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6

Transport in Ferromagnet/Superconductor spin valves

Evan Moen1, ∗ and Oriol T. Valls1, †

1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455(Dated: October 11, 2018)

We consider charge transport properties in realistic, fabricable, Ferromagnet/Superconductor spin valves hav-ing a layered structureF1/N/F2/S , whereF1 andF2 denote the ferromagnets,S the superconductor, andN thenormal metal spacer usually inserted in actual devices. Ourcalculation is fully self-consistent, as required toensure that conservation laws are satisfied. We include the effects of scattering at all the interfaces. We obtainresults for the device conductanceG, as a function of bias voltage, for all values of the angleφ between themagnetizations of theF1 andF2 layers and a range of realistic values for the material and geometrical param-eters in the sample. We discuss, in the context of our resultsfor G, the relative influence of all parameters onthe spin valve properties. We study also the spin current andthe corresponding spin transfer torque inF1/F2/Sstructures.

PACS numbers: 74.45.+c,74.78.Fk,75.75.-c

I. INTRODUCTION

Traditional spin valves1 consist of two ferromagnetic mate-rials where changing the relative orientation of their exchangefields is used to control the transport properties of the het-erostructure. They are based on the well-known and muchcelebrated2 Giant Magnetoresistive (GMR) effect. More re-cently, it has become possible to fabricate spin valves by lay-ering ferromagnetic (F) and superconducting (S ) materials.In this context, spintronic devices of various kinds3–5 havebeen proposed and considered. The fundamental propertiesof such devices arise from theF/S proximity effects6. Theseeffects lead to many new properties. In particular, spin valvedevices, having anF1/F2/S or (more typically in experi-mental situations)F1/N/F2/S , whereN is a normal spacer,have been extensively7–10 studied both theoretically and ex-perimentally. Research on these devices is furthered because,besides their great scientific interest, they have possibleappli-cations towards the creation of non-volatile magnetic memoryelements. The supercurrents can also be spin-polarized, andthis can then lead to a low energy spin transfer torque that canbe used to control the magnetization of nanoscale devices.

Ferromagnetism ands-wave superconductivity would ap-pear to be incompatible due to the opposite spin structure oftheir order parameters: the internal fields in the ferromag-nets tend to break the singlet Cooper pairs. Indeed, althoughproximity effects do exist inF/S heterostructures, they arevery different from those atN/S interfaces. The exchangefield leads to the Cooper pairs acquiring a center of massmomentum11 which results in damped oscillatory behavior ofthe singlet pair amplitudes in theF layer regions12–14. Thisbehavior is fundamentally important: it induces oscillations inmost of the physical properties of these structures, includingthe dependence of the transition temperature6 on the thicknessof the various layers. It also drastically changes the behaviorof transport quantities such as the the bias dependent conduc-tance, discussed below.

An even more noteworthy phenomenon arising from theF/S proximity effects is that in certainF/S heterostructurestriplet correlations may be induced, even though theS mate-rial is an s-wave superconductor15–17. These triplet correla-

tions are necessarily odd in frequency18 or, equivalently, oddin time16,19 as required by the Pauli principle. When the fer-romagnetic exchange fields are all aligned only themz = 0triplet component can be induced sinceS z, the z compo-nent of the Cooper pair spin, commutes with the Hamilto-nian. However, when there are two or moreF layers withnon-collinear exchange fields, as can happen for example inF1/F2/S structures,S z cannot commute with the Hamiltonianand themz = ±1 triplet states can also be induced. This is alsothe case with a singleF layer having a non-uniform magne-tization texture20–23. In contrast to the short-range proximity-induced singlet pair amplitudes, these oddmz = ±1 tripletstates are usually long ranged24–30 in the F layers. Their be-havior is also oscillatory. Because of this, the details of thegeometry of theF/S multilayers are crucial to determiningtheir equilibrium31 properties, including the oscillatory behav-ior of the transition temperature with layer thicknesses andwith the misalignment angleφ between the twoF layers in aspin valve32. The transport properties8 are also affected. As ina conventional spin valve, the relative exchange field orienta-tion of theF layers can have a large effect on the conductanceof the system. The introduction of triplet correlations canleadto a nonmonotonic dependence of the conductance onφ, justas for equilibrium quantities.

Ultimately, all superconducting proximity effects are gov-erned by Andreev reflection at the interfaces. Andreevreflection33 is the process of electron-to-hole conversion bythe creation or annihilation of a Cooper pair in the super-conducting layer. In conventional Andreev reflection, the re-flected electron/hole has opposite spin to the incident particle.However, it has been shown8,34–37 that inF/S interfaces tripletproximity effects are correlated with anomalous Andreev re-flection, in which the reflected quasiparticle has the same spinas the incident one. From this, it follows that the transportproperties are highly dependent on the proper considerationof Andreev reflection, as has been long recognized in bothN/S 38,39 andF/S 40–42 systems. These effects are particularlyimportant when examining the tunneling conductance in thesubgap bias regime where such systems can carry a supercur-rent.

In this paper, we are motivated by the increasing interest

2

in building actual, practical spin valve structures with poten-tial use as part of memory elements. We therefore inves-tigate the charge transport properties of a superconductingspin valve, anF1/N/F2/S structure which includes the nor-mal metal layer spacer, as used in spin valve devices. Thisnormal metal spacer is necessary in experiments in order tocontrol the relative exchange field of theF layers through theuse, for example, of a pinned and a soft ferromagnetic layer,in which the spacer decouples the ferromagnetic layers layers(see e.g. Ref.32). We will use typical values of the differentthicknesses, as in existing and planned devices, and realisticinterfacial scattering between the different layers. Parameterssuch as the exchange field and coherence length will be takento be in the range relevant to the materials actually used. Weare particularly motivated to identify the relevant experimen-tal transport features of actualF1/N/F2/S nanoscale systems.Thus, we investigate a geometry corresponding to experimen-tally realistic nanopillars with a normal metal layer spacerbetween two ferromagnetic layers. TheseF/N/F layers aregrown on top of a superconducting substrate. This substratemust be thick enough to allow for the sample to be supercon-ducting: its thickness must exceed the superconducting cor-relation length. Furthermore, experimental constraints do notallow for perfect interfaces. Although recent developments infabrication techniques4 have allowed for very clean interfaceswith ballistic transport properties, surface imperfections areunavoidable and even small interfacial scattering can havealarge effect on the transport properties, as we shall see, sincethey affect both ordinary and Andreev scattering. We willuse a self consistent solution of the Bogoliubov de Gennes(BdG) equations43 to calculate the conductanceG as a func-tion of bias voltage for realistic ranges of geometrical andma-terial parameters, and as a function of the angleφ. Temper-ature corrections, which we will show to be non negligible,will also be studied. The conductance will be obtained fromthe self consistent solutions of the Hamiltonian, via a transfermatrix procedure which makes use of the Blonder-Tinkham-Klapwijk (BTK) method38. In some previous calculations37,44

of the conductance, a non self-consistent, step-function pairpotential has been assumed. This neglects the very proxim-ity effects which act on the singlet and triplet pair amplitudes,and thus the pair potential. In order to properly take these intoaccount, one must use a self-consistent calculation of the pairpotential. Even more important, only a self-consistent solu-tion can guarantee that the conservation laws are satisfied8, aswe review in Sec.II below. The feasibility of the methodswe use here was demonstrated in previous work8 on simpleF/F/S heterostructures withoutN spacers or interfacial scat-tering, atT = 0. That work proved that the self-consistentBTK method embedded into a transfer matrix procedure canbe used to calculate the tunneling conductance as well as thespin transport quantities. Our work presented here exploitsthese methods with a broader focus on realistic experimentalparameters and sample compositions.

