Journal of MagneUsm and MagneUc Matermls 63 & 64 (1987) 435-438 435 North-Holland, Amsterdam

SUPERCONDUCTING INSTABILITY IN THE LARGE N-LIMIT OF THE COQBLIN-SCHRIEFFER MODEL

Marek G R A B O W S K I

Department of Physics, Umversay of Colarado, P 0 Box 7150, Colorado Spnngs, CO 80933, USA

A possible novel mechanism for superconductivity in heavy fermlon systems is proposed The Coqblln-Schneffer model [or a Kondo-hke impurity Is studied in the limit of large spin degeneracy The effective dynamical mteractmn between conduction electrons mediated by spin fluctuations is calculated and shown to be attractive at fimte frequencies

1. Introduction

It has been recently noted [1] that the contras- ting equlhbrium and transport experimental results [2] in the sohds which exhibit heavy fermlon behaviour [3] provide strong constraints on microscopic description of these systems These constraints lead to the conclusion that the heavy fernuons arise from ordinary conduction electrons via the exchange of spin fluctuations and that the effective interaction between the resulting quasipartlcles is strongly retarded in time but local in space Consequently, the physi- cal picture of the heavy fermlon system is that of magnetic impurities (locahzed f-electrons) immersed in a metallic matrix (conduction electrons) Below the Kondo temperature Tk the conduction electrons are scattered off the Kondo-compensa ted , localized magnetic moments The resulting polarization of the spin cloud around the impurity induces an effective e lect ron-elect ron interaction between conduc- tion electrons

A similar problem has been addressed by P Nozlbres [4] in his Fermi Liquid approach to the single impurity Kondo problem He has found [4] that the static effective interaction between con- duction electrons is repulsive for electrons with ant~parallel spin and essentially zero for electrons with parallel spin However , the superconducting instability would require an a t t ractwe Interaction between fermlons Consequently, a natural ques- tion to pose as whether the dynamical (retar- dation) effects can render the effective inter- action between parallel-spin electrons a t t ractwe

0304-8853/87/$03 50 © Elsevier S oence Publishers (North-Holland Physics Publishing Division)

To answer this question I have investigated the dynamical, finite frequency structure of such an interaction More specifically, to understand the basic nature of these interactions, I have restric- ted myself to the case of a single localized mag- netic moment described by the Kondo-hke model Hamlltonian leaving the problem of investigating the coherence effects between an array of such moments for future study

2. Functional integral formulation

To describe the effective e lectron-electron interaction between band electrons in the presence of a single Kondo-hke impurity I con- sider the model originally derived by Coqblin and Schrleffer [5] Thus I write the model Hamll- toman m the following form

~]f=~, EkC~,.Ck..+(J/hO ~_, C~'mf+f , .ck , . . k m k k r i m "

(1)

where N = 21 + l is the "spin" degeneracy of the localized f-level, Ek iS the energy of a conduction electron of a wave number k, the operators c~,. (Ck,.) create (annihilate) band electrons in a state of wave number k and z-component , m, of the total angular m o m e n t u m / , while f+(fm) create (annihilate) localized electrons on the impurity site The second term in eq (1) describes the exchange interaction between band and local electrons and couples only the spin-orbit elgen- states with the same ]

The Coqbhn-Schr lef fer (CS) Hamil toman, eq (l), is exactly lntegrable in the Bethe ansatz sense

B V

436 M Grabowskz / Superconductmay m the Coqbhn-Schneffer model

[6], however, it is not feasible to extract the dynamical behawour of the system from these exact solutions Since I am interested in the dynamical Interactions I have to approach the problem differently Following earlier works [7] concerning the CS model I will develop the mean field theory (exact In N - - ~ limit) for the Hamlltonlan eq (l) and then, by including the Gausslan fluctuations about the mean field broken symmetry state I will obtain the description of the dynamics of the system

Hence I begin by writing the partmon function as a thermal functional integral

Z(fl) = Tr e -°~e

= f @(c' c*)@(f' f * ) e x P l - I,~d'r~('r)] '

where 9 ( )'s are canonical measures for the Fermi fields represented by Grassmann variables (antlcommutlng at all times), ~- IS the imaginary t~me, /3 = 1/kBT and the Lagranglan is given by

:~('~) = Z d , . o,~,. + ~ f*.. o.f,. + krn rn

Subsequently, to ehmlnate the quartic fermlon term (second term in eq (1)), one introduces [7] the collective fields, b and b*, via the Hubbard- Stratonowch identity,

b('r) (J/N) ~ * - - ~ C k m f r o k m

It is very Important to note, that since the original CS Hamlltonlan, eq (1), commutes with

