Lowest energy Cooper pairing in the presence of antiferromagnetism

Solzd State Communzcatzons, Vol 54,No 7, pp 633-637, 1985 Prznted zn Great Brztaln

0038-I098/85 $3 00 + .00 Pergamon Press Ltd

LOWEST ENERGY COOPER PAIRING IN THE PRESENCE OF ANTIFERROMAGNETISM E.W. Fenton National Research Council of Canada, Ottawa, Canada KIA OR6 (Received

8 February 1985 by R. Barrie)

Cooper pairing of electron eigenstates of an antiferromagnet involves considerable complexity in spin space and in phases of the order parameters. With a BCS interaction the pairing scheme with lowest free energy has anisotroplc spln-matrix order parameter ~(~), however ~+(k)~(k) = IA12~. Magnitude of the anlsotropic order parameter satisfies the simple BCS gap equation but with electron energies of the antiferromagnet elgenstates appearing instead of Bloch state energies of the nonmagnetic crystal.

Consider a Hamiltonian for an antlferromagnetic metal with a BCS pairing interaction

H

=

[ k_,~

~kC~:Ck~-

-

-

the antiferromagnetlsm, i.e. cosQ.r (sinQ.r) magnetism causes slnQ_-r (cosQ°_r) odd-parity

i ~

l

(~'--n)~(MkC~+Q~Ck~+M~c~aCk+Q~) -

k,~,~

VC~+q~C~,-q~,Ck,~,Cka k_,k'

sq

(1)

=- H O + H M + H V

trlplet-spln Cooper pairing amplitude to occur. 15 In all theories to date for Cooper pairing in scheme (a), +Q and -Q components of the Cooper pairing amplltude implicit in pairing have had opposite phases. In the following we obtain very different results by taking scheme (a) with +Q and -Q components of the Cooper pairing having the same phase. For convenience in the following, we denote these new results as pairing scheme (c). The spin-matrlx gap function ~c(k) is anlsotropic,

Cooper palrlng for this Hamiltonlan has been treated extensively in two pairing schemes (a) H O + H M is dlagonalized and then pairing of electron elgenstates of the -- -anti~erromagnet is subject to the pairing interaction V wlth H V re-expressed in (a,a +) fteld operators for the eigenstates, Hv(C,C+) + Hv(a,a+),l-5 and (b) pairlng of normal-metal Bloch states is subject to both the magnetism field--and~e pairing interaction. 6-II The spin-matrlx pair potential or gap function for scheme (a) is anisotropic in k and r space, Aa(k) = Aa(k)(1~.) , whereas in scheme (b) the pai~ potential ~ ( k ) = Ab(l~v) is isotropic. Recently we performed a model calculation of the free energy in schemes (a) and (b). 12 Surprisingly, free energy is higher for scheme (a) Cooper pairing although this pairing is apparently symmetry-adapted to the anti~erromagnetism. Investigations have shown that in both (a) and (b) pairing schemes, Cooper pair amplitude F is a mixture of slnglet-spin even-parlty and trlplet-spln odd-parity components. 12-15 (The pair potential is A = VF). This mixing occurs because the one-electron spinor in the anti~erromagnet varies periodically in space on the scale of the lattice constant and within the size of the Cooper pair, so the pair spinor is not an S = 0 or S = 1 eigenstate. 16 For sinusoidal antiferromagnetlsm and with an even-parity Vkk , = V interaction, the

with mixed parity and mixed pair spinor, however

A~(~)Ac(k) : IAc121

and the gap equation has a very simple form. It will be evident that pairing scheme (c) is matched to the symmetry of the Vkk, = V interaction in the gap equation.

