ferromagnetism and reentrant superconductivity

*Corresponding author. E-mail: [email protected].

Journal of Magnetism and Magnetic Materials 187 (1998) 51—62

Ferromagnetism and reentrant superconductivity

E. Gos"awska, M. Matlak*Institute of Physics, Silesian University, 4 Uniwersytecka, PL-40-007 Katowice, Poland

Received 21 April 1997; received in revised form 19 February 1998

Abstract

We consider the extended two-band s—f model with additional terms, describing intersite Cooper pairs interactionbetween 4f (5f) and conduction electrons. Numerical calculations show that for a proper choice of model parameters thesystem possesses three critical temperatures: ¹

C(Curie temperature), ¹

S1and ¹

S2(lower and upper transition temper-

atures to the superconducting state). The critical temperatures fulfil the relation ¹S1(¹

C(¹

S2. The system is

ferromagnetic and normal for ¹(¹S1, ferromagnetic and superconducting for ¹

S1(¹(¹

C, paramagnetic and

superconducting for ¹C(¹(¹

S2, paramagnetic and normal for ¹'¹

S2, as well. The calculations of the Curie

temperature ¹C

and upper critical temperature ¹S2

as functions of the model parameters (3D plots) show the appearanceof the reentrant superconductivity when ¹

S2'¹

C. The calculated temperature dependence of the DC resistivity (3D

plots) is in many cases similar to the typical reentrant behaviour, experimentally found for magnetic superconductors (e.g.ErRh

4B4, HoMo

6S8). ( 1998 Elsevier Science B.V. All rights reserved.

Keywords: Ferromagnetism; Superconductivity; Two-band model; Exchange interaction

1. Introduction

The competition between magnetism and super-conductivity belongs to the most interesting phe-nomena in solid state physics. There is onlya relatively small number of elements, compoundsand alloys where superconductivity appears in itspure nature. The other superconducting materials,however, clearly show interplay between magnet-ism and superconductivity. The properties of thesematerials has been summarized in a number ofbeautiful articles (see e.g. Refs. [1—4]). Also, the

so-called high-¹C

superconductors belong to theclass of magnetic superconductors (cf. e.g. Ref. [5]).

Simplest examples of the competition betweenferromagnetism and superconductivity can befound among reentrant superconductors such asErRh

4B

4[6] and HoMo

6S8

[7]. Here the com-pounds transit to the superconducting state ata temperature ¹

S2(normal superconductor), at the

Curie temperature ¹C

(¹C(¹

S2) they enter the

superconducting and ferromagnetic phase, and,when the temperature further decreases, at ¹

S1they

enter the ferromagnetic (nonsuperconducting)phase which persists up to ¹"0 K. This reentrantbehaviour is strongly correlated with the temper-ature dependence of the DC resistivity o of these

0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 1 1 8 - 8

compounds. One observes [7,8] that for¹3[¹

S1, ¹

S2] o is equal to zero (superconducting

phase). For long there has been no experimentalindications that state about the temperature de-pendence of the superconducting order parameterD (energy gap) of these compounds. Recently, avery important experiment, however, using thepoint contact spectroscopy technique resolved thisproblem for HoMo

6S8

[8]. The temperature de-pendence of D [8] has a distinct maximum between¹

S1and ¹

S2and vanishes for ¹(¹

S1and ¹'¹

S2(disagreement with the BCS model).

Up to now there is no satisfactory microscopicexplanation of the properties of the reentrant fer-romagnetic superconductors. Therefore, it is rea-sonable to consider the extended two-band s—fmodel [9] with intersite Cooper pairs interactionsbetween 4f (5f) and conduction electrons [10]. Inthis way, we obtain qualitative agreement with theresults of paper [8] (Fig. 3 in [8]).

2. Model and calculations

We take the Hamiltonian of the extended s—fmodel describing the grand canonical ensemble ofinteracting electrons in the following form:

H"Hf#H

d#H

f,d#H

N, (1)

where

Hf"E

f+i,p

nfi,p#º+

i

nfi,

nfi,¬# +

i,j,ptfi,j

fì,p f

j,p

#+i,j

Ji,j

fì,

f`j,¬

fj,¬

fi,

, (2)

Hd"td

0+i,p

ndi,p# +

i,j,ptdi,j

dì,pdj,p

#+i,j

Ri,j

dì,

d`j,¬

dj,¬

di,

, (3)

Hf,d

"!

