Properties of tunnel Josephson junctions with a ferromagnetic interlayer

0.48 MB·English

by  A. S. Vasenko

Properties oftunnel Josephsonjunctions withaferromagnetic interlayer
A. S. Vasenko,1,2 A. A. Golubov,1 M. Yu. Kupriyanov,3 and M. Weides4
1Faculty of Science and Technology and MESA+ Institute for Nanotechnology,
University of Twente, 7500 AE Enschede, The Netherlands
2Department of Physics, Moscow State University, Moscow 119992, Russia
3Nuclear Physics Institute, Moscow State University, Moscow, 119992, Russia
4Center of Nanoelectronic Systems for Information Technology (CNI),
Research Centre Ju¨lich, D-52425 Ju¨lich, Germany
8 (Dated:February2,2008)
0
0 We investigate superconductor/insulator/ferromagnet/superconductor (SIFS) tunnel Josephson junctions in
2 thedirtylimit,usingthequasiclassicaltheory. Weformulateaquantitativemodeldescribingtheoscillationsof
criticalcurrentasafunctionofthicknessoftheferromagneticlayerandusethismodeltofitrecentexperimental
n
data. Wealsocalculatequantitativelythedensityofstates(DOS)inthistypeofjunctionsandcompareDOS
a
J oscillationswiththoseofthecriticalcurrent.

4
PACSnumbers:74.45.+c,74.50.+r,74.78.Fk,75.30.Et
1

n] I. INTRODUCTION to the decaycharacteristic lengthx f1. Further, in symmetric
o S/F/Sjunctions,theextensionoftheorytothecaseofnonho-
c mogeneousmagnetizationandlargemeanfreepathwasper-
Itiswellknownthatsuperconductivityandferromagnetism
- formedinRefs.3,18.
r aretwocompetingorders,howevertheirinterplaycanbereal-
p Thepurposeofthisworkistoprovideaquantitativemodel
izedwhenthetwointeractionsarespatiallyseparated. Inthis
u describing the behavior of critical current and DOS in SIFS
case the coexistenceofthe two orderingsis due to the prox-
s
. imityeffect1,2,3. Experimentallythissituationcanberealized junctionsas a functionofparameterscharacterizingmaterial
t propertiesoftheS,FlayersandtheS/Finterfacetransparency.
a in superconductor/ferromagnet(S/F) hybrid structures. The
m mainmanifestationoftheproximityeffectinS/Fstructuresis Themodelprovidesatooltofitexperimentaldatainexisting
SIFSjunctions.
- the damped oscillatory behavior of the superconductingcor-
d The paper is organized as follows. In the next section
relations in the F layers. Two characteristic lengths of the
n decayandoscillationsare,correspondingly,x andx . Un- we formulate the theoretical model and basic equations. In
o f1 f2 Sec.IIIwesolvenonlinearUsadelequations,applysolutions
usualproximityeffectinS/Flayeredstructuresleadstoanum-
c
forcalculationofcriticalcurrentinSIFSjunctionswithlong
[ berofstrikingphenomenalikenonmonotonicdependenceof
ferromagnetic layer, d x , and fit recent experimental
2 theircriticaltemperatureandoscillationsofcriticalcurrentin data.InSec.IVweperffor≫mnuf1mericalcalculationsforcritical
S/F/S Josephson junctions upon the F layer thickness. Neg-
v
currentinaSIFSjunctionwitharbitrarylengthoftheFlayer.
ative sign of the critical current correspondsto the so-called
5
6 p state. Spontaneousp phase shifts in S/F/S junctionswere InSec.VwenumericallycalculateDOSintheferromagnetic
3 observedexperimentally.4,5,6,7,8,9,10,11,12,13,14,15 interlayer,andthensummarizeresultsinSec.VI.
0 SIFSjunctions,i.e. S/F/Strilayerswithonetransparentin-
.
