Properties of tunnel Josephson junctions with a ferromagnetic interlayer

Properties of tunnel Josephson junctions with a ferromagnetic interlayer

A. S. Vasenko,1,2,

*

A. A. Golubov,1M. Yu. Kupriyanov,3and M. Weides4

1_{Faculty of Science and Technology and MESA}+_{Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands} 2_{Department of Physics, Moscow State University, Moscow 119992, Russia}

3_{Nuclear Physics Institute, Moscow State University, Moscow 119992, Russia}

4_{Center of Nanoelectronic Systems for Information Technology (CNI), Research Centre Jülich, D-52425 Jülich, Germany} 共Received 31 October 2007; revised manuscript received 17 December 2007; published 4 April 2008兲

We investigate superconductor/insulator/ferromagnet/superconductor tunnel Josephson junctions in the dirty limit using the quasiclassical theory. We formulate a quantitative model describing the oscillations of critical current as a function of thickness of the ferromagnetic layer and use this model to fit recent experimental data. We also calculate quantitatively the density of states 共DOS兲 in this type of junctions and compare DOS oscillations with those of the critical current.

DOI:10.1103/PhysRevB.77.134507 PACS number共s兲: 74.50.⫹r, 74.45.⫹c, 74.78.Fk, 75.30.Et

I. INTRODUCTION

It is well known that superconductivity and ferromag-netism are two competing orders; however, their interplay can be realized when the two interactions are spatially sepa-rated. In this case, the coexistence of the two orderings is due to the proximity effect.1–3 _{Experimentally, this situation can}

be realized in superconductor/ferromagnet 共S/F兲 hybrid structures. The main manifestation of the proximity effect in S/F structures is the damped oscillatory behavior of the su-perconducting correlations in the F layers. Two characteristic lengths of the decay and oscillations are, correspondingly,␰f1

and␰f2. Unusual proximity effect in S/F layered structures

leads to a number of striking phenomena such as nonmono-tonic dependence of their critical temperature and oscilla-tions of critical current in S/F/S Josephson juncoscilla-tions upon the F layer thickness. Negative sign of the critical current corresponds to the so-called ␲ state. Spontaneous ␲ phase shifts in S/F/S junctions were observed experimentally.4–15

Superconductor / insulator / ferromagnet / superconductor 共SIFS兲 junctions, i.e., S/F/S trilayers with one transparent interface and one tunnel barrier between S and F layers, rep-resent practically interesting case of␲junctions. SIFS struc-ture offers the freedom to tune the critical current density over a wide range and at the same time to realize high values of a product of the junction critical current Icand its normal

state resistance RN.14,15 In addition, Nb based tunnel

junc-tions are usually underdamped, which is desired for many applications. SIFS␲junctions have been proposed as poten-tial logic elements in superconducting logic circuits.16 _SIFS

junctions are also interesting from the fundamental point of view since they provide a convenient model system for a comparative study between 0-␲ transitions observed from the critical current and from the density of states共DOS兲. At the same time, despite such an interest, there is no complete theory yet of SIFS junctions which could provide quantita-tive predictions for critical current and DOS in such struc-tures. All existing theories dealt only with a number of lim-iting cases, when either linearized quasiclassical equations can be used for analysis17_{共e.g., temperature range near}

criti-cal temperature, small transparency of interfaces兲 or thick-ness of the F layer is small1–3_{compared to the decay}

char-acteristic length ␰_f1. Further, in symmetric S/F/S junctions, the extension of theory to the case of nonhomogeneous mag-netization and large mean free path was performed in Refs.3

and18.

The purpose of this work is to provide a quantitative model describing the behavior of critical current and DOS in SIFS junctions as a function of parameters characterizing material properties of the S and F layers and the S/F interface transparency. The model provides a tool to fit experimental data in existing SIFS junctions.

The paper is organized as follows. In the next section, we formulate the theoretical model and basic equations. In Sec. III, we solve nonlinear Usadel equations, apply solutions for calculation of critical current in SIFS junctions with long ferromagnetic layer, df
␰f1, and fit recent experimental

data. In Sec. IV, we perform numerical calculations for criti-cal current in a SIFS junction with arbitrary length of the F layer. In Sec. V, we numerically calculate DOS in the ferro-magnetic interlayer, and then summarize results in Sec. VI.

II. MODEL AND BASIC EQUATIONS

The model of an S/F/S junction we are going to study is depicted in Fig. 1 and consists of a ferromagnetic layer of thickness dfand two thick superconducting electrodes along

the x direction. Left and right superconductor/ferromagnet interfaces are characterized by the dimensionless parameters

␥B1 and␥B2, respectively, where ␥B1,B2= RB1,B2␴n/␰n, RB1,B2

are the resistances of left and right S/F interfaces, respec-tively,␴n is the conductivity of the F layer, ␰n=

冑

Df/2␲Tc,

S

F

S

-d /2f 0 d /2f x

g

B2

g

B1

FIG. 1. Geometry of the considered system. The thickness of the ferromagnetic interlayer is df. The transparency of the left S/F

in-terface is characterized by the␥_B1coefficient, and the transparency of the right F/S interface is characterized by␥B2.

