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Possible phase change between itinerant ferromagnetism and superconductivity
~
Solid State Communications, Vol.43,No.12, pp.899-902, 1 9 8 2 . Printed in Great Britain.
0038-I098/82/360899-04503.00/0 Pergamon Press Ltd.
POSSIBLE PHASE CHANGE B E T W E E N ITINERANT FERROMAGNETISM AND SUPERCONDUCTIVITY HaNakanishi Department of Physics) Faculty of Science and Technology, Keio University Yokohama 223, 3apan and K3vlachida and T.Matsubara Department of Physics) Kyoto University) Kyoto 606, 3apan (Received
26 May 1982 by
3. Kanamori )
It is pointed out theoretically that a single electron band model is able to exhibit the interchange of two phases: itinerant band ferromagnetism and superconductivity. Our theory is based on the molecular field approximation applied for a simplified electron-electron interaction. Possible phase changes are discussed in connection with two phase transitions of ferromagentism and superconductivity in the intermetallic compound Y~Co 3.
Recently i t is found [ I - 7 ] that the intermetallic compound Y.u~Co~~ exhibits an interesting interplay . between ferromagnetlsm (FM) and superconductivity (SC):Namely the system shows ferromagnetism below 6K and at a lower temperature ( . ~ 2.5K) superconductivity sets in. It is not yet clear
We s t a r t with the following single band electron model with the simplified effective interaction: H = Z (~P-~)CIsCpsv ps
+ "
experimentally whether or not the ferromagnetism persists in the superconducting state. The magnetism in this case may be itinerant (possibly weak ferromagnetism) in nature (the saturation moment 0.037 or 0.01 ~.(B/Co)[7,g], mainly originating from the d-electrons in Co atoms. The same conduction electrons also play an important role in forming the superconductivity [9]. This new situation should be compared with other previously known magnetic
+
+
"~-'-', V(plP2P3Pt~)CplsCp2s,Cp3s,Cpt+s (I) PI+P2=P3+P4 •
.
.
+
where }~- is the chemical potentlal) c s(C s) the annihilation (creation) operator of an eleU~tro~ri with momentum p and spin s and
V(PlP2P3Pt ~)
;for region l
( VC )
(2)
-[IVc _ VB ;for region 2.
superconductors) such as ternary rare earth compounds [ I 0 ] where rare earth atoms carry t~f localized permanent moments) or several organic superconductors i l l ] (TMTSF)2X (X=PF6) CIO~ etc.) where itinerant antiferromagnetism compete~ with superconductivity. The purpose of the present paper is to point out the possibility that even a single band interacting electron system is able to exhibit such an interchange of the two long range orders: FM and SC within the molecular field approximation. Our theory is based on the following two observations:(1) Even if the eleciron-electron interaction near the Fermi energy within the Debye frequency ~'Ih is repulsive) the effective electron-electron interaction) which is modified by the repulsive interaction outside the Debye frequency region) might be a t t r a c t i v e in some cases [12]. (2) It is known that in the band magnetism the saturation moment sensitively depends on a band structure. Therefore even if the ferromagnetic transition temperature T~ is high) the saturation moment (or equivalently t"ne magnetic energy) could be very small in certain cases. These two observations have led us to examine the interchange of the two phases) itinerant FM and SC) in connection with the above experimental finding) generalizing our idea [13-15] of the molecular field approximation applied to various magnetic superconductors.
The region 2 denotes the energy region for four momenta p. within the Debye frequency co~ about the Fermi ~nergy where the a t t r a c t i v e intera~-~ion Vj~ coming from the electron-phonon interaction operative in addition tu the Coulomb repulsion Vr.. The region I is the outside of the region 2. ThYs simplified form for the electron-electron interaction is often used in the calculation beyond the simple BCS theory [ i 6 ~ We employ the molecular field approximation to this Hamiltonian, assuming the presence of the relevant order parameters such as the ferromagnetic magnetization m arising from the repulsive part and the superconducting order parameter A from the a t t r a c t i v e part: Thus the eq.([) is rewritten as H = .7_C~MF
Cp
P + ½ Zw- V(Pi)(Cpl°" 3Cp4)'(Cp2°- 3 Cp 3) Pi -
S _ C +p ( - m p - npO-3 + ApO- I) Cp, P
~MF
= -
mp + ( ~p -
H--
%)¢3
+ t, po- l
(3)
(~)
where we have introduced the 2 x 2 Nambu space~ Cp~-= (c ~ , C p,) and r~"i is the Pauli matrix. We note
899
900
ITINERANT FERROM,AGNETISM AND SLTERCONDUCTIVITY
Vo[. 5.3, No.
