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Superconductivity and antiferromagnetism in heavy electron systems
Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 507 509 North-Holland, Amsterdam
SUPERCONDUCTIVITY
507
AND ANTIFERROMAGNETISM
IN HEAVY ELECTRON SYSTEMS
Kazuo U EDA Institute of Materials Science. University of Tsukuba, Tsukuba 305, Japan
and Rikio K O N N O Institute for Solid State Physics, Universi O' of Tokyo, Roppongi. Tol
The coexistence of superconductivity and magnetism in heavy electron systems is an interesting problem. Compared with magnetic superconductors, new feartures are, firstly, the same f-electrons are responsible to both phenomena and, secondly, the ordered moments are very small. The first example was URu 2Si2 [1-4]. The upper transition at T N = 17.5 K is to antiferromagnetic order and below T~ = 1.5 K superconductivity coexists with it. The antiferromagnetic moment is very small, (0.03 + 0.01)~ B. Recent neutron experiments [5] reported similar results for UPt s, Ty = 5 K, T~, = 0.5 K and the ordered moment is also small, ( 0 . 0 2 + 0 . 0 1 ) ~ . Although such a tiny moment itself is an interesting problem, here we use it as a fact. Then we can use a G i n z b u r g - L a n d a u expansion for both order parameters. An advantage of this approach is that we can use group-theoretical arguments. We start from residual interactions between quasiparticles in real space. As an example we consider the simple cubic lattice and nondegenerate orbitals. The first term is the on-site interaction
Von, rni * .
(1)
We expect that V0 is positive (repulsive) for heavy electron systems. For the nearest neighbor pairs, there are couplings between charge densities
Vini+ani
(2)
as well as spin densities
Jis,+~'s,.
(3)
We note that the couplings mediated by spin fluctuations are of this form.
In general, from given couplings, pairing interactions for Cooper pairs are obtained in the form of 1
l~vc,
vl',"?'V/,
1 '4"4t, + N"
3
E E l~Yt, =/
" " ' " + ' .~yi, "'J'I Vl,)..C,,y/.(
(4)
where the first term is for singlet pairings and the second term for triplet pairings. /" denote irreducible representations and ")'r stand for their basis functions. Vv is the coupling constant for the irreducible representation F. ,1, "~) is a field operTy/" ator for singlet Cooper pairs add ~P(~,I are field operators ( j = 1, 2, 3) for triplet Cooper pairs. For the gap matrix A(k), we use the standard 4-vector notation
-d,(k)+id2(k)
A(k)= -ido(k)+d3(k)
ido( k ) + d3( k ) ) d,(k) + i d 2 ( k ) " (5)
The order parameter di(k of the field operators
) is given by the average
Vc( +r,)e~r,(k). (i>
d , ( k l = ~ E1E
(61
I" Y/'
where ~v,.(k) is the basis function of the irreducible representation. For the example of the simple cubic lattice with up to the nearest neighbor couplings, eqs. (1)-(3), we have two one-dimensional representations, A 1, and one two-dimensional representation, E, for singlet pairings and one three-dimensional representation, T, for triplet pairings. The coupling constants and the basis functions are tabulated in table 1. We use the lattice constant as the unit of
0304-8853/88/$03.50 © Elsevier Science Publishers B.V.
508
K. Ueda, R. Konno / Superconductivi(v and antiferromagnetism
Table 1 Irreducible representations of Cooper pairs in the simple cubic lattice with up to the nearest neighbor couplings
nearest n e i g h b o r h o p p i n g . In general we expect that they lie in the range
I"
Vr
T 2>Ti>Tv.