1 terfaceandonetunnelbarrierbetweenSandFlayers,repre-
1 sentpracticallyinterestingcaseofp junctions.SIFSstructure II. MODELANDBASICEQUATIONS
7 offers the freedom to tune the critical currentdensity over a
0
wide range and at the same time to realize high values of a ThemodelofanS/F/Sjunctionwearegoingtostudyisde-
:
v productofthejunctioncriticalcurrentIc anditsnormalstate pictedinFig.1andconsistsofaferromagneticlayerofthick-
i resistanceR .14,15 Inaddition,Nbbasedtunneljunctionsare ness d and two thick superconducting electrodes along the
X N f
usuallyunderdamped,whichisdesiredformanyapplications. xdirection. Leftandrightsuperconductor/ferromagnetinter-
r SIFS p junctions have been proposed as potential logic ele- faces are characterized by the dimensionless parameters g
a B1
mentsinsuperconductinglogiccircuits.16 SIFSjunctionsare andg ,respectively,whereg =R s /x ,R are
B2 B1,B2 B1,B2 n n B1,B2
alsointerestingfromthefundamentalpointofviewsincethey the resistances of left and right S/F interfaces, respectively,
provide a convenient model system for a comparative study s is the conductivity of the F layer, x = D /2p T, D
n n f c f
between0-p transitionsobservedfromthecriticalcurrentand is the diffusioncoefficientin the ferromagneticmetaland T
c
p
from the density of states (DOS). At the same time, despite is the critical temperatureof the superconductor(we assume
suchaninterest,thereisnocompletetheoryyetofSIFSjunc- h¯ =k =1). We also assume that the S/F interfaces are not
B
tionswhichcouldprovidequantitativepredictionsforcritical magneticallyactive.Wewillconsiderdiffusivelimit,inwhich
currentandDOSinsuchstructures.Allexistingtheoriesdealt theelastic scatteringlengthℓismuchsmallerthanthedecay
onlywith a numberof limiting cases, wheneither linearized characteristiclengthx . Inthispaperweconcentrateonthe
f1
quasiclassicalequationscanbeusedforanalysis17 (e.g. tem- caseofaSIFStunnelJosephsonjunction,wheng 1(tun-
B1
peraturerangenearcriticaltemperature,smalltransparencyof nelbarrier)andg =0(fullytransparentinterface).≫Forcom-
B2
interfaces) or thicknessof the F layer is small1,2,3 compared parison, we also consider two other limiting cases: an SFS
2
g g
B1 B2 and the pair potential D (x) is determined by the self-
consistencyequation
S F S D (x)lnTc =p T (cid:229) 2D (x) F F . (5)
T w >0(cid:18) w − s↑− s↓(cid:19)
TheboundaryconditionsfortheUsadelequationsattheleft
andrightsidesofeachS/Finterfacearegivenbyrelations24
-d /2 0 d /2 x ¶ ¶
f f x g Gˆ Gˆ =x Gˆ Gˆ , (6a)
n f¶ x f s s¶ x s
FIG. 1: Geometry of the considered system. The thickness of the (cid:18) (cid:19)±df/2 (cid:18) (cid:19)±df/2
¶
ftfhaecerreorimigshactghFnae/rStaiccinteitnreitrzeferaldcaeybeiysrtcihshedargfa.Bc1TtehcreoizeterfdfiancbsiyepnagtrB,e2na.ncydothfethteralnesfptaSr/eFnicnyteorf- 2x ngB1(cid:18)Gˆf¶ x¶Gˆf(cid:19)−df/2=(cid:2)Gˆs,Gˆf(cid:3)−df/2, (6b)
2x g Gˆ Gˆ = Gˆ ,Gˆ , (6c)
n B2 f¶ x f f s df/2
(cid:18) (cid:19)df/2
junction(g =g =0)andaSIFISjunction(g ,g 1). (cid:2) (cid:3)
B1 B2 B1 B2≫ whereg =x s /x s ,s istheconductivityoftheSlayerand
Under conditions described above, the calculation of the s n n s s
x = D /2p T.
Josephson current requires solution of the one-dimensional s s c
Usadel equations.19 In the F layer the equations has the To complete the boundary problem we also set boundary
form20,21 condiptionsatx= ¥ ,
±
w
¶ ¶ G ( ¥ )= , (7a)
Df¶ x Gˆf↑(↓)¶ xGˆf↑(↓) s ± |D |2+w 2
(cid:18) (cid:19)
1 p
=(cid:2)(w ±ih)s z+2tms zGˆf↑(↓)s z,Gˆf↑(↓)(cid:3), (1) Fs(−¥ )= |D D|e2−+ij /w22, Fs(+¥ )= |DD |e2ij+/2w 2, (7b)
wherepositivesignaheadofhcorrespondstothespinupstate | | | |
( )andnegativesigntothespindownstate( ),w =2p T(n+ where j is thepsuperconductingphase dpifference between S
1↑)aretheMatsubarafrequencies,histheexc↓hangefieldinthe electrodes.
2
ferromagnetand s z is the Pauli matrix in the Nambu space. InMatsubaratechniqueitisconvenienttoparameterizethe
The parameter tm is the spin-flip scattering time. The influ- Green’sfunctioninthefollowingway,makinguseofthenor-
enceofspin-flipscatteringonvariouspropertiesofS/Fstruc- malizationcondition,Eq.(4),25
tures was considered in a number of papers.13,20,21,22,23,31,32
iWnewchoicnhsicdaesreththeefmerarogmneatgicnesctawttiethrinstgrodnogesunnoiatxcioaulpalneisthoetrsoppiyn, Gˆ = sincoq seq−ic sincqoesiqc . (8)
(cid:18) − (cid:19)
upandspindownelectronpopulations.
TheUsadelequationintheSlayercanbewrittenas19 Solving a system of nonlinear differential equations,
Eqs. (1)-(7), generallycan be fulfilled onlynumerically. We
¶ ¶ presentfullnumericalcalculation in Sec. IV. The analytical
Ds¶ x Gˆs¶ xGˆs = ws z+Dˆ(x),Gˆs , (2) solutioncan be constructedin case of oneS/F bilayer, when
(cid:18) (cid:19) we can set the phase c in Eq. (8) to zero. We can also set
(cid:2) (cid:3)
the phase to zero in case of long S/F/S junction, where the
whereD isthediffusioncoefficientinthesuperconductor.In
eEqqu.a(t2io)nGssˆsin≡suGˆpse↑(r↓c)onandducwtoerloomokitisduebnstcicriaplltys ‘f↑or(↓sp)’inbuepcaaunsde tthheickdneceassyoofftthheefCeroroopmeargpnaeitricwlaavyeerfudnfc≫tionx fi1n.fiIrnstthaaptpcroasxei-,
mation occurs independently near each interface. Therefore
spindownelectronstates.
wecanconsiderthebehavioroftheanomalousGreen’sfunc-
InEqs.(1)-(2)weusefollowingmatrixnotations(weomit
tionneareachS/Finterface,assumingthattheferromagnetic
‘f’,‘s’and‘ ( )’subscripts)
↑ ↓ interlayeris infinite. Thisanalyticalcalculationfor anS/F/S
G F 0 D (x) trilayerwithlongferromagneticinterlayerisperformedinthe
Gˆ(x,w )= F G , Dˆ(x)= D (x) 0 , (3) nextsection.
