PHYSICAL REVIEW RESEARCH 2, 033133 (2020)
Diso de ed hype uni o mi y in supe conduc ing o ex la ices
José Beni o Llo ens,1Isabel Guillamón ,1Ismael G. Se ano ,2,3Rosa Có doba ,2,3,4Ja ie Sesé ,2,3
José Ma ía De Te esa ,2,3M. Rica do Iba a ,2,3Sebas ián Viei a,1Miguel O uño,5and He mann Sude ow 1
1Labo a o io de Bajas Tempe a u as y Al os Campos Magné icos, Unidad Asociada UAM, CSIC, Depa amen o de Física de la Ma e ia
Condensada, Ins i u o Nicolás Cab e a and Condensed Ma e Physics Cen e , Uni e sidad Au ónoma de Mad id, 28049 Mad id, Spain
2Labo a o io de Mic oscopías A anzadas (LMA), Ins i u o de Nanociencia de A agón (INA),
Uni e sidad de Za agoza, 50018 Za agoza, Spain
3Depa amen o de Física de la Ma e ia Condensada and Ins i u o de Ciencia de Ma e iales de A agón (ICMA),
Uni e sidad de Za agoza-CSIC, 50009 Za agoza, Spain
4Ins i u o de Ciencia Molecula , Uni e si a de València, Ca ed á ico José Bel án 2, 46980 Pa e na, Spain
5Depa amen o de Física, CIOyN, Uni e sidad de Mu cia, Mu cia 30071, Spain
(Recei ed 24 Feb ua y 2020; e ised 11 May 2020; accep ed 22 June 2020; published 24 July 2020)
The cu en ca ying capabili y o ype II supe conduc o s unde magne ic ields is de e mined o a la ge
ex en by he in e ac ion o supe conduc ing o ices wi h pinning cen e s. Vo ices a e a anged in la ices
wi h a ying deg ees o diso de depending on he balance be ween he in e o ex in e ac ions and he pinning
s eng h. We analyze he e o ex a angemen s in diso de ed o ex la ices o di e en supe conduc ing sys ems,
single c ys als (Co-doped NbSe2, LiFeAs, and CaKFe4As4), and amo phous W-based hin ilms (wi h c i ical
empe a u es Tc om 4 K o 35 K and c i ical ields om 3.4 T o mo e han 90 T). We calcula e o each case
he s uc u e ac o and numbe a iance and compa e o calcula ions on an in e ac ing se o pa ially pinned
pa icles. We ind ha andom densi y luc ua ions appea when pinning o e comes in e ac ions and show ha
he supp ession o densi y luc ua ions is co ela ed o he p esence o in e ac ions. We discuss he esul s wi hin
he amewo k o hype uni o m dis ibu ions and ind ha all s udied la ices ollow a simila inc ease o he
numbe a iance wi h he de ec densi y.
DOI: 10.1103/PhysRe Resea ch.2.033133
I. INTRODUCTION
Pa icles occupying si es o a andom la ice p esen den-
si y luc ua ions a all leng h scales. I has been p oposed
ha inc easing in e pa icle in e ac ions educes long- ange
densi y luc ua ions, de ia ing om andom beha io [1,2].
The absence o densi y luc ua ions occu s in all o de ed
la ices, including hose wi h e y-long- ange o de such as
moi é pa e ns and quasic ys als. Densi y luc ua ions can be
desc ibed by discussing powe laws in he s uc u e ac o S(k)
and he numbe a iance σ2(R). The e m “hype uni o mi y”
was coined o ga he o de ed as well as diso de ed la ices
wi h educed densi y luc ua ions [1,2]. I has been shown ha
diso de ed hype uni o m la ices can be c ea ed by a i icially
designing diso de ed pa e ns wi hou densi y luc ua ions a
la ge leng h scales [1–12]. I is no ye ully clea howe e how
o link densi y luc ua ions wi h in e ac ions in a diso de ed
hype uni o m sys em. Recen ly, diso de ed hype uni o m be-
ha io has been ound in he con ac numbe be ween subsys-
ems o pa icles [7,8], al hough he o igin o he supp essed
densi y luc ua ions emains unde deba e [13].
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.
He e we add ess he beha io o o ices in ype II supe -
conduc o s. Vo ices a e whi lpools o cu en s, each ca ying
a lux quan um [14,15]. They epel each o he and o en o m
o de ed la ices wi h in e o ex dis ances a0∝1
√H. In e o -
ex in e ac ions a e sc eened abo e he pene a ion dep h λ,
which is mos o en much la ge han he in e o ex dis ance
a0[16–19]. In hin ilms, λs ongly inc eases wi h dec easing
hickness and o ices in e ac h ough hei s ay ield, which
leads o a long ange Coulomb-like 1
in e ac ion [20–22].
Vo ices a e pinned a de ec s in supe conduc ing c ys als
and hese de ec s a e o en andomly dis ibu ed. How he
o ex la ice eac s o pinning depends on pinning s eng h
and dis ibu ion as well as s i ness o he o ex la ice and
de e mines he capabili y o ype II supe conduc o s o ca y
la ge cu en s. The deg ee o o de in supe conduc ing o ex
la ices is de e mined by he balance be ween pinning and in-
e ac ion s eng hs and is ela ed o he c i ical cu en [14,15].
The ques ion we add ess he e is o wha ex en in e o ex
in e ac ions can elimina e densi y luc ua ions while allowing
he o ma ion o a diso de ed la ice due o pinning. The
in luence o o ex a angemen s wi h supp essed densi y
luc ua ions in he cu en ca ying capabili y o a supe con-
duc o was analyzed heo e ically in Re . [11]. A angemen s
o o ices in cup a e supe conduc o s a e y small magne ic
ields whe e weak in e o ex in e ac ions a e weak we e
discussed in Re . [23].