- E . , f ~ f ~ , i t the f-electron number operator ~ - + conserves the f-charge However, this con- servatlon symmetry ~s lost in the transformed Hamlltonlan Y(' To restore this symmetry one adds a Lagrange multiplier term hn~ to the Hamiltonlan and subsequently one constructs the constrained partition function [7,8],

f 2~n/~ Z(Q) = dh ec3~QZ(A) (2) do

This Laplace-like transformation constrains the f-occupation number to nf = O at least when the partition function IS evaluated exactly However, when Z(Q) is approximated via the I / N expan-

sion the above procedure is still quite controver- sial [9], especially in the context of so-called infinite U Anderson model Nevertheless, if one takes the large N hmlt properly [8], i e keeping the f-level filling factor q = Q / N constant while N---)o0, the f-charge conservation symmetry is restored order by order in I / N expansion on the expense of allowing [8] for multiple f-occupation of the impurity site It is then difficult [9] to reconcile the O > 1 occupation of the f-level with Infinite on-site Coulomb repulsion in the case of the Anderson model However, for the CS model, this procedure is still meaningful if one considers the exchange interaction as a generic one instead of one derived from the Anderson model in the appropriate limit

In preparation for the integration over the fermlon fields one notes that further proceeding with the calculation involves the expansion of the effective action functional m terms of the small fluctuations about the mean field or saddle point value of the collective field b(z) = bo+~b(~-) However, smce the mean field bo develops a nonzero value corresponding to the broken symmetry state, the Goldstone mode fluctuations lead to the unbounded phase fluctuations [8] Consequently, following Read and Newns [7], the phase degree of freedom is factored out of the collective b-fields via a simple gauge trans- formation

b(T) = r(~') e '°(~), f,.(-r) = f ' ( 7 ) e '°(~

This transformation removes the "unwanted" divergencies associated with the Goldstone mode

After some mampulations [8] with the collec- tive fields measure @(b, b*) one can rewrite the partmon function, eq (2), as

= f ~(c, c*)~(f, f*)~(r, ~) Z( Q)

with the effective Hamiltoman

~("(~') = ~ {~kc~mck,. + r(~')(c~,.f.. ÷ f*ck,.)} k m

+ (N/J)r2('r) + {el + ~(1-)}(n~ -- Nq),

M Grabowsk~ / Superconductway m the Coqbhn-Schneffer model 437

where f~(~-) = i(O.0 + Im A), ef = Re A and the in- tegration over A has been absorbed into the ~( r , f~) measure which in turn has been ap- proximated [8] as locally Cartesian The nf con- servatlon is now dynamically imposed by the integration over f l The local collective fields, r and ~, can be interpreted physically as representing the strengths of the hybridization between the band and f-electrons (r-field) and polarization of the f -moment (~-field), while ~t gives the position of the sharp, local resonance and q measures the extent to which this resonance is filled

Finally, the integration over the fermion fields yields the partition function of the form

Zo ~ @(r, ll) e -s( ' ~*), (3) Z ( Q )

where Z~) is the bare conduction electron par- tition function and S(r, fI) ~s the effective action for the collective fields

3. Effect ive interaction

T o extract the effective e lect ron-elect ron in- teraction f rom the partition function, eq (3), one can calculate the two-particle fermion pro- pagator by differentiating the generating func- tional, 1 e the logarithm of the partition function, with respect to the appropriately introduced fer- mionic sources and subsequently omitting the "external legs" of the single-particle propaga- tors To actually evaluate the partition function I expand the effective action functional to the second order in the fluctuations of the collective fields around their saddle point values This ap- proximation is equivalent to the 1 / N expansion for large N Hence

S[r, ~ ] ~ S(°)[ro, e,] + S(2)[Sr(r), 8,Q(T)],

where S (°)~ G(1) is the classical action, S (e)~ G(1/N) represents the Gaussian dynamical cor- recnons which are second order in the collective fields fluctuations while ro and Ef are determined by the mean field equations If one assumes a simple band of conduction electrons of band- width W and constant density of states p then

ef = Tk cos ~rq, A = Tk sln-rrq,

where A = ~rpr~ IS the width of the resonance at ef and Tk = W e -1/pj is the Kondo temperature

Without going into algebraic details [10] of the calculations I will present the results for the effective interaction in a particularly interesting and illuminating limit of very low temperature, T ~ Tk, and half-filled Kondo resonance, q = 1/2 In this case the resonance is centered at the Fermi energy and has the width of Tk For N = 2 this corresponds to the standard Kondo problem of a singly occupied impurity site and spin- l /2 fermlons, al though one has to r emember that the present results are valid only in the hmit of large N Thus the comparison to the Kondo problem has only a heuristic value Under the above stated conditions the effective interacUon between two " t ime- reversed" conduction electrons, t e ( to , - to ) pair scattered into ( to ' , - to ' ) state, in the low frequency hmit is dominated by the rater- action mediated by the f~-field which describes the magnetic susceptibility of the polarization cloud around the Kondo-hke impurity For the antlparallel "spin" electrons it has the following form

