On coexistence of superconductivity and ferromagnetism

Physica 101A(1980) 613-628 © North-Holland Publishing Co.

ON C O E X I S T E N C E O F S U P E R C O N D U C T I V I T Y AND FERROMAGNETISM N.N. BOGOLUBOV Jr., A.N. ERMILOV and A.M. KURBATOV V.A. Steklov Mathematical Institute o[ the Academy o[ Sciences of the USSR, Moscow, GSP-I, USSR

Received 30 October 1978 Manuscript received in final form 6 April 1979

A model for a superconductor with magnetic impurities is solved exactly on the basis of the method of approximating Hamiltonians. The model consists of a combination of the BCS model and the Zener s-d exchange interaction model. The thermodynamicallyequivalent Hamiltonian is constructed from which exact results are obtained in the thermodynamic limit for the energy gap and the magnetization. Thus it is proved rigorously that coexistence of superconductivity and ferromagnetism takes place in this model. The exact value for the Curie point of the Zener model is also obtained.

I. Introduction In the past few years there has been growing interest in the theoretical investigation of the thermodynamic systems with two types of phase transitions, i.e., the structural and the superconducting one. This interest is brought about first of all by the experimental evidence that most of hightemperature superconductors undergo structural transitions at temperatures Tp just above the temperatures of the superconducting transition Ts. This fact suggests that the high value of Ts in such c o m p o u n d s is connected with the o c c u r r e n c e of the structural phase transition. Thus the investigation of this problem is of great significance for the problem of high-temperature superconductivity. Besides, the p h e n o m e n o n of the coexistence of the superconductive and some other phase connected with structural changes in substance is of interest itself because of its important technical applications. The investigation of the simultaneous existence of superconductivity and ferromagnetism is worth considering in view of the search to increase the superconductive transition temperature Ts in these c o m p o u n d s and in view of its technical applications. However, up till now the problem whether superconductivity and ferromagnetic ordering m a y coexist in the same volume of a sample has not yet 613

614

N.N. BOGOLUBOV

JR. et al.

been solved definitively, for the results obtained by means of different methods j-s) are rather different. Therefore, a mathematically rigorous approach to this problem which is the subject of the present paper can be important. In the trend of exact treatments of statistical systems much progress has been made along the line of the method of approximating Hamiltonians6"7). Making use of this method one can obtain in the thermodynamic limit exact solutions for a number of nontrivial model systems, thus it enables us to separate the errors of a model, i.e., in setting up a problem, from the errors in solving. In the present paper it is, in fact, this method which will be used in the analysis of the p h e n o m e n o n under consideration.

2. T h e m o d e l

Let us consider a cristal consisting of N sites of a superconductor and c N sites of a magnetic impurity. L e t a k+., a ~ denote the creation and annihilation operators of the electron with k being the momentum and ~r the z-component of the spin, S~ (a = x, y, z) is the a - c o m p o n e n t of the localized magnetic m o m e n t of the impurity on site l, ST = S~ - iSr. For the volume of the system we have V = vN, where v is the average volume per site of the superconductor, ~(k) is the energy of an electron with momentum k, and/~ is the chemical potential. Then the following model Hamiltonian ~) enables us to take into account both the BCS interaction of the electrons 8"9) and the s-d exchange Zener interaction between the electrons and the localized magnetic moments ~0) I)

-- X

r v X .:÷a+. a . a.÷ cN

- a V ~ ~ [(a~+ak+- a-~-ak-)S~ + a~i+a~-S~ + a~_al+S{].

