Detection of small exchange fields in S/F structures

0.21 MB·

by  A. S. Vasenko

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Detectionof smallexchangefieldsinS/F structures
v
o
N A. S. Vasenkoa, S. Kawabatab, A. Ozaetac, A. A. Golubovd,e, V. S. Stolyarovf,g,h,e,
F. S. Bergeretc,i, F. W. J.Hekkinga,j
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aLPMMC, Universite´ Joseph Fourier and CNRS, 25Avenue des Martyrs, BP166, 38042 Grenoble, France

] bElectronics andPhotonics Research Institute (ESPRIT),
n National Institute ofAdvanced Industrial Science andTechnology (AIST), Tsukuba, Ibaraki, 305-8568, Japan
o cCentro de F´ısica deMateriales (CFM-MPC), Centro Mixto CSIC-UPV/EHU, Manuel de Lardizabal 5, E-20018San Sebastia´n, Spain
c dFaculty of Science andTechnology and MESA+ Institute for Nanotechnology, University ofTwente, 7500 AEEnschede, The Netherlands
r- eMoscow Institute ofPhysics andTechnology, 141700 Dolgoprudny, Russia
p fSorbonne Universite´s, UPMCUniv Paris 06, UMR7588, Institut des Nanosciences de Paris,F-75005, Paris, France
u gCNRS, UMR 7588, Institut des Nanosciences deParis, F-75005, Paris,France
s hInstitute of Solid State Physics RAS,142432, Chernogolovka, Russia
t. iDonostia International Physics Center (DIPC),Manuel de Lardizabal 4, E-20018San Sebastia´n, Spain
a jInstitut Universitaire de France, 103, bdSaint-Michel 75005 Paris,France
m
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d
n
Abstract
o
[c Ferromagnetic materials with exchange fields Eex smaller or of the order of the superconducting gap D are important for
applications of corresponding (s-wave) superconductor/ ferromagnet/ superconductor (SFS) junctions. Presently such materials

2 arenotknownbutthereareseveralproposalshowtocreatethem.Smallexchangefieldsareinprincipledifficulttodetect.Based
v onourresultsweproposereliabledetectionmethodsofsuchsmallEex.Forexchangefieldssmallerthanthesuperconductinggap
6 thesubgapdifferentialconductanceofthenormalmetal-ferromagnet-insulator-superconductor(NFIS)junctionshowsapeak
4 atthevoltagebiasequaltotheexchangefieldoftheferromagneticlayer,eV=Eex.Thusmeasuringthesubgapconductanceone
6 canreliablydetermine smallEex<D .IntheoppositecaseEex>D onecandeterminetheexchange fieldinscanning tunneling
0
microscopy (STM) experiment. The density of states of the FS bilayer measured at the outer border of the ferromagnet shows
.
1 a peak at the energy equal to the exchange field, E=Eex. This peak can be only visible for small enough exchange fields of
0 theorder of few D .
4
1
: Key words: exchange field, S/F hybrid structures, proximity effect
v
PACS:74.45.+c, 74.50.+r, 74.78.Fk, 75.30.Et
i
X
r
a
1. Introduction brid structures[1–4].Let us briefly reviewthe essence
ofthe S/Fproximityeffect.
Asweknowfromthequantumtheoryofmagnetism UponenteringoftheCooperpairintothe ferromag-
the ferromagnetic metal can be described by the pres- neticmetalitbecomesanevanescentstateandthespin
ence of the so called exchange field, Eex. This field upelectroninthepairlowersitspotentialenergybyEex,
is responsibleto manyinterestingphenomenain artifi- whilethespindownelectronraisesitspotentialenergy
ciallyfabricatedsuperconductor/ferromagnet(S/F)hy- bythesameamount.Inorderforeachelectrontocon-
serveitstotalenergy,thespinupelectronmustincrease
its kinetic energy, while the spin down electron must
decrease its kinetic energy, to make up for these addi-
Email address:[email protected]
tionalpotentialenergiesinF.Asaconsequence,thecen-
(A. S.Vasenko).
