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 167

depending on the type and strength of magnetic

anisotropy in the system, thus giving rise to the co-

existence of SC and magnetic order. This was first

pointed out in 1959 by Anderson and Suhl [9] for

the case when the exchange field is small (h/  1)

and later elaborated on by the present authors and

collaboratorsby including the exchange and electro-

magnetic interaction for any ratio of h/ [8,10]. In

the RE ternary compounds the competition of fer-

romagnetism and superconductivity is rather strong

and therefore these two ordering coexist in a limited

temperature interval T

< T < T

< T

only; the

re-entrant behavior. For instance, in ErRh

the co-

existenceregion is narrow with T

≈ 0.8K,T

≈ 0.7

K. The appearance of the modulated (non-uniform)

magnetic structure with the period L = 100 Å ob-

served in neutron scattering experiments [11], while

the superconductivity appears in the paramagnetic

state at much higher temperature T

=8.7K.By

further cooling the first order phase transition into

the ferromagnetic phase occurs at T

≈ 0.7Kwith

simultaneous destruction of superconductivity.The

transition at T = T

is called re-entran t transition

because the superconductor goes back into the nor-

mal phase.The temperature dependence of the resis-

tivity and magnetic susceptibility in the re-entrant

superconductor ErRh

isshowninFig.4.5,where

the hysteresis is realized around and slightly below

T = 1 K, which is a sign of the first order phase

transition. In HoMo

the coexistence interval of

the non-uniform magnetic structure with the period

L = 200 Å and superconductivity is even narrower,

with T

≈ 0.74 K, T

≈ 0.7K,whileT

=1.8

K [12]. It turns out that the exchange interaction in

HoMo

is weaker (than in ErRh

andHoMo

with T

=5.5K,T

≈ 0.8 and the coexistence of

SC and modulated ferromagnetism persists down to

T = 0 K; see the discussion below and in [6,8,10].

The theory of magnetic superconductors in sys-

tems with localized magnetic moments, such as the

RE ternary compounds, was developed in the last 30

years to a such level that nowadays there is a rather

good qualitativeand quantitativeunderstanding of a

number of relevant properties of these systems [8]. It

turns out that in the RE ferromagnetic ternary super-

conductors the exchange interaction domina tes over

Fig. 4.5. A.C. susceptibility and resistance vs. temperature

in ferromagnetic superconductor ErRh

[13]

the EM one in the competition of superconductivity

and mag netic order. The role of the electromagnetic

(EM) interaction in this competition is to make the

magnetic structure transverse. The reason is that the

rather large wave vector Q of the spiral or domain

structurereducesthe effect of the non-uniformmag-

netic field (due to the magnetization) on SC much

more than the effect of the non-uniform exchange

field. This will be discussed in the forthcoming sec-

tions.

A new research field in the physics of ferromag-

netic superconductors was opened in 1997 by Po-

bell’s group in Bayreuth [14], who discovered the

coexistence of superconductivity and nuclear mag-

netic order in the type-I superconductor AuIn

.In

this system SC appears at T

=0.207 K and the

magnetic ordering at T

=35K. Although in

this system there is a tendency to the nuclear fer-

romagnetic order in absence of SC, the theory [15]

predicts that below T

the superconductivity en-

forces appearance of a spiral or domain-like nuclear

magnetic ordering in the SC state, depending on the

strength of the nuclear dipole–dipole interaction.

Experimental studies of the Josephson effect in

systems with bulk magnetic superconductors are

rather scarce. However, recent theoretical studies of

Josephson junctions based on MS with the spiral or-

dering show that these junctions, if realizable, might

have fascinating properties, such as the existence of

168 M.L. Kuli´candA.I.Buzdin

-contacts [16], the combination of a magnetic ana-

log of the Josephson effect for spin current with the

ordinary Josephson effect for charge current [17].In

such a system the realization of two qubits (a super-

conducting and magnetic one) in a single Josephson

junction is possible in principle. Furthermore, all

these novel properties originate from the crucial fact

that in systems with non-uniformmagnetization the

triplet amplitude F

↑↑

= 

↑

 (and F

↓↓

= 

↓

)

of Cooper pairs is finite. It turns out that this model

is in fact a paradigm for superconductor/ferromagnet

superlattices with non-collinear orientation of mag-

netization. The importance of the triplet amplitude

was recently rediscovered in the SF

S weak links

with rotating magnetization [18]. The latter induces

the triplet amplitudes giving rise to so-called giant

proximity effects, and in thicker films the Joseph-

son current is dominantly transported via the triplet

amplitude.