(1)

Here the summation index k runs over the N wave vectors of the Bloch states in the conduction band, the parameters F and A describe the strength of the BCS and the Zener interactions. Introducing the following notations 1 cU

SZ

1 cU /';---I

M-

~

a ~i_a~+,

M+ = ( M - ) +, Mz

-~

(a +

- aLa~_),

(3)

COEXISTENCE OF SUPERCONDUCTIVITY

a

=

N~

AND FERROMAGNETISM

at+a-t-, + +

615

(4)

T = ~ [~(k) - / ~ ]

(5)

t,o"

one m a y rewrite the Hamiltonian (1) as (6)

H = T - F V A A +- c A V ( S _ M + + S + M _ + 2SZMS). V

1)

. The thermodynamically equivalent Hamiltonian To construct the approximating Hamiltonian we add to (6) the auxiliary terms _ E 2 A V_V_( S - S + + S W ~ ) , I)

where E is an arbitrary positive number. Then we obtain the Hamiltonian H~ = T - F

vV A A + -

1 c A V [(~2S- + M - ) ( e 2 S + + M +) -~

+ ( e 2 S s + M s ) ( E 2 S S + M s) - M - M

(7)

+- M Z M Z ] .

We take the approximating Hamiltonian in the f o r m H~o( a, s , s , m , m ~ ) =

T-FV[aA

++a+ A_aa+]_lcA

V

E-

V V

x [s(E2S + + M +) + s + ( e 2 S - + M - ) - ss + + (st + s+~)(~2S s + M z) - szs~] 1 V +M+ ~ c A --v [ m M + + m - m m + + (ms + m +~)Ms - m s m ~+],

(8)

where a, s, s~, m, ms are so far arbitrary complex p a r a m e t e r s and + denotes the hermitean (or complex) conjugation. We also consider the intermediate approximating Hamiltonian H ' o ( a , s, Sz) : T -

F V [aA+ + a+ A _ a a + ] _ l 1)

E-

c A V [ s ( e 2 S + + M +) V

+ S+(E2S - + M - ) - ss + + (sz + s+~)(~2S ~ + M s) - SzS+~],

(9)

where

1

V

T ' = T + -~ c A --v [ M - M + + MSMZ]"

(10)

H e r e and in the following for the sake of brevity we will use the vector notations s = {s, ss}, m = {m, mz}.

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N.N. BOGOLUBOV JR. et al.

The Hamiltonian H,o(a, s, ra) is the approximating one for H ~ ( a , s) with respect to operators M -+ and MZ: H~(a,s)=

H,o(a,s,m)+~cA

V [(M--m)(M+-m+)

- ( M ~ - m z ) ( M Z - mY)]

= H,0(a, s, m ) + H ' , l ( m ) . In order to estimate the difference of the free energies 0 f v [ ~ ( ] = - ~ in Tr e -~e° for the Hamiltonians H~o and /4,0 we take advantage of the Bogolubov inequality /)

/o~

\(V)

~

'/2 / c 1 ~ \ ( V )

We have _< /)

,

(V)

V (H',,(m))~,o.~) <~fv[H'~(a, s)] - fv[n,o(a, s, m ) l ~ -~ H,o(..... ~.

Since the operator [ ( M - - m ) ( M + - m +) + ( M z - m D ( M z - rn~)] is positive semidefinite we obtain 0 < ~ f v [ H ~ ( a , s ) ] - f v [ H , o ( a , s, m)] <~cA((M-

-

m)(M +

-

m+)+(M ~

-

m ~ ) ( M ~ _ _m + ~z\;(/ nv ,)o ( ..... ,. ,

(12)

The best approximation is obtained for the values m, m~ which are determined by the absolute maximum of the free energy function of the Hamiltonian H~o(a, s, m ) over the variables m and mz for fixed a, s, and s~: f v [ H , o ( a , s, rh("V)(a, s))] = max [ v [ H , o ( a , s, m)]. m

(13)

As has been shown in ref. 11 in the general case, rh("V)(a, s) and rfi7'V)(a, s) exist for all values of a, s, s~ and [tfi("V~(a, s)l <~ IIM-I[ ~< 1, [rhT"V'(a, s)[ <~ [[M~[[ ~< 1. The condition (13) results in the set of self-consistent field equations Ofv[H,o(a, s, m ) ] / O m = O,

Ofv[H,o(a, s, m)]/Om~ = 0.