12
that the order parameters m , A and the H a r t r e e f i e l d n have the momentum ~ r e~ergy) dependence, corresponding to the energy regions 1 or 2 to which t h e y belong. The thermal Green function for the mean field H a m i l t o n i a n 2CMF is readily obtained as
given by K. = V . ~ (i = C o r B) a n d R . = '. o 1,2) r e s p e c ~ v e l y ! o We have introducec~ heNet/J) (i =
iCC'n + mp + ~ n + &o°-I G(p, iu-'n) = . i rCr3 ' mp) 2 - E 2 with (i~'r, + p
We n~)te f i r s t from eq.(9) t h a t t h e e f f e c t i v e a t t r a c t i v e i n t e r a c t i o n K ~ is given by K ~ = K 8 _ Kr./(1 + K r - ~ ,) which is nothing but the form dertved b~" Morel ~'nd~ A n d e r s o n [12 ]. If we t a k e a c o n s t a n t d e n s i t y of s t a t e s ~ ( ~ ) - ~ (0), then
(5)
and
Ep=( P . P P P The self-consistent equation for determining Ai, mi, n i ( c o r r e s p o n d i n g t o t h e region i = t or 2) and /-£ is g w e n in t h e matrix form by T ~- S V( [p,~q)[ [G i l (q'iEn) - G22(q'i ~n)~Cr3
i~ q
- O-3G(q,i£ n)0- 3] + m p + n ~ - 3 - AD'1-~ = 0
(6)
where
V( ~p, ~q) = V(p,q,q,p) = V(p,q,p,q). A f t e r the sums over i ~_ and q are replaced by the a p p r o p r i a t e integrals, each m a t r i x element gtves rise to m I = ½VcAN , n I = _ ½VcN , •
•
n
m 2 = m I - ½VBAN2,
.
.
n 2 = nl + ½VBN2)
VC AI = - A 2
a2
(7)
1 + VcA 1
~'i = ½J'. d~>(~)tanh('~i)/~i , L
Ri
:
(10)
d
zd.
K~ = K B - KC[i + KCY(0) In(tle,D)] -t
l + KBC = (B2 + D)Kc/(I " KCB'I)
C = B2 +.~(~t+ n 2 + cc, Dlf(cc~)
- ~ ( F+ n 2 -~CD)~(- ~b), D = -~+n2+ u-~f(nz-n l+ccD) - f(tcD)] -ae(p+n2-LOD)[f(n2_nl) - f(_tc~)].
:Y(
c
with t h e t o t a l e l e c t r o n number N : N[ + N2, w h e r e A i = "Jld~a°(~)[tanh'~(-mi+E i) + tanh½B(mi+Ei)]/E i
Ni =,lidS) °(~l + 2~." f(-mi+Ei) - f(-mi-Ei) ] [
AN i : . i d ~ j ° ( ~ f ( E i - m i) - f(Ei + mi)]. l
(g) (i=l
or 2)
The density of states is denoted by..ff(.F) and f ( E ) is the Fermi d i s t r i b u t i o n function. These equations c o n s t i t u t e a complete set for the order parameters. We now examine the conditions on which the SC and FM should appear. A l t h o u g h it is possible t h a t t h e SC can d e v e l o p in t h e p r e s e n c e of FM via a s e c o n d o r d e r t r a n s i t i o n as will be s e e n l a t e r , we only i n v e s t i g a t e t h e c a s e in which o n e of t h e t w o o r d e r s a p p e a r s in t h e a b s e n c e of t h e o t h e r o r d e r . T h e r e f o r e t h e p h a s e c h a n g e s to b e e x p e c t e d a r e of a f i r s t order transition. The s u p e r c o n d u c t i n g p r o p e r t i e s in t h e a b s e n c e of FM a r e d e t e r m i n e d by p u t t i n g m i = 0 in eq.(7). We obtain n I = - ½ K c R , n 2 = n I + ½KBR2,
i
+
KC A 2, KCA I
(9)
KC I = ~2
(
KB
(13)
Note that
i + VcA l
At =-A'2
(12)
with
)
!