Contrary to bulk materials, the coexistence of su-

perconductivity and ferromagnetism may be eas-

ily achieved in artificially fabricated superconduc-

tor /ferromagnet heterostructures. Due to the prox-

imity effect the Cooper pairs penetrate into the F

layer giving the unique possibility to study proper-

tiesof superconducting electronsunder theinﬂuence

ofthehugeexchangefield.Moreover,byvaryingthe

thicknesses of the ferromagnetic and superconduct-

ing layers in a controllable manner it is possible to

change the relative strength of the two competing

orderings. The Josephson junctions with ferromag-

netic layers reveal many unusual properties that are

quite interesting for applications, in particular the

so-called -Josephson junction (with the -phase

difference in the ground state, [18–21]) which was

recently fabricated [22]. This interesting aspect of

physics is also discussed in the following sections.

We would like to point out that our discussion

is restricted to magnetic superconductors in the

ternary rare earth compounds with localized mo-

ments as well as to superconductor/ferromagnet het-

erostructures, while the coexistence problem of SC

with the bulk itinerant magnetism is discussed only

brieﬂy. However, one expects that the results dis-

cussed here hold also for itinerant magnetism if it

is treated in the mean-field approximation.

There are a number of other magnetic supercon-

ductors that will be discussed here only brieﬂy, if at

all. The quaternary borocarbides (RE)Ni

Cshow

the coexistence of superconductivity and antiferro-

magnetic order, which is a phenomenon understood

in principle, although details of related to particular

structures await further elaboration. Very interest-

ing compounds are heavy fermion superconductors,

URu

,UPd

,UBe

, etc., which show the co-

existence of superconductivity and (itinerant or lo-

calized) (anti)ferromagnetic order. While in the RE

ternary compounds and quaternary borocarbides

the superconductivity is due to the electron–phonon

interaction, in heavy fermions it is probably due to

spin ﬂuctuations and with (possible) triplet pair-

ing [23].

The question related to the interplay between

magnetism and superconductivity in high Tc super-

conductors is well beyond the scope of this chap-

ter. The mechanism of high temperature supercon-

ductivity (HTSC) still remains mysterious though a

number of theoretical approaches have been pro-

posed. From the theoretical point of view the elec-

tron–phonon interaction (EPI) and spinﬂuctuations

are the most serious candidates for pairing in HTSC,

while the former interaction is favored by the re-

cent ARPES experiments; see the reviews [24]. In

(RE) Ba

6+x

and La

2−x

)CuO

4+ı

the oxygen

concentration is a very important parameter con-

trolling the magnetism of Cu ions. For small x and

ı the compounds are insulators and Cu ions order

antiferromagnetically with T

≈ (300–500) K (see,

for example, [25]). For optimally doped compounds

the Cu antiferromagnetic transition is absent and

is around 40 K and 95 K for (La

2−x

)CuO

4−ı

and (RE)Ba

6+x

, respectively. The trivalent rare

earth magnetic ions (RE) have very little effect on

the superconducting properties of (RE) Ba

6+x

compounds. The (RE) spins order antiferromagnet-

ically at low temperature T

≈ (0.5−2) K [25],

which means that the exchange interaction between

(RE) spins and superconducting electrons (mainly

moving in the Cu-O

planes) is small. Note that

in these systems the magneto-dipole and RKKY

mechanisms give comparable contributions to the

antiferromagnetic ordering energy. The condition

4 Coexistence of Singlet Superconductivity and Magnetic Order 169

 T

ensures quite weak interplay of the AF

order and superconductivity in (RE) Ba

6+x

However, the coexistence of SC and itinerant anti-

ferromagnetism in high-temperature superconduc-

tors have been realized for some doping. For in-

stance in (La

2−x

)CuO

4−ı

one has: T

=42K,

≈ 42 K – for x =0,ı =0.14; T

=12K,

≈ 25 K (or T

=29K,T

≈ 25 K depend-

ing on the synthesis); for x =0.12, ı =0,thereisa

pronounced Bragg peak at Q ≈ Q

=(, )and

the static correlations are developed on the length

scale 

> 400 Å. Since 

 

≈ (20−25) Å

(the SC coherence length in the ab-plane) this means

that these static correlations can be considered ap-

proximately as a static AF order that competes with

SC. Similar results hold forYBa

6.5

=55K,

≈ 310 K ) and in YBa

(Cu

1−y

)

7+ı

93 K, T

≈ 330 K for y =0.013); see [26] and

references therein. The neutron scattering measure-

ments show that the effective magnetic moment in

all these superconductors were small  < 0.1 

in the La family and  ≈ (0.02−0.05) 

in the Y-

family. This small moment points to a rather small

spin-ﬂuctuation coupling constant in these systems,

which is apparently insufficientfor the pairing mech-

anism. Based on the theory of weak (  

)

itinerant antiferromagnetism this coupling is ex-

tracted in [26] to be g

< 0.1−0.2eV.Thisvalue

is much smaller than the ad hoc assumed value

≈ 0.65 eV in the spin-ﬂuctuation theories ofpair-

ing in high-temperature superconductors [27], thus

giving strong arguments against the spin-ﬂuctuation

theories of pairing.

