Interaction of Pancake Vortices with c-Axis Plasmon in Josephson-Coupled Layered Superconductors

(1)

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Interaction of Pancake Vortices with c-Axis Plasmon in Josephson-Coupled Layered Superconductors

L. Bulaevskii

To cite this version:

L. Bulaevskii. Interaction of Pancake Vortices with c-Axis Plasmon in Josephson-Coupled Lay- ered Superconductors. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.2355-2365.

�10.1051/jp1:1996222�. �jpa-00247317�

(2)

Interaction of Pancake Vortices with c-Axis Plasmon in Josephson-Coupled Layered Superconductors

L.N. Bulaevskii

(*)

Los Alamos National

Laboratory,

Los Alamos, NM 87545, USA

(Received

17 may1996,

accepted

1

July1996)

PACS.74.50.+r Proximity effects, weak links,

tunneling phenomena,

and Josephson effects

Abstract. The

power-law dependence

of

plasma

_resonance

frequency

_on

magnetic

field ap-

plied along

the c-axis is

explaiued

in the mortel of strong disorder in

pancake positions along

the c-axis. The fine width of

plasma

_resonance _is

explained by inhomogeneous broadening

caused

by

random positions ^of

pancake

vortices in the vortex

glass phase.

This leads to _a nearly tem-

perature

independent

fine width which is

inversely proportional

ta

magnetic

field. The

dynamic

interaction of vortices with

plasmon

_is calculated. It is ineffective _in fields studied but _may become strouger in

crystals

with weaker

pinning

_or in

higher magnetic

fields.

1. Introduction

Highly anisotropic high-Tc superconductors

_may be considered _as _a stack of

superconducting Cu02 layers coupled by Josephson

interactions

[1,2].

^Trie ^novel

properties

of these materials

as

compared

with _a

single Josephson junction

are associated with their

multilayer

structure and with the _presence of

pancake

vortices when _a

magnetic

field is

apphed along

trie c-axis.

It _was shown

previously

that

pancake

vortices induced

by

such _a field

strongly

_suppress the

interlayer

maximum

superconducting

current

by inducing

random

phase

diiferences between

layers

in the presence of disorder in

pancake positions along

the c-axis [3]. ^It was

predicted

_[4]

that this eifect leads to a decrease of the c-axis

Josephson plasmon frequency

1N-ith

magnetic

field

applied along

trie c-axis.

Recently,

_a

sharp magnetoabsorption

resonance was observed in trie vortex state of trie

highly amsotropic layered superconductor

Bi-2:2:1:2

by

Tsui et ai. [Si and Matsuda et ai. [6]

m trie

frequency

_range 30-90 GHz

depending

on

magnetic

^field and

temperature.

Trie

plasma

resonance was observed _as _a

peak

in

absorption

_m AC field and

temperature

^when

magnetic

field is

sweeped

at fixed

frequency,

_see

Figure

1. Trie field behavior of this _resonance

(decrease

of

frequency

with

magnetic

field

apphed along

trie

c-axis)

_as well _as its

angular dependence (sharp

decrease ofthe _resonance

frequency

_near orientations ofthe

strong magnetic

field

parallel

to

layers)

_was found to be _m

agreement

^with

predictions [4,7]

for the

Josephson plasmon

_m

layered superconductors.

Matsuda et ai. [6] and Tsui [Si confirmed that this _resonance is _a

maximum when _an AC electric field is oriented

along

trie _c-axis. This observation

provides

strong

^evidence ^that ^trie ^observed resonance is mdeed trie _c-axis

Josephson plasmon. Thus,

trie

plasma

resonance found in Bi-2:2:1:2

superconductor

_is trie extension of trie

Josephson plasmon

(*)

e-mail:

Inb@viking.tant.gov

@

Les

Éditions

de Physique 1996

(3)

49.7

46.1

41.1

38

ÀÎ 35

§

_33.3

30 GHz

é

ÎÎ

q

T ₌ 4.2 K

-1 0 2 3 4 5 6 7 8

~oH (T)

Fig.