(14)

Differentiation yields rn

/ ~1-\(v) /H~o(a,s,m),

~- \ l v l

/ A1~\(v~ /H~o(a,s,m ).

m z -~ \ z v a

(15)

COEXISTENCE OF SUPERCONDUCTIVITY AND FERROMAGNETISM

617

and t h e r e f o r e 0 <~.fv[H'.o(a. s)] - f v [ H . o ( a , s, rht"V)(a, s))]

- 1- c A ( ( M - +(m z

- ~v) + - ( M +)M.o(,,,,,,n)) ~v) (M)n.o~.,~.~))(M

/~Arz\(V) ~z\(v) - \~-* /H.0~.,s.,~)s/~.0~ ..... )-

(16)

N o t i n g that H.0(a, s, m) is quadratic with r e s p e c t to the o p e r a t o r s a L and ak~ and m a k i n g use o f the W i c k t h e o r e m due to B l o c h and de Dominicis to d e c o u p l e the e x p r e s s i o n s (a k~rak~a + + ~v) k,~,ak,~,) H.0~a.s.~) we obtain (( M - - ( M - ) ) ( M + - (M+)) + ( M ~ - ( M ~ ) ) 2 = ~

|

+

+

+

+

~ ((ak+ak,+)(ak-ak,-) - (ak+ak,-)(ak-ak,+l))

r

k,k'

+~

1

~ [((a ~+ak,+)(ak+a I,+) - (a ~+a I'+)(ak÷ak'+))

+ ( ( a k+- a k , _ ) ( a t - a k+, - ) - (a~ al,-)(ak_ak,_)) -

((aI+a~,_)(ak+a~,_) - (a~+a~,)(a~+ak,-))

-

((a I-ak,+)(ak-a ~,+) - (a I-ak,+)(ak-a ~,+))],

(17)

with the n o t a t i o n

(...)

=

(..

~

. ) H , o ( a , s , m ).

Since c o n s e r v a t i o n o f the total m o m e n t u m o f the electrons holds f o r the H a m i l t o n i a n H,0(a, s, m), the first term in each r o u n d b r a c k e t m a y differ f r o m zero, only if k = k ', and the s e c o n d o n e - o n l y if k = - k '. In view o f the inequality Ita~,a~k_+~[I~< 1 we h a v e ((M--

- ~v) +_ + + ~ z 4 (M)n.o(.,..,~))(M ( M ) n . o ( ..... )) ( M z - (M)n.0¢a,~.~)) ) ~< V '

hence, t

0 ~fv[H.o(a,

4cA s ) ] - m a x f v [ H . o ( a , s, m)] ~< cz V .

(18)

It is not difficult to s h o w f r o m the definitions (2)-(4) and (10) that

II,~s" + M~I[ ~ Z,, [[al[ ~< Z,,

tOrT',

+ M~][I ~< L~,

[lIT', A][[ <- t 2 ,

II[,~s ~ + M ~, ,~S~'+ M"']I[ <~ L3/V,

ILIA, A+]II ~ L3/V,

(19)

618

N.N. BOGOLUBOV JR. et al. [I[~2S" + M ", A]II ~ La/V,

I/v[T']l ~ L0, (~, a ' = +, - , z), where L1, L2, L3, L0 are constants independent of Vo As has been shown in ref. 6, the conditions (19) lead to the equality in the thermodynamic limit of the free energies of the Hamiltonians H',o(a, s) and H, provided that the constants a, s, s, are determined by the absolute minimum condition for the free energy fv[H',o(a, s)] (V~>O)O<~minfv[H',o(a,s)]-fv[H,]<-6,(V) a,s

V~

>0,

(20)

in which the minimum really exists. Furthermore according to ref. 11 the minimum min fv[H,o(a, s, dl~"V)(a, s))] = min max fv[H,o(a, s)] a,s a,s rd exists and occurs at Ig~'~[ <- L~. Then we the inequality (18), th~.v~(a~,.v), #,,v~) we

points ti <'v~, g¢,.v), g~.v~ such that [tW'v)I ~< L1, [g~,.v] <~ Lt, are able to take the minimum over the variables a, s, sz in Using the notation ~<,.v~= rh,.v~(a~,.v~,g<,.v~), r ~ , . ~ = obtain

4cA -(, v) ~2 v <~fv[H,o(a " , ¢,.v), ~ , . v ) ) ] _ m!n/v[H',o(a, s)] ~< O.

