Gross R., Sidorenko A., Tagirov L. Nanoscale Devices

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178 M.Yu. Kupriyanov, A.A. Golubov, M. Siegel

In the SF bilayer )( −∞→x modifications of DoS due to spin splitting of

energy levels in F were investigated in [21, 22]. The DoS per spin

projection in the F layer has the form (below we will suppose for simplicity

that F and N metals differs only due to existence of exchange field

0H ≠

in the F film)

)sgn(

)(

∆−

↓↑

(10)

γγ

εγεε

−∆

↓↑

,)(

m .

Equation (10) demonstrates the energy renormalization due to the exchange

field. From (10) it follows that there are two minigaps in the spectrum

↑g

and

↓g

, and

↑↓

≤≤

Spin resolved DoS for identical transport parameters on the F and N

metals at FN interface (x = 0) has the form

22 22 22

2()

Re ,

sgn( ) sgn( ) 2

NFN

εε ε

εε εε ε

↑↓

−− +

∆− + ∆− + ∆−

%% %

%%%

(11)

22 22 2

sgn( )sgn( )

FF F

εε εεεε

↑↓ ↑

∆− ∆− − −∆

%% %%

It follows from Eq. (11) that, similar to the case of the SF bilayer

considered above, the minigap exists if

. With increasing

exchange field the total DoS at

becomes nonzero if

≥

and is

given by the simple expression

πγ

)(

)0(

222

−

= .

(12)

Figure 2 shows the densities of states calculated far from FN interface

both in the SN region (open squares) and the spin resolved DoS in the SF

part (open circles and triangular). Shot dash, dash dot and solid lines show

Josephson Effect in Composite Junctions with Ferromagnetic Materials 179

the spin resolved and full DoS at the FN interface. All calculations were

done for

5.0

It is clearly seen from both (11) and the results of numerical calculations

(see Fig. 2) that there are four characteristic energies in the spectrum. They

are

↓↑ g

, and ∆. Here

↓g

is the minigap for the spin-down subband

in the SF bilayer at x→-∞. It follows from (11) that

↓F

↓

≤

εε

and becomes nonzero at

↓

εε

, i.e.

↓g

is the minigap for the spin-

down subband in S(FN) at x = 0. However, contrary to the SF case

)(

↓F

has no peak at

↓

but grows continuously from zero value.

For the spin-up subband, the minigap in

)(

↑F

N is not equal to the gap

↑g

in the spin-up subband in the SF bilayer at x→-∞. Instead, the gap

value in this subband is determined by

, the minigap in the SN bilayer at

x→∞. The formal reason is that for energies

εε

≥ both the numerator

and denominator in (11) become complex thus leading to a nonzero DoS in

this energy range. Similar to the spin-down case, )(

↑F

N has no peak at

the gap energy

= , while the peak occurs at

↑

(see Fig. 2). With

further increase of energy there is a sharp peak in DoS at

∆

followed by

saturation of full DoS at

N for

∆

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Hγ=0.5πT

ε)/N

ε/∆

SF down

SF up

S(FN) down

S(FN) up

S(FN) total

Fig. 1. Densities of states calculated far from the FN interface both in the SN

region (open squares) and spin resolved DoS in the SF part (open circles and

triangles). Short dash, dash dot and solid lines show the spin resolved and full DoS

at FN interface.

180 M.Yu. Kupriyanov, A.A. Golubov, M. Siegel

Critical Current of S(F/N)I(F/N)S Tunnel Junctions

The theory for the proximity effect in complex S(F/N) proximity

multilayers developed above allows for the direct examination of the

Josephson effect in the junctions composed of S(F/N) electrodes. To

calculate a dc current J across the tunnel barrier in tunnel junctions with

complex S(F/N) electrodes we may start with the general expression for the

current in

-parameterization

** *

sin cos sin cos ,

exx

ωωω ωωω

θθ θ θ θθ

=∞

−− −

=−∞

∂∂



=−



∂∂



∑

(13)

and making use of boundary conditions [17] at the isolator I of the

S(F/N)I(F/N)S tunnel structure and symmetry relations

}exp{sin}exp{sin,coscos

χθχθθθ

ωωωω

−−

=−= ,

(14)

we arrive at

[]

ϕθθ

ωω

sinsinsinRe

∑

∞=

−∞=

AeR

(15)

where

–

) is the phase difference of the order parameters across

the structure, R

is a normal junctions resistance, A

is the junction area and

the Usadel functions of the upper (U) and the bottom (B) electrodes are

defined by equations (4)-(7), (9).