We analyze he e o ex la ices in bulk supe conduc o s
and in hin ilms ob ained in Re s. [24–28] and p o ide
2643-1564/2020/2(3)/033133(9) 033133-1 Published by he Ame ican Physical Socie y
JOSÉ BENITO LLORENS e al. PHYSICAL REVIEW RESEARCH 2, 033133 (2020)
addi ional da a in a hin ilm. We mos ly ocus on la ices
wi h la ge amoun s o diso de . We calcula e S(k) and σ2(R)
and discuss he powe laws as a unc ion o kand R.We
discuss da a on a nanos uc u ed W-based hin ilm (W- ilm-1)
wi h s ong pinning and polyc ys alline o ex la ice a ange-
men s ha a e diso de ed a high magne ic ields. We show
esul s in Co-doped 2H-NbSe2, a sys em wi h s ong poin like
pinning cen e s ha also leads o polyc ys alline hexagonal
la ices which inc ease he le el o diso de when inc eas-
ing magne ic ields [25]. We discuss a W-based hin ilm
wi h weak one-dimensional (1D) diso de po en ial c ea ed
by he discommensu a ion be ween he o ex la ice and a
linea nanos uc u e. This ilm shows o de ed o ex la ices
consis en wi h a B agg glass a low magne ic ields ha a e
diso de ed a high magne ic ields (W- ilm-2) [24,29]. We
conside ha he o de ed la ices a e ep esen a i e o he
beha io expec ed in 2H-NbSe2o o he ma e ials wi h well-
o de ed la ices [30–34]. Finally, we discuss wo i on-based
supe conduc o s wi h s ongly diso de ed la ices and s ong
poin like in insic pinning a all magne ic ields, LiFeAs and
CaKFe4As4[26,27]. We ind a g adual appea ance o densi y
luc ua ions wi h inc easing diso de in he la ice ha suc-
cess ully desc ibes all s udied o ex la ices.
II. METHODS
To analyze in e ac ions in o ex la ice images we i s
ha e o ecall he meaning o he spa ial dependence o he
s uc u e ac o and numbe a iance. We show in Figs. 1(a)
and 1(b) images including ma ices o poin s dis ibu ed
on a wo-dimensional squa e. Poin s a e posi ioned an-
domly [Fig. 1(a)] and ollowing a hype uni o m dis ibu ion
[Fig. 1(b)]. Bo h images a e squa e and ha e he same size,
wi h a ea A, and he same numbe o poin s N. In a an-
dom dis ibu ion [Fig. 1(a)], he e a e densi y luc ua ions a
all leng h scales. In a hype uni o m dis ibu ion [Fig. 1(b)],
densi y luc ua ions disappea a la ge leng h scales. We
ha e gene a ed he hype uni o m dis ibu ion o Fig. 1(b) by
s a ing wi h poin s a anged in a squa e la ice o cons an
a0=√A/N. We hen add o each la ice poin a ec o 
wi h andom coo dina es, whose absolu e alue <2a0[inse
o Fig. 1(b)]. The esul is a andom dis ibu ion which is
spa ially uni o m o leng h scales la ge han he in e cell
dis ance. I is impo an o ealize ha he e a e no u he
signa u es o he o de ed la ice in Fig. 1(b) o he han he
la ge-scale uni o mi y.
The calcula ion o he numbe a iance is e y use ul o
discuss densi y luc ua ions [1,2,11]. I is gi en by σ2(R)=
N2(R)−N(R)2, whe e N(R) is he numbe o poin s in-
side a ci cle o adius R. The a iance σ2(R) inc eases wi h R
as a powe law σ2(R)∝Rβ. In a andom dis ibu ion o poin s
[Fig. 1(a)], σ2(R) g ows as he dimension (i.e., wi h he a ea,
acco ding o he la ge numbe law) so ha β=2 [Fig. 1(c)].
On he o he hand, in a hype uni o m dis ibu ion [Fig. 1(b)],
σ2(R) g ows as he dimension minus one (i.e., wi h he
pe ime e ) wi h β=1 [Fig. 1(d)]. O de ed la ices p o ide a
s ong decay o small wa e ec o s and an oscilla ing σ2(R)
wi h a dip each ime Ris somewha smalle han in ege
mul iples o he a e age in e pa icle dis ance [35].
FIG. 1. Se o poin s dis ibu ed in a 2D space, ob ained by
assigning some pixels he alue 1 ollowing (a) a andom dis ibu ion
and (b) a hype uni o m dis ibu ion. No e ha bo h dis ibu ions o
poin s a e ully diso de ed. In he inse o (b) we show schema ically
how we c ea ed he hype uni o m dis ibu ion. We s a ed om
a squa e la ice (g ay c osses) wi h la ice cons an a0and added
o each la ice poin a ec o wi h andom coo dina es 
=(x,y)
and magni ude <2a0. This gi es he dis ibu ion o black do s.
(c) and (d) Va iance o he dis ibu ion σ2as a unc ion o he adius
no malized o a0, calcula ed as explained in he ex . He e σ2(R)
ollows a powe law wi h he exponen β( ed do ed line). Resul
a e shown o (c) he andom dis ibu ion shown in (a) and (d) he
hype uni o m dis ibu ion shown in (b). The s uc u e ac o S(k)
no malized o one o la ge alues o he ecip ocal space ec o k
is shown o (e) he andom dis ibu ion shown in (a) and ( ) he
hype uni o m dis ibu ion shown in (b). He e S(k) also ollows a
powe law wi h exponen α[ ed do ed lines in (e) and ( )].
To calcula e he s uc u e ac o S(k) we use he Fou ie
ans o m o he image o he o ex posi ions and make he
adial a e age o e he Fou ie ans o m. In he Fou ie space,
he s uc u e ac o S(k) dec eases a small kwi h he powe
law S(k)∝kα. In he andom dis ibu ion α=0 [Fig. 1(e)],
while in he hype uni o m dis ibu ion α>1 [Fig. 1( )].
The e is a ela ion be ween he exponen s o S(k) and hose
o σ2(R). When 0 <α<1, β=2−α, and when α>1, β
emains locked a 1. O de ed la ices show a B agg peak a
he ecip ocal la ice wa eleng h and a s ongly decaying S(k)
o small k.
033133-2
DISORDERED HYPERUNIFORMITY IN SUPERCONDUCTING … PHYSICAL REVIEW RESEARCH 2, 033133 (2020)
In he Appendix we illus a e in g ea e de ail how o
ob ain S(k) and σ2(R). We i s Delaunay iangula e all
o ex posi ions, elimina ing missing bonds in he sides o he
image and ca e ully e i ying ha e e y o ex was co ec ly
iden i ied, as in Re . [24]. We hen use he esul ing ma ix o
poin s o pe o m he calcula ion o S(k) and σ2(R) and ob ain
he exponen s as desc ibed abo e (see Fig. 1). In he Appendix
we desc ibe in de ail how o calcula e σ2(R) om an image o
a diso de ed o ex la ice (Fig. 4in Appendix).
