Antiferromagnetic fluctuations and neutron scattering in high-Tc superconductors

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Antiferromagnetic fluctuations and neutron scattering in high-Tc superconductors

S. Maleyev

To cite this version:

S. Maleyev. Antiferromagnetic fluctuations and neutron scattering in high-Tc superconductors. Jour-

nal de Physique I, EDP Sciences, 1992, 2 (2), pp.181-192. �10.1051/jp1:1992132�. �jpa-00246471�

J. Phys. I France 2

(1992)

181-192 FEBRUARY 1992, PAGE 181

Classification Physics Abstracts

74.70V 74.20 75.40G

Antifierromagnetic fluctuations and _neutron scattering in high-Tc superconductors

S.V.

Maleyev

Leningrad

Nuclear Physics Institute, Gatchina, Leningrad 188350, U.S.S.R.

(Received

28 August 1991, ^revised 19 September _1991, accepted 8 November

1991)

Abstract. The magnetic neutron scattering data in the

high

Tc superconductors _are di~

cussed and _an attempt of their theoretical interpretation is given. It is shown that the _conven- tional Fermi liquid theory is incapable to explain the data above Tc, but they strongly support the nested Fermi liquid theory. This theory predicts strong anisotropy of the scattering in the

plane

perpendicular

to the

magnetic

rod. The observation of this anisotropy would be _a crudal experiment for this theory. Experimental ^data below Tc _are qualitatively explained in the &ame of the Abrikosov-Gorkov theory of the superconductivity with _a magnetic disorder.

1. Introduction.

Magnetic

fluctuations

play

_an

important

but not very clear role in

high-Tc superconductors.

Experimentally they

_are studied

mainly by

nuclear _resonance

(NMR

and

NQR)

dud neutron

scattering. Recently

results obtained

by

resonant methods _were

throughly

discussed _on the

ground

of

nearly antiferromagnetic

Fermi

liquid

model

[1- _4].

It _was shown that this

theory explains fairly

^well

existing experimental

data above n Tone

suggests

^that

antiferromagnetic

correlation

length

is

proportional

to

T~~/~.

_{The neutron}

scattering

data _were

published

in [5

-13].

The most detailed results have been obtained for

YBa2Cu306+~

[6

-12].

Here _we discuss them

briefly.

The main

findings

_are the

following. I)

The neutrons

inelastically

scatter in the

vicinity

of the direction

(1/2, 1/2,1)

and therefore there _are

strong antiferromagnetic

fluctuations in

Cu02 Planes.

The

corresponding

correlation

length f

> a, is temperature

independent

in contradiction with

assumption

made in

[1-

_4].

2)

There is _a drastic

change

of the _energy

dependence

of the

scattering intensity

between _z

= 0.45

(Tc

₌

45K)

dud _{z =}

0.5

(Tc

_" 50

K) [9,12].

In the first _case the

intensity

decreases with

increasing

_w, and in the

second _one it increases. At the _same time the AF correlation

length

^decreases

by

_an order of

magnitude.

A drastic

disruption

of

spin-spin

correlation

length

_upon hole

doping

_{[9] was}

suggested.

Below _we will discuss the

region

_z > 0.5

only. 3)

There is _{a gap} 26 in the spectrum of the scattered neutrons for T « Tc

[7,8,11,12].

This gap indicates that the

scattering

is

related to the carriers excitation. But the _gap value is rather small. If _we accept ^the

beginning

of the inelastic

scattering

in

figures

8 and 9 of reference [6] ^and in

figure

3 of reference [11] as

26 then _we obtain

26/Tc

₌

0.5, 1-1,

^and 2.0 for _{z =}

0.51(Tc

₌

47K),

_z

= 0.60

(Tc

₌

53K)

and _{z =} 0.69

(Tc

= So

K) respectively.

In all these _cases this ratio is small in

comparison

with the BCS value

2A/2~

= 3.5. At the _same time the AF correlation

length

decreases with

increasing

z [8], ^{and therefore the}

doping

_suppresses the AF fluctuations.