Because of the oscillatory nature of the superconductingsinglet (and triplet) amplitudes in theF layers, we will seethat, as expected, the transport results are highly dependenton the layer thicknesses, as they are on the exchange field.

FIG. 1. (Color online) Sketch of the structures studied. Thenotationfor thicknesses of the different layers is indicated, but the plot is notto scale. They axis is normal to the layers. The magnetizations ofthe outer magnetic layerF1 is along thez axis while inF2 it is in thex − z plane, forming an angleφ with thez axis, as indicated.

We report on theφ dependence of the tunneling conductanceas the angular spin valve effect of the system. We do so fora variety of thicknesses for the ferromagnetic and normal lay-ers. Furthermore, we investigate the dependence ofG on theinterfacial scattering strengths at all the interfaces. The depen-dencies that we find are, as a rule, nonmonotonic, and there-fore straightforward extrapolations are not possible. Ourgoalis to provide a better understanding on the full range of exper-imentally relevant results where the interfacial quality cannotbe perfectly controlled. From this, not only can one determinehow these parameters affect the spin valve effect, but one canalso provide the approximate set of parameters that can thenmaximize this effect: this has both experimental and techno-logical importance. We investigate also, in a more restrictedset of cases, the spin current and spin-transfer torque (STT).

After this Introduction, we briefly review our methods (bothfor equilibrium and transport calculations) in SecII . The re-sults are presented, chiefly in graphical form, in Sec.III , anddiscussed in the proper context. A summary Sec.IV closesthe paper.

II. METHODS

A. The basic equations

The basic methods and procedures used are straightforwardextensions of those discussed in Ref.8 and they need not tobe described again here. We merely sketch the main points,in order to establish notation and to make the paper under-standable. The geometry of the system under considerationis represented qualitatively in Fig.1. The layers are assumedto be infinite in the transverse direction. They-axis is normalto the layers: this somewhat unconventional choice turns outto be computationally convenient because only theσy Paulimatrix is complex. The magnetizations of the outer and innerlayers form an angleφ with each other.

3

The Hamiltonian appropriate to our system is,

He f f =

∫

d3r

∑

α

ψ†α (r)H0ψα (r)

+12

∑

α, β

(

iσy

)

αβ∆ (r)ψ†α (r)ψ†β (r) + H.c.

−∑

α, β

ψ†α (r) (h · σ)αβ ψβ (r)

, (1)

where∆(r) is the pair potential andh is the usual Stoner field,which we take to be along thez axis (see Fig.1) inside theouter magnetF1, while forming an angleφ with the z axisin the x − z plane inside the inner magnetF2. We assumeh1 = h2 ≡ h since in most experiments the same material isemployed. The field vanishes in the superconductorS and inthe normal spacerN. H0 is the single particle Hamiltonian,which we will take to include the interfacial scattering as ex-plained below. Performing a generalized Bogoliubov transfor-mation in the usual way, with the phase conventions of Ref.8,and taking advantage of the quasi one dimensional geometryone can recast the eigenvalue equation corresponding to theHamiltonian given by Eq.1 as:

H0 − hz −hx 0 ∆

−hx H0 + hz ∆ 00 ∆ −(H0 − hz) −hx

∆ 0 −hx −(H0 + hz)

un↑

un↓

vn↑

vn↓

= ǫn

un↑

un↓

vn↑

vn↓

, (2)

with theunσ andvnσ being the usual position and spin depen-dent quasiparticle and quasihole amplitudes involved in thetransformation. We use units such that~ = kB = 1. The quasione dimensional Hamiltonian isH0 = −(1/2m)(d2/dy2) +ǫ⊥ − EF(y) + U(y) whereǫ⊥ is the transverse energy, (so thatthe above Eq. (2) is a set of decoupled equations, one foreachǫ⊥), EF (y) is the layer dependent width of the band:EF(y) = EFS ≡ k2

FS /2m in the S layer andEF (y) = EFM

in the F layers. We define a mismatch parameter45Λ as

EFM ≡ ΛEFS . U(y) is the interfacial scattering. We takethis scattering, due to unavoidable surface roughness at theinterfaces, to be spin-independent and of the formU(y) =H1δ(y−d f 1)+H2δ(y−d f 1−dN)+H3δ(y−d f 1−dN −d f 2). Thedimensionless parametersHBi ≡ Hi/vF , wherevF is the Fermispeed inS , conveniently characterize the strength of the deltafunctions.

All calculations must be performed self-consistently, oth-erwise a large part of the proximity effect is eliminated fromthe problem. As previously shown8,46–48, and as reiterated inSectionII C, it is paramount to perform the transport calcu-lations self-consistently: not doing so jeopardizes the law ofconservation of change49. The self consistency condition is:

∆(y) =g(y)2

∑

n

′[

un↑(y)v∗n↓(y) + un↓(y)v∗n↑(y)]

tanh(

ǫn

2T

)

, (3)

where the sum is over all the eigenvalues and the prime in thesum denotes, as usual, that the sum is limited to states witheigenenergies within a cutoffωD from the Fermi level. Thesuperconducting coupling constantg(y), in the singlet chan-nel, is nonvanishing inS only. Self consistency is achieved bystarting with a suitable choice of∆(y) and iterating Eqs. (2)and (3) until the input and output values of∆(y) coincide. Thethermodynamic quantities can then be derived from the wavefunctions. The transition temperature itself can be most con-veniently obtained by linearization of Eq. (3) and an efficienteigenvalue technique10,16 as in previous32 work.

B. Transport: the BTK method and self-consistency

After the self consistent∆(y) function has been obtainedas reviewed above, one can proceed with the calculation ofthe transport properties. There are no fundamental difficultiesin extending the self consistent8 BTK method38 to the casewhere an extraN layer and interfacial scattering exists. Thisis because the only nontrivial part of the transfer matrix proce-dure is that which deals with the self consistent pair potentialinsideS and this is extensively discussed in previous8 work.For the rest, one has of course additional matching equationsat the two added interfaces. The matching equations are ofthe same basic form as those found previously8 except for theinterfacial scattering, which requires, as in elementary situ-ations, a modification of the derivative continuity condition.Again, it is not necessary to discuss here these relatively ele-mentary questions, although care is required to include themcorrectly in the computations. We confine ourselves to theminimum necessary to make the notation clear.