Dfis the diffusion coefficient in the ferromagnetic metal, and

Tc is the critical temperature of the superconductor共we

as-sumeប=kB= 1兲. We also assume that the S/F interfaces are

not magnetically active. We will consider diffusive limit, in which the elastic scattering lengthᐉ is much smaller than the decay characteristic length␰f1. In this paper we concentrate

on the case of a SIFS tunnel Josephson junction, when

␥B1
1 共tunnel barrier兲 and␥B2= 0 共fully transparent

inter-face兲. For comparison, we also consider two other limiting cases: an SFS junction 共␥B1=␥B2= 0兲 and a SIFIS junction

共␥B1,␥B2
1兲.

Under conditions described above, the calculation of the Josephson current requires solution of the one-dimensional Usadel equations.19 _{In the F layer, the equation has the}

form20,21 Df ⳵ ⳵x

冉

G ˆ f↑共↓兲 ⳵ ⳵xG ˆ f↑共↓兲

冊

=

冋

共␻⫾ ih兲␴z + 1 2␶m ␴zGˆf↑共↓兲␴z,Gˆf↑共↓兲

册

, 共1兲

where positive sign ahead of h corresponds to the spin up state 共↑兲 and negative sign to the spin down state 共↓兲, ␻ = 2␲T共n+1₂兲 are the Matsubara frequencies, h is the

ex-change field in the ferromagnet, and␴zis the Pauli matrix in

the Nambu space. The parameter␶mis the spin-flip scattering

time. The influence of spin-flip scattering on various proper-ties of S/F structures was considered in a number of papers.13,20–22,25,31,32 _{We consider the ferromagnet with}

strong uniaxial anisotropy, in which case the magnetic scat-tering does not couple the spin up and spin down electron populations.

The Usadel equation in the S layer can be written as19

Ds ⳵ ⳵x

冉

G ˆ s ⳵ ⳵xG ˆ s

冊

=关␻␴z+⌬ˆ共x兲,Gˆs兴, 共2兲

where Ds is the diffusion coefficient in the superconductor.

In Eq.共2兲, Gˆs⬅Gˆs_↑共↓兲and we omit subscripts “↑ 共↓兲” because

equations in superconductor look identically for spin up and spin down electron states.

In Eqs.共1兲 and 共2兲, we use following matrix notations 共we

omit f, s, and↑ 共↓兲 subscripts兲:

Gˆ 共x,␻兲 =

冉

G F

F* − G

冊

, ⌬ˆ共x兲 =

冉

0 ⌬共x兲

⌬*_{共x兲 0}

冊

, 共3兲 where G and F are normal and anomalous Green’s functions, respectively, and⌬共x兲 is the superconducting pair potential. The matrix Green’s function Gˆ satisfies the normalization condition

Gˆ2= 1, G2+ FF*= 1, 共4兲 and the pair potential ⌬共x兲 is determined by the self-consistency equation,

⌬共x兲lnTc

T =␲T_␻⬎0

兺

冉

2⌬共x兲

␻ − Fs_↑− Fs_↓

冊

. 共5兲

The boundary conditions for the Usadel equations at the left and right sides of each S/F interface are given by relations23 ␰n␥

冉

Gˆf ⳵ ⳵xG ˆ f

冊

⫾df/2 =␰s

冉

Gˆs ⳵ ⳵xG ˆ s

冊

⫾df/2 , 共6a兲 2␰n␥B1

冉

Gˆf ⳵ ⳵xG ˆ f

冊

−d_f/2 =关Gˆs,Gˆf兴−d_f/2, 共6b兲 2␰n␥B2

冉

Gˆf ⳵ ⳵xG ˆ f

冊

d_f/2 =关Gˆf,Gˆs兴d_f/2, 共6c兲

where␥=␰s␴n/␰n␴s,␴sis the conductivity of the S layer, and

␰s=

冑

Ds/2␲Tc.

To complete the boundary problem, we also set boundary conditions at x =⫾⬁, Gs共⫾⬁兲 = ␻

冑

兩⌬兩2_+␻2, 共7a兲 Fs共− ⬁兲 = 兩⌬兩e−i␸/2

冑

兩⌬兩2_+␻2, Fs共+ ⬁兲 = 兩⌬兩ei␸/2

冑

兩⌬兩2_+␻2, 共7b兲

where␸is the superconducting phase difference between S electrodes.