(cid:18) ∗ − (cid:19) (cid:18) ∗ (cid:19) Thegeneralexpressionforthesupercurrentisgivenby
TrwehhsepeermecatGitvreiaxlnyGd,raFenedanr’eDs(fnxuo)nrcmistiaotlhnaeGnˆdssuaapnteiosrfimcoeasnldothuuecstnGionrrgemepnaal’iisrzfaputoinotcentnicotoinansl-., Js= ip 4Tes nn= ¥+(cid:229) ,¥s = , (cid:18)Ffs ¶¶xFfs −Ffs ¶¶xFfs (cid:19), (9)
− ↑↓
dition, e e
whereF (x,w )=F (x, w )aretheanomalousGreen’s
Gˆ2=1, G2+FF =1, (4) functionfs↑(i↓n)theferromf∗a↑g(↓n)et. −
∗
e
3
III. CRITICALCURRENTOFJUNCTIONSWITHLONG wherex f = Df/handtheboundaryconditionq f(x ¥ )=
→
FERROMAGNETICINTERLAYER 0hasbeenused. InEq.(16)weusethefollowingnotations
p
WeneedtosolvethecompletenonlinearUsadelequations q= 2/h w ih+1/tm, (17a)
±
isnoltuhteiofnermroamyabgenefto,uEnqds.if(1d).ForxSIFaSnjdunwcetiocanns,saentathnealpyhtiacsael e 2=p(1/tm)p(w ±ih+1/tm)−1. (17b)
f f1
≫
ofthe anomalousGreen’sfunctionto zero(see discussionin Here we again adopt convention that positive sign ahead of
Sec.II). h corresponds to the spin up state ( ) and negative sign to
Setting c s = c f = 0 we have the following q - the spin down state ( ). Here and b↑elow we did not write
↓
parameterizationsofthenormalandanomalousGreen’sfunc- spinlabels‘ ( )’explicitlybutimplythemeverywherethey
tions, Eq. (8), G=cosq and F =sinq . In this case we can needed. ↑ ↓
writeEqs.(1)intheFlayeras Fortherightinterface(x=d /2),afirstintegralofEq.(10)
f
leadstoasimilarequation,
D ¶ 2q cosq
2f ¶ xf↑2(↓) =(cid:18)w ±ih+ tmf↑(↓)(cid:19)sinq f↑(↓). (10) x2f ¶q¶ xf =qsinq2f 1−e 2sin2q2f. (18)
r
IntheSlayertheUsadelequation,Eq.(2),maybenowwritten
as FollowingFaureetal.23weintegrateEq.(16),whichgives
D2s¶¶2xq2s =w sinq s−D (x)cosq s. (11) q11−ee 22ssiinn22qq2ff −+ccoossqq2ff =g1exp(cid:18)−2qdf/x2f+x(cid:19). (19)
Theself-consistencyequationintheSlayeracquirestheform − 2 2
q
The integration constant g in Eq. (19) should be deter-
D (x)lnTc =p T (cid:229) 2D (x) sinq sinq . (12) mined from the boundary con1dition at the left S/F interface,
T w >0(cid:18) w − s↑− s↓(cid:19) Ecaqn. (n1e3gble).ctSsimncaellwqe cinontshiederrigthhtehtaunndneslidliemoitf(EgqB.1(≫13b1)),awnde
f
In the case of c s = c f = 0, the boundary conditions, alsoassume,neglectingtheinverseproximityeffect,
Eqs. (6), for the functions q at each S/F interface can be
f,s
D
writtenas q s(−df/2)=arctan|w |. (20)
¶q ¶q
x ng ¶ xf =x s ¶ xs , (13a) ThenEq.(13b)becomes
(cid:18) (cid:19)±df/2 (cid:18) (cid:19)±df/2
¶q ¶q D
x ngB1(cid:18) ¶¶qxf(cid:19)−df/2=sin(cid:0)q f−q s(cid:1)−df/2, (13b) x ngB1(cid:18) ¶ xf(cid:19)−df/2=−G(n), G(n)= w 2|+||D |2. (21)
x ngB2(cid:18) ¶ xf(cid:19)df/2=sin(cid:0)q s−q f(cid:1)df/2. (13c) Fatroxm=E−qds.f/(126a)nadnsdu(b2s1ti)tuwtiengobittainintothEeqb.o(1up9n)dwareyfivnaalulleyogfetq f
Theboundaryconditionsatx= ¥ are G2(n)1 e 2 x 2
f
± D g1= 16gB21 −q2 (cid:18)x n(cid:19) . (22)
q s(±¥ )=arctan|w |. (14) Linearizing Eq. (19), we can now obtain the anomalous
Green’s function in the ferromagnetic layer of the SIF tun-
Intheequationforthesupercurrent,Eq.(9),thesummation
nel junction with infinite F layer thickness. Similar formula
goesoverallMatsubarafrequencies. Itispossibletorewrite for the FS bilayer with a transparentinterface (g =0) was
B2
the sum only overpositive Matsubarafrequenciesdue to the developed by Faure et al.23 [to obtain it one should inte-
symmetryrelation,
grateEq.(18)andthenlinearizetheresultingequation]. The
anomalousGreen’s function at the center of the F layer in a
q (w )=q ( w ). (15)
f(s)↑ f(s)↓ − SIFS junction may be taken as the superposition of the two
decaying functions, taking into account the phase difference
Inwhatfollows,wewilluseonlyw >0inequationscontain-
ineachsuperconductingelectrode,
ingw .
