Proximity effect in ferromagnet/superconductor hybrids: from diffusive to ballistic motion

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by  Jacob Linder

Proximityeffect inferromagnet/superconductor hybrids: from diffusive to ballisticmotion
Jacob Linder,1 Malek Zareyan,2 and Asle Sudbø1
1Department ofPhysics, NorwegianUniversityofScienceandTechnology, N-7491Trondheim, Norway
2Institute for Advanced Studies in Basic Sciences, 45195-1159, Zanjan, Iran
(Dated:ReceivedJanuary21,2009)
Wepresentananalyticalstudyoftheproximityeffectinferromagnet/superconductor(F/S)heterostructures,
allowingforanarbitrarymagneticexchangeenergyaswellasarbitraryimpurityandspin-flipscatteringrates
9 within a quasiclassical approach. While previous studies mainly have focused on the clean or dirty limits,
0 our resultsgrant access to the regimeof intermediate impurity concentrations, thus allowing us to probe the
0 crossoverfromthecleantodirtylimit.Wefindthatinthecrossoverregime,allpossiblesymmetrycorrelations
2 oftheproximity-induced anomalousGreen’sfunctionareinducedintheferromagnet. Wealsopointoutthat
thelocaldensityofstatesoscillatesspatially,notonlyforanF/Sbilayer,butalsoforanormal/superconductor
n
(N/S)bilayerinthediffusivelimit, afactwhichappearstohavegone unnoticed intheliterature. Withinthe
a
J weak-proximity effect regime, we present compact analytical expressions valid for arbitrary exchange fields
andimpurityscatteringratesfori)thelocaldensityofstatesinanF/Sbilayer,ii)theJosephsoncurrentinan
1
S/F/Sjunction,andiii)thecriticaltemperatureinanF/S/Fmultilayer. Forallcases,westudyinparticularthe
2
crossoverregimebetweendiffusiveandballisticmotion. Ourresultsmaybeusefulforanalyzingexperimental

dataincaseswhenthedirtylimitisnotfullyreached,thusinvalidatingtheuseoftheUsadelequation.
]
n
o PACSnumbers:74.25.Fy,74.45.+c,74.50.+r,74.62.-c
c
-
r
p I. INTRODUCTION space, while it may still retain a complicated spin-structure.
u From an experimental point of view, the diffusive regime is
s certainlyrelevant, butthere are neverthelesssome complica-
. The interest in ferromagnet/superconductor (F/S) het-
at erostructures has increased much during the last decade1,2,3. tions.Onepointbearsuponthetheoreticalframeworkusedto
studythe physicsin the diffusiveregime. The quasiclassical
m This may probably be attributed to advances in experimen-
Usadel16 equationiswidelyemployedtostudytheproximity
talfabrication/depositiontechniquesaswellasintriguingthe-
-
d oretical predictions. The main hope is that future devices effectinF/Sheterostructures,andisvalidundertwomainas-
n sumptions. Firstly,thattheFermienergyismuchlargerthan
and applications will rely on manipulation of not only the
o anyotherenergyscaleandtheessentialphysicsisgovernedby
electron charge but also its spin. Based on this idea, a
c fermionsatFermilevel,andsecondly,thattheinverseimpu-
[ new research area known as superspintronics has emerged,
aiming at utilization of charge and spin transport in ferro- rityscatteringrateismuchlargerthananyotherenergyscale

1 exceptfortheFermienergy. Forstrongferromagnetssuchas
magnet/superconductor heterostructures. For instance, sev-
v CoorNi,thesecondconditionmaybeviolated. Inthatcase,
eral authors have investigated the possibility of dissipation-
3 one must revert to the more general Eilenberger17 equation,
less currents of spin and charge in magnetically ordered
6
3 superconductors4,5,6,7,8,9,10,11. Alargenumberofotherstudies whichisonlysubjecttothefirstcondition.
3 relatedtospindegreesoffreedominsuperconductingsystems TheEilenbergerequationismorecomplicatedtosolveana-
. hasalsoappearedintheliterature12,13,14,15. lyticallythantheUsadelequation,althoughsomespeciallim-
1
0 Aconsiderableamountofattentionhasbeendevotedtothe its permit fairly simple analytical expressions. Let h denote
9 arguably most simple experimental laboratory where the in- the exchange-energy of the ferromagnet while τimp denotes
0 terplay between ferromagnetism and superconductivity may the inverse impurity scattering rate. The Usadel equation is
v: be studied, namely a F/S bilayer. The two long-range or- then obtained from the Eilenberger equation by demanding
i der phenomena mix close to the interface, giving rise to in- hτimp 1,whilethecaseofastrongandcleanferromagnet
X terestingeffectsbothfromabasicphysicsperspectiveandin is obta≪inedin the limit hτimp 1. We assume thath ∆
r terms of potentialapplications. These effectsinclude induc- is fulfilled. In Ref.18, some a≫spects of the DOS in F/S≫het-
a
tionofunusualsuperconductingsymmetrycorrelationsanda erostructureswere consideredto leadingorderin the param-
highly non-monotonicbehaviour of various physical quanti- eter(hτimp)−1, correspondingtoastrongferromagnetwhich
ties on the size of the system. The latter is a result of the fallsoutsidetherangeofapplicabilityoftheUsadelequation.
non-uniformsuperconductingcorrelationsthatareinducedin InRef.19,theJosephsoncurrentinanS/F/Sstructurewasalso
theferromagneticlayerbymeansoftheproximityeffect. investigatedforthecaseofastrongferromagnet,hτimp 1.