In Matsubara technique, it is convenient to parametrize the Green’s function in the following way, making use of the normalization condition关Eq. 共4兲兴:24

Gˆ =

冉

cos␪ sin␪e

i␹

sin␪e−i␹ − cos␪

冊

. 共8兲 Solving a system of nonlinear differential equations关Eqs. 共1兲–共7兲兴, generally can be fulfilled only numerically. We

present full numerical calculation in Sec. IV. The analytical solution can be constructed in case of one S/F bilayer, when we can set the phase␹in Eq.共8兲 to zero. We can also set the

phase to zero in case of long S/F/S junction, where the thick-ness of the ferromagnetic layer df
␰f1. In that case, the

decay of the Cooper pair wave function in first approxima-tion occurs independently near each interface. Therefore, we can consider the behavior of the anomalous Green’s function near each S/F interface, assuming that the ferromagnetic in-terlayer is infinite. This analytical calculation for an S/F/S trilayer with long ferromagnetic interlayer is performed in the next section.

The general expression for the supercurrent is given by

Js= i␲T␴n 4e _{n=−⬁,␴=↑,↓}

兺

+⬁

冉

F˜f␴ ⳵ ⳵xFf␴− Ff␴ ⳵ ⳵xF ˜ f␴

冊

, 共9兲

where F˜f_↑共↓兲共x,␻兲=Ff*_↑共↓兲共x,−␻兲 are the anomalous Green’s

functions in the ferromagnet.

III. CRITICAL CURRENT OF JUNCTIONS WITH LONG FERROMAGNETIC INTERLAYER

We need to solve the complete nonlinear Usadel equations in the ferromagnet关Eq. 共1兲兴. For SIFS junctions, an

analyti-cal solution may be found if df
␰f1 and we can set the

phase of the anomalous Green’s function to zero共see discus-sion in Sec. II兲.

Setting ␹s=␹f= 0, we have the following␪

parametriza-tions of the normal and anomalous Green’s funcparametriza-tions 关Eq. 共8兲兴, G=cos␪and F = sin␪. In this case, we can write Eq.共1兲

in the F layer as Df 2 ⳵2_␪ f_↑共↓兲 ⳵x2 =

冉

␻⫾ ih + cos␪f_↑共↓兲 ␶m

冊

sin␪f↑共↓兲. 共10兲

In the S layer, the Usadel equation 关Eq. 共2兲兴 may be now

written as Ds 2 ⳵2_␪ s ⳵x2 =␻sin␪s−⌬共x兲cos␪s. 共11兲

The self-consistency equation in the S layer acquires the form

⌬共x兲lnTc

T =␲T_␻⬎0

兺

冉

2⌬共x兲

␻ − sin␪s↑− sin␪s↓

冊

. 共12兲

In the case of ␹s=␹f= 0, the boundary conditions 关Eqs.

共6a兲–共6c兲兴 for the functions␪_f,sat each S/F interface can be written as ␰n␥

冉

⳵␪f ⳵x

冊

_⫾d f/2 =␰s

冉

⳵␪s ⳵x

冊

_⫾d f/2 , 共13a兲 ␰n␥B1

冉

⳵␪f ⳵x

冊

−d_f/2= sin共␪f −␪s兲−d_f/2, 共13b兲 ␰n␥B2

冉

⳵␪f ⳵x

冊

d_f/2 = sin共␪s−␪f兲d_f/2. 共13c兲

The boundary conditions at x =⫾⬁ are

␪s共⫾⬁兲 = arctan

兩⌬兩

␻ . 共14兲

In the equation for the supercurrent关Eq. 共9兲兴, the

summa-tion goes over all Matsubara frequencies. It is possible to rewrite the sum only over positive Matsubara frequencies due to the symmetry relation

␪f_共s兲↑共␻兲 =␪f_共s兲↓共−␻兲. 共15兲

In what follows, we will use only␻⬎0 in equations contain-ing␻.

For the left interface共tunnel barrier at x=−df/2兲, a first

integral of Eq.共10兲 leads to ␰f 2 ⳵␪f ⳵x = − q sin ␪f 2

冑

1 −⑀ 2_sin2␪f 2 , 共16兲

where ␰f=

冑

Df/h and the boundary condition ␪f共x→⬁兲=0

has been used. In Eq.共16兲, we use the following notations:

q =

冑

2/h

冑

␻⫾ ih + 1/␶m, 共17a兲

⑀2_=共1/␶

m兲共␻⫾ ih + 1/␶m兲−1. 共17b兲

Here, we again adopt convention that positive sign ahead of

h corresponds to the spin up state共↑兲 and negative sign to the

spin down state共↓兲. Here and below, we did not write spin labels ↑共↓兲 explicitly but imply them everywhere they needed.

