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4 Coexistence of Singlet Superconductivity and Magnetic Order

in Bulk Magnetic Superconductors and SF Heterostructures

M. L. Kuli´c Brookhaven Natl. Lab., Upton, New York, USA and Max Planck Institute – Physics of Complex

Systems, Dresden, Germany

A. I. Buzdin Inst. Univ. de France and Univ. Bordeaux, Talence, France

Abstract. The basic physics of bulk magnetic superconductors (MS) related to the problem of the coexistence of singlet

superconductivity (SC) and magnetic order is reviewed. The interplay of the exchange (EX) and electromagnetic (EM)

interaction between conduction electrons and localized magnetic moments in the formation of the coexistence phase is

discussed.Itis shown thatthesinglet SC anduniform ferromagnetic (F) order practically nevercoexist in bulk materials.In

the case of their mutual coexistence the F order is modified into a domain-like or spiral structure depending on magnetic

anisotropy. It turns out that the domain-like structure is realized in several superconductors such as ErRh

,HoMo

HoMo

with electronic order and (probably) the spiral structure in AuIn

with nuclear magnetic order. The latter

problem is also discussed. The coexistence of SC with antiferromagnetism (AF) is more favorable. A rich phase diagram

is realized in AF systems with SC and weak-ferromagnetism. A number of properties of magnetic superconductors in

the magnetic field are very peculiar, especially near the (ferro)magnetic transition temperature, where the upper critical

field becomes smaller than the thermodynamical critical field.

The Josephson junctions based on MS with spiral magnetic order is discussed, in which the induced triplet pairing

amplitude gives rise to  -contact (with the  -phase difference in the ground state). The interplay of the superconducting

and the magnetic phase, ' , , respectively, in such a contact renders possibilities for an interplay of the charge and spin

current, and for a new type of coupled Josephson-qubits in a single Josephson junction.

The coexistence of superconductivity and ferromagnetism may be easily achieved in artificially fabricated supercon-

ductor/ferromagnet heterostructures. Due to the proximity effect, the Cooper pairs penetrate into the F layer and one

has the unique possibility to study properties of superconducting electrons under the inﬂuence of a large exchange field.

Moreover,varying in a controllablemanner the thickness of the ferromagnetic and superconducting layers it is possibleto

change the relative strength of two competing orderings. The Josephson junctions with ferromagnetic layers reveal many

unusual properties that are quite interesting for applications, in particular the recently fabricated so-called -Josephson

junction.

4.1 Introduction ..............................................................................164

4.2 Ferromagnetic Superconductors ............................................................169

4.2.1SpiralMagneticOrderinSuperconductors...............................................170

4.2.2 Sinusoidal Magnetic StructureDue to SC for T  T

......................................171

4.2.3DomainMagneticStructureDuetoSuperconductivity....................................175

4.2.4DomainMagneticStructureinThinSuperconductingFilms...............................179

4.2.5CoexistenceofNuclearMagnetismandSuperconductivity.................................179

4.3 Antiferromagnetic Superconductors (AFS) ...................................................180

4.3.1WeakFerromagnetisminAntiferromagneticSuperconductors.............................182

4.4 Magnetic Superconductorsin the Magnetic Field ..............................................183

4.4.1FerromagneticSuperconductors........................................................183

4.4.2AntiferromagneticSuperconductorsintheMagneticField.................................187

4.5 JosephsonEffect with Bulk Magnetic Superconductors ........................................188

164 M.L. Kuli´candA.I.Buzdin

4.6 Superconductor/Ferromagnetic Heterostructures .............................................189

4.6.1Fulde–Ferrell–Larkin–OvchinikovStateandProximityEffectinS/FStructures...............190

4.6.2OscillatorySuperconductingTransitionTemperatureinS/FMultilayerandBilayers..........193

4.6.3 Superconductor-Ferromagnet-Superconductor -Junction................................194

4.7 Conclusion ................................................................................196

Appendix ....................................................................................197

A.1Gor’kovEquationsforMagneticSuperconductors..........................................197

A.2Eilenberger-Larkin-OvchinikovEquationsforMagneticSuperconductors....................197

References................................................................................199

