Theory of Kondo Effect in Type II Superconductors
Progress of Theoretical Physics
Theory of Kondo Effect in Type II Superconductors
Shin'ichi ICHINOSE 0
0 Department of Physics, 1\fagoya University , 1\fagoya 464
Ve study the equilibrium properties of type II superconductors with Kondo effect. The Kondo effect associated with the impurity spins is taken into account within the interpolation approximation, which was used previously in our calculations of the superconducting transition temperature and the specific heat jump at the transition temperature. Using this approximation, the general GinzburgLandau equations are derived for superconductors with Kondo effect and the GinzburgLandau parameter IC,(T) is calculated. One finds that the initial slope of the GinzburgLandau parameter shows a continuous change with TK/T,o and approaches the BCSvalue at the end of an infinite Kondo temperature TK/T,,=oo in contrast to Maki's theory.

/Cz (T), which describes the magnetization in this region. Summary is gi\"en in § 4.
§ 2. Generalized GinzhurgLandau equations
In the previous papers1l,zl we have assumed that the order parameter is constant
1n space. There are, however, other classes of interesting phenomena, which in
volve a spatially varing order parameter. To describe the spatial variation of the
order parameter it is convenient to start with the generalized GinzburgLandau
equations. In the following we will consider only the gapless region, where the
order parameter .d (r) is small. Then the selfconsistent equation is given up to
third order in .d(r) by5l
J+(r)=[g[T~ sd 3r'Q"'(r,r').:J+(r')+lgiT~ s... sd 3r1 ...
d 3rsBw (r, rh .. ·, rs) .:J+ (r1) L1 (rz) .:J+ (r3),
(2·1)
where Qw(r, r') and Bw(r, r~> ... , r 3) are the twoparticle and the fourparticle
Green's functions, respectively.
1)
The first term of Eq. (2·1)
Introducing the vertex correction r(?/, w) and the bare twoparticle Green's
function Clw0 (q) in the presence of magnetic fields, the first term of Eq. (2 ·1)
can be written as
T ~
w
sd 3r'Q"' (r, r') .:J+ (r') = T ~ r (q2, w) (_Lo (7J) .:J+ (r).
w
Using the renormalized oneparticle Green's function in the presence of magnetic
field G" (w), 6l the bare twoparticle Green's function Clw0 (q) is given as
(2·2)
(2·3)
(2· 4)
Q~wo(q ~) ..::."... G k (W)Gkq (  oJ )_ 2rrNptanl(vF!7JI) ,
k vFI7JI 2!&5!
where p is the density of states of conduction electrons per atom per spin, 1V is
the number of atoms and wis the renormalized frequency. q is defined as q 2eA
or q + 2eA depending on whether it operates on J+ (r) or .d (r) using the external
momentum q.
Gk (cu) = [icu ev ·A ~k .S(w)] \
G_k( cJJ) = [ iw+ev·A~kl:(w)]\
where OJ= (2n +1) rrT, ~k is the oneelectron energy of conduction electrons and
1: (w) is the selfenergy correction clue .to both the magnetic and the nonmagnetic
impurities. A is the vector potential and v the Fermi velocity.