For the right interface共x=df/2兲, a first integral of Eq. 共10兲

leads to a similar equation,

␰f 2 ⳵␪f ⳵x = q sin ␪f 2

冑

1 −⑀ 2_sin2␪f 2. 共18兲

Following Faure et al.,25 _{we integrate Eq. 共16兲, which}

gives

冑

1 −⑀2sin2 ␪f 2 − cos ␪f 2

冑

1 −⑀2sin2 ␪f 2 + cos ␪f 2 = g1exp

冉

− 2q df/2 + x ␰f

冊

. 共19兲 The integration constant g1 in Eq. 共19兲 should be

deter-mined from the boundary condition at the left S/F interface 关Eq. 共13b兲兴. Since we consider the tunnel limit 共␥B1Ⰷ1兲, we

can neglect small␪f in the right hand side of Eq.共13b兲 and

also assume, neglecting the inverse proximity effect,

␪s共− df/2兲 = arctan

兩⌬兩

␻ . 共20兲

Then, Eq.共13b兲 becomes ␰n␥B1

冉

⳵␪f ⳵x

冊

_−d f/2 = − G共n兲, G共n兲 =

_冑

兩⌬兩 ␻2_+兩⌬兩2. 共21兲

From Eqs.共16兲 and 共21兲, we obtain the boundary value of␪f

at x = −df/2 and substituting it into Eq. 共19兲, we finally get

g1= G2共n兲 16␥_B12 1 −⑀2 q2

冉

␰f ␰n

冊

2 . 共22兲

Linearizing Eq. 共19兲, we can now obtain the anomalous

Green’s function in the ferromagnetic layer of the SIF tunnel junction with infinite F layer thickness. Similar formula for the FS bilayer with a transparent interface共␥B2= 0兲 was

de-veloped by Faure et al.25 _{关to obtain it one should integrate}

Eq. 共18兲 and then linearize the resulting equation兴. The

anomalous Green’s function at the center of the F layer in a SIFS junction may be taken as the superposition of the two decaying functions, taking into account the phase difference in each superconducting electrode,

␪f= 4

冑

1 −⑀2

冋

冑

g1exp

冉

− q df/2 + x ␰f − i␸ 2

冊

+

冑

g₂exp

冉

qx − df/2 ␰f + i␸ 2

冊

册

. 共23兲

The expression for g2 was obtained in Ref. 25for the rigid

boundary conditions at the transparent FS interface,

g₂= 共1 −⑀

2_兲F2_共n兲

关

冑

共1 −⑀2_兲F2_{共n兲 + 1 + 1兴}2, 共24a兲

F共n兲 = 兩⌬兩

␻+

冑

␻2+兩⌬兩2. 共24b兲 Using the above solutions and Eqs.共9兲 and 共15兲, we arrive at

sinusoidal current-phase relation in a SIFS tunnel Josephson junction with the critical current

IcRN= 16␲T e Re

冋

兺

_n=0 ⬁ G共n兲F共n兲exp共− qdf/␰f兲

冑

共1 −⑀2_兲F2_{共n兲 + 1 + 1}

册

. 共25兲

Here and below, we fix positive sign in the definition of q and ⑀2 _{in Eqs. 共17a兲 and 共17b}兲: q=冑2/h冑_{␻+ ih + 1/␶}

m and

⑀2_=共1/␶

m兲共␻+ ih + 1/␶m兲−1. It is possible since we already

performed summation over spin states and have to define now spin-independent values. In Eq.共25兲 and below, RNis a

full resistance of an S/F/S trilayer, which include both inter-face resistances of left and right interinter-faces and the resistance of the ferromagnetic interlayer. In case of SIFS and SIFIS junctions, the F layer resistance can be neglected compared to large resistance of the tunnel barrier.

At this point, we define the characteristic lengths of the decay and oscillations␰_f1,2as

q/␰f= 1/␰f1+ i/␰f2, 共26a兲 1 ␰f1,2 = 1 ␰f

冑

冑

1 +

冉

␻ h + 1 h␶m

冊

2 ⫾

冉

␻ h + 1 h␶m

冊

. 共26b兲 The critical current in Eq.共25兲 is proportional to the small

exponent exp共−df/␰f1兲. The terms neglected in our approach

are of the order of exp共−2df/␰f1兲 and they give a tiny

second-harmonic term in the current-phase relation.

The critical current equation关Eq. 共25兲兴 can be simplified

in the limit of vanishing magnetic scattering,␶m−1Ⰶ␲Tc,

IcRN= 16␲T e n=0

兺

⬁

冤

G共n兲F共n兲exp

冉

− df ␰f1

冊

cos

冉

df ␰f2

冊

冑

F2共n兲 + 1 + 1

冥

. 共27兲

Equation共25兲 also simplifies near Tcand may be written as

共for TcⰆh兲 IcRN= ␲兩⌬兩2 2eTc exp

冉

− df ␰f1

冊

cos

冉

df ␰f2

冊

. 共28兲

The damped oscillatory behavior of the critical current can be clearly seen from this equation. With increasing df, the

junction undergoes the sequence of 0-␲ transitions when positive values of the IcRNproduct correspond to a zero state

and negative values correspond to a␲state.