This leaves little doubt that this theory describes the lowest-free-energy Cooper pairing for the Hamiltonlan of eq. I, however we present as well an explicit comparison of free energies in a model calculation. Because treatment of the Hamiltonlan in eq. I is controversial, we present explicitly the important steps in showing how ~c(k) results instead of ha(k) when phases of the magnetization and Cooper pairing, and the spln-space structure, are treated carefully. When 2Q = G (a reciprocal lattice vector of the host lattice), a canonical c-operator to a-operator transformation exactly diagonallzea H O + HM

slnglet-spln Cooper pairing amplitude component is uniform in space and the trlplet-spln amplitude component varies sinusoidally. The sinusoidal Cooper pairing is out of phase with

a+,_ka : UkCk= + ~(~'n)~Vk__Ck_+Q ~

633

LOWEST ENERGY COOPER PAIRING

634

a_,k =

=

-

~(~'n)Sav~ck_~ + u~_Ck+Q~

(2)

The reverse transformation is

c_].~o~ u~a+,k= =

Ck+Q a = l(=.n)~v~a ~ = -- v~ = +,kw~ + u k a - , k a

first term of HM represents -Q_magnetlzatlon and the second term represents +Q magnetization. These relations mean M_k = M~ and with u k = lUkl in eq. 4, V_k = v~. The magnetization may be expressed as--either cosQ.~ (M~ = M k and

~(~°£)~vka_,k~

-

Vol. 54, No. 7

(3)

With a = Ac and ~ = A - I ~ , ~ - 1 = ~ when lUkl2~+ I~k 12 = 1. P a r a m e t e r s o f t h e transforma-t'ion are

v~ = Vk) , or slnQ-r (M~ = -Mk--and ~ = -Vk), d~pendTng on the arbitrary origin of r-spa--ce. In representing HV in terms of (a,a +) operators, a+,k__~a+,_k__ = and a_,k_~a__k_ a pairs representing degenerate eigenstates o~ the antiferromagnet are most important. Using eqs. 3, the following cc pair combinations and their conjugates in Hv(c,c+ ) lead to the most important terms in the pairing part of Hv(a,a +)

Ck+Q~C_k..a + CkaC_k+Q..a = a l u k I ( v k* - v k )~+ a+_ , k a a -+,-k_-a + ( t e r m s i n a+a )

(7)

CkaC k_a + Ck+Q~C_k+Q_~ = (lUkl2 - IVkl2) ! a+,k_aa_+,_k_a + (terms in a+a¥)

(8)

,uki 2 = i -

l I ,~ki 2 = ~ + z ($~ + ,Mkl2)~

Mksign~ k UkVk = 2 -(~

f % ,+ ,Mk,2)~ ' Ck_ = ~1[ e k - g k + Q J

(4)

and H 0 + HM = k , +~, a

E.(k)a~ _ a. z - z,z_~ z,k__~

(5)

with I k + ~k+Q) ¥ ($~+,Mk,2)½ E+(k) = ~(e ~airing

=-

~1 V

Here we have specialized the antiferromagnetlsm polarization vector n to ~ with (g'~)~8 = ~6=8" Similar but algebrai~ally more complicated results are obtained for n = ~,~. We retain all electron scattering in ~v--which conserves momentum only modulo the reciprocal lattice vectors G of the host lattice, and only modulo the magnetlc reciprocal lattice vectors +Q including their spln-space structure. The magnetic vectors in the electron scattering are not optional but are required for self consistency since from eqs. 2 all terms of Hv(a,a+) conserve momentum only modulo the +Q magnetlc reciprocal lattice vectors. Uslng relations (7) and (8) in HV, retaining only a+a+ degenerate-palr terms, and then as usual reducing H V to a one-partlcle form using a quasiaverage for Cooper pairing, we obtain

(6)

~ [lUk, 12-1Vk,12 + =lUk, l(vk, - v *k,)][lUkl2-1Vkl2+~ lu k I ( v ~ - v k ) ] k,k' ,a . . . . .

x ~ ~ a+,kaa_+,_k_¢ + h.c. 1 = - ~

~ Ac(k)_a, a+,k a+,_k_~ + h.c. k,~ ,-+

The phase of u k is trivial if we require the (a,a +) represen~atlon to become the (c,c +) representation when Mk ÷ O, so that u k = u~ = lukl. In ~ both k and k + Q are

