Electromagnetic absorption of light in superconducting Kondo alloys and superconductini alloys with strong electron —magnetic-impurity interaction

PHYSICAL REVIEW

8

VOLUME 21,NUMBER 1 1JANUARY 1980

Electromagnetic

absorption

of

light in

superconducting

Kondo

alloys

and

superconductini

alloys with

strong

electron

—

magnetic-impurity

interaction

C.

B.

Cuden

Universidade Federal da Bahia, Departamento deFisica, Salvador, Bahia, Brasil

(Received 22February 1979)

E}ectro~agnetic absorption oflight in superconducting Kondo alloys and in superconducting

alloys with strong electron

—

magnetic-impurity interaction isdiscussed without recourse to the

quasiparticle approximation. The electromagnetic (em) absorption oflight defined in terms of

Shiba-Rusinov equilibrium one-particle Green's functions isexamined in the limit ofsmall

momentum transfer (London limit) and in the Eliashberg approximation ofneglecting the momentum dependence ofthe electronic self-energy. Forthe specific case ofthe Fe-In alloy numerical calculations have been made for the ratio ofem absorption in the superconducting

and normal states as afunction ofreduced frequency and temperature for different impurity

concentrations.

I.

INTRODUCTION

In 1965tunneling experiments were reported by

Woolf and Reif' on

Fe-In,

Mn-Pb, and Gd-Pb alloys. By using Abrikosov-Gor'kov2

(AG)

theory

of

su-perconducting alloys, Skalski, Betbeder-Matibet, and Weiss have shown that the theoretical conductances for the Fe-In and Mn-Pb systems show poor agree-ment with experiment.

On the other hand the AG theory seems to account for the data, at least for a rare-earth impurity like ga-dolinium.

In

Ref.

3 the absorption

of

light in a superconduct-ing alloy with low concentration

of

randomly distri-buted paramagnetic impurities was calculated, The

interaction

of

conduction electrons by a magnetic im-purity was assumed weak, arid first-order Born ap-proximation was used to treat the exchange

scatter-ing.

The advantage

of

the electromagnetic (em)

absorp-tion'

or Raman-scattering technique'

'

isthat one

can estimate the value

of

the energy gap directly from the experimental results.

In this work we calculate the absorption

of

light in a superconducting alloy containing low concentration

of

paramagnetic impurities for the case

of

strong

conduction-electron

—

magnetic-impurity interaction.

In

Sec.

IIwe give the system Hamiltonian. In

Sec.

IIIwe present the equilibrium one-particle Green's

functions

of

a superconducting alloy. In

Sec.

IVwe discuss a general formalism for the em energy ab-sorption by the isotropic system. In

Sec.

Vwe calcu-late the em energy absorption in the London limit for

various concentrations and temperatures. And final-ly, in

Sec.

VIwe give our concluding remarks.

II. SYSTEM HAMILTONIAN

We take the Hamiltonian

of

the system to have the following form: H

=

d'r

_{yt(r)e(p)y. (r)}

d3r

_{Pt(r)P)(r)p&(r)P}

(r)

+

_X

_J

d'r

_Pt(r)

V.

&(r

—

K,

)p&(r)

J ~2

p= —

i

7,

~(p)

=

2m

The first term represents the kinetic energy

of

the electrons in aconduction band measured from the chemical potential p,

.

The second term represents

the usual phonon-induced attraction between the

electrons introduced by Gor'kov, and the third term represents the interaction

of

an electron with a paramagnetic impurity

of

spin

S

at the position

R,

. The short-ranged potential V

_&(r)

istaken to be

of

the form

where o.isthe Pauli-spin-matrix vector, and

Vl(r)

and V2(r) are the strengths

of

the potential and

ex-change interaction, respectively.