Equation共28兲 in the absence of spin-flip scattering

coin-cides with the corresponding equation关Eq. 共37兲兴 from Ref.

17, taken in the limit of long dfⰇ␰f1 in case of ␥B1Ⰷ1,

␥B2= 0.

Using the same approach, we can obtain the equation for the critical current in a SIFIS structure with two strong tun-nel barriers between the ferromagnet and both superconduct-ing layers共␥_B1,2Ⰷ1兲, IcRN= 4␲T␰f e␰n ␥B1+␥B2 ␥B1␥B2 Re

冤

兺

n=0 ⬁ G2共n兲exp

冉

− qdf ␰f

冊

q

冥

. 共29兲 This formula coincides with corresponding expression 关Eq. 共39兲兴 for the critical current in a SIFIS structure in Ref.25

for␥B1,2=␥BⰇ1 and dfⰇ␰f1. Equation共29兲 near Tcmay be

written as共for TcⰆh兲 IcRN= ␲兩⌬兩2_␰ f2 2eTc␰n ␥B1+␥B2 ␥B1␥B2 cos共⌿兲exp

冉

− df ␰f1

冊

sin

冉

⌿ − df ␰f2

冊

, 共30兲 where⌿ is defined by tan共⌿兲=␰f2/␰f1. Equation共30兲 in the

absence of spin-flip scattering coincides with the correspond-ing equation关Eq. 共35兲兴 from Ref. 17, taken in the limit of long dfⰇ␰f1.

We also provide here equation for the critical current in an SFS junction关see Ref.25, Eq.共74兲兴, written in our notations,

IcRN= 64␲Tdf e␰f Re

冋

兺

n=0 ⬁ F2共n兲q exp共− qdf/␰f兲 关

冑

共1 −⑀2_兲F2_{共n兲 + 1 + 1兴}2

册

. 共31兲 We compare critical current dependencies over dffor SFS

关Eq. 共31兲兴, SIFS 关Eq. 共25兲兴, and SIFIS 关Eq. 共29兲兴 structures in

Fig.2. Each of above junction types undergoes the sequence of 0-␲ transitions with increasing thickness of the F layer. From the figure, we see that the transition from 0 to␲state occurs in SIFS tunnel junctions at shorter df than in SFS

junctions with transparent interfaces, but at longer dfthan in

SIFIS junctions with two strong tunnel barriers. This ten-dency can be qualitatively explained by the fact that in struc-FIG. 2.共Color online兲 The F layer thickness dependence of the critical current for SFS 共␥_B1,2= 0兲, SIFS 共␥_B1= 102_{, ␥}

B2= 0兲, and

SIFIS共␥B1,2= 102兲 junctions in the absence of spin-flip scattering.

Red dashed lines correspond to the modulus of the analytical results 关Eqs. 共31兲, 共25兲, and 共29兲兴 and black solid lines correspond to the

tures with barriers 共SIFS, SIFIS兲 part of the ␲ phase shift occurs across the barriers. Therefore, a thinner F layer in a SIFS junction compared to an SFS one is needed to provide the total shift of␲due to the order parameter oscillation. For the same reason, 0-␲transition in a SIFIS junction occurs at a smaller thickness than in a SIFS junction. We note that in Fig. 2, we plot both analytical and numerical calculated

Ic共df兲 dependencies, where numerical calculation was

per-formed for full boundary problem关Eqs. 共1兲–共7兲兴 共see further

discussion in Sec. IV兲.

In Fig.3, we plot the F layer thickness dependence of the critical current in a SIFS junction for different values of spin-flip scattering time. For stronger spin-spin-flip scattering, the pe-riod of supercurrent oscillations increases and the point of 0-␲ transition shifts to the region of larger df. The same

tendency exists for SFS and SIFIS junctions.25

In Fig.4, we plot the F layer thickness dependence of the critical current in a SIFS junction for different values of the exchange field h. We see that for large exchange fields h Ⰷ␲Tc, the critical current scales with the ferromagnetic

co-herence length␰f.

From comparison with numerical results presented in Fig.

2, we can conclude that the results for the critical current in

SIFS junctions presented in Figs.3 and4 give correct mag-nitude of the IcRN product for dfⲏ␰n/2.