(It}
w h e r e t is t h e band width. The condition on which t h e SC is s t a b i l i z e d is given by K ~ > 0. Similarly from eq.(7) we can d e r i v e t h e self-consistent equations which determine the stability c o n d i t i o n for FM w i t h o u t SC by p u t t i n g A = 0. From t h e s e e q u a t i o n s along with t h e limiting p r o c e s s m. • 0 we obtain t h e implicit e q u a t i o n to d e t e r m i n e ~he FM t r a n s i t i o n t e m p e r a t u r e T F as
VC l = A2( VB -
i )f(gi)
)' l + KC~ l
R l + R 2 = R, w h e r e we have k e p t only t h e l o w e s t o r d e r o f A to d e t e r m i n e t h e t r a n s i t i o i n point. The d e n s i t y of s t a t e s Y( ~ ) i s n o r m a l i z e d so as f f ~ ) = f f o ~ ( f f ) andAJ3(~ )all: = 1. Then t h e r e n o r m a l i z e d i n t e r a c t i o n c e n s t a n t s and p a r t i a l e l e c t r o n n u m b e r s in t h e e n e r g y region i a r e
when T=0. case is
:
rig'
: D
~(~+ %)-y(kL+
:
o and
nz -%)
Tt-,erelore the Stoner condition in this
Keg(Ix+ n 2) - K B [ Y ( P + n2) - Z (P+ n2 -tX"D)] = I , (l#) which is hardly a f f e c t e d by t h e a t t r a c t i v e i n t e r a c t i o n around t h e Fermi e n e r g y in c o n t r a s t with t h e BCS c a s e . This is b e c a u s e t h e BCS i n t e r a c t i o n is assumed a l w a y s to s t a y around t h e Fermi e n e r g y . In o r d e r to d e r i v e t h e phase diagram . in t h e K L. ~ vs K B plane) we must c h o o s e an a p p r o p r i a t e form for t h e d e n s i t y of s t a t e s . In t h e following we a d o p t
•7(~) = 3(t 2 - ~2)/#t3 ( ~ St)" The e l e c t r o n number is f i x e d so t h a t t h e F e r m i e n e r g y in t h e p a r a m a g n e t i c normal s t a t e a t T = 0 is s i t u a t e d a t ½ cO D . The s t a b l e region for SC is d e t e r m i n e d by K B >__KC/(I + KcAI)~re show this boundary as t h e d o t t e d line in Fig.l. On t h e o t h e r hand it is easy t o s e e a f t e r some manipulation t h a t a t T = 0 t h e FM is s t a b i l i z e d when F'C > 1 (~. = V. (-'(½cC~)), which is i n d e p e n d e n t of ~B b e c a u s e o~ thetJcance~lation in t h e a t t r a c t i v e par~ under t h e i n f i n i t e s i m a l shift of t h e up- and down-spin Fermi e n e r g i e s . In Fig.l t h e h a t c h e d region i n d i c a t e s t h e o v e r l a p p e d a r e a o f s t a b i l i t y for both SC and FM phase w h e r e t h e t w o o r d e r s a c t u a l l y c o m p e t e e a c h o t h e r . In w h a t f o l l o w s we c o n f i n e our discussions to this r e g i o n . We c a l c u l a t e d n u m e r i c a l l y t h e SC and FM t r a n s i t i o n t e m p e r a t u r e s T c and T F. The region (A) in Fig.1 c o r r e s p o n d s t o T > T F w h e r e SC o r d e r a p p e a r s at a higher temperatuCre t h a n t h e FM. The region (B) c o r r e s p o n d s t o T < T F. in t h e ground s t a t e a t T = 0, on t h e o t h e r ~.and, t h e r e is c e r t a i n l y a b o u n d a r y
Vol. 43, No.
12
901
ITINERANT FERROMAGNETISM AND SUPERCONDUCTIVITY T c --T~ I J
"E# E_,
,
,
,ii
09
-- K-
(a)
(b)
OB FM IOI Kc Fig.1 Phase diagram in t h e K B vs K c plane in the case of t = 3 c o D" The Hatched region indicates the possible area for both FM and SC state. The d o t t e d line is the boundary ol SC and FM, and the straight line at ~ C = l denotes FM instability. (A) is t h e region for T c > T F and (B) is for Tc < TF" (c) hne m the same K c. vs K_ plane where the ground s t a t e energies E~C and EF~/ of two phases are equal. This should star'~ at t h e it5lersecting point of t h r e e lines in Fig.l and does not coincide with the line T = T_ in general. Thus t h e r e may appear severa~ posslblhties for the e x p e c t e d phase diagram as depicted schematically in Figs.2a-2d. In Fig.2 the dotted line is the boundary b e t w e e n anffS C > EFM (right hand side of the d o t t e d line) ~ C e < E E M (left hand side). region (I) should exhibit the successive transition: normal paramagnetic s t a t e -- FM -, SC from high t e m p e r a t u r e s . Similarly t h e region (If) in Figs.2b, 2c and 2d exhibits the successive transitions: pars normal -> SC -> FM. In Fig.3 we also plot s c h e m a t i c t e m p e r a t u r e variations of two order p a r a m e t e r s in t h e case (I) where the d o t t e d curves correspond to t h e cases in the absence of the other order. From eq.(7) it is possible to see t h a t in c e r t a i n c a s e s t h e SC can appear in the FM s t a t e via a second order transition. This is physically because if t h e magnetization is small enough, t h e SC order is able to develop continuously by neglecting small Fermi e n e r g y shift between the up and down spin bands which is already present when the SC s t a r t s to appear. In such a case immediate below T t h e r e might be a t m p e r a t u r e interval in which thec SC and FM coexist [17]. It is r e p o r t e d that in t h e i n t e r m e t a l l i c compound Y#Co~ t h e r e exist successive phase transitions such t h a t t~aramagnetic normal s t a t e - 7 FM-7 SC. Because of the above reasoning it is net totally unphysical to e x p e c t such successive phase change of two long range orders when t h e ferromagnetism is weak enough (i.e. small saturation moment and relatively low transition t e m p e r a t u r e TF). Careful t h e r m o dynamic and transport measureinents near t h e superconducting onset t e m p e r a t u r e T are highly desirable. C • Especially t h e specific heat e x p e r i m e n t near T is
(d)
Fig.2 Possible schematic phase diagrams near Kc" = I and F,~ = 0.8 ir, t h e ~ vs F,~ plane. T h e dotted line corresponds tobthe bo~undary of the equal ground s t a t e energies ESC =.E. FM" The Iollowing successive phase transLnons take place in (I): Normal -7 FM -7 SC, and in (II): Normal ~* SC -> FM.