4.1 Introduction

The physics of magnetic superconductors (MS) is

a very intriguing subject due to the pronounced

competition of the ferromagnetic order and sin-

glet superconductivity in bulk materials. Two fun-

damental properties of the singlet superconducting

state are zero electrical resistance and the Meissner–

Ochsenfeld effect; expulsion of the magnetic field

from the volume of a superconductor due to screen-

ing currents. On the other hand, superconductivity

can be destroyedbyapplying a magnetic fieldthat ex-

ceeds some critical value and acts on the orbital mo-

tion of superconducting electrons.A magnetic order

is characterized by magnetizationthat createsa dipo-

lar magnetic field B

inthe sample.This fieldinduces

the screening (the Meissner–Ochsenfeld effect) su-

perconducting current j

scr

and the corresponding

magnetic field B

scr

making the total field (magnetic

induction)in the sample B = B

scr

= ∇×A.This

type of interactionbetween the orbital motion of su-

perconducting electrons and (ferro)magnetic order

is called the electromagnetic (EM) interaction,the

EM model.The question regardingthe coexistence of

SC and ferromagnetism was first addressed theoreti-

cally byVitaliiLazarevich Ginzburgin 1956[1] in the

framework of the EM model. The latter is schemat-

ically shown in Fig. 4.1. In [2] it was found that in

type-I SC with 4M < H

the coexistence of ferro-

magnetic order and superconductivity is possible in

the Meissner phase, where the superconducting or-

der parameter  and the magnetization M are con-

Fig. 4.1. The Meissner superconducting-ferromagnetic

phase. Schematic mechanism of electromagnetic (EM) in-

teraction in ferromagnetic superconductors. Dipolar field

dip

, created by the localized moments with the magneti-

zation M(r), induces screening current j

scr

(r)andfieldB

scr

The total induction is B = B

dip

+ B

scr

stant in the sample. The magnetic induction B =0,

which is due to the surface screening current. In the

case when 4M > H

the coexistence is possible

only in a very thin film where the orbital effect of

the magnetic field is suppressed, or by applying an

external field in the opposite direction to M.

After the advent of the microscopic theory of su-

perconductivity, by Bardeen, Cooper and Schrieffer

in 1957 [3], it became clear that the singlet super-

conductivity may be also destroyed by the Zeeman

(paramagnetic) effect of the effective magnetic field,

4 Coexistence of Singlet Superconductivity and Magnetic Order 165

which is due to the exchange interaction (EX)ofelec-

tronic and localized spins; the EX model.Theresult-

ing effective exchange field, which acts on conduc-

tion electrons in the ferromagnetic state, tends to

align spins of Cooper pairs in the same direction,

thus preventing pairing. The BCS theory predicts

that a Cooper pair is formed from two electrons with

opposite momenta and also opposite spins, singlet

superconductivity, and when the gain in the energy

| E

|= 

H (

is the Bohr magneton) due to the

electronic spin reorientation becomescomparable to

the Cooper pair superconducting coupling energy

(which is of the order of the superconducting criti-

cal temperature T

),then the singlet superconductiv-

ity is destroyed. One obtains the following estimate

for the paramagnetic limit H

≈ 1.24T

/

(or in

Tesla H

[T] ≈ 1.8 T

[K]) above which the singlet su-

perconductivity is destroyed.While the orbital effect

in non-magnetic superconductors is more impor-

tant for superconductivity destruction by the mag-

netic field, i.e. the EM effects dominate, the situation

is quite different in ferromagnetic superconductors

where the EX effects usually dominate over the EM

ones. The internal magnetic induction (B

≈ 4M)

due to the spontaneous magnetization M does not

exceed several kOe in the rare earth (RE) ternary

magnetic superconductors that are discussed in this

chapter, and it is smaller than the typical upper crit-

ical field H

in type II superconductors. Therefore

the EM interaction is usually not harmful for su-

perconductivity in magnetic superconductors as it

is the case with the EX one.. On the other hand, the

effective internal exchange field h

(which acts on

electronic spins and plays the role of the Zeeman

energy 

H in non-magnetic superconductors) is

very large. Typically, if expressed in the energy units

≈ (100−1000) K, which substantially exceeds the

superconducting critical temperature T

.Thismakes

the exchange interaction the main destruction mech-

anism of the superconductivityin most magnetic su-

perconductors. The strong exchange interaction is

at the origin of very strong suppression of super-

conductivity by magnetic impurities. Early experi-

ments [4] demonstrated that even very small (sev-

eral percent) concentration x of magnetic atoms is

enoughto completely destroy superconductivity.The

Fig. 4.2. The critical temperature variation versus the con-

centration n of the Gd atoms in La

1−x

alloys. points:

