Phonon mediated interaction for triplet pairing in Pd

Solid State Communications, Vol. 28, pp. 157—160.

0038—1098/78/1001—0157 $02.O0/0

© Pergamon Press Ltd. 1978. Printed in Great Britain. PHONON MEDIATED INTERACTION FOR TRIPLET PAIRING IN Pd

J. Appel” Physics Department, Purdue University, Lafayette, IN 47907, U.S.A. and D. Fay Abteilung für Theoretische Festkorperphysik, Universität Hamburg, Hamburg, West Germany (Received 15 May 1978 by H. Suhi)

The pairing interaction due to phonon exchange is calculated for the triplet p-state of Pd, using the atomic-site representation for the heavy d electrons. It is found that this interaction is repulsive, X’~”<0, and that its magnitude is small compared with the singlet s-state interaction of BCS, X’~.The net interaction due to both phonons and spin-fluctuations is attractive but it is so small that the observation of a triplet p-state in Pd appears unlikely at the lowest accessible temperatures. RECENTLY we have calculated the spin fluctuation contribution to the p-state (1= 1) pairing interaction in Pd [1], employing Scbrieffer’s model [2] for the irreducible particle—hole interaction. We find that the tram. sition temperature is of the form =

~exp

~SF + xi~) (— I + 1x8i’

(1)

where TF is the Fermi temperature, S is the Stoner factor and I X~and X’~account for the electron-mass enhancement due to spin fluctuations and phonons, respectively. In equation (1) the effect of phonons °‘~ the electron—electron interaction is ignored. However, the electron—phonon self-energy effect is taken into account, X~’= 0.35. The resulting transition temperature is only of the order of 1 O~—i06 K, indicating that the occurence of an observable p-state transition depends on the phonon interaction, XI~h.For experimentalists, who attempt to observe triplet pairing in Pd and in certain other metals at very low temperatures, the question of whether or not the ordinary electron—phonon interaction favors this type of pairing is of considerable current interest [11]. It is the purpose of this comment to evaluate X~’jh for Pd. Ignoring spin fluctuation pairing, we depart from the standard vertex equation in wave vector space, transform this equation into the site representation [3], and determine the p-state kernel, and thereby XP 1h using the methods of group theory. In carrying out this procedure, we assume a short-range electron—phonon

*

Permanent address: Umversität Hamburg, Hamburg, West Germany.

interaction (cf. [31)which appears justified for the heavy d-electrons at the Fermi surface of Pd. The transition temperature T~is the eigenvalue of the linearized integral equation for the vertex function, Foe(k)

=

— T~

I~’(Ick’)Fa(k’)Fa(k’).

(2)

Here k (k, A.~,) and I~’(k,k’) is the irreducible interaction between electrons scattering from k, — k in subband a to k’, k’ in subband a’; F~.is the anomalous Green’s function for the subband a~We assume here that I~’is due to the exchange of virtual phonons and we parameterize its frequency dependence in the manner of BCS. To obtain the vertex equation for triplet pairing, we transform equation (2) from the Bloch wave (k) representation to the atomic-site (Wannier or n) representation. To this end we write 4~’ in the following form: —

J’(k k’)

=

~

e~k

_k’.n’)j,(~

n’)

n,

=

~

(cos k n cos k’ n’

~

+ sin k n sin k’• n’)I~’(ii, n’) =

JM1~(k,k’) +

I~”(k,k’).

(3)

The singlet and triplet interactions are even and odd fo~ k -~ k (or k’ -+ — k’), respectively [41.In the discussion of singlet pairing in the site representation [3], emphasis is given to the contact model (n = n’ = 0), corresponding to scattering processes where the two electrons are initially both on one site and finally again both on one

157

—

158

PHONON MEDIATED INTERACTION FOR TRIPLET PAIRING IN Pd

Vol. 28, No. 1

z

et 1

e12\

110

~J

/

/1/

/4

Fig. 1. Electrons 1 and 2 in a triplet state are initially separated by a nearest neighbor vector a1 and finally by a1.