(21)

Summing the inequalities (20) and (21) term by term we find 4cA - ~ <~ m!n max fv[I"I,o(a, s, m)] - f v [ H A <- ~,(V).

(22)

L e t us now show that (VV) fv[H,] -fv[H]

~

>O.

(23)

Application of the Bogolubov inequality (11) to the Hamiltonians H and H , = H - c A ( W v ) E 2 ( S - S + + SZS z) gives - e 2 ( c A ( S - S + + SZS*))tn~ <~f v [ H , ] - f v [ H ] <~2(c A ( S - S + + SzSZ))(ttv).

Taking into account the inequality HS~II~< 2 s where S is the magnitude of the impurity spins and the fact that the operator S - S + + S z S " is positive semidefinite we obtain - 8 ~ 2 c A S 2 <~flv[H,] - fly[H] <~O.

(24)

The parameter ~ appears in a number of terms of the Hamiltonian H.0,

COEXISTENCE OF SUPERCONDUCTIVITY AND FERROMAGNETISM which does not the free energy the free energy i n e q u a l i t i e s (21)

619

e x i s t f o r E = 0. N e v e r t h e l e s s w e still a r e a b l e t o m a k e u s e o f o f t h e H a m i l t o n i a n H,0 f o r ~ > 0 t o o b t a i n t h e e x p r e s s i o n o f o f t h e H a m i l t o n i a n H = H,l,=0. I n f a c t , a f t e r s u m m i n g t h e a n d (24) w e h a v e

4cA -~, v) -8cAS262 - ~ <~fv[H,o(a " , ~,.v~, ffi ~.v))] _ f v [ H ] <~ 8 , ( V ) . T a k i n g the l i m i t V--> ~ w e o b t a i n -8cAS2e

2 <~ lim

V-w*

{fv[H,o(~ t'v), i t.'v), ~t,.v))] _ f v [ H ] } <~O,

(25)

hence lim lim {fv[H,o(~ ~" v3,/~,.v3, ~t,.v))] _ fv[H]} = O,

(26)

,~-,0 V ~ . ~

t h e c o n s t a n t s ti t'v~, i t'v), ~t,.v3 b e i n g d e t e r m i n e d b y t h e m i n i m a x c o n d i t i o n

fv[H,o(a, s, ~t"V3(a, s))] = m am x fv[H,o(a, s, m)], fv[H,o(~,.v), i~,.v), n~,.v)(~(,.v), g~,.v)))] = m i n fv[H~o(a, s, tfi~'v))],

(27)

a,s

~ . , v ) = ~.,V)(~.,v),/~.,v~), which leads to the set of self-consistent equations (V) a = (A)n,0to.,.m),

- ~v) s = ( S ) n , 0 t ..... ),

v ) ..... ~, m = i\ M - \ (//4#

z (v)

Sz = ( S ) n , ot..... ),

mz = ( M )zn , 0 ( ..... )-

U n d e r a c a n o n i c a l t r a n s f o r m a t i o n the a p p r o x i m a t i n g H a m i l t o n i a n m a y be r e d u c e d to a d i a g o n a l form. A f t e r s o m e c a l c u l a t i o n s ~2) w e arrive at the result

H,o(a, s, m ) = ~ (- A ) { E~,~, - [ EI2 + -~ (E/2)21} T(k) - [ T 2 ( k ) + F2aa+]'/2+ F a a + - - ~ c A ( m m w h e r e t h e o p e r a t o r s fit a n d ak~ a r e i n d e p e n d e n t

+ + mzm+)},

(28)

and

E = N/ss + + szs z, + T ( k ) = ~(k) - Ix, E ( k + ) = [ T2(k) + F2aa +]in + c A [ le-~(m - s)(m + - s *) + l-7(m~ - s D ( m : - s:)]u2 ,

(29)

620

N.N. B O G O L U B O V JR. et al.