21 ELECTROMAGNETIC ABSORPTION OF LIGHT IN 139

III. EQUILIBRIUM ONE-PARTICLE

GREEN'S FUNCTION

We assume that the magnetic impurities are ran-domly distributed in a superconductor, and that their

concentration is low enough sothat impurity-impurity interaction can be neglected. The effects

of

the

ex-change interaction, which lead to large changes in

or-der parameter and the attractive electron-electron in-teraction can be taken into account simultaneously by using four-component space introduced by Eliash-berg. Namely, the Green's-function matrix

charac-terizing the equilibrium properties

of

the supercon-ducting alloy can be defined in this case as

GS(x(,

x&)

=

i

_(T

[(—_{I(;(x()(i()(x3)}

_])

i

j=1,

2,

3,

4,

where the four-component annihilation and creation operators are given by

'(i(t(x)'

(i(t(x)

y()

(i(t(x)

(1(t(x)

=

_{((1(f(x)(i(J(x) (1(t(x)}

_(i(t(x)),

x

=

r,

(,

(—

4)

and

₍

₎

denotes the thermal average aswell as the average over the possible impurity positions and spin configurations.

In the energy-momentum representation the

Dyson's equation for the 4

x

4Green's-function ma-trix

of

the superconducting alloy averaged over the possible positions and the spin directions

of

the im-purities, can be written

G(p,

it„)

=

Gp(p,

is„)

+c;Gp(p,

i

_e„)

x

_X

(p,

p;i

p„)G(

p,i

_p„)

trices operating on the space composed

of

the

elec-tron and hole states, and on the ordinary spin states,

respectively,

e(p)

is the single-particle kinetic energy measured from the chemical potential,

e(p) =

p3/2m

—

p, and Ii,

(T,

O) is the temperature-dependent order parameter

of

a pure superconductor.

The self-energy matrix

X;(p, p',

i

p„)

satisfies the following integral equation:

X((p3p

'(e»)

=

V(p

p

)

+X

V(p

—

p")G(p",

( p)

x

X,.

(p",

_p';i

_p„)

where the matrix V(p

—

p

')

isthe Fourier transform

of

the electron-impurity interaction potential in the

four-component notation and is given by

V(P P

)

=

₃ V((P

—P')(T3x

1)+

₄ V3(p

—

P

)

x

[S„(T3

x

(r])

+

Sy(T3

x

o'3)

+ S

(T3

x

(T3)] S

=

(S„,

Sy,

S,)

Graphical representations for G

(

p, i

s„)

and

X((p,

p',i

p„)

are shown in Figs. 1and 2, respectively.

Neglecting the Kondo effect, Shiba' and

Rusi-nov"'3

(SR)

have solved the above self-consistent

equations for

G(p,

ie„),

namely,

[G(p,

(e»)]

'

=(

p»(1

x1)

—

e(p)(T3 x 1)

+

(((„(T3X_(r3)

with

where Gp( p,i

e„)

is the Green's function

of

the pure superconductor given by

x(u„+1)'I (u„+

p ) '

(10)

[Gp(p ip») ] '

=i

e»(1 X

1)

—

e(

p

)

(T3X

₁₎

+I

6(T,

O)(T3

x

o3) a»

=(2n

+1)

Tr/p, n

=0,

+1,

+2, +

P=

1/k((T

1is the unit 2

x

2 matrix, v; and o.; are Pauli

ma-(6)

u»

=

p»/It» (T»() '

=

c([2(TN(0)]

'(1

—

s(

)

p(

=

cos(8(

—

8()

a=(T,

p) '

.

(11)

(12)

(13)

(14)

FIG. 1. Diagramatic representation ofthe Dyson's equa-tion for the Shiba-Rusinov 4x4 Green's-function matrix.

FIG.2. Diagramatic representation ofthe self-energy ma-trix ofasuperconducting alloy with strong

140 C.

8.

CUDEN 21

In Eqs.