With v~ = vk, H~ alrlng is the same as scheme (a) expres~ions--ln the literature. With v~ = -v k so that +Q and -Q Cooper pairs represented-in oairlng have the same phase (eq. 7), H~atrlng is much dlfferent and

[ Ac(k)_=,~__ = V~,,±l [lUkl2-1Vkl2_ _ + 2=lUklV~][lUk, 12-1Vk, 12 + 2=lUk, lVk,] _a;,k, _

within the first Brillouin zone of the host lattice. Q modulo G with k < 0 means the first term of HM--represenFs +Qmagnetizatlon and the second term represents -Q (or Q-G) magnetization. In contrast, wlth k > 0 the

(9)

The Ao(k)_~ ~ with k-dependeuce given by the k-dependen~'~actor o~ the right hand side of eq. 10 are elements of a spln-matrix gap function

Vol. 54, No. 7

LOWEST ENERGY COOPER PAIRING

Ac(k) = Ac[(lUk12-1Vk12)(i~y)

,

+ 21UklV~(i~.n)]nffi ~

Here we have used (ui~v)_ = ¢ = (io,~.~)_= ¢, and G = ~xx ~v~ ~z %. ~he order parameter is a m-ix turf of even-parlty singlet-spin (i~y) and odd-parity triplet-spln (i~y_=.n) components. The relation v_ k ffi v~ (discussed following eq. 61 combined--with--v~ ffi-vk results in

[lUkl2-1Vkl2

+ 2{xluklVk][lUk12

--

-

+ 2~lUklV~]

Ivkl2

-v k

(11)

= 0

interaction--'8. This does not mean that F o(k,k+Q) pair amplitude does not exist and was to be expected, because F(k_,k+~) is odd parity and in this case integration over VF in the gap equation must yield a null result. We have shown explicitly that ~ 0 components occur in both scheme Ca) and scheme (b) (transforming back to in the former). 12-16 In the present scheme (c) theory~ inclusion of all phase and spin structure in the derivation of Hv(a~a+ ) in eq. 9 from Hv(c~c +)

antlferromagnetism causes an--electron energy gap in k space and with lUk12 * I and IVkl2 * 0 far Using t-he relation

v~ =

scheme (b). 6,8 It was shown that A_o(k,k~) with the Vkk.(C,C+) = V even parity =P ---

V_k = -v k. Both lUkl2--and IV-kl2 are highly ani-sotropic, with lu-k[2 = lVkT2 = ½ where

from that gap.

635

= 1

,

q

I =-v

k

(12)

I I

we obtain A+(k)A(k) ffi IA 121

, v~ = - v k

results in an interaction Vkk,(a,a +) for eigenstates of the antiferromagnet which has mixed parity and some spin structure. Components of the resulting gap function in eq. II have mixed parity and mixed pair-splnor structure. It is possible to transform the entire scheme (c) theory in terms of (a,a + ) operators to a form in terms of (c,c +) operators. Evidently a (c,c +) representation of

(13)

Using eqs. 5 and 9, H(a,a +) is expressed in very standard form and the usual equations of motion or Bogoliubov-Valatln transformation of superconductivity theory result immediately in

<

+

+

>

a_+,_k_~a_+,k¢

Ac(k)-=,=

=

+

E~(~)) ½ tanh [(A~(~)-¢'=Ac(~)-~'=~T

Using eqs. II to 14, the magnitude A c of the anisotroplc matrix gap function ~c(k) is given by a simple gap equation

+

..

Ac

..

[-

in the host lattice by E+(k) energies of the eigenstates of the antiferromagnet, just as though only the host-lattlce electrostatic potential had been changed. This is clearly the minimum change of the gap equation and minimum reduction of the gap function by the antlferromagnetism, and it is manifest in eqs. 9 to 15 that Cooper pairing has been symmetry matched to the V interaction. There has been some controversy regarding existence of ffi F~(k,k+~) pairing in

Aa(lUkl2 - IVk12)2

.