AI AL E T
Yr
Ov~-
V0 Vi - ]J1 Vi - ]Jt
u
1 2~2/3 (cos k 1+cos k 2 +cos k3) (1/~/3)(cos k 1+cos k 2 2 cos k3)
U
COS k 1 - - COS k
V 1 + t4J I
x
f2sin k t
y z
~2sin k 2 (2sin k 3
length, a = 1, a n d k = ( k t, k 2, k3). T h e first one of the two A~ r e p r e s e n t a t i o n s is the c o n v e n t i o n a l singlet s-wave superconductivity. T h e second A 1 is the so-called e x t e n d e d s-wave a n d the E representation c o r r e s p o n d s to a d-wave (de) a n d T to a p-wave. W h e n a n t i f e r r o m a g n e t i c fluctuations are d o m i n a n t , ,/1 is positive and we expect either the extended s-wave or the d-wave state. If the s p i n - o r b i t coupling is strong we have to include it to discuss the T - r e p r e s e n t a t i o n a n d this has been d o n e in refs. [6-8]. The present forms of the basis functions are c o m p a t i b l e with the t r a n s l a t i o n of the lattice. W e would like to p o i n t out that q~v,,(k) for the second A1, E a n d T change their sign when k is shifted by the a n t i f e r r o m a g n e t i c wave vector Q = (,~, ,~, 4 ) [9]. The coupling term between the two o r d e r p a r a m e t e r s is derived either b y e x p a n d i n g the gap function in terms of the staggered m o m e n t or by e x p a n d i n g the staggered susceptibility b y the s u p e r c o n d u c t i n g o r d e r p a r a m e t e r . Of course the two m e t h o d s give the same result: r~,+, = (7, + V 2 ) ( d * ( k ) . d ( k ) + d,~'( k ) d o ( k ) } B Q " BQ
+Tz(d,~" ( k + Q ) d o ( k ) ) n o . s
2
(8)
where TF* is the effective F e r m i temperature. In the h e a v y f e r m i o n systems, TF* is small, typically ten or t w e n t y degrees. T h e r e f o r e we can expect sizable interference p h e n o m e n a between antiferrom a g n e t i s m a n d s u p e r c o n d u c t i v i t y in heavy fermion systems. F r o m the e q u a t i o n we can see that the o r d i n a r y s-wave state is the most destructive a n d the a n i s o t r o p i c s- or d-wave state is the least destructive since d , ( k + Q ) = - d , ( k ). The lack of time reversal s y m m e t r y shows up firstly in the positiveness of 71 + Y2 a n d secondly for the p-wave as an a n i s o t r o p y for the d-vector. F o r the p-wave state the spin o r b i t c o u p l i n g is i m p o r t a n t to discuss the relative stability b u t in general it is in the m i d d e l of the a b o v e two cases. The n e u t r o n e x p e r i m e n t s of U P t 3 [5] r e p o r t e d that the staggered m o m e n t s t o p p e d grow below T~. The analysis of the G L free energy reveals that in this case the c o u p l i n g c o n s t a n t Y is of the o r d e r of (TcTN) 1, j u s t m i d d l e of the range given b y eq. (8), since T N ad TF* are the same o r d e r of magnitude. O n e of the c o n s e q u e n c e s of such analysis is that the j u m p of the specific heat at T N should be small, less than the ratio T J T N - 0.1. A m y s t e r y of the a n t i f e r r o m a g n e t i s m of U P t 3, if it is true, is that no a n o m a l y at T N is r e p o r t e d in specific heat data. The p r e s e n t result can be a possible e x p l a n a tion for that since such a tiny a n o m a l y m a y be s m e a r e d out by small a m o u n t of impurities. On the o t h e r h a n d , for U R u 2 S i 2 a big X-type a n o m a l y is o b s e r v e d at T N. F r o m the fact it can be conc l u d e d that the c o u p l i n g is - ( T N) 2 which means it lies a l m o s t in the lower end of the region, eq. (8). In this case we can not expect a significant interference effect. As a result the staggered mom e n t will c o n t i n u e to grow below T,,.