Tc
Tc
TT
Fig.3 Possible t e m p e r a t u r e d e p e n d e n c e s of t h e two order p a r a m e t e r s , which corresponds to (I) in Fig.2, when a first order transition is assumed to take place at T = T' C" •
g:
crucial to clariiy t h e nature of t h e phase transitions, t h a t is, il the second order transition is realized at T , then we can a n t i c i p a t e the c o e x i s t e n c e of the it~Cnerant ferromagnetism and superconductivity in lower temperatures. From t h e above argument we can e x p e c t t h a t other existing weak f e r r o m a g n e t s might also exhibit superconductivity at lower t e m p e r a t u r e s and t h a t certain superconductors are r e e n t r a n t t o normal ferromagnets.
902
ITINERANT FERROMAGNETISM
# ~D SUPERCONDUCTIVITY
Vol. 43, No.
12
REFERENCES ~° 2.
3.
t,. 5. 6.
7. 8. 9.
10.
E.Gratz and E.P.Wohlfarth, 3. Magn. Magn. Mat. 15-18 903 (1980). E ~ ' ~ t z , H. R. Kirchmayr, V. Sechovsky and E. P. Wohlfarth, 3. Magn. Magn. Mat. 2.~_1 191 (1980). A. Kolodziejczyk, B. V. B. Sarkissian and 15. R.
Coles, 3. Phys. F 10 L333 (1980). E.Gratz, 3. Magn. , ~ g n . Mat. 24 1 (1981). 3.Sebek, 3.Stehno, V.Sechovs-~y and E.Gratz, Solid State Commun. 40 457 (1981). E.Gratz, 3;O.Strom-O'~en and M.3.Zuckerman, Solid State Commun. 40 833 (1981). W.Cheng, G.Creuzet,--lS.Garoche, LA.Campbell and E.Gratz, 3. Phys. F 12 475 (1982). S. Ogawa, private communication. The problem of the interplay between itinerant magnetism and superconductivity is reviewed by E. P. Enz in Superconductivity in d- and f-band metals ed. H. 'Suhi and' ~{."B.-'~{ap-p-p-~ ~-~"d~m~'c, New York, 1980) p.l$1. See for review, M.B~Maple, 3. de Phys (Paris). C6 1374 (1978), M.Ishikawa, ~.Fischer and 3"~uller, ibid 1379.
II. 12. 13. 14. 15. 16. 17.
See for review, S. S. P. Parkin, M. Ribault, D.3erome and KdSechgaard, 3. Phys. C 14 5305 (1981). P.Moreland P.W.Anderson,Phys. Rev. 125 1263 (1962). K. Machida, K. Nokura and T. Matsubara, Phys. Rev. B 22 2307 (1980). K~lachi~'a, 3. Phys. Soc. 3pru 50 2195 (19gl) and 51 1 4 2 0 (1982). K. Mac~ida and T. MatsuS-ara, ibid 50 3231 (1981). K~lachida, subm]~ed to 3. Phys. So(:. 3pn. See /or example, M. L. Cohen, in Superconducti~it~ ed. R. D. Parks (Marcel Dekker, New York, 1969) Chap.12. In fact in the rare earth ternary compound ErRh4B. the coexistence of ferromagnetism and superconductivity is found in neutron scattering experiment: See S. K. Sinha, G. W. Crabtree, D. G. Hinks and H. Mook, Phys. Rev. Lett. 48 950 (1982).