The vertex correction r(7J2, w) is given by solving the follmving equation:
=1+ 2rrNpT"' r, (w w')~(~q2 cu')tan1 (vFICll)
VF I ~q~ Lw....'J II ' I ' 21~U)'1 '
where T 11 (o), cu') is the irreducible vertex of the effective interaction between elec
trons due to both the magnetic and the nonmagnetic impurities. Introducing the
electronic relaxation time due to nonmagnetic impurities alone r0 and the approxi
mation used in Ref. 1), vve get an approximate expression for rH (w, w') as
r, (u) w') = _ oro,ro'
'1 ' 2rrroTNp
+!.:..[2. r (w)o
N T 1
ro,ro
,_f(w)f(())_')_]
4TxP2
'
where
f(cu) = ( 1 + :~~) 2,
and n is the magnetic impurity concentration. If we neglect the effect of an
applied magnetic field on impurity spins, rl (w) Is independent of the magnetic
field and is given in Ref. 1) as
(2· 5)
(2· 6)
(2·7)
(2·8)
(2·9)
Using the expansiOn
the vertex correction r (qZ, w) is finally obtained as
r (q2, U)) = ______ lcul±:_l I (u)) I
jciJI +na(u)) + (1/2)D1 (o))q2
. [1
n . [Juj) fff1 (ql ],
4TxP 1+ (n/4Txp)(})2(q2)
(2·10)
and a(cu) is the pairbreaking parameter which appears in Ref. 1). Two kinds
of diffusion constants D 1 (rJ)) and D 2 (w) are written by the diffusion constant Do
which is determined by nonmagnetic impurities alone, 1.e.,
D 1 (w) =Do·1+2rrronPT1 ~W),
[1 + !'o/rJ]
D2 (w) =
IJ_o_
1 + ro/r1
vvhere D 0=r0v//3 and T1 is the electronic relaxation time due to magnetic impurities
alone. When the electron mean free path is so short that it is hardly affected
by a small amount of magnetic impurities, we can approximate them by D 0 ,
Finally substituting Eq. (
2 10
) into Eq. (2 · 2), we arrive at
T~ sd 3r'Qw(r,r')LJ+(r')
=Np[i§o(7/)"'
n . {i§ 1 (q2)}~
4TxP 1+ (nj4TKp){]) 2 (q 2)
2) The second term of Eq. (2 ·1)
N mv \ve turn to the analysis on the second term o£ Eq. (2 1). It is conven
ient to rewrite it as follows:
T ~
S···Sd 3r1· ··d 3rsBw (r, rh · • ·, r 3) J+ (r1) L1 (r,) J+ (r3)
= lim T ~ lJw (qi) Ji· (r1) L1 (r,) J+ (rs).
;r!,} ~r w
The quantity lJ, (qi) can be calculated by means of the same perturbation method
that was used previously by the present author in the calculation of the specific
heat jump at the transition temperature,") though in this case the dependence of
the external momentum 7j is introduced into the selectron Green's function Gk (1u)
and vertex corrections r(q2, w). Since it is tedious to vvrite down various contri
butions of diagrams for lJ"' (q_J, here we show only the results.
jJ (?f)= _rrNp .lo)[?Js(w) + (1/8)D2(w) [(q_~?fsr+C?fz?f,2_"l
w ' 2 [u)[ 4 {ID=117z(?f/, w)}
'
(
2 17
)
vvhere ~;=l?fi = 0 and ?ji operates only on L1 (r;). Here use has been made of the
fact that in the dirty limit, the term associated with the noncommutative nature
of the operators ?ji only gives rise to a higher order correction in lj !;0 even at
low temperature,6) vvhere l is the electronic mean free path and !;0 is the BCS
coherence length. Thus we arrive at a quasilocal equation even at low temper
ature. Here [wft?s(W) is giVen in Ref. 2) as
where the magnitude o£ the impurity spm 1s assumed 1/2.
The renormalization
(2·14)
]J+(r).
factor 1!2(qj, w) is the extension of 1J2 (w), which appears in Ref. 2), into a magnetic
field dependent case and is given by
,
Y(q·) = 7r T 2.:: [cu[·1s(o))+ (1/S)~z(w) [_(?J.J?J.s) 2 + (q2 ~q4L]_.
' 2 w !cunn~~J1/z(?J./, w)]
3)
GinzburgLandau current density
The second GinzburgLandau equation which relates the current to the vector
potential is obtained by a similar procedure. Since the reduction involves methods
already described, we \vill not take the space to give more than the final result6l
\vhere
Equations (2 · 21) and (2 · 24) are the basic equations for our discussion. As we
will see later, the temperature dependence of the GinzburgLandau parameter /C2 (T)
is quite different from that of Maki's theory, mainly due to the behavior of ;.:: (cu)
at low temperature and at low frequency. However, there remains some ambiguity
in the form of a(w) an~ ;.::(cu) in the region w<TK. Fortunately no sharp transi
tion between two limiting regwns, and so we can believe such interpolation gives
semiquantitatively correct results.
§ 3.
Magnetic properties
Using the basic equations in the preceding section, we can discuss quite gener
ally the magnetic properties of superconductors near the upper critical field. In
the following we describe several applications of the above equations.
1) The ujJper critical field Hc2 (T) D
If the transition from the superconducting state to the normal state 1n the
presence of a magnetic field is of second order, the upper critical field Hc2 (T)
is determined from the first term in Eq. (2 · 21) by
where q02 1s the lo~west eigenvalue of the following equation:
D,(w)7'tL1+(r) =D,(cu)q02Ll+(r),
qo2 = 2eHc, (T).
If we confine ourselves to the case of alloys where the electronic mean free path
is extremely short, then we can obtain the previous resultll using D0 = r 0v//3
instead of D2 ((})).
