alen
powe s es ima ed o be 1´1012W=ffiffiffiffiffiffi
Hz
pwi h a oom
empe a u e ol age amplifie . On he o he hand, p omising
sensi i i ies up o ∼2×10
3A/W and noise equi alen powe s
down o 1´1019W=ffiffiffiffiffiffi
Hz
pha e been es ima ed o configu a-
ion (i) wi h oom- empe a u e cu en amplifie s (see sec ion III
o he Supplemen a y In o ma ion o de ails on he analysis).
Such e y low NEP can be limi ed by he he mal
Johnson–Nyquis noise wi h an uppe bound o 
1´1016W=ffiffiffiffiffiffi
Hz
pes ima ed a 100 mK. This is a s ep owa ds
he de elopmen o de ec o s in he THz egion con ibu ing o
he e ahe z gap closu e.
Besides de ec ion and ec ifica ion, his de ice can be used also
o o he con en ional diode unc ionali ies, bu a much lowe
ol age and he eby much lowe dissipa ion le els han con en-
ional semiconduc o -based diodes. Such applica ions include
mixe s, e e se cu en egula o s, ol age clamping, and mo e
passi e supe conduc ing elec onics1. Fu he unc ionali ies can
also be expec ed wi h mo e complica ed s uc u es con aining
se e al EuS o Al laye s31.
Me hods
Sample ab ica ion and anspo measu emen s. The samples a e c oss-ba s
made by elec on-beam e apo a ion employing an in-si u shadow mask. The
s uc u es consis o a glass subs a e on which he laye s o Cu(20)/ EuS(2)/ Al(4)/
Al
2
O
3
(13) a e deposi ed sequen ially ( hicknesses in nm). The o e lap be ween he
Al and he Cu s ip has an a ea o 300 × 300 μm2. The unneling spec oscopy is
ca ied ou a c yogenic empe a u es down o 50 mK in a fil e ed c yogen- ee
dilu ion e ige a o . The I(V) cha ac e is ics a e ob ained om DC ou -wi e
measu emen s, as ske ched in Fig. 1(a), and a e used o calcula e he di e en ial
conduc ance ia nume ical di e en ia ion.
Diode equa ion. The I(V) cha ac e is ic o he spin-pola ized junc ion o a spin-
spli supe conduc o is gi en by (he e, e=k
B
=ℏ=1 o b e i y)
IðVÞ¼∑
σGσZdϵNσðϵÞ½ 0ðϵVÞ 0ðϵÞ;ð2Þ
whe e σ=± 1 o spin up/down, G
σ
=G
0
(1+σP) is he spin-dependen unneling
conduc ance, N
σ
=(N
0
+σN
z
)/2 is he spin-dependen densi y o s a es, 0ðϵÞ¼
½expðϵ=TÞþ11is he Fe mi unc ion, G
0
is he no mal-s a e unneling con-
duc ance, N
0/z
is he spin a e age/di e ence densi y o s a es, and P∈[−1, 1] is he
spin pola iza ion. Ca ying ou he sum o e he spin yields
IðVÞ¼G0Zdϵ½N0þPNz½ 0ðϵVÞ 0ðϵÞ:ð3Þ
The dis ibu ion unc ion ac o can be simplified as
ðϵVÞ ðϵÞ¼ 1
eðϵVÞ=Tþ11
eðϵÞ=Tþ1¼1eV=T
1þeV=Tþeϵ=TþeðϵVÞ=T:
Because o he gap in he N
0
and N
z
unc ions, his needs o be e alua ed only o
ϵ>Δ−hand o ϵ<−Δ+h.I V≪Δ−h, o he posi i e ene gies he las e m
in he denomina o is la ge han he o he s so we may app oxima e
ðϵVÞ ðϵÞð1eV=TÞeðϵVÞ=T:
On he o he hand, o nega i e ene gies he hi d e m in he denomina o is la ge
han he o he s and we may app oxima e
ðϵVÞ ðϵÞð1eV=TÞeϵ=T:
In he absence o spin elaxa ion o o bi al depai ing, he spin-dependen DoS is
N0þPNz¼Re ϵþhj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðϵþhÞ2Δ2
p
1þP
2þjϵhj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðϵhÞ2Δ2
p
1P
2
"#
;
we ge he cu en o he o m
I¼G0
2ð1eV=TÞR1
ΔhðϵþhÞð1þPÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðϵþhÞ2Δ2
peðϵVÞ=TdϵþR1
ΔþhðϵhÞð1PÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðϵhÞ2Δ2
peðϵVÞ=T