intFegorratlhoeflEefqt.i(n1t0e)rflaecaeds(ttuonnelbarrier at x=−df/2), a first q f = √14−e 2(cid:20)√g1exp(cid:18)−qdf/x2f+x−ij2(cid:19)
x d /2 j
x f ¶q f = qsinq f 1 e 2sin2q f, (16) +√g2exp q −x f +i2 . (23)
2 ¶ x − 2 − 2 (cid:18) f (cid:19)(cid:21)
r
4

= 0
-1 -1
10 SFS 10 = 0.5
Tc Tc
= 1
e/ e/
RN SIFS RN = 1.5
Ic Ic

-4 -4
10 SIFIS 10
-7 -7
10 10
0 2 4 df / n 6 0 2 4 6 df / n 8
FIG. 2: (Color online) The F layer thickness dependence of the FIG.3:(Coloronline)TheFlayerthicknessdependenceofthecriti-
criticalcurrent forSFS(gB1,2=0), SIFS(gB1=102, gB2=0)and calcurrentinaSIFSjunction[modulusoftheEq.(25)]fordifferent
SIFIS(gB1,2=102)junctionsintheabsenceofspin-flipscattering. valuesofa =1/p Tctm,h=3p Tc,T =0.5Tc.
Reddashedlinescorrespondtothemodulusoftheanalyticalresults
(31),(25)and(29)andblacksolidlinescorrespondtotheresultsof
numericalcalculationinSec.IV,h=3p Tc,T =0.5Tc. ThecriticalcurrentinEq.(25)isproportionaltothesmall
exponent exp d /x . The terms neglected in our ap-
f f1
−
proach are of the order of exp 2d /x and they give a
The expression for g2 was obtained in Ref. 23 for the tinysecond-ha(cid:0)rmonicter(cid:1)minthe−currefnt-fp1haserelation.
rigid boundary conditions at the transparent FS interface, (cid:0) (cid:1)
The critical currentequation (25) can be simplified in the
q (d /2)=arctan(D /w )andreads
f f | | limitofvanishingmagneticscattering,tm−1≪p Tc,
(1 e 2)F2(n)
g2= [ (1−De−2)F2(n)+1+1]2, (24a) IcRN = 16ep T (cid:229)¥ G(n)F(n)Fex2p(n(cid:16))−x+df1f1(cid:17)+co1s(cid:16)xdff2(cid:17). (27)
F(n)= p | | . (24b) n=0
w + w 2+ D 2  p 
| | Eq. (25) also simplifies near T and may be written as (for
c
UsingtheabovesolutionsapndEqs.(9),(15)wearriveatsinu- T h)
c
soidalcurrent-phaserelationinaSIFStunnelJosephsonjunc- ≪
tionwiththecriticalcurrent p D 2 d d
f f
16p T (cid:229)¥ G(n)F(n)exp qdf/x f IcRN = 2|eT|c exp(cid:18)−x f1(cid:19)cos(cid:18)x f2(cid:19). (28)
I R = Re − . (25)
c N e "n=0 (1 e 2)F2(cid:0)(n)+1+1(cid:1)# Thedampedoscillatorybehaviorofthecriticalcurrentcanbe
−
clearlyseenfromthisequation. Withincreasingd thejunc-
Here and below we fixppositive sign in the definition tionundergoesthesequenceof0-p transitionswhefnpositive
of q, e 2 in Eqs. (17): q = 2/h w +ih+1/t , e 2 =
m valuesoftheI R productcorrespondtoazerostateandneg-
c N
(1/tm)(w +ih+1/tm)−1. Itisppossibplesincewealreadyper- ativevaluescorrespondtoap state.
formed summation over spin states and have to define now Eq.(28)intheabsenceofspin-flipscatteringcoincideswith
spin-independentvalues. In Eq. (25) and below RN is a full thecorrespondingequation,Eq.(37),fromtheRef.17,taken
resistance of an S/F/S trilayer, which include both interface inthelimitoflongd x incaseofg 1,g =0.