≫
SomeauthorshavealsoconsideredF/Sheterostructureswhere
AsanaturalextensionoftheF/Sbilayer,therehasalsobeen
theimpurityscatteringratewasdisregardedorassumedtobe
much focus on S/F/S systems and F/S/F systems, where the
small,correspondingtotheballisticregime.20,21,22,23,24,25,26,27
influenceofferromagnetismontheJosephsoncurrentandthe
criticaltemperaturehasbeenstudied,respectively. Thelarge Althoughtheagreementbetweentheoryandexperimentin
majority of works related to these systems assumed that the thisresearchareahasproventobesatisfactoryinmanycases,
diffusive limit was reached. In this case, elastic scattering therearestilldiscrepanciestobeaccountedfor. Forinstance,
on impurities rendersthe Green’s function to be isotropic in the Usadel equation has failed to account quantitatively for
2
the critical temperature in F/S/F spin-valves. Furthermore, II. THEORETICALFORMULATION
anomalousfeaturesintheDOSforaverythinF/Sbilayerthat
could not be accounted for even qualitatively, were reported TheEilenbergerequationreads17
inRef.28. Moreover,theUsadelequationapproachfailsfrom
thestartwhenaddressingsystemswithstrongferromagnets. ıv gˆ+[ερˆ +Mˆ Vˆ Sˆ +∆ˆ,gˆ]=0, (1)
F 3 imp flip
·∇ − −
Allofthispointstotheneedoftakingtheroleofimpurity
scatteringmoreseriously. Inthispaper,weaimatdoingpre- where gˆ gˆR(R,ε,pF) is the retarded part of the Green’s
≡
ciselysobysolvingtheEilenbergerequationanalyticallyand function. Here,εisthequasiparticleenergy,Risthecenter-
studyingthecrossoverregimebetweenballisticanddiffusive of-masscoordinate,andpF (vF)istheFermimomentum(ve-
motion(seeFig. 1). Toillustratehowvariousphysicalquan- locity)vector.Theself-energiesthatenterEq.(1)arethemag-
tities behave in this crossover regime, we study i) the local neticexchangeenergyMˆ =hdiag τ ,τ ,theimpurityscat-
3 3
{ }
densityofstatesinanF/Sbilayer,ii)theJosephsoncurrentin tering Vˆ = [ı/(2τ )] gˆ , the (uniaxial) spin-flip scat-
imp imp
anS/F/Sjunction,andiii)thecriticaltemperatureinanF/S/F tering Sˆ = −[ı/(2τ )]ρˆh igˆ ρˆ , and the superconducting
flip flip 3 3
multilayer for arbitrary valuesof h and τ (within the quasi- − h i
orderparameter
classicalapproach).Ineachcase,wepresentcompactanalyt-
ical formula to facilitate comparison to experimentaldata in 0 ıτ ∆
∆ˆ = 2 .
caseswherethediffusivelimitmaynotbefullywarrantedor ıτ ∆∗ 0
2
wherestrongferromagnetsareinvolved. (cid:18) (cid:19)
All matrices used above(ρˆ,τ ) are defined in the Appendix
i i
Ballistic regime [Eq. (41)]. Thebrackets ... denoteanangularaverageover
h i
the Fermi surface. Also, h is the exchange splitting while
τ isthe scatteringtime associatedwithimpurity(spin-
imp(flip)
Bulks-wave flip)scattering.WemayconvenientlyrewriteEq. (1)as:
superconductor
ı
Ferromagneticlayer ıv g +[(ε+σh)τ +σ∆+ g
F σ 3 σ
·∇ 2τ h i
imp
Intermediate regime ı
+ τ g τ ,g ]=0, σ = , = 1 (2)
3 σ 3 σ
2τ h i ↑ ↓ ±
sf
Impurities wherethesuperconductingorderparametermatrix∆reads
0 ∆
Electronicmotion ∆= ∆∗ 0 , ∆=∆0eıχ, (3)
(cid:18)− (cid:19)
Diffusive regime
upon letting χ denote the phase corresponding to the glob-
allybrokenU(1)symmetryinthesuperconductingstate. The
brackets ... denote angular averaging over the Fermi sur-
face. WehempiloytheRicattiparametrization29oftheGreen’s
function:
x=0 x=d gσ =Nσ 1−2baσσbσ 12+aaσσbσ , Nσ =(1+aσbσ)−1.
(cid:18) − (cid:19)
(4)
FIG.1: (coloronline)Overviewofthesuperconductor/ferromagnet Here, a and b are two unknown functions used to
σ σ
heterostructure we will study in this paper. We take into account parametrize the Green’s functions. They will be determined
an arbitrary strength of the exchange field as well as an arbitrary by solvingthe Eilenbergerequationwith appropriatebound-
rateofnon-magneticandmagneticscatteringwithinaquasiclassical aryconditions. A generaltreatmentoftheEilenbergerequa-
approach.
tioncallsforanumericalsolution.Inthecaseofaweakprox-
imity effect, however, the Eilenberger equation may be lin-
Thispaperisorganizedasfollows. InSec. II,weestablish earized in the anomalouspartof the Green’s functionwhich
thetheoreticalframeworkwhichisemployedinthiswork. In permits an analytical approach. The assumption of a weak
Sec. III, we presentour main results with belongingdiscus- proximity effect corresponds mathematically to a scenario
sion: theDOSofan F/SbilayerinSec. IIIB, theJosephson where higher order terms of a ,b are disregarded in the
σ σ
{ }
current in an S/F/S multilayer in Sec. IIIC, and finally the Eilenbergerequation,i.e. oneassumesthat a 1, b
σ σ
| |≪ | |≪
criticaltemperatureinanF/S/FmultilayerinSec. IIID. Eqs. 1. In an experimental situation, a weak proximity effect in
(25),(32),and(37)arethemainanalyticalresultsofthiswork. F/Sheterostructuresmaybeexpectedwheneverthetunneling
WeconcludeinSec.IV. Below,wewilluseboldfacenotation limitisreachedandthenumberofconductingchannelsatthe
forvectors,...for2 2matrices,and.ˆ..for4 4matrices. interfaceislow. Also,assumingasuperconductingreservoir,
× ×
The reader may consult the Appendixfor a definition of the the proximity effect becomes weaker in magnitude upon in-
generalizedPauli-matricesweemployinthispaper. creasingthethicknessoftheferromagneticlayer.
