P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19

Coexistence of Ferromagnetism and Singlet Superconductivity via Kinetic Exchange

Mario Cuoco,1,2 Paola Gentile,1 and Canio Noce1,3

1Unita I.N.F.M. di Salerno, Dipartimento di Fisica ‘‘E. R. Caianiello,’’ Universita di Salerno, I-84081 Baronissi (Salerno), Italy2Centre de Recherches sur les Tres Basses Temperatures associe a l’Universite Joseph Fourier,

C.N.R.S., BP 166, 38042 Grenoble-Cedex 9, France3Unita I.N.F.M. di Salerno-Coherentia, Salerno, Italy

(Received 10 April 2003; published 3 November 2003)

197003-1

We propose a novel mechanism for the coexistence of metallic ferromagnetism and singlet super-conductivity assuming that the magnetic instability is due to kinetic exchange. Within this scenario, theunpaired electrons which contribute to the magnetization have a positive feedback on the gain of thekinetic energy in the coexisting phase by undressing the effective mass of the carriers involved in thepairing. The evolution of the magnetization and pairing amplitude and the phase diagram are firstanalyzed for a generic kinetic exchange model and then are determined within a specific case with spindependent bond-charge occupation.

DOI: 10.1103/PhysRevLett.91.197003 PACS numbers: 74.20.Mn, 74.70.Tx, 75.10.Lp

exchange coupling. Theoretical studies along those direc- tential energy gain.We will analyze this possibility in two

The problem of the interplay between ferromagnetic(FM) and superconducting (SC) long range order hasbeen recently attracting new interest due to the discoveryof superconductivity in ferromagnetic metals, UGe2[1], ZrZn2 [2], URhGe [3], and in rutheno-cuprateRuSr2RECu2O8 compounds, with RE � Eu or Gd [4].

The investigation of ferromagnetic superconductorsstarted by analyzing the case of two interacting subsys-tems: one formed by localized spins or aligned magneticimpurities which produces the ferromagnetic background,and the other composed by itinerant electrons which givesrise to the superconductivity. Within this framework,early works [5–7] focused on singlet superconductivityin the presence of a spin-exchange field, showing that itcan exist only below a critical value of the magneticcoupling. Hence, with the purpose to increase the thresh-old of the critical spin exchange, it was suggested that afinite-momentum pairing state coexist with the ferromag-netic order [8,9]. In this pairing configuration, commonlyindicated as the Fulde-Ferrell-Larkin-Ovchinnikov state,the subtle balance between the condensate energy of theCooper pairs with a finite center-of-mass momentum andthe Zeeman energy, related to the magnetic moments ofdepaired itinerant electrons, was able to account for asuperconducting-ferromagnetic (SF) phase in the pres-ence of spin exchange higher than the zero-momentumpairing state.

Nevertheless, in the above-mentioned materials, a newphenomenology seems to arise if compared to the conven-tional case of a metal with magnetic impurities. Indeed, ithas been suggested that (i) ferromagnetism and super-conductivity are cooperative phenomena, (ii) the FMstate is due to itinerant electrons, (iii) the same electronsparticipate in both the FM and SC orders, and (iv) for thecase of systems with two types of carrier responsible forthe SC and FM phases separately, there are interestingcooperative effects due to competing charge- and spin-

0031-9007=03=91(19)=197003(4)$20.00

tions have been recently performed by considering theoccurrence of ferromagnetism and superconductivity ineither singlet [10–13] and/or triplet [14–16] channels ofpairing. In particular, as far as the SC state with singlets-wave pairing is concerned, it has been shown [17] thatits coexistence with weak itinerant ferromagnetism canbe obtained within a single band model, where the ferro-magnetic order is driven by the same electrons that par-ticipate in the formation of Cooper pairs. One crucialaspect of such analysis concerns the stability of the SFstate. The loss of condensation energy due to the depairedelectrons which produce a nonzero total magnetization isnot compensated by the energy gain of the magneticexchange; thus the SF state turns out always to be ener-getically unfavorable against the nonmagnetic SC oneeven if s- and d-wave symmetry or the finite-momentumpairing state are considered [18].

In this Letter, we propose a novel mechanism for thecoexistence of superconductivity and ferromagnetismwithin a single band model. The new and crucial ingre-dient is that the metallic ferromagnetism is not due, as inthe previous studies, to a rigid shift in the positions of themajority and minority spin bands (i.e., Stoner model), butit is a consequence of a change in the relative bandwidthof electrons with up and down spin polarization. In thiscircumstance, the gain of energy comes from the undress-ing of the mass for the majority spins which induces abandwidth enlargement and in turn lowers the kineticenergy [19]. When the pairing interaction is switchedon, the interplay between the gain in kinetic energy,due to the relative change of the majority and minorityspin bands, and the condensation energy which wouldtend to pair all the electrons becomes crucial for thestability of the SF state. It turns out that such a phase isthe most favorable only if a suitable tuning of the ratiobetween the density of depaired and paired states isreached, to optimize the balance in the kinetic and po-

2003 The American Physical Society 197003-1

P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19

steps: (I) the case in which the variation between the massof up and down spin electrons is arbitrarily modifiedwithout referring to any microscopic mechanism is firstconsidered; (II) a specific tight-binding model in whichoff-diagonal Coulomb interactions are responsible for anasymmetric bond-charge distribution for each spin chan-nel is then investigated. For both cases, the attractiveinteraction will be assumed to be of BCS type.