In particular, for relatively small transparency of SN and SF interfaces

≥

the equations (4)-(7), (9), (15) may be linearized resulting in

the fully symmetric structure

[]











≤

≥

∑

∞

0,)(Re

)(

AeR

(16)

where

Re 1 2 exp exp ,

()

CNN

ωωγ π ζ ζ







∆



=+−+−























(17)

Josephson Effect in Composite Junctions with Ferromagnetic Materials 181

πγω

=−−

=−

,21

)1(

)1(2

Re 1 2 exp exp ,

()(1)

CNN

ax xa

qU V

ωωγ π ζ ζ









∆



=−++

























()cos()sin, ,

NN C

xb xb

Uq q

µη µη ζ ξ

ζζωγπ

=+ + − =

()sin2cos, 1, ,

xb xb

Vaqbbqaq

µη µη µ η

ζζ

=− − =−+ =−+

−+

It follows fom (16) and (17) that for positive

0>x the critical current

density increases exponentially with x and saturates at the value previously

derived for SNINS tunnel structures. For negative

x the critical current

density exhibits damping oscillations and tends to the value previously

obtained for SFIFS tunnel junctions. For

1>q

the critical current may

change sign far from the FN interface. Therefore the expressions (16) and

(17) describe the continuous transition from "0" to "π" junction in this

structure.

182 M.Yu. Kupriyanov, A.A. Golubov, M. Siegel

-4 -3 -2 -1 0

-0,02

0,00

0,02

0,04

0,06

0,08

0,10

(x)R

x/ξ

Fig. 3. Critical current density calculated at T=0.1T

and as a function of the

distance from F/N interfaces into F metal for various values of

HTgp

-4 -3 -2 -1 0

1E-5

1E-4

1E-3

0,01

0,1

Hγ=15πT

Hγ=10πT

Hγ=5πT

Hγ=2πT

Hγ=30πT

Hγ=50πT

e|J

(x)|R

x/ξ

Fig. 4. Magnitude of critical current density calculated at T=0.1T

and as a

= 1;3;5;8;10;15 (curves 1-6, respectively).

function of the distance from F/N interfaces into F metal for various values of H.

Josephson Effect in Composite Junctions with Ferromagnetic Materials 183

At low temperatures this transition is illustrated in Fig. 3 and Fig. 4. At

TH < , there is only the "0" state and )(xJ

falls monotonically with

0≤x in the SFIFS part of the structure. At

, the transition to

the "π" state occurs and the critical current density tends to zero as

−∞→x . At larger H the oscillation evolves near FN boundary and the

amplitude of this oscillation can be of the order of magnitude larger

compared to

)( −∞→xJ

for

TH ≥ . The number of oscillations of

)(xJ

increases with increasing H.

Boundary Conditions for the Sine-Gordon Equation

If the size of the S(FN)I(FN)S tunnel junction in x direction is large or

comparable with the Josephson penetration depths

of its SFIFS and

SNINS parts located at

≤

x and 0≥x , respectively, than the spatial

distribution of phase difference

across the structure should obey the so-

called sine-Gordon equation:

0sin))/(1(

=+−

±±±±±

ϕξδϕλ

(18)

where

for 0≥x and 1

depending on whether we have a 0 or π

junction on the left hand side of the structure (

≤

x ). The functions

)/(

±±

decay exponentially with x at the distances

- which are much

smaller than the Josephson penetration depths

. It follows from our

consideration that

1)( ∝

xj , while )(xj

−

may be at least one order of

magnitude larger than unity. Despite of this fact in typical experimental

situation

±±±

<< j

λξ

and a solution of Eq. (18) may vary in space only

on a scale much larger

. Taking this into account we can integrate (18)

along x and find the effective boundary conditions which relate the

functions

<<|x|<<

)0(sin)0()0(),0()0()0(

ϕηϕϕϕϕϕ

=−−+=+=−

(19)

∫∫

∞

∞−

−

−=

)()( dzzjdzzj

δη

184 M.Yu. Kupriyanov, A.A. Golubov, M. Siegel

Below we apply these boundary conditions for the examination the

properties of periodic arrays of 0 -π Josephson junctions in S(F/N)I(F/N)S

structures.

