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a r X i v :0801.3091v 1 [c o n d -m a t .s u p r -c o n ] 20 J a n 2008

Cryptoferromagnetism in Superconductors with Broken Time-Reversal-Symmetry

N.A.Logoboy and E.B.Sonin

Racah Institute of Physics,The Hebrew University of Jerusalem,Jerusalem 91904,Israel

(Dated:February 2,2008)The cryptoferromagnetic state (the state with intrinsic domain structure)in superconducting fer-romagnets subjected to external magnetic field is studied theoretically.Ferromagnetism originates either from electron spin or the intrinsic angular momentum of Cooper pairs (chiral p -wave super-conductors like Sr 2RuO 4).The phase transitions towards the Meissner and the mixed states are investigated,and the magnetic phase diagrams are obtained.Cryptoferromagnetism,as a form co-existence of superconductivity and ferromagnetism,can be detected by observation of magnetization curves predicted in the present analysis.

PACS numbers:74.25.Ha,74.90.+n,75.60.-d

In recent years numerous experimental evidences of superconductivity-ferromagnetism coexistence in various materials were reported [1,2,3,4,5,6].Two types of such coexistence are possible:(i)The phase transi-tions to the ferromagnetic and the superconducting (SC)states occurs at different temperatures,so the coexistence starts below the lower from the two transitions.Rutheno-cuprates [1]belong to this type:the superconductivity onset occurs at the temperature much lower than the temperature of the magnetic transition.Normally differ-ent elements of the crystal structure are responsible for ferromagnetism and superconductivity,and spontaneous magnetization (ferromagnetic order parameter)is related to spin.Later we call them spin superconducting ferro-magnets (spin SFs).(ii)The magnetic and the SC transi-tions occur simultaneously.This can take place in uncon-ventional superconductors with triplet Cooper pairing.An example of them is strontium ruthenate Sr 2RuO 4[3,4,7].The theory connects spontaneous magnetiza-tion in this material not with spin but with the orbital intrinsic angular moment of the p -wave Cooper pair with the wave function in the momentum space proportional to p x +ip y (chiral p-wave superconductivity).We shall call them orbital superconducting ferromagnets (orbital SFs).

Whereas proof of superconductivity in SF materials is quite straightforward,a clear-cut detection of the fer-romagnetic order parameter is much more problematic.The internal magnetic field is screened out by the SC Meissner currents and can be present only near sample borders and defects,in particular,domain walls (DWs).This strongly suppresses the stray magnetic fields around the sample,which are most convincing evidence of ferro-magnetism.Especially worrying is situation with stron-tium ruthenate Sr 2RuO 4,where Kirtley et al.[8]could not detect any stray field from DWs or sample edges at all.This is a challenge for the theory and for the very sce-nario of chiral p-wave pairing.Difficulties with direct de-tection of ferromagnetism coexisting with superconduc-tivity lead to the question whether one may use the term ferromagnetism at all.Indeed in the literature on uncon-ventional superconductors sometimes they prefer to tell about superconductivity with broken Time-Reversal Sym-metry (TRS).We used this name in the title of the paper following this more cautious semantics though one can-not imagine broken TRS without at least some features of ferromagnetism (see further discussion).

Among possible explanations why experimentalists cannot see stray fields from DWs is the presence of do-main structure with a period essentially smaller than a distance between a sample surface and a probe used by experimentalists.There were some experimental ev-idences of domains in SFs both in the spin [9]and the orbital SF [10].The theoretical investigations of the do-main structure in SFs were restricted with the case of zero external magnetic field [11,12,13,14].One must discern two possible types of equilibrium domain struc-ture.The first one is well known for normal ferromag-nets [15].The domain structure results from competi-tion between the energy of DWs and the magnetostatic energy of stray fields generated by the magnetic flux exit-ing from the sample surface.The period of the structure depends on the shape and the size of the sample going to infinity when the sample size grows.One can call these domains extrinsic ferromagnetic domains .Since in SFs the Meissner effect expels the magnetic field,it is impossible to benefit from decreasing the bulk magne-tostatic energy in comparison with the DW energy,and extrinsic domains cannot appear at equilibrium [12].But also long ago there was known another type of domains,which decrease the bulk magnetostatic energy at the ex-pense of destroying the Meissner state [11,13,14].The size of these domains is roughly of the order of the Lon-don penetration depth λand does not depend on either shape or size of the sample.Strictly speaking the state with this domain structure at the macroscopic scales is not ferromagnetic but antiferromagnetic though with a rather large period.We shall call such a state cryptofer-romagnetic ,the term introduced by Anderson and Suhl [16]for another model of ferromagnetism and supercon-ductivity,in which crystal anisotropy was neglected and spiral structure appeared instead of domains.

