Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

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4 Coexistence of Singlet Superconductivity and Magnetic Order 187

Fig. 4.15. The temperature dependence (qualitative) of:

(a) critical fields H

(T), H

(T)and(b) surface

energy 

(FS)

[62]

the second term is the gain in the magnetic energy

due to the penetration of magnetic field inside the

non-magnetic (∼ ) and ferromagnetic supercon-

ductors (∼ 

eff

), respectively. Since H

(T) < H

(T)

this leads to an increase of the surface energy in FS

withrespect to the non-magnetic superconductor.So,

below some temperature T

but near T

the surface

energy of FS becomes positive, i.e. 

(FS)

(T < T

) >

0. However, in spite that 

(FS)

(T < T

) > 0the

vortex line is thermodynamically stable object in FS

for these temperatures! This is contrary to the case

of non-magnetic superconductors where the vortex

line is unstablewhen thesurface energy 

(S)

> 0.The

order of magnitude of the energy (per unit length)

of the vortex line defect is given by

(S)

= 

8

(1 − 

) , (4.65)

(FS)

= 

8



1−

eff



. (4.66)

One can see that the vortex energy E

(FS)

depends on



and 

eff

whilethesurfaceenergyislinearin and



eff

.This means that the vortex line defect is energet-

ically stable for T

< T < T

where T

is obtained

from the condition E

(FS)

)=0.

4.4.2 Antiferromagnetic Superconductors

in the Magnetic Field

IAs a rule the appearance of the AF ordering be-

low T

is at the origin of an anomalous temperature

dependence of the upper critical H

(T) in antiferro-

magnetic superconductors.Some typical examples of

the H

(T)curvesmeasuredintheREternarycom-

pounds Tb

1.2

, Dy

1.2

,intheREternary

compounds Tb

1.2

, Dy

1.2

,haveinmany

respects similar properties.For instance,the increase

of H

below T

in SmRh

(and also in ErNi

C),

presented in Fig. 4.16, may be related to a decrease of

the magnetic scattering below T

.On the other hand

the decrease of H

(T) immediately below T

is par-

ticulary pronounced in Dy

1.2

;seeFig.4.16.This

effectis duetotheinternalexchangefieldh

∼ 

which is created by the polarized magnetic atoms

and which is proportional to the susceptibility 

The later has maximum at T

Fig. 4.16. Upper critical field H

(T) in antiferromagnetic

superconductors: (a)NdMo

and (b)SmRh

[49]

188 M.L. Kuli´candA.I.Buzdin

4.5 Josephson Effect with Bulk Magnetic

Superconductors

Before discussing the coexistence of superconductiv-

ity and ferromagnetism in heterostructures we con-

sider a theoretical prediction of the Josephson effect

in magnetic superconductors with spiral ordering

of magnetic moments; the MS/I/MS junction [16].

Besides a number of potentially very interesting ef-

fects the MS/I/MS junction is a paradigm for some

novel phenomena in heterogeneous SF superstruc-

tures. For instance, recent theoretical proposals [18]

on the giant proximity effect in the SFFS hybrid sys-

tems is based on the existence of the triplet amplitude

whichappearsinsuperconductorswithnon-uniform

orientation of magnetization. The triplet amplitude

↑↑

= 

↑

 (and F

↓↓

= 

↓

) was discovered

in 1979 in studying magnetic superconductors with

spiral magnetic ordering [10]. Their importance in

Josephson junctionswith magnetic superconductors

was recognized in 2001 [16].

(i) -contact in the MS/I/MS junction.Webrieﬂy

discuss the tunneling Josephson junction based on

magnetic superconductors, where the left-L and

right-R bulk magnetic superconductors are with spi-

ral magnetic ordering [16], see Fig. 4.17. The spiral

magnetic order is characterized by the wave vector

L,R

and the exchange fields h



L(R)

= h

L(R)

i

L(R

)

,re-

spectively, while the superconductivity in the banks

is described by the order parameter 

L,R

= |

L,R

. For simplicity reasons one assumes | 

|

|= , h

= h

= h, |Q

|= |Q

|= Q.Thespiral

wave vectors Q

L,R

= 

L,R

Qˆz are orthogonal to the sur-

face of the tunneling barrier where the spiral helicity



L(R)

= ±1forQ

L,R

along (+) or opposite (–) to the z-

axis.Note that such a junction is characterized by the

standard superconducting phase' = '

−'

=0and

additionally by the magnetic phase  = 

− 

=0.