Nowadays, several type of layered compounds,

where superconducting and magnetic layers alter-

nate, are known. For example in Sm

1.85

0.15

CuO

[28], which reveals superconductivity at T

=23.5K,

the superconducting layers are separated by two

ferromagnetic layers with opposite orientations of

the magnetic moments and the Neel temperature

is T

=5.9 K. Several years ago, a new class of

magnetic superconductorsbased on the layered per-

ovskite ruthenocuprate compound RuSr

GdCu

comprising CuO

bilayers and RuO

monolayers was

synthesized; see the review [29]. In RuSr

GdCu

the magnetic transition occurs at T

≈ 130−140 K

and superconductivity appears at T

≈ 30−50 K.

Though the magnetization measurements give evi-

dence for the small ferromagnetic component, the

neutron diffraction data on RuSr

GdCu

revealed

the dominant antiferromagneticordering inall three

directions. Later, the presence of the in-plane fer-

romagnetic component of about (0.1−0.3) 

was

confirmed by neutron scattering on iso-structural

RuSr

GdCu

; see [30].

Finally, let us mention that the ferromagnetic or-

der is much less detrimental in superconductors with

triplet pairing, since the parallel electronic spins are

weakly affected by the exchange field. In this case

these two orderings can coexist in bulk materials.The

first truly ferromagnetic superconductorsUGe

[31]

and URhGe [32] were discovered only recently and

have attracted a lot of interest. Apparently these sys-

tems display the triplet superconductivity with par-

allel spin orientation of electrons in Cooper pairs,

thus withdrawing the paramagnetic limit. However,

the physics of these materials is far from understood

and therefore it is not the subject of this chapter.

4.2 Ferromagnetic Superconductors

In the ferromagnetic superconductors the magnetic

ordering of the localized 4f moments (LM) is due to

the indirect exchange interaction (RKKY) going via

the conduction electrons. As will be demonstrated

below the effective (characteristic) exchange energy

(

) between localized moments is of the order of



≈ N(0)h

,whereN(0) is the density of states at

the Fermi level (per LM) and h

(= (g

−1)nJ

(0)

)

is the maximal exchange field acting on conduction

electrons. Here, g

is the Lande factor, n is the den-

sity of localized magnetic moments (LMs), J

(0) is

the direct (local) exchange energy between conduc-

tion electrons and LMs, 

 is the averaged total an-

gular moment of localized moments. Note that the

exchange field acting on electrons is h

(r)=h

S(r)

(note that very often we use h instead of h

), where

S(r)(= 

/J) is the localized relative spin normal-

ized to one. Let us mention in advance that in most

RE ternary superconductors the maximal exchange

field h

is rather large, h

∼ 10

K and it is much

170 M.L. Kuli´candA.I.Buzdin

larger than the bare superconducting gap 

,i.e.

 

 10 K. We shall see below that in spite

of the fact that h

is much larger than the Clogston

paramagnetic field h

,i.e.h

 h

≈ 0.7

,theco-

existence of SC and modified ferromagnetic order

can be realized. The energy parameter 

is a mea-

sure of the gain in the exchange energy (per spin)

in the ferromagnetic ground state, i.e. | E

|∼ 

In the ferromagnetic superconductors that will be

discussed here one has 

∼ 1 K. On the other hand

thegain in the superconducting condensation energy

per one electron is very small, i.e. | E

|≈ N(0)

∼ T

) ∼ 10

−3

,whereE

is the Fermi energy.

This means that even in the case T

 T

one has

| E

|| E

|,i.e.theferromagnetismisanenerget-

ically very robust phenomenon in comparison with

the singlet superconductivity. Therefore, the singlet

superconductivity cannot prevent the ferromagnetic

orderingbutmayonlymodifyit.Onthecontrary,

ferromagnetism can easily destroy superconductiv-

ity, as will be discussed below.

As we have already mentioned, in addition to the

exchange mechanism the electromag netic (EM) in-

teraction of superconductivity and ferromagnetism

is always present in real compounds. It is related to

the Meissner screening of the magnetic induction

created by the magnetization. Namely, the magne-

tization M(r)=nS(r) creates a dipolar magnetic

field B(r)=rotA(r), which on the other hand in-

duces the screening current j

of conduction elec-

trons; the Meissner effect. The Fourier transformed

is related to the vector potential A by the kernel

(q), i.e. j

(q)=−K

(q)A(q). This mechanism also

favors a non-uniform magnetic structure instead of

a ferromagnetic one [33].

In the study of the RE ternary superconductors

the mean-field approximation for both the SC and

the magnetic subsystem will be assumed. It is a

good approximation for SC and for ferromagnetism

it is discussed below. This means that in the Hamil-

tonian we replace

S(r) → S(r)=<

S(r) > and

g ˆ

↑

(r) ˆ

↓

(r) → (r)=g < ˆ

↑

(r) ˆ

↓

(r) > where

↑,↓

are spin up and down annihilation operators.