Solid State Communications, Vol.43,No.12, pp.899-902, 1 9 8 2 . Printed in Great Britain.
0038-I098/82/360899-04503.00/0 Pergamon Press Ltd.
POSSIBLE PHASE CHANGE B E T W E E N ITINERANT FERROMAGNETISM AND SUPERCONDUCTIVITY HaNakanishi Department of Physics) Faculty of Science and Technology, Keio University Yokohama 223, 3apan and K3vlachida and T.Matsubara Department of Physics) Kyoto University) Kyoto 606, 3apan (Received
26 May 1982 by
3. Kanamori )
It is pointed out theoretically that a single electron band model is able to exhibit the interchange of two phases: itinerant band ferromagnetism and superconductivity. Our theory is based on the molecular field approximation applied for a simplified electron-electron interaction. Possible phase changes are discussed in connection with two phase transitions of ferromagentism and superconductivity in the intermetallic compound Y~Co 3.
Recently i t is found [ I - 7 ] that the intermetallic compound Y.u~Co~~ exhibits an interesting interplay . between ferromagnetlsm (FM) and superconductivity (SC):Namely the system shows ferromagnetism below 6K and at a lower temperature ( . ~ 2.5K) superconductivity sets in. It is not yet clear
We s t a r t with the following single band electron model with the simplified effective interaction: H = Z (~P-~)CIsCpsv ps
+ "
experimentally whether or not the ferromagnetism persists in the superconducting state. The magnetism in this case may be itinerant (possibly weak ferromagnetism) in nature (the saturation moment 0.037 or 0.01 ~.(B/Co)[7,g], mainly originating from the d-electrons in Co atoms. The same conduction electrons also play an important role in forming the superconductivity [9]. This new situation should be compared with other previously known magnetic
+
+
"~-'-', V(plP2P3Pt~)CplsCp2s,Cp3s,Cpt+s (I) PI+P2=P3+P4 •
.
.
+
where }~- is the chemical potentlal) c s(C s) the annihilation (creation) operator of an eleU~tro~ri with momentum p and spin s and
V(PlP2P3Pt ~)
;for region l
( VC )
(2)
-[IVc _ VB ;for region 2.
superconductors) such as ternary rare earth compounds [ I 0 ] where rare earth atoms carry t~f localized permanent moments) or several organic superconductors i l l ] (TMTSF)2X (X=PF6) CIO~ etc.) where itinerant antiferromagnetism compete~ with superconductivity. The purpose of the present paper is to point out the possibility that even a single band interacting electron system is able to exhibit such an interchange of the two long range orders: FM and SC within the molecular field approximation. Our theory is based on the following two observations:(1) Even if the eleciron-electron interaction near the Fermi energy within the Debye frequency ~'Ih is repulsive) the effective electron-electron interaction) which is modified by the repulsive interaction outside the Debye frequency region) might be a t t r a c t i v e in some cases [12]. (2) It is known that in the band magnetism the saturation moment sensitively depends on a band structure. Therefore even if the ferromagnetic transition temperature T~ is high) the saturation moment (or equivalently t"ne magnetic energy) could be very small in certain cases. These two observations have led us to examine the interchange of the two phases) itinerant FM and SC) in connection with the above experimental finding) generalizing our idea [13-15] of the molecular field approximation applied to various magnetic superconductors.
The region 2 denotes the energy region for four momenta p. within the Debye frequency co~ about the Fermi ~nergy where the a t t r a c t i v e intera~-~ion Vj~ coming from the electron-phonon interaction operative in addition tu the Coulomb repulsion Vr.. The region I is the outside of the region 2. ThYs simplified form for the electron-electron interaction is often used in the calculation beyond the simple BCS theory [ i 6 ~ We employ the molecular field approximation to this Hamiltonian, assuming the presence of the relevant order parameters such as the ferromagnetic magnetization m arising from the repulsive part and the superconducting order parameter A from the a t t r a c t i v e part: Thus the eq.([) is rewritten as H = .7_C~MF
Cp
P + ½ Zw- V(Pi)(Cpl°" 3Cp4)'(Cp2°- 3 Cp 3) Pi -
S _ C +p ( - m p - npO-3 + ApO- I) Cp, P
~MF
= -
mp + ( ~p -
H--
%)¢3
+ t, po- l
(3)
(~)
where we have introduced the 2 x 2 Nambu space~ Cp~-= (c ~ , C p,) and r~"i is the Pauli matrix. We note
899
900
ITINERANT FERROM,AGNETISM AND SLTERCONDUCTIVITY
Vo[. 5.3, No.