(9)

—

(13)

the various symbols have the fol-lowing meaning:

h(T,

n) is the

temperature-dependent order parameter

of

the impure

supercon-ductor, ~,Irepresents the time ittakes for an electron spin to flip during scattering from the impurity in a state with orbital momentum I, SI—are the phase shifts describing the scattering

of

an electron by the impurity with orbital momentum Iand spin

projec-tions

+

₂ in the normal metal,

N(0)

is the density

of

states

of

the pure superconductor at the Fermi

ener-gy, and el specifies the position

of

the bound state in-side the gap brought about by treating the scattering

of

electrons by the magnetic impurity in an exact

way,

In this work we shall consider only the physical si-tuation when the orbital momentum

of

the impurity isquenched. In that case the electron scattering by the impurity is isotropic.

For

1=0

Eq.

(10)

reduces to

II II

=u

[1

—((u

+I)'/'(u'+e')

n 0

'],

(1S)

the following relation.'

f 't — 1

co=

ln TK ln TK

+n~S(5+1)

T,

, T

(19)

~(T,

)

=2~gN(0)

—

_{X (I+u„')-'/',}

n)0

where u„satisfies Eq.

(15).

The integral form

of

Eq.

(20)

iso

h(T,

n)

_{=gN(0) J}

I'"o

de„(1+u„2)

'/2

—

_2gN(0)

(20)

d~ e Im 1

—

u' -'/" 21 where coo is the Debye frequency,

f

(p) is the

Fermi-I3irac function, and u isdetermined by the equation

where in the Born approximation one has T/Tx E'0

—

1.

The self-consistent equation for the order parame-ter isgiven

by"

g

=

[~/~(T.

~)],

(r„)

-'

=

c,

[2~N

(O)]-'(I

—

&',

),

(16)

tp

=cos(gp

gp)

(18)

One may note that in SRtreatment the position

of

the local level ~0isindependent

of

temperature and

the sign

of

the exchange interaction.

Muller-Hartmann" and Zittartz' treated the

Kon-do effect in superconductors by means

of

the pole ap-proximation and calculated the transition temperature

of

a Kondo alloy.

In the pole approximation, for antiferromagnetic coupling between the impurity spin and the conduc-tion electrons, ~p as afunction

of

temperature Tand the Kondo temperature TK

of

the alloy is given by

ln '

= —

ln[uo+

(1+

uo)'/']

5(0,

0)

+

g up(up

+

Eo) Ep(1

—

ep)

xarctan

—

up

—

—

m (1

+

eo)—1

IEp 2

+ (I —

fp) 'arctanup with up assuming the following real values:

(23)

=u[l

—

((I

—

u')'

'(eo2

—

u')

'],

₍₂₂₎

A(T,

a)

which isobtained from Eq.

(IS)

by doing a usual analytical continuation to real frequencies e.

At T

=

0

'K,

the order parameter can be calculated from the first term.

of

Eq.

(21)

and isgiven by

y2 &/2[g2

₂₆₂₊

_g(g2 4&2+4)1/2] I'or _g

)

&2 up(T

=0

K g)

[0

f

(24)

n„=

lim

a

~(p,~)

-0+

From Eqs.

(23)

and

(25)

one finds

(25)

At T

=O'K,

the critical value

of

o.at which the su-perconductivity is completely destroyed isdetermined by taking the limit

The gaplessness sets in at

u(rug

=0)

=mph(0, u)

which combined with Eqs.

(23)

and

(26)

gives

-~"I2(1+.

) u(cog

=0)

=

eo25(0,0)e (2g)

(29)

n„=

A(0,0)/2 (26)

(C;)„=

vrN(0)h(0,

0)(1

—

eo2) '

(27)

By using Eqs.

(14),

(17),

and (26) one obtains the critical concentration

of

impurities

IV. ELECTROMAGNETIC ABSORPTION

We are primarily interested in the photon energy absorption in the range

0

(

co&6A, i.

e.

, in the

re-21 ELECTROMAGNETIC ABSORPTION OFLIGHT

IN.

.

. 141 gion

of

low-lying electronic excitations with small

momentum transfer where VF~q ~

«1.

