Coexistence of superconductivity and magnetic order

Solid State Communications, Printed in Great Britain.

Vol.51,No.3,

0038-1098/84

$3.00 + .00 Pergamon Press Ltd.

1984.

pp.135-138,

COEXISTENCE OF SUPERCONDUCTIVITY AND MAGNETIC ORDER

J. Ashkenazi,

C.G. Kuper, M. Revzen, A. Ron and D, Schmeltzer

Department of Physics, Technion, Haifa 32000, Israel (Received 2 March 1984 by S. Alexander) The microscopic theory of simple antiferromagnetic superconductors is extended to structures where the magnetic order is described by a spindensity wave. For the case where the magnetic system is inherently ferromagnetic, the free energy of the combined system is minimized with respect to the amplitude and the wave vector of the spin-density wave. It is found that the wave vector depends very weakly on temperature. We apply the theory to the coexistence region in ErRh4B4, finding a firstorder re-entry transition.

Several ternary compounds are knownl,which despite a high concentration of (localized)magnetic moments, become superconducting 2 (SC) at low temperatures. When the magnetic ions order below the SC transition temperature Tcl, the SC is generally weakened. When the order is antiferromagnetic 2 (AF), the weakening of SC can be understood in terms of the electron-ion exchange coupling 3'~. But when the ground state of the magnetic lattice is ferromagnetic 2 (FM), SC may be quenched -- the phenomenon of re-entry 5. In ErRh, B~ , there is a region close to the reentry temperature Tc2 where SC coexis~with some kind of magnetic order: probably ~ a linear spindensity wave 2 (SDW). Current theoretical analyses of the coexistence region in re-entrant SC's have been phenomenological 7,8, based on a Ginzburg-Landau (GL) approach (which includes the effect of the magnetic induction ~ on the orbital motion of the Cooper pairs). However, the GL theory only holds if ~ changes slowly on the scale of the SC coherence length ~ . But experiment s shows that near TO2 , ErRh4B ~ is Type I in its a-b plane; i.e. the penetration depth ~L is shorter than ~ . It therefore seems natural to make the opposite assumption, i.e. to assume a linear SDW of wavelength 2w/Q short compared with the Cooper-pair radius ~. That the material can tolerate such a short-wavelength structure was first suggested by Anderson and Suhl I° (AS). We shall generalize the method of refs. 3,4 (where ~ = 1/2 ~, with ~ a reciprocal lattice vector) to an SDW of arbitrary~, which for the moment we regard as externally imposed; later we find its amplitude and wave vector semi-phenomenologically. Well above the magnetic ordering temperature Tm, we shall find that there is no SDW solution, and the system will be a paramagnetic SC with uncorrelated spins. For T >> T m , the Abrikosov-Gorkov 11 (AG) theory holds and the observed T c is, therefore, already renormalized by AG scattering. Near Tm , we may no longer use the AG theory, but Ramakrishnan and Varma 12 show that the pair breaking is normally insensitive to spin-spin correlations. We shall therefore not take explicit account of the AG mechanism.

The s y s t e m i s d e s c r i b e d =

+ H.

*

H

*

H

by the H a m i l t o n i a n

+ H

H He P s e-p e-s , w h e r e t h e s p i n - s p i n H a m i l t o n i a n Hs and t h e p h o n o n H a m i l t o n i a n Ho w i l l n o t be w r i t t e n

(1)

explicitly. The electron Hamiltonian and the electron-phonon and electron-spin terms are respedtively He

:

,

He-; )<[b(k-k ) + b (k'-k)]

He_s=

,

(2b)

~- J(q)S[q) "so<~,c~, LK-q) cff(~) • , C)', 0 "l

(2c)