~
+v2((d*(k +
-(d*(k+Q)×ne).(d(k)×ne) )
References
(7)
where BQ is the staggered exchange field a n d ( - - - ) m e a n s the average on the F e r m i surface. The expressions of the coefficients Y1, 72 will be p u b l i s h e d elsewhere [10,11] a n d they can be calculated analytically for a b i p a r t i t e lattice with a
[1] W. Schlabitz, J. Baumann, B. Pollit, U. Rauchschwalbe, H.M. Mayer, U. Ahlheim and C.D. Bredl, Z. Phys. B62 (1986) 171. [2] T.T.M. Palstra, A. Menovsky, J. van den Berg, A.J. Dirkmaat, P.H. Kes, G.J. Nieuwenhuis and J.A. Mydosh, Phys. Rev. Lett. 55 (1985) 2727. [3] M.B. Maple, J.W. Chen, Y. Dalichaouch, T. Kohara, C.
K. Ueda, R. Konno / Superconductivity and antiferromagnetisrn Rossel, M.S. Torikachvili, M.W. McElfresh and J.D. Thompson, Phys. Rev. Lett. 56 (1986) 185. [4] C. Broholm, J.K. Kjems, W.J.L. Buyers, P. Matthews, T.T.M. Palstra, A. Menovsky and J.A. Mydosh, Phys. Rev. Lett. 58 (1987) 1467. [5] G. Aeppli, E. Bucher, C. Broholm, J.K. Kjems, J. Baumann and J. Hufnagl, Phys. Rev. Lett. 60 (1988) 615. [6] G.E. Volovik and L.P. Gor'kov, Pis'ma Zh. Eksp. Teor. Fiz. 39 (1984) 550.
509
[7] K. Ueda and T.M. Rice, Phys. Rev. B31 (1985) 7114. [8] E.I. Blount, Phys. Rev. B32 (1985) 2935. [9] M. Kato, K. Machida and M. Ozaki, J.J.A.P. Suppl. 26-3 (1987) 1245. [10] R. Konno and K. Ueda, unpublished. [11] For related work of magnetic superconductors, see for example P. Fulde and J. Keller, in: Superconductivity in Ternally Compounds (Springer-Verlag, Berlin, 1982).
SUPERCONDUCTIVITY
507
AND ANTIFERROMAGNETISM
IN HEAVY ELECTRON SYSTEMS
Kazuo U EDA Institute of Materials Science. University of Tsukuba, Tsukuba 305, Japan
and Rikio K O N N O Institute for Solid State Physics, Universi O' of Tokyo, Roppongi. Tol
The coexistence of superconductivity and magnetism in heavy electron systems is an interesting problem. Compared with magnetic superconductors, new feartures are, firstly, the same f-electrons are responsible to both phenomena and, secondly, the ordered moments are very small. The first example was URu 2Si2 [1-4]. The upper transition at T N = 17.5 K is to antiferromagnetic order and below T~ = 1.5 K superconductivity coexists with it. The antiferromagnetic moment is very small, (0.03 + 0.01)~ B. Recent neutron experiments [5] reported similar results for UPt s, Ty = 5 K, T~, = 0.5 K and the ordered moment is also small, ( 0 . 0 2 + 0 . 0 1 ) ~ . Although such a tiny moment itself is an interesting problem, here we use it as a fact. Then we can use a G i n z b u r g - L a n d a u expansion for both order parameters. An advantage of this approach is that we can use group-theoretical arguments. We start from residual interactions between quasiparticles in real space. As an example we consider the simple cubic lattice and nondegenerate orbitals. The first term is the on-site interaction
Von, rni * .
(1)
We expect that V0 is positive (repulsive) for heavy electron systems. For the nearest neighbor pairs, there are couplings between charge densities
Vini+ani
(2)
as well as spin densities
Jis,+~'s,.
(3)
We note that the couplings mediated by spin fluctuations are of this form.