The total rate

of

em energy absorption by the

iso-tropic system subjected to a time-varying perturbation

of

the form

H, „,

= —

M~E_sino)t

can be written'

(30)

M

(q,

t)

=e'+'M (q,

0)e

' '

₍₃₂₎

The integral in Eq.

(31)

is related to the imaginary

W(q,

ru)

=(o(I

e&"—

)E'

x

_J)

dt e

'"'(M (q,

t)

M

(

—

q,

0)),

(31)

where M

(q,

0)

isthe crystal electric dipole moment operator and

part

of

the generalized susceptibility by the fluctuation-dissipation theorem

dt e

'"'(M

(q,

t)

M

(

—

q,

0)

)

=6

V(1

—

e

t'")ImX(q,

cu),

(33)

where Vis the volume

of

the Fourier integration. Combining Eqs.

(31)

and

(33)

one obtains the fol-lowing expression for the total rate

of

energy absorp-tion'.

W(q,

(o)

=

—

Vo)E~ImX(q, o))

(34)

Next we shall calculate the generalized

susceptibili-ty by using the diagrammatic technique suggested by Abrikosov et al.2

The generalized susceptibility in the complex

fre-quency plane can be written

3

X(q

i'„)

=

—

_J)

₃

XTr[(G(p,

iet)I (p,

tet,

p+q

iel

+i co„)G(p+q

i

et+i

~„))]

P

2vr', ,

co„=,

n

=0,

1, 2,.

..

(35)

Vacuum polarization processes and interactions between the electrons in the excited states created in the

ab-sorption process are taken into account by the renormalized vertex part I which can be determined by solving the following integral equation'.

I

_(p,

i et,

p+q,

ie't+i

t«„)

=

y(p,

iet ,

p+q,

i

et+'ice„)

'd3'

XTr[

Vs(q)(r3

x

1)G(p',

i ')et

P

"

(2vr)';,

_,

x

I'(p',

iet',

p'+q,

ie~

+ice„)

G(p'+q,

iet

+ice„)].

d p

+

—

QTr[

V,tt(p

p',

ie~

—

i et

—

)(r3

x 1)

G

(

p',iet)

P

"

(2n)';,

_,

x

r

(

p',i

e(,

p'

+

qiet

+

i,co„)G

(

p'

+

q,i et

+

ico„)]

(36)

where

2

y(

p, i et,

p+q,

i

e~+i

cu„)

=

(r3 x

I)

2m

1 0 1 0

O-1

'

O1

Ve(q)

is the sum

of

bare Coulomb and bare longi-tudinal phonon interactions and V,rt(q, co) is an ef-fective Coulomb potential which includes all vacuum polarization processes. eis the electric charge, and m

denotes the bare-electron mass.

The graphical representation

of Eq. (36)

is shown in Fig.

3.

In metallic superconducting alloy the electron den-sity is high enough to effectively screen the

interac-tion between electrons in the excited states created by em radiation. In fact, for high-electron densities the vacuum polarization processes are dominant. The in-stantaneous nature

of

Coulomb interaction allows

one to neglect the dependence

of

I' on iet in Eq.

(36)

to a good approximation.

The renormalized vertex part in the presence

of

vacuum polarization processes only and for the case

of

instantaneous interaction takes the simple form

r(q,

ir«,

)

=

e(q,

i

«)„)

(3g)

where

e(q,

i

t«„)

is the analytically continued dielectric

constant to Matsurbara frequencies and is related to the susceptibility by the well-known relation

142 C.B.CUDEN 21

t

eff

k+q

~

&+a

FIG. 3. Schematic representation ofthe integral equation for the renormalized vertex matrix I. The vacuum polarization processes and the interaction between the electrons created in the absorption process are considered.

e

=

e(q,

i

c»„0+)

(40)

where the arro~ indicates the analytical continuation As we are looking for absorption in the frequency region far below the collective plasma mode we can safely approximate the dielectric constant in Eq.