As an application of the developed formalism, we present in Fig. 5 the theoretical fit of the experimental data for a Nb/Al₂O₃/Ni_0.6Cu_0.4/Nb junctions by Weides et al.14

mak-ing use of Eq.共25兲. We used following values of parameters:

RB= 3.9 m⍀, Df= 3.9 cm2/s, T=4.2 K,14 and Tc= 7.2 K

共damped critical temperature in Nb兲. Good agreement was obtained with the following parameters: h/kB= 950 K and

1/␶m= 1.6 h共see Fig.5兲. These parameters can be compared

with parameters obtained by Oboznov et al.13 for similar ferromagnetic material, Ni0.53Cu0.47: h/kB= 850 K and 1/␶m

= 1.3 h. Higher Ni concentration in the NiCu alloy in the experiment of Weides et al. results in higher exchange field. In Ref.13, it was suggested that a “dead” layer exists in the ferromagnet near each S/F interface, which does not take part in the “oscillating” superconductivity. Other authors also include into consideration the existence of nonmagnetic lay-ers at the interface of the ferromagnet and the supercon-ductor or normal metal.26,27,32 _{Thickness of the dead layer}

cannot be calculated quantitatively in the framework of our model and also cannot be directly estimated from the experi-ment. In the experiment of Weides et al.,14 _{the range of F}

layer thicknesses was rather narrow and only the first 0-␲ transition was observed. Due to these reasons, we did not take into account the existence of a nonmagnetic layer in our fit. This question deserves separate detail experimental and theoretical study.

We should mention that the above estimates of exchange field and spin-flip scattering time could be different if we consider magnetically active S/F interfaces. It was shown in Ref.28that the effect of spin-dependent boundary conditions on the superconducting proximity effect in a diffusive ferro-magnet results in the change of the period of critical current oscillations.

IV. CRITICAL CURRENT OF JUNCTIONS WITH ARBITRARY LENGTH OF THE FERROMAGNETIC

INTERLAYER

In the previous section, we derived the expression for the critical current of a SIFS junction in case of considerably FIG. 3.共Color online兲 The F layer thickness dependence of the

critical current in a SIFS junction 关modulus of the Eq. 共25兲兴 for

different values of␣=1/␲T_c␶_m, h = 3␲T_c, and T = 0.5T_c.

FIG. 4.共Color online兲 The F layer thickness dependence of the critical current in a SIFS junction 关modulus of the Eq. 共25兲兴 for

different values of exchange field h in the absence of spin-flip scat-tering, T = 0.5Tc.

FIG. 5.共Color online兲 Fit to the experimental data from Ref.14

for the critical current in a Nb/Al2O3/Ni0.6Cu0.4/Nb junction. The fitting parameters are h/kB= 950 K and 1/␶m= 1.6 h.

long F layer thickness, dfⰇ␰f1. For arbitrary F layer

thick-ness in the absence of spin-flip scattering, general boundary problem关Eqs. 共1兲–共7兲兴 was solved numerically using the

it-erative procedure.29_{Starting from trial values of the complex}

pair potential⌬共x兲 and the Green’s functions Gˆs,f, we solve

the resulting boundary problem. After this, we recalculate

Gˆ_s,f and⌬共x兲. We repeat the iterations until convergency is reached. The self-consistency of calculations is checked by the condition of conservation of the supercurrent across the junction.

In Fig.2, we compare numerically and analytically calcu-lated Ic共df兲 dependencies in case of SFS, SIFS, and SIFIS

junctions. We see that, as expected, the numerical method provides correction only for small length of ferromagnetic layer. We note that for SFS and SIFS junctions, analytical curves关Eq. 共31兲 and 共25兲兴 practically coincide with

numeri-cal results in the region of the first 0-␲ transition. For a SIFIS junction, this transition occurs at smaller df, where the

assumptions of Sec. III are not valid. However, in the pres-ence of strong spin-flip scattering the first 0-␲ transition peak in a SIFIS junction shifts to the region of larger dfand

Eq.共29兲 describes the transition accurately.

The main result of this section is that Eq. 共25兲 for the

critical current of a SIFS junction can be used as a tool to fit experimental data in SIFS junctions with good accuracy.

V. DENSITY OF STATES OSCILLATIONS IN THE FERROMAGNETIC INTERLAYER

It is known that in a ferromagnetic metal attached to the superconductor the quasiparticle DOS at energies close to the Fermi energy has a damped oscillatory behavior.33–35

Experi-mental evidence for such behavior was provided by Kontos

et al.36_{In SIFS junctions, we can compare the DOS}

oscilla-tions with the critical current oscillaoscilla-tions.

We are interested in the quasiparticle DOS in the F layer in the vicinity of the tunnel barrier共x=−df/2+0 in Fig. 1兲.

Below, we will refer to the local DOS at this point. For the case of strong tunnel barrier共␥_B1Ⰷ1兲, left S layer and right FS bilayer in Fig. 1 are uncoupled. Therefore, we need to calculate the DOS in the FS bilayer at the free boundary of the ferromagnet. Solving numerically Eqs.共10兲–共14兲, we set

to zero the␪f derivative at the free edge of the FS bilayer,

x = −df/2, 共⳵␪f/⳵x兲−df/2= 0.