In general, from given couplings, pairing interactions for Cooper pairs are obtained in the form of 1
l~vc,
vl',"?'V/,
1 '4"4t, + N"
3
E E l~Yt, =/
" " ' " + ' .~yi, "'J'I Vl,)..C,,y/.(
(4)
where the first term is for singlet pairings and the second term for triplet pairings. /" denote irreducible representations and ")'r stand for their basis functions. Vv is the coupling constant for the irreducible representation F. ,1, "~) is a field operTy/" ator for singlet Cooper pairs add ~P(~,I are field operators ( j = 1, 2, 3) for triplet Cooper pairs. For the gap matrix A(k), we use the standard 4-vector notation
-d,(k)+id2(k)
A(k)= -ido(k)+d3(k)
ido( k ) + d3( k ) ) d,(k) + i d 2 ( k ) " (5)
The order parameter di(k of the field operators
) is given by the average
Vc( +r,)e~r,(k). (i>
d , ( k l = ~ E1E
(61
I" Y/'
where ~v,.(k) is the basis function of the irreducible representation. For the example of the simple cubic lattice with up to the nearest neighbor couplings, eqs. (1)-(3), we have two one-dimensional representations, A 1, and one two-dimensional representation, E, for singlet pairings and one three-dimensional representation, T, for triplet pairings. The coupling constants and the basis functions are tabulated in table 1. We use the lattice constant as the unit of
0304-8853/88/$03.50 © Elsevier Science Publishers B.V.
508
K. Ueda, R. Konno / Superconductivi(v and antiferromagnetism
Table 1 Irreducible representations of Cooper pairs in the simple cubic lattice with up to the nearest neighbor couplings
nearest n e i g h b o r h o p p i n g . In general we expect that they lie in the range
I"
Vr
T 2>Ti>Tv.
AI AL E T
Yr
Ov~-
V0 Vi - ]J1 Vi - ]Jt
u
1 2~2/3 (cos k 1+cos k 2 +cos k3) (1/~/3)(cos k 1+cos k 2 2 cos k3)
U
COS k 1 - - COS k
V 1 + t4J I
x
f2sin k t
y z
~2sin k 2 (2sin k 3
length, a = 1, a n d k = ( k t, k 2, k3). T h e first one of the two A~ r e p r e s e n t a t i o n s is the c o n v e n t i o n a l singlet s-wave superconductivity. T h e second A 1 is the so-called e x t e n d e d s-wave a n d the E representation c o r r e s p o n d s to a d-wave (de) a n d T to a p-wave. W h e n a n t i f e r r o m a g n e t i c fluctuations are d o m i n a n t , ,/1 is positive and we expect either the extended s-wave or the d-wave state. If the s p i n - o r b i t coupling is strong we have to include it to discuss the T - r e p r e s e n t a t i o n a n d this has been d o n e in refs. [6-8]. The present forms of the basis functions are c o m p a t i b l e with the t r a n s l a t i o n of the lattice. W e would like to p o i n t out that q~v,,(k) for the second A1, E a n d T change their sign when k is shifted by the a n t i f e r r o m a g n e t i c wave vector Q = (,~, ,~, 4 ) [9]. The coupling term between the two o r d e r p a r a m e t e r s is derived either b y e x p a n d i n g the gap function in terms of the staggered m o m e n t or by e x p a n d i n g the staggered susceptibility b y the s u p e r c o n d u c t i n g o r d e r p a r a m e t e r . Of course the two m e t h o d s give the same result: r~,+, = (7, + V 2 ) ( d * ( k ) . d ( k ) + d,~'( k ) d o ( k ) } B Q " BQ
+Tz(d,~" ( k + Q ) d o ( k ) ) n o . s
2
(8)