As he esul s in W- ilm-1 ha e no been published, we
p o ide a ew expe imen al de ails. We acqui e he image in
ze o- ield cooled condi ions a 100 mK using he sys em de-
sc ibed in Re s. [24,28,36]. The sample has been made using
a ocused ion beam assis ed deposi ion and has a composi ion
simila o he composi ion o W- ilm-2, which has pe ec ly
o de ed la ices in a la ge ange o magne ic ields, desc ibed
in Re . [24]. The c i ical empe a u e is 5 K [37,38]. Howe e ,
con a y o W- ilm-2, he e he subs a e has s ong andom
hickness modula ions a a leng h scale o abou 200 nm,
which conside ably enhance pinning. The o ex la ice ea -
anges acco dingly, showing a polyc ys alline pa e n, which
we discuss in he Appendix (Fig. 5).
We pe o m he nume ical simula ion using he ollowing
p ocedu e, which ies o mimic he dynamics o eal o ices
as much as possible. Vo ices a e ep esen ed by in e ac ing
poin s. We dis ibu e Npoin s a andom on a squa e sample
wi h pe iodic bounda y condi ions. We choose N=2000,
which is a numbe la ge enough o a oid undesi able bound-
a y e ec s and small enough o con e ge quickly unde ou
dynamics. We ix a ce ain pe cen age o poin s (be ween
10% and 50%) in hei ini ial andom posi ions, while he
es poin s a e ee o mo e and end o minimize hei
po en ial ene gy. We use a 1/ in e ac ion, which con e ges
as e han he usual in e o ex in e ac ion and p oduces
a simila spa ial dis ibu ion. We do no ake in o accoun
he ac ha he in e o ex in e ac ion is sc eened abo e λ,
al hough his should no g ea ly in luence he compa ison o
he expe imen . The p ocedu e selec s a poin a andom and
chooses a possible new posi ion displaced by a small dis ance
om he o iginal posi ion in a andom di ec ion. I he new
posi ion minimizes he sys em ene gy, he poin is mo ed o
he new posi ion; o he wise is kep in he old posi ion. These
ene gy elaxing displacemen s a e epea ed i e a i ely. A e a
numbe o displacemen s equal o 1000 imes he numbe o
poin s, changes in posi ions a e minu e, since he sys em has
eached a me as able s a e, and we s op he algo i hm. Once
a inal con igu a ion has been eached, we un a Delaunay
iangula ion algo i hm o de e mine he nea es neighbo s
o any poin and calcula e he p opo ion o poin s wi h a
numbe o neighbo s di e en om 6, in o de o be able o
compa e di ec ly wi h expe imen . Finally, we calcula e S(k)
and σ2(R), ob aining β om he slope o σ2(R), as discussed
p e iously o he expe imen al esul s.
III. RESULTS
In Fig. 2we show ou esul s. We s a wi h la ices in Co-
doped 2H-NbSe2[Figs. 2(a) and 2(b)][25]. We ind S(k)∝kα
wi h α⩾1 and β=1. We hen show esul s in he amo phous
W-based ilm (W-based-1) in Figs. 2(c) and 2(d). We ind a
FIG. 2. The le column shows he s uc u e ac o S(k) ob ained
o o ex la ices in many di e en ma e ials as a unc ion o he
ecip ocal la ice ec o kin uni s o 2π
a0in each image. The igh
column shows he a iance σ2(R) ob ained in he same ma e ials. A
ew images o o ex posi ions a e shown be ween g aphs. (a) and
(b) Resul s in Co-doped NbSe2wi h da a aken om Re . [25]. The
la e al size o he images is 375 nm. (c) and (d) Resul s ob ained in a
W-based hin ilm wi h s ong pinning (W- ilm-1). The la e al sizes o
he images a e 1700, 1000, and 750 nm. (e) and ( ) Resul s in LiFeAs
wi h da a om Re . [26]. The la e al size o he images is 500 nm.
(g) and (h) Resul s in a highly o de ed W-based hin ilm (W- ilm-2)
wi h a e y weak 1D diso de po en ial, wi h da a om Re . [24].
The la e al sizes o he images a e 1000 nm (2.5 and 3 T), 550 nm (4
and 5 T), and 500 nm (5.5 T). (i) and (j) Resul s in pu e CaKFe4As4
om Re . [27]. The la e al sizes o he images a e 400 and 470 nm.
Poin s a e joined by lines as a guide. We plo all da a in loga i hmic
scale and p o ide he powe -law dependences wi h exponen s α=1
o S(k)andβas shown in he legends o he igu es o σ2(R). See
also Table I.
033133-3
JOSÉ BENITO LLORENS e al. PHYSICAL REVIEW RESEARCH 2, 033133 (2020)
TABLE I. Supe conduc ing pa ame e s o he sys ems s udied
he e, c i ical empe a u e Tcand c i ical magne ic ield Hc2. Pinning
is by poin de ec s in Co-NbSe2[25] and by s ong su ace co uga-
ion in W- ilm-1 and is in bo h cases s ong. In he W- ilm-2, he e is
an ex emely weak in e ac ion be ween he o ex la ice and he ilm,
due o a weak linea nano ab ica ed s uc u e [24]. In LiFeAs pinning
is by poin de ec s [26]. In CaKFe4As4pinning is e y s ong [27,39–
42]. The exponen βo he a iance σ2(R) is ob ained om he
da a in Fig. 2. Random beha io is cha ac e ized by β=2 and no
c ys alline o de . The de ia ion om andom owa ds diso de ed
hype uni o m is seen by βdec easing om 2 owa ds 1 in he
absence o c ys alline o de . C ys alline o de shows he p esence o
absence o a isible oscilla ing pa e n in σ2(R) (las ow).
Pa ame e Co-NbSe2W- ilm-1 W- ilm-2 LiFeAs CaKFe4As4
Tc(K) 5.7 4 4 17.5 35
Hc2(T) 3.4 6.4 6.4 15 ≈90
β1 1 1.3 1.1–1.5 1.7
oscilla ing σ2(R) yes yes yes-no yes-no no
e y simila beha io , wi h α⩾1 and β=1. Bo h o hese
sys ems hus show close o hype uni o m beha io . They also
bo h show ha he e is sho - ange hexagonal o de in all
s udied images a leng h scales well abo e a0. We can see his
in he oscilla ions appea ing in σ2(R) close o in ege s o a0.