Unfortunately,

there is almost total lack of the

experimental

data for _{z ci} I.

Only

_one

point

in

figure

2 of reference [10] ^for z = 0.9

(Tc

= 80

K)

indicates that

26/Tc

_ci 6. This value is in _a

good agreement

with

macroscopic

measurements

(see

for

example [14]).

From these results _{one can} draw the

following

conclusions:

I) appearing

of the _gap below Tc indicates that neutrons scatter _on carriers.

2)

There is _an enhancement of the

scattering along

^the

magnetic

rod

[Q

₌

(1/2, 1/2,1)]

due to the AF fluctuations.

3)

^{Low value of the ratio}

26/Tc

and its

increasing

with

suppression

of the AF fluctuations supports a

suggestion

that the

pair-breaking scattering

of the carriers _on the AF fluctuations is _very

important

at low _z, and that in this _case the system ^is near the threshold of the

gapless superconductivity

[15].

One extra

important

observation should be noted. In the normal state the

imaginary

part

of the

spin susceptibility

is _a

decreasing

function of T [7,

8,10].

If this function is determined

by

carriers such _a behaviour is in _a contradiction with the conventional

Fermi-liquid theory.

In the last _case

Im,

_x is T

independent

^if _w < EF.

In this _{paper we} discuss the

following problems.

I. The enhancement of the carrier-neutron

scattering

due to the AF fluctuations in the frame of one-fluid and twc-fluid models

[ii (Sect.

2).

2. In section 3 _we consider the

scattering

in the normal state and demonstrate that

experimental

data

strongly

support the nested Fermi

liquid theory [16].

3. The neutron

scattering

below Tc is discussed in section

3,

where _we show that the Abrikosov-Gorkov

theory

of the

superconductivity

with _a

pair-breaking

interaction [15] ^{is in} a

qualitative

_agreement with the

experimental

data.

2. The models.

It is well known that the neutron cross section for _a

magnetically isotropic

systems has the form

i~~w

~~~°~~~ ~~~~~~~

x[I ~~li~j~T)]

~~~

where _{ro =}

e~/mc~,

_{7 =}

1.91, Q

_" ki

kf,

_w

= E; Ef and

x(Q,W)

₌

(i13) /~ dte'~'(isi(t), _{SfQ(°)i) (2)}

where

S(

^{is the Fourier transform of the}

spin density.

2. I ONE-FLUID _MODEL. As it has been noted in

Introduction,

Q and _w

dependence

of the

cross section could be

explained

Tone takes into account both the AF fluctuations and carrier

excitations.

Recently proposed

the

nearly antiferromagnetic

Fermi

liquid

model

(NAFL)

_{[1 4]}

provides

_a natural

description

of such _a behaviour. We

begin

with the consideration of this model. It is based _on the

following assumptions [ii. I)

There is

only

_one S

=

1/2

electronic

degree

of freedom _per

Cu02

unit ceu.

2) Spins

_are

antiferromagneticauy

correlated due to

strong exchange

interaction

Jq.

^{In the mean-field}

approximation

from these

assumptions

the

N°2 ANTIFERROMAGNETIC FLUCTUATIONSIN SUPERCONDUCTORS 183

following expression

^{has been obtained} [1,3]

x(Q,W)

₌

(~(((

~~, ⁽³⁾

where

K(Q, w)

describes the

dynamical properties

^{of the} system ^{without AF} correlations. For

an ideal

Fermi-gas K(Q, w)

is the Pauli

susteptibility.

If the Stoner criterion is

nearly

fulfiued:

JrK(r,o)

_~

i, (4)

where _r is the twc-dimensional

reciprocal

vector

corresponding

to 2D AF

fluctuations,

_we have strong enhancement of the

susceptibility

_near 2D AF

Bragg points,

for

example

at

Qo

₌

(xla)[1/2,1/2, (a/b)fl.