(38)

by its static limit value

to the real zero frequency.

In the static limit approximation for the-vertex part I the vacuum polarization processes can be

con-veniently taken into account by replacing the Gor'kov

bare-phonon-induced attractive interaction between the electrons, g, by the effective screened interaction,

gle.

The susceptibility is then given simply by

d

X(qic»„)

=

—,

_X

T

r[ (G (p,iec

)

G(p

+

qi ec

+

i_c»„))]

P

~ 2rr

',

_,

(41)

The fact that the average

of

the product

of

Green's functions is not equal to the product

of

averaged Green's

functions can be taken conveniently into account by introducing vertex corrections.

The analytical expression for the generalized susceptibility in the complex frequency plane corresponding to the two diagrams

of

Fig.4 can be written in the form

X(q,

ic»„)

=

X'(q,ic»„)

+

X"

(qic»„),

where

3

X'(q,

ic»„)

= —

XTr[G(p, ie()G(p+q, iec+i

)c]»

2P"

2n

';,

(42)

(43)

d

"

d

X"(q,ic»„)

= —

_{3 3}

_X

Tr[G(p

i ec)

G(p+q,

i

ec+ic»„)

&cp(p,i ec, p

+

q,i ec

+i

c»„,p

+

q,i ec

+

'i

c»„,

p,

iec')

&&

G(p'+q

i_ec

+ic»„)

G(p',

iec

)]

(44)

In the ladder approximation the average vertex part

4

satisfies the following integral equation:

4(p

ie,

;p+q

ie,

+ic»„;p'+q

iec

+i

c»„,

p',

iet

)

=

(

X(p,

p';i0+)

(

3 tt

+,

_4P"

_2m';,

XTr[(X;(p,

p";i0')('G(p"

+q,

ie;+ic»„)

,

„

x

4(p",

i

ec,

p"

+q,

ie&

+i

c»„,

p'+q,

i

ec'

+i

c»„,p',iec

)

6

(

p

",

iec"u)']

₍₄₅₎

ELECTROMAGNETIC ABSORPTION OF LIGHT

IN.

.

. 143

(a)

FIG.4. Diagramatic representation ofthe susceptibility.

FIG.5. Schematic representation ofthe integral equation for the vertex matrix

4

resulting from the averaging the product oftwo Shiba-Rusinov Green's function matrices.

Green's-function matrix

G(

p,i

el) of

a superconducting alloy given by Eqs.

(9)

—

(14)

and the shaded quare

corresponds to the four vertex part

4.

An overall factor ₂1 in Eqs.

(43)

and

(44)

and the factor _~]. in the second term

of

Eq. (45) come from the redundancy

of

the four-component representation given by Eq.

(4).

In fact itcan be demonstrated that the major effect

of

the vertex corrections isto replace the inverse collision time by the correct transport value v'".

In what follows, for simplicity, we shall neglect the vertex corrections. This will limit the validity

of

our results

for low impurity concentrations. The important contribution to the em absorption will come then only from the first diagram in Fig.

4.

As a matter

of

convenience let us divide the spectral representation

of

the Green's-function matrix

G(

p, icl

)

into two parts characterizing the positive and negative energy spectrum

of

a superconducting alloy in the follow-ing manner: G(p, iei

)

=

G+(p,i si

)

+

G

(

p, ia~

)

where G+( .

)

J& da

a(p, e)

2''I

I KI

—

6

G'(

.

)

"de

a(p,

—

e)

2vol IKI

+

6

and the spectral density matrix

a(p,

a)

is defined by

a(p,

a)

=2ImG(p,

e

—

i0+)

=i

[G(p,

e+i0+)

—

G(p,

e

—

i0+)]

By using Eq.

(46)

the susceptibility given by Eq.