g

2+i

( fì,

fi,¬

dì,¬

di,#f`

i,¬fi,

dì,

di,¬

#12(nf

i,!nf

i,¬)(nd

i,!nd

i,¬))

#»+i

( fì,pdi,p#d`

i,p fi,p), (4)

and

HN"!k+

i,p(nf

i,p#ndi,p). (5)

The 4f (5f) subsystem is described by the Hamil-tonian (2), where E

fdenotes the position of the 4f

(5f) atomic level, º is the Coulomb intrasite repul-sion and tf

i,jis the hopping integral of the narrow 4f

(5f) band (tfi,i"E

fis the center of gravity of this

band). The last term in Eq. (2) describes the inter-site Cooper pairs interaction between 4f (5f) elec-trons [10]. The Hamiltonian (3) describes theelectrons of the conducting band, td

i,jis the hopping

integral (td0,td

i,i) and the last term in Eq. (3) is

responsible for the intersite Cooper pairs interac-tion of the conduction electrons. The first term inEq. (4) describes the local exchange interaction be-tween 4f (5f) electrons and conduction electrons(g is the intrasite exchange integral) and the secondterm is responsible for the hybridization of the 4f(5f) and conduction electron orbitals with the hy-bridization parameter ». In Eq. (5) k denotes thechemical potential.

The Hamiltonian (1) is very complicated andtherefore we perform the following simplifications.To the fourth term in Eq. (2) and to the third termin Eq. (3) we apply the standard approximation:

+i,j

Ji,j

fì,

f`j,¬

fj,¬

fi,

++i,j

Ji,j

(S fì,

f`j,¬

T fj,¬

fi,#S f

j,¬fi,

Tfì,

f`j,¬

), (6)

+i,j

Ri,j

dì,

d`j,¬

dj,¬

di,

++i,j

Ri,j

(Sdì,

d`j,¬

Tdj,¬

di,#Sd

j,¬di,

Tdì,

d`j,¬

). (7)

These approximations produce the intersite pairingmechanism, leading to the reentrant superconduc-tivity. When we introduce

D(f)i,j"J

i,jS f

j,¬fi,

T, (8)

D(d)i,j"R

i,jSd

j,¬di,

T, (9)

52 E. Gos!awska, M. Matlak / Journal of Magnetism and Magnetic Materials 187 (1998) 51—62

we can rewrite

+i,j

Ji,j

fì,

f`j,¬

fj,¬

fi,

++i,j

[D*(f)i,j

fj,¬

fi,#D(f)

i,jfì,

f`j,¬

], (10)

+i,j

Ri,j

dì,

d`j,¬

dj,¬

di,

++i,j

[D*(d)i,j

dj,¬

di,#D(d)

i,jdì,

d`j,¬

].

(11)

Other approximations which allow to keep themathematics on a tractable level are similar to thatalready used in [9]. We can also use the mean fieldapproximation to the first term in Eq. (4). We willhave

fì,p f

i,~pdì,~p di,p"!f`

i,pdi,pdì,~p f

i,~p

+!S fì,pdi,pTd`

i,~p fi,~p!Sd`

i,~p fi,~pTf`

i,pdi,p,(12)

(nfi,

!nfi,¬

)(ndi,!nd

i,¬)

+2mf(nd

i,!nd

i,¬)#2m

d(nf

i,!nf

i,¬), (13)

where

mf,d

"12(Snf,d

T!Snf,d

¬T) (14)

are the magnetizations of 4f (5f) and conductionselectrons, respectively. Introducing Eqs. (10)—(13)into Eq. (1) the final form of the approximatedHamiltonian is

H"+i,p

Efpnfi,p#º+i

nfi,

nfi,¬# +

i,j,ptfi,j

fì,p f

j,p

#+i,j

D*(f)i,j

fj,¬

fi,#+

i,j

D(f)i,j

fì,

f`j,¬

#+i,p

tpndi,p

# +i,j,p

tdi,j

dì,pdj,p#+

i,j

D*(d)i,j

dj,¬

di,

#+i,j

D(d)i,j

dì,

d`j,¬

#+i,p

»p fì,pdi,p#+

i,p»*pdì,p f

i,p,

(15)

where

Efp"Ef!p

gmd

2!k, (16)

tp"!

gmf

2!k (td

i,i"td

0"0), (17)

»p"»#

g

2Sd`

~p f~pT. (18)

We consider first the case º"0. The Hamiltonian(15) can be rewritten in the standard k