In this _case

i

JqK(Q,o)

=

(ta/t)211

+

(qt)21, (5)

where _q

=

Qjj

+ r, qa <

I, to

~- a and

(

»

to

is the

antiferromagnetic

correlation

length.

^If

we now

neglect

the _w

dependence

of Re

K(Q,w)

then from

equations (3)

^and

(4)

we have

~~~~q ~~

_~

(f/fo)~ImK(q,W)

'

ii

+

(qf)~l~

+

(f/fo)~iJrIm K(Q, _W)i~'

~~~

If in the

nonsuperconducting

state

ImK(Q,w)

~- w

following _[I]

_we write

ImK(Q,w)

₌

K(q,o)1, (7)

where

rq

is the characteristic energy of

spin

fluctuations at _wave vector

Q. Using equations (5)

and

(7)

_we obtain

[ii:

(f/fo)~(W/WSF)

~'~~X(~'°~)

_"

(8)

ii

₊

_(qf)~i~

₊

_(W/WSF)~'

where _wsf

"

(rq/x)(fo/()~

_<

(rq/~)

is the

typica1energy

^scale for the

antiferromagnetic

paramagnons that describe AF

spin dynamics

_[1,

_3].

From

equations (7)

and

(8)

_{we see} that _near 2D

antiferromagnetic Bragg position

the _neutron

cross section is enhanced

by

the factor

(f/fo)4.

Due to this enhancement the neutron-carrier

scattering

becomes

observable,

because

rq

_~- EF and the ratio

w/rq

is smau.

2. 2 TWO-FLUID MODEL. This model is based _on the

assumption

that there _are two differ- ent

spin degrees

of freedom _one is related to localized

Cu~+ _spills

_and the other with delocalized holes

[ii.

It is

argued

that this model is _not

adequate

to the real situations in the

high-(

system. Nevertheless _we discuss it below because in _our

opinion

the situation is not clear yet.

The

exchange

interaction in the twc-fluid model has the form

v =

L f _{drsji(B~ r)s(r), (o)}

i

where Si _are the localized

Cu~+ spins

and

s(r)

is the

spin density

of the delocalized holes. In this

theory

there _are

fouowing susceptibflities:

_{xcu;cu # xA, xo,o #} K and xcu,o " xo,cu # xI.

The interaction

(9)

mixes them and _we have the

following equations

xA = x~°> +

12x(~K(°ixA

K =

K(°I

₊

I~K(°lxflK ₍₁₀₎

~~ =

j~(°)

~

Hence,

for the total

susceptibflity

ofthe

system

we have

~(o)

~

~(o)

₊

2Ixf~K(°1

X ^~

(11)

i

I~xf~K(°1

Here the AF fluctuations _are described

by

_xA. Therefore

xA(Q)

has _a maximum at

Qjj

= r.

In the

simplest approximation

K i8 _a Pauli

susceptibility. Really

the interaction

(9)

must be taken into account in

evaluating Xf~

^and

^K(°I

as weu. For

example,

if _we

neglect it,

the system of

Cu~+ spins

would be

antiferromagnetic.

The AF fluctuations in systems ^with a

short-range

AF order have been the

Subject

of intense

study during

the last several _years [17

20].

It _was

proposed

that it i8 _a gap A

~-

Ja/f

at

T ₌ 0 in the spectrum ^{of the AF}

fluctuations,

where J is the Cucu

exchange

interaction and

f

is the correlation

length.

Below _{we assume} that in the whole

w,T

_range of interest the AF

subsystem

is not excited.

Therefore in this _{range we can}

negleit Imxf~

and from

(10)

_we have

Imx(Q,w)

-

~i~~ilii~~i,i~iilliii~ ⁽¹²⁾

Near the AF

Bragg point

the

susceptibility xfl(Q, ₀₎

_may be

represented

in the

following

form

xf~(Q,0)

=

f(qf)w(fla)/J, (13)

where

f(0)

₌ and

p(fla)

_» I

If

» a.