Model I. Let us start from a model Hamiltonian whichcontains a local attractive potential and itinerant elec-trons with a spin dependent mass:

HI � �tXhij;i

�2w��cyicj � H:c:� � g

Xi

cyi"cyi#ci#ci"

��Xi

cyici; (1)

where cyi �ci� creates (destroys) an electron with spin at the site i, w is a positive term that controls therenormalization of the mass for electrons with spin (the factor 2 being introduced for convenience), � is thechemical potential, t is the hopping amplitude whichdefines the bare bandwidth, and g is the pairing couplingbeing effective only in a shell of amplitude 2!c aroundthe Fermi surface as in the usual BCS theory. It is worthpointing out that in this case the mass dressing andundressing can be generated both via the coupling to abackground of spin, as in double-exchange systems, or todynamical processes intrinsically generated by electroncorrelations.

The introduction of a pairing amplitude, after themean-field decoupling of the attractive term, brings thefollowing diagonal expression for the Hamiltonian:

HI-MF �Xk

�E�k �yk�k � E

�k �

yk�k� � E0; (2)

E0 �Xk

��E�k � �2w#�k ���� ��2

g; (3)

where �k � �tP� exp�ik �� is the bare dispersion with

� being a vector connecting a site to its nearest neighbors,and � �

Pkghck"c�k#i is the pairing amplitude, respec-

tively. The field operators �k, �k correspond to fermionicexcitations with quasiparticle dispersion

E�;�k � �a�k ��������������������������������������b�k ���2 � �2

q: (4)

b � w" � w# and a � w" � w# being the average and halfthe difference of the spin mass renormalization. It isimmediately apparent that the kinetic exchange ampli-tude is proportional to a while the value of b determinesthe order of magnitude of the average kinetic energy. Still,the sign of the total magnetization (M) follows that ofa, while b is always positive. Thus, without any lossof generality one can focus only on the case witha � 0, the other one being derived just by inverting thedirection of M.

197003-2

The equations for the pairing amplitude and the mag-netization have the following form:

� �g�2

Xk

1� n�k � n�k������������������������������������

�b�k ���2 � �2

p ; (5)

M ��B2

Xk

�n�k � n�k �; (6)

where n�;�k are the Fermi distributions in the momentumspace of the fermionic fields having E�;�k as energy dis-persion relations, respectively, and �B is the Bohr mag-neton. Let us now discuss under what conditions thesystem can accommodate a coexistence of SC and FMorder. The analysis will be restricted to the zero tempera-ture limit, and furthermore we will assume that thedensity of electrons n is self-consistently fixed by a posi-tive value of the chemical potential �. The results can besymmetrically extended within the same procedure to thenegative � case. Hence, considering that b; a;�; � � 0and b � a, it is possible to show that n�k � 0 as E�k isalways positive, while n�k � 1 when �k;� � �k � �k;�,where �k;� � �b��

��������������������������������������������a2��2 ��2� � b2�2

p�=�b2 � a2�.

The �k;� define a range on the positive side �k, which isdifferent from zero only when a and b are linked to theamplitude of � and � in a way to fulfill the relation a

b >�=

��������������������2 ��2

p. If the previous inequality holds, single

particle states form within the gap and contribute togive a finite magnetization. Then, depending on theirdensity, the system can allow for a state with coexistingSC and FM orders. In the limit of�! 0 (n� 1) or a! 0(M� 0), there are no real solutions for �k;� so that oneends up with the usual BCS-like state with zero magne-tization. In the other cases, the self-consistent equationsfor the gap amplitude and the number of electrons have tobe solved to determine the conditions for the existence ofthe SF state. For such purpose, it has been used as a modeldensity of states given by N��� � 1=2w (with w � 2t) if�w � � � w. We notice that, contrary to the case of theStoner model, a nontrivial ferromagnetic solution can beobtained without the need of a curvature in the density ofstates close to the Fermi level [19]. The analysis of thesolutions is performed by fixing the value of b and n andstudying how the pairing amplitude, the energy, and themagnetization are modified by changes of a. Because ofthe formation of unpaired electrons within the gap, thevalue of the effective average mass in the SF state (bsf) ismodified with respect to the case of zero magnetization(bsc). Hence, there might occur two distinct cases de-pending on the microscopic mechanism which controlsthe relative change of the majority and minority spinbandwidths: (1) undressing of the average effectivemass, i.e., �b > 0; (2) dressing of the average effectivemass, i.e., �b < 0, �b � �bsf � bsc�=bsc being the rela-tive percentage variation.