Here, (r)isthesinglet superconducting order pa-

rameter and g is the electron–phonon coupling con-

stant.The total Hamiltonian of the system is given by

H =

BCS

imp



[−B(r

)g

] ,

BCS





†

(r)







ˆp −



− 



ˆ (r)

+ ˆ

†

(r)

(r) ˆ (r)+

| (r) |

epi

(4.2)

(r) ˆ

†

(r)i

†

(r)−



∗

(r) ˆ (r)i

ˆ (r)



Here,

(ˆp−

A) is the band energy of electrons in the

magnetic field B = ∇×A,  = {

,

} are Pauli

matrices, while the matrix of the exchange field in

the acting mean-field approximation is given by

(r)=



(r) h

⊥

(r)

⊥,∗

(r)−h

(r)

. (4.3)

For the moment we proceed in advance by informing

the reader that in the SC phase the ferromagnetic or-

der is modifiedinto a spiral ordomain-like structure

with the wave vector Q. As to which magnetic struc-

ture is realized depends on the magnetic anisotropy

contained in the operator

.Ifitissmall(orifitis

of the easy-plane type) then the spiral structureis re-

alized with h

⊥

(r)=he

iQz

and h

(r)=0,whileinthe

opposite case with an easy axis anisotropy one has

⊥

(r)=0andh

(r)=h

(r + L), L =2/Q.Theef-

fectof non-magneticimpuritiesis described by

imp

whose effect is characterized by the mean-free path

l and time .

4.2.1 Spiral Magnetic Order in Superconductors

The question is: how is the ferromagnetic order

modified in the presence of SC and do they coex-

ist at all? The first hint was given by Anderson and

Suhl [9] who studied the case h   and calculated

the spin susceptibility (q) of conduction electrons.

They found that (q)ispeakedatQ

≈ (a



)

−1/3

wherethelengtha ∼ k

−1

andk

is the Fermi momen-

tum and 

is the superconducting coherence length,

see Fig. 4.6. This magnetic order, called the crypto-

ferromagnetic order by Anderson and Suhl, has the

wave-length L where one has a  L  

. The prob-

lem of the coexistence of the spiral magnetic order

4 Coexistence of Singlet Superconductivity and Magnetic Order 171

and superconductivity for any ratio of h/ was stud-

ied in [10] and solved exactly. Because this solution

is a paradigm not only for the competition of su-

perconductivity and magnetic order but also for S-F

superlattices,we discuss some relevant results of this

theory by assuming (i) that the bare superconducting

transition temperature is much larger than the bare

magnetic criticaltemperature,i.e.T

 T

and (ii)

large exchange energy h

 . In an isotropic sys-

temwith negligible magnetic anisotropy and electro-

magnetic interactionthe natural choiceformagnetic

ordering is the spiral (helicoidal) spin structurewith

=0andh

⊥

=0

⊥

(r)=h



cos(Q ·r)+e

sin(Q · r)



. (4.4)

Here, e

and e

are unit vectors along the x-axis and

the y-axis, respectively. In isotropic systems in the

absence of EM interaction there is no preferable di-

rection of Q. Later we show that EM makes the struc-

ture transversal, i.e. S · Q = 0. By solving the Gorkov

equations (given in Appendix A) one obtains the nor-

mal and anomalousGreen’s functions G



andF



respectively. We shall not give details here (for de-

tails, see [10]) because of simplicity but stress only

some important properties. First, besides the singlet

pairing amplitude, which is responsible for the sin-

glet pairing  = gF

↑↓

(r, r), the triplet a mplitudes

↑↑

= 

↑

 and F

↓↓

= 

↓

 appear in the pres-

ence of the spiral exchange field

,↑↑

(k + Q/2, k



− Q/2) =

−ı(k − k



)2h

− 

+ E

1,k

][!

− E

2,k

]

, (4.5)

with F

↓↓

= F

↑↑

(Q → −Q, h → −h). Curi-

ously, F

↑↑

↓↓

)isanodd function of the com-

bined inversion k → −k, ! → −!, F

↑↑

(k, !)=

−F

↑↑

(−k, −!). Moreover, the quasiparticle spec-

trum is given by

1,2,k

= "

+ 

+ h

+ 

∓ 2



+ 

)+

(4.6)

where "

=(

k+Q/2

+ 

k−Q/2

)/2and

=(

k+Q/2

−



k−Q/2

)/2. The important fact is that for h >  the

branch E

1,k

= 0 is gapless for "

= h

− 

and



= 0. In the case of the parabolic band the gapless

spectrum is realized on the line that is the intersec-

tion of the plane v

· Q = 0 with the Fermi surface.