RΔh
1 ðϵþhÞð1þPÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðϵþhÞ2Δ2
peϵ=TdϵRΔþh
1 ðϵhÞð1PÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðϵhÞ2Δ2
peϵ=Tdϵ
Shi ing he ene gies by he spin-spli ing field up and down, and e e ing he sign
o he ene gy in he la e wo in eg als yields
I¼G0ð1eV=TÞ
2½ð1þPÞeðhþVÞ=Tþð1PÞeðVhÞ=Tþð1þPÞeh=Tþð1PÞeh=T
´Z1
Δ
ϵeϵ=T
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ϵ2Δ2
pdϵ
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
¼ΔK1ðΔ=TÞ
;
whe e K1ðΔ=TÞ ffiffiffiffiffiffiffiffi
π=2
peΔ=TffiffiffiT
Δ
qis he Bessel K unc ion and he la e
ab
-0.8 -0.4 0 0.4 0.8
0
0.05
0.1
0.15
0.2
0.25
0.3
)
S( ecn
a cu
d
n
oC lai n
e e iD
VAl - VCu (mV)
T (K)
0.05
0.5
1
1.4
1.6
1.9
0 0.1 0.2 0.3 0.4
0
10
20
30
40
ega
l
o
V
ci
emmyS
Applied Cu en (mA)
T (K)
0.2
0.8
1
1.3
1.6
1.7
1.9
Fig. 4 Tempe a u e dependence o he supe conduc ing ec ifie . a Di e en ial conduc ance s. ol age o he junc ion measu ed o di e en
empe a u es om 50 mK o 1.9 K. bTempe a u e e olu ion o he ans e se ec ifica ion ol age s. biasing cu en . Bo h measu emen s a e pe o med a
B=0.1 T.
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app oxima ion is alid o Δ≫T. Le us define
ISG0ΔK1
Δ
T

eh=T:ð4Þ
Now ea anging e ms in he exp ession o he cu en allows us o w i e i as
IðVÞ¼ISðeV=T1ÞþISe2h=Tð1eV=TÞ
þISð1e2h=TÞcosh V
T

1

ðP1Þ:ð5Þ
This also yields he ec ifica ion
R¼P anh h
T

anh V
2T

:ð6Þ
Fo h≫Twe ge Eq. (1) and he co esponding simplified exp ession o Rquo ed
in he main ex .
Co ec ions o he cu en due o subgap s a es. Inelas ic sca e ing in oduces
subgap s a es, which can be well desc ibed wi hin he Dynes model32. A low
ene gies (ϵ<Δ−h), a weak Dynes pa ame e Γ≪Δ−hin oduces a co ec ion o
he supe conduc ing densi y o s a es gi en as δNσðϵÞ¼Γ
ρσð1þϵ2
σ
ρ2
σÞ. He e
ϵ
σ
=ϵ+σhand ρσ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Δ2ϵ2
σ
q. Combining his wi h Eq. (2), we find he ol-
lowing co ec ion o he cu en , alid a low empe a u es and o ol ages
V<Δ−h:
δI¼ΓG0½FasymðeV;hÞþPFsymðeV;hÞ:ð7Þ
He e we in oduced he unc ions FasymðeV;hÞ¼1
2½FðeV þhÞþFðeV hÞ,
FsymðeV;hÞ¼1
2½FðeV þhÞFðeV hÞ2FðhÞ, wi h FðxÞ¼x=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Δ2x2
p. Fo
small ol ages and weak exchange field, h,eV ≪Δ, we may app oxima e
δIΓeV
Δ½1þ3
2
eVh
Δ2.
Taking in o accoun he co ec ion δI oge he wi h Eq. (1), he exp ession o
he ec ifica ion coe ficien Rbecomes (h≫k
B
T)
R¼P2sinh2eV
2kBTþξFsymðeV;hÞ
sinh eV
kBTþξFasymðeV;hÞ;ð8Þ
whe e ξ¼G0Γ
ISΓ
ΔeΔ=ðkBTÞ. I he empe a u e is high-enough, kBTΔ=logðΔ
ΓÞ,we
ha e ξ≪1, and inelas ic sca e ing can be neglec ed. In his case we ob ain he
exp ession shown in he main ex : R¼P anh½eV=ð2kBTÞ. Howe e , in he
opposi e egime o e y low empe a u es, kBTΔ=logðΔ
ΓÞ,wefind ha δIac ually
p o ides he dominan con ibu ion o he cu en . In ha case, R=PF
sym
/
F
asym
. No e ha in bo h egimes he maximal ec ifica ion coe ficien is gi en by
R
max
=P. In he fi s egime, he maximum is eached a eV ∼k
B
T, whe eas in he
second i is a eV ∼Δ−h.
Model o he densi y o s a es (DoS). In he calcula ion o he I(V) cha -
ac e is ics he densi y o s a es o he supe conduc o , NσðϵÞ, plays a cen al ole.
We ob ain i om he quasiclassical G een’s unc ions (GFs), 
g,defined in he
Nambu ⊗spin space. These a e 4 × 4 ma ices ha sa is y he no maliza ion
condi ion, 
g2¼1. He e he “check”symbol, , indica es 4 × 4 ma ices.
In he bulk o a di y supe conduc o wi h a cons an exchange field aligned
along a gi en axis, he e a ded quasiclassical GFs ulfill he ollowing Usadel
equa ion28,31,33:
iðϵþiΓÞ^
τ3þih^
τ3
^
σzΔ^
τ1