f f1 B1 B2
resistancesofleftandrightinterfacesandtheresistanceofthe Using the same app≫roach we can obtain≫the equation for
ferromagneticinterlayer. IncaseofSIFSandSIFISjunctions thecriticalcurrentinaSIFISstructurewithtwostrongtunnel
theFlayerresistancecanbeneglectedcomparedtolargere- barriers between the ferromagnet and both superconducting
sistanceofthetunnelbarrier. layers(g 1),
B1,2
Atthispointwedefinethecharacteristiclengthsofthede- ≫
cayandoscillationsqx/fx1,f2=as1,/x f1+i/x f2, (26a) IcRN = 4peTx nx f gBg1B1+gBg2B2Ren(cid:229)=¥ 0G2(n)exqp(cid:16)−xqfdf(cid:17). (29)
 
This formula coincides with corresponding expression
1 1 w 1 2 w 1
= 1+ + + . (26b) Eq.(39)forthecriticalcurrentinaSIFISstructureinRef.23
x f1,2 x fvuus (cid:18)h htm(cid:19) ±(cid:18)h htm(cid:19) for gB1,2 =gB ≫1 and df ≫x f1. Eq. (29) near Tc may be
t
5

)
A
m
(
-1 I c
10
Tc
/
e
N
R -1
Ic 10
h= Tc
-4
10 h= Tc
h= Tc
h= Tc
h= Tc
-7 -4
10 10
0 2 4 6 df / f 8 3 4 5 6 7 df (nm)
FIG.4: (Coloronline)TheFlayerthicknessdependenceofthecrit- FIG.5: (Coloronline)FittotheexperimentaldatafromRef.14for
icalcurrentinaSIFSjunction[modulusoftheEq.(25)]fordiffer- thecriticalcurrentinaNb/Al O /Ni Cu /Nbjunction.Thefitting
2 3 0.6 0.4
entvaluesofexchangefieldhintheabsenceofspin-flipscattering, parametersare:h/kB=950Kand1/tm=1.6h.
T =0.5Tc.
In Fig. 3 we plot the F layer thickness dependence of the
writtenas(forTc h) criticalcurrentinaSIFSjunctionfordifferentvaluesofspin-
≪
flip scattering time. For stronger spin-flip scattering the pe-
p D 2x f2gB1+gB2 riodofsupercurrentoscillationsincreasesandthepointof0-p
I R = | |
c N 2eTcx n gB1gB2 transitionshiftstotheregionoflargerdf. Thesametendency
d d existsforSFSandSIFISjunctions.23
×cos(Y )exp −x f sin Y −x f , (30) In Fig. 4 we plot the F layer thickness dependence of
(cid:18) f1 (cid:19) (cid:18) f2(cid:19) the critical current in a SIFS junction for differentvalues of
whereY isdefinedbytan(Y )=x /x . Eq.(30)intheab- the exchange field h. We see that for large exchange fields
f2 f1 h p T thecriticalcurrentscaleswiththeferromagneticco-
senceofspin-flipscatteringcoincideswiththecorresponding he≫renceclengthx .
equation,Eq.(35),fromtheRef.17,takeninthelimitoflong f
d x . FromcomparisonwithnumericalresultspresentedinFig.2
f f1
≫ wecanconcludethattheresultsforthecriticalcurrentinSIFS
Wealsoprovidehereequationforthecriticalcurrentinan
junctionspresentedinFigs.3-4givecorrectmagnitudeofthe
SFSjunction[seeRef.23,Eq.(74)],writteninournotations,
I R productford &x /2.
c N f n
As an application of the developedformalism, we present
I R = 64p Tdf Re (cid:229)¥ F2(n)qexp −qdf/x f . (31) in Fig. 5 the theoretical fit of the experimental data for a
c N ex f "n=0[ (1−e 2)F2(cid:0)(n)+1+(cid:1)1]2# Nb/Al2O3/Ni0.6Cu0.4/Nb junctions by Weides et al.14 mak-
ing use of Eq. (25). We used following values of parame-
Wecomparecriticalcurrpentdependenciesoverd forSFS ters: R =3.9mW ,D =3.9cm2/s,T =4.2K,14 T =7.2K
f B f c
[Eq.(31)],SIFS[Eq.(25)]andSIFIS[Eq.(29)]structuresin (dampedcriticaltemperatureinNb).Goodagreementwasob-
Fig.2. Eachofabovejunctiontypesundergoesthesequence tainedwiththefollowingparameters: h/k =950K,1/t =
B m
of 0-p transitions with increasing thickness of the F layer. 1.6h (see Fig. 5). These parameters can be compared with
From the figure we see that the transition from 0 to p state parameters obtained by Oboznov et al.13 for similar ferro-
occursinSIFStunneljunctionsatshorterd thaninSFSjunc- magneticmaterial,Ni Cu :h/k =850K,1/t =1.3h.
f 0.53 0.47 B m
tionswithtransparentinterfaces,butatlongerd thaninSIFIS HigherNiconcentrationin the NiCu alloyin the experiment
f
junctionswith twostrongtunnelbarriers. Thistendencycan ofWeidesetal. resultsinhigherexchangefield.