Near this line the spectrum is approximated by

1,k

≈





+ 

g,k

; 

g,k





+ 

, (4.7)

which leads to the linear density of states N(E)

N(E)

N(0)

h



, E   . (4.8)

The latter gives rise to the power law behavior of a

number of transport and thermodynamics proper-

ties.For instance, the specific heat is quadratic in T

(T)=

Q

. (4.9)

The model with the spiral order predicts a generic

phase diagram for ferromagnetic superconductors

that is qualitatively correct also for systems with

magnetic anisotropy, see Fig. 4.8 below. For, instance

at T

( T

) there is a second order phase transition

into the superconducting state with paramagnetic

magnetic phase, while at T

= T

(1 − ˛a/) ≈

there is a second order transition to the co-

existence phase; SC with the spiral magnetic order

with the wave vector Q

≈ (a



)

−1/3

.Bylowering

T the exchange field h grows, reaching the value

h  

,andatT

the superconductivity disap-

pears in the first order transition where the ferro-

magnetic phase is realized. At this point one has

≈ Q

(

)

1/6

(ln(

))

1/3

,i.e.Q

< Q

The phase diagram resembles that in Fig. 4.8. The ef-

fect of impuritiesis similar to those in the sinusoidal

and domain structures, which will be discussed be-

low.

4.2.2 Sinusoidal Magnetic Structure Due to SC

for T  T

In the case when there is a large easy axis mag-

netic a nisotropy D, S(r) is parallel to the z-axis. Be-

low but near T

the magnetic order parameter is

small, S(r)  1, and the non-uniform sinus struc-

ture S(r) ≈ S(T)e

sin qr is realized. In this case

172 M.L. Kuli´candA.I.Buzdin

(T)=h

S(T)  h

,  and the magnetic free-

energy can be calculated by the perturbation theory.

The general structure of the free-energy is given by

F{S(r), (r), A(r)} = F

{S} + F

{}

+ F

int

{S, , A} +



(B −4M)

8

. (4.10)

Here, F

(= E

− TS

+ E

anis

)andF

is the magnetic

and SC free-energy functional in the absence of mu-

tual interaction, respectively.The energy of the mag-

netic anisotropy is E

anis

=−



D | S

(q) |

,while

the magnetic energy E

and the entropy are obtained

straightforwardly from the microscopic theory based

on the Hamiltonian in Eqs. (4.2)–(4.3). For E

it gives

{S} =−







n,q



n,0

| S

{S}, (4.11)

where S(q) is the Fourier transform of S(r). The first

term is the lowering of the energy due to the indi-

rectRKKY exchangeinteraction via the spin polariza-

tion of the conduction electrons in the normal state.

The RKKY coupling is proportional to the spin sus-

ceptibility 

(q) of the conduction electrons. Since



 1 K, the dipole-dipole energy E

between mo-

ments must inevitably be taken into account. It has

the form [8]

{S} =







(˛

−1)| S

(qS

)(qS

−q

)



(4.12)

Here, the characteristic dipole–dipole energy



(

/3) + 



,where

=(B

/8n)=2n

and





characterize the long-range and short-range part

of the dipole–dipole interaction between LMs, re-

spectively.Their values are given in Table 4.1. The pa-

rameter ˛

characterizes the stiffness of the dipole–

dipole interaction [34]. It is evident from Eq. (4.12)

that E

is minimized when the structureis transverse

=0,i.e.q ⊥ S

. The length (magnetic stiffness)

a, which is of the order of the lattice constant, de-

pends on the lattice structure and the Fermi surface

of conduction electrons. The bare (in the absence of

superconductivity) ferromagnetic critical tempera-

ture is T



with



= 

+ 



,where



and 



characterize the long-range and short-

range parts of the EX interaction [8]. Proceeding in

advance we inform the reader that both E

and E

are renormalized (screened) in the superconducting

state. Since D

> 0, the magnetic anisotropy orients

spins along the z-axis. The last term in Eq. (4.10)

is the magnetic energy for a given magnetization

M(r)=nS(r) and in the equilibrium the magnetic

induction in absence of SC is given by B =4M.At

temperatures near T

( T

)onehas| S(r) | 1

and the exchange field is small, h  .Inthiscase

the SC free-energy density F

is weakly temperature-

dependent and is given by

{(r)} =−

N(0)

e



. (4.13)

It is minimized for  ≈ 

since at T near T

SC it is weakly affected by the ferromagnetism.

Therefore, near T

the SC order parameter is de-

termined solely by the pairing mechanism. The part

int

= F

int,ex

+ F

int,em

describes the EX and EM in-

teraction between SC and the magnetic order. Since

int,em

=−



/2andbyusingtherelation

=−K

s,q

one obtains

int









n,q

− 

s,q



| S

s,q

| A



(4.14)

The first term, which is positive, describes the in-

crease of the free-energy due to the decrease of the

spin susceptibility 

s,q

in the SC state,see Fig.4.6.The

EM kernel K

s,q

describes the Meissner screening of

the dipole–dipole interaction by the superconduct-

ing electrons.