For an incident particle with spin up the wavefunction inF1

is:

ΨF1,↑ ≡

eik+↑1y+ b↑e

−ik+↑1y

b↓e−ik+↓1y

a↑eik−↑1y

a↓eik−↓1y

. (4)

where we have include the appropriate amplitudes for the or-dinary and Andreev reflection processes, which we must cal-culate. If the incident particle has spin down, the correspond-ing wavefunction inF1 is

ΨF1,↓ ≡

b↑e−ik+↑1y

eik+↓1y+ b↓e

−ik+↓1y

a↑eik−↑1y

a↓eik−↓1y

. (5)

with appropriate amplitude coefficients, numerically differentfrom those for the spin up incident particle. One has, in theabove equations:

k±σ1 =[

Λ(1− ησh1) ± ǫ − k2⊥

]1/2, (6)

4

whereησ ≡ 1(−1) for up (down) spins, andk⊥ is the length ofthe wavevector corresponding to energyǫ⊥. All wavevectorsare understood to be in units ofkFS and all energies in termsof EFS .

All of the amplitudes are then determined from the trans-fer matrix procedure discussed in Ref.8, where the self-consistent pair potential determines the wavevectors in the Slayer. The transfer matrix matches the continuity conditionsfor each layer. The outcome of the calculations includes thereflection amplitudesaσ and bσ of the incoming wavefunc-tions for the different (ordinary and Andreev, spin up and spindown) reflection processes. From these the conductance is ex-tracted as explained below.

C. Conservation laws and conductance

In transport calculations great care has to be taken not toviolate49 the conservation laws. Consider the equation forcharge densityρ(r, t) which arises from the Heisenberg equa-tion:

∂

∂t〈ρ(r)〉 = i

⟨[

He f f , ρ(r)]⟩

. (7)

We are considering here steady state situations, so the timederivative vanishes and we simply should have a zero a diver-gence condition for the current. In our quasi two dimensionalgeometry, the only non-vanishing component of the current isjy, and it depends only ony. Hence we need to ensure that∂ jy/∂y = 0. Upon computing the commutator in the right sideof Eq. (7) under these conditions we find, however:

∂ jy(y)

∂y= 2eIm

∆(y)∑

n

[

u∗n↑vn↓ + u∗n↓vn↑

]

tanh(

ǫn

2T

)

(8)

In transport calculations the wavefunctions cannot be takento be real, as is possible for the evaluation of static quanti-ties in a current-free situation. Hence it is not necessarily truethat the right side of Eq. (8) will vanish. However, it is easyto see9,47 that it will be identically zero when the self consis-tency condition Eq. (3) is satisfied. Therefore, the importanceof performing the calculations self consistently, despitethecomputational simplifications inherent to non-self-consistentmethods, cannot be overemphasized.

D. Extraction of the conductance

From the results of the previous subsection, one can extractthe conductance. The current is related to the applied bias38 Vvia the expression:

I(V) =∫

G0(ǫ)[

f (ǫ − eV) − f (ǫ)]

dǫ, (9)

where f is the Fermi function. The bias dependent tunnelingconductance isG(V) = ∂I/∂V. The functionG0 in Eq. (9)

is the conductance in the low-T limit or, more generally, theconductance obtained by replacing the derivative of the Fermifunction by aδ function. It is related to the scattering ampli-tudes by:

G0(ǫ, θi) =∑

σ

PσGσ(ǫ, θi) (10)

=

∑

σ

Pσ

1+k−↑1

k+σ1

|a↑|2+

k−↓1

k+σ1

|a↓|2 −

k+↑1

k+σ1

|b↑|2 −

k+↓1

k+σ1

|b↓|2

,

in the customary natural units of conductance (e2/h). InEq. (10) the differentk symbols are as defined in Eq. (6).The angleθi is the angle of incidence: for spin up it is givenby tanθi = (k⊥/k+↑1), and similarly for spin down. Thusone hasθi = 0 for the forward conductance. The factorsPσ ≡ (1 − h1ησ)/2 are included to take into account the dif-ferent density of incoming spin up and spin down states. Theenergy dependence ofG(ǫ) arises from the applied bias volt-ageV. It is customary and convenient to measure this bias interms of the dimensionless quantityE ≡ eV/∆0 where∆0 isthe value of the order parameter in bulkS material. We willrefer to the dimensionless bias dependent conductance simplyasG(V) or G(E) usually omitting the angular argument.

One can not always assume that the experiments are per-formed in the lowT limit. At finite temperature there are twosources ofT corrections. The first and more obvious is thatarising from theT dependence of∆(y), that is, theT depen-dence of the effective BCS Hamiltonian. This is of coursestraightforward to include: one just calculates the self con-sistent∆ at finite T (see Eq. (3) and uses it as input in thetransfer matrix calculations. But there is also a temperaturedependence arising from the Fermi function in Eq. (9). If thetemperature is not too close toTc0, the transition temperatureof the bareS material, which sets the overall scale, one canuse a Sommerfeld type expansion. Because the energy scaleover whichG(V) varies is of order∆0, the relevant expansionparameter isT/Tc0, notT/TF, and hence not necessarily neg-ligibly small in all experimental situations. One finds usingelementary50 methods:

G(V, T ) = G0(V) + a1

(

T∆0

)2 (

∂2G(V)∂ǫ2

)∣

∣

∣

∣

∣

∣

ǫ=V

+ O

(

T∆0

)4

(11)

wherea1 can be expressed50 in terms of a Bernoulli number.Alternatively, one can use the general form:

G(V, T ) =1

4T

∫

dV ′1

cosh2[(1/2T )(V − V ′)]G0(V ′). (12)

In Eqs. (11) and (12) G0(V) means the result of Eq. (10) eval-uated with the self consistent pair potential at temperatureT .The second form turns out to be more useful as most relevanttemperatures turn out to be too high for the Sommerfeld ex-pansion.