.

.

.

k,_ [IAal2(lUkl2-1Vkl2)2 + E2+_(k)]½ tanh[

(14)

scheme (c) pairing would be more complicated algebraically but would lead to the same eq. 15 for A . To express the phase degree of freedom for ~ a n d - ~ C o o p e r pairing components implicit

(15)

2kT

Eq. 15 is identical to the BCS equation except for replacement of e k energies of Bloch states

=vX.

]

(mc12 + Z2+(k))½ _]

.

k,~ 2(IAc12 + Z2+(k)) ½ tanh

Aa

+

i n p a i r i n g (eq, 7), i t has been n e c e s s a r y t o v a r y p h a s e s of t h e a n t i f e r r o m a g n e t i s m t h r o u g h Mk and v k. This i s r e q u i r e d s i n c e t h r e e Cooper pa--iring p--hases i n t h e ( c , c +) r e p r e s e n t a t i o n a r e r e p l a c e d by a s i n g l e p h a s e i n t h e ( a , a +) representation. (The c - o p e r a t o r t o a - o p e r a t o r transformation simplifies equations and perhaps clarifies the underlying physics, but does so at a certain price.) The gap equation for A a in ~a(k) = A a ( ~ k l 2 - IVk12)(i~y) of scheme (a) pairing is: --

[IAal2(lUkl2-1vkl2)2 2kT

+ E2(k)] ½

]

636

LOWEST ENERGY COOPER PAIRING

The transition temperatures are determined by llnearlzlng eqs. 15 and 16. The only difference between these llnearlzed gap equations is that in eq. 16 for scheme Ca) the factor V of eq. 15 for scheme (c) is replaced by

6F (b) = - y1

Vol. 54, No. 7

N~

4 (o)a2( MÈ ) o 3A4 o

6F (c) :

_

i N~(0)~2(7M") 2

V(lukl2-}Vk{2) 2 = V ( ~

+~Mk{~ ) < V and therefore

T c(c) > T (a) Since t~ansi~ion temperatures of schemes ~a) and (b) are the same, 3'12 T (c) > T c(b) as well " At sufficiently high c temperatures, scheme (c) has lowest free energy as eKpected. To make an explicit comparison of free energies near or at T = 0, we take a model as in ref. 12 where the Fermi surface (F.s) consists of two regions in region I the F.s. is gapped by the antiferromagnetlsm with ~k+Q = -ek' the

For T = 0 and M 2 >> A 2, the l e a d ~ , e~r,m ~n the free energy differ by a factor e I(0~7~2(0~.

model the free energy for s~heme (b) was found in all cases to be lower than that of scheme (a), 12 we compare scheme (c) only with scheme (b). For T = 0 and M 2 << ~2, the gap functions are given by NI(0) M 2 Ab

=

Ao[1

2N(0) A~

Ac

=

Ao[l

NI(O) M 2 2N(0) A2 c

I NI(0) M 4 +

+

3 N(0) A~ 7 NI(0) M 4 8 N(0) A4 c

+ ...]

+

...]

(17)

where A O = 2c0 exp[-I/N(O)V] and we have taken IAil + A i and~IMl ÷ M. For T = 0 and M 2 >> A 2, the gap functions are

NI(0) Nt(0) A2 A b = AO(~) - ~

[I + 6N(0) M2

+ ...]

Nt(0) NI(0) A2 Ac

"

N(0)

[I

-

~. 2+

...]

(18)

Determination of free energies proceeding from isotropic or anisotrople gap functions is discussed In detail in ref. 12. For T = 0 and M 2 << A 2, free energy expansions are the same for schemes (b) and (c) up to terms fourth-order in M/A, where the third terms in A b and A c of eq. 17 contribute

1

F(b)_F s

A

2NI(0)/N2(0)

normal = -

F(C)_F s normal

+...