1-representa-

tion to be

H(º"0)"+k1 ,p

Efpnfk1 ,p#+

k1 ,pefk1nfk1 ,p

#+k1

D*(f)k1

f~k1 ,¬

fk1 ,

#+k1

D(f)k1

f`k,

f`~k1 ,¬

#+k1 ,p

tpndk1 ,p

#+k1 ,p

edk1ndk1 ,p

#+k1

D*(d)k1

d~k1 ,¬

dk1 ,

#+k1

D(d)k1

d`k1 ,

d`~k1 ,¬

#+k1 ,p

»p f`k1 ,p

dk1 ,p

#+k1 ,p

»*pd`k1 ,p

fk1 ,p

, (19)

where

ef,dk1

" +(i~j)

tf,di,j

e~*k1 > (R1 i~R1 j), (20)

D(f)k1"

1

N+k1 {

J(k1!k

1@) S f

~k1 {,¬fk1 {,

T, (21)

and

J(k1)" +

(i~j)

Ji,j

e~*k1 > (R1 i~R1 j). (22)

The expressions for D(d)k1

and R(k1) can be obtained

from Eqs. (21) and (22) by replacing JPR andfPd. Applying the standard Green’s function tech-nique to the Hamiltonian (19) we obtain the exact

E. Gos!awska, M. Matlak / Journal of Magnetism and Magnetic Materials 187 (1998) 51—62 53

system of equations:

C1!ef

k1G(0)

(E) !»

G(0)

(E) !D(f)

k1G(0)

(E) 0

!»*

E!t!ed

k10 !D(d)

k1!D*(f)

k1G(1)

¬(E) 0 1#ef

k1G(1)

¬(E) »*

¬G(1)

¬(E)

0 !D*(d)k1

»¬

E#t¬#ed

k1D C

SS fk1 ,

DATT

SSdk1 ,

DATT

SS f`~k1 ,¬

DATT

SSd`~k1 ,¬

DATTD"CSM f

k1 ,,ANTG(0)

(E)

SMdk1 ,

,ANT

SM f`~k1 ,¬

,ANTG(1)¬

(E)

SMd`~k1 ,¬

,ANT D,(23)

and

C1!ef

k1G(0)

¬(E) !»

¬G(0)

¬(E) D(f)

k1G(0)

¬(E) 0

!»*¬

E!t¬!ed

k10 D(d)

k1D*(f)k1

G(1)

(E) 0 1#efk1G(1)

(E) »*

G(1)

¬(E)

0 D*(d)k1

»

E#t#ed

k1D C

SS f~k1 ,¬

DBTT

SSd~k1 ,¬

DBTT

SS f`k1 ,

DBTT

SSd`k1 ,

DBTT D"CSM f

~k1 ,¬,BNTG(0)

¬(E)

SMd~k1 ,¬

,BNT

SM f`k1 ,

,BNTG(1)

(E)

SMd`k1 ,

,BNT D, (24)

with A"( f`k1 ,

,d`k1 ,

), B"( f`~k1 ,¬

,d`~k1 ,¬

), and

G(0)p (E)"1

E!Efp, (25)

G(1)p (E)"1

E#Efp, (p"C,B). (26)

The propagators G(0)p (E) and G(1)p (E) are nothingelse but free propagators SS fpD f`p TT(0) andSS f`p D fpTT(0), respectively, when we put ef

k1"

Dk1"tp"ed

k1"»p"0 in Eq. (19) it is when

H"+i,pEfpnfi,p. Thus, the simplest way to take into

account the presence of the Coulomb repulsion º isto replace in Eqs. (23) and (24)

G(0)p (E)PSnf

~pT(0)

E!Efp!º

#

1!Snf~pT(0)

E!Efp, (27)

G(1)p (E)PSnf

~pT(0)

E#Efp#º

#

1!Snf~pT(0)

E#Efp, (28)

where the propagators (27) and (28) are exactpropagators when H"+

i,pEfpnfi,p#º+infi,

nfi,¬

,and renormalize Snf

~pT(0)PSnf~pT for self-consist-

ency. A similar procedure of renormalization hasbeen applied in Ref. [9]. The described procedureallows the calculation of all necessary Green’s

functions, which determine the thermodynamicproperties of our system. Finally, we obtain

SS fk1 ,p

D f`k1 ,p

TT"6+a/1

¸pf,f

(Epa)M@p(Epa)

)1

E!Epa, (29)

SSdk1 ,p

Dd`k1 ,p

TT"6+a/1

¸pd,d

(Epa)M@p(Epa)