E(k-)

= [T2(k) + F2aa+] '12 - c A [1~ ( m - s ) ( m + - s

+) +l(m~,4 - Sz)(m +z-S+)] 112.

H e n c e , the free e n e r g y f u n c t i o n is given by

fv[H,o(a,s,m)]= cA [-~(ss+ + s~s+~)+(ss+ + s~s +)'/2] 1

- cA ~ (turn ++ m~m ~)

+ ~ ~ {(~(k)- ~ ) - [(~(k)- ~)2+ r 2 a a + l , a }

- O V ~ In { 1 + e x p [ -

[(g(k)

- tz )2 + F ' a a

- sign(~r)cA [ ~ ( s

- m)(s +- m +)

÷ -~' (st -

m+)]ll2lo]}.

mz)(s +~-

+1'/2/01

(30)

T h u s f r o m (26), (27) we h a v e o b t a i n e d the e x p r e s s i o n (30) f o r the free e n e r g y f u n c t i o n of the s y s t e m u n d e r c o n s i d e r a t i o n (1), w h i c h is e x a c t in the t h e r m o d y n a m i c limit,

4. The equations for order parameters It is not difficult to c o n v i n c e o n e s e l f that f o r a r b i t r a r y e > 0, and f o r all a, s, sz, m, mz the t h e r m o d y n a m i c limit of the f u n c t i o n (30) exists and is equal to

cA f~[H,0(a, s, m)] = 7

[(ss+ + s z s D - (ram + ÷ mzm~)] ÷ c A ( s s + ÷ szs~) '12

,f

{(g(k) - ~x) - [ ( g ( k ) - ~)2 + F2aa+l,/2} dk

+ --

~Br

VBr

r2A

VBr

+

COEXISTENCE OF SUPERCONDUCTIVITY AND FERROMAGNETISM

cA

•

+ s,gn(o-)~

[(m -

621

s)(m +- s +)

+ (m~ - s~)(m +- s f)l"Z]]} dk,

(31)

w h e r e VBr d e n o t e s (the v o l u m e of) the Brillouin zone. T h u s , following refs. 11 and 13 the e x a c t t h e r m o d y n a m i c values of the o b s e r v a b l e s of the s y s t e m should be d e t e r m i n e d in the following way. F o r a finite • > 0 o n e should first find the solutions t o g e t h e r with the e q u a t i o n f o r the c h e m i c a l potential f o r the free e n e r g y for the a p p r o x i m a t i n g H a m i l t o n i a n (31)

f c~f~[H(,~)( a, s, m ) ]/ c~a = 0, ,g[~[H~,'~)(a, s, m ) ]/ ~s = 0,

(32) (33)

8f=[H(,$)(a, s, m)]/am = 0,

(34)

O[=[HC,$)(a, s, m)]/ alz = - I/v,

(35)

T h e values ti t'), ~'), ,hi'),/2 t') are then o b t a i n e d using the condition that tic,), gt,), n~(,) are the solutions of the m i n i m a x p r o b l e m f o r the f u n c t i o n f~[H(,g"))(a, s, m)]. As has b e e n s h o w n in ref. 14, outside the points of p h a s e transitions the limits as • ~ 0 of the quantities fit,), s-t,), ~ht,) exist and their following c o m b i n a t i o n s are the e x a c t values of the o b s e r v a b l e s of the s y s t e m : lira (A)(ff ~ = lira ~(o, V-~.~

~--*0

lira ( M - ) ~ v) = lira n~ (o = lira g('), V-~

e-.O

~-~0

lira (MZ)~v~ = lim n~z~) = lim ( S - ) ~ v) = lira 1 [g(,) _ n~(,)], V ~'~

e--~

I

limw~0. Furthermore d can be chosen to be a real non-negative quantity, which also may be obtained by a proper choice of an infinitely small term with sources of electron pairs.

622

N.N. B O G O L U B O V JR. et al.