The esul ing polyc ys alline o ex la ices a e shown in he
Appendix (Figs. 5and 6).
In Figs. 2(e) and 2( ) we show esul s in W- ilm-2, om
Re . [24]. A small magne ic ields, hexagonal o de is nea ly
pe ec wi h jus a ew disloca ions. This p o ides oscilla ions
close o in ege s o he la ice cons an and β≈1. When
inc easing he magne ic ield, he numbe o de ec s in he
la ice inc eases, pa icula ly abo e abou 4 T. The co e-
sponding angula and posi ional co ela ion unc ions show
he p oli e a ion o opological de ec s in he la ice [24]. A
5 T he la ice has no long- ange posi ional no o ien a ional
o de [24]. He e we show ha S(k) and σ2(R) emain wi h
he same powe -law dependences, wi h β=1.1 and α⩾1
o magne ic ields below o equal o 5 T. A 5T, al hough
he la ice has nei he long- ange posi ional no o ien a ional
o de (see Re . [24]), he leng h scale o o ien a ional o de
is su icien ly la ge o p o ide β≈1, i.e., nea diso de ed hy-
pe uni o m beha io . When eaching 5.5 T, he decay leng h
o o ien a ional o de goes om abou i e imes a0down
o a couple o a0[24]. A he same ime, he oscilla ions in
σ2(R) anish o ally a 5.5 T. Howe e , he e is also a s ong
de ia ion om hype uni o mi y, wi h an inc ease o β o 1.6.
Thus, he onse o s ong diso de leads o a endency o o m
a andom dis ibu ion o o ices.
In Figs. 2(g) and 2(h) we show esul s in LiFeAs, om
Re . [26]. The o ex la ice is highly diso de ed abo e abou
2 T, wi h no clea hexagonal pa e ns obse ed a any leng h
scale. As discussed in Re . [26], he s uc u e ac o has a
squa e-shaped o ien a ional dependence, which shows ha
o e all he e is a endency o he o ex la ice o lock i s
shape o he squa e c ys al la ice. No e ha oscilla ions in
σ2(R) a e much less p onounced han in o he cases, al hough
hese a e clea ly isible a 2 T. The coe icien αis sligh ly
smalle han one and βis close o one a 2 T bu inc eases
FIG. 3. (a) Exponen o σ2(R), β, as a unc ion o he s anda d
de ia ion no malized by he in e o ex dis ance SD
a0. (b) Exponen
βas a unc ion o he de ec densi y. We show esul s (ci cles) o
he sys ems o Fig. 2, wi h colo s simila o hose in Fig. 2: G een
is Co-NbSe2, o ange is W- ilm-1, blue is LiFeAs, ed is W- ilm-2,
and iole is CaKFe4As4. Open ci cles indica e esul s whe e σ2(R)
has an oscilla o y beha io , showing he p esence o hexagonal o de .
Closed ci cles show si ua ions wi h a ully diso de ed o ex la ice
and no oscilla ions in σ2(R). G ay poin s show he esul o he
nume ical simula ion discussed in he ex ( he g ay dashed line is
a guide o he eye).
wi h he magne ic ield. This si ua ion is close o a diso de ed
hype uni o m a angemen . No e ha , despi e he p esence o
diso de , he o ex in e ac ion is no negligible since he e
he endency o he diso de ed o ex la ice is o show squa e
a angemen s a small dis ances [26].
Nex we analyze highly diso de ed o ex la ices in
CaKFe4As4, omRe .[27]. We ind [Figs. 2(i) and 2(j)]a
conside able de ia ion om diso de ed hype uni o m beha -
io , wi h βclose o o la ge han 1.5 and αless han 1. We
ind no signa u es o oscilla ions in σ2(R).
We summa ize all esul s in Table I. The ac ual posi ions
o o ices in all discussed la ices a e gi en in he Appendix
(Fig. 7). We can now plo (Fig. 3)βas a unc ion o he s an-
da d de ia ion (SD) in he nea es -neighbo o ex posi ions
no malized o he in e o ex dis ance a0,SD
a0, and he densi y
o de ec s in he o ex la ice (we call a de ec a o ex wi h
coo dina ion numbe di e en om 6).
I we s a om an o de ed la ice, we a e close o β=1
and SD
a0as well as he de ec densi y close o ze o. We see
ha when he e a e oscilla ions in σ2(R) (open poin s in
Fig. 3), β=1 al hough he SD
a0and he de ec densi y can
be qui e la ge. No ice ha he e a e no da a wi h β=1 and
SD
a0la ge han abou 30–40% o he in e o ex dis ance. This
essembles a Lindemann c i e ion. Abo e a ce ain luc ua ion
ampli ude (SD
a0), he o de ed la ice is uns able.
When we ha e a andomly diso de ed o ex la ice, we
expec β ending owa ds 2 and la ge alues o SD
a0and o
he de ec densi y. This indeed occu s o he ully diso de ed
la ices o CaKFe4As4, wi h LiFeAs a 11 T and wi h W-
ilm-2 a 5.5 T (closed ci cles in Fig. 3). Fo ully diso de ed
hype uni o m o close o hype uni o m beha io , we expec β
close o 1. Ei he SD
a0o he densi y o de ec s o bo h should
be la ge. In LiFeAs a 2 T we obse e βclose o 1 and a small
SD
a0bu a la ge numbe o de ec s. As shown in Re . [26], he
diso de ed la ice is locked o he c ys al la ice, ollowing i s
033133-4
DISORDERED HYPERUNIFORMITY IN SUPERCONDUCTING … PHYSICAL REVIEW RESEARCH 2, 033133 (2020)
o ien a ion. In W- ilm-2 a 5 T we obse e simila pa ame e s,
βclose o 1 and a small SD
a0bu a la ge numbe o de ec s.
The e a e oscilla ions in σ2(R) highligh ing ha o ien a ional
o de is main ained up o se e al a0. Thus, he appea ance
o diso de ed hype uni o mi y, o he dec ease in densi y
luc ua ions in diso de ed la ices, is linked o he p esence o
in e ac ions, ei he in he o m o locking o he c ys al la ice
(LiFeAs) o in sho - ange o ien a ional o de (W- ilm-2).