)1

E!Epa, (30)

SS fk1 ,p

Dd`k1 ,p

TT"6+a/1

¸pf,d

(Epa)M@p(Epa)

)1

E!Epa, (31)

SS f`~k1 ,¬

D f`k1 ,

TT"6+a/1

¸f`

¬,f`(Ea )

M@(Ea )

)1

E!Ea, (32)

SSd`~k1 ,¬

Dd`k1 ,

TT"6+a/1

¸d`¬,d`

(Ea )M@

(Ea )

)1

E!Ea, (33)

SSd`~k1 ,¬

D f`k1 ,

TT"6+a/1

¸d`¬,f`

(Ea )

M@(Ea )

)1

E!Ea, (34)

where

¸pf,f

(E)"[E!Efp!º(1!Snf~pT)]

]M[(E#Ef~p)(E#Ef

~p#º)

#efk1(E#Ef

~p#º(1!SnfpT))]

][(E!tp!edk1)(E#t

~p#edk1)

!DD(d)k1

D2]!D»~pD2[E#Ef

~p#º(1!SnfpT)](E!tp!ed

k1)N, (35)


¸pd,d

(E)"[(E!Efp)(E!Efp!º)

!efk1(E!Efp!º(1!Snf

~pT))]

]M[(E#Ef~p)(E#Ef

~p#º)

#efk1(E#Ef

~p#º(1!SnfpT))]

](E#t~p#ed

k1)!D»

~pD2(E#Ef~p

#º(1!SnfpT))N!DD(f)k1

D2(E#t~p#ed

k1)

](E!Efp!º(1!Snf~pT))

](E#Ef~p#º(1!SnfpT)), (36)

¸pf,d

(E)"[E!Efp!º(1!Snf~pT)]

]M»p(E#t~p#ed

k1)][(E#Ef

~p)

](E#Ef~p#º)#ef

k1(E#Ef

~p#º(1!SnfpT))]!(E#Ef

~p#º(1!SnfpT))[D(f)

k1D*(d)k1

»*~p

#»pD»~pD2]N, (37)

¸f`

¬,f`(E)"(E!Ef

!º(1!Snf

¬T))

](E#Ef¬#º(1!Snf

T))

]MD*(f)k1

[(E!t!ed

k1)(E#t

¬#ed

k1)

!DD(d)k1

D2]!»*»*

¬D*(d)k1

N, (38)

¸d`¬,d`

(E)"D*(d)k1

[(E!Ef)(E!Ef

!º)

!efk1(E!Ef

!º(1!Snf

¬T))]

][(E#Ef¬)(E#Ef

¬#º)

#efk1(E#Ef

¬#º(1!Snf

T))]

![(E!Ef!º(1!Snf

¬T))

](E#Ef¬#º(1!Snf

T))]

](DD(f)k1

D2D*(d)k1

#»»¬D*(f)k1

), (39)

¸d`¬,f`

(E)"(E!Ef

!º(1!Snf

¬T))

]M»*D*(d)

k1[(E#Ef

¬)(E#Ef

¬#º)

#efk1(E#Ef

¬#º(1!Snf

T))]

!»¬D*(f)k1

(E#Ef¬#º(1!Snf

T))

](E!t!ed

k1)N, (40)

where Epa (a"1,2,2,6) is the solution of the alge-braic equation of degree 6:

Mp(E)"[(E!Efp)(E!Efp!º)


~pT))]

][(E#Ef~p)(E#Ef

~p#º)

#efk1(E#Ef

~p#º(1!SnfpT))]

][(E!tp!edk1)(E#t

~p#edk1)

!DD(d)k1

D2]!D»~pD2(E!tp!ed

k1)

][(E!Efp)(E!Efp!º)


~pT))]

][E#Ef~p#º(1!SnfpT)]

!D»pD2(E#t~p#ed

k1)][(E#Ef

~p)

](E#Ef~p#º)#ef

k1(E#Ef

~p

#º(1!SnfpT))]][E!Efp

!º(1!Snf~pT)]#[E!Efp

!º(1!Snf~pT)]][E!Ef

~p

#º(1!SnfpT)]]M!DD(f)k1

D2

][(E!tp!edk1)(E#t

~p#edk1)!DD(d)

k1D2]

#»*p»*~pD(f)

k1D*(d)k1

#»p»~pD*(f)k1

D(d)k1

#D»pD2D»~pD2N. (41)

and

M@p(Epa)"MpE K

E/Epa

. (42)


1The other symmetries of the superconducting order para-meter D(f),(d)

k1as e.g. p-symmetry or dc-symmetry we have to

exclude because they lead to the labourous calculations of thetriple integrals which enter nine implicit equations which haveto be solved (see later). Only s* symmetry allows to avoid tripleintegration.