Here cq(~), b(~), are, respectively, annihilation operators for conduction electrons and

phonons, EC~) are the electron energies,g(~,~') is the electron-phonon coupling, JC~) is the Fourier component of the exchange interaction between conduction electrons and localized spins, ~(~) is the ~ component of the spin operator and ~ , , are Pauli matrices. We assume that the magnetic ions are aligned in an easy direction (the z-direction) and are almost static 13. For a sinusoidal SDW of wave vector ~, f2c) simplifies to H =

½J~)s

zl

~

-.

e-s

where S = , We apply the canonical transformation ~ , i% :

:

cos +&) -~[%&+Q)

cos

_

sin~+(~) + %(~-Q) sin~ (~)], -

(3)

to diagonalize H e + H e _ to first order in J(~)S. (In the de_~engrate regions (where E(~) - E(]~ , ~) < J(Q)S) , the diagonalization is only to zero order, but since they constitute only a fraction O(J(Q)S) of the Brillouin zone, we preserve overall accuracy to first order.) In (3), tan 2 e + ~ ) = J(~Q)S/[E(~) - E([ ± Q)]. Under this transformation, Eq. (2b) takes the form: 135

136

where y : 8~J(O)v(g - l)/~BgaB. Minimizing with respect to Q, and substitutlng the optimal value in (i0) gives Q = {If~] I/F21MQI}¥ 3 , and

X{b(~-~") + bt(~ ' - ~ ) } + h.c.}]

+O(J((~)S) +~

+ bt"(~'-~-Q')']

_

o

3

fs --)fsl

(4)

where g and g are functxons of g and the angles ( ~ . ( ~ ) , ~ + ( ~ ) . The t r a n s f o r m e d e l e c t r o n - p h o n o n i n t e r a c t i o n (4) c o n t a i n s a r e d u c e d F r B h l i c h t e r m , and a s p i n - d e p e n d e n t Umklapp p a r t , w h i c h f u r t h e r weakens t h e e l e c t r o n p a i r i n g , s i n c e the phase factor g makes for electron-electron repulsion. After eliminating the phonons Is, the effective electron-electron interaction parameter is altered from its usual value g2(~,~,) to ggff(-~,~') = ~2(~,~,) _ ~ 2 ~ , ~ , ) . With the simplifying assumptions of isotropy of the conduction electrons and the absence of dispersion in the phonon spectrum or the electron-phonon interaction, we may ave~rage g~#f over the FS. If, f u r t h ~ , we take E(k) - E(~ ± Q~ to be linear in k near the FS (an approximation which is accurate for free electrons), the effective phonon-mediated electron-electron attraction h is renormalized, to order J(~)S, to: ~ 2 2 % = %kcFS = %kcFS

%{(E2-E])- I~E 2dE [1 + ( 'E 1

~

) 2]-12} 2 '

X/% = 1 - 41J(O)S~T/aBQI

+ O(J(O)S) z

IMQI

4/3'

1

~mlMql

+ ~

2

(ll)

+ ~1Bm ] M Q) 4 + 61 Ym [MQIS '

where A = I/Ff°1 • Minimizing also with respect to MQ yields t~e a~gebraic equation

-I%llMql

+ ~mlMql ~ + ~m]Mql" + 2A=~IMQI w

:

(12)

o.

We h a v e s o l v e d (12) n u m e r i c a l l y , t a k i n g S : Ym = 1 . 0 x 1 0 - l ° G - 4 , ~ = O, F = 10 -7 cm, f~ = " " _ 0 -i.0 X I0 5 erg cm - 3 ' e~m - ~ ( T m - T)ITm ' ~m0 =~ -S.S and where Tm is the "bare" Curie temperature. Fig. 1 shows a few selected solutions: in (a), = l. OxlO 4, in (b) ~ = 2.0xlO 4 , and in (c) = 1 . 2 x l O 4 cm-IG -I I

I

I

I

(a)

f~ I

!

f~

~

!