In Fig. 1 is shown the comparison between the energyof the SF and the SC states, together with the behavior of

197003-2

P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19

the magnetization and the pairing amplitude as a functionof the relative shift between the mass of up and down spinelectrons. For simplicity, we have fixed bsc � 1, and wehave studied how the difference of energy is modified incases (1) and (2). Changes in bsc produce only quantitativebut not qualitative differences in the results. As shown inthe top panel of Fig. 1, only when �b is positive and for asuitable density of the depaired states, controlled by theamplitude of a, can the system be stabilized in a SF phase.When the average mass increases, though the effectiveamplitude pairing grows and in turn lowers the conden-sation energy, the loss in the kinetic energy of theelectrons that participate in the pairing cannot becoun-terbalanced to get a stable SF state. It is worth pointingout that the transition from the SF state to the SC one is of

FIG. 1. From the bottom to the top: superconducting gap,total magnetization, and relative energy between the SF andthe SC phases, respectively, as a function of a, for differentvalues of the average effective mass renormalization. Thedensity of electrons has been fixed to n � 1:2 and the couplingstrength is g=t � 1.

197003-3

first order for the magnetization but it is continuous forthe pairing amplitude by moving from small to largevalues of a, while it is reversed in the opposite a direction.Nevertheless, depending on the amplitude of the undress-ing �b, the transition from the SF state can be first ordertype in both the order parameters. Finally, due to thepeculiar link between � and�, we notice that the changein the density of electrons modifies the region where theSF solution exists. In particular, the interval of solutionsshrinks if one moves towards the half-filling case,whereas a large band undressing is required to stabilizethe SF phase.

Model II. Now we consider an explicit tight-bindingmodel where the bandwidth change depends on the bond-charge occupation in each spin channel [19]. The modelHamiltonian has the following form:

HII � � tXhiji

�cyicj � H:c:� �UXi

ni"ni# � VXhiji

ninj � JX

hiji0

�cyicyj0ci0cj � H:c:� � J0

Xhiji

�cyi"cyi#cj#cj" � H:c:�

� gXi

�cyi"cyi#ci#ci" � H:c:� ��

Xi

cyici; (7)

where U and V represent the on-site and nearest-neighbor

Coulomb repulsions, respectively, and the parameters Jand J0 describe nearest-neighbor exchange and pairhopping processes, while g is the BCS pairing. By apply-ing the Hartree-Fock decoupling, one can obtain theexpression of the bare quasiparticle dispersion in termsof a bond-charge quantity I � hcyicji that in the mo-mentum space is given by I �

Pknk�

��kw � in a way that

the kinetic part of the Hamiltonian (7) (neglectingthe Stoner exchange) reads HII-MF �

Pk��1� 2j1I �

2j2I���k ���nk. It is possible to show that the valuesof j1 and j2 are linked to the V, J, and J0 interactions bymeans of the following relations: j1 � �J� V�=w andj2 � �J� J0�=w [19]. Hence, by following the same pro-cedure as in model I, it is convenient to introduce theparameters a and b, which now has to be determined self-consistently via the following relations: b � �1� �j1 �j2��I" � I#��, a � �j2 � j1��I" � I#�. Still, I depends on aand b by means of the occupation number of the � ���bands, via the following relations:

nk" � v2k � u

2k�n

�k � n

�k � � n

�k ;

nk# � v2k � u

2k�n

�k � n

�k � � n

�k ;

(8)

where u2k � �12�f1� ��b�k ���=�������������������������������������b�k ���2 � �2

p�g

and v2k � 1� u2k are the coefficients of the Bogoliubovtransformation.

Few comments are worth mentioning on the possiblesolutions and on the effect of depaired electrons in theamplitude of the average mass b within the SF state. Firstof all, the amplitudes of a and b are determined by j� ��j2 � j1� and j� � �j1 � j2�, respectively. Moreover, thesign of j� controls whether the starting value in the SCstate for b is larger or smaller than 1, while j� sets thesign of the magnetization. As far as the stability is con-cerned, as in model I, it is crucial to see whether theintroduction of depaired electrons in the SF phase reducesor increases the average effective mass with respect tothat of the SC state. Hereafter, we will assume that j� arepositive, which is consistent, in the weak coupling re-gime, with realistic values of the microscopic couplings(J; J0; V) above introduced.