In the strong-coupling limit ~(k) = ~

(36)

the limiting free energy function of the approximating Hamiltonian takes the form f~[Ht~$)(a, s, m)] = c--~-A( s 2 - m2)+ cAs + ~ - Ix

- [(~ - it) 2 + F2a2] 1/2+ Fa 2 - Oc ln[1

+

[2A

s)]

- 0 ~] In{1 + e x p [ - 1 [ [ ( ~ - / x ) 2 + F2a2]l12+sign(tr)cA 1) (s¢- m)]]}i37 2 In this case eq. (35) yields /2(')= ~.

(38)

Taking into account that a/> 0 we have / ~ [-"(a~'))¢a n,0 , , s , m )] = cA l

E2

(s_

m ) ( s + m ) + F a 2 - 0 ( c + 2 ) 1n2

A cA 1 - Oc l n c o s h ( ~ - s ) - 0 1 n c o s h [ F a + - 2 ~ ( s - m ) ]

- 0 In cosh I F ~-~ a - ~cZ- ~l (s - m) ] ,

(39)

and the equations (31) take the form

- tanh[ a-CZ~I(s - m) ]= 0,

(40)

F a cA 1 2s - ~1 tanh [~--~ + ~--~~-~(s - m)]

+ i t a n h [ F a _ ~__@ ¢ 2~ (s c _Am)] 1 _

t a n h ( A s) = 0,

(41)

+ ~--( ~ (s - m ) ] - ~lt a n h [ ~ -F~ a - ~cA 1 (s . - m ) ] = 0. - ~-~

(42)

Let us now investigate the criterion for selecting the physical solution of 'the set (40)-(42).

COEXISTENCE OF SUPERCONDUCTIVITY AND FERROMAGNETISM

623

First, any solution ~t'J, #c'J, r~ t'j of this set satisfies in particular eq. (42) which has the form O[~[Ht~')J(a, s, m)]/~gm = 0. Noting that

02¢ rlr...1.cac,:~),~ =_ 1 1 cA 1 ~-.o t.,s,m)] 2-~ 0 cosh ~ a + ~ - ~ 1 lcA

4 ~ "~ 0 cosh2[~0

(s-m)

1

cA20E-~ l(s-

m)a] < 0 ,

we obtain that for any root of the set of equations (40)-(42) n~t° is the solution of the absolute maximum problem /~[Ht,~'~J(~ ~'), §~J, n~'J)] = max f~[H~'~J(~ ~'j, §~J, m)],

(43)

i.e., in our notations A ~'j = m~'J(,Y"j, ~'J).

(44)

For the solution of the minimax problem of the limiting free energy function that is for sure present among the roots of the set of equations (40)-(42) for all a, s we have f~[H~")(tit,J, g~,), #ht,J(ti~,J, g~o))] ~
(45)

and in particular by virtue of (44)

I~[H~C")(d t'j, gt'j, n'i c'))] ~
(46)

for any solution a t'j, #t'J, n~t'j of the system (40)-(42). Thus, the proper solution of the system of equations (40)-(42) is determined by the condition that the value of the free energy function for this solution should be minimal. Let us now rewrite the set (40)-(42) as follows I (s - m) = ~ t a n h ( A s ) ,

a -s =½tanh[Fa

- - ~ca- t a n h ( ~A- s ) - , 2 tanh ~,~A s/j,

a + s : ~ t a n h [ F a + ~cA -~tanh\A ~s/+e2tanh\os/].

(47) (48) (49)

At the points which are solutions of eqs. (40)-(42) the limiting free energy function has the form

f$'(a, s)= - O ( c + 2)In 2 + cZs t a n h ( A s ) + Fa 2- 0c In c o s h ( A s) \o/ \o]

624

N.N. BOGOLUBOV JR. et al. a + ~-ff tann ~-ff s ) ]

I

-O,ncos [ 0o - cA tan

From eqs. (49) and (50) it is clear that the equations, the expression for the free energy and the final result are not singular as • ~ 0, the terms containing the parameter • are infinitely small and may be omitted, as will be proved later on. Differentiating the function f~')(a, s) we obtain

Of~)(a, s) Oa Of~')(a, s) Os

Of~[H~,ao~"~(a,s, m)] C~all/,2(s_m)=l/2tanh(AlOs)' ~2A Of~[H,,cat,))(a, s, m)] 20 cosh2( A s) Oml'/''-'~-m'='/2ta'hc''°''"

