Theory of U impurities in metals in a j-j coupling scheme

208

Journal of Magnensm and Magnenc Materials 63 & 64 (1987) 208-210 North-Holland, Amsterdam

THEORY

OF U I M P U R I T I E S IN M E T A L S IN A j - j C O U P L I N G S C H E M E

A C NUNES*, J W R A S U L and G A G E H R I N G Department of Theoreucal Physics, 1 Keble Road, Oxford OXI 3NP, UK We examine v a n a u o n a l l y the ground state of an A n d e r s o n model of relevance to u r a n i u m systems m which the lowest bare configuratmns are f2 and f3 for strong 1-1 c o u p h n g We also derive an integral equation for the 5f G r e e n ' s f u n c t m n and discuss possible extensmn to periodic systems

Although a number of promising approaches to the cermm Anderson lattice have been developed [1-5], there has been comparatwely httle effort mvested m problems where the f electron occupation fluctuates between two magnetic configuratmns, for mstance f~ and f2 as m T m or Pr The single impurity problem, where these configurauons couple to spin 1/2 band electrons have been thoroughly investigated by the Barfloche group [6] and by Schlottmann [7] Indeed th,s problem has recently yielded to the Bethe-ansatz method by these authors [8] Other work has been concerned wtth the case where local and band states of equal orbital degeneracy are coupled For the fl_f2 Anderson model, variational [9] and 1 / N expansion methods [10] have been employed to show that in this case the impurity ground state is a slnglet Recently [11], we have extended the latter approach to the f2f~ Anderson model m order to describe the ground state propernes of dilute U systems Experimentally (see ref [11] for references), a wide range of bulk measurements indicate that the U lmpurmes are m a slnglet ground state We obtained the same result theoretically m a number of different coupling schemes Here we extend the vananonal method to the case of strong I-I couphng and discuss the results for the binding energies We also derive an integral equation for the local 5-f Green's function and indicate extensions for the method to concentrated U systems In the hmlt of large 1-1 couphng the largest 12 and /3 angular * Permanent address Department of Physics, Umverslty of Braslha, 70910 Brasflm, DF, Brazil

momenta are selected (4 and 9/2, respectwely, for the approprmte s~gn of the spin-orbit couphng) The Anderson Hamlltonmn then becomes

Ho = Z ~kC~,.Ck,.+ E2 E IJ2M2)(J2M21 km

M~

M~

Hm,x = x/3 V

~

{IJ3, M2 + m)

k m,m' m'

× (J3 M313 ram'm")(2mm'lJ2M2) x (JzM21ck,,,, + H C },

(1)

m terms of the appropriate e~genstates and C l e b s c h - G o r d o n coefficients We hst two of the trml ground states namely the "no-hole triplet" and slnglet counterparts of those discussed previously [11] T h e other relevant trial wave functtons are obtained in a s~mflar manner In what follows 10) represents the filled Fermi sea (1) The no-hole J = 4 magnetic state

14,J),. = alJ2M2)10) + E

E [3(~)IJ3M3)Ck~-~IO)

(2)

k<,Er M~

By using the appropriate sum rules [12] we obtain [13] 3(6j - 5)~ l/z (tkjIHr.,xi~b,)M = 2 V (--~-/2T~ / a ~k/3(k),

(3)

and by acting on the wavefunctton (2) with

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209

A C Nunes et al / Theory of U tmpuntles m metals m a I-1 couphng scheme

H0 + H .... and ehmlnatlng /3(k) we obtain E2 -- E s = F ( ] )

I;

dek {1/(E3 - ek - Es)},

(4)

D

where F(I) = 3 ( 6 / - 5)/(41 - 1) V2po and po Is the density of conduction states at the Fermi level Apart from the constant multiplying the hybrl&zatlon width this equation is identical to that obtained In other coupling schemes A similar feature appears in the energy elgenvalue equation arising for the smglet state wave function [~bj=o)= ~

~

a ( k , k')

k,k' raM2

x IJ2M2>Ck'mCk~=-mlO>

+ Z k k',k"

~ rrl

/3(k, k', k")

M2,M 3

× ]J3M3)Cv, M , - - ~ C v M ~ _ , . C k m [ O )

(5)

which, after elimination of the /3 amphtude becomes a ( k , k')(E2 - ek - e k , - Es) = P(1)

I f dek, {a(k, k') + a(k, k") + a(k', k")} o

(6) Thus the strong l - I coupling results may be expressed in terms of those in the other coupling schemes simply by rescallng the hybrl&satlon matrix element The overall physical conclusion remains the same, namely the slnglet state lies lower in energy than the other possible magnetic states, although the energy separation In the intermediate valence regime IS two orders of magnitude lower than in the corresponding f0_f Anderson model Turning now to the f-level Green's function, we define (7)

and g<(z) = (~bolg~+[z - Eo(N) + H I -1 ~bvl~bo)

[~bo)= A{~-~f ~ ot(e,e')b,,o,b,~[O) I)'< i)