3
Thespatialdepletionofthesuperconductingorderparam- bergerequationsintheferromagneticregiontaketheform:
eterneartheS/Finterfacewillbedisregarded. Thisisanex-
ı
cellentapproximationinthecorrespondinglow-transparency αıv ∂ a +2a (ε+σh)+ (aα a−α)
regime,whichwillbeconsideredthroughoutthispaperexcept F x σ σ 2τimp σ − σ
ı
forinSec. IIID,wherethisissueisdiscussedfurther. Atthe + (3aα+a−α)=0
S/F interface(x = 0) we use Zaitsev’sboundaryconditions. 2τsf σ σ
Define the symmetric and antisymmetric part of the Green’s αıv ∂ b 2b (ε+σh) ı (bα b−α)
functionas F x σ− σ − 2τimp σ − σ
ı
(3bα+b−α)=0, (8)
= 1(g ++g −), = 1(g + g −), (5) − 2τsf σ σ
σ,i σ,i σ,i σ,i σ,i σ,i
S 2 A 2 −
whereα = denotesright-andleft-goingquasiparticles,re-
±
spectively.Itisnecessarytotakeintoaccountthedirectionof
where the superscript on the Green’s function denotes the quasiparticles at Fermi level due to the term v gˆ in
right/left-go±ing quasiparticle excitations and the subscript i Eq. (1). Thus, σ denotesthe spin directionwhile αF d·e∇notes
denotes the ferromagnetic or superconducting region. The the direction of motion in aα and likewise for bα. The im-
σ σ
first of Zaitsev’s boundary conditions30 demands continuity purity and spin-flip scattering self-energies enter Eq. (8) by
of the antisymmetric part Aσ,i of the Green’s function. The means of the matrices Vˆimp and Sˆflip in Eq. (1), which both
secondonerelatestheGreen’sfunctionsintheferromagnetic dependontheFermi-surfaceaveragedGreen’sfunction. For
andsuperconductingregionstotheinterfacetransparency.We aweakproximityeffect,wehave
obtain
1 a++a−
g = σ σ . (9)
h σi b++b− 1
[ (1 2)+ T ( )2] (cid:18) σ σ − (cid:19)
σ,F σ,F σ,S σ,F
A R −A 4 S −S
Forabulkferromagnet,thesolutionisa± =b± =0.
σ σ
= T [ , ] , (6) InRef.18,theDOSinaS/Fbilayerwasstudiedbyneglect-
σ,F σ,S −
4 S S
inbbothspin-flipscattering(τ )andthecouplingterm
sf
→∞
betweentheright-andleft-goingexcitationsinEq.(8).Inthis
where and arethereflectionandtransmissioncoefficients case,onefindsthatEq. (8)reducesto
R T
satisfying + =1,and[...] denotesacommutator.High
−
and low trRanspTarency interfaces correspond to 1 and ıv ∂ a±+[2(ε+σh)+ ı ]a± =0,
1, respectively. Although Eq. (6) is exprTess≃ed rather ± F x σ 2τimp σ
T ≪
compactly, a general solution for arbitrary and is very ı
hard to obtain. In the experimentally relevaTnt situaRtion, one ±ıvF∂xb±σ −[2(ε+σh)+ 2τ ]b±σ =0. (10)
imp
mayassumethat . Foralow-transparencybarrierand
aweakproximityTeff≪ectR,Eq. (6)simplifiesgreatlyto The decaying solution for x of the above equations
→ ∞
reads
Aσ,F|x=0 =γ[Sσ,F,Sσ,S]−|x=0, (7) a+σ =kaσexp[−κσx/l], b−σ =kbσexp[−κx/l],
κ =1 2ı(ε+σh)τ , l =v τ . (11)
σ imp F imp
−
where γ = /(4 ) is a measure of the barrier trans-
T R while a− = b+ = 0. Above, k and k are constants to
parency. At the end of the ferromagnetic layer, we demand σ σ aσ bσ
bedeterminedfromtheboundaryconditionatx =0,andthe
=0.