E. Spin transport

We will consider also spin transport across the junction. Inour quasi one-dimensional geometry the tensorial spin current

5

becomes a vector in spin space, while spatially it depends onlyon y. Denoting this vector as~S (y) it can be written8 in termsof the wavefunctions, as:

S i ≡iµB

2m

∑

σ

⟨

ψ†σσi∂ψσ

∂y−∂ψ†σ∂y

σiψσ

⟩

. (13)

It is not difficult to write the componentsS i in terms of theun andvn wavefunctions. In theT = 0 limit, the result is:

S x =−µB

mIm

∑

n

(

−vn↑

∂v∗n↓∂y− vn↓

∂v∗n↑∂y

)

(14a)

+

∑

ǫk<eV

(

u∗k↑∂uk↓

∂y+ vk↑

∂v∗k↓∂y+ u∗k↓

∂uk↑

∂y+ vk↓

∂v∗k↑∂y

)

S y =µB

mRe

∑

n

(

−vn↑

∂v∗n↓∂y+ vn↓

∂v∗n↑∂y

)

(14b)

+

∑

ǫk<eV

(

u∗k↑∂uk↓

∂y+ vk↑

∂v∗k↓∂y− u∗k↓

∂uk↑

∂y− vk↓

∂v∗k↑∂y

)

S z =−µB

mIm

∑

n

(

vn↑

∂v∗n↑∂y− vn↓

∂v∗n↓∂y

)

(14c)

+

∑

ǫk<eV

(

u∗k↑∂uk↑

∂y− vk↑

∂v∗k↑∂y− u∗k↓

∂uk↓

∂y+ vk↓

∂v∗k↓∂y

)

,

where the first terms in the right side are the spin current com-ponents in the absence of bias. A static spin transfer currentmay exist near the boundary of two magnets with misalignedfields. The above results are valid at lowT , we will not con-sider temperature corrections for this quantity. In the steadystate the conservation laws require:

∂

∂yS i = τi, i = x, y, z (15)

whereτ is the torqueτ ≡ 2m × h with m being the localmagnetizationm = −µB

∑

σ〈ψ†σσψσ〉. The expression form

in terms of the wavefunctions is given in Ref.8.

III. RESULTS

In this section we present our results. As discussed in theIntroduction, our emphasis is in exploring a range of valuesofexperimental interest for the relevant parameters. This, in ad-dition to helping us meet our goal of helping experimentalistsunderstand their data, will keep the discussion within reason-able bounds: otherwise, with a more than ten-dimensional pa-rameter space to be investigated, this work would completelylose its focus. We do have an extensive and growing databaseof results for many other cases. As mentioned above, we usedimensionless parameters in our plots: all lengths are givenin units of kFS and all energies in units ofEFS except, asalready stated, for the bias. Dimensionless lengths will be

0.5

1

1.5

2

HB3=0

G

E

0o

30o

60o

90o

120o

150o

180o

HB3=0.1

0.5

1

1.5

2

0 0.5 1 1.5

HB3=0.2

0 0.5 1 1.5

HB3=0.3

FIG. 2. Effect on the conductance of the barrier between the su-perconductor and the inner ferromagnetHB3. The four panels showresults forG in natural units, as a function of bias voltageE ≡ eV/∆0

at seven values of the misalignment angleφ as indicated in the leg-end. The panels correspond to different values ofHB3 ranging from0.0 to 0.3 withHB1 = HB2 ≡ HB = 0. The thicknesses areDF1 = 20,DN = 40, DF2 = 12 andDS = 180. The internal field parameter ish = 0.145

denoted by capital letters with the appropriate subscript.Theunits for the dimensionless barrier height parametersHBi havebeen explained before. Values close to unity would representa strong tunneling limit: these would be experimentally veryundesirable as the proximity effects would be very small. Zerovalues represent an ideal interface, which is unlikely to beat-tainable experimentally. Since the first and second interfacesare both betweenF andN materials, one can fairly safely as-sume that these two barrier strengths are similar, and we willusually take them to be identical,HB1 = HB2 ≡ HB. In ourdimensionless units a field parameter value ofh = 1 wouldcorrespond to a half metal. The results forG presented arefor h = 0.145 a value previously found adequate32 in fittingCo static properties in similar devices. As in Ref.32 we setΛ = 1, which subsumes some of the wavevector mismatch ef-fects with the phenomenologialHBi parameters. We will alsoassume a value ofΞ0 = 115 for the dimensionless correlationlength inS , a value used in the same context32 for Nb. Wewill vary the thicknesses of all layers, keepingDF2 relativelysmall, which is necessary to obtain good proximity effect, andallowing DN andDF1 to be somewhat larger. As toDS , thethickness of the superconducting layer, it must of course bekept aboveΞ0: otherwise the sample tends to become non-superconducting, for rather obvious reasons. We will focushere on forward conductance results, which can be obtainedfrom point probes and involve trends much easier to under-stand.

6

0.5

1

1.5

2

HB=0.1

G

E

0o

30o

60o

90o

120o

150o

180o

HB=0.2

0.5

1

1.5

2

0 0.5 1 1.5

HB=0.3

0 0.5 1 1.5

HB=0.4

FIG. 3. Effect on the conductance of the barriers between thenormalspacer and the ferromagnetsHB1 = HB2 = HB. The four panels showresults for the same arrangement as in Fig.2 and the same geometri-cal and field parameters except in this caseHB3 is held constant andthe value of the barrier parameter at the other two interfaces is variedbetween 0.1 and 0.4.

A. Barrier effects

The effects of interfacial scattering are very strong and im-portant. Recall that even in standard normal-superconductorinterfaces the zerto bias conductance (ZBC) can vary betweena value of two for a perfect interface, and an exponentiallysmall value for the tunneling limit. One should recall herethat even in the case where a certain barrier parameter van-ishes, there is still scattering at the correspondent interface:this is because it is impossible for the two Fermi wavectorsin the ferromagnets to match the Fermi wavevector of eitherthe N or theS materials. This has to be kept in mind in thediscussion below.

In Fig.2 we show the effect of increasingHB3 assuming thatthe other interfaces have zero interfacial potential, althoughscattering due to wavevector mismatches is present. Four val-ues ofHB3 are studied, one in each panel, and curves for sevenvalues of the misalignment angleφ are plotted. The geomet-rical parameters areDF1 = 20, DN = 40, DF2 = 12 andDS = 180. The overall trend on increasingHB3 is a markeddecrease of the low bias conductance and a much smaller de-crease of the high bias limiting value. The critical bias (CB)is the value of the bias at whichG sharply changes behaviorand begins trending towards its normal state limit. In general,the critical bias is smaller than unity, and smaller values areassociated with stronger proximity effects since the CB is as-sociated with the saturated value of∆(Y) well insideS . Wesee that the CB tends to increase withHB3, while the valueof G at critical bias (the critical bias conductance, CBC) re-mains nearly the same. On the other hand, the CB is in allcases a strong function ofφ, decreasing asφ increases, up toaboutφ = 100◦ and then flattening, for this geometry. The

dependence is less marked at higher barrier values. The ZBChowever, is monotonically decreasing inφ. This dependenceon φ is different from that of the CB or CBC, and it leadsto a crossover in the conductance values. Remarkably, thiscrossover tends to occur with a ”nodal” behavior at a singlebias value in the subgap region: this can best be seen in thethird and fourth panels. Monotonic behavior in the ZBC alsooccurs for other values ofDF2 that we have studied, but thedirection (increasing or decreasing inφ) is reversed in an os-cillatory way: for example the ZBC increases withφ at valuesof DF2 of 7 and 10 and again at 16,17. This is one more exam-ple of the multiple oscillatory behavior found in this problemand an illustration of how much care one has to take beforeextrapolating results.