Σ;
g

¼0:ð9Þ
He e, ϵis he ene gy, Γis a small ene gy e m known as he Dynes pa ame e 32,h
s ands o he s eng h o he exchange field, Δis he sel -consis en supe conduc ing
o de pa ame e and ^
τiand ^
σalabel he Pauli ma ices spanning Nambu and spin
space, espec i ely. Di ec p oduc be ween Pauli ma ices spanning di e en spaces is
implied, and iden i y ma ices, ^
τ0and ^
σ0, a e ob ia ed. The squa e b acke s, [⋅,⋅],
s and o commu a ion ope a ion and 2 × 2 ma ices a e indica ed wi h a ^symbol. A
ypical alue o he Dynes pa ame e is Γ∼10−3Δand i s impo ance is wo old: fi s
i a oids analy ical p oblems in he compu a ion o he GFs and second i models he
e ec o non-elas ic p ocesses in he supe conduc o . The

Σma ix is he sel -ene gy
ha consis s o h ee con ibu ions:

Σ¼

Σso þ

Σs þ

Σo b:ð10Þ
F om le o igh , hese a e he spin elaxa ion due o spin-o bi coupling, he spin
elaxa ion due o spin-flip e en s and he o bi al depai ing due o ex e nal magne ic
fields, espec i ely. Explici ly, each con ibu ion wi hin he elaxa ion ime
app oxima ion, eads:

Σso ¼
^
σa

g^
σa
8τso
;
Σs ¼
^
σa
^
τ3

g^
τ3
^
σa
8τs
;
Σo b ¼
^
τ3

g^
τ3
τo b
:ð11Þ
He e τ
so
,τ
s
and τ
o b
s and o spin-o bi , spin-flip and o bi al depai ing elaxa ion
imes, espec i ely, and we sum o e epea ed indices. We es ima e he o bi al
depai ing in he supe conduc ing laye due o an applied magne ic field as34,35:
τ1
o b ¼πdξ0B
ffiffiffi6
pΦ0

2
Δ0;ð12Þ
whe e Φ
0
is he quan um o magne ic flux, ds ands o he wid h o he
supe conduc ing laye , Bis he applied magne ic field, Δ
0
is he supe conduc ing gap
a ze o field (T=0andh=0) and ξ
0
is he supe conduc ing cohe ence leng h.
In addi ion o Eq. (9), he alue o he supe conduc ing gap is ela ed o he
quasiclassical GFs ia he sel -consis en equa ion,
Δ¼λ
8iZΩD
ΩD
dϵT ^
τ1

g

;ð13Þ
whe e he ace uns o e he Nambu ⊗spin space, λis he coupling cons an and
Ω
D
is he Debye cu o ene gy.
F om Eqs. (9), (13) and he no maliza ion condi ion we compu e he alue o 
g,
om which he he spin a e age/di e ence densi y o s a es, N0=z, can be di ec ly
calcula ed:
N0=zðϵÞ¼1
2Re ½T ð^
τ3
^
σ0=z