Periodic Arrays of 0 - Π Josephson Junctions in

S(F/N)I(F/N)S Structures

Consider now a periodic structure composed of alternating 0- and π-

Josephson junctions of lengths d

and d

πi

respectively. The ground state

phase distribution in the array is determined by the equations

ioii

xxdx

≤≤−=− ,0sin

ϕϕλ

(20)

iii

dxxx

ππ

ϕϕλ

+≤≤=− ,0sin

(21)

where x

is the position of one of FN interfaces. At these interfaces, the

phase difference

and its derivative have to satisfy the conditions (19).

Suppose now that

<<d

, d

πi

. In this situation

(x) may vary in

space on different scales. Following the method described in [15] for the

part )(x

(x) varying slowly on the scale d

, d

πi

we get

)(

,sin

effeff

ππ

λλ

λψψλ

−

=Φ=

(22)

−

+=Φ cos

)(12

))((

)(

222

ππ

ππππ

λλ

λλλλληλ

iiiieff

eff

dddd

In the limit

iiii

dddd

ππ

≈<<−

, and

(x)=0 this equation

transforms into the Buzdin and Koshelev result [15].

Several assumptions were made to derive Eq. (22). First, it was supposed

that the scale of

)(x

variations is larger than d

, d

πi













))((

)(12

)(

,min,

222

iiii

effii

dddd

ππππ

πππ

λλλλ

λλ

(23)

Josephson Effect in Composite Junctions with Ferromagnetic Materials 185

Second, it was also supposed that the array is periodic with the period

+ d

πi

. It means that local size variations of the contiguous items of the

array should be smaller than the accuracy of the consideration, namely

{}

(1) (1)

000

min , , , ( ) max , .

ii ii

dd dd

dd dd dx

ππ ππ

ππ

πππ

λλ

−+

−−



>> +





(24)

Generally, the parameter

in (22) is small and can be neglected if

)(10

))((

222

iiii

dddd

ππππ

ξλλ

λλλλ

+≥

−

(25)

or, in other words, if the lengths of the segments d

, d

πi

are not too small.

It follows from Eq. (22) that if

is not a function of x than the ground

state in alternating 0- and π-Josephson array made of S(F/N)I(F/N)S tunnel

junctions may have the phase difference

determined by equation

λλ

λλλλ

ππ

ππππ

−

222

)(

cos

)(

))((

iiii

dddd

(26)

The phase difference

may take any value in the interval –π ≤

≤ π

depending on the properties of 0 and π segments in the array.

Conclusion

In conclusion, we have investigated theoretically the proximity effect in the

S(F/N) structure and calculated the density of states and the spatial

distribution of critical current in the S(F/N)I(F/N)S tunnel junction.

We have shown that the density of states near the FN interface depends

on four characteristic energies and has a form essentially different from that

in the SN or SF bilayers. This fact should be taken into account in the

calculation and interpretation of the current voltage characteristics of the

S(F/N)I(F/N)S tunnel devices.

The critical current density exhibits damped lateral oscillations in SFIFS

part of S(F/N)I(F/N)S structure. Moreover, the amplitude of these

oscillations

may become an order of magnitude larger compared to the

bulk value of

J far from the FN interface. This fact modifies the form of

186 M.Yu. Kupriyanov, A.A. Golubov, M. Siegel

the boundary conditions which previously have been used at the interfaces

of “0” and “

” segments of alternating arrays studied in [10]-[15] and

should be taken into account if the sizes of the segments d

, d

πi

are small

enough (see Eq. (25). The conditions for realization of “

- junctions”, Eqs.

(23)-(25), have been formulated.

Acknowledgments

This work has been supported by the Russian Ministry of Education and

Science and RFBR Grant N 04 0217397-a.

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Gross R., Sidorenko A., Tagirov L. Nanoscale Devices - Fundamentals and Applications

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