We ha e pe o med a nume ical simula ion o ge low-
ene gy con igu a ions o in e ac ing o ices wi h a ixed p o-
po ion o ixed o ices, as desc ibed in Sec. II.InFig.3(g ay
poin s) we show he esul s o he exponen βas a unc ion o
he s anda d de ia ion no malized by he in e o ex dis ance
[Fig. 3(a)] and o he de ec densi y [Fig. 3(b)]. The ag eemen
wi h expe imen al esul s is ai ly good, gi ing suppo o he
idea ha he deg ee o hype uni o mi y is associa ed wi h he
numbe o de ec s in he la ice. Nume ical simula ions also
se e o ob ain an indica ion o he numbe o de ec s in e ms
o he numbe o pinning poin s included in he calcula ion.
In he Appendix (Fig. 8) we show how hese wo quan i ies
a e ela ed. We no e ha he numbe o de ec s is clea ly
co ela ed wi h he numbe o pinning si es and i is a bi
la ge han he la e , especially a low densi ies o pinning
si es.
In o ex la ices, he numbe o de ec s is a consequence
o he balance be ween he pinning ene gy and he in e o ex
in e ac ions. Fo example, in LiFeAs, diso de appea s ela-
i ely a om Hc2, wi h a o ex la ice ha in e ac s s ongly
h ough he c ys alline la ice, as shown by he ou old sym-
me y in he s uc u e ac o discussed in Re . [26]. This leads
o βclose o 1, wi h a la ge numbe o de ec s, al hough
SD
a0is main ained o ela i ely small alues. When inc easing
he magne ic ield he e is a s ong endency owa ds andom
beha io . In W- ilm-2 wi h βclose o 1, s ong diso de
appea s a 5 T (which is 0.78Hc2), al hough wi h a smalle
de ec densi y and wi h enough o ien a ional o de o p esen
maxima in σ2(R). A a sligh ly la ge magne ic ield, a 5.5 T
(which is 0.85Hc2), he la ice o W- ilm-2 yields o andom
diso de and βinc eases o 1.5, wi h p ac ically he same SD
a0
bu a la ge numbe o de ec s.
We no e ha he model p oduces la ices ha ha e e y
small alues o SD
a0in Fig. 3(a). These la ices do no appea in
he expe imen (no poin s below SD
a00.1). When he o ex
la ice s ill has sho - ange posi ional o de (open ci cles in
Fig. 3), β=1. In his g oup o da a, we ha e la ices ha
show de ec s and polyc ys alline a angemen s (W- ilm-1 and
Co-NbSe2, shown in Fig. 3), as well as o de ed la ices wi h
a e y small numbe o de ec s (open ed ci cles o W- ilm-
2inFig.3). In he la e he posi ional co ela ions decay
exponen ially wi h dis ance [24], which explains why he
SD
a0 emains abo e 0.1. Fully o de ed hexagonal la ices (o
he o ex B agg glass wi h algeb aically decaying posi ional
co ela ion) p o ide β≈1 o close o ze o SD
a0in he ep e-
sen a ion o Fig. 3[24,29,43–46].
IV. DISCUSSION
The o ex la ice a angemen s a e a consequence o he
balance be ween elas ic and pinning ene gies. In Co-doped
NbSe2and in he W- ilm-1 hin ilm, pinning is s ong bu
s uc u ed, leading o hexagonal o ex clus e s obse ed a
all magne ic ields. On he o he hand, in CaKFe4As4pinning
cen e s a e so s ong and andomly dis ibu ed ha he o ex
la ice is essen ially andomly diso de ed in he whole ange
o magne ic ields s udied [27,39–42]. In W- ilm-2, whe e he
diso de po en ial is e y weak, he o ex la ice is only diso -
de ed when i is e y so , e y close oHc2, bu hen he o ex
dis ibu ion shows a s ong endency o diso de andomly,
because in e o ex in e ac ions a e e y weak. In LiFeAs we
ha e quali a i ely he same beha io as in CaKFe4As4,bu
wi h weake pinning. Fu he mo e, he in e o ex in e ac ion
wi h nonlocal con ibu ions due o he in luence o he c ys al
la ice symme y [18,47] is s ill impo an and esponsible o
dec easing densi y luc ua ions.
The o ex la ice o LiFeAs depends s ongly on he em-
pe a u e ange whe e he magne ic ield is applied. The e a e
measu emen s showing hexagonal o ex la ices in he same
magne ic ield ange [48], whe eas he ones we ha e used
he e [26] and neu on sca e ing expe imen s [49] p o ide dis-
o de ed la ices. No ice ha he diso de ed la ices discussed
he e a e locked o he c ys al la ice. This is a a he peculia
combina ion o long- ange in e ac ion and diso de . Locking
can be explained by nonlocal co ec ions o he London model
ha a o a ou old o ex la ice [18,47].
I is ele an o no e ha he only diso de ed la ices wi h
close o hype uni o m beha io (β≈1, poin s in Fig. 3co e-
sponding o LiFeAs a small magne ic ields and o W- ilm-2
a 5 T) ha e ela i ely small SD
a0. In he case o LiFeAs,
da a ollow closely he calcula ions. The e o e, we expec ha
ully diso de ed o ex la ices will no all o he beha io
o la ices showing luc ua ions in σ2(R) as a consequence o
sho - ange o de (open ci cles wi h β≈1 in he ep esen a-
ion o Fig. 3), bu a he ollow he smoo h inc ease o βwi h
he amoun o diso de p edic ed by he model.
Calcula ions show ha he o ex glass can p esen dis-
o de ed hype uni o mi y in a ange o magne ic ield and
empe a u es in he p esence o s ong epulsi e in e ac ions
and quenched diso de [11]. Those au ho s p oposed a phase
diag am wi h he close o diso de ed hype uni o m beha io
in be ween he B agg glass and he andom o ex glass.
Ou esul s con i m indeed he p esence o his in e media e
s a e and show ha i can be ob ained as a balance be ween
in e ac ion and pinning.
In a ecen wo k, o ex la ices a e y small magne ic
ields we e analyzed in iew o hei hype uni o m p ope -
ies. Those au ho s analyzed images o he high-Tccup a e
supe conduc o Bi2S 2CaCu2O8+δwi h magne ic Bi e dec-
o a ion in he p esence o diso de [23]. A small magne ic
ields, o ices a e e y a apa and hei mu ual epulsion
is small [50,51]. Vo ex a angemen s a e hen s ongly in lu-
enced by hei in e ac ion wi h pinning cen e s. Fu he mo e,
a high empe a u es, close o he ansi ion o he no mal s a e,
he o ex la ice mel s, leading o he o ex liquid which is
a dynamic angle o o ices [52,53]. The e o e, expe imen s
a small magne ic ields a e made by cooling om he liquid
phase, which esul s in quenched o ex a angemen s [54].