1

(8)

/3( E, ~', e"'b ) .,o,,b. ~ b.olO )} ,

(9) where b.,o, denotes $.,~, ~o+ and A is a normahsing factor T o evaluate g<> the time dependent method used by Gunnarsson and Schonhammer [14] for the f0_fl_f2 Ce problem could be followed Such procedure IS particularly suited for including higher order electron-hole pair effects Here, however, we are concerned with obtaining other results and shall consent ourselves with obtamlng integral equations for quantities relevant to g>, g< T h e method Is an extension of a procedure adopted for the fl_f2 problem (Rasul, unpublished) We define the auxiliary Green's function g(z,

(E2 - U s - ek -- e k ' - ek")

g>(z) = (4~o1~b~[z - Eo(N) + H]-l~b~+l~bo)

Calculating (1/~r)lm gX(x - iS) yields the spectral density appropriate to PES and BIS measurements T h e basic functions In the 1 / N expansion can be written

rF

t

It

t

el, el, el, e , e,

iE, 1))

-. . .(el, . el, e l v l ( z - E o + H ) - l l e " , e ' , e ,

v),

(10)

where [e"'e"e', e v ) = b.,,v,,b.,o,~.dO) and we suppress the v", v' dependence (these follow the appropriate energy index) Acting on ]E"e', ev) with z - Eo(N) + H ymlds (z - Eo + H)le", e', ev)

= (z - e - e' - e " - AE + E2)

× I~"~', ,v>- v Z {le"'~"e', ,~> + le","',', co)+ le"e',"', ev)},

(11)

where [e'"e"E', e v ) = b.,,,v.,le"e', ev) can, m turn, be related to permutations of le"e', E.) by acting on the left-hand side by (z - E0 + H ) and lgnormg higher order electron-hole pairs Now, acting on each of these with the

210

A C Nunes et al / Theory of U tmpunnes m metals m a 1-1 couphng scheme

resolvent operator ( z - E o + H ) ~ and taking the scalar products wtth le~, e~, e~v) and IE'~', e]", e'l, ely) ytelds (z - e - e' - e " - A E + E e ) g ( z , •~, e'l, el, en •'•l.))

- V ~ {(e'(e'~e,vl(z - E,,+ H)-'[l•"e'•"'ev) + Ie"e"'•'ev) + I•"'e"e'ev>]}

(12)

T h e quantmes m parenthesis can be expressed m terms of g's and may therefore be e h m m a t e d Combining (10), (11) and (12) we obtain

g~(z) = a 2 ~

[/31•, e', •,,)]2

×~,(z-AE+E~-e-e'-•"),

(13)

where ~(z, e, e', e") = g ( z , e"e'e, • " # e v ) sattsfies g(Z, • , e', • " ) [ Z -- e -- • ' - - • " - - A E + E2]

= l+f"

de"' D

[~(z, e, e', • " ) + g(z, e, e', e'")+ ~,(z, e, e", e"')] (z - • - e' - •" - •"' + E~ - AE) (14) where AE is the ground state energy shift On c o m p a r m g this mtegral equation with (6) we find the Kernels to be the same However, the prohferatlon of energy variables m (14) has so far prevented a direct solution The approach presented here may be de-

veloped (a) by mcludmg higher electron-hole parrs [14] or (b) by extenstons to the lattice either in a t-matrix formulation [15] or by mc o r p o r a t m g translational mvartance into the wavefunctlon (5) A C N acknowledges the C N P Q (Brazlhan agency) for a grant during the course of this work J W R acknowledges the S E R C for financial support References [1] N Read, D M Newns and S Domach, Phys Rev B ~1) (1984) 3841 [2] N Read and D M Newns, Sohd State Commun '~2 (1984) 993 [3] P Coleman Int Conf Valence Fluctuations J Magn Magn Mat 47&48 (1985) 32 ~, [4] T M Rice and K Ueda Phys Rev Lett '55 11985) 995 [5] B Brandow, Phys Rev B 33 11986) 21'; [6] C R Proetto, A A Ahgla and C A Balselro Phvs Len 107A 11985) 93 [7] P Schlottmann, Z Phys B 59 11985) 391 [8] The ground state of the impurity is found to be magnetic [9] Y Yafet, ( M Varma and B Jones, Phys Rev B 32 11985) 36/) [10] N Read, K Dharmmr, J W Rasul and D M Newns, J Phys C 19 11986) 1597 [11] A C Nunes, J W Rasul and G Gehrmg, J Phys C 18 11985) L873 19 (1986) 11117 [12] A C Nunes, D Phil Thesis, Oxford Umverslty (1986) [13] B R Judd, Operator Techmques m Atomic Spectroscopy (McGraw-Hdl, New York, 1963) chaps I and 3 [14] O Gunnarsson and K Schonhammer, Phys Rev B 31 11985) 4815 [15] N Grewe, Z Phys B 52 11983) 193