be qualitatively explained by the fact that in structures with InRef.13itwassuggestedthata“dead”layerexistsinthe
barriers(SIFS,SIFIS)partofthep phaseshiftoccursacross ferromagnetneareachS/Finterface,whichdoesnottakepart
the barriers. Therefore a thinner F layer in a SIFS junction inthe “oscillating”superconductivity. Otherauthorsalso in-
comparedtoanSFSoneisneededtoprovidethetotalshiftof cludeinto considerationthe existenceof nonmagneticlayers
p duetotheorderparameteroscillation.Forthesamereason, attheinterfaceoftheferromagnetandthesuperconductoror
0-p transition in a SIFIS junction occurs at a smaller thick- normalmetal.26,27,32 Thicknessofthe“dead”layercannotbe
ness than in a SIFS junction. We note that in Fig. 2 we plot calculated quantitativelyin the frameworkof our model and
bothanalyticalandnumericalcalculatedI (d )dependencies, also can not be directly estimated from the experiment. In
c f
wherenumericalcalculationwasperformedforfullboundary the experimentof Weides et al.14 the rangeof F layerthick-
problemEqs.(1)-(7)[seefurtherdiscussioninSec.IV]. nesseswasrathernarrowandonlythefirst0-p transitionwas
6
) ) ) )
E E E E
N( N( N( N(
1
1.0
2 1.5

0.5
0 0 0.0
0 1 E/ 0 1 E/ 0 1 E/ 0 1 E/
a) b) a) b)

) )
E E 1.05
( (
N N
1.0
) )
E E
1.0 1.05 N( N(

0.9 1.00
0.8
1.00
0 1 E/ 0 1 E/ 0 1 E/ 0 1 E/
c) d) c) d )
FIG.6: DOSonthefreeboundary of theFlayerintheFSbilayer FIG. 7: DOS N(E) on the free boundary of the F layer in the FS
calculatednumericallyintheabsenceofspin-flipscatteringfordif- bilayercalculatednumericallyfora =1/p Tctm=0(solidline),a =
ferentvaluesoftheFlayerthicknessdf:N (E)(dashedline),N (E) 0.5(dashedline)anda =1(dottedline)fordifferentvaluesofthe
(dotted line) and N(E) (solid line), Eex =↑ 3p Tc, T =0.5Tc.↓(a): Flayerthicknessdf,Eex=3p Tc,T =0.5Tc. (a): df/xn=0.4,(b):
df/xn=0.4,(b):df/xn=1,(c):df/xn=1.6,(d):df/xn=2.2. df/xn=1,(c):df/xn=1.6,(d):df/xn=2.2.
observed. Dueto these reasonswe did nottake intoaccount lated I (d ) dependencies in case of SFS, SIFS and SIFIS
c f
theexistenceofanonmagneticlayerinourfit. Thisquestion junctions.Weseethat,asexpected,thenumericalmethodpro-
deservesseparatedetailexperimentalandtheoreticalstudy. videscorrectiononlyforsmalllengthofferromagneticlayer.
We should mention that the above estimates of exchange WenotethatforSFSandSIFSjunctionsanalyticalcurves(31)
field and spin-flip scattering time could be different if we and(25)practicallycoincidewithnumericalresultsinthere-
considermagneticallyactiveS/F interfaces. Itwas shown in gionofthefirst0-p transition.ForaSIFISjunctionthistransi-
Ref.28thattheeffectofspin-dependentboundaryconditions tionoccursatsmallerd ,wheretheassumptionsofthesection
f
on the superconductingproximityeffect in a diffusiveferro- IIIarenotvalid.However,inpresenceofstrongspin-flipscat-
magnetresults in the change of the periodof critical current teringthefirst0-p transitionpeakinaSIFISjunctionshiftsto
oscillations. the region of larger d and Eq. (29) describes the transition
f
accurately.
ThemainresultofthissectionisthatEq.(25)forthecrit-
IV. CRITICALCURRENTOFJUNCTIONSWITH ical current of a SIFS junction can be used as a tool to fit
ARBITRARYLENGTHOFTHEFERROMAGNETIC experimentaldatainSIFSjunctionswithgoodaccuracy.
INTERLAYER
In the previous section we derived the expression for the V. DENSITYOFSTATESOSCILLATIONSINTHE
criticalcurrentofaSIFSjunctionincaseofconsiderablylong FERROMAGNETICINTERLAYER
Flayerthickness,d x . ForarbitraryFlayerthicknessin
f f1
≫
theabsenceofspin-flipscattering,generalboundaryproblem It is known that in a ferromagnetic metal attached to the
(1)-(7)wassolvednumericallyusingtheiterativeprocedure.29 superconductorthequasiparticleDOSatenergiesclosetothe
Starting from trial values of the complex pair potential D (x) Fermi energy has a damped oscillatory behavior.33,34,35 Ex-
andtheGreen’sfunctionsGˆs,f,wesolvetheresultingbound- perimentalevidenceforsuchbehaviorwasprovidedbyKon-
ary problem. After this we recalculate Gˆ and D (x). We tosetal.36 InSIFSjunctionswecancomparetheDOSoscil-
s,f
repeat the iterations until convergencyis reached. The self- lationswiththecriticalcurrentoscillations.
consistencyofcalculationsischeckedbytheconditionofcon- We are interested in the quasiparticle DOS in the F layer
servationofthesupercurrentacrossthejunction. inthevicinityofthetunnelbarrier(x= d /2+0inFig.1).
f
−
In Fig. 2 we compare numerically and analytically calcu- BelowwewillrefertothelocalDOSatthispoint.Forthecase
7

)df Fermienergyincreasesandthedampedexponentialdecayoc-
(N cursfaster.