Therefore if

I/J

< I _we have

Imx(Q,w)

₌

If xfl~(Q,0)ImK(°I(Q,w))I I)x(°l~(Q, 0)K(°I(Q,w))~~ (14)

K(°I(Q, 0)

_~-

1/EF

where EF is the

largest

_energy of _our

problem.

Then if

(I) /EF)xf~(Q, ₀₎

_«

1 _{we can}

replace

the denominator in

(14) by unity.

In this _{case we} have enhancement of

Imx

due to the factor

(Jr xA)~.

Let _us compare now

expressions (6)

and

(14).

We _see for both models _near 2D AF

Bragg point

and at small _w that Im _x is _a

product

of two factors. The first _one is _a function of

(qf)

and describes _an enhancement connected with AF fluctuations. The second factor determines the

spectrum

^{of the carriers. Such} a factorization has been

reauy

observed in neutron

experiments [7,8,10],

where it1vas demonstrated that the _q

dependence

of the

scattering intensity

is the

same for different values of _w.

It should be noted that _an enhancement of neutron-carrier

scattering

takes

place

also in the

small-Q region.

This enhancement is connected with the _neutron

scattering

_on the current

density

fluctuations

[21].

N°2 ANTIFERROMAGNETIC FLUCTUATIONS IN SUPERCONDUCTORS 185

3. Neutron

scatterblg

b1normal _state

(T

>

Tc).

We _see from

equations (6)

and

(14)

^{that the} spectrum ^{of the scattered} neutrons is determined

mainly by

the factor

ImK(Q,w)

which describes carrier

spin density

fluctuations without

exchange

enhancement. This function has been examined in _many different _cases

(see

for

example

Refk.

[10 -12,14]).

Here _we discuss it _{once more}

bearing

in mind the conditions of the neutron

experiments.

We

begin

with the _case of the conventional Fermi

liquid theory,

when _one could

neglect quasipartical

interaction. In this _{case we} have weu-known

expression:

Im

K(Q,W)

_"

] _{Lin(£k) n(£k+Q)1 6(£k+Q}

_{Sk W),}

_(IS)

where

n(z)

is the Fermi-fiJnction _Sk

=

Ek

_p is the

quasi-particle

_energy and _p is the chemical

potential. Usuauy

in the

theory

of metals it is

supposed

^that

Q

4C

*F

and

Im

K(Q, w)

₌

(3~n3nW)/(16 EFkFQ)

^D ₌ 3

(n2nw)/(4EFQkq)

D ₌

2, ~i~~

where EF »

T,

_(w( <

(QkF/m),

and

(k(

₌

kJ Q~/4)~/~

^and _nD is the number of carriers in the unit cell. But in _{our case}

Q

_"

xvfla

is not small and

equation (16)

is not

applicable.

If for

example

_we consider _an ideal Fermi gas, then for

Q

>

2kF

instead of

(16)

nom

(16)

at T _- 0 _we

get

Im

K(Q, w)

_~-

l~p~

exp

[(Q~/4) kJ] /2mT)

sinh (w

/2T). (17)

This

expression

is

exponent1ally

small. The _reason for _a strong

decreasing

of ImK at

large Q

becomes evident if _we rewrite

equation (is)

in the

following

form:

~~

~~~'~~

~~~~

~'~~~~iii [

cash

~~~~12T)~osh)~k/2Tl'

~~~~

Here if _{w -} 0 the

argument

^{of the b-function is} zero at k _ci

-Q/2.

