Bound State Due to a Pair of Interacting Magnetic Impurities in Metals. II

Progress of Theoretical Physics, Feb 1975

The bound states of conduction electrons due to a pair of magnetic impurities which interact with each other by an exchange interaction are investigated by the Yosida theory. Calculations of the previous work by the present authors are generalized by including the vertex correction. The binding energy is calculated within the most divergent approximation for some limiting cases of the strength of the exchange interaction W between impurities, and its behavior as a function of W is obtained by interpolation. Some unphysical features of the previous results caused by the neglection of the vertex correction are corrected. It is found that the binding energy is essentially given by the Kondo temperature defined as an energy where a perturbational series of the most divergent terms diverges.

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Bound State Due to a Pair of Interacting Magnetic Impurities in Metals. II

Progress of Theoretical Physics Bound States Due to a Pair of Interacting Magnetic Impurities in Metals. II Koi SATO 0 Yosuke NAGAOKA 0 0 Department of Physics, Nagoya University , Nagoya The bound states of conduction electrons due to a pair of magnetic impunties which interact with each other by an exchange interaction are investigated by the Yosida theory. Calculations of the previous work by the present authors are generalized by including the vertex correction. The binding energy is calculated within the most divergent approximation for some limiting cases of the strength of the exchange interaction W between impurities, and its behavior as a function of W is obtained by interpolation. Some unphysical features of the previous results caused by the neglection of the vertex correction are correct~d. It is found that the binding energy is essentially given by the Kondo temperature defined as an energy where a perturbational series. of the most divergent terms diverges. - § 2. Vertex correction H = :E fkCkaCka- __!___ :E :E {ck-k's + isk-k'4} . (1"aa'CkaCk'a'- wsl. 82' (2 ·1) ka 2N ka k'a' where ck = cos(k ·R/2), sk=sin (k·R/2), -·~~§X_(2·4) (2·5) (2·6) Wz~Wi _,- JKL..........,._ {3.,...W-Wz Fig. 2. The two-particle irreducible vertex. (t[a-·S[t)= -2, +~-.iJ::."., {Sk-k•r<k•'lk' + Sk'-k*r<k•kl' } 2N k' -I:; {A~~.(ek,E-ek-ek,; ek.,E-ek·-fk•)F}.')k, k' (ek+ ek,+ W,-E)F}.'k, + -3i-J "~" {Sk-k•r<kt)'k' + Sk'-k'r<kt)k'} 2N ~<' +I:; {A):~.(ek,E-ek-ek'; ek.,E-ek·-ek,)FJ:)k, k' + A}:~k, (ek,, E- ek -ek'; ek., E- ek- ek.)Ff.'k·} A k(tt)k'-- A<tt)O kk' - 2A(kttk)I' . § 3. Calculation of irreducible vertex Following Abrikosov,Sl we put (2·9) (2 ·10) (2·11) (2 ·12) (3·1) (3·2) Then after some manipulation we find m the MD approximation. L~~~(w) as Further we put the wave vector dependence of (3 ·6) (3·3) (3 ·4) (3·5) (3·7) (3·8) where L~2,(w) =0, L~~~(w) =0 91(w)=- 2JN+pac ac_- <Ck2) -_ -1 (1 + sin kFR) 2 kFR (4·1) z.<ckc.k'r<k•kl'>-- ---'-"<-J;-2-C'f-k=,f-k-=,)-:.-fk+fk'+ w.-E Then equations for them are obtained as + _!_r ~Ddf"{. ac¢1 (f, f11) 2 Jo f + f" + W,- E as<J;2(f11 , f1) f" + f' + W.- E _ _!_raca.fndf"I(f+f'+f"+W.-E)'{ 4 Jo ¢1(f,f") f + 611 + Wt- E ¢1(f",f') f 11 + f 1 + Wt- E 2 Jo <J;1(f, f") f + f" + W. - E <J;2 (f", f') f 11 + f' + W.- E } =0 (4·2) </Jl(f,f')+~ra. rndf"{. 