2) The GinzburgLandau parameter !C2 (T)
The general GinzburgLandau parameter tC2 (T) introduced by Maki, which
may be determined from the slope of the magnetization curve near He,, is defined
as follovvs :6l
1
where
IC z(T)  1 . ~Y(qo2)
2  4e Z (q02) X' (q02) '
of /C2 (T) at the transition temperature:
,,~ 1
(3 ·9)
where a,= na (T,) /2r.T,. The calculation of JC* can be simplified by the use of
the approximations
a(T,o):::::: 37r(ln TJ(_)2,
8p T, 0
dn
dt, t,~l
::::::3?pT,a(ln TK)'
37:"2 T, 0
c):::::n:!2_ ( ln T K_) 2•
6 Teo
,
(3 ·10)
The result 1s given by
IC* = IC* +
AG
n4
252( (
3
)
( In _T___!£_) 2
T co '
where (J:.G = 0.091. One easily finds that IC* approaches the AG value "XG in
the limit Tco/Tx>;oo.
b)
On the other hand, if T<f:..Tx, we can put
mo (qo2) :::.:::_@1 (qo2) :::.:::_@2 (qo2) :::.:::_~'
jgjNp
where jgj = lgln/4TxNp2• Using the approximations
we can simplify the calculations of /C2 (T) as
( !C2 (t)) 2 ~ n4 ( n ) 1J!C2l((1/2)+X)
 , .  56((
3
). 1 + 4TxP . [1J!C1l((1/2) +X)J2'
where
When T = Tc, this result reduces to the simple formula,
!wlna(w):::::. (1+n) jwj,
4TxP
jwj +na(w)::::::. (1+ _77_) jwj,
4TxP
X= eDoHc2 (T) .
2nT
1
1+ (nj4TxP)
('IC2(tc)) 2:::::.1+_n_.
" 4TxP
Then the Tx/Tc0dependence of IC* is given by
Here use has been made of the approximation
dn:'
dtc 1 tc=l
:::::.4TxP ( In___T_!£_)2•
Teo
One easily finds that IC* approaches the BCSvalue ~Cics = 0 in the limit of high
Kondo temperature, T x/Tc0>;oo, in contrast to Maki's theory. The behavior of
/C2 (tc) / /C as a function of Tc/Tco is given by the numerical calculation for the
wide region of T x/Tco· Results are given in Fig. 1. Here the parameter used
for the numerical calculation is wDjnTco = 1169. Results of numerical calculations
for IC* are given in Fig. 2. The initial slope of the GinzburgLandau parameter
In this paper we have discussed how the Kondo effect affects the magnetic
properties of typeIl superconducting alloys based on the interpolation approxi
mation. Though our results contain the already known limiting cases: the AG or
BCSlike behavior, they are different from those of Maki's theory. In particular,
the initial slope JC* of the GinzburgLandau parameter tends to the BCSvalue
at the end of an infinite Kondo temperature. Therefore we can say that the
study of the magnetic properties near H,2 in typeIl superconductors with Kondo
effect will also provide important information on the Kondo effect in superconduc
tors. Throughout this paper we have neglected completely the effect of the mag~
*> The definition of the Kondo temperature in the YamadaYosida theory is different from
that of the most divergent approximation. Since we made an interpolation between them, our
calculation contains some ambiguity about TK. Therefore, quantitative comparison of our results
with experiments does not seem so much meaningful.
S. Iclzinose
netic field on magnetic impuntles. Since the magnetic field characteristic of the
Kondo effect is given by Hx = kBTx/flB, this approximation is allowed only when
Hc2<f::_Hx. In general, the magneticfield effect on magnetic impurities is expected
to appear as (H/ Hx) 2 in the lowest order. Therefore, the GinzburgLandau param
eter /C2 Ctc) at Tc and IC* are not affected by this effect, and our approximation of
neglecting it is justified.
Experimentally, the Kondo effect in type II superconductors has not been
studied extensively. Most of the previous work'J was carried out in alloy systems
with Tx/Tco<l. For the most interesting case of TK/Tco>l there exist no reliable
experimental results. However, if alloy systems with TK/Tco>l are found in
the future, \Ve belieye that a continuous change in the quantity IC* from the
negati\e yalue to the positive ones can be observed experimentally.
Acknowledgement
Thanks are due to Professor Y. Nagaoka for critical reading of the manuscript.
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