31

We use the self-consistent two step iterative procedure.29–31_{In the first step, we calculate the pair}

poten-tial coordinate dependence ⌬共x兲 using the self-consistency equation in the S layer关Eq. 共12兲兴. Then, by proceeding to the

analytical continuation in Eqs.共10兲 and 共11兲 over the

quasi-particle energy i␻→E+i0 and using the ⌬共x兲 dependence obtained in the previous step, we find the Green’s functions by repeating the iterations until convergency is reached. We define the full DOS N共E兲 and the spin resolved DOS

N_↑共↓兲共E兲, normalized to the DOS in the normal state, as N共E兲 = 关N_↑共E兲 + N_↓共E兲兴/2, 共32a兲 N_↑共↓兲共E兲 = Re关cos␪_↑共↓兲共i␻→ E + i0兲兴. 共32b兲

The numerically obtained energy dependencies of the DOS at the free F boundary of the FS bilayer are presented in Figs.6and7. Figure6demonstrates the DOS energy depen-dence for different df. At small df, full DOS turns to zero

FIG. 6. DOS on the free boundary of the F layer in the FS bilayer calculated numerically in the absence of spin-flip scattering for different values of the F layer thickness df: N↑共E兲 共dashed line兲,

N_↓共E兲 共dotted line兲, and N共E兲 共solid line兲, E_ex= 3␲T_c, and T = 0.5Tc. 共a兲 df/␰n= 0.4, 共b兲 df/␰n= 1, 共c兲 df/␰n= 1.6, and 共d兲 df/␰n

= 2.2.

FIG. 7. DOS N共E兲 on the free boundary of the F layer in the FS bilayer calculated numerically for ␣=1/␲Tc␶m= 0 共solid line兲, ␣

= 0.5共dashed line兲, and␣=1 共dotted line兲 for different values of the F layer thickness d_f, E_ex= 3␲T_c, and T = 0.5T_c. 共a兲 d_f/␰_n= 0.4, 共b兲 df/␰n= 1,共c兲 df/␰n= 1.6, and共d兲 df/␰n= 2.2.

inside a minigap, which vanishes with the increase of df.

Then, the DOS at the Fermi energy N共0兲 rapidly increases to the values larger than unity and with further increase of dfit

oscillates around unity, while its absolute value exponentially approaches unity共see also Fig.8兲. In Fig.6, we also plot the spin resolved DOS energy dependencies N_↑共E兲 and N_↓共E兲. Figure7 demonstrates full DOS energy dependence for dif-ferent values of flip scattering time. For stronger spin-flip scattering, the minigap closes at smaller df, the period of

the DOS oscillations at the Fermi energy increases, and the damped exponential decay occurs faster.

In case of long F layer 共dfⰇ␰f1兲 it is also possible to

obtain an analytical expression for the DOS at the free boundary of the ferromagnet,

N_↑共↓兲共E兲 = Re关cos␪b↑共↓兲兴 ⬇ 1 −

1

2 Re␪b↑共↓兲

2 _{, 共33兲}

where␪b↑共↓兲is a boundary value of␪f at x = −df/2. It can be

obtained by the mapping method, similar to the one used in the electrostatic problems. We consider the FS bilayer where

x苸关−df/2,df/2兴 stands for the ferromagnetic metal and x

⬎df/2 stands for the superconductor; the point x=−df/2

cor-responds to the free F layer boundary. For infinite F layer 共df→⬁兲, the solution for␪f_↑共↓兲far from the interface is given

by the exponential term in Eq.共23兲, written in the real energy

space, ␪ឈf↑共↓兲= 4

冑

1 −␩2

冑

g2exp

冉

p x − df/2 ␰f

冊

, 共34兲 where p =

冑

2/h

冑

− iER⫾ ih + 1/␶m, 共35a兲 ␩2_=共1/␶ m兲共− iER⫾ ih + 1/␶m兲−1, 共35b兲 g2= 共1 −␩2_兲F2_共E兲 关

冑

共1 −␩2_兲F2_{共E兲 + 1 + 1兴}2, 共35c兲 F共E兲 = 兩⌬兩 − iER+

冑

兩⌬兩2− ER 2, ER= E + i0. 共35d兲

Here, as above, positive sign ahead of h corresponds to the spin up state in Eq.共34兲 and negative sign for the spin down

state. By using the arrow “from right to left” in ␪ឈf↑共↓兲, we

want to stress that this solution is induced in the ferromagnet from the right FS interface.

In the case of finite ferromagnet length, the boundary con-ditions at the free F layer boundary, x = −df/2, become

␪f↑共↓兲共− df/2兲 =␪b↑共↓兲,

冉

⳵␪f_↑共↓兲

⳵x

冊

_−d

f/2

= 0. 共36兲

To ensure these conditions, we add another exponential so-lution, ␪ជf↑共↓兲= 4

冑

1 −␩2

冑

g2exp

冉

− p 3df/2 + x ␰f

冊

, 共37兲

resulting from the mirror image of the F layer with respect to the point x = −df/2. At x=−df/2 both exponential terms are

equal to each other and the final solution, ␪b↑共↓兲

=␪ឈf_↑共↓兲共−df/2兲+␪ជf_↑共↓兲共−df/2兲, is two times larger than the

so-lution for infinite ferromagnetic layer at this point and reads

␪b↑共↓兲= 8F共E兲

冑

共1 −␩2_兲F2_{共E兲 + 1 + 1}exp

冉

− p df ␰f

冊

. 共38兲 This equation coincides with the result obtained in Ref. 32

by direct integration of the Usadel equation.