In this _case sk+q t Sk t

s(qj2)

^{is not small in}

comparison

with T and _we have

ImK(Q,w)

~- exP

i-I£(q/2>1/Tl. (lo)

Obviously equation (17)

is _a

special

_case of this

expression. Equation (lo)

is rather _a

general

one and if it takes

place

the neutron-carrier

scattering

is _very small in _a contradiction to the

experimental

data [5 9]. ^{But this}

equation

does not

hold,

if both _Sk and sk+q~ where

Qo

_"

(«la) (I, I)

_may be _near the Fermi surface and the

following

condition takes

place

Sk, sk+q~ ^{< T.}

(20)

Particularly,

this condition takes

place

in the _case of sc-called

nesting, (see [16,22,23]

and references

therein)

when

Sk + sk+q~ ci 0.

(21)

An

example

of

nesting

is

given by

Sk "

-2t(cos

k~a + cos

kya) 6p, (22)

«la

-W/ «la

-«la

a) b) c)

Fig. 1. Nested Fern~i surfaces

(solid line)

in three _cases:

a)

perfect nesting, bp

= 0;

b)

nesting with bp _< 0, ^and

c) nesting

with bp > 0.

where (6p( « t and the Fermi surface is determined

by

the condition _Sk

" 0. It is shown in

figure

I for three _cases

a) 6p

₌

0, b) 6p

< 0 and

c) 6p

> 0.

The nested Fermi systems _are

extensively

studied _now in relation _to the

high-Tc problem [16, 22, 23].

^In

particular

it has been shown [16] ^{that the}

nesting

_may

provide

_an

explanation

for linear T variation of the

resistivity

and is

roughly

consistent with the

predictions

of the

phenomenological "marginal"

Fermi

liquid theory [24].

It should be noted also that the ^" nest-

in~' parametrization

of the neutron

scattering

data has been discussed in [12].

Nevertheless,

the

theory

of the nested Fermi

liquid

is far from the

completeness.

Below _we

analyse

the sim-

plest expression

for ImK. It must be considered _{as a very}

rough approximation.

Its

possible generalizations

_are discussed in [16]. ^If q = 0 from

equations (18)

and

(22)

we obtain

sInh

(w/2T)

~~

~~~°'"~

"

2COSh

1(6»

+

w/2)/2Tl

COSh

1(6» w/2)/2Tl

^{~ "}

F(w)

₌

(x/2N) £

_6[w

_{4t(cos k~a}

₊

cos

kya)]. (23)

k

The temperature

dependence

^of

ImK(Qo,w)

is determined

by

the first factor in

equation (23).

In the _case of the

perfect nesting (6p

₌

0)

it is

equal

to

tan(h w/4T).

In reference [16] ^{it is}

argued

that if the interaction is taken into account, ^{thi8 factor should be}

replaced by

tanh

(w/7T)

where ₇

~- 1. If

)6p(

> T _we have ImK

~- exp

(-(6p)/T)

and the

scattering

is very small.

Hence,

_{we see} that

experimental

results [7

-12] strongly

support ^almost

perfect nesting

in

YBa2Cu306+~

for 0.51 < _z < 0.69.

It is _easy to evaluate the function

F(w)

in

equation (23)

and if _w « 4t _we have

Im

K(Q°'")

₌

16~t cash

j&(~j~ji(~jiii)oil(ij

w

/2) /2T~ (24)

The similar

expression

is in

[20]. Comparing equations (16)

and

(24)

_{we see} that for _w

~-

T, 6p

< T and

EF

_'~ t the nested

expression (24)

is much

larger

than the conventional Fermi

liquid

_one

(16).

Therefore the

nesting provides

the additional enhancement of the

magnetic

neutron

scattering.

In

equation (24)

_we have _a factor In

(4t/w)

which increases _{as w}

- 0. At the _same time

ReK(Qo,0)

~-

In~(t/T).

But there _{are many reasons} which prevent ^{from such} a behaviour.

N°2 ANTIFERROMAGNETIC FLUCTUATIONS IN SUPERCONDUCTORS 187

First of all the real 3D nature of the

high

Tc systems ^{should be mentioned. Below} we don't consider this

problem.

In [16] ^{it i8}

supposed

that the

density

of states near Fermi surface is _a

constant instead of In

(t/w)

_as in the ideal

nesting

_case. ^This

assumption

does not

change

_our

principal

observation about the enhancement of Im K at _w < t.

For q

#

0 if _qa < I from

equations (18)

and

(22)

we have

ImK(q,w)

₌ sinh

(w/2T)(~/8Nt) £ _6[cos

_{k~a +}

cos

kya-

k

(q~ a/2)

sin

k~a

_(qy

a/2)

sin kga w

/4t]

lcosh

^(w

^/2T)

^{+ cosh}

^(q~a

^{sin k~a + qya sin kga +}

^{6p It) (25)}

T

^~

This

expression

can be

analysed analytically

in the two

linfiting

cases: (6p( < T and (6p( » T

[25].

^{In the first} case we have

~

sinh

(w/2T)In (4t/w) Ii

~

~~'"~

16xt cosh

(w/2T)

+ cosh

(taq+ IT)

^~

~ cosh

(w/2T) /cosh _(taq- IT)1'

^~~~~

where _q~

= ]q~) + (qy(. ^This

expression

exhibits _{a very}

strong anisotropy

in the

plane

_perpen-

dicular _to the

magnetic

rod.

For

example,

_{if (q~( = (qy( =}

q/vfi

we have