1. Curve-lits

(sohd fines)

_to data obtained in reference [8] ^for ^Bi-2:2:1:2 ^at ^T = 4.2 K assuming

inhomogeneous broadening

with B

jj ^c

(see text). Experimental

data of different

frequeucies

_are

represented by

different

symbols.

Trie vertical scale of individual

curve has been normalized and

shifted with _an offset for the

ease of comparison.

discovered in trie

single Josephson junction by

Dahm et ai. [8] ^to trie

multilayered _system

with

Josephson interlayer couphng.

In _a

single Josephson junction

the

plasma

mode is the

charge

oscillation between two

superconductors forming

trie

junction,

with trie current between trie

superconductors being

trie

Josephson tunnehng

current. In

layered superconductors,

trie

plasma

mode _is _a

charge

oscillation between trie

top

and bottom

layers

of trie

sample,

and trie

corresponding

currents flow between all

layers forming

trie

crystal.

Trie

experimental

studies

[5,6]

bave established trie

following properties

of trie _c-axis

plasma

resonance:

a)

Below trie

irreversibility

hne in trie

(B,T) plane

the

dependence

of trie

plasma frequency

Q

on trie

magnetic

field B ^j _c and trie

temperature

^T ^bas trie form:

Q~(B, T)

=

AIB~~ exp(T/To), ₍₁₎

where

Ai

is a constant, _~ m 0.7-0.8 is

temperature independent

and

To

" 12.5 K in fields 0.3-7 T and at

temperatures

3-16 K [Si. ^Above ^trie

irreversibility

line trie

power-law

field

dependence

holds with ~ m ù-g-1 and here Q

drops

with

temperature

at fixed

magnetic

field [6].

Thus,

trie

temperature dependence

of

plasma frequency

exhibits _a cusp ^at trie

irreversibility

line.

hi

At low

temperatures,

when trie

magnetic

field B is tilted

by

the

angle

with respect to trie ab

plane,

at

high angles

> 10° trie

only perpendicular

_component of trie field

Bz

= B sin is

(4)

effective,

and

equation (1)

with B

=

Bz

describes trie data. In

high magnetic

fields above 2

T,

at smaller than 5° trie

plasma frequency

decreases

sharply

_as

approaches

_zero _[Si.

c)

In _a

magnetic

^field B j c, and below trie

irreversibility line,

trie relative _resonance line

width, r/Q,

decreases

weakly

with

magnetic

field [Si. ^Here ^{r is trie} ^line ^width. ^At ^T = 4.2 K trie relative _resonance line width,

r/Q, changes

^from ^0.12 at

Q/2~

= 50 GHz to 0.087 at

Q/2~

₌ 30 GHz. Trie hne width of _resonance with

respect

to variable

magnetic

field at fixed

frequency

_was found to be almost

temperature independent

in trie vortex

glass

state [Si. ^Trie

dependence

of the relative line width _on

magnetic

field below trie

irreversibility

line may be

described _as

r/Q

=

A2B~",

where

A2

is _a constant and _v _m 0.25. Matsuda et ai. found the relative line width _m 0. là 0.2 below trie

irreversibility

line and

they

observed its increase with

temperature

above trie

irreversibility

line in trie vortex

liquid

state up to ù-à at T

= 63 K. As

for trie _resonance

frequency,

trie relaxation rate bas _a _cusp at trie

irreversibility

line [Si.

2.

Field, Angular

and

Temperature Dependence

of Plasma

Frequency

The

power-law dependence

of the

plasma frequency

_on B at B j

c was

explained

_assummg

that trie vortex lattice is

strongly

disordered

along

trie _c-axis due to pmning in the vortex

glass

state _or

by

thermal fluctuations _m trie

liquid

vortex state

[9].

^Deviations ^of

pancake

vortices from

straight

lines in

equilibrium

induce _a _nonzero

phase

diiference

çJn,n+i(r)

between

neighboring layers

_n and _n +1 at coordinate _r

=

(x, y).