PreprintsubmittedtoElsevier 4November2014
ter of mass motion is modulated and superconducting for 5.5% of Ni fitting gives E =0.11 meV, which is
ex
correlationsin the Flayer havethe dampedoscillatory 0.1 D , for 6% of Ni it gives E =0.45 meV, which is
ex
behavior[5,6].Ifweneglecttheinfluenceofotherpos- 0.4D ,for7%ofNiitgivesE =2.8meV,whichis2.2
ex
sibleparametersofferromagneticmetal(likemagnetic D , andfor11.5%ofNiE =3.9meV,whichis3 D .
ex
scatteringrate,etc.)thecharacteristiclengthsofthede- Another promising alloy with small exchange field,
cay and the oscillations are equal to x = D /E , Pd Fe ,was studiedin [24–26].
f f ex 0.99 0.01
where D is the diffusion coefficient in the ferromag- Finally, a small exchange field can be induced by a
f
p
neticmetal[1]. ferromagnetic material into the adjacent normal metal
Thelengthx isalsothelengthofdecayandoscilla- layer.Inrecentproposal[27],athinnormalmetallayer
f
tionsofthecriticalcurrentinJosephsonS/F/Sjunctions wasplacedontopoftheferromagneticinsulator.Itwas
[7,8]. Negative sign of the critical current corresponds shown that the ferromagneticinsulator may induce ef-
to the so-calledp -state [9–13].S/F/S p -junctionshave fectiveexchangefieldin thenormalmetallayer[27],
been proposed as potential elements in superconduct- Eeexff=h¯DGf r /d, (1)
ing classical and quantum logic circuits [14–16]. For
whereDisthediffusioncoefficientinthenormalmetal,
instance,S/F/Sjunctionscanbeusedascomplementary
elements (p -shifters) in RSFQ circuits (see Ref. [17] Gf isasurfaceconductancelikecoefficientforthenor-
mal metal/ ferromagnetic insulator interface, r is the
and referencestherein).S/F/S based deviceswere also
resistivity of the normal metal, and d is the thickness
proposed as elements for superconducting spintronics
of the normal metal layer in the direction, perpendic-
[18]. Finally, S/F/S structures have been proposed for
the realization of so called j -junctions with a j drop ular to the ferromagnetic insulator surface. The field
inthe groundstate, where0<j <p [19,20]. Eeexff is expectedto be much smaller than the exchange
field inside standard ferromagnets. Interestingly, such
Presently known ferromagnetic materials have large
exchange fields, E D and therefore short char- exchange field is possible to tune at the sample fab-
ex
acteristic length of os≫cillations, x x , where x = rication stage since it is inversely proportional to the
f s s
D /2D is the superconductingcoh≪erencelength and normalmetallayerthicknessd. Thisgivesa flexibility
s
D is the diffusion coefficient in the superconductor. with respect to material constraints. We also note that
Tphsis requires very high precision in controlling the F as Eeexff(cid:181) Gf , inducingthe tunnelbarrierat the normal
metal/ ferromagnetic interlayer interface, one can fur-
layerthicknessinthefabricationprocessoftheJoseph-
sonp -junctions.InalreadyexistingS/F/Sstructuresthe therreducethevalueoftheeffectiveexchangefield.
Belowwesuggestdirectmeasurementsofsuchsmall
roughnessisoftenlargerthanthedesiredprecision.The
exchangefields.Thedetectionmethodsaredifferentin
way to solve this problem is to invent ferromagnetic
case of the exchangefield smaller, E <D , and larger
materialswith smallexchangefields. ex
thanthe superconductinggap,E >D .
In this paper we review several proposals for ferro- ex
Weshouldmentionthatweproposemethodsofsmall
magneticmaterialswithexchangefieldsE smalleror
ex
ofthe orderofthe superconductinggapD . Thenbased exchangefield detectionin the ideal case of ferromag-
netic layer with homogeneous magnetization and ab-
on our results we propose reliable detection methods
sence of magnetic and spin-orbit scattering in contact
of such small exchangefields in experiments.Another
withasuperconductor.However,incaseofrealisticfer-
detectionmethodwasrecentlysuggestedin [21].
romagnetssituation can be more complicated. We dis-
cuss some possible limitations of the detection in the
2. Waystogeneratesmallexchangefields endofthe twofollowingsections.
The easiest way to create small exchange field is to 3. Detection of exchange fields smaller than the
applyanexternalmagneticfieldB to thenormalmetal superconductinggap
lead, in which case E =m B, where m is the Bohr
ex B B
magneton. InthissectionweconsiderthefollowingSIFNstruc-
It may be also an intrinsic exchange field of weak ture: a ferromagnetic wire F of a length d (smaller
f
ferromagnetic alloys. For example, in Ref. [22] were than the inelastic relaxationlength[28,29])is attached
reportedexchangefieldsforPd Ni withdifferentNi at x=0 to a superconducting (S) and at x=d to a
1 x x f
concentration,obtainedbyafitti−ngprocedure(seealso normal(N)electrode.Theinterfaceatx=0isatunnel
[23]).ConsideringNbasasuperconductorwithD =1.3 barrierwhileatx=d wehaveatransparentinterface.