The magnetic phase  is a novel property giving rise

to very interesting physics.As was mentioned above,

besides the singlet amplitude F

↑↓

↓↑

)theinduced

(by the non-uniform spiral ordering) triplet pairing

amplitudes, F

↑↑

andF

↓↓

,play an importantrole in the

Josephson effect [16]. In this case F

↑↑

and F

↓↓

give

rise to an additional, and in some cases dominant,

channel for tunneling of superconducting electrons.

It turns out that the Josephson current J

(', )con-

tains two contributions

(', )=(J

− J

−

cos )sin' , (4.67)

where the standard term J

∼ 

isdue to the left and

right singlet amplitudes F

†

↑↓

, !

), F

†

↑↓

, !

while the “triplet” current J

−

is due to the triplet

amplitudes

−

∼ −T



| T

†

↑↑k

)

× [F

†

↑↑k

(−!

)]

∗

+ F

†

↓↓k

) (4.68)

× [F

†

↓↓k

(−!

)]

∗

Fig. 4.17. The Josephson junction with the insulating

contact.S

and S

are superconductors with spiral mag-

netic order.The exchange fields h

L,R

at the surface make

angles 

L,R

with the y-axis. Q

L,R

are along the z-axis [16]

4 Coexistence of Singlet Superconductivity and Magnetic Order 189

The calculations give J

−

∼ 

+(



(, h)],

where the functions f

1,2

(, h) are calculated in [16].

The Josephson current also depends on the total he-

licity  = 



. It was shown in [16] that in some

parameter region the effects of the triplet ampli-

tude dominate since one has | J

−

|> J

,thusgiving

rise to the -J osephson junction.Ifsuchajunction

is placed in a superconducting ring with the suffi-

ciently large inductance L,i.e.withL > L

,thena

spontaneous current ﬂows in the ring producing the

half-ﬂux quantum in the hole of the ring [63]. From

Eq. (4.67) one can see that by changing the magnetic

phase  and helicity  one can tune the system from

the 0-junction to the -junction. These new degrees

of freedom in the junction,themagnetic phase  and

chirality, first proposed in [16], open a new possibil-

ity for switching elements and qua ntum computing.

From the physical point of view the above model is a

paradigm for the S/F



/S structures with rotating

magnetization. In this case  is the angle between

rotating magnetization in neighboring layers [17].

(ii) Charge and spin Josephson currents in MS/I/MS

contacts. In ferromagnetic singlet superconductors

with rotating magnetization (e.g. as spiral) besides

the standard Green’s function G

↑↑

↓↓

), F

↑↓

↓↑

the non-diagonal functions in spins G

↑↓

↓↑

)and

the triplet amplitudes F

↑↑

↓↓

) are of immense im-

portance [16,17], since they produce the spin current

spin

through the junction even in absence of voltage

spin

= J

+ J

= ˛h

(1 + ˇ

cos ')sin , (4.69)

where

∼ h

sin  ∼



| T |



↑↓,L

↓↑,R

− G

↓↑,L

↑↓,R



and

∼ h



cos ' sin 

∼



| T |



†

↑↑k

)



†

↑↑k

(−!

)



∗

− F

†

↓↓k

)[F

†

↓↓k

(−!

)

∗

]



The part J

,which is a magnetic analog ofthe Joseph-

son junction was also discussed thoroughly in [64],

while the term J

describes the spin current due to

superconducting triplet amplitude. The physics of

the MS/I/MS (and also S/F



/S)junctionismore

transparent if one writes the energy E(' , )ofthis

combined magnetic and superconducting Josephson

junction. Since E(, ') must be an even function on

' and  it has the form

E(, ' )=−Ah

cos  − 

(B + C



cos )cos' .

(4.70)

A(, h), B(, h), C(, h) are a cumbersome expres-

sion, which is not given here. Note that both, the

spin J

spin

(, ') ∼ @E/@ and the charge J

(', ) ∼

@E/@' Josephson current, depend on ' and .Thus,

by tuning  and ' one can tune these currents. For

instance, one can obtain the superconducting -

junction, with spontaneous charge currents in the

superconducting ring. Analogously, the magnetic -

contact can be realized and accordingly the sponta-

neous spin current ﬂows in the ring from magnetic

superconductors.