The au ho s o Re . [23] concluded ha long-wa eleng h luc-
ua ions a e sys ema ically supp essed in he o ex la ice a
small magne ic ields, as a consequence o he hyd odynamic
033133-5

JOSÉ BENITO LLORENS e al. PHYSICAL REVIEW RESEARCH 2, 033133 (2020)
(b)(a)
FIG. 4. Example o he calcula ion o σ2(R)andS(k)ina o ex
la ice image. (a) Image ob ained in CaKFe4As4a 4 T. Vo ices
a e he yellow colo ed pa ches. The o ex la ice is diso de ed.
(b) Black poin s show he o ex posi ions o (a). These a e ob ained
by calcula ing he cen e o mass o each whi e pa ch in (a), ob ained
as desc ibed in he ex . In he inse we show he Fou ie ans o m
o he image o he o ex posi ions. To calcula e σ2(R), we p oduce
andomly loca ed ci cles wi h a ying adius R, such as hose shown
schema ically in di e en colo s. Ci cles wi h he same adii a e
shown in he same colo .
p ope ies o he liquid phase, which leads o de ia ions om
ully andom o ex dis ibu ions.
V. CONCLUSION
In summa y, we ha e analyzed he condi ions o he
o ma ion o diso de ed hype uni o m o ex la ices in supe -
conduc o s a high magne ic ields. The o ex la ice shows
a endency away om andom beha io and owa ds hype-
uni o mi y when he numbe o pinning cen e s is be ween
30% and 40%. We ind ha he leng h scale o he in e ac ion
plays a mino ole in de e mining he s eng h o densi y
luc ua ions. Ins ead, he balance be ween pinning and o ex
la ice s i ness con ols densi y luc ua ions, wi h a con in-
uous a ia ion be ween densi y luc ua ions and diso de in
he la ice. We show ha he dec eased densi y luc ua ions
equi e in e o ex in e ac ions. We conclude ha we can
iden i y eme gen co ela ions in a o ex la ice using he
s uc u e ac o S(k) and numbe a iance σ2(R) and show
ha diso de ed o ex la ices a e cha ac e ized by supp essed
densi y luc ua ions.
ACKNOWLEDGMENTS
We a e e y g a e ul o Te suo Hanagu i o sha ing wi h
us da a in aw o m. We also acknowledge discussions wi h
Cha les Reichha d , who di ec ed ou a en ion o his p ob-
lem, and wi h Yanina Fasano and Ma ia Ia a one. This wo k
was suppo ed by he Spanish S a e Agency o Resea ch
(G an s No. FIS2017-84330-R, No. MAT2017-82970-C2-1-
R, No. MAT2017-82970-C2-2-R, No. RED2018-102627-T,
No. RYC-2014-15093, and No. CEX2018-000805-M), he
A agón Regional Go e nmen (Cons uyendo Eu opa desde
A agón) h ough P ojec s No. E28_20R wi h Eu opean So-
cial Fund unding, he Comunidad de Mad id h ough p o-
g am NANOMAGCOST-CM (G an No. S2018/NMT-4321),
and EU P og am No. CA16218 (Nanocohyb i). I.G. ac-
knowledges suppo om he Eu opean Resea ch Council
(e)
FIG. 5. Vo ex la ice in he W- ilm-1 hin ilm o di e en alues o he magne ic ield: (a) 0.03 T, (b) 0.07 T, (c) 0.09 T, (d) 0.11 T, and
(e) 5 T. Vo ices a e shown as black egions. Red ba s a e 120 nm long. The posi ion o each o ex, iden i ied by he me hod men ioned in he
ex , is shown by blue do s. Vo ices wi h ewe han i e nea es neighbo s a e shown as g een do s and wi h se en nea es neighbo s as whi e
do s. Pai s o such o ices p o ide one disloca ion. Black lines p o ide he Delaunay iangula ion o he o ex la ice. Hexagons show he
o ien a ion o he la ice in di e en pa s o he images. La ices showing di e en o ien a ions a e sepa a ed by yellow dashed lines.
033133-6
DISORDERED HYPERUNIFORMITY IN SUPERCONDUCTING … PHYSICAL REVIEW RESEARCH 2, 033133 (2020)
FIG. 6. Resul s in Co-doped NbSe2, omRe .[25], o (a) 2.5 T, (b) 2.7 T, and (c) 3.3 T. We ep esen o ices, iangula ion, la ice
o ien a ion, and ed scale ba s as in Fig. 5.
FIG. 7. Vo ex posi ions in all he images analyzed in his wo k.
(a) Vo ex posi ions in Co-doped NbSe2, om Re . [25]. (b) Resul s
ob ained in a W-based hin ilm wi h s ong pinning (W- ilm-1).
(c) Resul s in a highly o de ed W-based hin ilm (W- ilm-2) wi h a
e y weak 1D diso de po en ial, wi h da a om Re . [24].(d) Resul s
in LiFeAs, ob ained om da a om Re . [26]. (e) Resul s in pu e
CaKFe4As4, om Re . [27]. Magne ic ields a e gi en a each panel.
Scale ba s in ed a e 200 nm long.
PNICTEYES (G an Ag eemen No. 679080). M.O. acknowl-
edges suppo om Fundación Séneca h ough G an No.
19907/GERM/15. We also acknowledge SEGAINVEX a
UAM. R.C. acknowledges he suppo o a ellowship om
“la Caixa” Founda ion (ID 100010434). The ellowship code
is LCF/BQ/PR19/11700008.
APPENDIX
To calcula e σ2(R) and S(k) we s a by inding o ex
posi ions ollowing Re . [24]. We maximize he con as in
he image, inse ing a h eshold ha gi es a clea iew o
o ices as single colo ed and ex ended disks. We calcula e he
cen e o mass o each disk and use his o iden i y he posi ion
o each o ex. This leads o he ma ix o poin s shown in
Fig. 4(b). We hen make he Fou ie ans o m o ind S(k). To
calcula e σ2(R) we ollow Re . [11]. We gene a e ci cles o
size Rcen e ed a andomly gene a ed posi ions and inc ease
R om he a e age in e o ex dis ance a0 o nea ly he size
o he image. We use wo condi ions. Fi s , ci cles ha e o
be comple e and wi hin he image. Second, ci cles canno
FIG. 8. Densi y o de ec s ( o ices wi h coo dina ion numbe
di e en om 6) s he densi y o ixed poin s (pinned o ices) in
he ma ix posi ions om he calcula ions desc ibed in he ex .