(43)

can be split into four parts as follows: 3

X'( q, ico„)

= —

_2P

_J

2m';,

3

X

Tr [ G+(p,iel

)

G+(p

+

q,i al

+

icu„)

+

G (p,ieI

)

G+(p

+

q,ie&

+

iao„)

+6

(p

icI)G

(p+qi

el+i

co„)

+G

(p

ill)G

(p+q

i~&+ice„)]

(46)

(47)

(48)

(49)

(50)

The first two and the last two terms in Eq.

(50)

determine the Stokes and anti-Stokes part

of

the absorption spectrum, respectively.

In what follows we shall be considering only the Stokes part

of

the spectrum. Then after combining Eqs.

(47),

(48),

and

(50)

one obtains the following expression for the susceptibility:

and

x'(qi~„)

=

x,

'"(q,

ia„)

+

x'

(q,

si

~„),

x'"(q,

i~„)

= —

_Jt

_Jl '

Jt

Tr[a

(p,

a~)

a

(p+q,

a2)]

X

IKg 6] I KI

+

IM+

—

6p

(51)

Xts

(q,ice„)

—

=

—

1

'I,

_Ji

_J

Tr[a(p, —

at)a(p+q,

e2)]

_X

~ d3p ~_d~] ~_{de2 1 1}

2P

"

(2')3

2n 2m. _(» I6I

+

t)

I

61+

IOJg 62

(53)

The frequency summation in Eqs.

(52)

and

(53)

can now be easily performed, and one obtains

x

(q

)=-'Jt

"

JI''")&""

anh

"-tanh

n+

61 &2

C.

8.

CUDEN 21 and

f

X

(q,ice„)

= —

₄

& tanh

+tanh

Tr[a(p, —

p~)a(p+q,

pq)] .

(55)

$3p dat dpg Peg PE)

2

The susceptibility

X(q,

ru) can be obtained by performing the analytical continuation

of X(qic,

o„)

from the discrete points

of

the imaginary frequency axis (n &

0)

to the real axis.

The imaginary parts

of

X'~(q,co) and X's(q,pp) can then be found to be

of

the form

d d

ImX'"(q,

pp)

=

_{4 3} tanh

—

tanh

Tr[a(p,

p)a(p+q,

p+pp)]

(56)

ImX

_(q,

co)

= —

—

_„

i tanh

+tanh

Tr[a(p, —

p)a(p+q,

p+

pp)]

~s I I rrp pp

P(p+Ql)

PE'

2~

By using Eqs.

(9), (11),

and

(49)

and performing the trace in Eqs.

(56)

and

(57),

leads to the following form

for the imaginary part

of

the susceptibility: y3~ +b ImX'(q, rp)

=

(2n)'

"

~ f r 't

P(p+

col) Pp El Q dp tanh 2

—

_tanh 2 m

(I

„s)tp

Im

(I

„r)tp

+I

(1-u')'

'

_(I

-

u')'~'

r 1 I 1 for GJQo)& 1

ImX

(q,

co)=& 3 de tanh

—

tanh

Im,

Im

2m

' "

~ _{2 I}

—

u''"

_I

—u"

'"

i

(58)

1

I

1

(1-

u')'~'

(1-

u')'"

for

M(Mg

(59)

where I

gion

0

(

u

«0,

and is determined by the condition

u

=u(p,

p;T, A)

u'=u(p+q,

a+a,

T,

a)

(60)

1 8e

5(T, a)

tlu

(63)

and, depending on the parameter g the integration

limits in Eqs.

(58)

and

(59)

assume the following values:

(o)p,

~)

for g &pp'

a,b

(0,

~)

for

g~p~p,

'(ppg, pp

—

_a)g) for

(

&e~p

c, d

=

~(0,

)

for)

pp .

(61)

(62)

co~is defined as the maximum value

of

e in the

re-At T

NO'K

the value

of

u when a=co~ satisfies the equation

(pp'

—

up')'(I

—

up') '

'

—

g(op+

ug'

—

2ppug')

=0

for

0(u~(~o

(64)

After u~ iscalculated by solving Eq.

(64)

numeri-cally, a&, is easily found from Eq.