gÞ:ð14Þ
By fi ing he expe imen al I(V) cu es wi h Eqs. (3) and (14) we a e able o ob ain
he di e en pa ame e s used in he model.
Model o ans e se ec ifica ion. In Fig. 3(c), we calcula e he ec ifica ion
ol age om he expe imen ally measu ed I(V) cu es a di e en hea ing cu en s
I
H
using he ollowing heo e ical model. The open ci cui ol age V
s
o he
ans e se ec ifie configu a ion shown in Fig. 3(a) can be de e mined by imposing
ha he o al cu en c ossing he unnel junc ion is ze o:
I¼ZL
0
iðVðxÞÞdx ¼0:ð15Þ
He e, he unnel-cu en densi y (i(x)) is in eg a ed om 0 o Lalong he leng h o
he junc ion. VðxÞ¼x
LIHRxþVsþVins is he ol age d op be ween he N and S
a x(0 < x<L), R
x
is he la e al esis ance o he junc ion, and V
ins
is he ins u-
men al o se , which is ob ained om I(V
ins
)=0a I
H
=0. The fi s wo pa s
con ains wo con ibu ions: a la ge i ial Ohmic con ibu ion due o he hea ing
cu en (x
LIHRx), and a smalle con ibu ion due o he ec ifica ion e ec (V
s
). The
o me is an isymme ic in I
H
, whe eas he la e is symme ic. The e o e, he
symme ized ol age
Vsym ¼1
2½VsðIHÞþVsðIHÞ ð16Þ
comes om he ec ifica ion e ec only.
Da a a ailabili y
The da a ha suppo he findings o his s udy a e a ailable om he co esponding
au ho upon easonable eques .
Code a ailabili y
The code o ma hema ical algo i hm used o gene a e esul s ha a e epo ed in he
pape and cen al o i s main claim a e a ailable om he co esponding au ho upon
easonable eques .
Recei ed: 30 Sep embe 2021; Accep ed: 11 Ap il 2022;
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Acknowledgemen s
This wo k was mainly suppo ed by he EU’s Ho izon 2020 esea ch and inno a ion
p og am unde G an Ag eemen No. 800923 (SUPERTED) unding all he au ho s. E.S.
and F.G. acknowledge he Eu opean Resea ch Council unde G an Ag eemen No.
899315 (TERASEC), and he EU’s Ho izon 2020 esea ch and inno a ion p og am unde
G an Ag eemen No. 964398 (SUPERGATE) o pa ial financial suppo . M.S. and E.S.
acknowledge pa ial unding om he Eu opean Union’s Ho izon 2020 esea ch and
inno a ion p og am unde he Ma ie Skłodowska Cu ie Ac ion IF G an No. 101022473
(Supe CON ac s). J.M. acknowledges financial suppo in he USA by he A my Resea ch
O fice (g an ARO W911NF-20-2-0061), ONR (g an N00014-20-1-2306), NSF (g an
DMR 1700137) and NSF C-Accel T ack C G an No. 2040620. The wo k o F.S.B., C.R.
and M.I. was suppo ed by he Spanish Minis e io de Ciencia e Inno acion (MICINN)
h ough P ojec PID2020-114252GB-I00 (SPIRIT). F.S.B. acknowledges financial suppo
by he A. . Humbold Founda ion.
Au ho con ibu ions
E.S., N.L., and M.S. pe o med he expe imen s and analyzed he da a. S. I., M.R, P.V.,
T.H., and F.S.B. p o ided heo e ical suppo . J.M. ab ica ed he Cu/EuS/Al de ices and
C.G.O, M.I, and C.R. he EuS/Al/AlO
x
/Co ones. E.S. concei ed he expe imen oge he
wi h F.G. who supe ised he p ojec . E.S. and M.S. w o e he manusc ip wi h eedback
om all au ho s.
Compe ing in e es s
Wi h he Ins i u e Nanoscienze-CNR, he ollowing au ho s: E.S., M.S., F.G., P.V., T.T.H.,
S.I, and F.S.B., ha e filed a pa en (In e na ional Applica ion N. PCT/IT2021/000038
“APPARATUS AND METHOD FOR SUPERCONDUCTING DIODE”, s a us: pending,
aspec o manusc ip co e ed in pa en applica ion: ec ifica ion and diode-beha io o
he ma e ial combina ions, a chi ec u e and measu emen configu a ions p esen ed). The
emaining au ho s decla e no compe ing in e es s.
Addi ional in o ma ion
Supplemen a y in o ma ion The online e sion con ains supplemen a y ma e ial
a ailable a h ps://doi.o g/10.1038/s41467-022-29990-2.
Co espondence and eques s o ma e ials should be add essed o E. S ambini,
M. Spies o F. Giazo o.
Pee e iew in o ma ion Na u e Communica ions hanks Nicola Pa adiso and he o he ,
anonymous, e iewe (s) o hei con ibu ion o he pee e iew o his wo k. Pee
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