For further calculations we consider a simple cubiclattice, where (see Eqs. (20) and (22))

edk1"!

¼

6f (k

1), (43)

efk1"de

k1, (44)

Jk1"!

J03

f (k1), (45)

Rk1"!

R0

3f (k

1), (46)

and

f (k1)"cos k

xa#cos k

ya#cos k

za, (47)

where ¼ is the bandwidth of the conduction band,d is a parameter (d @1) determining the bandwidthd¼ of the 4f (5f) narrow band and J

0"

+j*/./.+

Ji,j

(J0'0), R

0"+

j*/./.+R

i,j(R

0'0).

Furthermore, we assume that (see Eq. (21))

D(f),(d)k1

"D(f),(d)f (k1), (48)

implying that we consider (for simplicity) the s*-type superconductivity.1 We see that ed

k1, ef

k1and

D(f),(d)k1

possesses the same dispersion relations andtherefore we can introduce ‘unperturbed’ density ofstates. We choose (similar to Ref. [9]) the rectangu-lar density of states

o(e)"G1/¼, e3[!¼/2, ¼/2],

0, otherwise.(49)

With the use of Eqs. (49), (29)—(34), we finally get

SnfpT"1

¼PW@2

~W@2

de6+a/1

¸pf,f

(Epa)M@p(Epa)

) f (Epa), (50)

SndpT"1

¼PW@2

~W@2

de6+a/1

¸pd,d

(Epa)M@p(Epa)

) f (Epa), (51)

Sd`p fpT"1

¼PW@2

~W@2

de6+a/1

¸pf,d

(Epa)M@p(Epa)

) f (Epa), (52)

D*(f)"!

2J0

3¼2PW@2

~W@2

de e6+a/1

¸f`

¬,f`(Ea )

M@(Ea )

) f (Ea ),

(53)

D*(d)"!

2R0

3¼2PW@2

~W@2

de e6+a/1

¸d`¬,d`

(Ea )M@

(Ea )

) f (Ea )

(54)

S f`

d`¬

T"1

¼PW@2

~W@2

de6+a/1

¸d`¬,f`

(Ea )

M@(Ea )

) f (Ea ), (55)

where

f (x)"(ebx#1)~1, b"1

k¹. (56)

The expressions for ¸pf,f

(Epa), ¸pd,d

(Epa), ¸pf,d

(Epa),¸f`

¬, f`(Ea ), ¸d`¬,d`

(Ea ), ¸d`¬,f`(Ea ) and Mp(Epa) are the

same as Eqs. (35)—(41) but we have to replace there-in ed

k1Pe, ef

k1Pde and D(f),(d)

k1P!6eD(f),(d)/¼ (cf.

Eqs. (48), (45) and (46)). To the system of 8 (p"C,B)implicit equations (50)—(54) we have to add theequation for the chemical potential k

+p

(SnfpT#SndpT)"n, (57)

where n is the average number of electrons peratom.

Looking at formula (55) we see that the s*-typesuperconductivity present in the 4f (5f) and conduc-tion electrons subsystems induces also the mixedpairing between 4f (5f) electrons and conductionelectrons, as well.

Assuming D(f),(d)"0 we obtain exactly the sameequation for SnfpT, SndpT and Sd`p fpT (p"C,B) (seeEqs. (50)—(52)) as in Ref. [9]. For D(f),(d)"0 we canalso adopt the formulae for the DC conductivityfrom [9]

pp0

"

b¼4 P

W@2

~W@2

deA1!A2e¼B

2

B3@2

]6+a/1

+p

[ M̧ pd,d

(EM pa)]2#2d[ M̧ pf,d

(EM pa)]2#d2[ M̧ pf,f

(EM pa)]2[MM @p(Epa)]2

) f (EM pa)(1!f (EM pa)), (58)


Fig. 1. Plot of the Curie temperature ¹C

(continuous line) andupper superconducting transition temperature ¹

S2(dashed line)

versus 4f (5f) level position Ef

for n"1. Case a: J0

[eV]"0.3,R

0[eV]"0.3; case b: J

0[eV]"0.15, R

0[eV]"0.15. The

other parameters are: ¼[eV]"2, º[eV]"10, g[eV]"0.2,»[eV]"0.1, d"0. The inset shows the temperature depend-ence of the magnetizations m

f,dand superconducting order

parameters D(f),(d) [eV] for n"1. J0

[eV]"0.3 and R0

[eV]"0.3 (a), 0.5 (b), 0.7 (c). The other parameters are as above.