Aσ,F|x=d structureoftheGreen’sfunctionbecomes
Weconsiderhereaneffectiveone-dimensionalcalculation,
which should provide sound results due to the isotropic na- 1 a+ 0 a+
= σ , = σ . (12)
tureoftheferromagneticandsuperconductingorderparame- Sσ,F b− 1 Aσ,F b− 0
(cid:18) σ − (cid:19) (cid:18)− σ (cid:19)
ters. Wedonotexpectanyqualitativedifferencesfromatwo-
This shows how the decay length of the proximity-induced
dimensionalorthree-dimensionalmodel,sincethesupercon-
anomalousGreen’sfunctionintheferromagnetisgovernedby
ducting gap and the magnetic exchange field do not depend
onthequasiparticlemomenta,andsincetherearenosurface- themeanfreepathl,andthatitisindependentoftheexchange
bound states31 at the interfaces of the systems we consider. fieldinthismainapproximation. Wenowpresentamorerig-
oroussolutionbyfullytakingintoaccountthecoupling-term
Thus,itshouldbepossibletocapturetheessentialphysicsby
inEqs. (8). Tosolvethisproblem,wenotethatEqs. (8)may
studyingan effectiveone-dimensionalmodel, which permits
bewrittenasamatrixdifferentialequation:
us to proceed analytically. This point of view is supported
by the fact that, as seen later in this work, we reproduce in
∂ a =M a , a =[a+a−]T,
limitingcasespreviousresultsobtainedintheliteraturewhich x σ aσ σ σ σ σ
employedatwo-dimensionalcalculation. 1 r g
M = σ , (13)
Undertheassumptionofaweakproximityeffect,theEilen- aσ vF (cid:18)−g −rσ(cid:19)
4
where T denotes matrix transpose and we have defined the and p± = p . Note that in the diffusive limit where
σ 1,2σ
auxiliaryquantities g h,ε,∆ ,g , onewouldexpectthatthe distinction
imp 0 sf
≫ { }
betweenright-goingandleft-goingparticlesisremoved,such
rσ =2ı(ε+σh)−(gimp+3gsf)/2, thatgσ,F+ =gσ,F−. Thisiseasilyshownbyexploiting
g =(g g )/2, g τ−1 . (14)
imp− sf imp(sf) ≡ imp(sf)
Diagonalizing M according to D = P −1M P , we ob- lim (vFλσ+rσ)= gimp/2, (20)
tainthetrivialsetσofdecoupleddiffσerentialσequatσionσs gimp≫{h,ε,∆0,gsf} −
∂x˜aσ =Dσ˜aσ, ˜aσ =Pσ−1aσ. (15) as seen from the previous equations. We also want to com-
paretheresultsforg h,ε,∆ ,g withthoseobtained
Fromtheabove,wefindthat imp ≫{ 0 sf}
whenusingthelinearizedUsadelequation.TheUsadelequa-
a˜± =C± e±λσx, λ =v−1 r2 g2, (16) tioninadiffusiveferromagnetthenreads
σ a,σ σ F σ −
whilethediagonalizationmatrixP readps
σ D∂2f +2ı(ε+ıg h)f =0, (21)
x ± sf± ±
p p
P = 1σ 2σ , G =g/(v λ +r ),
σ p p σ F σ σ
(cid:18) 2σ 1σ(cid:19) wheref± =ft fsandftistheodd-frequencytripletanoma-
±
p1σ =Nσ, p2σ = NσGσ, Nσ =(1+ Gσ 2)−1/2. lous Green’s function while fs is the even-frequencysinglet
− | | anomalousGreen’sfunction(bothareisotropicinmomentum
(17)
space). Weobtainthattheonlyphysicallyacceptable(decay-
Inthesuperconductingregion,weemploythebulksolution ingforx )solutionis
→∞
under the assumption that the interface transparency is low
andthattheferromagneticlayerismuchmoredisorderedthan
f =f eık+xifε>0, f =f e−ık−x ifε<0,
thesuperconductor1.Inthismainapproximation,wemayem- + 0 − 0
ploythebulksolutionoftheGreen’sfunctioninthesupercon- k = 2ı(ε+ıg h)/D, (22)
± sf
±
ductor:
p
g± = c(θ) σs(θ) , where f0 is a constant to be determined from the boundary
σ σs(θ) c(θ) conditions. Above, D is the diffusion constant. For consis-
(cid:18)− − (cid:19)
tency,weshouldbeabletoobtainthesamedecayingsolution
with the definitions c(θ) = cosh(θ), s(θ) = sinh(θ), θ =
fromEq. (19)wheng h,ε,∆ ,g . Focusingonthe
imp 0 sf
atanh(∆/ε). OncetheexpressionfortheGreen’sfunctionin ≫ { }
wavevector,weseethatinthislimit:
theferromagnethasbeenobtained,onemaycalculatevarious
physicalquantitiesofinterest. By approximating τ
σ,F 3
S ≃
in Eq. (7) in accordance with a weak proximity effect, we λ v−1 2ı(ε+σh)g +2g g
obtainforthecasewheretheimpurity-scatteringcouplingbe- σ → F − imp imp sf
q
tweentheRicatti-equationsisignored: = 2ı(ε+σh+ıgsf)/D, (23)
−
g± =τ +2γσs(θ)exp( κ x/l)(τ ıτ ), (18) p
σ,F 3 − σ 1± 2 where D = v2τ is the diffusion constant in one dimen-
F imp
which is precisely the result of Ref.18 for two semi-infinite sion (in three dimensions, D = vF2τimp/3). Eq. (23) is then
superconductingand ferromagnetic layers in contact. When consistentwiththeformofEq. (22).