I~P(k,k’)

=

~ I~,.(a1,a,) sin k a, sin k a~1, .

(4)

/

I

site. For triplet pairing in a non-degenerate band [5], we must go beyond the contact approximation because the two electrons with parallel spin cannot be simultaneously at one and the same site. The leading contribution to the triplet interaction is given by

2

/

5~/ / //

9x Fig. 2. The nearest neighbor configuration in an f.c.c. lattice.

(cf. Fig. 1), where a1 (i = 1, 2,. 12) is a nearest neighbor vector in the f.c.c. lattice of Pd. We substitute (4) Ff,f (a,, aaj), transform according to the reducible into equation (2), write r’a(k) = ~ r~(a,) sin k a1, and representation, F1 + f\2 + r15 + r25 + F~,of the get the vertex equation for triplet pairing in the site cubic point group. respresentation, Let us consider what corresponds to p-state pairing in an isotropic system, namely the pairing in the F15 = ~ K~’(a,,a,)Fa’(a,). (5) representation of a cubic crystal. The vertex equation a’ I in this representation is given by . .

,

—

Here, the triplet kernel is given by Kaa’(ai, a1)

=

~ L~~(a1, aj’)Fa•~aj’,a1),

rK

—T~ ~

(KIKIK’)rg’,

(8)

(6) where K l5 the basis index of the vertex function, I’,~, and where the 1’15-kernel is given by

where Fa(a,, a)

3 =

=

~ sin k a3 sin k a, Fa(k).

(7)

(KIKIK’>

=

~3K’

k

The vertex equation (4) consists of a system of 12 x 3 equations when the 12 vectors a, are combined with the 3 subband indices a. We use here, as in [1], the heavy-hole subband model of Doniach where the d-holes are equally distributed among the three t2g orbitals of symmetry xy, yz, zx (a = 1. 2. 3) according to whether the Bloch wave vector k is in the region ofI or Yor Z of the f.c.c. Brillouin zone. The necessity of a group theoretical reduction of (4) is evident; it is carried out in the manner described previously [6]. By virtue of the symmetry properties of a pair orbital for two electrons centered at nearest neighbor sites, wa(rj — 0)wa(r2 — a.), we have 12 different vertex functions Fa(ai); e.g. for a = 1, only the four functions F1(a,) ‘ 0, where a, is one of the four vectors in the x—y plane (Fig. 2). The twelve functions

~ AKfKfr’AfK’.

(9)

f,f’

The transformation matrix AKf (yielding the sym. metrized pair states of [6]) is readily determined [7]; the basis of the F15 vertex function is given by FK = ~~‘AK! K = 1, 2, 3. The final result for the phonon-pairing interaction in the triplet state of the F15 representation can be written in terms of the coupling constant Xf”

xr’.

‘5

This parameter takes the place of the BCS parameter ~Ph in the BCS formula for 7’,,. We finally get

~‘~l”

=

N(0) (1 — sinbkF/M0 bkF’~ 1

x

~{

~

70 ~c~,

(aj xy IV~V(0)I a1, yz) ,

x (0,xzj V~V(0)la1,yz) —

2 }.

I(ai xy IV~V(0)I0, xy)1 ,

dw

(10)

Vol. 28,No. 1

PHONON MEDIATED INTERACTION FOR TRIPLET PAIRING IN Pd

Here N(0) is the total density of states at the Fermi surface, b = y’~a,where a is the lattice constant, kF is the

159

By comparing equations (10) and (12) it is evident that I~PhI

IA

subband Fermi momentum, g(w) is the phonon density of states with the cut-off frequency ~ and the matrix element between atomic-like orbitals Wa(r — a,) is givenby