In Fig.8, we plot analytically and numerically calculated function

␦N共df兲 = 兩1 − N0兩, N0= N共E = 0兲, 共39兲

together with the Ic共df兲 dependence for a SIFS junction. We

see that the point of 0-␲transition on the Ic共df兲 plot does not

coincide with the first minimum of␦N共df兲 corresponding to

sign change of 1 − N0. This difference can be qualitatively

explained as follows. The transition from 0 to␲ state in a junction, seen as sign change of Ic共df兲, is the result of

inter-ference of solutions for␪foriginating from two S electrodes.

0-␲ transition in Ic共df兲 occurs approximately at such

thick-ness df when the boundary value of ␪f in Eq. 共23兲 at x

= −df/2 becomes negative, i.e., when ␪f acquires the phase

shift␲. On the other hand, sign change of 1 − N₀ occurs at such df when the boundary value␪bin Eq.共38兲 becomes an

imaginary number, i.e., when␪facquires the phase shift␲/2.

It occurs at smaller df compared to 0-␲ transition in the

critical current. Corresponding 0 and␲states defined from Ic

and from the DOS are indicated in Fig.8.

It is also seen from Fig.8 that the DOS oscillations have the period approximately twice smaller than those of the critical current. This fact is easy to see from the analytical expression for␦N共df兲. Using Eqs. 共32兲–共39兲, we obtain

FIG. 8. 共Color online兲 The F-layer dependence of the function ␦N共df兲 in the absence of spin-flip scattering, h=3␲Tc, T = 0.5Tc.

Black solid line is a result of the numerical calculation; blue dashed line is calculated with the use of Eq.共41兲. Red line shows

normal-ized critical current for a SIFS junction. Zero and␲ states defined from I_care indicated by red color, while zero and␲ states defined from the DOS are indicated by black color.

␦N共df兲 = 32

冏

Re

冋

1 共

冑

2 −␩02+ 1兲2exp

冉

− p0 2df ␰f

冊

册

冏

, 共40兲 where␩0=␩共E=0兲 and p0= p共E=0兲 in Eqs. 共35a兲 and 共35b兲.

At vanishing magnetic scattering, ␶m−1Ⰶ␲TC, this equation

can be simplified, ␦N共df兲 = 32 3 + 2

冑

2

冏

exp

冉

− 2df ␰f1

冊

cos

冉

2df ␰f2

冊

冏

, 共41兲

where characteristic lengths of decay and oscillations ␰f1,2

are given by Eq.共26b兲 with the substitution i␻→E+i0. This

equation can be compared with Eq. 共27兲. We see that the

period of the DOS oscillations is approximately twice smaller than the period of the critical current oscillations and the exponential decay is approximately twice faster than the decay of the critical current.

VI. CONCLUSION

We have developed a quantitative model, which describes the oscillations of the critical current as a function of the F layer thickness in an SIFS tunnel junctions with thick ferro-magnetic interlayer, dfⰇ␰f1, in the dirty limit. We justified

this model by numerical calculations in general case of arbi-trary df: for all values of parameters characterizing material

properties of the ferromagnetic metal numerical and analyti-cal results coincide in physianalyti-cally important region of the first 0-␲transition. Thus, the derived analytical expression for the critical current can be used as a tool to fit experimental data in various types of SIFS junctions. We have discussed the details of the damped oscillatory behavior of the critical cur-rent for diffecur-rent values of the F layer parameters.

We also studied the superconducting DOS induced in a ferromagnet by the proximity effect. We showed that the oscillation pattern of DOS at the Fermi energy in the ferro-magnet共at location of the tunnel junction兲 does not coincide with that of the critical current in a SIFS junction and its period is approximately twice smaller. Therefore, the DOS oscillations do not reflect the 0-␲ transition in Ic共df兲. We

calculated the quasiparticle DOS in the F layer in the close vicinity of the tunnel barrier which can be used to obtain current-voltage characteristics for a SIFS junction. These cal-culations will be presented elsewhere.

Finally, we used our results to fit recent experimental data for SIFS tunnel junctions and extracted important parameters of the ferromagnetic interlayer.

ACKNOWLEDGMENTS

The authors thank W. Belzig, E. V. Bezuglyi, A. I. Buzdin, T. Champel, S. Kawabata, and F. Pistolesi for useful discus-sions. This work was supported by NanoNed program under Project No. TCS7029 and RFBR Project No. N08-02-90012.

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