V+~(to, to') = r ~ ( o J ) ~ ( - t o )

x ~ ( o ~ - to') %(to') .%(- to ' ) ,

w h e r e 1~(to) = {ioJ - ~f + iA sgn to } - i is the s ta t i c - a l l y dressed f-electron propagator ,

~ ( t o ) = d~(to)sq(-to)) = (¢rTk/N)O3(~ + 2)/in{1 + O3(O3 + 2)}

stands for the ~-field propagator and o3 = Itol/Tk For electrons close to the Fermi surface, o3, a3',~ 1, this interaction reduces to

p V t t ( t o , ~o') ~- ( N r r p T k ) - ' { 1 + [to - to'I/Tk}

and is obviously repulsive However , the effective interaction in the parallel "spin" chan- nel obtained f rom Vtt by adding the "twisted" diagram contribution Vii(to, to') = Vii(to , t o ' ) - Vii(to , - to ' ) , is given, in the same approxima- tnon, by

pVt t ( t o , to') ~- ( N a r p T 2 ) - ' { l t o - o9' I - I t o + to'I} (4)

Therefore , the effective interaction between electrons of the same "spin" on the same side of

438 M Grabowskl / Superconducamry m the Coqbhn-Schneffer model

the Ferret surface becomes attractive at finite frequencies thus allowmg for the posslbthty of a superconductmg mstabihty of a qutte unusual kmd The mteractton, eq (4), ts purely local and thus momen tum Independent but strongly fre- quency dependent This behawour of the effective mteract ton mdtcates a possible superconducttng mstabthty of a quite exottc type

4. Conclus ions

There are several questions that remain to be answered before one can clatm that the rater- actions proposed in this paper can lead to a novel mechanism for superconducttvt ty Some of them are presently being studted [10] Prehmmary calculations indtcate that the superconducting transition tempera ture associated wtth the just described mteract ion is of the order of one tenth of the Kondo temperature However , one still needs to esttmate the strength of the renor- mahzatton effects, m particular the vertex cor- recttons which are known [11] to effecttvely suppress superconduct ing lnstabthty For more conventional mteract ions the spin fluctuatton vertex depresses Tc most effect tvely[ 11 ] and thus for an mteract ton medtated by spm fluctuations ~t mtght be not so destructtve Also, ~t ~s ~mportant to ask how the p recedmg results will be modtfied for q # 1 (see [10]) when the finite quastparttcle hfetlme becomes important or if one considers a finite concentrat ion of Kondo-hke lmpurlttes In the latter case one should expect that the strong spattal coherence charactertsttc of the Kondo latttce wtil even enhance the superconducting mstabdtty However , 1 would hke to retterate

once agam that the very existence of the Kondo lattice ts not a precondtt ton for the mechamsm described above to be ettecttve

1 wish to thank J R Schrteffer, D J Scalapmo, P Coleman and H Ketter for many valuable dtscussions during my stay at the Untverstty of Cahfornta, Santa Barbara where this work was begun and where I was supported by the National Science Foundation grant No DMR82-16285 I also acknowledge the support of the NSF travel grant No DMR85-19904 and the grant of the UCCS Commtt tee on Research and Creat ive Works

References

[1] C M Varma, Phys Rev Lett 55 (1985) 2723 [2] For a rewew of the experimental results, see G R

Steward, Rev Mod Phys 56 (1984)755 N B Brandt and V V Moshchalkov, Advan Phys 33 (1984) 373

[3] For a rewew of the theoretical situation, see C M Varma, Commun Sohd State Phys 11 (1985) 221

[4] P Nozl~res, J Low Temp Phys 17 (1974) 31 P Nozlbres and A Blandm J de Phys 41 (1980)193

[5] B Coqbhn and J R Schrleffer, Phys Rev 185 (1969) 847

[6] A M Tsvehck and P B Wlegmann, J Phys C15 11982) 1707

[7] N Read and D M Newns, J Phys Cl6 11983) 3273 [8] P Coleman, J Magn Magn Mat 47&48 11985)323

and m Proc 8th Tamguchl Symp on Mixed Valence (Sprmger-Verlag, Berhn 1985)

[9] H Kelter and M Grabowskl, lnst Theor Phvs preprlnt NSF-ITP-85-90

[10] M Grabowsh, to be pubhshed [11] M Grabowskl and LJ Sham, Phys Rev B29 11984)

6132