Next we consider, in Fig.3, the effect of increasingHB1 =

HB2 ≡ HB while keepingHB3 = 0 at theF2/S interface.Again, four barrier values are considered, in an arrangementvery similar to that in the previous figure. The effects of inter-facial scattering are now more pronounced. This is not neces-sarily due to the presence of two barriers: as in well known sit-uations in elementary one-dimensional quantum mechanics,we find that having more barriers does not necessarily leadto less transparency. This analogy is imperfect: our systemis not one-dimensional, there are multiple scattering mecha-nisms (interfacial imperfections, wavevector mismatch, An-dreev reflection, etc). Still, we find that having two barriersdoes not always reduce transmission. A clear example of thiscan be seen in the ZBC value which, for the chosen valuesof DF2 = 12 andDN , is nearly independent ofHB. This isbecause of resonance-like behavior in this geometry. Further-more, changing the values ofDF2 = 12 andDN leads to ZBCbehavior more similar to that in Fig.2, which we discuss inthe next subsection in connection with Fig.6. The behavior ofthe CB with angle is nonmonotonic, in a way similar to thatfound in Fig.2. The minimum is now somewhat less shallow,particularly at higherHB. At low bias,G decreases as the biasis increased, although an upturn does occur as the CB is ap-proached albeit at a lower value of the CBC for increasingHB.This is in contrast to Fig.2 where the CBC was unaffected byHB3.

B. Geometrical Effects

We have mentioned in the previous discussion that thethickness of the different layers may have a strong and oftennonmonotonic effect onG. The thickness of the inner mag-netic layer,DF2 turns out to be the more important of thesegeometrical variables. In the six panels in Fig.4 we considerincreasing values ofDF2 while keeping the other geometricaland material parameters fixed to their values in the previousfigures. The three interfacial barrier parameters are set toin-termediate values (see the caption).

Consider in detail the first panel, whereDF2 = 7. Onenotices immediately the reduction in ZBC, as opposed to theresults forDF2 = 12 in the third panel or to those in the pre-vious figures. The behavior of this reduction occurs, as hasbeen mentioned above, in an oscillatory manner withDF2: it

7

0.5

1

1.5

2

DF2=7

G

E

0o

30o

60o

90o

120o

150o

180o

DF2=10 DF2=12

0.5

1

1.5

2

0 0.5 1 1.5

DF2=15

0 0.5 1 1.5

DF2=16

0 0.5 1 1.5

DF2=17

FIG. 4. Effect on the conductance by varying the thicknessDF2 of the inner ferromagnetic layer. The values of the other thicknesses, field, andcorrelation length are as in the previous two figures, and thebarrier values are set to 0.3, 0.3, and 0.1 respectively, which are representative ofpossible experimental values. The six panels showG vs bias voltage for several angles, at six values ofDF2 = 7, 10,12, 15, 16, and 17. Thespin valve effect varies significantly in both the CB and the ZBC.

can be seen again atDF2 = 15 (fourth panel). In this panel, asin the second and the fifth, the minimum value of the CB withangle is atφ = 90◦, and this minimum is very well marked– this is an optimum situation for valve effects. The ZBCvalue depends somewhat onφ but not in the same way asthe CB: hence, the crossing conductance curves near a biasof 0.2. The second panel exhibits similar behavior, but theZBC is markedly higher. On further increasingDF2 to 12(third panel) the CB becomes monotonic inφ while the lowbias conductance does not change: indeed the node where thelines cross barely moves. The caseDF2 = 15 (fourth panel) isyet different: the CB is larger and there is a marked “bump”in the low bias conductance, the height of which increaseswith φ. Resonance in the ZBC is observed again in the fifthpanel, and the angular dependence of the CB returns to havinga marked minimum atφ = 90◦ although with a weaker depen-dence. Furthermore, the node noticeably moves to a higherbias value. Finally, atDF2 = 17 (last panel) the ZBC dropsagain, the angular dependence of the CB is reversed, and thenode disappears. Thus we see that the thickness of the innermagnetic layer is a very important variable in determining theconductance properties.

On the other hand, the effect of varyingDF1, the thicknessof the outer ferromagnetic layer, is much weaker than that ofvaryingDF2. This is illustrated in the first two panels of Fig.5.There we display, in each panel, results forG at fixedφ = 0.In the first panel we do this for several values ofDF1 rang-ing from 12 to 30 and, in the second panel, forDF2 values

from 7 to 17 at fixedDF1. In both panelsDN = 40. Barrierheights and other parameters are as in Fig.4. The difference isobvious: while in the first panel the results barely change (al-though the change is nonmonotonic), in the second one everyrelevant quantity (CB, ZBC, high bias and low bias behav-iors etc) changes, in obvious and very strongly nonmonotonicways. Thus, in the fabrication process, the precise thicknessof DF1 is less critical than that ofDF2. As to the normal spacerthickness, in the last two panels of Fig.5 we consider the de-pendence ofG on DN . We again plotG at fixedφ = 0 forseveral values ofDN at two values ofDF2 (see caption). Onecan see that while quantities such as the CB do not dependvery much onDN , the low and high bias behaviors vary quiteappreciably overall, the former rather dramatically. Hence weconclude thatDF2 is the crucial geometrical parameter in theproblem, followed in importance byDN and withDF1 beingmuch less relevant.

Careful examination of the above results yields insights onthe combined effects of interfacial scattering and on geometry,particularly onDF2: how geometry and interfacial strength arerelated follows ultimately from the oscillatory nature of theCooper pairs and from quantum mechanical interference. Wenow display, in Fig.6, these combined effects in a more directway. As in Fig.5 we study results for fixedφ = 0. We con-sider four values ofDF2, one in each panel, ranging from 7 to17, and plot results for several values ofHB at HB3 = 0. In thefirst panel we see a large and monotonic dependence onHB ofthe entire conductance dependence. In the next case shown,

8

0.5

1

1.5

2

DN=40 DF2=12

G

E

DF1=12DF1=15DF1=20DF1=25DF1=30

DN=40 DF1=20

DF2=7DF2=10DF2=12DF2=15DF2=16DF2=17

0.5

1

1.5

2

0 0.5 1 1.5

DF1=20 DF2=12

DN=30DN=35DN=40DN=45DN=50

0 0.5 1 1.5

DF1=20 DF2=7

DN=30DN=35DN=40DN=45DN=50

FIG. 5. Effects of varyingDF1 or DN , compared with dependence onDF2. All panels are forφ = 0, barrier values of 0.3, 0.3, and 0.1 andthe field parameter, correlation length, andDS are as in Figure2. Thefirst two panels contrast the effect on the conductance of varying thethicknessDF1 of the outer ferromagnetic layer withDF2 of the innerferromagnetic layer. In the first panel,DF1 is varied, as indicatedin the legend, atDF2 = 12, while in the second oneDF2 is variedat DF1 = 20. The last two panels show the effect of varyingDN atDF1 = 12 andDF2 = 7 respectively. The dependence of the resultson DF1 is much weaker than that onDF2 or DN . Both DF2 andDN

have a large impact on the ZBC, meanwhileDF2 has a much largereffect on the CB.