033133-7
JOSÉ BENITO LLORENS e al. PHYSICAL REVIEW RESEARCH 2, 033133 (2020)
o e lap. We schema ically show a ew ci cles in Fig. 4(b).In
each ci cle, we coun N(R), he numbe o o ices inside he
ci cle, and ob ain σ2(R)=N2(R)−N(R)2by a e aging
o e many ci cles. When Ris small, we ob ain in one ial
many ci cles spanning he whole image. When Ris la ge, we
ob ain jus a ew ci cles. We make he calcula ion in such
a way as o inc ease he numbe o andom ies gi ing he
cen e o he ci cles wi h R, aking ca e ha we a e age o e
a leas 100 ci cles o all R.
In W- ilm-1, o ices ge pinned by di e ences in he hick-
ness o he hin ilm, as desc ibed in de ail in Re s. [37,55].
In Figs. 5(a)–5(e) we show esul s wi h inc easing magne ic
ields in W- ilm-1. We see ha he o ex la ice emains wi h
egions showing hexagonal o de a all magne ic ields. We
obse e ha egions wi h di e en o ex la ice o ien a ion
a e sepa a ed by egions wi h a la ge numbe o disloca ions.
The smalles o de ed egions appea a ela i ely la ge mag-
ne ic ields. A u he inc ease o he magne ic ield owa ds
Hc2leads o a diso de ed andom con igu a ion, shown in
Fig. 5(e), whe e we can s ill iden i y hexagonally o de ed
egions o ini e size. This la ice is acco dingly s ill hype -
uni o m, bu wi h a la ge densi y o de ec s (abou 40%).
In Fig. 6we show he o ex la ice in Co-doped NbSe2.
Regions wi h hexagonal o de a e obse ed a all magne ic
ields [Figs. 6(a)–6(c)] and he numbe o de ec s can be qui e
la ge, o abou 40% [25].
Fo comple eness we show all o ex posi ions used o
calcula e he s uc u e ac o and numbe a iance in Fig. 2
in Fig. 7. We also show he densi y o de ec s s he pinned
poin s om he model calcula ions in Fig. 8.
[1] S. To qua o and F. H. S illinge , Phys. Re . E 68, 041113
(2003).
[2] S. To qua o, Phys. Rep. 745, 1 (2018).
[3] W. Man, M. Flo escu, E. P. Williamson, Y. He, S. R.
Hashemizad, B. Y. C. Leung, D. R. Line , S. To qua o, P. M.
Chaikin, and P. J. S einha d , P oc. Na l. Acad. Sci. USA 110,
15886 (2013).
[4] Y. Jiao, T. Lau, H. Ha ziki ou, M. Meye -He mann, J. C. Co bo,
and S. To qua o, Phys.Re .E89, 022721 (2014).
[5] R. Ku i a and E. R. Weeks, Phys.Re .E84, 030401(R) (2011).
[6] R. D ey us, Y. Xu, T. S ill, L. A. Hough, A. G. Yodh, and S.
To qua o, Phys. Re . E 91, 012302 (2015).
[7] D. Hexne , A. J. Liu, and S. R. Nagel, Phys. Re . Le . 121,
115501 (2018).
[8] D. Hexne , P. U bani, and F. Zamponi, Phys.Re .Le .123,
068003 (2019).
[9] C. E. Zacha y, Y. Jiao, and S. To qua o, Phys. Re . Le . 106,
178001 (2011).
[10] J. H. Weijs, R. Jeanne e , R. D ey us, and D. Ba olo, Phys. Re .
Le . 115, 108301 (2015).
[11] Q. Le Thien, D. McDe mo , C. J. O. Reichha d , and C.
Reichha d , Phys. Re . B 96, 094516 (2017).
[12] I. A. Sado skyy, Y. L. Wang, Z.-L. Xiao, W.-K. Kwok, and A.
Gla z, Phys. Re . B 95, 075303 (2017).
[13] H. Diaman , J. Club Condens. Ma e Phys. (2019).
[14] G. Bla e , M. V. Feigel’man, V. B. Geshkenbein, A. I. La kin,
and V. M. Vinoku , Re . Mod. Phys. 66, 1125 (1994).
[15] E. H. B and , Rep. P og. Phys. 58, 1465 (1995).
[16] J. F iedel, P. G. De Gennes, and J. Ma icon, Appl. Phys. Le .
2, 119 (1963).
[17] P. G. de Gennes and J. Ma icon, Re . Mod. Phys. 36, 45 (1964).
[18] V. G. Kogan, Phys. Re . B 24, 1572 (1981).
[19] A. I. Buzdin, A. S. Mel’niko , and A. V. Samokh alo ,
J. Supe cond. No el Magn. 26,2853 (2013).
[20] J. Pea l, Appl. Phys. Le . 5, 65 (1964).
[21] E. H. B and , Phys.Re .B48, 6699 (1993).
[22] E. He e a, I. Guillamón, J. A. Gal is, A. Co ea, A. Fen e, S.
Viei a, H. Sude ow, A. Y. Ma yno ich, and V. G. Kogan, Phys.
Re . B 96, 184502 (2017).
[23] G. Rumi, J. A agón Sánchez, F. Elías, R. Co és Maldonado,
J. Puig, N. R. Cejas Bolecek, G. Nie a, M. Konczykowski,
Y. Fasano, and A. B. Kol on, Phys.Re .Res.1, 033057
(2019).
[24] I. Guillamón, R. Có doba, J. Sesé, J. M. De Te esa, M. R. Iba a,
S. Viei a, and H. Sude ow, Na . Phys. 10, 851 (2014).
[25] M. Ia a one, R. Di Capua, G. Ka ape o , A. E. Koshele , D.
Rosenmann, H. Claus, C. D. Malliakas, M. G. Kana zidis, T.
Nishizaki, and N. Kobayashi, Phys. Re . B 78, 174518 (2008).
[26] T. Hanagu i, K. Ki agawa, K. Ma subayashi, Y. Mazaki, Y.