thecouplingisproperlytakenintoaccount,inadditiontothe WithacompletedescriptionofthebehaviouroftheGreen’s
vacuumboundaryconditionatx=d,wefindthat functionin the ferromagneticregion, we now investigatethe
influenceoftheproximityeffectonthelocaldensityofstates
1 2a±
g± = σ , upondefining (LDOS),and also studythe singletand tripletsuperconduct-
σ,F 2b± 1
(cid:18) σ − (cid:19) ingorderparametersinducedintheferromagnet.Thenormal-
ized LDOS as obtained from the solution of the Eilenberger
a± =p±C (λ )+p∓C (λ ), equationmaybewrittenas
σ σ 1σ σ σ 2σ σ
b± =p±C ( λ )+p∓C ( λ ),
σ σ 1σ − σ σ 2σ − σ 1
N(ε,x)= Re 1+4a (ε,x)b (ε,x) (24)
σ σ
2 h { }i
2γσs(θ)eλσx eλσd Xσ
C = 1 ,
1σ p+ p− − 2sinh(λ d)
σ σ σ
− h i foraweakproximityeffect. Inthenormalstate, thenormal-
γσs(θ)eλσ(d−x)
C = , (19) izedDOSisN =1. Insertingtheexpressionsfora± andb±
2σ −(p+ p−)sinh(λ d) 0 σ σ
σ σ σ intotheaboveequationyields
−
5
2γ2s2(θ) cosh(2λ x)
N(x,ε)=1 Re 1+G2 2G 2sinh(λ d 2λ x)+ σ (25)
− ( sinh(λσd)(1+Gσ)2 ×" σ− σ σ − σ sinh(λσd) !#)
σ
X
Eq. (25)isthefirstofourthreemainanalyticalresultsinthis Here, the abbreviationsare explainedin Tab. I. Note that in
work. Within the weak proximity effect regime, it provides the general case of finite h and τ , all possible symmetry
imp
a generalexpressionforthe DOS, taking into accountan ar- components of the anomalous Green’s function are induced
bitrary exchangefield and impurity scattering rate. As seen, in the non-superconducting region. In the case of h = 0,
the correction to the normal state DOS N = 1 is zero for one may confirm from Eq. (19) that a± σa±, where a±
0 σ →
a vanishinginterfacetransparency(γ = 0). While the weak is independent of σ, such that f = f = 0. Physi-
OTE ETO
proximityrestrictiononlyallowsaccesstovariationsfromthe cally,theinductionofothersymmetrycomponentsthanf ,
ESE
normal-state of DOS of around 10%, this seems to be suffi- correspondingto the bulk superconductor,may be explained
cient for the experimentally relevantsituation. For instance, as follows33,34. In a normal metal/superconductor junction,
thedeviationfromthenormal-stateDOSduetothesupercon- thetranslationalsymmetryisbrokenatthe interfaceseparat-
ductingproximityeffectwasoforder1%inRef.32. ing the two regions. This causes even-parityand odd-parity
In order to study the superconducting correlations inside components of the Green’s function to mix near the inter-
the ferromagnetic region, first note that the full structure of face. SincethePauli-principlemustbesatisfiedatalltimes,a
theretardedGreen’sfunctionis change in the parity-symmetryof the Green’s function must
be accompanied by a change in either spin- or frequency-
g f
gˆR = , (26) symmetry.Intheabsenceofanexchangefield,nothingbreaks
f˜ g˜ the spin symmetry, such that only the frequency-symmetry
(cid:18)− − (cid:19)
maybealteredindirectlybythebrokentranslationalsymme-
wherethespin-structurereads
try. However,ifthespin-symmetryisalso brokenbyreplac-
ingthe normalmetalwith a ferromagnet,the spin-symmetry
f f
f = ↑↑ ↑↓ , (27) oftheGreen’sfunctionmayalsobealtered.Theseconsidera-
f f
↓↑ ↓↓ tionsaresummarizedinTab. I. Thepossibilityofabulkodd-
(cid:18) (cid:19)
frequency superconductingstate was discussed in Refs.35,36,
andwe havedefinedfαβ = fαβ(pF,ε,x)andf˜(pF,ε,x) = andtherehasveryrecentlybeenmadesomepredictionscon-
f( p , ε,x)∗. From the Ricatti-parametrization, we may
F cerningcharacteristictransportpropertiesofsuchabulkodd-
− −
define the differentsymmetry-componentsof the anomalous frequencysuperconductingstate.33,37,38,39.
Green’sfunctionsasfollows:
f = σ(a++a−), f = σ(a+ a−),
ESE σ σ OSO σ − σ
σ σ
X X
f = (a+ a−), f = (a++a−). (28)
ETO σ − σ OTE σ σ
σ σ
X X
TABLEI:Proximity-inducedanomalousGreen’sfunctionsinanormalmetalincontactwithaconventionalBCS-superconductor,whichhas
aneven-frequencyspin-singleteven-paritysymmetry. Below,thequasiballisticlimitregimeischaracterizedbyavanishingorsmallvalueof
theimpurityscatteringrate,whilethediffusivelimitischaracterizedbyanimpurityscatteringratewhichdominatesallotherenergyscalesin
theproblem(exceptfortheFermienergy).
Symmetry h=0,quasiballistic h=0,quasiballistic h=0,diffusive h=0,diffusive
6 6
Even-frequencyspin-singleteven-parity(ESE) √ √ √ √
Odd-frequencyspin-singletodd-parity(OSO) √ √ - -
Even-frequencyspin-tripletodd-parity(ETO) √ - - -
Odd-frequencyspin-tripleteven-parity(OTE) √ - √ -
III. RESULTSANDDISCUSSION quasiparticletransport,andhencedependenceofthedifferent
A. AnomalousGreen’sfunctions
ThelinearizedEilenbergerequationsallowustostudythe
direct crossover from the diffusive to the ballistic regime of
6
B. Densityofstates
To demonstrate the applicability of Eq. (25), we study
in particular how the DOS depends on the crossover from
the ballistic (g = 0) to the diffusive limit (g
imp imp
≫
h,ε,∆ .g ). We will fix γ = 0.05 and h/∆ = 15 to
0 sf 0
{ }
model a realistic experiment, corresponding to a weak fer-
romagnetic alloy like Cu Ni or Pd Ni . The setup is
1−x x 1−x x
showninFig. 3. Itis well-knownthattheDOSoscillates in
spaceuponpenetrationdeeperintotheferromagneticregion40
due to the presence of an exchange field, a feature which is
robustbothinthecleananddirtylimit. However,theenergy-
dependence of the DOS in the presence of an arbitrary im-
purity concentration has not received much attention so far.
This is because most works concerned themselves with the
simplifiedUsadelequation(diffusivelimit)ortheEilenberger
equationintheabsenceofimpurities(cleanlimit).