After minimization of F{S(r), (r), A(r)} with re-

spect to A(r) one obtains F{S(r), (r)} in the follow-

ing form [8]

F{S, } =





[(T − T

)+a

] | S

− D

| S





n,q

− 

s,q



n,0

| S



4K

s,q

| S

+(qS

)(qS

−q

)

+4K

s,q



. (4.15)

4 Coexistence of Singlet Superconductivity and Magnetic Order 173

Fig. 4.6. Schematic spin susceptibility in

the normal (continuous line)andSC

state (dotted line ) 

n,s

(q): (a) for the fer-

romagnetic order in the normal state,

peak at Q =0;(b) for the antiferromag-

netic order, peak at Q

Here,  = 

+ ˛





+ ˛





= 

; for more details

see [8,34].Due to the singlet SC pairing the spin sus-

ceptibility 

s,q

is reduced significantly at small wave

vectors q < 

−1

,where

is the SC coherence length.

In the clean limit (l →∞)andatT =0onehas



s,0

= 0, which means that the ferromagnetic order

cannot coexist with the singlet superconductivity.In

Fig. 4.6 we show 

s,n,q

schematically for the cases

when the ferromagnetic (a) or antiferromagnetic or-

der (b) is realized in the normal state.It is seen from

Fig. 4.6 that the singlet superconductor behaves as a

normal metal at large momenta, i.e. 

s,q≈k

is weakly

affected by SC.Therefore AF competes with SC much

less than the ferromagnetic order does.

We stress that in the case of the singlet s-wave SC



s,0

= 0 and falls off exponentially at finite tempera-

tures, i.e. 

s,0

∼ exp(−/T), while in d-wave SC one

has 

s,0

≈ 1.4

n,0

(T/

).In the presence of the spin-

orbit (SO) scattering 

s,0

is also finite. The general

expression for 

s,q

in this case is [35]



s,q

=1−T



(1 + u

)(P

!,q

−1/2

)

, (4.16)

where u

= !

/ and

!,q

2arctan



2



1+u

+1/2

−



. (4.17)

Here !

= T(2n +1)and

−1

= 

−1

−

−(4/3)

−1



−1

−

= 

−1

+ 

−1

,where

−1

and 

−1

is the relaxation

rate due to non-magnetic impurities and spin-orbit

scattering, respectively. Later, we shall discuss effects

of the SO interaction on the coexistence phase. The

effect of the exchange scattering is similar, i.e. 

s,0

is finite for finite exchange scattering time 

.Con-

cerning this point, as well as other theoretical ques-

tions related to magnetic superconductors see the

review [7].

In the following we shall also study the competi-

tion between SC and ferromagnetism at low temper-

atures and in the clean limit where the EM kernel has

the form

s,q



mcqv



1−x

arcsh



xqv

2







xqv

2



. (4.18)

The expression for K

s,q

in the case of finite mean-

free path l is rather complicated and is omitted here,

see [7,8].However,somelimitingexpressionsforK

s,q

which are relevantfor real magnetic superconductors

with impurities, will be given below.

By knowing 

s,q

and K

s,q

one can minimize

F{S(r), (r)}with respect to the wave vector q,whose

equilibrium value Q

depends on microscopic pa-

rameters a, 

, 

(the London penetration depth),



, 

. From Eq. (4.15) we conclude that the EM in-

teraction is minimized for q · S(q)=0,i.e.themag-

netic structure in SC is transverse.

We give explicit expressions for 

s,q

and K

s,q

the interesting range of parameters.In the clean limit

and for q

 1onehas

s,q

4

; 

n,q

− 

s,q

= 

n,0



1−

(q

)



(4.19)

while for q

 1

s,q

4

q

; 

n,q

− 

s,q

= 

n,0



2q

. (4.20)

174 M.L. Kuli´candA.I.Buzdin

The minimum of F if Eq. (4.15) gives the transition

temperature into the sinusoidal magnetic structure

(r) ≈ S sin(Q

r)(withQ

⊥S

)

= T



1−3(

a/4

)

2/3



. (4.21)

Since a  

and 

∼  one gets that T

≈ T

i.e. the spiral order is very favorable since it lowers

the exchange energy and critical temperature very

little. In the case when 

 a

, the wave vector

is determined by the EX interaction only and for



/

 (a/

)

it is given by [34]





a





1/3

. (4.22)

For the extremely low exchange energy, i.e.

for 

/

 (a/

)

, the EM interaction pre-

vails with Q

=(1/a

)

1/2

. In the opposite limit



 a

, the EX interaction dominates for



/

 (a



/

)

2/5

, which again gives Q

(

4a



)

1/3

.For

/

 (a



/

)

2/5

the

EM interaction dominates, which gives Q

≈

(1/a





)