197003-3

FIG. 2. Phase diagram relative to model II, showing thetransition (full line) from the SF to the SC state, and thatone from a paramagnetic to FM state when � � 0 (dashedline), as the difference �j� � j�� and the density of electronsare varied. Because of the particle-hole symmetry, the part forn in the range �0; 1� is just symmetrically related.

P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19

Let us now analyze the dependence of b in the case ofthe SF and SC phases. One can show that bsf gets un-dressed in the coexisting phase; thus its value is largerthan bsc when n�k � 0. This result can be deduced bywriting down the explicit expression of the ground stateenergy of the SF and SC phases. The expectation value ofthe bond-charge term on the unpaired electrons gives anegative contribution to I" � I# which is not present in thenonmagnetic SC case, thus increasing the average kineticenergy and consequently the value of E0 in the coexistingstate.

In Fig. 2 is reported the phase diagram for j� � 0:5,g=t � 0:25 obtained by varying j� with respect to thedensity. Of course, by modifying the amplitude of j� onecan span the phase diagram for all the possible values ofj1 and j2. We have checked that a change in j� yields ashift in the critical line but does not give any qualitativechange in the phase diagram. The phenomenology of thetransition between the SC and SF states is the same as thatobserved for model I. Indeed, when one goes through thecritical line, the magnetization has a jump to its possiblemaximal value which depends on � and �, while the SCgap grows continuously from zero. For completeness, inFig. 2 it is also reported the line of transition from aparamagnetic to a FM state in the case of absence ofsuperconductivity. The shape of the region of stability inthe ��j� � j��; n� diagram can be understood by noticingthat the kinetic exchange is given by kex � �" � �# � 2a�and a � 2j�m�n� 1� (for the pure ferromagnetic case).Thus approaching the half-filling limit (n! 1) one hasthat a! 0, and consequently the kinetic exchange goesto zero so that an infinite value of j� is needed to get aspin polarized state. This consideration explains theasymptotic behavior of the critical line as we get closeto the half-filling limit where it becomes more and moredifficult to have a magnetized state because the difference

197003-4

in the bond-charge occupation for opposite spin becomesclose to zero thus requiring high values of the couplingconstants to create a charge imbalance.

In conclusion, we have studied the occurrence of fer-romagnetism and s-wave singlet superconductivity withina single band model where the magnetic moments are dueto a kinetic exchange mechanism, both for a generic andfor a specific case with bond-charge coupling. It has beenshown that the depaired electrons play a crucial role inthe energy balance and that only when their dynamicaleffect is such to undress the effective mass of the carrierswhich participate in the pairing can the SF phase bestabilized. As far as the specific case of a bond-chargekinetic mechanism is concerned, we have shown that thephase diagram has a peculiar dependence on the densityof carriers, and that only when the system is far from thelimit of exact particle-hole symmetry, can the SF phasebe obtained without going to very large couplings.

M. C. acknowledges support from the European pro-gram ‘‘Improving Human Potential.’’ The authors thankMaria Teresa Mercaldo for helpful comments and valu-able discussions.

[1] S. S. Saxena et al., Nature (London) 406, 587 (2000).[2] C. Pfeiderer et al., Nature (London) 412, 58 (2001).[3] D. Aoki et al., Nature (London) 413, 613 (2001).[4] J. Tallon et al., IEEE Trans. Appl. Supercond. 9, 1696

(1999); C. Bernhard et al., Phys. Rev. B 59, 14 099(1999).

[5] A. A. Abrikosov and L. P. Gorkov, Zh. Eksp. Teor. Fiz. 39,1781 (1960) [Sov. Phys. JETP 12, 1243 (1961)].

[6] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).[7] B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962).[8] P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964).[9] A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz.

47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)].[10] K. Blagoev, J. Engelbrecht, and K. Bedell, Phys. Rev.

Lett. 82, 133 (1999).[11] H. Suhl, Phys. Rev. Lett. 87, 167007 (2001).[12] A. A. Abrikosov, J. Phys. Condens. Matter 13, L943

(2001).[13] M. Cuoco, P. Gentile, and C. Noce, Phys. Rev. B 68,

054521 (2003).[14] T. R. Kirkpatrick, D. Belitz, T. Vojta, and R. Narayanan,

Phys. Rev. Lett. 87, 127003 (2001).[15] K. Machida and T. Ohmi, Phys. Rev. Lett. 86, 850 (2001).[16] M. B. Walker and K.V. Samokhin, Phys. Rev. Lett. 88,

207001 (2002).[17] N. I. Karchev, K. B. Blagoev, K. S. Bedell, and P. B.

Littlewood, Phys. Rev. Lett. 86, 846 (2001).[18] R. Shen, Z. M. Zheng, S. Liu, and D.Y. Xing, Phys. Rev. B

67, 024514 (2003).[19] J. E. Hirsch, Phys. Rev. B 40, 2354 (1989); 40, 9061

(1989); 59, 6256 (1999).

197003-4