This

phase

diiference suppresses trie average

interlayer Josephson

_energy and maximum

possible interlayer superconducting

current,

Jm

=

Jo(cosçJn,n+i(r)).

Trie

parameter Jo(T)

=

c4lo/8~~sÀ(~(T)~

charactenzes trie

Josephson interlayer couphng

at B

= 0 and

.)

means

averaging

_over _space and disorder. Here

Àab(T)

is trie London

penetration length,

s is trie

interlayer

_spacing and _~ is trie

amsotropy

ratio. Trie

suppression

of

plasma frequency by pancake

vortices _was described _assummg that

plasma frequency

is

proportional

to trie maximum

interlayer

current

[4,9]:

Here _ec _is trie

high frequency

dielectric function for _an electric field

along

trie c-axis.

Equa-

tion

(2) corresponds

to averaging over trie

Josephson

interaction. In this _mean field

approach, pancake

vortices _are assumed to be fixed and eifect of the

mhomogeneity

of

Josephson

interactions due to random

positions

of

pancakes

is not accounted for. It is the

phase

diiference

çJn,n+i(r)

which

depends

_on trie vortex structure and leads to the field

dependence

of Q. The diiference induced

by pancake

vortices _is:

ç2n,n+i(rl

₌

~lf(r _rn,vl f(r rn+i,vil, (31

v

where

f(r)

=

arctan(x/y)

and rn,v =

(xn,v, yn,v)

are trie

positions

^of

pancakes

labeled

by

trie index _v _in trie n-th

layer

_[3]. Trie expression in trie _sum _is obtained

by neglecting interlayer Josephson coupling,

and it is correct at distances smaller than trie

Josephson length Àj

₌ _~s from

neighbonng pancakes

at rn,v and rn+i,v in trie ab

plane. Equation (3)

^shows that many

terms contribute

eifectively

to trie

phase

^diiference at given r since trie

phase

diiference for given close _rn,v and rn+i,v ^is ^a

slowly decreasing

function of _r: it vanishes _as

1/jr rn,vj

far away from these vortices. As _a

result,

trie

phase

_~gn,n+i

(ri obeys

_a Gaussian distribution with

(çJn,~+i (r))

= 0.

Therefore, plasma frequency

_may be

expressed

in terms of _a

Debye-Waller

factor:

Q~

"

QI

_eXp

[-(§ÙÎ,n+1(~))/21

'

(41

(5)

where

Qo

"

c/Àabi@

is the

plasma frequency

in trie absence of

magnetic

^field.

Using

_equa-

tion

(3)

_we _see that

(çJ(,~~~ (ri diverges logarithmically

at

large distances;

trie cut-off is determined

by Àj.

As result, when

Àj

is

larger

than trie intervortex distance _a

=

(4lo/B)~/~

and trie

long

_range order of trie vortex lattice in trie ab

plane

is

absent,

trie power law holds:

Q(B)~

_m

Q((Bje~ /B)~, (5)

where

Bj

=

4lo/~~s~.

Trie

exponent

_~ is determined

by

trie correlation function

K(r)

=

(pn(0)pn(r) pn(0)pn+i(r))

of

pancake density pn(r)

=

£~ ô(r rnv) by

trie relation:

~ =

(~/4) /

_dr

_r~A(r).

(6)

For

strong

disorder

along

trie c-axis _we obtain

(pn(0)pn+i(r))

_m

(pn(r))~,

and _~ becomes field

independent

and of trie order

unity

because

Ii(r)

oscillates _on trie scale _a and

decays

at

large

r in trie ab

plane

_on trie _same scale _or

slowly.

Trie parameter _~

depends

_on trie correlations of

pancakes

in trie ab

plane

and may be

slightly

diiferent in trie vortex

glass phase

and in trie

hquid phase. Thus,

the field

dependence

of the

plasma frequency

characterizes trie arrangement of

pancakes along

trie c-axis: field

independent

_~ of order

unity

_means that vortices _are disordered

along

trie c-axis, while smaller _~

decreasing

with field indicates in trie

pancake ordenng along

lines at distances

larger

than trie

interlayer spacing. Note,

that _~ may

depend

_on

history

_as

well.