Another interesting aspect of the two-phase

Josephson junction might be realized in small

Josephson contacts. In such a case the smallness of

the charge and “spin” capacitance brings the system

to the quantum regime, thus providing a possibility

for a novel Josephson qubit.In fact, the latter consists

from two qubi ts, the superconducting and magnetic

one,which might be also of apotentialinterestforap-

plications in hardware for quantum computing [17].

4.6 Superconductor/Ferromag net

Heterostructures

The coexistence of singlet superconductivity with

ferromagnetism was not observed in bulk materials,

but it may be easily achieved in artificially fabricated

layered ferromagnet/superconductors (F/S) systems.

Due to the proximity effect,the Cooper pairs can pen-

etrate into the F layer and induce superconductivity

there. This provides the unique possibility to study

the properties of superconducting electrons under

the inﬂuence of a large exchange field. Furthermore,

it is possible to study the interplay between super-

conductivity and magnetism in a controlled manner.

For instance,byvarying the layerthickness and mag-

netic moment of F layers the relative strength of two

190 M.L. Kuli´candA.I.Buzdin

competing orderingsis changed.The behavior of the

superconducting condensate under these conditions

is quite peculiar. This field is covered in several ex-

cellent reviews articles [18,19,21,65].

4.6.1 The Fulde–Ferrell–Larkin–Ovchinikov State

and the Proximity Effect in S/F Structures

AlongtimeagoFuldeandFerrell[56]andLarkinand

Ovchinnikov [57] demonstrated that in a pure su-

perconductor at low temperature the paramagnetic

effect leads to a non-uniform superconductivity (r)

(thisistheso-calledFFLOorLOFFstate).Themod-

ulation of the superconducting order parameter in

the FFLO state is related to the Zeeman’s splitting of

electronic levels under magnetic field that acts on

electron spins. To elucidate this effect qualitatively,

we consider the simplest case of the 1D supercon-

ductor. In the absence of the field, the Cooper pair is

formedby two electrons with opposite momenta +k

and −k

and opposite spins (↑) and (↓), respectively.

The resulting momentum of the Cooper pair is zero,

+(−k

) = 0. In the magnetic field, because of the

Zeeman’s splitting, the Fermi momentum of the elec-

tron with spin (↑) will shift from k

to k

= k

+ ık

where ık

= BH/v

and v

is the Fermi velocity.

Similarly, the Fermi momentum of an electron with

spin (↓)willshiftfrom−k

to k

=−k

+ık

,seeFig.

4.18. Then, the resulting momentum of the Cooper

pair will be k

+ k

=2ık

= 0, which just implies

the space modulation of the superconducting order

parameter with the resulting wave-vector 2ık

Due to the incompatibilityof ferromagnetismand

superconductivity it is not easy to verify this predic-

tion on experiment. Moreover the electron scatter-

ing on the impurities destroys the FFLO state very

quickly and its observation is possible only in the

clean limit [66]. It turns out that in a ferromagnet

in contact with a superconductor the Cooper pair

wave function has damped oscillatory behavior [20],

which may be considered in some sense as an anal-

ogy with the decaying non-uniform FFLO supercon-

ducting state. Indeed, when a superconductor is in a

contact with a normal metal the Cooper pairs pene-

trate across the interface at some distance inside the

metal.A Cooper pair in a superconductor comprises

Fig. 4.18. (a) Energy band of 1D superconductor near the

Fermi energy in the presence of Zeeman splitting. In the

FFLO state electrons (−k

+ ık

, ↓)and(k

+ ık

, ↑)are

paired with total momentum ık

= h/v

; see the text.

(b)The(h, T) phase diagram for the 3D superconductor.

At T < T

∗

the second order transition occurs between nor-

mal and FFLO state. The bold line is the first order tran-

sition from FFLO to uniform superconducting state [21]