~~~~~'"~ ~~~

~~~~~~~ ~~~~~i

2cosh~(w/4T~

+

cosh (~)

_{+ cosh}

_~~fl ^~ ₍₂₇₎

Here the first _term is _q

independent

and the second has the width _qo

=

T/(atvfi).

_For

larger

_q it is

negligibly

small. If for

example

t ₌ 0.5 eV

[22,

23] ^{and T} = 300K _we have go = 3.7

x10~~«~~.

As _a result for all

experimentally

measurable _q this term may be

neglected.

In the other direction where (q~(yj = q and qy(q = 0 both terms have the width

qovfi

and _may be

neglected.

Therefore _{one can} observe the

scattering

in the first _case

only.

It should be noted that

really

in both _cases of

La2-~Sr~Cu04

and

YBa2Cu307

_{IS 12]} the

scattering

has been measured

exactly

at this direction. The

investigation

of the

scattering

at (q~(y~( = q, qy(~~ ^{= 0} would be _a crucial

experiment

for the

nesting theory.

If _w

~-

T,

T «

)6p]

« t and

tqa

» _w from

equation (24)

_we

_get

~~~~~ ~~

_~

Sinll(w/2T)

In

((2t)/16wl)

~

' 32«t _(COSII

(W/2T)

+ CC6l1

1(q+at 16Pl)/Ti

~ COSII

(W/2T)

+ CC6l1

1(q-

at +

16»1)/Ti

^{~ COSh}

(W/2T)

+ COSII

I(q-

at

16Pl)/Ti~

~ cc&h

(w/2T)

+

cojh _j(q+

_{at +}

6p)/T~ ^(~~)

Here the second and the last terms should be omitted _as

proportional

to

exp(-)6p( IT).

This

expression displays

rather weak

anisotropy compared

with

equation (27).

But in this _case Im K has

sharp

maximum at _q

~- (SW

)tat

with the width of order of

T/ta.

It would be considered _as

"incommensurate behaviour" observed for

scattering

in

La2-~Sr~Cu04

_[5]. But

really

such _a behaviour must be masked

by

the rather strong

q-dependent prefactor given by equations (8)

and

(14)

in the small

q-region. Nevertheless,

the "incommensurate behaviour" should _appear in _a rather broad _range of parameters and

experimenta1results

for

La2-~Sr~Cu04

_may be

explained

_on the base of

equation (24)

for )6p( ~- T. But relevant

fitting

of the

experimental

data should be done,

4. Neutron

scattering

below

Tc.