Experimental

observation of _a field

independent

_~ _m

(0.7- 0.8)

confirms that

strong

^disorder

m

pancake positions along

trie c-axis _is present ⁱⁿ ^trie

samples

studied. This conclusion is indi-

rectly supported by

trie fact that in

high magnetic

field

(above

ù-1

T)

and at low

temperatures

trie

Bragg peaks

bave not been observed in _neutron

scattering experiments [10].

^This can be also

explained by

trie absence of

pancake ordering along

trie c-axis.

Strong angular dependence

of trie

plasma

_resonance in

high magnetic

fields B _»

4lo/sÀj

_was

predicted

_I?i _as _a result of interactions of

pancake vortices, produced by

trie

Bz-component

of trie

field,

with trie dense lattice of

Josephson vortices,mduced by

the field component

parallel

to the ab

plane.

In trie

magnetic

field

parallel

to trie

layers Josephson

structure can move as a whole under the eifect of electric field

applied along

trie

c-axis,

and trie

only

low

laymg

resonance is at zero

frequency. Pinning

shifted this _resonance to _nonzero

frequency.

When

component of

magnetic

^field

along

the c-axis

presents, pancake

vortices

provide

such pmmng.

They

form _an almost

hexagonal

_structure in trie ab

plane

and _a

zigzag

structure

along

trie

c-axis, see

Figure

2. Trie

zigzag

structure of

pancakes

minimizes trie

Josephson interlayer

interaction and induces _pmmng for

Josephson

vortices. This results _m _a

sharp

enhancement of trie

plasma frequency

when increases from _zero and

pancake

vortices appear:

~ Î~Î ^~~ ÎÎÎ

~

Î~~~~~~~ ÎÎ

^~

' ~~~

where G

=

(8~~BsinÙ/4loVà)~/~

is trie

reciprocal

vector of

pancake triangular

lattice formed

by

trie

perpendicular

_component of

magnetic field,

h

=

2~sB/4lo

is trie

reciprocal

vector of the

Josephson

lattice formed

by

trie

parallel

_component of trie

magnetic

field and

g(z)

_is trie

minimum value of

jm

+

xj

with

respect

to

integer

m.

Next _~ve discuss trie

temperature dependence

of

plasma frequency

_m the vortex

glass phase.

It mcludes the

temperature dependence

of

Jo

and that due to

phase

fluctuations

(thermally

_ex-

cited

phase

collective modes and

pancake

oscillations _in pmmng

wells). Jo(T)

_may be extracted

from measurements of trie

plasma frequency

_at B

= 0. Trie renormahzed

plasma frequency

ù(T)

=

uJc(T)Jo(0)/Jo(T)

charactenzes trie eifect of

phase

fluctuations

only.

(6)

n+1

,

n

'

' n- J

~, ~

~

₈

~

, , ,

'

Fig.

2. The

zigzag

structure of

pancake

vortices at

high magnetic

fields _near ab orientation at h

= G.

Oscillations of

pancakes

around

equihbrium positions

ⁱⁿ pinning wells renormalize trie _param- eter ~

according

to

equation (5).

Trie maximum

thermally

induced _meau square

displacement

of

pancakes

_may be estimated

by

_use ofthe classical expression,

(it(~)t

"

T/saL.

Here _aL is trie

Labusch

parameter

^estimated as aL "

Jc,ab4lo/c(ab [11,12].

^This

gives

_aL " 5 x

10~ g/cm

x s~

for

Jc,ab

m 5 _x

10~ A/cm~.

Then

(it(~)t

_" 3 _x

10~~~ cm~

_and

((unv ~n+i,v)~)t

is about trie

same. Trie

pinning

mduced mean square relative

displacements (m ^a~

for

strong disorder)

_are

about two orders of

magnitude larger.