(51)

Then each solution d ~'), §~') of the set of equations (40)-(42) satisfies also the set of equations

oof~)(a,s)/Oa=O,

(52)

f~)(a, s)]cgs = 0

(53)

and the other way round. Therefore, on the basis of the criterion for selecting the proper solution of eqs. (40)-(42) and taking into account that the function (50) has a minimum we conclude that the problem of determining the quantities a ~'~, gc,), rfic,) is reduced to the determination of this absolute minimum of the function

f~(a, s). Let us introduce the function

f~(a, s)= - O ( c + 2)In 2 + cAs tanh( A s ) + Fa z - Oc In c o s h ( A s)

o,ncoshf o + ~ - t a n h ~

s)]

- ~-~-tann ~ - s ) ] .

(54)

As has already been mentioned above, it has been proved in ref. 14 that outside the points of phase transitions the limits of the quantities ~¢'~, g~o as •---> 0 exist and are equal to the corresponding exact thermodynamic values A = l i m v - ~~ n( V ) and M = l i m v ~ ( M ~ ) ~ ) lim d (')--- ~i - A,

lim g~')= g -- M.

(55)

COEXISTENCE OF SUPERCONDUCTIVITY AND FERROMAGNETISM

625

Let us now show that these values ti, g are determined by the absolute minimum condition for the limiting function L(a, s) independent of e. By virtue of the continuity of f~(a, s) and the definition of the quantities ti, g (55) we have (V6 > 0)(3El > 0)(rE E (0, Ei))f~(a, g) - f ~ ( a ~'), g~')) < &

(56)

Since (cA~4)tanh2(A/O)s) is a bounded function of a and s, [~')(a, s) converges to ]'~(a, s) as • ~ 0 uniformly in every domain of complex variables a, s, hence (Va, s)(V6 > 0)(:IE2 > 0)(rE E (0, e2)),

fl~')(a, s) - L ( a , s) < 6,

L(a'",

~"') - f'~"(a ~'',

(57)

e'") <

a.

(58)

As we have shown, the function ]~')(a, s) assumes the absolute minimum at a ~'), ff~'), i.e., ('ca, s)F~')(a ~'), g~')) - ]~)(a, s) ~< 0.

(59)

Summing the inequalities (56)-(59) term by term and taking into consideration the fact that L ( a , s) is independent of e we find ('Ca, s)(V6 >

0)L(a, e) -

L ( a , s) ~< 3a.

Taking the limit •--->0 we obtain

('ca, s)L(a, g)<~(a, s).

(60)

Thus, in the thermodynamic limit the exact values of the magnetization of the electrons M=limv_~(MX)~ ) and the energy gap parameter A = limw=(A)~v) are found respectively as the solution g¢0)= g and a ¢°)= a of the absolute minimum problem for the function L ( a , s) (54): ]~(a~0), g~0))= min ]~(a, s),

(61)

a,s

i.e., as the solution of the set of equations Ji

-

t =a ~1tanh[~-~ n h (F Aa s cA ) ] , s

+ s = ~tanh[~0 a +4--0-cAt a n h ( A s ) l ,

(62)

(63)

at which the function [®(a, s) becomes a minimum among all the solutions of this set. In the thermodynamic limit the exact value of the impurity mag-

626

N.N. BOGOLUBOV JR. et al.

netization $ = limv_~(SX)~v~ is given by $ : ½tanh(A g ) = , tanh (~A M).

(64)

5. Exact solution of Zener model

In the case of the Zener s-d exchange model (F = 0) g can be determined from the equation 1

cA

A

s = ~ tanh [~-~- tanh (--ff s) ].