FIG. 2: (color online) Plot of the proximity-induced anomalous
Green’sfunctionsinthemiddleoftheferromagneticregion(x/d=
0.5)usingh/∆=15andε/∆0 =0.5. Superconductor
Ferromagnet
Insulator
symmetrycomponentsonthe impurityscattering. In the ex-
perimental situation, one usually probes the DOS at the F/I
FIG.3:(coloronline)Setupforourstudyofthedensityofstates.
interface x = d, although it in principle is possible to ob-
tain a spatially resolved DOS in the entire ferromagneticre-
InRef.18,correctionstothenormal-stateDOSasinduced
gion by using local scanning tunneling microscopy (STM)-
bytheproximityeffectwerecalculatedundertheassumption
measurements. Letus first focuson x = d and considerthe
thathτ 1.Thiscasecorrespondstoaferromagnetwhere
ballistic limit in which case simple and transparent analyti- imp
≫
theexchangefieldisconsiderablylargerthantheself-energy
calexpressionsmaybeobtainedfromEqs. (28). Inthecase
associatedwiththeimpurityscattering.Thismaydescribeei-
h=0,weobtain
6 therastrongferromagnet(onemuststilldemandh ε )or
F
≪
a weak ferromagnetwith weak impurity scattering. Neither
ofthese casesare possible to treatwith the Usadelequation.
f = [ 2γs(θ)]/sinh(λ d), f =0,
ESE σ OSO In the presentwork, however,we do notimpose any restric-
−
Xσ tionsontheparameterhτimp,whichallowsustostudythefull
f =0, f = [ 2σγs(θ)]/sinh(λ d) (29) crossoverregime. Thismaybeimportantinordertoobtaina
ETO OTE σ
−
σ largerdegreeofconsistencybetweentheoryandexperimental
X
datainthecasewhenthediffusivelimitisnotfullyreached.
In Fig. 4a), we study the energy-resolvedDOS for an in-
Note that for h = 0, λ becomesindependentof σ, leading termediaterangeofimpurityscattering. Asameasureofthe
σ
to f = 0. Atfirstglance, thisappearsto bein contradic- junctionwidth,weusethesuperconductingcoherencelength
OTE
tiontoTab.Isincetheodd-paritycomponentsareabsenteven inthecleanlimitξ = v /∆ . Toisolatetheroleoftheim-
S F 0
in the ballistic limit. However, evaluation of Eqs. (28) for purityscattering,wefixthejunctionwidthatd/ξ =0.5.For
S
x = drevealsthatthese componentsarein generalinduced, a superconductorwith v = 105 m/sand∆ = 1 meV,this
F 0
6
astheyshouldbe. Itisremarkablethattheodd-paritycompo- correspondstod 30nm,whichisexperimentallyrelevant.
≃
nentsvanishexactlyrightattheF/Iinterface. Inthepresence Asseen, theDOSexhibitsaslightlyoscillatingbehaviouras
ofafiniteexchangefieldh = 0,however,theodd-frequency afunctionofenergywhentheimpurityscatteringrateg is
imp
6
componentf survivesatx=d,anditsinfluenceonphys- comparable in magnitude to the superconducting gap. This
OTE
icalquantitiessuchastheDOSmaybedirectlyprobedthere. effectbecomesmoreobviousforwiderjunctionsd/ξ 1,
S
≫
Theseresultssuggestthatinordertoinvestigatetheinfluence and is attributed to bound states appearing in the ferromag-
of the odd-frequencysuperconducting correlations f and netic film. We discuss this in more detail below. As g
ETO imp
f ,onewouldhavetomeasuretheDOSatseveralpositions increases,however,theDOSbecomesfeaturelessforsubgap
OSO
in the ferromagneticregionandnotonlyat the F/I interface. energies although one may still observe an alternating pos-
InFig. 2, we plotthe differentsymmetrycomponentsofthe itive and negative correction to the zero-energy DOS upon
anomalousGreen’sfunctionintheferromagnetandtheirde- increasing g . In Fig. 4b), we plot the spatially-resolved
imp
pendenceontheimpuritylevel. DOSatε = 0forvariousratesoftheimpurityscattering,in-
7
cluding the case when hτ 1. As seen, the oscillations functionintheferromagneticregion. InanS/Njunction,this
imp
∼
ofthezero-energyDOSarereducedwithincreasingimpurity decayismonotonous,andhenceonewouldnotexpecttosee
scattering. We have also investigated the effect of spin-flip any oscillations in the DOS. However, we underlinethat the
scattering for an intermediate value of the impurity concen- impurityscatteringplaysanimportantroleinthisrespect. In
tration. The spin-flip scattering, here taken to be uniaxial, theballisticcaseg 0,theproximityofthesuperconductor
→
is pair-breakingand thus suppresses the proximity-effectin- induces Andreev-bound states with well-defined trajectories
ducedbythesuperconductor.ThisaspectagreeswithRef.41, whichpropagateinthenormalpartofthesystem. Thestatis-
whichfoundthatboththe tripletandsingletcomponentsare ticaldistributionofallpossibletrajectoriesispeakedatgiven
suppressedwithuniaxialand/orisotropicspin-flipscattering. lengths,typicallyattrajectoriescorrespondingtothefirstand
Forothertypesofmagneticscattering,suchasplanarspin-flip second reflection processes at the interface. As a result, the
orspin-orbitscattering,thesingletandtripletcomponentsare DOS in a clean S/N junction acquires oscillations both as a
affectedverydifferently41. functionofenergyandcoordinateinsidethenormalregionas
The oscillations of the DOS in S/F junctions are usually seen in Fig. 5 upon averaging over all possible trajectories.
attributed to the oscillating decay of the Cooper pair wave- ThiseffectisknownasTomasch-oscillations42.