This _means that thermal motion of

pancakes

_may be

neglected

below trie

irreversibility

line, and trie

parameter

_p is

practically temperature

inde-

pendent

there in

agreement

^with ^the ^results [Si.

Thus, temperature dependence

_m

equation (ii

for

plasma

_resonance cannot be

explained by

thermal

depinning.

As _was

argued [9],

^enhance-

ment of

plasma frequency

with

temperature

may be caused

by

trie eifect low

frequency

thermal fluctuations of

phase

^diiference

(thermally

excited

phase

collective

modes).

These smooth out

rapid changes

of the

phase

diiference

produced by

disordered

pancake

vortices and result in lin-

ear in T increase of the average

Josephson

interaction and

plasma frequency

with temperature at low temperatures. ^Their ^effect on the

plasma frequency

_may be

compared

^with ^the eifect of

spm waves on trie

magnetic susceptibility

of _an

antiferromagnet

below trie Neel temperature:

magnetic susceptibility

_mcreases because _spm _waves make the

antiferromagnet

softer.

3.

Equation

for Phase Dilference in Presence of Mobile Pancakes

The

relatively large

hne width of trie _c-axis

plasma

_resonance

là,

_6] is

surprising.

In _a _sm-

gle Josephson junction

the hne width of the

Josephson plasmon

_is ^associated ^with ^incoherent

dissipative tunnehng

of

quasiparticles.

In _a

superconductor

with gap A in the

quasiparticle

spectrum

^trie concentration of

quasiparticles

is

exponentially

^small at low

temperatures

T < A and vamshes

lmearly

_m T for d-wave

superconductors.

Pancake vortices increase trie number

(7)

of

quasiparticles linearly

in B in _s-wave

superconductors

and _as

B~/~

_in _d-wave

superconduc-

tor

[13].

^Thus we can

imaging

that _a

relatively large weakly

temperature and field

dependent

line width _can not be

explained by

_an

interlayer

current of normal

quasiparticles. Rather,

it may be caused

by

trie effect of

pancake

vortices _on

Josephson interlayer interaction; randomly positioned they

mduce

inhomogeneity

of trie

Josephson

interaction which _may lead _to _a

sig-

nificant

broadening

of trie

Josephson plasma

_resonance. In

addition, plasma

oscillations _may mix with vortex oscillations and this effect also results in

broadening

of

plasma

_resonance. To

descnbe these effects

quantitatively,

_we need

time-dependent equations

for the

phase

difference

(like

sine-Gordon

equation

for

single Josephson junction)

in trie _presence of mobile

pancake

vortices.

Trie

equation

for

phase

difference _at

equilibrium

_m trie presence of

pancakes

is [14]

~ Lnmv2çJ()~~i

+

Ài~ Sinlç2ÎÎ+1

+

ç2ÎÎ+il

= 0.

(8)

~

Here

çJn,n+i(r)

=

çJ((~~(r)

+

çJ (r)

is trie total

phase difference, Lnm

=

(Àab/s)(1- s/Àab)Î"~~Î

_is trie mutual inductance of

layers,

and

çJ~(~~(r,rnv)

is trie

phase

difference mduced

by pancakes

at

positions

_rnv in trie absence of

Jolephson

interaction

(at

infinite

Àj).

It is given

by equation (3).

Trie function

çJ~(~~(r)

is

singular

at vortex

positions

_rnv, while trie function

çJ((~~(r)

is

regular everywhere

and descnbes the effect of three-dimensional

screening

causeÀ by interlayer Josephson

currents

[14].

^For ^mobile

pancake

vortices _we insert

m

equation (3) time-dependent

coordinates

rnv(t)

and in

equation (8)

_we need to add terms with time derivatives of the

phase

difference. These terms _were obtained in trie framework of

time-dependent Ginzburg-Landau (TDGL) equation (see

_[15]

corresponding

to trie Lawrence- Doniach functional

[16].