(S)

where E 2 and E~ are the extreme values of E(k) E(k±Q) when k~FS. If a zs the lattice constant ~G, and B is the width of the band containing the FS, we may take E 2 - E~ = aBQ/~ , whence 12

2/3

+ ~ A

-2 =

Vol. 51, No. 3

COEXISTENCE OF SUPERCONDUCTIVITY AND MAGNETIC ORDER

"~ f

? I= u

f

i

i !

0

~ 11

•

Coexistence i

I

1

! .-

7 6

I

(b)

B

~-

7

--_-" ~

. -1

E u

(6) 0

the superconducting free-energy density fs is proportional to ~%2 ; hence ~ it is reduced by the SDW : fs = f: (I-8~IJ(O)S/aBQI) + O(J(O)S)2"

;B I

Fig. i:

I~m)MqI~

YmlMql,,G ,

(9)

and the total free-energy density is f = fsM: f~. MQ is proportional ~3,~s to S = _~ = pBgS/v(g - l). Hence: 1 f : f:(1-Y IMQI/Q) + ~ (lO)

(am+F~q2)lMqla

1 Bm i~Q

1

IMQI~

8

i

! -o.5

I

i -0.3

6

IT-TIn) /Tm

+ ~

F2Q2))~Q)2

1

fo

7 -] -o.7

where t h e n o t a t i o n i s t h a t o f KRRH~, and w h e r e , f o l l o w i n g B e h r o o z i e~ a%. ~7, we h a v e i n c l u d e d an M6 t e r m . F o r an SDW, (8) r e d u c e s t o i 1 f m = 2 (C~m + + 4 +

7

I

C0exis. :, ~.p

-2

F m = fd3rf m = fd3r{21-~zmlM 1

~ml~l ~

I

;

(7)

The I/Q dependence is reminiscent of AS ~°. However, its origin is quite different; it arises from the SDW-induced modification of the electronic states, while AS treats the effect of a BCS condensation on the magnetic interactions. Moreover, unlike AS, we find a ~ n ~ J ~ dependence on the exchange integral J. We now relax the constraint that the SDW is externally imposed. We assume a phenomenological free-energy functional for the magnetic system:

+ ~

I

(C)

The (i) The phase is cm-IG -I

Free-energy density of the pure SC phase (f~), of the homogeneous 0 FM phase (fm), and of the SDW coexistence phase (f), and the wave vector Q of the SDW. In (a), (b), and (c) we have taken / (i.0, 2.0, 1.2)xlO cm-IG -I , r : s pectively. Stable regions are indicated by solid curves, and metastable regions by broken curves. Note the spinodal temperature,above which there is no SDW solution.

solutions show the following features: range of stability of the coexistence sensitive to ft. The choice ff = 1.2×104 reproduces the observed range of co-

Vol. 15, No. 3

COEXISTENCE OF SUPERCONDUCTIVITY AND MAGNETIC ORDER

existence in ErRh4B g quite well s . Taking IG,18 2/3x5.6 < g/(g - i) < 2/3x8.5 , this ~] corresponds to 0.8x10 -2 < J/B < 1.3x10 -2 , gCrange which is reasonable 19 since (J/B) 2 should be kTm/B ~ i0 -4 (ii) The wave vector ~ is insensitive to temperature 2°. For .; = 1.2x104 cm-iG -I, its value, Q = 7x10 ~ cm -I, is close to the observed value s for ErRhgB4. (iii)The solution exhibits a first-order transition at the re-entry temperature Tc2, as well as another at the onset of coexistende T_. The latent heat at Tc2 is 0.31 J/mol, cf. Re~. i. The first-order transition 2~ at T s is not confirmed experimentally. (iv) A modest increase in ~ will destroy coexistence entirely. HoMo6S 8 seems to be on the borderline (cf. Lynn6). On the other hand, a modest decrease can stabilize coexistence down to T = 0, i.e. even though the magnetic system alone would be FM, the SC can stabilize an SDW. This possibility may perhaps characterize 22 HoMo~Se 8.