FIG.4: (coloronline)Plotofthea)energy-resolvedDOSatx=dandb)spatially-resolvedDOSatε=0forseveralvaluesoftheimpurity
scatteringrate.Here,theexchangefieldissettoh/∆0 =15andd/ξ =0.5.
However,thereisanotherpointwhichhasappearstohave thattheinducedsuperconductingGreen’sfunctioninthenor-
beenoverlookedin theliterature: namelythatthe spatialos- malregionhasafinitecenter-of-massmomentumq =2ε/v .
F
cillationsoftheDOSinaS/Njunctionatfiniteenergiesdonot This is typically much smaller than the center-of-mass mo-
vanishinthediffusivelimit. Hence,theoscillatingDOSasa mentumacquiredinaferromagnet,q =2h/v ,whichmeans
F
functionofdistancepenetratedintothenon-superconducting that the corresponding oscillation length is much larger, but
region is not a feature pertaining uniquely to F/S junctions, stillpresent.
as have been implied in some works43. To see this, we plot
the spatially-resolved DOS both for a F/S and N/S junction Having stated this, it should be noted that the oscillating
inFig. 6inthediffusiveregime. Thecurvesareobtainedby natureoftheanomalousGreen’sfunctiondoesnotnecessarily
using the frameworkof Ref.41, and thuscorrespondto a full implythatthecriticaltemperaturedependenceortheJoseph-
numericalsolutionoftheUsadelequationwithoutrestricting soncurrentinN/Smultilayersisnon-monotonuous,e.g. dis-
ourselves to the weak proximity effect regime. The oscilla- playing 0-π oscillations, since the energy-dependenceof the
tionsoftheDOSintheN/Scasemaybeunderstoodbynoting Green’s functions is integrated out when obtaining the criti-
caltemperatureorcriticalcurrent.ForanF/Sjunction,onthe
8
FIG.5: (coloronline)Plotofthea)energy-resolvedDOSatx = dandb)spatially-resolvedDOSatε/∆0 = 0.5forseveralvaluesofthe
impurityscatteringrate.Here,theexchangefieldissettozero,correspondingtoanormalmetal,andd/ξ=5.0.
otherhand,the Cooperpairwave-functionmayretainits os- center-of-massmomentumdependsontheexchangefieldh.
cillatingcharacterevenafter theenergy-integrationsince the
FIG.6: (coloronline)PlotofthespatiallyresolvedDOSforadiffusiveN/SandF/Sjunction,respectively. Inbothcases,oscillationsofthe
DOSareseenatfiniteenergies.Wehaveherefixedd/ξ =3.0andτ =0.2,usingthenotationofRef.41(here,ξ=pD/∆0whileτ denotes
thebarriertransparency).
C. Josephsoncurrent inFig.7. Denotingthephaseattheleft(right)superconductor
as +χ ( χ), the total phase differenceis given by ϕ = 2χ.
−
WenowevaluatetheJosephsoncurrentinanSFSjunction
foranarbitraryimpurityconcentration,withasetupasshown
9
hτ 1,whilethemajorityofstudiessofarconsideredex-
imp
≫
clusively the limiting case of diffusivemotion. We here pay
Superconductor
Ferromagnet particularattentiontothecrossoverbetweentheballisticand
Superconductor diffusive sector, which has not been investigated previously.
Tomodelinelasticscattering,weaddasmallimaginarynum-
bertothequasiparticleenergy,ε ε+ıδwhereδ =10−3.
→
FIG.7:(coloronline)SetupforourstudyoftheJosephsoncurrent.
Thecurrentthroughthejunctionisevaluatedby
N S ev
I = F 0 F dεtanh(βε/2)Tr ρˆ e (gˆR gˆA)
J 3 F
4 {h − i}
Z
(30)
under the assumption of equilibrium distribution functions.
Here, S is the effective area of the contact through which
0
thecurrentflows,whileβ = 1/T isinversetemperature. Ex-
perimentally,onemeasuresthecurrentthatflowsthroughthe
junction, correspondingto the x-direction here. We employ
thefollowingboundaryconditions:
=γ[ , Left] ,
Aσ,F|x=0 Sσ,F Sσ,S −|x=0
= γ[ , Right] , (31)
Aσ,F|x=d − Sσ,F Sσ,S −|x=d
andapproximate = τ asintheprevioussection,inac- FIG. 8: (color online) Plot of the critical current as a function of
σ,F 3
cordancewithourSassumptionofaweakproximityeffect.Af- junctionwidthd.WehaveusedT/Tc =0.2.
ter some calculations, we arrive at the following expression
fortheJosephsoncurrent: InFig. 8,weplotthewidth-dependenceofthecriticalcur-
rentfor a temperatureT/T = 0.2. As seen, increasing im-
c
I =4γ2N S ev I sinϕ, withthedefinition purityscatteringsuppressesthemagnitudeofthecurrentand
J F 0 F c
also reduces the oscillation length l . The dependence of
∞ s2(θ)(1 G )tanh(βε/2) osc
I = dε Re − σ . (32) thelatteronimpurityscatteringisshownexplicitlyinFig. 9.
c
Z−∞ σ ( ı(1+Gσ)sinh(λσd) ) UsingtheUsadelequation,itispredictedthattheoscillation
X
lengthof the critical currentin the dirty limitshoulddepend
Thereaderisremindedofthedefinitions
on the impurity scattering rate like hlimp √τimp (for a
∼
discussion of the characteristic decay and oscillation lengths
Gσ =g/( rσ2 −g2+rσ), inthecleananddirtylimit,seeTab.pIinRef.2). We obtaina
rσ =2ı(εp+σh)−(gimp+3gsf)/2, goodfitwiththisinFig.9whengimp ≫∆. Forvaluesofgimp
g =(g g )/2, g τ−1 . (33) comparableto∆, however,theoscillationlengthsaturatesat
imp− sf imp(sf) ≡ imp(sf) a finite value. In the ballistic limit, the oscillation length is
Eq. (32) isthesecondofourthreemainanalyticalresultsin known to depend on the exchange field like 1/h. We have
thiswork. Itisprobablythemostcompactwayofexpressing alsoconfirmedthisforseveralvaluesofhwheng ∆.