^In ^trie _presence of trie

homogeneous

external electric field

Dz (t) (for

a

sample placed

in _a

cavity acting

_as _a

capacitor)

trie

equation

^for

çJ$( _~~(r, t)

is:

Î~ ~~~~~~

^~

^~~~~~~ ~ ~~~~~~~~~~~

_~

Î

~~~

~~~~~~

^~

~~~~~~

11eo t~~~~~' ~ Î

~

~~~'

_~~~

where _co

# cs

/Àab@ plays

trie role of trie Swihart

velocity,

_fa

#

4l(s _/16~~À(~

is trie character- istic energy of

pancake vortices,

_uJab

#

c/Àab

is the

in-plane plasma frequency

and

Pc

₌

4~ac lec,

where _oc is trie c-axis

conductivity

due to

quasiparticles.

Here trie function

dn,n+i (r, t)

is determined

by solving

TDGL

equation

^for

given

vortex coordinates

rnv(t).

This function comcides with

çJ~( ~~(r,r(t))

when relaxation of trie

superconducting

order parameter is

neglected.

Trie _nonzero difference

[dn,n+i(r, t) çJ(( _~~(r, _t)]

is caused

by

retardation in trie time _varia- tions of order parameter and

supercondulting

_currents _when

_pancakes

are moving. Trie effect of retardation becomes

negligible

far _away from _vortex _center.

Respectively,

trie difference

[ôn,n+i(r, t) çJ~[ ~~(r, t)]

vamshes

exponentially

far _away from vortices with trie character- istic

length

of

decay Aab

₊

(&aab/Be~~GLÎ4/~j~)~/~,

where _~GL characterizes trie relaxation of trie

superconducting

order parameter and _may be estimated _as _~GL >

&uJ(~/4e~s~j4/nj~a[(~.

Here _aab _is trie

conductivity

due to

quasiparticles, aÎ~

_is _trie

conductivity

m the normal _state and

(lin(

is the

amplitude

of the

superconducting

order parameter. For

typical

parameters trie

mequahty Aab

< a

holds,

and trie difference

[dn,n+i(r,t) çJ~(~~(r,t)]

_is of trie order

unity

m trie region of trie radius

Aab

around each

pancake

and

rapidly

_vanishes _outside. _It

(8)

is this difference which determines trie

dynamic

interaction of

plasmon

with

pancakes.

Trie

equation

of motion for

pancake

coordinates

rnv(t)

should be added to

equation (9)

to obtain trie

complete

set of

equations

for

coupled phase

difference and

pancake

coordinates.

4.

Inhomogeneous Broadening

of Plasma Resonance

We consider _now trie solution for

çJ(( _~~(r,t)

and vortex coordinates

rnv(t)

in trie presence of DC

magnetic

field B

jj c and

an'oscillatory

_weak external field

Dz(t).

Then trie

phase

collective modes _are excited and in addition vortices oscillate _near

equihbrium positions rÎÎ

due to

phase

variations. Then _we obtain

equations

which describe small variations of trie

phase

difference and small

amplitude

vortex oscillations. We denote

pancake

deviations

by unv(t)

=

rn~(t) rÎÎ

_and

_expand çJ(( _~~(r, _t)

_in

_unv(t):

i~ÎÎ+1(~> t)

"

~lf(~ ~fl) _f(~ _~ÎÎI,u)1

₊

#ÎÎ+1(~>

t)>

(io)

u

#ÎÎ+1(~> t)

"

~ D(~ ~fll'~nu(t) D(~ ~ÎÎI,ul'~n+1,u(t)> (ii)

u

where

D(r)

=

(-y/r~, x/r~)

for _a _< _r <

Àj

and

D(r) drops

much faster with _r for _r _>

Àj.

We also

expand ôn,n+i(r, t)

in

unv(t).