two electrons with opposite spins and momenta. In

a ferromagnet the up spin electron (with the spin

orientation along the exchange field) decreases its

energy by h , while the down spin electron increases

its energy by the same value. To compensate this en-

ergy variation, the up spin electron increases its ki-

netic energy, while the down spin electron decreases

its kinetic energy. In the result, the Cooper pair ac-

quires a center of mass momentum 2ık

=2h/v

which implies the modulation of the order parame-

ter with the period v

/h. The direction of the mod-

ulation wave vector must be perpendicular to the in-

terface, because only this orientation is compatible

with the uniform order parameter in the supercon-

4 Coexistence of Singlet Superconductivity and Magnetic Order 191

ductor. This phenomenon, however, is quite general

and must be present in both the clean and dirty lim-

its [67]. It results in many interesting effects: the spa-

tial oscillations of the electronicdensity of states,the

non-monotonous dependence of the critical temper-

ature of S/F multilayers and bilayers on the ferro-

magnet thickness, the realization of the Josephson

-junctions in S/F/S systems. The interplay between

superconductivity and magnetism in S/F structures

occurs at the nanoscopic range of layer thicknesses

and the observation of these effects became possible

only recently due to the great progress in the prepa-

ration of high-quality hybrid S/F systems.

Let us consider the question of the proximity ef-

fect in a weak ferromagnet (h

∼ ) in contact with

a superconductor (x < 0). To get an idea of the

proximity effect in S/F structures, as well as of the

FFLO state,we may start from a description based on

the Ginzburg–Landau functional, which takes into

account the paramagnetic (Zeeman) effect in the

magnetic field. As is known the standard Ginzburg–

Landau functional F, which describes the behavior

of the superconducting order parameter ¦ (r)near

,isgivenby

F = a | ¦ |

+ |∇¦ |

| ¦ |

. (4.71)

The orbital effect of the magnetic field is taken into

account by the replacement ∇→∇−(2ie/c)A.Near

the coefficients a,  > 0, b > 0donotdepend

on B(= ∇×A). For T < T

and B =0onehas

a < 0 and the uniform superconductivity occurs

with | ¦ |

=−a/b. However, when the paramagnetic

effect becomes dominant for 

B ∼  this approx-

imation fails, since near the point (B

∗

, T

∗

)theco-

efficient  changes sign. This means that the in the

presence of the paramagnetic effect the minimum of

F is realized for the oscillating ¦ (r). To describe this

situation it is necessary to add higher derivatives in

the expansion of Eq. (4.71). This is also confirmed

by the microscopic derivation in the presence of the

paramagnetic effect [68–70].It turns out that in such

a case the generalized G-L functional is given by [70]

F = a(B, T) |¦ |

+ (B, T) |∇¦ |

(4.72)

(B, T)

|∇

¦ |

|¦ |

| ¦ |

wherea(B, T)=a



(T −T

(B)) and T

(B) is thecriti-

cal temperature into the uniform(¦ = const.) super-

conducting state. In fact in the FFLO state the coeffi-

cient b can be small and even negative and therefore

the sixth order term ∼ c must be added. The effect

of the orbital motion is accounted for by the replace-

ment ∇→∇−(2ie/c)A; for moredetails see [69,70].

The critical temperature of the second order phase

transitionandthenon-uniformstructureisobtained

by solving the linearized equation (4.72)

a¦ −  ¦ +





¦ =0. (4.73)

Ifweseekthesolutionintheform¦ = ¦

exp(iq · r),

then after the minimization of F with respect of q

oneobtainsthemodulationvectorq

and the crit-

ical temperature of the non-uniform (FFLO) state

(> T

)

=−





a ≡ a



− T





. (4.74)

This means that the FFLO state is obtained simply

by changing sign of  in the phenomenological G-L

equation. The more detailed microscopic theory [69]

givesthatthe order parameter near the point (B

∗

, T

∗

)

is not the plane-wave but the one-dimensional sinu-

soidal structure ¦ ∼ sin q · r.

Let us apply this theory to the proximity effect

in the S/F metal surface which is perpendicular to

the x-axis. It is known that in the case of the S/N

(superconductor-normal) interface Cooper pairs can

penetrate into the normal metal at some distance

(T), which is (D/T)

1/2

for diffusive electron motion

and v

/T for pure normal metal. In the framework

of the standard G-L theory (with  > 0) for the S/N

ideal contact and by assuming perfect transparency

of the interface, one obtains that the superconduc-

tivity is induced in the normal metal and decaying

exponentially ¦ = ¦

exp[−x/(T)]; the proximity

effect. On the other hand, the leakage of the SC elec-

trons weakens SC near the surface (in verse proximity

effect) as can be seen in Fig. 4.19. In the case of the

S/F interface being the solution of Eq. (4.73) one ob-

tains that the exponential decay of the induced su-

192 M.L. Kuli´candA.I.Buzdin

perconducting order parameter in the ferromagnet

is accompanied by its oscillation

¦ = ¦

exp(−x/

)cos(x/

) ,



−2

|  |

2

(

√

1+ −1), (4.75)



−2

|  |

2

(

√

1+ +1),

where  =(T − T

)/(T

− T

). We stress that the

oscillation of the order parameter in the ferromag-

netic metal F is the fundamental difference between

the S/F and S/N systems.