In present publication we extend the theory of intrinsic domain structure (cryptoferromagnetic state)on nonzero external magnetic field and analyze competition of the cryptoferromagnetic state with the pure Meissner state

2 and the mixed state with vortices.This allows to ob-

tain the full phase diagram of both spin and orbital SFs. We demonstrate that the measurement of magnetization curves in various areas of the phase diagram can provide evidence of ferromagnetic or cryptoferromagnetic order in superconductors with broken TRS.

Let us consider a stripe domain structure with1800 DWs in a sample subjected to external magneticfield H0=(0,H0,0).The DWs are parallel to the yz-plane separating domains with alternating magnetization M= (0,±M0,0)along the+y or−y direction.Since the H0 orientation is preferable the width d↑of domains with the magnetization M parallel to H0(↑-domains)exceeds the width d↓of the domains with M antiparallel to H0 (↓-domains).We restrict ourselves to the simplest case when the London penetration lengthλexceeds the DW thickness.Then the surface energyσand the internal structure of DW is not affected byfields and currents at scales ofλ.

The Gibbs potential inside domains is

G= d3x h2c2j2−h·M−h·H0

coshξ↑,↓

,(2)

where x is the distance from the DW andξ↑,↓=d↑,↓/2λare reduced domain widths.

Application of the Gibbs potential Eq.(1)to orbital SFs requires some

comments.As

shown

in

Ref.[17],for

orbital ferromagnetism related to the intrinsic angular momentum of Cooper pairs the spontaneous magnetiza-tion cannot be defined unambiguously and therefore the Landau-Lifshitz theory of ferromagnetism[15]based on this definition is not valid.Nevertheless,interaction of magnetization currents inside narrow DW with the mag-neticfield can be reduced to the expression looking like the standard Zeeman energy−h·M0.However,here M0is not a magnetic moment inside the domain but is defined so that8πM0would be the jump8πM0of the magneticfield on the DW[see Eq.(2)].So“magnetiza-tion”M0is determined by the DW structure and cannot be used for other phenomena connected with ferromag-netic ordering,e.g.,analyzing the magnon spectrum[17]. Substituting Eq.(2)into Eq.(1),adding the sur-face energyσof DWs,and averaging over the domain-structure period d=d↑+d↓we arrive to the following expression for reduced energy density E=G/2πM20V(V is the sample volume):

E=2w−(1+h0)2tanhξ↑−(1−h0)2tanhξ↓

2

tanh

ξ↑+ξ↓

(coshξ↑+coshξ↓)2

=w.(4)

The magnetic induction B= h is determined by re-duced magnetic induction b=B/4πM0:

b=−

1

∂h0

=

(1+h0)tanhξ↑−(1−h0)tanhξ↓

h c(1+h c)−

(1−h c)2h c

h c

,(6) the critical size of the↑-domain being

ξ↑c=ln

1+

1−

h c ln1+√1−√

4

(3w)2/3,ξ↑c=(3w)1/3.(9) Aside from the critical line Eqs.(4)yield:

h0=

ξ2↑−ξ2↓

3

.(10) For small h0≪h c one can solve Eqs.(10)analytically:ξ↑,↓=41/3 44/3h2c ±h0h c.(11)

Let us consider the magnetization curve in the crypto-ferromagnetic state.The linear magnetic permeability is

3

w 2

w 2

=

w 2

b)h 0h 0

h 0

a)c)FIG.1:(color online)Phase diagram for various values of the reduced lower critical field h c 1:(a)h c 1→∞;(b)h c 1=2;(c)h c 1=0.83.The lighter (yellow)shaded area is the cryptofer-romagnetic state.The darker (blue)shaded area is the mixed state.The rest is the Meissner state.The horizontal and the vertical arrows show the processes of cool-down across the SC critical temperature of spin and orbital SFs respectively.

determined from two relations connecting µand w with the period ξ=ξ↑+ξ↓≈2ξ↑

µ=

db

ξ

,

w =tanh ξ−

ξ

4πM

2

=−

1+h 0−h c 1w 2 2,(13)

where the field H ∗inside the mixed state differs from H c 1by another logarithm factor,but we neglect it assuming H ∗≈H c 1.The phase transition to the mixed state may occur either from the Meissner state being determined by the condition E m =0,or from the cryptoferromagnetic state crossing the critical line on which E m =E .At zero external field h 0and small w the phase transition be-tween the mixed state and the cryptoferromagnetic state occurs at w m ≈

4

0.00.10.20.30.4

0.00.10.20.30.4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

b

h 0

a )

b )

3

2

1

,

h

FIG.2:(color online)(a )Magnetization curves and (b )mag-netic field dependencies of ξ=ξ(h 0)(solid line)and δ=δ(h 0)(dash line)are shown at different values of parameter w .Ver-tical lines correspond to critical fields above which the intrin-sic domain structure collapses.

this case the “magnetization”M 0∼Φ/λ2∝τ,and the

DW surface energy is a product of the condensation en-ergy H 2

c (τ)∼[Φ0/λ(τ)ξ0(τ)]2an

d th

e coherence length ξ0(τ)∼ξ0/

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