1/5

[34]. The microscopic parameters of

some rare earth ferromagnetic superconductors are

given in Table 4.1. The latter gives evidence that the

wave-vector Q

of the oscillatory structure is deter-

mineddominantlyby the EX interaction,i.e.itisinde-

pendent of the EM parameter 

.It turns out that the

EM interaction is dominant only for an extremely

small EX interaction (

 

(a/

)

or 





/

)

2/5

), i.e. for (

/

)  10

−4

−10

−5

However, in typical ferromagnetic superconductors,

such as ErRh

,HoMo

and AuIn

the

EX interaction dominates since 

> 0.1 

and

a  

 

In real compounds non-magnetic impurities are

always present and the knowledge of 

s,q

(l)and

s,q

(l) as a function of the mean-free path l is

needed. The corresponding calculations [34] show

that if (l





)  1 and for 

/

 a



the EX interaction dominates and one has Q

(

/4a



)

1/3

, while for (a



)  

/



/

),a

l/

one obtains Q

≈ (

/)(1/la



)

1/4

Inthecasewhen(

/

)  (a



), or

(

/

)  l

/

, the EM interaction dominates

and Q

≈ (l/a





)

1/4

Let us stress some interesting theoretical predic-

tions for ferromagnetic superconductors.

(i) The ferromagnetic critical temperature T

ferro

strongly reduced in the presence of SC due to the for-

mation of Cooper, i.e. one has T

ferro

= T

[1 − (



)/]  T

. Note that for the spiral state one has

≈ T

, which means that the oscillatory mag-

netic structure is more favorable than the uniform

ferromagnetic one.In fact, this result is more general

and holds also for the coexistenceof SC and itinerant

ferromagnetism (F) in the mean-field approximation

(MFA). Namely, in systems with the BCS-like pairing

and when the EX interaction dominates the EM one

the singlet superconductivity and fer romagnetism do

not coexist in the MFA.Inthissenseanumberof

recent papers claiming that the itinerant F and SC

coexist in the MFA should be abandoned [36]. How-

ever, in some itinerant ferromagnets such as Y

(with T

=4.5 K) the microscopic parameters favor

the spiral or domain-like magnetic structure in the

SC statewith T

=2.5K,aswasproposedin[37].Un-

fortunately, this interesting compound has not been

investigated sufficiently.

(ii) In isotropic magnetic systems near the critical

temperature T

the inverse scattering time due to

magnetic ﬂuctuations (at small q)candivergeand

thus destroy SC. However,this effect is suppressed in

the real RE ternary superconductors due to the long-

range dipole–dipole interaction. The interaction of

SC with magnetic ﬂuctuations is described by the

free-energy contribution

sc,ﬂ





S

z,q

z,−q





n,q

− 

s,q



n,0

, (4.23)

where

S

z,q

z,−q

∼

 + a

+(

/)q

, (4.24)

with  =(T − T

)/. Due to the large dipole–dipole

temperature with 

∼  these ﬂuctuations look

four-dimensional, thus giving a rather small value

for the inverse scattering time 

−1

∼   T

;

4 Coexistence of Singlet Superconductivity and Magnetic Order 175

(iii) the relative strength of the EX and EM interac-

tionsiscontrolledbytheparameterr

r =

(EM)

Int

(EX)

Int





. (4.25)

In the RE ternary compounds the case r  1isal-

ways realized, due to the large value of Q



 1.

Therefore, practically in all RE ternary superconduc-

tors the EX interaction dominates in the formation of

the magnetic structure in the superconducting state,

while the EM interaction, which contributes signifi-

cantly to T

, makes the structure transversal; see the

exception in the case of weak ferromagnets below.

(iv) In the RE ternary superconductors the ferro-

magnetism is stronger phenomenon than SC, since

the gain in the ferromagnetic energy (per LM and

at T =0K)E

≈ N(0)h

is larger than the gain

in the SC condensation energy E

≈ N(0)

,since

(∼ 10

K)  

( 10 K). Nevertheless, the fer-

romagnetic order is more “generous” since it varies

spatially in the SC state, while the SC order param-

eter stays practically homogeneous. The reason for

this peculiar phenomenon lies in the fact that the

(ferro)magnetic stiffness (∼ a)ismuchsmallerthan

the superconducting stiffness (∼ 

), since a  

We stress that the sinusoidally modulated magnetic

structure in the SC state perfectly resolves the an-

tagonism between superconductivity and ferromag-

netism. Indeed, from the“point of view of supercon-

ductivity”such a magnetic structure is like the anti-

ferromagnetic one since Q



 1 (for the EM in-

teraction) and Q



 1 (forthe EX interaction).On

the other hand,it islikea ferromagnetic one from the

“point of view of ferromagnetism”, since Q

a  1.

Then it is really a compromised structure.