In real ferromagnets the exchange field may be

much larger than the superconducting gap, which

causes that the gradients of the superconducting

order-parameter variations are also large. The lat-

ter cannot be treated by the generalized G-L func-

tional and we need to use a microscopic approach,

for instance Bogoliubiv–de Gennes equations or the

quasiclassical Eilenberger and Usadel equations; see

[21] and references therein. If the electron scatter-

ing mean free path l is small (which is usually the

case in S/F systems), the most natural approach is to

use the Usadel equations for the Green’s functions

averaged over the Fermi surface; see Appendix A.2.

The linearized Usadel equation for the anomalous

Green function F

in the ferromagnet reads (note in

Appendix A.2 we use f for F

)



| !

| +ih · sgn(!





−

=0,

(4.76)

where !

=(2n +1)T are the Matsubara frequen-

cies, D

=(1/3)v

l is the diffusion coefficient in the

ferromagnet and h(≡ h

). The parameter 1/

de-

scribes the magnetic scattering in the F layer. Note

that this form of theUsadelequation in theferromag-

net implies a strong magnetic uniaxial anisotropy

when the magnetic scattering in the plane (xy)per-

pendicular to the anisotropy axis is negligible [21].In

the F region,we may neglect the Matsubara frequen-

cies compared to the large exchange field (h  T

Also assuming first that the magnetic scattering is

weak, we readily obtain the decaying solution for F

Fig. 4.19. Schematic behavior of the superconducting or-

der parameter ¦ near (a) superconductor-normal and

(b) superconductor-ferromagnet interface. The continuity

of the order parameter at the interface is assumed [21]

(x, !

> 0) = A exp



−

1+i





, (4.77)

where 



/h is the characteristic length of

the superconducting correlation decay (with oscilla-

tions) in the F-layer. In systems with h >> T

,this

length is much smaller than the superconducting co-

herence length 



/h (where D

isthe diffusion

constant in the superconductor), i.e. 

 

.Inafer-

romagnet, the role of the Cooper pair wave function

is played by ¦ , which decays and oscillates as

¦ ∝



(x, !

) ∝  exp



−





cos







(4.78)

The different behavior of the superconducting or-

der parameter in S/F and S/N systems is illustrated

schematically in Fig. 4.19.

As was said before,the damping oscillatory behav-

ior of ¦ is the fundamental difference between the

proximity effect in S/F and S/N systems, and it is at

the origin of manypeculiar characteristics of S/F het-

erostructures. In the absence of magnetic scattering

the scale for the oscillation and decay of the Cooper

pair wave function in a ferromagnet is the same. If

4 Coexistence of Singlet Superconductivity and Magnetic Order 193

we take into account the magnetic scattering, then

the decaying length 

f 1

becomes smaller than the

oscillating length 

f 2

,i.e.

f 1

= 

√

1+˛

+ ˛)

1/2

and 

f 2

= 

√

1+˛

− ˛)

1/2

, where the parameter

˛ =1/

h characterizes the relative strength of the

magnetic scattering. The damped oscillatory behav-

ior of the order parameter may lead to the electronic

density of states oscillations in a ferromagnet in con-

tact with a superconductor [71]. This prediction has

been confirmed experimentally [72], and up to now

this remains the only experimental observation of

the density of states oscillations in the F layer. The

magnetic scattering effect complicates this type of

experiments strongly reducing the amplitude of the

oscillations.