0.5

1

1.5

2

DF2=7

G

E

HB=0HB=0.1HB=0.2HB=0.3HB=0.4HB=0.5

DF2=10

0.5

1

1.5

2

0 0.5 1 1.5

DF2=12

0 0.5 1 1.5

DF2=17

FIG. 6. Combined effect ofDF2 and barriers. The behavior at fixedφ = 0 andHB3 = 0 is studied. Each of the four panels corresponds toa fixed value ofDF2: 7, 10, 12, and 17 and the curves correspond tovalues ofHB1 = HB2 ≡ HB as indicated in the legend. A nonmono-tonic feature in the ZBC is observed as a function ofDF2, owing tothe oscillatory behavior of the Cooper pairs.

DF2 = 12, the ZBC depends only very weakly onHB. In thenext panel, the spread in the ZBC withφ increases somewhat,as compared to the previous panel, and it does so even morein the last panel. This resonance-like behavior is not the sameas in the one-dimensional two barrier problems in basic quan-tum mechanics, where a resonance feature is observed in the

0

0.5

1

1.5

2

φ=0 HB=0 HB3=0.9

G

E

G for T=0G0 for T=0.1G for T=0.1

0.5

1

1.5

2

φ=0 HB=0 HB3=0

G for T=0G0 for T=0.1G for T=0.1

0.5

1

1.5

2

0 0.5 1 1.5

HB=0.3 HB3=0.1

0o

30o

60o

90o

120o

150o

180o

FIG. 7. Temperature dependence of the conductance. In the first twopanels we considerG at fixedφ. The thicknesses and fields are as inFig. 2. TemperaturesT = 0.1, in units ofTc0, are compared toT = 0results. The result of including onlyG0, the correction toG arisingfrom theT dependence of∆(y) is also shown. The first panel is for avery high barrier (HB3 = 0.9) betweenS andF2 andHB1 = HB2 = 0,while in the second allHBi = 0. TheG0 result atT = 0.1 is nearlyidentical to theG at T = 0. The last panel illustrates (for the samevalues as the first panel in Fig.4), a case where the CB varies verynonmonotonically with angle, and shows how little this behavior isaffected byT .

transmission coefficients as a function of the distance betweenthe barriers. This analogy might apply better toDN , but notto the inner ferromagnetic thicknessDF2. Instead, this reso-nance is due to the oscillatory behavior of the Cooper pairs.We see then that certain values ofDF2 make the system, orat least its ZBC, partly “immune” to the effects of fairly highsurface barriers. Although this holds only to a limited extent,it may be worthwhile to attempt to exploit this effect to pal-liate the existence of unfavorable interfaces with unavoidablylarge scattering.

C. Temperature dependence

Experiments in these systems are not performed at zerotemperature, nor, in practice, at ultralowT . Therefore theinfluence ofT must be examined. There are two transitiontemperatures to consider: the transition temperatureTc0 ofpure bulkS material, and the transition temperatureTc of thedevice, which is typically considerably lower. In our discus-sion we will use a dimensionless temperatureT in units ofTc0

9

-0.02

-0.01

0.00

0.01

0.02

F1/F2 F2/S

Sx(

10-3

)E=0.6

Y

E=1.0 E=2.0

-3.00

-2.00

-1.00

0.00

1.00

2.00

F1/F2 F2/S

Sy(

10-3

)

0o

30o

60o

90o

120o

150o

-0.02

-0.01

0.00

0.01

0.02

-40 -30 -20 -10 0 10

Sz(

10-3

)

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

F1/F2 F2/S

Sx(

10-3

)

-3.00

-2.00

-1.00

0.00

1.00

2.00

F1/F2 F2/S

Sy(

10-3

)

-0.06

-0.04

-0.02

0.00

0.02

0.04

-40 -30 -20 -10 0 10

Sz(

10-3

)

-0.20

0.00

0.20

0.40

0.60

F1/F2 F2/S

Sx(

10-3

)

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

F1/F2 F2/S

Sy(

10-3

)

-0.40

-0.20

0.00

0.20

0.40

0.60

-40 -30 -20 -10 0 10

Sz(

10-3

)

FIG. 8. The three components of the spin current are shown as afunction ofY for several values ofφ, as indicated, and three values of the biasvoltage. We haveh = 0.1, DF1 = DS = 250= 5Ξ0, DF2 = 30, DN = 0. Only the central region ofY is plotted:Y = 0 is at theF2/S interface.All components of the spin current are zero forφ = 180◦.

sinceTc varies as the geometry is changed.As explained in Sec.II D one has to consider two sources of

T dependence. The first is that arising from the self-consistentpair potential,∆(y), that is, theT dependence in the effec-tive Hamiltonian. This leads to the functionG0 defined belowEq. (9) and in Eq. (10) beingT dependent. The second is thatoriginating in the Fermi functions in Eq. (9). As discussed inconnection with Eq. (11) the latter is not negligible since thescale of the variation ofG with bias is∆0, not the Fermi en-ergy. We have found that, in practice, Eq. (12), which is notdependent on any expansion, is much more useful than theSommerfeld method in the relevant temperature range. Thisis because the conductance has large, and even discontinuousderivatives, which the Sommerfeld expansion does not handlewell.

Representative results are shown in Fig.7. In the first twopanels we consider a fixedφ = 0 and we show results forGboth atT = 0 and at a reduced temperatureT = 0.1. Sincefor the size ranges considered in this section we have foundthat Tc/Tc0 values are in the 0.5 to 0.6 region, these corre-spond toT/Tc of about 0.2. The first panel shows results ina strong tunneling limit regime, with high barriers, and the

second for zero barrier heights. Plots ofG0 , i.e. the resultsobtained by using the∆(y) correction only are also included:these are obviously inadequate in both cases, and the full re-sult is needed. We have found this to be invariably the caseexcept at unrealistically lowT . The overall effect of the tem-perature is, otherwise, that of rounding up and softening thesharp features of the lowT results. A consequence of this isthat at finiteT one has to redefine more carefully the CB asthe bias value at whichG has a peak or a high derivative. Theproper redefinition is the bias value at whichG varies fastest.