As it _was

pointed

out in introduction the

scattering intensity

below Tcⁱⁿ

YBa2cu306+~ displays

two remarkable features [7 12]: ^{1. It has} a gap 2b and for 0.51 _{< z} < 0.69 the ratio

26/Tc

is less than the BCS value

equal

to 3.5. 2. With

increasing

w the

intensity

becomes

larger

than it is above Tc. ^Below _we

give

_a

qualitative explanation

of both these features _on the

ground

of the well-known Abrikosov-Gorkov

theory (AG)

of the

superconductivity

_[15] in the _presence of the

spin

fluctuations

(see

also

[26]).

It should be noted however that

strictly speaking

this

theory

is not

applicable

to the

high-(

_systems because it does not take into _account the

nesting

discussed in the

preceding

section. But at present ^relevant

theory

has not been

developed

and

we use the conventional _one

pretending

to _a

qualitative description

of the

experimental

data

only.

The

principal

idea of the AG

theory

is the

following.

The

Cooper pairs

_are

singlets

and if in the system we have _a

spin-dependent

interaction of the carriers with _any other

objects

such

as

paramagnetic impurities

AF fluctuations etc. it

destroys

the

Cooper pairing

and diminishes

Tc,

the order parameter AT

(Cooper pairs

_wave

function)

and

superconducting

_gap ST. In this

case ST _< AT in contrast with the

nonspin-flipping

_case. Moreover with the

increasing

^{of the}

spin-flipping

interaction the zerc-temperature _{gap So} decreases faster than

ho

and it is _a

region

of the

gapless superconductivity

where ho

# 0,

but So = 0. In this _case the carrier excitations

are

overdamped.

In the

simplest approximation

the

principal

parameter ^{of this}

theory

is _a time _ra of the

spin-flipping

carrier

scattering

in the normal state. In the weak

coupling

limit the

gapless region

is determined

by

the condition

(«/7)exp (-«/4)

_Ci °.8° <

(rsToo)~~

<

(«/27)

_#

°.88, (29)

where Too is the transition temperature for _{ra = co} and _{In 7}

= 0.577 is the Euler _constant.

Below Tc ^there are two Green functions

Gk(w)

^and

Fk(w). They

_are determined

by

the Gorkov

equations [15, 27]. Using

these functions for the carrier

susceptibility

K _we have

~~~'"~ ~~ ~ ~~+~ ~"~

~

~~ ~~

~)

'

(30)

~~~+Q ~~~

~

~i)

^~~

^~~~ ~i)j

where _w dud _{wi are} Matsubara

frequencies.

This

expression

must be

averaged

_over the

spatial

distribution of the

spin-flipping

scatterers.

In _a

general

_case, it is _a rather cumbersome

problem.

But if the dondition

Qvfra

>

(31)

N°2 ANTIFERROMAGNETIC FLUCTUATIONS IN SUPERCONDUCTORS 189

is fiJlfilled _{one can} show

by

^the method of the reference [27] ^that

W(f7)

_m

©©(fl+fl), ₍₃₂₎

where the bars denote this

averaging.

In [15] ^the

following expressions

have been obtained

Gk(iwn)

₌

~;~

)fl~

~~

(33)

F~(~w»)

=

Fz(~w»)

₌

~

~j~

~,

(34)

where

fk

_{# Sk p #}

(k kF)vF,

_wn i8 _a Matzubara

frequency

and

an

and

In

are connected with _wn and the order parameter ^A

by

un =

£inliin ₍₃₅₎

wn/A

_{= un}

Ii «(vi +1)-1'2]

,

(36)

where

a =

I/(raA).

After

analytical

continuation

equation (33)

determines

u(w IA):

~ "

"(~) ""(~)li "~(~)l~~'~, (37)

where

@M

=

-isgn (u)@0

for u2 _> 1. As i8 shown in [15] ^if a < I the fiJnction

u(u)

has _{two extrema at u}

= +uo where _{uo =}

(1- a~'2)V2

Near these

points

we have

u(z)

= uo +

;«i'(jf/3vo)i'(jj _zo)i'(j~

^{z > zo} _~~~~

-uo + ia

(2/3uo) (z

₊

zo)

_{z < -zo,}

where _{zo =}

6/A

and 6 ₌

A(1- a2'~)~'2

^is the

superconducting

_gap. If _a > I _we have the

gapless superconductivity.