IncaseoflongFlayer(d x )itisalsopossibletoob-
f f1
0 ≫
-1 tainananalyticalexpressionfortheDOSatthefreeboundary
10
oftheferromagnet,
0
1
0 N (E)=Re[cosq ] 1 Req 2 , (33)
10-3 ↑(↓) b↑(↓) ≈ −2 b↑(↓)
0
whereq is a boundaryvalue of q at x= d /2. It can
beobtainb↑e(d↓)bythemappingmethod,simf ilartot−heofneusedin
-5
10 0 theelectrostaticproblems. WeconsidertheFSbilayerwhere
x [ d /2, d /2] stands for the ferromagnetic metal and
f f
∈ −
x>d /2standsforthesuperconductor;thepointx= d /2
f f
−
-7 correspondstothefreeFlayerboundary. ForinfiniteFlayer
10 0 1 2 3 4 df / n b(dyft→he¥ex)ptohneesnotliuatlitoenrmfoirnqEf↑q(.↓)(2fa3r),frwormitttehneiinnttehrefarceealisengeivrgeny
FIG. 8: (Color online) The F-layer dependence of the function space,
d N(df)intheabsenceofspin-flipscattering,h=3p Tc,T =0.5Tc.
Blacksolidlineisaresultofthenumericalcalculation;bluedashed 4 x d /2
lcirniteicisalccaulcrurelanttefdowraitShItFhSeujusnecotfioEnq..Z(4er1o).aRndedplisntaetsehsodwefisnneodrmfraolmizeIdc ←q−f↑(↓)= 1−h 2√g2exp(cid:18)p −x ff (cid:19), (34)
areindicatedbyredcolor,whilezeroandp statesdefinedfromthe
where p
DOSareindicatedbyblackcolor.
p= 2/h iE ih+1/t , (35a)
R m
− ±
ofstrongtunnelbarrier(gB1 1)leftSlayerandrightFSbi- h 2=p(1/tm)p( iER ih+1/tm)−1, (35b)
≫ − ±
layerinFig.1areuncoupled. Thereforeweneedtocalculate (1 h 2)F2(E)
trhoemDagOnSet.inSotlhveinFgSnubmilaeyreicraalltytEheqsf.r(e1e0)b-o(1u4n)d,awryeosefttthoezfeerro- g2= [ (1 h−2)F2(E)+1+1]2, (35c)
−
theq derivativeatthefreeedgeoftheFSbilayer,x= d /2, D
¶q f/¶ x =0.31 − f F(E)= p | | , ER=E+i0. (35d)
Wfe us−edf/2the self-consistent two step iterative −iER+ |D |2−ER2
(cid:0) (cid:1) q
procedure29,30,31. In the first step we calculate the pair
Here, as above, positive sign ahead of h corresponds to the
potential coordinate dependence D (x) using the self-
spinupstate in Eq.(34) andnegativesignforthespin down
consistency equation in the S layer, Eq. (12). Then, by
state. By using the arrow ‘from right to left’ in ←q− we
proceeding to the analytical continuation in Eqs. (10), (11) f ( )
wanttostressthatthissolutionisinducedintheferrom↑↓agnet
overthequasiparticleenergyiw E+i0andusingtheD (x)
→ fromtherightFSinterface.
dependenceobtainedinthepreviousstep,wefindtheGreen’s
In the case of finite ferromagnetlengththe boundarycon-
functions by repeating the iterations until convergency is
ditionsatthefreeFlayerboundary,x= d /2,become
reached. We definethefullDOSN(E)andthespinresolved − f
DOSN (E),normalizedtotheDOSinthenormalstate,as
( ) ¶q
↑↓ N(E)= N↑(E)+N↓(E) /2, (32a) q f↑(↓)(−df/2)=q b↑(↓), (cid:18) ¶f↑x(↓)(cid:19)−df/2=0. (36)
N↑(↓)(E)=R(cid:2)e cosq ↑(↓)(iw (cid:3)→E+i0) . (32b) To ensurethese conditionswe add anotherexponentialsolu-
Thenumericallyobt(cid:2)ainedenergydependenc(cid:3)iesoftheDOS tion,
atthefreeFboundaryoftheFSbilayerarepresentedinFigs.6
4 3d /2+x
adnifdfe7re.nFtidgf..6AdtesmmoalnlsdtrfatfeuslltDheODSOtuSrnesnteorgzyerdoeipnesniddeenacmeifnoi-r −→q f↑(↓)= 1−h 2√g2exp(cid:18)−p fx f (cid:19), (37)
gap,whichvanisheswiththeincreaseofd . ThentheDOSat
f resulting from tphe mirror image of the F layer with respect
the Fermi energy N(0) rapidly increases to the values larger
to the point x = d /2. At x = d /2 both exponential
thanunityandwithfurtherincreaseofdf itoscillatesaround termsare equalto−eacfh otherand th−e fifnal solution, q =
unitywhileit’sabsolutevalueexponentiallyapproachesunity b ( )
↑↓
(esneeergaylsodeFpiegn.d8e).ncInieFsiNg.(6Ew)eaanldsoNpl(oEt)t.heFsigp.in7redseomlvoendstDraOteSs s←q−olfu↑t(i↓o)n(−fodrfi/n2fi)n+ite−→qfefr↑r(o↓)m(−agdnfe/t2ic),laisytewroattitmhiessploairngtearnthdarneathdes
fullDOS energydepen↑dencefor d↓ifferentvaluesof spin-flip
scatteringtime. For strongerspin-flip scatteringthe minigap q = 8F(E) exp pdf . (38)
closesatsmallerdf,theperiodoftheDOSoscillationsatthe b↑(↓) (1−h 2)F2(E)+1+1 (cid:18)− x f(cid:19)
p
8
ThisequationcoincideswiththeresultobtainedinRef.32by VI. CONCLUSION
directintegrationoftheUsadelequation.