(22).

The two limiting cases

of

interest can now be dis-tinguished:

(a) Pippard limit or extreme anomalous skin-effect

21 ELECTROMAGNETIC ABSORPTION OF LIGHT

IN.

.

.

145

I/~,

p&&wt;~q~ and one can neglect

I/r,

p

.

Then the

effect

of

the impurities disappears and one can treat the superconducting alloy as apure metal. Obviously

one then gets the results

of

Skalski, Betbeder-Matibet, and

%eiss.

3

(b) London limit, or the normal skin-effect region

of

small momentum transfer ~here

I/r,

p

)

)

vF~q~.

One can then neglect q dependence in Eqs.

(58)

and

(59).

This is the case we are interested in.

It

isconvenient to relate the rate

of

em energy ab-sorption in the superconducting alloy to that in the normal metal.

If

one makes the weak momentum approximation introduced by Eliashberg, that is,

if

one neglects the momentum dependence

of

uand u', the integration

Imx'

_(q,

tp)

+ImXts(q,

tp) ImX'~(q,

~)

+~s

for Ctl

)

GJg ~w for cd

4

_oJg

(65)

where

over moments in Eqs.

(58)

and

(59)

can be easily carried out and one obtains the following expression for the relative absorption rate:

t t r

g(p +

tp) PE u u 1 1

l~=

da tanh

—

tanh Im —

2»2

Im,

2,

i2

+Im

z,

t

Im,

2,

t~,

(66)

t i t 1 T 1 t t t

(67)

I„=

'l _{da tanh} (E

+

QJ) 13' It

_Jp

2

—

tanh 2 u

=

u

(e;

T,

n),

u'

=

u(p

+

tp,T,

a)

(69)

stead, we introduce new variables

I

u

=coshz,

z

=x+iy

For

sufficiently low temperatures, the explicitly

temperat'ure-dependent factors in Eqs.

(66)

and

(67)

can be replaced by corresponding combinations

of

ex-ponential factors

of

the form e

]'".

Because

of

the rapidly decreasing exponential fac-tors a good approximation is then obtained by using the value

of

the integrands in the vicinity

of

eu

=

cog for

_f

&pp or in the neighborhood

of

tp

=0

for g

~

pp.

However the results

of

integration assume arather

cumbersome form and will not be repeated here.

In-u

=coshx

cosy

+i

sinhx siny and put

s=siny, t

=coshx

Then one has

u

=

t(1

—

s')'

'+is(t'

—

I)'

'

.

Equation (22) can be written in terms

of

new variables sand tas

(71)

[t(1

—

s )'i2

—a+is(t —

1)'

][(t

—

s)

—

a

+2ist(1

—

s )'i2(t2

—

I)'i2]

+

&0s(t'

—

I)'"

—

t4't(I

—

s')'"][(t2 —

I)'"(I

—

s')'"+ist]

=0,

(73)

p=a/5(T, a)

Equating to zero separately the real and imaginary part

of

Eq.

(73)

leads to the following two equations which have to be solved self-consistently for s and

t:

C.B.CUDEN

t'(1

—

s')'/'

(1

—

4s')

+

t'[2gs

(1

—

s')'

'

—

e(1

—

2s')]

+ t(l —s')'

'(3s'

—

eo)

+

e(eo

—

s')

—

Ps(1

—s')'

'

=0

The solution

of

Eqs.

(74)

and

(75)

gives s

=

s(E;T,(x), t

=

t(E;T, 0.')