4.2.3 Domain Magnetic Structure Due

to Superconductivity

By lowering T the higher order term in the free-

energy, ∼ S

(r), makes the change of the modulus of

S(r) unfavorable.Asa result,the sinusoidal-structure

is transformed into the striped domain structure

(DS), see Fig. 4.7(b). The exchange field grows h

S(T)byloweringT,butifh

(T) <  the mu-

tual interaction of the magnetic order and SC can

be treated by the perturbation theory. In such a case

the free-energy density is completed by the energy

density of domain walls, QE

/,whereE

is the

wall-energy per unit surface.In the case of the suffi-

ciently large magnetic anisotropy, D

> ,therota-

tion of magnetic moments in the wall is unfavorable

and the linear domain wall with S

(x)=Sth(x/l

= S

= 0 is realized. Here, l

= a/

√

 is the

domain-wall thickness [8]. The domain wall energy

perunitsurfaceisgivenby



√

2/3



nS

a

1/2

≡ nS

˜a , (4.26)

where  =(T

− T)/, see [8].

The free-energy density

in the DS phase is

given by

= n



−

S





+ n

7(3)

2

Q

(4.27)

Fig. 4.7. (a) Spiral structure with the period L

. (b) Striped

domain magnetic structure S(x)=S

(x)e

with the period

=2/Q

. The wave vector of the DS structure Q

along the x-axis

176 M.L. Kuli´candA.I.Buzdin

depends on S and Q only, since SC is practically

unaffected by the magnetic order near T

.Notethat

if the anisotropy energy is small, i.e.  > 2D

/,

the rotating domain wall is realized with the wall

thickness l

≈ a(/D

)

1/2

and the wall energy is

∼ nS

a(D

)

1/2

.In the case of an extremely small

magnetic anisotropy (D

/) < (a/

) ∼ 10

−2

−10

−3

the spiral structure is favorable. Minimizing F

Eq. (4.27) with respect to Q oneobtainstheequilib-

rium wave vector of the st riped DS phase

 2





˜a



1/2

. (4.28)

By comparing Eqs. (4.22) and (4.28) one concludes

that the period of the DS structure (L

=2/Q

)

is larger than that of the sinus one.We point out that

(i) the striped domain structureis due to SC and it is

a property of the bulk material,and (ii)thestructure

of the free-energy of the DS phase is mathematically

similar to the problem of the domain structure in a

normal ferromagnetic plate with the magnetization

perpendicular to the surface of the sample; see also

Sect. 4.2. In this case the role of F

int

plays the mag-

netic energy dissipated out of the plate.Generally,the

domain structure is realized when the wall thickness

√

 is much smaller than the domain thickness

/Q,thusimplyingthat  (a/

)

2/3

∼ 10

−2

At lower temperatures when the exchange field

is large, h

(T) > , the problem appears to be

non-perturbative. Since the period L

of the domain

structure is much larger than the atomic length a,

i.e. L

 a, the problem is studied by the quasi-

classical ELO equations, see the Appendix A.2. In the

presence of non-magnetic impurities these equations

are solved for the striped domain structure with the

domain-likeexchangefieldh

(r)=h

(r) where the

relative magnetization in the DS phase is given by

(r)=

4S(T)



∞



k=0

sin(2k +1)Qr

2k +1

≡



iqr

(4.29)

Since S · Q = 0, it follows that Q is along the x-axis.

The solution for the quasi-classical Green’s functions

f (v, x), (x)andg(v, x) is searched in the form

f (x, v)=f

(v)+



(v)e

ikx

, (4.30)

g(x, v)=g

(v)+



(v)e

ikx

, (4.31)

(x)= +



ikx

, (4.32)

where k =(2m +1)Q and | f

|| f

|, | 

||  |

and | g

|| g

|. In order to solve the problem the

self-consistent equations for the SC order parameter

are needed. The calculations were done in [8,34] and

here we present only the final result in the dirty limit

(l  

). It turns out that in this case the quasiclassi-

cal equations are reduced to those for

= f

(v)and

¯g

= g

(v) (... is the average over the Fermi sur-

face) which are the solutions of the following equa-

tions

+ ¯g

=1, !

− ¯g

=−2

¯g



−1

. (4.33)

The structure of these equations is the same as those

for SC in the presence of magnetic impurities with

the inverse scattering time 

−1

.Inthecaseofthe

magnetic domain (due to SC) one obtains



−1





h

z,q

z,−q

(ql)

·B

−q

16

nN(0)v

(ql)



(4.34)

where h

z,q

= h

z,q

and 

is the London penetration

depth. The functions L

and L

are given by [34]

(y)=

2y arctan y

(y −arctany)

(4.35)

and

(y)=







arctan y −



. (4.36)

The minimization of F with respect to the vector po-

tential A gives the magnetic induction B

which is

related to S

by [34]

4n[q

− q(qS

)]

+ K

s,q

(1 − 4/3

)

, (4.37)

where the EM kernel K

(q) in the dirty limit has the

form

s,q

3

16v



(ql) . (4.38)