2 Jo ~;(f,f") f+f + Wt-E + "¢2'(f",f') f +f + Wt-E -~ra,2 fndf"I(f+f'+f"+Wt-E){ 2 Jo ¢1(f,f") f + f" + Wt- E ¢2(f",f') f 11 + f 1 + Wt - E (4·3) </Jl(f, f1) =<J;l(f', f), </J2(f, f1) =<J;2(f1 , f), ¢1(f, f1) =¢2(f1 , f). (4·4) E=tW, f -tW -De!fr for W> -D e11r, for W< -De11r. The equation for ¢ is obtained as where A (f) = (1 + rlog E = - 2D e11r • § 5. Solution for general cases Now we divide the region of variables into three: (4·5) (4·7) ( 4 ·8) Here the Region I. Region II. f, f 1 <F, Region III. F<f, f 1 , where and .dE rs the binding energy defined by F=IWI-.dE, E=Wt+.dE E= W.+ .dE W>O, for W<O. (5 ·2) W>O, D~ W~ I.dEl + a.[¢/II> (f", f') + ~ 1/J·JYI> (f", f') ]} Jo ¢1 (f, f") f + f"- LIE (5·4) If f<F<f', (5·7) where we put _1_ya, lD d~;' I(E' +f") {asql1(II)(f, f11) _ _!_acc/J1(Il)(f, f")} 2 JF f 2 Comparing these equations, we can easily see W<O, D~IWI~IJEI _1_ya, lD d:' I(f'~) {asql1(II)(f, f11) _ _!_acc/J1(Il)(f, f11) 2 JF f .• 2 IWI~IJEI (5 ·10) (5 ·11) (5·12) (5 ·13) (5·15) (5 ·16) (5 ·17) (5 ·18) (5 ·19) W<O, D~IWI~I.JEI ¢1(Il)(f, f1 ) =0. :_ (5·20) W>O, D> W~ I.dEl where r c = - - -ra-e- - - , 1+rlogDjF -r •= - - -r·a·.- - - . 1+r logD/F A A -f- E-=1-+ f' - E 2 ' E2= -F exp(-J-), rc A fictitious JWJ~JJEI Jc [¢h<1l(f, f11) + _!_C/~I<1l(f, f")] 2 Jo [¢<ll(f" f')+_!_¢<ll(f" f')]=o 1 ' 2 2 ' where They are solved by putting - rae r c = _l_+_r---=-1-'--og'-'-'D~/c-::IE=-1 ' -r.= _ _ __,_ra_._._ _ 1+r log D/IEI (5. 21) (5 ·22) (5 ·23) (5·24) (5·25) (5·26) (5 ·27) where From Eq. (5 · 25) g IS determined by ¢1(!) ( f' f1 ) = ¢1(l) (f' f') = ¢2(!) (f, f1 ) = f-AE1 + f'-AE2 ' f-BE1 + f'-BE1 ' 1+g--!g2=0' This relation can be satisfied only when Thus we obtain the ground-state energy as exp(+) +exp(~) =1. 2rc , 2r. E~ -De1fr (5·28) (5·29) § 6. Summary and discussion 1) for W>O and W rvD, 3) for IWI <tTxo, 4) for W<O and IWI ~Txo, E= -2De11r; E = - _!_ W- D e11r · 4 ' (6-2) 3 In cases where the concentration of Appendix equations for f(x) and g (x) are given by ¢h<IIl(f, f1 ) =f(x), ¢1<IIl(f, f1) =g(x), f(x) -ac Idx'[f(x') + ~ g(x')] -~acas ["'dx' 2 Jo x' , [~ f(x') + g (x;) J 1-x 2 -~acas 1~x s:dx'[~f(x')+g(x')] = -ac(1+~a.-x-}m~-~ac(1+a. , 4 1-x, 2 (A·4) where We can solve these equations by putting g(x) _l._a. rtdx' f(x') 2 Jo _l._a. f"'dx' 2 Jo x' , [a.!(x') -~acg(x')] 1-x 2 _l._a. _x_ rtdx'[a.J(x') -~acg(x')] 2 1-xJ, 2 (A·1) (A·2) (A·3) (A·5) (A·6) t -1-{(1+~)A, _ __!!:_c_B·} =0 i=l p, + 1 p, 2p, ' t -1-{(1+~~)A, +_!_(1+~)B,}'=0. •=1 p, +1 \ 4p, 2 p, where (A·7) (A·S) (A·9) (A·lO) (A·ll) (A·12) (A ·13) (A·15) (A·17) (A·18) (A·19) (A·20) References put 1) As a review of the Kondo effect , see, e.g., ]. Kndo, Solid State Phys . 23 ( 1969 ), 183. H. Suhl, ed., Magnetism V ( Academic Press , 1973 ). 2) M. T .. Beal-Monod , Phys. Rev .. 178 ( 1969 ), 874 . K. Matho and M. T. Beal-Monod , J. de phys. 32 ( 1971 ), 213 ; Phys. Rev . B5 ( 1972 ), 1899 . 3) K. Matho and M. T. Beal-Monod , J. of Phys. F3 ( 1973 ), 136 . 4) K. Sato and Y. Nagaoka , Prog. Theor. Phys. 47 ( 1972 ), 348 ; 49 ( 1973 ), 1377 . 5) K. Sato and Y. Nagaoka , Prog. Theor. Phys. 50 ( 1973 ), 37 , referred to as I. 6) H. Ishii , Prog. Theor. Phys. 50 ( 1973 ), 1777 . 7) Y. C. Tsay and M. W. Klein , Phys. Rev. B7 ( 1973 ), 352 ; preprint. 8) As a review of the Yosida theory, see K. Yosida and A. Yoshimori , Magnetism V , Suhl ed. (Academic Pres~, 1973 ), ·p. 253 . 9) A. A. Abrikosov , Physics 2 ( 1965 ), 5 . 10) S. Nakajima , Prog. Theor. Phys. 39 ( 1968 ), 14Q2 . 11) I. Okada and K. Yosida , Prog. Theor. Phys. 49 ( 1973 ), 1483 .


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Koi Sato, Yosuke Nagaoka. Bound State Due to a Pair of Interacting Magnetic Impurities in Metals. II, Progress of Theoretical Physics, 1975, 366-385, DOI: 10.1143/PTP.53.366