4.6.2 Oscillatory Superconducting Transition

Temperature in S/F Multilayers and Bilayers

The damped oscillatory behavior of the supercon-

ducting order parameter in ferromagnets may pro-

duce the commensurability effects between the pe-

riod of the order parameter oscillation (which is of

the order of 

f 2

) and the thickness of the F layer.This

results in the striking non-monotonous supercon-

ducting transition temperature dependence on the

F layer thickness in S/F multilayers and bilayers. In-

deed, for the F layer thickness smaller than 

f 2

,the

pair wave function in the F layer changes a little and

the superconductingorder parameter in the adjacent

S layers must be the same. The phase difference be-

tween the superconducting order parameters in the S

layers is absent and we call this state the 0-phase. On

the other hand, if the F layer thickness becomes of

the order of 

f 2

thepairwavefunctionmaygotrough

zero at the center of F layer providing the state with

the opposite sign (or shift of the phase) of the super-

conducting order parameter in the adjacent S layers;

the so-called -phase. The increase of the thickness

of the F layers may provoke the subsequent transi-

tions from 0 to -phases, which superposes on the

commensurability effect and results in a very special

dependence of the critical temperature on the F layer

thickness [67,73].Theexperimental observation [74]

of this unusual dependence in Nb/Gd was the first

Fig. 4.20. Oscillation of T

of Nb/Gd multilayers vs. thick-

ness of Gd layer d

for (a) d

= 600Å and (b) d

= 500 Å

[74]

strong evidence in favor of the -phase appearance;

see Fig. 4.20a.

For the S/F bilayers,the transitions between 0 and

-phases are impossible; nevertheless the commen-

surability effect between 

f 2

and F layer thickness

also leads to the non-monotonous dependence of T

on the thickness of the F layer. Processes of the nor-

mal quasiparticle reﬂection at the free F layer bound-

Fig. 4.21.Var iat ion of T

ofNb/Cu

0.43

0.57

bilayervs.F layer

thickness [75]. The theoretical fit for h ≈ 130 K [76]

194 M.L. Kuli´candA.I.Buzdin

ary and Andreev reﬂection at SF-interface interfere

and this results in T

minima being reached when

the thickness of the F layer is close to a quarter of

the spatial oscillation period. The dependence of T

of the Nb/Cu

0.43

0.57

bilayer on the thickness of the

F layer [75] is presented in Fig. 4.21.

4.6.3 Superconductor-Ferromagnet-Superconductor

-Junction

The experiments on the critical temperature of the

S/F multilayers and bilayers attracted a lot ofinterest

in the proximity effect in S/F systems but their inter-

pretations were controversial due to the very small

value of the characteristic length 

f 2

(only several

nanometers). The most direct proof of the -phase

observation would be the observation, following the

theoretical predictions [20,67], [73] of the vanishing

of the critical current at the 0 to -phase transition.

The first experimental evidence of the 0 to  transi-

tion in S/F/S (Nb-Cu

1−x

-Nb) Josephson junction

was obtained in [77] from the measurements of the

temperature dependence of the critical current. The

0to transition was signalled by the vanishing of

the critical current with the temperature decrease.

Such a behavior is observed for a F layer thickness

close to some critical value d

. In fact, this simply

means that the critical thickness d

slightly depends

onthe temperature.The temperature variation serves

as a fine tuning and permits the study of this transi-

Fig. 4.22. j

vs. the F-layer thickness d

for Nb/Cu

0.47

0.53

Nb junction at T =4.2 K. Open circles experimental data

and dashed line model calculations [80]

tion in detail. The damped oscillations of the critical

current as a function of the thickness of the F layer

were observed later in Nb/Al/Al

/PdNi/Nb [78]

and Nb/Cu/Ni/Cu/Nb junctions [79].Very recent ex-

periments ( [80] have enabled the observation of the

two-node thickness dependence of the critical cur-

rent in Josephson SFS junctionswith a ferromagnetic

interlayer, i.e. both direct transition into the -state

and the reverse one from  into the 0-state,as shown

in Fig. 4.22.

ThecompletequantitativeanalysisoftheS/F/S

junctions is rather complicated, because the ferro-

magnetic layer may strongly modify superconduc-

tivity near the S/F interface. In addition, the bound-

ary transparency and electron mean free path, as

well as magnetic scattering, are important param-

eters affecting the critical current. In the case of

small conductivity of the F layer or small interface

transparency, the “rigid boundary” conditions [81]

are applied and the inﬂuence of the ferromagnet on

the superconducting order parameter in the elec-

trodes may be neglected. This solution of Eq. (4.76)

describes the F(x) behavior near the critical temper-

ature and gives the sinusoidal current-phase depen-

dence I

(')=I

sin('), where ' is the phase differ-

enceonthejunction.ThecriticalcurrentI

passes

through zero and change its sign with the increase of

the thickness of the F layer.