experiment [6]. Line: the A-G theory. T

=3.24 K and

=0.590 atomic percent Gd

inﬂuenceof magnetic impuritieson superconductiv-

ity is described by the Abrikosov–Gorkovtheory [5],

see Fig. 4.2. This theory predicts that the slope of

the curve T

(x)isgivenby(dT

/dx) ≈ −

,where



≈ N(0)h

and N(0) is the density of states per

electron spin. So, if one puts even a very small con-

centration of Fe atoms, they destroy SC since h

and



are very large in Fe. However, the allowed small

concentration of magnetic atoms in superconduct-

ing alloys is not enough to produce ferromagnetic

ordering and there are no possibilities to investigate

its competition with SC.

The breakthrough in the physics of MS came with

the discovery of rare earth (RE) ternary c ompounds,

such as borides (RE)T

with transition elements

T =Rh, Ir, chalcogenides (RE)Mo

(X =S, Se),

silicides (RE)

and stannides (RE)T

[6–8].

In most of these type-II superconductivity is real-

ized and in all of them the localized RE magnetic

moments are r egularly distributed in the crystal lat-

tice. The basic crystallographic structure, which is

shown in Fig. 4.4 for RERh

, contains localized

magnetic moments (LMs) placed far a way from the

Rh and B blocks which deliver conduction electrons.

Due to this spatial separation the conduction elec-

trons rarely jump onto magnetic ions, thus mak-

ing the effective direct exchange interaction (EX) J

muchsmallerthanintransitionmetallicmagnets,i.e.

166 M.L. Kuli´candA.I.Buzdin

Fig. 4.3. Schematic mechanism of the indirect RKKY ex-

change (EX) interaction H

RKKY

in rare earth superconduc-

tors. The spin S

at R

polarizes spins of conduction elec-

trons with the spin susceptibility (R

− R

). The polariza-

tion is transferred to the spin S

at R

(10

K).Inthe rare earth compounds the 4f shell

is responsible for localized moments,since the f -level

lies much below the Fermi energy, i.e. E

 E

.In

these compounds various magnetic ordering are re-

alized. Since the distance between the neighboring

magnetic ions is large and the direct exchange inter-

action of localized moments is negligible,this means

that the magnetic ordering in these system must be

due to some other mechanism. In RE metals this

mechanism is provided by conduction electrons in

the following way. The magnetic moment S

at place

polarizes the spins of conduction electrons (via

the direct exchange interaction J

)andthispolar-

ization is transferred to electrons surrounding the

spin S

at R

. The latter interacts via the direct ex-

change interaction J

with the polarization cloud of

conduction electrons. Since this is a second order

process the characteristic effective exchange interac-

tion between localized moments at distance R

− R

is proportional to J

and to the spin susceptibility

(R

− R

) of the conduction electrons. This type of

indirect exchange between localized spins, which is

shown in Fig.4.3,iscalled the RKKY interactionwith

the Hamiltonian

Fig. 4.4. Crystal structure of tetragonal ferromagnetic su-

perconductor ErRh

phase [8]

=−J





− R



·S

. (4.1)

In the normal state the susceptibility  can give rise

either to a ferromagnetic order or to a non-uniform

one such as antiferromagnetic or spiralordering,de-

pending on the topology of the Fermi surface. In the

superconducting state the susceptibility  is signif-

icantly changed by the Cooper pairs, what shall be

discussed below.In a number of the RE earth ternary

compounds, such as (RE)Rh

(RE = Dy, Sm,...),

the coexistence of superconductivity and antiferro-

magnetism has been realized; they are called antifer-

romagnetic superconductors (AFS). In most of them

the Ne´el (AF) transition temperature T

is smaller

than the superconducting one T

,i.e.T

 T

.The

reason for the peaceful coexistence of superconduc-

tivity and antiferromagnetism is well understood

and is based on the fact that the averaged exchange

and orbital fields of the AF order is zero on the su-

perconducting coherence length 

. This will be dis-

cussed in the following sections.

Since the late 1970s much experimental and the-

oretical research has been devoted to MS systems

in which ferromagnetic (F) order and singlet SC

compete due to their strong antagonistic characters.

These systems are called ferromag netic superconduc-

tors (FS) and they are one of the main subjects of

this review. It will be demonstrated below that su-

perconductivity modifies the F order into a mod-

ulated structure such as spiral or domain-like one,