12
that the order parameters m , A and the H a r t r e e f i e l d n have the momentum ~ r e~ergy) dependence, corresponding to the energy regions 1 or 2 to which t h e y belong. The thermal Green function for the mean field H a m i l t o n i a n 2CMF is readily obtained as
given by K. = V . ~ (i = C o r B) a n d R . = '. o 1,2) r e s p e c ~ v e l y ! o We have introducec~ heNet/J) (i =
iCC'n + mp + ~ n + &o°-I G(p, iu-'n) = . i rCr3 ' mp) 2 - E 2 with (i~'r, + p
We n~)te f i r s t from eq.(9) t h a t t h e e f f e c t i v e a t t r a c t i v e i n t e r a c t i o n K ~ is given by K ~ = K 8 _ Kr./(1 + K r - ~ ,) which is nothing but the form dertved b~" Morel ~'nd~ A n d e r s o n [12 ]. If we t a k e a c o n s t a n t d e n s i t y of s t a t e s ~ ( ~ ) - ~ (0), then
(5)
and
Ep=( P . P P P The self-consistent equation for determining Ai, mi, n i ( c o r r e s p o n d i n g t o t h e region i = t or 2) and /-£ is g w e n in t h e matrix form by T ~- S V( [p,~q)[ [G i l (q'iEn) - G22(q'i ~n)~Cr3
i~ q
- O-3G(q,i£ n)0- 3] + m p + n ~ - 3 - AD'1-~ = 0
(6)
where
V( ~p, ~q) = V(p,q,q,p) = V(p,q,p,q). A f t e r the sums over i ~_ and q are replaced by the a p p r o p r i a t e integrals, each m a t r i x element gtves rise to m I = ½VcAN , n I = _ ½VcN , •
•
n
m 2 = m I - ½VBAN2,
.
.
n 2 = nl + ½VBN2)
VC AI = - A 2
a2
(7)
1 + VcA 1
~'i = ½J'. d~>(~)tanh('~i)/~i , L
Ri
:
(10)
d
zd.
K~ = K B - KC[i + KCY(0) In(tle,D)] -t
l + KBC = (B2 + D)Kc/(I " KCB'I)
C = B2 +.~(~t+ n 2 + cc, Dlf(cc~)
- ~ ( F+ n 2 -~CD)~(- ~b), D = -~+n2+ u-~f(nz-n l+ccD) - f(tcD)] -ae(p+n2-LOD)[f(n2_nl) - f(_tc~)].
:Y(
c
with t h e t o t a l e l e c t r o n number N : N[ + N2, w h e r e A i = "Jld~a°(~)[tanh'~(-mi+E i) + tanh½B(mi+Ei)]/E i
Ni =,lidS) °(~l + 2~." f(-mi+Ei) - f(-mi-Ei) ] [
AN i : . i d ~ j ° ( ~ f ( E i - m i) - f(Ei + mi)]. l
(g) (i=l
or 2)
The density of states is denoted by..ff(.F) and f ( E ) is the Fermi d i s t r i b u t i o n function. These equations c o n s t i t u t e a complete set for the order parameters. We now examine the conditions on which the SC and FM should appear. A l t h o u g h it is possible t h a t t h e SC can d e v e l o p in t h e p r e s e n c e of FM via a s e c o n d o r d e r t r a n s i t i o n as will be s e e n l a t e r , we only i n v e s t i g a t e t h e c a s e in which o n e of t h e t w o o r d e r s a p p e a r s in t h e a b s e n c e of t h e o t h e r o r d e r . T h e r e f o r e t h e p h a s e c h a n g e s to b e e x p e c t e d a r e of a f i r s t order transition. The s u p e r c o n d u c t i n g p r o p e r t i e s in t h e a b s e n c e of FM a r e d e t e r m i n e d by p u t t i n g m i = 0 in eq.(7). We obtain n I = - ½ K c R , n 2 = n I + ½KBR2,
i
+
KC A 2, KCA I
(9)
KC I = ~2
(
KB
(13)
Note that
i + VcA l
At =-A'2
(12)
with
)
!