(65)

For this equation there exists only one zero solution M = g = 0 and hence $ = 0 or two solutions, one of which corresponding to non-zero values for the order parameters, depending on the values of the quantities A, c, and 0. As the derivative of the right hand side of eq. (65) 1 A 2(2c) 1 16 0 2 cosh 2[~_fftanh(__ffs)]cosh(As) cA A

f~'rhs(S )

is a decreasing function of s, the second possibility takes place when 1

/'rhs(O) > or

1 A2(2c) _ 16 ~T -~1 whence O < Oc = ~A(2c) I/2.

Let us now verify whether the non-zero solution for 0 < 0c satisfies the minimum condition for the function L(s): f~(s) = - O(c + 2) In 2 + c A s tanh ~- s - O c In c o s h ( A s ) -

20 In cosh[-~0 t a n h ( A s ) ] .

Differentiation yields f ' ( s ) = C A 2 1 { s - ~ t a n h [ ~ - ~ t a n h ( ~ s ) ] } . cA _

_

0 c o s h ( A s)

I

A

(66)

COEXISTENCE OF SUPERCONDUCTIVITY AND FERROMAGNETISM

627

Now it is easy to see that ~'(s)~<0

for

O<~s<-g,

)r'(s)>0

for

g
where g is the non-zero solution of eq. (65). Therefore, at 0 < 0c for s ~< g, L ( s ) becomes a minimum and, hence, at 0 < 0c the order parameters M = g and $ = ~ tanh((A/O)g) differ from zero, i.e., the temperature 0c = ~A (2c) I/2

(67)

is the critical temperature. To find the magnetizations of the electron and spin subsystems as functions of temperature it is easy to solve eq. (65) numerically.

6. Coexistence of superconductivity and ferromagnetism The set of equations (62)-(64) and the minimum condition (54) of the function L ( a , s) coincide with those obtained in ref. 1. For this reason the results of the present paper and those of ref. 1 are the same. So, the analysis of the eqs. (62), (63) together with (54) at zero temperature shows that at 0 = 0 the system is a pure superconductor for F > 2cA and a pure ferromagnet for F < 2 c A . Thus, the coexistence of superconductivity and ferromagnetism is impossible at zero temperature for the system under consideration (1), (36), and the phase diagram at 0 = 0 is a straight line. Nevertheless, by solving the problem at arbitrary temperature numerically with the aid of a computer the ranges of the temperatures have been found for certain values of the constants of interactions r and A and the concentration of the magnetic impurity c for which both the energy gap G=

and

the

d-½cA t a n h ( A g ) = A - c A S , magnetization

of

the

electron

(68)

M = g and

the

impurity

$=

~tanh((A/O)g) subsystems differ from zero, i.e., the coexistence of superconductivity and ferromagnetism occur. Finally, it should be emphasized that the method of the present paper is mathematically rigorous and all the results obtained are exact. Thus, in our work it has been proved rigorously that the coexistence of superconductivity and ferromagnetism take place.

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N.N. BOGOLUBOV JR. et al.

Acknowledgment The authors wish to thank Dr. H.W. Capel for his attention and notes.

R eferences 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ll) 12) 13) 14)

W.A. Smit, G. Vertogen and J. Kraak, Physica 74 (1974) 97. W. Baltensperger, Helv. Phys. Acta 32 (1959) 197. L.P. Gorkov and A.L. Rusinov, JETP 46 (1964) 1363. A.A. Abrikosov and L.P. Gorkov, JETP 39 (1960) 1781. A.I. Larkin and D.E. Khmelnitzkiy, JETP 58 (1970) 1789. N.N. Bogolubov (Jr.), Physica 32 (1966) 933. N.N. Bogolubov (Jr.), A Method for Studying Model Hamiltonians, (Pergamon, Oxford, 1972). J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. N.N. Bogolubov, D.N. Zubarev and Ju.A. Tzerkovnikov, Doklady 117 (1957) 788. C. Zener, Phys. Rev. 81 (1951) 440. A.N. Ermilov and A.M. Kurbatov, JINR IL%10237, Dubna, 1976. N.N. Bogolubov (Jr.), A.N. Ermilov and A.M. Kurbatov, JINR R17-9773, Dubna, 1976. A.N. Ermilov and A.M. Kurbatov, JINR R5-10245, Dubna, 1976. A.N. Ermilov and A.M. Kurbatov, JINR R17-10240, Dubna, 1976.