f
meV, we can estimate the exchange field in Pd Ni : We will show that the subgap differentialconductance
1 x x
−
2
ofsuchastructurehasa peakatthebiasvoltageequal
3
totheexchangefieldoftheferromagneticmetalincase
when E <D [30,31]. Thus we propose to determine
ex
smallE <D inexperimentsbymeasuringthesubgap
ex 2
differentialconductanceofNFISjunctionsatlowtem- V)
peratures. (A
G’
In this paper we consider the diffusive limit, i.e. we 1
assumethattheelasticscatteringlengthismuchsmaller
than the decay length of the superconducting conden-
sate intotheFregion.Hereandbelowwe considerfor 0
0,0 0,2 0,4 0,6 0,8 1,0
simplicity D =D D and h¯ =k =1. In order to
f s B
≡ eV/
describethetransportpropertiesofthesystemwesolve
Fig. 1. (Color online) The bias voltage dependence of differential
tqh-epaUrsaamdeetlriezqautiaotinorneiandsth[e32F,3la3y]er,thatinthesocalled c(bolnadcukcstaonlicdeliante)z,eEroex/tDem=p0er.4atu(grereefnordaesxhcehdanlignee),fiEeledxs/,D E=ex0/.D7(=red0
D dash-dot-dotted line),andEex/D =0.9(bluedash-dottedline).Here
2i¶ x2xq f↑(↓)=(E±Eex)sinhq f↑(↓). (2) G′A=2RTGA,W=0.014 and df =10xs.
Here the positive and negativesigns correspond to the
spin-up andspin-down states,respectively.Because
↑ ↓
of the high transparencyof the F/N interface the func-
tionsq =0atx=d .Whileatthetunnelinginter- 6
f ( ) f
face at x↑=↓ 0 we use the Kupriyanov-Lukichevbound-
arycondition[34] 4 V)
(
G’A
R 2
¶ q = F sinh[q Q ], (3)
x f↑(↓)|x=0 dfRT f↑(↓)|x=0− s 0
ocQwtlhaufhysereetr=hferrueenanatRrpcntctahdFtiraroaonaSnmunhsFg(deqDhtirRnf/it↑zthTE(ea↓er))tafijaoriauescnrneetct,ahhtonieerobegntsnslaue.poipeInrinmceentrdpiacvtalohoernrenlteyidecsSuui(ccsRlalttaaainTrnyngwecc≫reoeb,smuRaqolprsFkef.u)tt,viOehnaanetltenuhcrFdeee- 1,0 0,e8V/0,6 0,4 0,2 0,0 0,000,100,200,300,400,500,06,007,008,009,009,E599e/x
estedintheAndreevcurrent,i.e.thecurrentforvoltages
smaller than the superconducting gap due to Andreev
Fig. 2. (Color online) The bias voltage dependence of dif-
processesattheS/F interface. ferential conductance at zero temperature for exchange fields,
Due to the tunneling barrier at the S/F interface the Eex/D =0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.95and 0.99.
proximityeffectisweakandhencewelinearizeEqs.(2- Here G′A=2RTGA,W=0.014 and df =10xs.
3) with respect to R /R 1. After a straightforward
F T
≪
calculationwe obtainthe Andreevcurrentatzerotem-
peratureinthislimit[35,36], D x sRF (cid:229) arctanh(c+j )+arctan(c−j )
I = , (5)
A ed R2 √D +jE
WD 2 (cid:229) eV dE f T j=± ex
I =
A 4eRT j=±ZiD0 D 2−E2 E+jE d c+j =seDV++jjEEeexx, c−j =seDV+−jjEEeexx.
ex f
Re tanh , (4)
× "sE+jEex r iD x s!# In Fig. 1 we plot the Andreev differential conduc-
tance G =dI /dV which is equal to the full differ-
A A
where W =x R /d R is the diffusive tunneling pa- ential conductanceof the junctionatzero temperature.
s F f T
rameter[34,37,38].InthetunnelinglimitW 1. Theconductanceshowstwowelldefinedpeaks,oneat
We evaluate Eq.(4) in the long-junction≪limit, i.e. eV =E andtheotherateV =D .Thedetailedphysical
ex
when d x , and E .eV <D . We obtain for the explanationof the peakatE is givenin [30].Itturns
f f ex ex
≫
Andreevcurrent outthatitisthezerobiasanomaly(ZBA) peakforthe
3
diffusiveNISjunction,shiftedin FIScase by E . The
ex )
E
ZBApeakinNISisshowninFig.1byblacksolidline; (
Nf
E =0 correspondstothe normalmetalcase.