In terms

of

variables s and tone finds

(7&)

(76)

~b

_P(~+

₎

Pg t(/2

1)

t/2t'(t'2 1)1/2+(/2

1)

1/2(1 s2)1/2(t'2 1)1/2(1 s'2)1/2

(t'+

s'

—

₁₎

_{(t'+ s'}

—

1) , I'» _/3(

₊

₎

_g

_t(t2

—

_1)'2t'(t

—

_{1)' —}

_(t

—

_1)'

₍₁

—

)'

_(t'

—

_1)'2(l

—

'2)'/2

('+

'

—

_1)('+

'

—

₁₎

(78)

1.0

—

T =

00

E.~=0.6 Tcp

t=/(6;T,

A), t

=t(c+Ql;T,

iX)

s

=

s

(e;

T,

a),

s'

=

s

(e

+

co,T,n)

0.0 I.

0—

I.O T

—

=0.5 Tcp 2.0 6=0.6 B.O

Combining the results for the order parameter with

Eqs.

(22),

(61), (62), (64),

(74),

(75), (77), (78),

(79),

and

(65)

the frequency and temperature depen-dence

of

the relative absorption rate was obtained nu-merically for different concentrations

of Fe

impurities in quenched In films. The results are plotted in Fig.

6.

In numerical calculations the BCSexpression for

the order parameter

6(0,

0)

was used, namely, jk(0,

0)

=

1.

76ksTb where Tb is the transition tem-perature for In at c;

=0.

The energy level

of

the bound state inside the gap

of

the In-Fe superconduct-ing alloy was taken to be ep

=0.

6.

VI. DISCUSSION

00

l.O 2.0 3.0 4.0 I.O

—

T =_0.9 Cp 0.5

00

I.O 2.0

40

awto,

o)

FIG. 6. em absorption rate (relative to its value in a pure

material) for In

(T,

_z

=3.

37'K)

containing Feimpurities asa

function ofreduced frequency (relative to the order parame-ter ofa pure metal at T=O'K) for different temperatures

and concentrations.

Itisevident from our results that the threshold in the absorption spectrum is much less pronounced than the one found by Skalski et ai.

'

The appearance

of

the em energy absorption at en-ergies much less than the order parameter /b.

(T,

a)

can be attributed to three different mechanisms: (a) The formation

of

bound states inside the gap brought

about by treating the self-energy resulting from the strong electron-magnetic impurity interaction in an

exact way. (b) The spin-dependent interactions which are not invariant under the time-reversal operation cause the finite lifetime

of

Cooper pairs and consequently the broadening

of

the energy levels.

(c)

Thermal breaking

of

Cooper pairs.

Itwould be interesting to check the AG and SR

theories with the systematic study of transition-metal and rare-earth impurities in superconductors.

In fact, one should compare the temperature and

concentration dependence

of

the energy gap deduced from the tunneling experiments by

Reif

and Woolf'

tech-21 ELECTROMAGNETIC ABSORPTION OF LIGHT

IN.

.

.

147 niques like, the heat capacity and critical-field

meas-urements, infrared absorption, infrared transmission

on thin films, surface Raman scattering

of

light, ther-mal conductivity, ultrasonic attenuation, spin suscep-tibility, and nuclear spin relaxation rate.

Measurements

of

the transition temperature as a function

of

impurity concentration close to the critical

concentration indicate systematic deviation from the theory. ' '~ The origin

of

this discrepance may be partially attributed to the onset

of

aferromagnetic or

antiferromagnetic ordering among the impurity spins.

In this work we do not consider the effect

of

the indirect impurity-spin interaction brought about by the polarization effect

of

conduction electrons which may lead to magnetic phase atsufficiently high-impurity concentrations.

Spin fluctuations

of

a magnetic phase near the criti-cal temperature should play an important role in

understanding the experimental results. The

com-plete theory should discuss the dynamics

of

the im-purity spins coupled with the conduction electrons. Moreover, at higher concentrations the proper averaging procedure over different. impurity

confi-gurations taken into account by the vertex matrix

4

should lead to significant corrections.

All that one may expect is that the qualitative

features

of

the presented theoretical calculations should agree reasonably well with the experimental results at sufficiently low temperatures and

concen-trations

of

magnetic impurities.

ACKNOWLEDGMENT

This work was supported in part by the

Financia-dora de Estudos e Projetos

(FINEP)

of

Brasil.

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F.

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J.

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