where p0

is a numerical constant,

M̧ pd,d

(E)"(E!Efp)(E!Efp!º)

!de(E!Efp!º(1!Snf~pT)), (59)

M̧ pf,d

(E)"»p(E!Efp!º(1!Snf~pT)), (60)

M̧ pf,f

(E)"(E!tp!e)(E!Efp!º(1!Snf~pT)),

(61)

MM p(E)"[(E!Efp)(E!Efp!º)

!de(E!Efp!º(1!Snf~pT))]

](E!tp!e)!D»pD2(E!Efp!º(1!Snf

~pT)), (62)

MM @p(Epa)"LMp(E)

LE KE/Epa

(63)

and EM pa (a"1,2,3) are solution of the equationMM p(E)"0.

The formula (58) will be later used to calculatethe DC resistivity of system o ("p~1) in the tem-perature range where D(f),(d)"0 (for D(f),(d)O0 weshould have o"0 (superconducting phase)).

3. Numerical calculations and comments

The thermodynamic properties of the model canbe found after solving nine implicit equations:(50)—(52) (p"C,B), Eqs. (53), (54) and (57) assumingthat the superconducting order parametersD(f),(d) are real. In such a way, we can numericallycalculate the temperature dependence of the mag-netizations m

f,d"m

f,d(¹), and the superconduct-

ing order parameters D(f),(d)"D(f),(d)(¹). Weshould expect, as usually, that m

fand m

ddisappear

for ¹*¹C, where ¹

Cis the Curie temperature of

the system. Having in mind the reentrant supercon-ductivity, we should expect a more complicatedtemperature behaviour of the order parametersD(f),(d) when compared with a standard, paramag-netic superconductor. In the temperature range¹)¹

S1()¹

C) the superconducting properties of

the model cannot be expected (D(f),(d)"0) becausestrong ferromagnetism acting similarly to the ex-

ternal magnetic field will break intersite 4f (5f) andconduction electron pairs. In this range we canspeak about pure ferromagnetic behaviour of thesystem. The superconductivity should, however,exist (D(f),(d)O0) in the temperature interval¹3(¹

S1, ¹

S2). In the small interval ¹3(¹

S1, ¹

C)

a coexistence between ferromagnetism and super-conductivity should be expected (m

f,dO0,

D(f),(d)O0). In the interval ¹3(¹C, ¹

S2) the system

should be paramagnetic and superconducting(m

f,d"0, D(f),(d)O0) and for ¹'¹

S2paramag-

netic and normal (mf,d

"0, D(f),(d)"0). Thus, theorder parameter D(f),(d) should have two criticaltemperatures ¹

S1and ¹

S2(lower and upper critical

temperatures (¹S1)¹

C(¹

S2)) and D(f),(d)O0 for

¹3(¹S1, ¹

S2). The inset in Fig. 1, obtained as a nu-

merical solution of the nine transcendental equa-tions (50)—(52) (p"C,B), Eqs. (53), (54) and (57),corresponds exactly to the situation describedabove in accordance with the experimentalmeasurement for HoMo

6S8

[8] (cf. Fig. 3 therein).We can observe that the temperature behaviour ofD(f) and D(d) is similar (they have the same criticaltemperatures ¹

S1and ¹

S2). Besides, the anisotropic

s*-pairing for 4f (5f) electrons is dominant even inthe case when R

0'J

0and the value of R

0has


Fig. 2. 3D Plot of the Curie temperature ¹C

(continuous line)and the upper superconducting transition temperature¹

S2(dashed line) versus 4f (5f) level position E

ffor n"0.2, 0.4,

0.6, 0.8 and 1. The other parameters are: ¼ [eV]"2, º [eV]"10, g [eV]"0.2, » [eV]"0.1, J

0[eV]"0.15, R

0[eV] "0.15.

Case a: d"0, case b: d"0.002.

practically no influence on D(f). It means that the‘almost’ localized 4f (5f) electrons form intersiteCooper pairs much easier than the more mobileconduction electrons belonging to a broad conduc-tion band. We can also see that the difference¹

C!¹

S1is small which means that the coexistence

between ferromagnetism and superconductivitytakes place only in a small temperature intervalnear ¹

C.