In the third panel of Fig.7, we replotG for the same caseconsidered in the first panel of Fig.4, which, as we have re-marked before, shows good spin valve effects in its CB prop-erties, but now atT = 0.1 instead of at zero temperature. Thetwo results should be carefully compared. We see that whilethe curves are now much smoother the behavior of the dif-ferent features with angle are robust. In particular the sharpminimum of the critical bias atφ = 90◦ remains unchanged.We have found this to be the the situation in all the cases wehave checked. Hence, spin valve properties are only weaklydependent onT .

10

-0.01

0.00

0.01

F1/F2 F2/S

τ x(1

0-3)

E=0.6

Y

E=1.0 E=2.0

-16.00

-12.00

-8.00

-4.00

0.00

4.00

8.00

12.00

16.00

F1/F2 F2/S

τ y(1

0-3)

-0.01

0.00

0.01

-40 -30 -20 -10 0 10

τ z(1

0-3)

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

F1/F2 F2/S

τ x(1

0-3)

-16.00

-12.00

-8.00

-4.00

0.00

4.00

8.00

12.00

16.00

F1/F2 F2/S

τ y(1

0-3)

-0.03

-0.02

-0.01

0.00

0.01

0.02

-40 -30 -20 -10 0 10

τ z(1

0-3)

-0.20

-0.10

0.00

0.10

0.20

0.30

F1/F2 F2/S

τ x(1

0-3)

0o

30o

60o

90o

120o

150o

-16.00

-12.00

-8.00

-4.00

0.00

4.00

8.00

12.00

16.00

F1/F2 F2/S

τ y(1

0-3)

-0.30

-0.20

-0.10

0.00

0.10

-40 -30 -20 -10 0 10

τ z(1

0-3)

FIG. 9. The three components of the spin transfer torque plotted for the same situation as in the previous figure. The torque is identically zerofor φ = 0 andφ = 180◦. The discontinuities at the interface reflect those of the internal fields.

D. Spin Currents

We present here some results for the spin current and thespin transfer torque. We restrict ourselves to the case wherethere is no spacer, and the barrier parameters are zero. How-ever, we consider in this paper a range of bias voltages and allvalues of the angleφ. Very limited results for onlyφ = 90◦

value were given in Ref.8. We use units such thatµB = 1and takeh = 0.1. We consider a superconductor thicknessof five times the coherence length (DS = 250= 5Ξ0) so thatthe saturated value of∆(y) is essentially the same as the bulkS value∆0. We assume a rather thickF1 layer (DF1 = 250)while DF2 = 30.

The main quantities we will focus on are the three com-ponents of the spin currents and of the spin transfer torques(STT) as a function of position. For the charge current, theconservation law entails that the current is independent ofpo-sition. But for spin, the derivative of the current is the STT(see Eq. (15)) and the latter quantity is of great physical in-terest. As usual8,19 we normalizem to −µB(N↑ + N↓). Thenormalization for the spin current follows from these conven-tions. There are two alternative methods to calculate the spin

currents: one is directly from the expressions in Eqs. (14). Theother method is to calculate the torque first, from the expres-sion below Eq. (15) and then integrate over they variable. Thetwo methods agree when the calculations are done self con-sistently, as was conclusivelly shown in Ref.8. The secondmethod is computationally much easier, but it yields resultsonly up to a constant of integration. We have therefore usedthe direct method: it requires obtaining wavefunction resultsover a very fine mesh, so that the derivatives in Eq. (14) canbe calculated to sufficient accuracy.

In the following discussion it is well to recall the meaningof the indices and coordinates. The spin current is in generala tensor, each element having two indices, one correspondingto the spatial components and the other to spin. In a quasi-dimensional geometry, the only spatial component is in theydirection, normal to the layers in our convention (see Fig.1).The spin current is then simply a vector in spin space: the in-dices inS i denote spin components, with all transport beingin the spatialy direction. Recalling Eq. (15) and the definitionof the torqueτ = 2m×h we see thatτy tends to twist the mag-netization in the plane of the layers, but of course it can onlydo so in regions near the interfaces, wherem andh are notparallel due to magnetic proximity effects. We also see that

11

each component of the torque vanishes in theS layer wherethe internal field parameterh is zero.

We can now discuss the plots in Figs.8 and9. These twofigures show results for the three components of the spin cur-rent and of the STT respectively, each under the same con-ditions (see captions). These quantities are shown for threevalues of the bias,E, ranging from below to well above∆0:for each component, there is a panel corresponding to eachvalue ofE. The curves correspond to different values ofφ asindicated in the legend. Atφ = 0 andφ = 180◦ the sameconservation laws that preclude singlet to triplet pair conver-sion imply that the torques vanish. It is evident that there isno point in including the regions of the sample deep insideS or even well insideF1, so the region plotted is that whichincludes both interfaces: theS/F2 interface at the origin andthat between ferromagnets atY = −30, whereY is the dimen-sionless position.

The y-components results are easiest to understand: thecomponent of the torque has very sharp peaks, with oppo-site signs, near theF1/F2 boundary where it vanishes. Thesepeaks reflect the existence of a strong but short-ranged mag-netic proximity effect. InF2 and inF1, τy is small and oscilla-tory. It reaches its maximum value atφ = 90◦. It depends onlyweakly on the bias, since it basically reflects a static effect:the two magnets interacting with each other. This behavioris of course reflected inS y as both quantities are related viaEq. (15).

The behavior of the in-plane components,x andz, is similarto each other (they are related by spin rotations) and quite dif-ferent from that ofy. Now currents and torques are transport-induced and one sees immediately that they markedly dependon bias. Since inF1 the internal field always points alongz, wefind thatS z is a constant inF1, its value increasing with bias.As a function ofφ its behavior is complicated, the maximumvalue is not precisely atφ = 90◦ and it is dependent on bias.For this value ofφ the field points along thex direction inF2

(it is always alongz in F1). ThereforeS z is always spatiallyconstant inF1 and this applies also toS x in F2 at φ = 90◦.For other values of the mismatch angleS x oscillates in bothmagnetic layers, and so doesS z in F2. The amplitude of theoscillations ofS x decays slowly deep into theF1 layer. Inall cases the period of the spatial oscillations is approximately1/h indicating that the oscillations are due to the behavior ofthe Cooper pairs. As to the corresponding components of thetorque, one notes at once that their maximum value is muchsmaller than that of theτy peak but, away from theF1/F2

interface, the values are not all that different. This reflectsthe geometry, as explained above. We see that thex and zcomponents of the torque are also nonmonotonic withφ, withpeaks that are not necessarily atφ = 90◦, depending on thebias. For lower biases, the peak values appear to shift away tosmaller values, more closely aligned with thez direction, dueto the increasing static effect from theF1 layer. In our coor-dinate system,τz vanishes inF1 for all φ and oscillates inF2.Correspondingly,τx is oscillatory in bothF1 andF2 except atφ = 90◦ where it is zero inF2. We have not plotted the magne-tization itself, but its components exhibit damped oscillationswhich reflect the well known51precessional behavior of the

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 30 60 90 120 150 180

Sz(

10-3

)

φ

E=2.0

E=0.6

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 30 60 90 120 150 180

Sz(

10-3

)

FIG. 10. Thez component of the spin current in the outerF regionas a function ofφ, at two different bias values.

magnetization around the internal fields. Such precessionalbehavior is then reflected in the current oscillations discussedabove.