ex
InFig.2weplottheAndreevdifferentialconductance
for many different values of 0 Eex <1 to show the 1
≤
evolution of the peak with increasing Eex. Detecting
this peakone can carefullymeasure the valueof small
exchangefield E <D inthe ferromagneticmetal.
ex
Wewouldliketomentionthatthepeakwillbevisible
ateV =E forasingledomainferromagnetincontact
ex
withasuperconductor.Incaseofamulti-domainferro- 0
0 2 4 6
E/
magnetthepeakofthedifferentialconductanceoccurs

at eV equals to the “effective field”, which is the field E)
(
acting on the Cooper pairs in the multi-domain ferro- Nf
magneticregion,averagedoverthedecaylengthofthe
superconductingcondensateintoaferromagnet[39,40]. 1,05
4. Detection of exchange fields larger than the
superconductinggap
In this section we consider just a simple FS bilayer 1,00
with a transparent interface: wire F of a length d
f
(smaller than the inelastic relaxation length [28,29])
0 2 4 E/ 6
is attached at x=0 to a superconductingelectrode by
Fig.3.(Coloronline)DOSNf(E)a ttheouterborderoftheFlayer
a transparent interface. We will show that the density
in the FS bilayer calculated numerically for different values of the
of states (DOS) measured at the outer border of the exchangefieldEex.ParametersoftheF/Sinterfaceareg =gB=0.01,
ferromagnet(x=df) showsa peakattheenergyequal T=0.1Tc. Upper panel: df/xn=1; lower panel: df/xn=3. Solid
tios othfetheexcohradnegreoffiefledwfoD r[d8f,4≫1].xTfhiunscwaesepwrohpeonseEteox banladckdalsinhe-dcoottrerdespbolunedslintoeEtoexE/Dex/=D 2=,d6a.shedredline toEex/D =4,
determine E > D in experiments by measuring the
ex
DOS at the outer border of the ferromagneticmetal in
Attheouterborderoftheferromagnet(x=d )wehave
f
correspondingSF bilayerstructure,whichcanbe done ¶ q =0. At x =0 we use Kupriyanov-Lukichev
x f ( )
byscanningtunnelingmicroscopy(STM). boun↑d↓ary conditions which in case of the transparent
TheDOSN (E)normalizedtotheDOSinthenormal
f interfaceisconvenienttowrite as
state, canbewrittenas
g¶ q =¶ q , (9a)
Nf(E)= Nf↑(E)+Nf↓(E) /2, (6) x ngB¶ xxq ff|xx==00=sixnhs(|xq=f0 q s)x=0. (9b)
where Nf ( )(E) are(cid:2)the spin resolved(cid:3)DOS written in | − |
termsofs↑p↓ectralangleq f, Here g =s f/s s, s f(s) are the conductivities of the F
N (E)=Re coshq . (7) (S) layers correspondingly, x n = D/2p Tc, Tc is the
f↑(↓) f↑(↓) critical temperature of the superconductor, and gB =
ToobtainNf,weuseaself-co(cid:2)nsistenttw(cid:3)o-stepiterative d R /x R =x /x W.Theparampeterg determinesthe
f T n F s n
procedure[8,42].In the first step we calculate the pair
strength of suppression of superconductivity in the S
potential coordinate dependence D (x) using the self-
layer near the interface (inverse proximity effect). No
consistency equation in the S layer in the Matsubara suppressionoccursforg =0,whilestrongsuppression
representation. Then, using the D (x) dependence, we takesplaceforg 1.Inournumericalcalculationswe
solvetheUsadelequationsinthe Slayer, assumesmallg ≫1.Sinceweconsiderthetransparent
D ≪
2i¶ x2xq s=Esinhq s+iD (x)coshq s, (8) Winterfa1c,ethReFre≫forReTg and1c.oNntoratircyettohathteinptrheeviEoquss.s(e8c)t-i(o9n)
B
≫ ≪
together with the Usadel equations in the F layer wehaveomittedthesubscripts ( )becauseequations
↑ ↓
[Eq. (2)] and corresponding boundary conditions, re- for both spin directions are identical in the supercon-
peating the iterationsuntil convergencyis reached [8]. ductor.