A 2

2NI(0)/N2(0) +'"

(20)

F.s. density of states is Nl(0),--an~d -M k = M I = M, zn region 2 with density of states NT(0) = N(0) - NI(0), small effects of magnetism with a magnetic energy gap far from the F.S. are approximated as zero with M k = 0. Since In this

(19)

o SAg o

In all cases, the free energy of scheme (c) pairing is lower than that of scheme (b) pairzng, which in turn is lower than that of scheme (a) pairing. Although A~(k)Ac(~) = IAci2 ~ in scheme (c) pairing, we expect that with the anisotropic ~c(k) of eq. 11 the transition temperature would be depressed by nonmagnetic impurities as for the anisotropic Aa(k) in scheme (a) pairing and as observed in e~pe~iment. 18 However that is not known because theory for this and many other aspects of scheme (c) pairing has not yet been done. Evidently triplet spin components in the Cooper pairing will affect the critical magnetic field for cases where Hc2 is determined by Zeeman energy effects, as we have found for scheme (b). 12-16 The relative magnitude of trlplet-spln Cooper pairing in scheme (c) is probably not small, since in pairing schemes (a) and (b) all of the pairing is triplet at the edge of the electron energy gap due to antlferromagnetism. 12,15 In pairing scheme (c), slnglet-spln and triplet-spln pairing each have both uniform and sinusoldal components, in contrast to schemes (a) and (b) where singlet-spln palflng is uniform and trlplet-spln pairing is sinusoldal. Finally, mere sophisticated theories of Cooper pairing where the frequency-dependent phonon-medlated interaction and Ellashberg equations are developed, whether for pairing scheme (a) or scheme (b), should be re-examined in light of our result that for the BCS interaction those schemes do not have lowest free energy and omit important physics in spin space and in phase of the order parameter.

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W. Baltensperger and S. Strassler, Phys. Kondens. Mat. ~, 20 (1963).

5.

2.

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J. Ashkenazi, C.G. Kuper and A. Ron, Phys. Rev. B28, 468 (1983).

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J. Ashkenazi, C.G. Kuper and A. Ron, Solid State Sciences 52, 112 41984) (SpringerVerlag). K. Machlda, K. Nokura and T. Matsubara, Phys. Rev. B22, 2307 (1980). K. Machida, J-~'Phys. Soc. Japan 50, 2195 (1981). M.J. Nass, K. Levln and G.S. Grest, Phys. Rev. Lett. 46, 614 (1981).

VOl. 54, No. 7 9. 10.

II. 12. 13.

LOWEST ENERGY COOPER PAIRING

M.J. Nass, K. Levln and G.S. Grest, Phys. Rev. B25__, 4541 (1982). M.J. Nass, K. Levin and G.S. Grest, in "Ternary Superconductors", ed. Shenoy, Dunlap and Fradln (Elsevier North Holland) 1981, p. 277. K. Levln, M.J. Nass and G. Ro, Solid State Sciences 52, 104 (1984) (Springer-Verlag). E.W. Fento--n, Solid State Comm. 50, 961 (1984). E.W. Fenton and G.C. P s a l t a k i s , S o l i d S t a t e Comm. 45, 5 ( 1 9 8 2 ) .

14. 15. 16. 17.

18.

637

E.W. Fenton and G.C. Psaltakls, J. de Physique C3 44___, 1129 (1983). G.C. Psaltakis and E.W. Fenton, J. Phys. C 16, 3913 (1983). E.W. Fenton, Solid State Sciences 52, 136 (1984) (Springer-Verlag). Eq. 16 herein was used for scheme (a) pairing throughout ref. 12 however in eq. I0 of ref. 12 three (lUk12-1Vkl2)2 factors on the right hand s~de wer--e omitted during publication. G. Zwicknagl, private comunication.