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Submitted on 1 Jan 1978
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SPATIAL VARIATIONS OF ORDER PARAMETER
NEAR ANDERSON IMPURITY IN
SUPERCONDUCTORS
E. Šimánek, S. Yoksan
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-874
SPATIAL VARIATIONS OF ORDER PARAMETER NEAR ANDERSON IMPURITY I N SUPERCONDUCTORS +
E. Simanek and S. Yoksan
Department of Physios, University of California, Riverside, California 92521, U.S.A.
Résumé.- L'objet de ce travail est de calculer dans le modèle d'Anderson les variations spatiales du paramètre d'ordre A(r) autour des impuretés-3d dans l'aluminium. On montre que l'effet de la diffu-sion résonante est en général beaucoup plus fort que l'effet de la diffudiffu-sion inélastique. Pour I « k„r « e_/w_ le comportement de (A(r)-A ) _ .. est donné par A (5-10 sin^-or) (k„r)- 2.
F F D o résonant r o * F
Abstract.- The purpose of this work is to calculate, using Anderson model, the spatial variations of the order parameter A(r) near the 3d-impurities in kl. It is shown that the effect of the resonant scattering is much stronger than the effect of the inelastic scattering. For 1 « k r « e-,/wn the
behavior of (A(r)-A ) is given by A (5-10 sin2k r)(k r )- 2.
1. INTRODUCTION.- Since the first attempt by Tzuzuki and Tsuneto /l/ several calculations have been per-formed of the spatial dependence of the order para-eter A(r) in superconductors containing a single ma-gnetic impurity (see Schlottmann/Z/ and references therein). For a nonmagnetic impurity A(r) has been calculated by Fetter /3/ taking into account the nonresonant scattering of conduction electrons. The importance of resonant scattering, caused by tran-sition metal impurities, for the decrease of T , was first demonstrated by Zuckermann /hi using the Anderson's model for localized states in metals /5/. The purpose of the present paper is to describe a calculation of A(r) using the same model. The moti-vation for our work is the relaxation study on Ai, nuclei in Aft-Mn alloys by Daugherty, et al. /6/. These authors have presented evidence for an anoma-lous broadening of the density of states below T . In their alloys the average distance of the resona-ting nuclei from the impurities is of the order of 100 A and thus one expects that short range varia-tions of A(r) could provide a possible explanation of the above broadening.
2. THEORY.- We employ the Hamiltonian of a BCS su-perconductor containing a single impurity at the origin :
H = Hn - d3r [A*(r)>Pt(r)'i'^(r) + h . c ] (1)
Research supported by the U.S. Energy Research and Development Administration under Contract EY-76-S-03-0034 P.A..
where H is the Anderson Hamiltonian / 5 / and n
A(r) = g < ¥,(r)>i,+ (r) > is the local parameter. Near
T the anomalous average can be evaluated to first order in A yielding
M r ) = gjd
3r'A(r*)
dT < TT^(r,o>¥*(r,o)¥+(r* . T ^ f r ' . T ) >p (2)
Jo
where 6 = 1/T and the average of the time ordered operator product is to be taken over the states of H = H . Explicit evaluation of this average bears certain similarities to the calculation of the d e -pression of T in superconducting alloys by Sakurai /7,8/. Performing the angular average of the kernel of the gap equation, we obtain, from equations (1) and ( 2 ) ,
M r ) = gT Z | d V A ( r ' ) { G (r-r',e)G (r-r',-e) ej o o
+ ( 2 4 + 1 ) ^ - [ 6o( n ' , 6 ) Gl( t)- e ) Gd( - e ) G1( i ' , T e )
o
+ Go(r-r',-e)GJl(r,e)Gd(e)Gf(rI,e)] + ( 2 4 + 1 ) ( ^ - )2
o (G)l(r,e)GJt(r,-e)Gd(e)Gd(-e)6A(r',e)GJl(r',-e)]
- (2J!,+ 1 ) ( ^ - )2T I Jpa(r,e)GA(r,-e)Gd(e)Gj(-e)
o e'
r
++
(£'
e'
>Gd
(s'
)G<i
(-
e')
GS.(
r,'
e,>
GJl
(-
r,'-"
e'lI
} (3;)where the summation over discrete frequencies e = (2n+ 1)TTT involves a Debye-cutoff at |e| = O L . G0= G ^= o, where G^ is the 8- = 2 component of the
con-duction electron propagator / 9 / . G, is the Green's, function for d-electrons of energy width T. The
symbol T++ (E,E') stands for the vertex part /7/due
to inelastic scattering of up and down spin elec-
trons via the repulsive interaction
U.For a symme-
tric Anderson model the E-dependence of both r++and
G can be determined (for
Eand T
<<Kondo tempera-
d
ture T
)via the microscopic Fermi liquid theory
k
/7,8,10/. Consequently for transition-metal impu-
rities with Tk large compared to wD the kernel of
the integral equation (3) can be calculated as an
expansion in uD/Tk. TO estimate the spatial varia-
tions in the vicinity of the impurity we iterate
equation
(3)by putting A(r')
=.
A
and evaluate the
kernel to the leading order in wD/Tk. For distances
such that
1<<
k r
<<
E ~ / w ~
the &-summations in
F
equation (3) can be performed yielding
where
z++
is the odd-part of the normalized d-elec-
tron susceptibility /lo/. For g - M n alloy we put
T
=1.4 x
lo4
K, wD
=4
xlo2
K and
z++
1.10 /8/.
With these parameters the third term of equation(4),
representing the inelastic scattering, turns out to
be only about 15 x
lo-'
times the magnitude of the
first resonant scattering term. For the nearest
neighbor AR-shell (kFrl=5) equation (4) predicts
(A(rl)-Ao)/Ao
=
1showing that resonant scattering
leads to a strong change of the local order parame-
ter in the vicinity of the Mn-impurity. This is in
contrast with the calculations of the change of Tc
/ 7 , 8 /
where the inelastic contribution is 1arger.In
general, for alloys with Tk large compared to
"D
(such as Fe group in AR) the inelastic contribution
to equation (4) is small compared to the resonant
contribution. It is interesting that Parvin and
MacLaughlin /11/ recently found an evidence that Mn
and V in AR produce a density of states broadening
of about the same size. This independence of theim-
purity T provides an additional argument to antici-
k
