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 {ckk's + iskk'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.,...WWz
Fig. 2. The twoparticle irreducible vertex.
(t[a·S[t)= 2,
+~.iJ::."., {Skk•r<k•'lk' + Sk'k*r<k•kl' }
2N k'
I:; {A~~.(ek,Eekek,; ek.,Eek·fk•)F}.')k,
k'
(ek+ ek,+ W,E)F}.'k,
+ 3iJ "~" {Skk•r<kt)'k' + Sk'k'r<kt)k'}
2N ~<'
+I:; {A):~.(ek,Eekek'; ek.,Eek·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;2C'fk=,fk=,):.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 + WtE
+ "¢2'(f",f')
f +f + WtE
~ra,2 fndf"I(f+f'+f"+WtE){
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 =   rae   ,
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 ) =
fAE1 + f'AE2 '
fBE1 + f'BE1 '
1+g!g2=0'
This relation can be satisfied only when
Thus we obtain the groundstate 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 '
(62)
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
1x 2
~acas 1~x s:dx'[~f(x')+g(x')]
= ac(1+~a.x}m~~ac(1+a.
, 4 1x, 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')]
1x 2
_l._a. _x_ rtdx'[a.J(x') ~acg(x')]
2 1xJ, 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 .. BealMonod , Phys. Rev .. 178 ( 1969 ), 874 . K. Matho and M. T. BealMonod , J. de phys. 32 ( 1971 ), 213 ; Phys. Rev . B5 ( 1972 ), 1899 .
3) K. Matho and M. T. BealMonod , 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 .