In the most interesting limit from the practical

point of view, when the F layer thickness d

> 

f 1

the universal expression for the I

) dependence

is [21]

∝ exp



−



f 1



sin





f 2

+ ˇ



, (4.79)

where the angle /4 < ˇ < /2 depends on the

magnetic scattering amplitude and the boundaries

transparencies.

In [82] it was pointed out that the -junction in-

corporated into a superconducting ring (with large

inductance L > L

) would generate a spontaneous

current and a corresponding magnetic ﬂux would be

half of the ﬂux quantum ¥

= hc /2e.The appearance

of the spontaneous current is related to the fact that

thegroundstateof the -junction correspondsto the

phase difference  and thus this phase difference will

4 Coexistence of Singlet Superconductivity and Magnetic Order 195

generate a supercurrent in the ring, which short cir-

cuits the junction.Naturally,the spontaneous current

is generated if there is an odd number of -junctions

in the ring. This circumstance has been exploited in

an elegant way in[77] toprovideunambiguous proof

of the -phase transition. The observed half-period

shift of the external magnetic field dependence of the

transport critical current in triangular S/F/S arrays

was observed on the 0– transition occurring with

temperature variation.

The current-phase relation for a metallic S/NS/

Josephson junction is sinusoidal I

(')=I

sin(')

only near the critical temperature T

.Atlowtemper-

ature the higher harmonic terms appear. However in

the diffusive limit at d

> 

f 1

they are very small and

in the usual junctions their presence is hardly ob-

served.In S/F/S junctions in general in the dirty limit

the first harmonic contribution is I

∼ exp(−d

/

f 1

)

and the second harmonic contribution is very small

and positive ∼ exp(−2d

/

f 1

). The peculiarity of the

situation with the 0- transition is that in the transi-

tion region the first harmonic term changes its sign

passing trough zero and then the role of the sec-

ond harmonic contribution becomes predominant.

To study the scenario of the 0– transition let us

address to the general current-phase relation

j(' )=I

sin(')+I

sin(2') , (4.80)

which corresponds to the following phase dependent

contribution to energy of the Josephson junction

(')=

2



−I

cos(')−I

cos(2')



. (4.81)

If we neglect the second harmonic term, then the

0 state occurs for I

> 0. Near the 0– transition

→ 0 and the second harmonic term becomes im-

portant.The critical current atthe transitionj

=| I

and if I

> 0, the minimum energy always occurs at

' =0or' = .

Intheoppositecase(I

< 0) the transition from

the 0 to the -state is continuous and there is re-

gion where the equilibrium phase difference takes

any value 0 < '

< . The characteristics of such

a“' -junction” are very peculiar but at the moment

there is no experimental evidence of its observation.

F/S/F Spin-Valve Sandwiches

The strong proximity effect in superconductor-

metallic ferromagnet structures may lead to the spin-

orientation-dependent superconductivity in F/S/F

spin-valve sandwiches. A long time ago de Gennes

[83] considered theoretically the system consisting

of a thin superconducting layer between two ferro-

magnetic insulators. He argued that the parallel ori-

entation of the magnetic moments is more harmful

for superconductivity because of the presence of the

non-zero averaged exchange field acting on the sur-

face of the superconductor.This prediction has been

confirmed experimentally in an In film sandwiched

between two Fe

films[84]andonanInfilmbe-

tween oxidized FeNi and Ni layers [85].

A similar effect has been predicted for metallic

F/S/F sandwiches [86,87] and observed on experi-

ment in CuNi/Nb/CuNi [88] and Ni/NbNi [89]. The

spin-valve effect was proposed also for semiconduct-

ing F/S/F sandwiches in [90], which could be also

used to determine the spin structure of Cooper pairs

in triplet superconductors.

In the diffusive regime the proximity effect in S/F

structures with a spatial rotation of the magnetiza-

tion may be rather special [18].As was shown in [10]