(It}
w h e r e t is t h e band width. The condition on which t h e SC is s t a b i l i z e d is given by K ~ > 0. Similarly from eq.(7) we can d e r i v e t h e self-consistent equations which determine the stability c o n d i t i o n for FM w i t h o u t SC by p u t t i n g A = 0. From t h e s e e q u a t i o n s along with t h e limiting p r o c e s s m. • 0 we obtain t h e implicit e q u a t i o n to d e t e r m i n e ~he FM t r a n s i t i o n t e m p e r a t u r e T F as
VC l = A2( VB -
i )f(gi)
)' l + KC~ l
R l + R 2 = R, w h e r e we have k e p t only t h e l o w e s t o r d e r o f A to d e t e r m i n e t h e t r a n s i t i o i n point. The d e n s i t y of s t a t e s Y( ~ ) i s n o r m a l i z e d so as f f ~ ) = f f o ~ ( f f ) andAJ3(~ )all: = 1. Then t h e r e n o r m a l i z e d i n t e r a c t i o n c e n s t a n t s and p a r t i a l e l e c t r o n n u m b e r s in t h e e n e r g y region i a r e
when T=0. case is
:
rig'
: D
~(~+ %)-y(kL+
:
o and
nz -%)
Tt-,erelore the Stoner condition in this
Keg(Ix+ n 2) - K B [ Y ( P + n2) - Z (P+ n2 -tX"D)] = I , (l#) which is hardly a f f e c t e d by t h e a t t r a c t i v e i n t e r a c t i o n around t h e Fermi e n e r g y in c o n t r a s t with t h e BCS c a s e . This is b e c a u s e t h e BCS i n t e r a c t i o n is assumed a l w a y s to s t a y around t h e Fermi e n e r g y . In o r d e r to d e r i v e t h e phase diagram . in t h e K L. ~ vs K B plane) we must c h o o s e an a p p r o p r i a t e form for t h e d e n s i t y of s t a t e s . In t h e following we a d o p t
•7(~) = 3(t 2 - ~2)/#t3 ( ~ St)" The e l e c t r o n number is f i x e d so t h a t t h e F e r m i e n e r g y in t h e p a r a m a g n e t i c normal s t a t e a t T = 0 is s i t u a t e d a t ½ cO D . The s t a b l e region for SC is d e t e r m i n e d by K B >__KC/(I + KcAI)~re show this boundary as t h e d o t t e d line in Fig.l. On t h e o t h e r hand it is easy t o s e e a f t e r some manipulation t h a t a t T = 0 t h e FM is s t a b i l i z e d when F'C > 1 (~. = V. (-'(½cC~)), which is i n d e p e n d e n t of ~B b e c a u s e o~ thetJcance~lation in t h e a t t r a c t i v e par~ under t h e i n f i n i t e s i m a l shift of t h e up- and down-spin Fermi e n e r g i e s . In Fig.l t h e h a t c h e d region i n d i c a t e s t h e o v e r l a p p e d a r e a o f s t a b i l i t y for both SC and FM phase w h e r e t h e t w o o r d e r s a c t u a l l y c o m p e t e e a c h o t h e r . In w h a t f o l l o w s we c o n f i n e our discussions to this r e g i o n . We c a l c u l a t e d n u m e r i c a l l y t h e SC and FM t r a n s i t i o n t e m p e r a t u r e s T c and T F. The region (A) in Fig.1 c o r r e s p o n d s t o T > T F w h e r e SC o r d e r a p p e a r s at a higher temperatuCre t h a n t h e FM. The region (B) c o r r e s p o n d s t o T < T F. in t h e ground s t a t e a t T = 0, on t h e o t h e r ~.and, t h e r e is c e r t a i n l y a b o u n d a r y
Vol. 43, No.
12
901
ITINERANT FERROMAGNETISM AND SUPERCONDUCTIVITY T c --T~ I J
"E# E_,
,
,
,ii
09
-- K-
(a)
(b)
OB FM IOI Kc Fig.1 Phase diagram in t h e K B vs K c plane in the case of t = 3 c o D" The Hatched region indicates the possible area for both FM and SC state. The d o t t e d line is the boundary ol SC and FM, and the straight line at ~ C = l denotes FM instability. (A) is t h e region for T c > T F and (B) is for Tc < TF" (c) hne m the same K c. vs K_ plane where the ground s t a t e energies E~C and EF~/ of two phases are equal. This should star'~ at t h e it5lersecting point of t h r e e lines in Fig.l and does not coincide with the line T = T_ in general. Thus t h e r e may appear severa~ posslblhties for the e x p e c t e d phase diagram as depicted schematically in Figs.2a-2d. In Fig.2 the dotted line is the boundary b e t w e e n anffS C > EFM (right hand side of the d o t t e d line) ~ C e < E E M (left hand side). region (I) should exhibit the successive transition: normal paramagnetic s t a t e -- FM -, SC from high t e m p e r a t u r e s . Similarly t h e region (If) in Figs.2b, 2c and 2d exhibits the successive transitions: pars normal -> SC -> FM. In Fig.3 we also plot s c h e m a t i c t e m p e r a t u r e variations of two order p a r a m e t e r s in t h e case (I) where the d o t t e d curves correspond to t h e cases in the absence of the other order. From eq.(7) it is possible to see t h a t in c e r t a i n c a s e s t h e SC can appear in the FM s t a t e via a second order transition. This is physically because if t h e magnetization is small enough, t h e SC order is able to develop continuously by neglecting small Fermi e n e r g y shift between the up and down spin bands which is already present when the SC s t a r t s to appear. In such a case immediate below T t h e r e might be a t m p e r a t u r e interval in which thec SC and FM coexist [17]. It is r e p o r t e d that in t h e i n t e r m e t a l l i c compound Y#Co~ t h e r e exist successive phase transitions such t h a t t~aramagnetic normal s t a t e - 7 FM-7 SC. Because of the above reasoning it is net totally unphysical to e x p e c t such successive phase change of two long range orders when t h e ferromagnetism is weak enough (i.e. small saturation moment and relatively low transition t e m p e r a t u r e TF). Careful t h e r m o dynamic and transport measureinents near t h e superconducting onset t e m p e r a t u r e T are highly desirable. C • Especially t h e specific heat e x p e r i m e n t near T is
(d)
Fig.2 Possible schematic phase diagrams near Kc" = I and F,~ = 0.8 ir, t h e ~ vs F,~ plane. T h e dotted line corresponds tobthe bo~undary of the equal ground s t a t e energies ESC =.E. FM" The Iollowing successive phase transLnons take place in (I): Normal -7 FM -7 SC, and in (II): Normal ~* SC -> FM.