However, the way of looking for the temperaturedependence of m

f,dand D(f),(d) (to find ¹

S1, ¹

Cand

¹S2) is numerically very complicated (nine transcen-

dental equations to solve) and ineffective, espe-cially, when we want to study the properties of ourmodel in a broad model parameter range. Because¹

S1and ¹

Cdiffer only very small (about 1—3 K) the

ferromagnetism and superconductivity can betreated independently and the most interestingthing is to investigate the influence of the modelparameters and n (the average number of electronsper site) on ¹

Cand ¹

S2. Such investigations give us

the necessary information about the appearance ofthe different solutions of the model (ferromagneticand normal phase, ferromagnetic and supercon-ducting phase, paramagnetic and superconductingphase, paramagnetic and normal phase). In thefollowing we describe the way how to find theequation determining ¹

Cand ¹

S2. From the inset in

Fig. 1 we can see that D(f),(d) are very small near ¹C.

Therefore we can put D(f),(d)"0 into Eqs. (50)—(52)and expand the right-hand sides of them (p"C,B)with respect to m

fand m

d(very small near ¹

C) up to

linear terms and combining these equations we canobtain a homogenous linear system of equationsfor m

fand m

d. The determinant D of this system

will exactly coincide with the denominator of themagnetic susceptibility calculated in [9]. From thecondition D"0 the Curie temperature ¹

Ccan be

calculated with quite a satisfactory accuracy. Theequation for ¹

S2can be found in a similar way.

Because ¹S2'¹

Cwe can put m

f,d"0 into

Eqs. (53) and (54) and expend the right-hand sidesof them with respect to D(f) and D(d), leaving onlythe linear terms in D(f),(d). We obtain again a linearhomogeneous system of equations for D(f),(d). Thedeterminant DM of this system should be equal tozero for ¹"¹

S2. Thus, the solution of the equation

DM "0 determines the upper superconducting

transition temperature ¹S2. The equations D"0

(¹C) and DM "0 (¹

S2) have to be solved together

with the system of equations (50)—(52) (p"C) andEq. (57) where we should insert m

f,d"D(f),(d)"0.

Thus, in both cases (to find ¹C

or ¹S2) we need only

4 implicit equations to solve what is numericallymuch easier than the solution of nine implicit equa-tions (general case). A special attention should bepaid to the results presented in Fig. 1 (n"1). Herewe have plotted the dependence of ¹

Con E

f(the

position of the 4f (5f) level) and also the dependenceof ¹

S2for the case J

0[eV]"0.3, R

0[eV]"0.3

(curve a) and J0[eV]"0.15, R

0[eV]"0.15 (curve

b). The reentrant superconductivity clearly appearsin case a for E

f[eV]3[!2,!1.05] and for E

f


Fig. 3. 3D Plot of the reduced DC resistivity versus temperature ¹ for different Ef

[eV]"!2, !1.75, !1.5, !1.25, !1, !0.75,!0.5, !0.25 and different n"0.2 (a), n"0.4 (b), n"0.6 (c), n"0.8 (d), n"1 (e). Continuous line: d"0; dashed line d"0.002.

[eV]3[!0.94,!0.48] and we can check it bysolving the whole system of 9 equations, mentionedabove, and the result for E

f[eV]"!2 can be

seen in the inset of Fig. 1. However, we have

to be careful of jumping to the conclusions whenwe try to interpret the results for the curve b inFig. 1. Here ¹

S2'0 (¹

C'¹

S2) in the broad

range of Ef

suggesting that superconductivity and


Fig. 4. The same as in Fig. 2 but for n"1.2, 1.4, 1.6, 1.8, 2. Casea: d"0, case b: d"0.002.

ferromagnetism coexist, in fact, it is not the case.When we try to solve again the system of 9 equa-tions to look for the behaviour of m

f,dand D(f),(d) on

the temperature we find that when ¹C'¹

S2the

system is ferromagnetic and normal (D(f),(d)"0) for¹(¹

Cand paramagnetic and normal above ¹

C.