Ph

1I’~Xo

13

Furthermore < 0

14

XPh 1

since, by inspection, the squared matrix element in the curly bracket of(10) is larger than the first term. The 3r. (11) 0.63 b = 5.5 A (the lattice constant of Pd is r factorA-’ (1 and — sin bkF/bkF) is of the order of 1 for kF = = crystal j w~,(r— ai)V~V(r — 0)w~~(r —0) d volume a=3.88A). In summary: When the phonon interaction correFor comparison we give the corresponding result for sponding to p-state pairing, i.e. the F, 5 phonon kernel, singlet pairing m the F1 representation,. is taken into account the net interaction of both spin fluctuations and phonons is still attractive, = N(0)-~~ dw XSf’_.. IX’7’I>O, with X~F~0.09 [8]. However, T,,, is M o even than the upperpairing boundalone. of 1 0~ K estimated 2 in [1]smaller for spin-fluctuating Hence, on the basis of our model calculation, the experimental obserx ~{I(aixy IV~V(0)I a1, xy)1 + I(a 2 vation of a triplet p-state is unlikely at the lowest 5,xyIV~V(0)Ia5,xy)I x Ia 2 accessible temperatures. To confirm this result, we are presently working on the evaluation of the ratio 9,xyIV~V(0)Ia9,xy)I + 2[la 2 XP 1,xyIV~V(0)IO,xy)I 1h/X~~0~~ using Anderson’s ASA wave functions [9] and 2 taking into account the phonon dispersion responsible for displacement correlations between nearest neighbor + I(a,, xylV~V(0)l0,xy)1 + I(a 2 atoms. In conclusion we would like to point out that 5,xyIV~V(O)I0,xy)I + I(a 2 recently Pinski, Allen, and Butler [10] have presented a result for X’7’ based on KKR wave functions that is 5,xyIV~V(0)Ias,yz)I + T7t’fl\I \12 (a1 ,xyl V~V(0)I0, xy)

$

I~a

2 9,xyIV~Y~v)Ia9,yz)l

negative, too.

+ (0, xyIV~V(0)Ias,yz)I + 1(0, xy I V~ V(0)I a 2

Acknowledgements

9, yz)I

+ I(a

2 9, xylV~V(0)I0,yz)1 + I(a 2]}. 5,xyIV~V(0)I0,yz)l

—

One of us (J.A.) would like to

thank Prof. A. Overhauser for constructive discussions and the Deutsche Forschungsgemeinschaft for its (12) support. REFERENCES

1.

FAY D. & APPELJ.,Phys. Rev. B16, 2325 (1977).

2.

SCHRIEFFER J.R.,J. AppL Phys. 39, 642 (1968).

3. 4.

APPEL J. & KOHN W.,Phys. Rev. 84,2162(1971). LUTTINGER J.M., Phys. Rev. 150, 202 (1966).

5.

The three heavy-hole subbands are parts of a nondegenerate band.

6.

APPELJ.&Kohn W.,Phys. Rev. B5, 1823 (1972).

7.

The coefficients A ~ apart from a common factor, 1 ~ 1

1 2

3 8.

2

1—1 —1 1 0 0

3

4

5

11 0 1—1—1 0 0—1

6

7

8

are given by (cf. Fig. 2): 9

0 0 0 1 1 1—1 0 1—1 1—1

10

11

12

—1—1 0 0 1—1

1 0 1

By comparing equations (10) and (12) it is evident that I Xc?,I is of the order of a few per cent of F~’ 0.35.

160

PHONON MEDIATED INTERACTION FOR TRIPLET PAIRING IN Pd

Vol. 28, No. 1

9. 10.

ANDERSON O.K., Phys. Rev. B8, 3060 (1975). PINSKI FJ., ALLEN P.B. & BUTLER W.H., Bull. Am. Phys. Soc. 23, 275 (1978).

11.

LEGGETT A.J., Proc. Physics at Ultralow Temperatures (Edited by SUGAWARA T.), p. 318. Hakoni, Japan (1977).