4
InFig.3weplottheDOSN (E)attheouterborder Detecting this peak one can carefully measure the
f
of the F layer in the FS bilayer calculated numerically valueofsmallexchangefiledE >D intheferromag-
ex
for different values of the exchange field E and for netic metal.
ex
different F layer thicknesses d . At large enough d Wementionthatinthepresenceofmagneticscatter-
f f
(d /x =3) we see the peak at E =E [see Fig. 3, ingtheDOSpeakatE=E donotchangetheposition
f n ex ex
lower panel]. For small d (d /x =1) the peakis not butgetssmearedatlargeenoughscatteringrate[8].We
f f n
visibleandDOStendsmonotonouslytounityforE>D did not consider the effect of domain structure of the
[see Fig. 3, upperpanel].Theamplitudeof the peakis F layer on this peak, but we can expectsimilar behav-
decreasingwithincreasingE :peakisonlyvisiblefor iorasdiscussedinprevioussection,i.e.thepositionof
ex
E ofthe orderof few D (see [8] fordetails). We also the DOS peak will move to the value of the “effective
ex
needtomentionthatincaseofE <D thereisnopeak filed” [40].
ex
inthe DOS.
To better illustrate the conditions when the peak at
E=E isvisibleinexperimentsweconsiderananalyt- 5. Summary
ex
icallimitingcase. Ifthe Flayer isthickenough(d
f
x ) and g =0 in Eq. (9), the DOS at the outer bord≫er We propose reliable methods to measure small ex-
f
oftheferromagnetcanbewritten as[7,8,43] change fields in weak ferromagnet/ superconductor
structures.ForE <D thesubgapdifferentialconduc-
1 ex
N (E)=Re[cosq ] 1 Req 2 . (10) tance of the normal metal - ferromagnet - insulator -
f↑(↓) b↑(↓) ≈ −2 b↑(↓) superconductor (NFIS) junction shows a peak at the
Hereq b ( ) istheboundaryvalueofq f atx=df,given voltage bias equal to the exchange field of the ferro-
by ↑↓ magnetic layer, eV =E . Thus measuring the subgap
ex
conductanceonecanreliablydeterminesmallE <D .
8F(E) d ex
q = exp p f , (11) In the opposite case E > D one can determine the
b↑(↓) F2(E)+1+1 (cid:18)− ↑(↓)x f(cid:19) exchangefieldinscanninexgtunnelingmicroscopyexper-
whereweuspe thefollowingnotations, iment.ThedensityofstatesoftheFSbilayermeasured
at the outerborderof the ferromagnetshowsa peakat
p↑(↓)= 2/Eex√D −iER±iEex, (12a) theNeenxetrgwyeeaqruealpltaontnhiengextcohsaenagrechfieflodr,sEm=allEeexx.change
F(E)=p , E =E+i0. (12b)
iE + D 2 E2 R fields Eex >D in the experiments, using the ultrahigh
− R − R vacuum Scanning Tunneling Microscopy (STM) and
From Eqs. (10)-(1q1)we obtain for the full DOS the Spectroscopy (STS) technique, recently developed by
followingexpressioninthelimitd x andforE D , oneofthe authors(Stolyarovetal.)[44].
f f
≫ ≥ We alsohopethatourresultswilltriggerfurtherex-
(cid:229) 16D 2cos(bj)exp( bj) perimental activity in finding ferromagnetic materials
N (E)=1+ − , (13)
f (E+e )(√E+e +√2e )2 with small exchange fields. Good candidates for such
j=
± materialscanbedilutedferromagneticalloys(likePdNi,
PdFe,CuNi,etc.)andnormalmetalsinproximitywith
2d E+jE
bj= x ffs| Eex ex|, e = E2−D 2. tfhererofemrraogmneatgicneitniscuilnastourlamtoarys (iFndI)u.cIentthheeelxatcthearncgaesefitehlde
p
We can clearly see the exponential asymptotic of the in the thin normalmetal layer, placed on top of the FI
peak at E = E from Eq. (13). We should keep in material.
ex
mind that Eq. (13) is valid for large d /x , but nev- This work was supported by European Union Sev-
f f
ertheless we may qualitatively understand why we do enth Framework Programme (FP7/2007- 2013) un-
not see the peak at E =E for small ratio of d /x : der grant agreement “INFERNOS” No. 308850, by
ex f f
if this factor is small the variation of the exponent Ministry of Education and Science of the Russian
exp 2(d /x ) E E /E near the point E = Federation, grants No. 14Y.26.31.0007, No. 2014-14-
f f ex ex
{− | − | }
E is also small. The peak is observable only for E 588-0010-061,RFBR No. mol a 14-02-31798,and by
ex ex
of the order of apfew D . For larger exchange fields the French National Agency for Research ANR (ELEC-
peakisdifficulttoobserve,sincetheenergydependent TROVORTEX). A.S.V. acknowledgethe hospitality of
prefactoroftheexponentinEq.(13)decaysasE 2 for Superconducting electronics group, AIST, during his
−
E D . stay inJapan.
≫
5
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