Tc
Tc
TT
Fig.3 Possible t e m p e r a t u r e d e p e n d e n c e s of t h e two order p a r a m e t e r s , which corresponds to (I) in Fig.2, when a first order transition is assumed to take place at T = T' C" •
g:
crucial to clariiy t h e nature of t h e phase transitions, t h a t is, il the second order transition is realized at T , then we can a n t i c i p a t e the c o e x i s t e n c e of the it~Cnerant ferromagnetism and superconductivity in lower temperatures. From t h e above argument we can e x p e c t t h a t other existing weak f e r r o m a g n e t s might also exhibit superconductivity at lower t e m p e r a t u r e s and t h a t certain superconductors are r e e n t r a n t t o normal ferromagnets.
902
ITINERANT FERROMAGNETISM
# ~D SUPERCONDUCTIVITY
Vol. 43, No.
12
REFERENCES ~° 2.
3.
t,. 5. 6.
7. 8. 9.
10.
E.Gratz and E.P.Wohlfarth, 3. Magn. Magn. Mat. 15-18 903 (1980). E ~ ' ~ t z , H. R. Kirchmayr, V. Sechovsky and E. P. Wohlfarth, 3. Magn. Magn. Mat. 2.~_1 191 (1980). A. Kolodziejczyk, B. V. B. Sarkissian and 15. R.
Coles, 3. Phys. F 10 L333 (1980). E.Gratz, 3. Magn. , ~ g n . Mat. 24 1 (1981). 3.Sebek, 3.Stehno, V.Sechovs-~y and E.Gratz, Solid State Commun. 40 457 (1981). E.Gratz, 3;O.Strom-O'~en and M.3.Zuckerman, Solid State Commun. 40 833 (1981). W.Cheng, G.Creuzet,--lS.Garoche, LA.Campbell and E.Gratz, 3. Phys. F 12 475 (1982). S. Ogawa, private communication. The problem of the interplay between itinerant magnetism and superconductivity is reviewed by E. P. Enz in Superconductivity in d- and f-band metals ed. H. 'Suhi and' ~{."B.-'~{ap-p-p-~ ~-~"d~m~'c, New York, 1980) p.l$1. See for review, M.B~Maple, 3. de Phys (Paris). C6 1374 (1978), M.Ishikawa, ~.Fischer and 3"~uller, ibid 1379.
II. 12. 13. 14. 15. 16. 17.
See for review, S. S. P. Parkin, M. Ribault, D.3erome and KdSechgaard, 3. Phys. C 14 5305 (1981). P.Moreland P.W.Anderson,Phys. Rev. 125 1263 (1962). K. Machida, K. Nokura and T. Matsubara, Phys. Rev. B 22 2307 (1980). K~lachi~'a, 3. Phys. Soc. 3pru 50 2195 (19gl) and 51 1 4 2 0 (1982). K. Mac~ida and T. MatsuS-ara, ibid 50 3231 (1981). K~lachida, subm]~ed to 3. Phys. So(:. 3pn. See /or example, M. L. Cohen, in Superconducti~it~ ed. R. D. Parks (Marcel Dekker, New York, 1969) Chap.12. In fact in the rare earth ternary compound ErRh4B. the coexistence of ferromagnetism and superconductivity is found in neutron scattering experiment: See S. K. Sinha, G. W. Crabtree, D. G. Hinks and H. Mook, Phys. Rev. Lett. 48 950 (1982).