137 O

(v) At r e - e n t r y , fs = 1/2 fs , i,e. although o u r e x p a n s i o n i n J ( 0 ) S w i l l n o t be q u a n t i t a t i v e , it should still be q u a l i t a t i v e l y correct. To s u m m a r i z e , we p r e s e n t a u n i f i e d t h e o r y of superconductivity in the presence of a weakly magnetic lattice, applicable whether the lattice a l o n e w o u l d b e FM o r /IF. The t h e o r y i s s e n s i t i v e to only the one (lumped) parameter J. For suitable ~ , it allows a direct tragsition from SC t o FM, a t r a n s i t i o n f r o m SC t o FM v i a an intermediate SDW p h a s e o f c o e x i s t e n c e , o r an SDW s t a b l e down t o T = 0. The c h o i c e ~ = 1 . 2 x 1 0 4 cm-lG -1 f i t s t h e c o e x i s t e n c e p h a s e i n ErRh B4 A c k n o w l e d g e m e n t -- C.G.K. a n d M.R. a c k n o w l e d g e s u p p o r t f r o m t h e Fund f o r P r o m o t i o n o f R e s e a r c h a t T e c h n i o n , M.R. a n d A.R. s u p p o r t f r o m Kernforschungszentrum Karlsruhe, and J.A. and D.S. s u p p o r t f r o m t h e U . S . - I s r a e l Binational Science Foundation.

References i.

2.

3.

4.

See, e.g., M.B. Maple, Yeccna~ Sc~peTtcondcte2~o~cs, Ed. G.M. Shenoy, B.D. Dunlap and F.Y. Fradin, Amsterdam: North Holland (1981), p. 131; M. Ishikawa, ibid., p.43, and Contemp. Phys. 23, 443 (1982). We use the following abbreviations: AF for antiferromagnet (-ic) , (-ism) ; FM for ferromagnet (-ic), (-ism), SC for superconductor, (-ing), (-ivity) ; SDW for spindensity wave, and FS for Fermi surface. G. Zwicknagl and P. Fulde, Z. Phys. B 43, 23 (1981); see also P.G. de Gennes and G. Sarma, J. Appl. Phys. 34, 1830 (1963), where these effects are discussed qualitatively. J. Ashkenazi, C.G. Kuper and A. Ron, Phys.

9.

i0. Ii.

12.

Rev. B 2S, 418 (1983). 5.

6.

7.

8.

W.A.Fertig, D.C. Johnston, L.E. Delong, R.W. McCallum, M.B. Maple and B.T. Matthias, Phys. Rev. Lett. 38, 987 (1977); M. Ishikawa and ~)Fischer, Solid State Commun. 23, 37 (1977); S.K. Sinha, G.W. Crabtree, D.G. Hinks and H. Mook, Phys. Rev. Lett. 43, 950 (1982). D.E. Moncton, G. Shirane and W.Tomlinson, J. Mag. & Mag. Mat. 14, 172 (1979); J.W. Lynn, Ternary S~perc~ducto~, Ed. G.K. S h e n o y , B.D. D u n l a p a n d F.Y. F r a d i n , Amsterdam: North Holland (1981), p. 51. E . I . B l o u n t a n d C.M. Varma, P h y s . Rev. L e t t . 4 2 , 1079 ( 1 9 7 9 ) ; H . S . G r e e n s i d e , E . I . B l o u n t a n d C.M. Varma, P h y s . Rev. L e t t . 466, 49 ( 1 9 8 0 ) ; C.G. K u p e r , M. R e v z e n a n d A . R o n , P h y s . Rev. L e t t . 4 4 , 1545 ( 1 9 8 0 ) , S o l i d S t a t e Commun. 36, 533 ( 1 9 8 0 ) ; M. T a c h i k i , ft. M a t s u m o t o , T. Koyama a n d H. Umezawa, S o l i d S t a t e Commun. 3_44, 19 ( 1 9 8 0 ) , a n d references contained therein; C.-R. Hu and T.-E. Han, Physica 108, B, 1041 (1981). C.G. Kuper, M. Revzen, A. Ron and C.-R. Hu, Phys. Rev. B (1983), hereafter KRRH. KRRH o o conclude that the parameter ~ * fm/fs lies in the range 3 < ~ < 17, exhibiting a difficulty of the GL-based theories where re-entry from SDW requires ~ >> i00.