This example allows us to state that only in the casewhen ¹

S2'¹

C(¹

C'0) reentrant superconductiv-

ity can be expected. In the case ¹C"0 and ¹

S2'0

the system is paramagnetic and superconducting.When, however, ¹

C"¹

S2"0 the system is para-

magnetic and normal. We have investigated ourmodel on the possibility of the appearance of super-conducting and magnetic solutions in the range ofthe average occupation numbers n3[0,2]. TheCurie temperature ¹

Cand the upper superconduct-

ing transition temperature ¹S2

of the system asfunctions of the 4f (5f) level position E

fare depicted

in Fig. 2 (n"0.2, 0.4, 0.6, 0.8, 1) and in Fig. 4(n"1.2, 1.4, 1.6, 1.8, 2). A survey of the correspond-ing DC resistivity dependence on temperature ¹,calculated from (58) (we put o"0 for¹3[¹

S1, ¹

S2] (D(f),(d)O0)), is presented in Fig. 3 (a:

n"0.2, b: n"0.4, c: n"0.6, d: n"0.8, e: n"1)and in Fig. 5 (a: n"1.2, b: n"1.4, c: n"1.6, d:n"1.8, e: n"2). From Figs. 2 and 4 we can easilyfind that the reentrant superconductivity appearsfor n"0.4 (d"0), n"0.6 (d"0, d"0.002),n"0.8 (d"0, d"0.002), n"1.2 (d"0,d"0.002), n"1.4 (d"0, d"0.002), n"1.6 (d"0,d"0.002) and n"1.8 (d"0), because one canfind there a range of E

fwhere ¹

S2'¹

C. We can

also find that the ferromagnetic and normal solu-tions of the model exist as well as the paramagneticand normal. Many curves, presenting the DC resis-tivity dependence on temperature, are similar inshape to the experimental curves for the typicalreentrant superconductors as ErRh

4B4

[6] andHoMo

6S8

[7], see e.g. Fig. 3b (Ef

[eV]"!2,!1.75), Fig. 3c (E

f[eV]"!2, !1.75, !1.5,

!1.25, !1), Fig. 3d (Ef

[eV]"!2, !1.75,!1.5, !1.25, !0.75), Fig. 3e (E

f[eV]"!0.5),

Fig. 5a (Ef

[eV]"!0.5) and Fig. 5c (Ef

[eV]"0).

4. Conclusions

We have seen that the extended s—f model withintersite Cooper pairs attraction between 4f (5f)electrons and conduction electrons is able to de-scribe the reentrant superconductivity for a properchoice of the model parameters and the averageoccupation number of electrons per site n. Thehybridization term in Eq. (1) produces also a mixedintrasite 4f (5f) — conduction electron pairing (55)which, however, is negligibly small in comparisonwith the intersite 4f (5f) — 4f (5f) pairing or theintersite pairing between conduction electrons. Inthe real reentrant superconductors as e.g. ErRh

4B

4or HoMo

6S8, which form complicated structures

(many ions in the elementary cell) the additionalpairings between electrons can be possible. Thus, inthese substances the nature of the reentrant super-conductivity is certainly much more complicatedbecause of the necessity to take into account a large


Fig. 5. The same as in Fig. 3 but for n"1.2 (a), 1.4 (b), 1.6 (c), 1.8 (d), 2 (e).


number of superconducting order parameters ori-ginating from the presence of the large number ofions in the elementary cell.

References

[1] M.B. Maple, Appl. Phys. 9 (1976) 179.[2] ". Fisher, M.B. Maple (Eds.), Superconductivity in Ter-

nary Compounds I. Topics in Current Physics, vol. 32,Springer, Berlin, 1982.

[3] M.B. Maple, ". Fisher (Eds.), Superconductivity inTernary Compounds II. Topics in Current Physics, vol. 34,Springer, Berlin, 1982.

[4] M.B. Maple, Physica B 215 (1995) 110.[5] A. Golnik, C. Bernhard, I. Budnick, M. Kutrowski, C.

Niedermayer, T. Szumiata, Proc. 1st Polish—US Conf.,Wroc"aw and Kudowa 14—15 Sept. 1995, Springer, Berlin,1996, p. 337.

[6] W.A. Fertig, D.C. Johnston, L.E. De Long, R.W.McCalum, M.B. Maple, B.T. Matthias, Phys. Rev. Lett. 38(1977) 387.

[7] M. Ishikawa, ". Fisher, Solid State Commun. 23 (1977) 37.[8] F. Morales, R. Escudero, A. Briggs, P. Monceau, R. Horyn,

F. Le Berre, O. Pen8 a, Physica B 218 (1996) 193.[9] M. Matlak, M. Pietruszka, J. Magn. Magn. Mater. 153

(1996) 115, 347.[10] R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod.

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ferromagnetism and reentrant superconductivity

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