spiral rotation of spins (magnetization) in super-

conductors generates the triplet component of the

anomalous Green’s functions F

↑↑

=< ¦

↑

> and

↓↓

=< ¦

↓

>. The importance of the triplet am-

plitudes in the Josephson junction based on mag-

netic superconductors was pointed out in [16]. In

these systems F

↑↑

and F

↓↓

were responsible for the

-contact, for the spin current and the dependence

of the current on the spiral helicity. In a metallic S/F

structure with rotating magnetization F

↑↑

and F

↓↓

penetrate into the ferromagnet at distances much

larger than 

. However, it is not the triplet super-

conductivity itself because the corresponding triplet

order parameter is zero in these systems, unlike the

superﬂuidity in He

,for example.An important find-

ing [18] was that the decay length-scale of triplet

component is insensitive to the exchange field and

can generate the triplet long-range proximity effect

(with 

f ,↑↑

∼ v

/T  

f ,↑↓

∼ v

/h,



/h.The

special long range triplet proximity effect was pre-

196 M.L. Kuli´candA.I.Buzdin

dicted to exist in the dirty limit. In the clean limit

it disappears [91], the spin-orbit and magnetic scat-

tering also destroy this effect. Up to now there have

been no reports on its experimental observation.

The proximity effect is related to the passing of

electronsacross the superconductor/ferromagnet in-

terface.In addition to this effect,if the magnetic field,

created by the ferromagnet, penetrates into the su-

perconductor, it switches on the orbital interaction

between the superconductivity and ferromagnetism.

This situation is naturally realized in the case when

the ferromagnet is an insulator or the buffer oxide

layer separates the superconductor and the ferro-

magnet. In such a case the ferromagnet plays the

role of an additional source of the local magnetic

field.In particular the nucleation of superconductiv-

ity in the presence of a domain structure occurs near

the domain walls [92].Hybrid S/F systems have been

intensively studied in connection with the problem

of the controlled ﬂux pinning. Enhancement of the

critical current has been observed experimentally

for superconductingfilms with arrays of sub-micron

magnetic dots and antidots. There is an excellent re-

view on these questions related to the electromag-

netic interaction of superconductivity and magnetic

dots [65].

During the last five years an enormous progress in

the controllable fabrication of the superconductor-

ferromagnet heterostructures has been achieved.

The peculiar effects predicted theoretically were ob-

served experimentally resulting in the qualitative

and semiquantitative understanding of the mecha-

nism of the superconductivity and ferromagnetism

interplay in S/F systems.Now,this domain of research

enters into a period when the design of the new types

of the devices becomes feasible and we may expect a

lot of interesting finding in the near future.

4.7 Conclusion

The rare earth ternary compounds are rich physical

systems that allow the study of the coexistence of

singlet superconductivity and various magnetic or-

ders,such as ferromagnetic, antiferromagnetic,weak

ferromagnetism. It turns out that in bulk materials

superconductivity and ferromagnetism practically

never coexist.However,the singlet superconductivity

modifies the ferromagnetic order into a spiral or do-

main structure (depending on magnetic anisotropy),

thus allowing their coexistence. This is realized not

only in rare earth ternary compounds but also in

AuIn

, where a modified nuclear ferromagnetism,

spiral or domain-like structure,coexists with singlet

superconductivity.

Although the antiferromagnetic order and super-

conductivitycoexist much easier,these systems show

a peculiar behavior in the presence of non-magnetic

impurities, which surprisingly act as pair-breakers.

In the case when the antiferromagnetic order is ac-

companied by the weak ferromagnetism new coexis-

tence phases appear; the Meissner and spontaneous

vortex states. Magnetic superconductors show pecu-

liarbehavior in the magneticfield.Near the magnetic

critical temperature the upper critical field tends to

zero faster than the thermodynamical field,thus giv-

ing rise to the first order transition. Various phases

are possible in the H − T diagram depending on the

purity and demagnetization effects of real samples.

The lower critical field is weakly affected by the ex-

change field (which is due to localized moments). A

specificity of ferromagnetic superconductors is that

the surface energy can be positive but the vortex state

is still thermodynamically stable.

The Josephson junctions based on bulk ferromag-

netic superconductors with spiral order are charac-

terized by the superconducting and magnetic phase.

This opens possibilities for a new kind of coupled

qubits in a single Josephson junction. The triplet

pairing amplitude, arising in systems with rotating

magnetization, gives rise to the -Josephson junc-

tion and spin current, which can be tuned by chang-

ing the magnetic phase and chirality.

The coexistenceof singlet superconductivity with

ferromagnetism may beeasily achieved inartificially

fabricated layered ferromagnet/superconductors

systems.Dueto the proximity effect,the Cooper pairs

can penetrate into the F layer and induce supercon-

ductivity there. The peculiarity is that in the ferro-

magnetic metal the induced superconducting ampli-

tude oscillates spatially giving rise to a number of

novel effects (not existing in S/N systems) such as