imp
∼
theJosephsoncurrentforarbitraryexchangefieldsandimpu- Also,onenotesfromFig.8thatthedecaylengthofthecur-
rityscatteringrateswithinthequasiclassicalframework. Itis rentincreaseswiththe concentrationofimpurities. Itshould
thus suitable both for the case of a weak ferromagnet (such benotedthatthemeasureξ usedasalengthunitinthiscon-
asthealloyCu Ni ),andforstrongferromagnets(likeCo text is independent of the impurity scattering rate, since we
1−x x
orFe)regardlessofwhethertheyarecleanordirty. Inexper- are using ξ = v /∆ . This way, we ensure that the effects
F 0
iments performed with such strong ferromagnets, where the observed are really due to the increased impurity scattering.
exchangefieldmaybeoforder100meV( T ),theUsadel Ifweforinstancehadusedthemeanfreepathl = v τ
c mfp F imp
≫
equationis notvalidat the same time asthe clean limitmay asameasureforthejunctionwidth,thescalewouldhavebeen
notbefullyreached.Inthiscase,onehastouseanexpression differentfor each valueof g in Fig. 8. We also underline
imp
validforthe crossoverregime,whichemphasizesthe impor- thatthedirtylimitconditionisthatξ/l 1,whilethesize
mfp
≫
tanceofEq. (32). dofthesamplemaybeeithersmallerorlargerthanξaslong
Below, we willstudy how impurityscattering affectsboth asthatconditionisfulfilled.
the width- and temperature-dependence of the critical cur- InFig. 10,wepayparticularattentiontothecasehτ =1
imp
rent,aswellasitsbelonging0-π phasediagram. Bergeretet whichisinaccessibleintheUsadelframework.Asseen,noth-
al.19 investigatedthisinthelimitingcasesofhτ 1and ingqualitativelynewshowsupinthed-dependenceortheT-
imp
≪
10
FIG. 9: (color online) Plot of the oscillation length of the critical
current as a function of the impurity scattering strength g . We
imp
haveusedT/Tc =0.2.
dependence of the critical current as compared to the diffu-
sivelimit, althoughthedecayrateisconsiderablylower. We
alsoinvestigatehowthe0-π phasediagramoftheJosephson
junctionisaffectedbyimpurityscattering. Thisismostcon-
venientlyplottedin the d-T plane. In Fig. 11, one observes
severalfeatures. First,itisclearthattheareaoccupiedbythe
0andπphases,respectively,diminisheswithincreasingg ,
imp
in agreementwith the shortenedoscillationlengthof Fig. 9.
Secondly,itisseenthatthermal0-πtransitionsarepractically
speakingimpossibletoobserveforscatteringratessatisfying
g h. As the scattering rate is increased, however, the
imp
≤
thermaltransitionsbecomepossiblewheng h,orequiv-
imp
≫
alentlyhτimp 1.Inthisregime,theUsadelequationisvalid FIG. 10: (color online) Plot of the d-dependence and the T-
≪
andwe obtainconsistencywith previousresults. Atall scat- dependenceofthecriticalcurrentforgimp/∆0 =15,corresponding
teringrates,thewidth-inducedtransitionsarepossible. tohτ =1.
imp
D. Criticaltemperature
predictedeffect. Recently,itwasproposedandinvestigated51
if an asymmetry in the interface transparencies of the F/S/F
Finally, we investigate an F/S/F layers where the critical
junctionscouldberesponsibleforthis,ineffectoneofthein-
temperatureof the superconductoris sensitive to the relative
terfaceswasmuchlesstransparentthantheother.Theauthors
orientationof magnetizationof the two F layers. Thiseffect
ofRef.51concludedthatthiswasnotthecase. Atpresent,the
isusuallydubbedtoaspin-switcheffectintheliterature. Our
singleferromagnetF/S/Fdevicestohavebeenexaminedsofar
setup is shown in Fig. 12. Tagirov44 was the first to point
haveusedstrongferromagnets,whichfallsoutsidethe range
outtheinterestingopportunityto”activate”superconductivity
ofapplicabilityoftheUsadelequation45,46,47. Inlightofthis,
simplybymeansofswitchingthedirectionofthemagnetiza-
itwouldbeinterestingtogobeyondtheusualtreatmentwith
tion in one of the ferromagnetic layers. Since then, a num-
the Usadel equation and solve the more general Eilenberger
ber of works have elaborated on the spin-switch effect both
equationtoinvestigatetheroleoftheimpurityscattering.
experimentally45,46,47 andtheoretically40,48,49,50. Inparticular,
aconvincingnumericalapproachwasdevelopedinRef.49. So Ageneralanalyticalsolutionforarbitraryproximityeffect
far,however,almostalltheoreticalworksfocusedonthedirty and barrier transparencyis hardly achievable, as pointed out
limit, in which the critical temperature may be conveniently previously. Nevertheless, it is reasonable to expect that one
calculatedbyusingtheUsadelequationintheMatsubarafre- maycapturetheessentialphysicsintheweakproximityeffect
quencyrepresentation.Althoughtheobtainedresultscompare regime.InordertocalculatethecriticaltemperaturefortheP
wellqualitativelywith experimentaldata, an unsolvedfactor and AP alignment, we assume that the temperature is close
so far is the discrepancy of two orders in magnitude of the to T , which allows us to write the Green’s function in the
c