In Fig. 8 we plot analytically and numerically calculated
function We have developeda quantitative model, which describes
the oscillations of the critical current as a function of the F
d N(df)= 1 N0 , N0=N(E=0), (39) layer thickness in a SIFS tunnel junctions with thick ferro-
| − |
magneticinterlayer,d x ,inthedirtylimit. Wejustified
f f1
togetherwiththeI (d )dependenceforaSIFSjunction. We ≫
c f thismodelbynumericalcalculationsin generalcase ofarbi-
seethatthepointof0-p transitionontheI (d )plotdoesnot
c f traryd : for allvaluesof parameterscharacterizingmaterial
coincidewiththefirstminimumofd N(d )correspondingto f
f propertiesof the ferromagneticmetal numericaland analyti-
signchangeof1 N .Thisdifferencecanbequalitativelyex-
0 calresultscoincideinphysicallyimportantregionofthefirst
plainedasfollow−s. Thetransitionfrom0top stateinajunc- 0-p transition. Thusthederivedanalyticalexpressionforthe
tion,seenassignchangeofI (d ),istheresultofinterference
c f criticalcurrentcanbeusedasatooltofitexperimentaldatain
of solutions for q originating from two S electrodes. 0-p
f varioustypesofSIFSjunctions.Wehavediscussedthedetails
transitioninI (d )occursapproximatelyatsuchthicknessd
c f f of the dampedoscillatory behaviorof the critical currentfor
whentheboundaryvalueofq inEq.(23)atx= d /2be-
f f differentvaluesoftheFlayerparameters.
comesnegative,i.e. whenq acquiresthe phases−hiftp . On
f
theotherhand,signchangeof1 N0occursatsuchdf when WealsostudiedthesuperconductingDOSinducedinafer-
theboundaryvalueq binEq.(38)−becomesanimaginarynum- romagnetbytheproximityeffect.Weshowedthattheoscilla-
ber, i.e. when q f acquiresthe phase shift p /2. It occuresat tionpatternofDOSattheFermienergyintheferromagnet(at
smaller df comparedto 0-p transition in the criticalcurrent. locationofthetunneljunction)doesnotcoincidewiththatof
Corresponding 0 and p states defined from Ic and from the thecriticalcurrentinaSIFSjunctionandit’speriodisapprox-
DOSareindicatedinFig.8. imatelytwicesmaller. ThereforetheDOSoscillationsdonot
ItisalsoseenfromFig.8thattheDOSoscillationshavethe reflect the 0-p transition in I (d ). We calculated the quasi-
c f
periodapproximatelytwice smaller than those of the critical particleDOSintheFlayerintheclosevicinityofthetunnel
current.Thisfactiseasytoseefromtheanalyticalexpression barrierwhichcanbeusedtoobtaincurrent-voltagecharacter-
ford N(df). UsingEqs.(32)-(39)weobtain isticsforaSIFSjunction.Thesecalculationswillbepresented
elsewhere.
1 2d
d N(d )=32 Re exp p f , (40)
f (cid:12) (cid:20)( 2 h 2+1)2 (cid:18)− 0 x f (cid:19)(cid:21)(cid:12) Finally,we usedourresultsto fitrecentexperimentaldata
(cid:12) − 0 (cid:12) forSIFStunneljunctionsandextractedimportantparameters
whereh =h (E(cid:12)(cid:12) =0)qandp =p(E=0)inEqs.(35a)(cid:12)(cid:12)-(35b). oftheferromagneticinterlayer.
0 0
At vanishing magnetic scattering, t 1 p T , this equation
m− ≪ C
canbesimplified,
32 2d 2d
d N(d )= exp f cos f , (41)
f 3+2√2(cid:12) (cid:18)−x f1(cid:19) (cid:18)x f2(cid:19)(cid:12)
(cid:12) (cid:12)
wherecharacteristicleng(cid:12)(cid:12)thsofdecayandoscillation(cid:12)(cid:12)sx f1,2are Acknowledgments
given by Eq. (26b) with the substitution iw E+i0. This
→
equationcanbecomparedwithEq.(27). Weseethatthepe-
riod of the DOS oscillations is approximately twice smaller TheauthorsthankW.Belzig,E.V.Bezuglyi,A.I.Buzdin,
thantheperiodofthecriticalcurrentoscillationsandtheex- T. Champel, S. Kawabata and F. Pistolesi for useful discus-
ponentialdecayis approximatelytwice fasterthan the decay sions. ThisworkwassupportedbyNanoNedprogramunder
ofthecriticalcurrent. ProjectNo. TCS7029andRFBRprojectN08-02-90012.
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