The condition _{a =} I is the _{same as}

given by (29) _[15].

If z2 _>

z(

we have

Imu(z)

> 0.

Using equations (30)

and

(31)

^{in the} 2D _case if

kF

>

Q/2

_we have

K~Q>

_{lW~ =}

8Eiikq

^T

li~uli"il

+

i~

i

~3°~

where

k(

₌

k( Q~/4

_> 0 and _u&

=

u(iwn

+

iw/2).

This

expression

has been obtained

by

the method used in [27] ^{for evaluation of the}

electromagnetic

_response. After

analytical

continuation of

equation (39)

_we

get

ImK(q,w)

₌

(m/8Epqkq)J«(w) (40)

Jo(w)

₌ A

/ dzidz2[n(z2A) n(ziA)]6(zi

_z2

w/A)

(Im)~(Im)21u(zi)u(z2) ii jjl u~(zi)iii u~(z2)ii~~'~ (41)

2

3 4

a=00

2 a=0.244

/ 3 a=0.724

/ 4 a=1.0

1' 1' 1'

G~/A

Fig. 2.-The function

Ja(w)/A

for different values of _a

=

I/(rsA):

_{ai =}

0(rs

=

oc);

_a2

=

0.244; _{a3 "} 0.724 and _a4

" 1.0

(the

threshold of gapless

superconductivity).

Dashed line corresponds

to the normal state.

Using (34)

for

Ja(w)

_we have

Ja(w)

₌

(Ala~) / dzidz2[n(z2A) n(ziA)]6(zi

_{z2 w}

IA)

(1 ziz2)Imu(zi)Imu(z2)(u(zi)u(z2)(~~ ₍₄₂₎

Here the

region

of

integration

is divided into four parts:

I)

_{zi, z2 > zo;}

2)

_ziz2 < -zo;

3)

zi > zo, z2 < -zo and

4)

_zi < -zo, z2 > zo. The first two

regions correspond

to the neutron

scattering

with

absorption

of _one carrier and emission of other and the last two

correspond

to

absorption

_or emission of _two excitations. For T « 6 the contribution to

Ja(w)

from the first two

regions disappears.

In the normal state _we have

Jn(w)

_{= w.}

(43)

It is _easy to show that in the absence of the

spin-flip

interactions when _a

= 0 and T

= 0

~~~~~

£oo ~~l(~~ A?o)1(~ ~)~ ~loll~'~

0 w <

2Aoo

= ~Aoo _{w =}

2a~~ (44)

w +

O(In

_w

/w)

_{w »}

2Aoo,

where

Aoo

is the

superconducting

_gap at T ₌ 0 in the lirr~it _{ra - co.} We _see that in this

case at the threshold _w

=

2Aoo

the

scattering intensity

for T ₌ 0 i8

by

the factor

~/2

= 1.57

larger

than in the normal state. But at _{w - co} both intensities coincide.

N°2 ANTIFERROMAGNETIC FLUCTUATIONS IN SUPERCONDUCTORS 191

If _o

#

0 from

equations (34)

^dud

(38)

at T ₌ 0 _we have _near the threshold

In figure 2

w

of _o(w)

for _dilerent

values of

a

at

T

is ^{own. pendence} h in _a qualitative agreement

with

the experimental

data^resented

in [7

-

Acknowledgements.

am very

grateful

to

J.Rossat-Mignod

for

stimulating

discussions and _very kind

hospitality

in Laboratoire Leon Briuouin and'CEN in Grenoble. I also thank D.N.Aristov for numerical calculations and G.V.

Stepanova

for great

help

with

preparing

this _paper.

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