In our coordinate system,S z is a constant in the outer layer,F1. Also, all the components of the spin current are triviallyconstant in theS layer, since there are no torques there. Ascan be seen in Fig.8, all spin current components vanish inS unless the bias exceeds the bulkS gap,∆0. This confirmsthe remarkable fact8 that, in this respect, spin currents behavelike charge currents in anN/S junction. It can rather eas-ily be shown via standard spin rotation matrix arguments thatthe constant values ofS z andS x deep in theS material, inthe limit of large bias, should be approximately related to thevalue ofS z in theF1 layer by factors of cosφ and sinφ respec-tively, and this can be seen in the last column of Fig.8 to holdrather accurately atE = 2. On the other hand, the dependenceof the constant value ofS z in the outer layer onφ is nontriv-ial as one can see in Fig.8. We display this more clearly inFig. 10, where we plot the value ofS z in F1 at two differentbias values. We see that for values below the CB the behavioris nonmonotonic: it cannot be, sinceS z vanishes at bothφ = 0andφ = 180◦. The maximum value is nearφ = 90◦. On theother hand, when the bias is well above the CB,S z, which inthis case is non-vanishing at zero angular mismatch, decreasesmonotonically withφ. It becomes slightly negative when thetwo magnets are aligned in opposite direction. The behaviorisnot described by a simple trigonometric function and a simpleargument leading to the behavior found seems elusive.

12

IV. CONCLUSIONS

The focus of this paper is on the prediction of the chargetransport properties of superconducting spin valves with aF1/N/F2/S layered structure. The emphasis is on studyingsystems having material and geometrical characteristics cor-responding to samples that can realistically be experimentallyfabricated. Our main results pertain to the conductanceG asa function of bias, particularly with respect to the misalign-ment magnetization angleφ between theF layers: variationof this angle produces the desired spin valve effects. The con-ductance is the basic information which is experimentally ob-tained from charge transport measurements: it is the derivativeof the current-voltage relation. To further our objective wehave used values of the material parameters (such as the inter-nal magnetic field and the superconducting coherence length)which have been previously shown32 to fit with great accu-racy the transition temperatures of such valve structures whenthe actual materials are Co, Cu and Nb. We have also usedthickness values which encompass the available and desirableexperimental ranges and have stayed away from idealistic as-sumptions, such as ideal interfaces, which are essentiallyirrel-evant to actual experimental conditions. We have also studiedthe often neglected temperature dependence of the results.Wehave used a fully self consistent approach, which is absolutelynecessary to ensure that charge conservation is satisfied.

Our results are summarized in Sect.III . The most importantconclusion to be learned from the figures presented is that sim-ple extrapolations are inadequate. There are several interfer-ing oscillatory phenomena involved – the center of mass oscil-lation of the Cooper pairs in ferromagnets, the transmissionsand reflections (ordinary, Andreev, and anomalous Andreev)at the three interfaces, and the usual quantum mechanical ef-fects. As a result, the dependence of the relevant quantitiesthat characterize the conductance (examples are the criticalbias, the zero bias conductance, and the low and high biasfeatures) have nonmonotonic behavior when just about anyparameter in the problem varies. From this it follows that thevalve effects, that is, the variation ofG with φ, vary quantita-tively and qualitatively depending on parameter values. Thelack of monotonicity makes it extremely difficult to predictbyextrapolation the measurable features expected for any givenset of conditions. The only thing that makes sense is to builda database of conductance plots for different sets of parametervalues, and compare the plots in the database with experimen-tal results as they become available. We have built such adatabase–the results included here are a representative subset.

As far as the geometry dependence we have found that re-sults depend most strongly on the thickness of the inner ferro-magnetic layer, with a large dependence on the normal spacerthickness as well and a relatively weaker one on that of theouterF electrode. This is however an overall, general state-

ment: specific details may be different. We have also foundthat the interfacial scattering specifically due to surfaceimper-fections (the barriers), does not severely affect the valveef-fects for typical experimentally accessible values. Of course,scattering strong enough to destroy the proximity effect wouldbe another matter. Another important conclusion we havereached is that temperature effects are not negligible in typicalexperimental situations. Furthermore, because of high deriva-tive regions in theG vs. bias curves, a Sommerfeld expan-sion does not work well. However, an exact calculation canbe performed numerically and it reveals that the shape of theconductance curve changes, becoming much smoother as biasvaries, where as the valve effects as a function ofφ remainunaffected.

We have also studied, in a much more limited way, the spintransfer torque and the spin currents in structures lackingtheN layer. The results are analyzed in Sec.III D . We have found,in our geometry, that they-component of the spin torqueshave sharp peaks at theF1/F2 interface, nearly independentof applied bias. These are due to the strong, static magneticproximity effects. The greatest peak occurs for a mismatchangleφ of 90◦. The spin torque components in thex and zdirection are bias dependent and more complex, with higherpeaks at angles smaller thanφ = 90◦ for lower biases. Weattribute this to static effects from theF1 layer magnetization.We have calculated the spin currents using the direct methoddescribed in Eq.14. We find a nonmonotonic behavior in thespin current amplitudes similar to that of the spin torque. Theoscillation amplitudes tend to peak for angles slightly belowφ = 90◦ for lower biases. TheS z component is constant inthe F1 layer and monotonic with angle for high bias values(above∆0) only. In theS layer, the spin currents are zeroexcept for at high bias when both theS x andS z componentsattain nonzero values for most values ofφ. The consistencybetween the torques and spin current gradients, imposed bythe conservation laws, is ensured in our approach.

To conclude, the measurable quantities have complex be-havior, often nonmonotonic as experimental parameters andinputs vary. Our plots provide an wide spectrum of features tostudy, many of which are not yet fully understood. We expectthat the results we have obtained will provide a very importantguide to experimentalists building real world superconductingspin valves in nanoscale heterostructures.

ACKNOWLEDGMENTS

The authors thank I.N. Krivorotov (University of Califor-nia, Irvine) for many illuminating discussions on the exper-imental issues. They are very grateful to Chien-Te Wu (Na-tional Chiao Tung University) for many helpful discussionsonall aspects of this problem. They also thank Yanjun Yang fortechnical help with the spin current calculations. This workwas supported in part by DOE grant No. DE-SC0014467

∗ moenx359@umn.edu † otvalls@umn.edu; Also at Minnesota Supercomputer Institute,

13

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