Uwa oko, and H. Takagi, Phys.Re .B85, 214505 (2012).
[27] A. Fen e, W. R. Meie , T. Kong, V. G. Kogan, S. L. Bud’ko,
P. C. Can ield, I. Guillamón, and H. Sude ow, Phys. Re . B 97,
134501 (2018).
[28] I. Guillamón, H. Sude ow, P. Kulka ni, S. Viei a, R. Có doba,
J. Sesé, J. D. Te esa, M. Iba a, G. Shaw, and S. Banne jee,
Physica C 503, 70 (2014).
[29] T. Giama chi and P. Le Doussal, Phys. Re . Le . 72, 1530
(1994).
[30] H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles, and J. V.
Waszczak, Phys. Re . Le . 62, 214 (1989).
[31] I. Guillamón, H. Sude ow, S. Viei a, L. Ca io, P. Diene , and P.
Rodiè e, Phys. Re . Le . 101, 166407 (2008).
[32] O. Fische , M. Kugle , I. Maggio-Ap ile, C. Be hod, and C.
Renne , Re . Mod. Phys. 79, 353 (2007).
[33] H. Sude ow, I. Guillamón, J. G. Rod igo, and S. Viei a,
Supe cond. Sci. Technol. 27, 063001 (2014).
[34] A. M. T oyano ski, J. Aa s, and P. H. Kes, Na u e (London)
399, 665 (1999).
[35] M. A. Kla , J. Kim, and S. To qua o, Phys.Re .E101, 032118
(2020).
[36] H. Sude ow, I. Guillamon, and S. Viei a, Re . Sci. Ins um. 82,
033711 (2011).
[37] I. Guillamón, H. Sude ow, S. Viei a, A. Fe nández-Pacheco, J.
Sesé, R. Có doba, J. M. D. Te esa, and M. R. Iba a, New J.
Phys. 10, 093005 (2008).
[38] R. Có doba, T. I. Ba u ina, J. Sesé, A. Y. Mi ono , J. M. De
Te esa,M.R.Iba a,D.A.Nasimo ,A.K.Gu ako skii,A.V.
La yshe , I. Guillamón, H. Sude ow, S. Viei a, M. R. Baklano ,
J. J. Palacios, and V. M. Vinoku , Na . Commun. 4, 1437 (2013).
[39] S. J. Singh, M. B is ow, W. R. Meie , P. Taylo , S. J. Blundell,
P. C. Can ield, and A. I. Coldea, Phys. Re . Ma e . 2, 074802
(2018).
033133-8
DISORDERED HYPERUNIFORMITY IN SUPERCONDUCTING … PHYSICAL REVIEW RESEARCH 2, 033133 (2020)
[40] S. Pyon, A. Takahashi, I. Veshchuno , T. Tamegai, S. Ishida, A.
Iyo, H. Eisaki, M. Imai, H. Abe, T. Te ashima, and A. Ichinose,
Phys. Re . B 99, 104506 (2019).
[41] N. Habe ko n, M. Xu, W. R. Meie , J. Schmid , S. L. Bud’ko,
and P. C. Can ield, Phys.Re .B100, 064524 (2019).
[42] S. Ishida, A. Iyo, H. Ogino, H. Eisaki, N. Takeshi a, K.
Kawashima, K. Yanagisawa, Y. Kobayashi, K. Kimo o, H. Abe,
M. Imai, J.-i. Shimoyama, and M. Eis e e , npj Quan um Ma e .
4, 27 (2019).
[43] T. Klein, I. Jouma d, S. Blancha d, J. Ma cus, R. Cubi , T.
Giama chi, and P. Le Doussal, Na u e (London) 413, 404
(2001).
[44] T. Giama chi and P. Le Doussal, Phys. Re . B 52, 1242
(1995).
[45]D.S.Fishe ,M.P.A.Fishe ,andD.A.Huse,Phys.Re .B43,
130 (1991).
[46] J. Kie eld, T. Na e mann, and T. Hwa, Phys. Re . B 55, 626
(1997).
[47] V. G. Kogan, M. Bullock, B. Ha mon, P. Mi ano i´
c, L.
Dob osa lje i´
c-G uji´
c, P. L. Gammel, and D. J. Bishop, Phys.
Re . B 55, R8693 (1997).
[48] S. S. Zhang, J.-X. Yin, G. Dai, H. Zheng, G. Chang, I.
Belopolski, X. Wang, H. Lin, Z. Wang, C. Jin, and M. Z. Hasan,
Phys. Re . B 99, 161103(R) (2019).
[49] D. S. Inoso , J. S. Whi e, D. V. E ushinsky, I. V. Mo ozo ,
A. Came on, U. S ocke , V. B. Zabolo nyy, T. K. Kim,
A. A. Ko dyuk, S. V. Bo isenko, E. M. Fo gan, R. Klingele ,
J. T. Pa k, S. Wu mehl, A. N. Vasilie , G. Beh , C. D. Dewhu s ,
and V. Hinko , Phys.Re .Le .104, 187001 (2010).
[50] R. P ozo o , A. F. Fidle , J. R. Hobe g, and P. C. Can ield, Na .
Phys. 4, 327 (2008).
[51] J. B. Llo ens, L. Embon, A. Co ea, J. D. González, E. He e a,
I. Guillamón, R. F. Luccas, J. Azpei ia, F. J. Mompeán, M.
Ga cía-He nández, C. Munue a, J. A. Sánchez, Y. Fasano, M. V.
Milose ic, H. Sude ow, and Y. Anaho y, Phys. Re . Resea ch 2,
013329 (2020).
[52] R. Cubi , E. M. Fo gan, G. Yang, S. L. Lee, D. M. Paul, H. A.
Mook, M. Ye hi aj, P. H. Kes, T. W. Li, A. A. Meno sky,
Z. Ta nawski, and K. Mo ensen, Na u e (London) 365, 407
(1993).
[53] E. Zeldo , D. Maje , M. Konczykowski, V. B. Geshkenbein,
V. M. Vinoku , and H. Sh ikman, Na u e (London) 375, 373
(1995).
[54] M. Ma che sky, P. Kes, and J. Aa s, Physica C 282–287, 2083
(1997).
[55] I. G. Se ano, J. Sesé, I. Guillamón, H. Sude ow, S. Viei a, M.
Iba a, and J. M. De Te esa, Beils ein J. Nano echnol. 7,1698
(2016).
033133-9