where TF* is the effective F e r m i temperature. In the h e a v y f e r m i o n systems, TF* is small, typically ten or t w e n t y degrees. T h e r e f o r e we can expect sizable interference p h e n o m e n a between antiferrom a g n e t i s m a n d s u p e r c o n d u c t i v i t y in heavy fermion systems. F r o m the e q u a t i o n we can see that the o r d i n a r y s-wave state is the most destructive a n d the a n i s o t r o p i c s- or d-wave state is the least destructive since d , ( k + Q ) = - d , ( k ). The lack of time reversal s y m m e t r y shows up firstly in the positiveness of 71 + Y2 a n d secondly for the p-wave as an a n i s o t r o p y for the d-vector. F o r the p-wave state the spin o r b i t c o u p l i n g is i m p o r t a n t to discuss the relative stability b u t in general it is in the m i d d e l of the a b o v e two cases. The n e u t r o n e x p e r i m e n t s of U P t 3 [5] r e p o r t e d that the staggered m o m e n t s t o p p e d grow below T~. The analysis of the G L free energy reveals that in this case the c o u p l i n g c o n s t a n t Y is of the o r d e r of (TcTN) 1, j u s t m i d d l e of the range given b y eq. (8), since T N ad TF* are the same o r d e r of magnitude. O n e of the c o n s e q u e n c e s of such analysis is that the j u m p of the specific heat at T N should be small, less than the ratio T J T N - 0.1. A m y s t e r y of the a n t i f e r r o m a g n e t i s m of U P t 3, if it is true, is that no a n o m a l y at T N is r e p o r t e d in specific heat data. The p r e s e n t result can be a possible e x p l a n a tion for that since such a tiny a n o m a l y m a y be s m e a r e d out by small a m o u n t of impurities. On the o t h e r h a n d , for U R u 2 S i 2 a big X-type a n o m a l y is o b s e r v e d at T N. F r o m the fact it can be conc l u d e d that the c o u p l i n g is - ( T N) 2 which means it lies a l m o s t in the lower end of the region, eq. (8). In this case we can not expect a significant interference effect. As a result the staggered mom e n t will c o n t i n u e to grow below T,,.
~
+v2((d*(k +
-(d*(k+Q)×ne).(d(k)×ne) )
References
(7)
where BQ is the staggered exchange field a n d ( - - - ) m e a n s the average on the F e r m i surface. The expressions of the coefficients Y1, 72 will be p u b l i s h e d elsewhere [10,11] a n d they can be calculated analytically for a b i p a r t i t e lattice with a
[1] W. Schlabitz, J. Baumann, B. Pollit, U. Rauchschwalbe, H.M. Mayer, U. Ahlheim and C.D. Bredl, Z. Phys. B62 (1986) 171. [2] T.T.M. Palstra, A. Menovsky, J. van den Berg, A.J. Dirkmaat, P.H. Kes, G.J. Nieuwenhuis and J.A. Mydosh, Phys. Rev. Lett. 55 (1985) 2727. [3] M.B. Maple, J.W. Chen, Y. Dalichaouch, T. Kohara, C.
K. Ueda, R. Konno / Superconductivity and antiferromagnetisrn Rossel, M.S. Torikachvili, M.W. McElfresh and J.D. Thompson, Phys. Rev. Lett. 56 (1986) 185. [4] C. Broholm, J.K. Kjems, W.J.L. Buyers, P. Matthews, T.T.M. Palstra, A. Menovsky and J.A. Mydosh, Phys. Rev. Lett. 58 (1987) 1467. [5] G. Aeppli, E. Bucher, C. Broholm, J.K. Kjems, J. Baumann and J. Hufnagl, Phys. Rev. Lett. 60 (1988) 615. [6] G.E. Volovik and L.P. Gor'kov, Pis'ma Zh. Eksp. Teor. Fiz. 39 (1984) 550.
509
[7] K. Ueda and T.M. Rice, Phys. Rev. B31 (1985) 7114. [8] E.I. Blount, Phys. Rev. B32 (1985) 2935. [9] M. Kato, K. Machida and M. Ozaki, J.J.A.P. Suppl. 26-3 (1987) 1245. [10] R. Konno and K. Ueda, unpublished. [11] For related work of magnetic superconductors, see for example P. Fulde and J. Keller, in: Superconductivity in Ternally Compounds (Springer-Verlag, Berlin, 1982).