13.

14.

15.

16.

17.

G.W. Crabtree, F. Behroozi, S.A. Campbell and D.G. Hinks, Phys. Rev. Lett. 49, 1342 (1982); F. Behroozi, G.W. Crabtree, S.A. Campbell and D.J. Hinks, Phys. Rev. B 27, 6849 (1982). In view of the strong anisotropy of ErRh~B~ revealed in this work, we assume that the vector ~ a l w a y s points along an easy direction. P.W. Anderson and H. Suhl, Phys. Rev. 116, 896 (1959). A.A. Abrikosov and L.P. Gorkov, Zh. Eksp. i Teor. Fiz. 39, 1781 (1960), Sov. Phys. JETP 12, 1243 (1961). T.V. Ramakrishnan and C.M. Varma, Phys. Rev. B 24, 137 (1981). They show that if FS nesting is strong, this conclusion is modified. In such exceptional cases, AF can even strengthen SC. However, for a reasonably long SDW, the I/Q in Eq. (7) guarantees that the present effect will dominate. Since 2~/Q >> a (cf. Re~. 16), we have put J(O) in place of J(Q) in Eq. (6) et seq. Near Tm, where magnetic fluctuations become important, we must replace the operator Sz(~) not by its expectation value S, but by a fluctuating moment. The exact diagonalization for free electrons will involve Mathieu functions, unlike Refs. 3,4. The extra complication arises from the inequivalence of the wave vectors ~ - ~ ~ n d ~ + ~ w h e n ~ # (i/2)~. However, if J(Q)S is small, the degenerateperturbation-theoretical treatment given in the text is adequate. H. F~6hlich, Proc. Roy. Soc. A 215, 291 (1952). Note that electron mass renormalization is determined b y e 2 + 72 (= g2 for isotropic electrons). For ErRh4B4, a = 5.29 A; J.M. Vandenberg and B.T. Matthias, Proc. Nat. Acad. Sci. 71, 1336 (1977). F. Behroozi, G.W. Crabtree, S.A. Campbell, M. Levy, D.R. Snider, D.C. Johnston and B.T. Matthias, Solid State Commun. 39, 1041 (1981).

138 18.

19.

20.

COEXISTENCE OF SUPERCONDUCTIVITY AND MAGNETIC ORDER There is experimental ambiguity regarding the gyromagnetic ration g. While the spin of Er +++ is 3/2, from Hund's rule, its magnetic moment sometimes appears as 5.6 ~B, and sometimes as 8.5 ~B (see e.g. Ref. 9). T. Jarlborg, A.J. Freeman and T.J. WatsonYang, Phys. Rev. Lett. 39, 1032 (1977); O.K. Andersen, W. Klose and H. Nohl, Phys. Rev. B 17, 1209 (1978). F. Behroozi (private communication) has suggested that commensurability locking may occur; in the direction with Miller indices (7,0,5), the half wavelength ~/Q is close to the lattice period.

21.

22.

Vol. 51, No. 3

Near T m, where fluctuations are large, the replacement of S by a fluctuating moment (cf. footnote 13) will tend to "smear out" the transition at T s. Inhomogeneous strains may act similarly (M. Tachiki, G.W. Crabtree and B.D. Dunlap, preprint) . J.W. Lynn, J.A. Gotaas, R.W. Erwins, R.A. Ferrell, J.K. Bhattacharjee, R.N. Shelton and P. Klavins, Phys. Rev. Lett. 52, 133 (1984). Q is a slowly-increasing function of T, as our theory would predict (cf. Fig. la).