We

present

trie

phase

difference _as

i~n,n+1(~>

~nu>

t)

_"

&~ÎÎ+1(~)

⁺

#ÎÎ+1(~> t)

+

#ÎÎ+1(~'~)> ⁽¹²⁾

where

çJ(((

_~~ ^is ^the

^phase

^difference ^at

^eqmhbrium

^when

^Dz

= 0. It is determined

by

_equa-

tion

(8)

at rnv =

rÎÎ.

_The contribution

@~[

_~~ ^accounts

^directly

^for ^vortex ^motion ^and ^is

^given by equations (10,

ii

),

while

@~[

~~ accounts for the rest of the

phase

variations.

Both, @((

~~

and

@~(

_~~ are small in _a weak external electric field. Thus _we _can

expand

in

equation (9)

and obtain _a linear

equation

for

@~~

~~.

(~ ^~ _~~~+l ~ ~"m~~~ÎÎÎm+1

~

/j

Î~~~

~Î~+1(~)Î~Î~+1

~ÎÎoec

~

Ît~~

~~~

Îj

^~~°~

^~~~+~ ^~~~~~~~+~ ⁽ ^~ Ît~"'"~~'

^~~~~

The last two terms _in trie

right

hand side describe trie excitation of trie

phase

collective mode due to vortex oscillations

un~(t).

Trie

equation

of motion for vortex

deviations, un~(t),

which

couple

with

phase oscillations,

will be

presented

later. After

solving

these two

coupled equations

for

@((

~~ and

un~(t),

_we obtain trie electric field between

layers

_n and _n +1:

~~"~'~~~

~~'~~

1s Ît _Î~~~+~

~~'~~ ^~

~'Î

'

~~~~

and then _we calculate trie _inverse dielectric function

Ile(uJ)

=

(Ez(r,uJ))/Dz(uJ),

which describes trie

plasma

resonance.

Now _we solve

equation (13)

with fixed

vortices,

when

unv(t)

₌ 0. Then the two last terms

in trie

nght

hand side of this

equation

are absent. We _assume that trie

inhomogeneous

part of

@((~~,

which is

8n(r,t)

=

@([~~(r,t) Ô(t),

_is much smaller than trie

homogeneous

(9)

part à(t)

=

(@~(~~(r, t)).

We _use here trie

perturbation theory

_m trie continuous spectrum,

see

iii]. Averaging

_over _space in

equation (13)

we obtain:

ÎÎÎ

^~

_Îj

^~~~~ ^~

_Îj ~°+ "'~ 8~eo

t~~~~~'

~~~~

where b

=

(cos çJf(~~(r)).

Trie

inhomogeneous

_part is determined

by

(~(~~~

^~

~jj ~"(~'~) ~ ~"m~~~m(~'~)

~(Î~°S~~~+1(~) ^Î~). ^(Î~)

~

The solution of this

equation

in trie Fourier

representation

_uJ,

k,

_q with respect ^to

t,

_r, n is:

°~~°'~'~~

"

)Î~IÎÎI~~Î2(kÎ(~°~~°~

~°

_"

l'

~~~~

' ~

~

~2~2

2(1 cosl)

₊

_s2/À[~'

^~~~~

Inserting

this

expression

into

equation (là)

_we obtain trie solution for

à(uJ):

~~~°~ ÎÎO

uJ(uJ

ire) Îb

uJ(uJ)]Q( ~~~~°~'

_~~~~

~~~°~

Î _ÎÎÎ

uJ(uJ

ÎÎÎ~~12(k, _ql'

~~~~

where

F(k, q)

is trie Fourier

component

^of ^trie correlation function

F(r r',

n

m)

=

([cos ~gf[ ~~(r)

_b]

_[cos ~g(~~

~~

(r')

_b])

(21)

The

dispersion

relation for the

phase

collective mode.

equation (18),

is valid for

frequencies

&Q(k, q)

«

A,

and thus

integration

_over k in

equation (20)

_is restricted

by

k <

A/co.

In calculation of trie correlation function

F(r,n)

_we follow trie

approach

_[9] used above to calculate b. We consider _zero

temperature

hmit. In trie Fourier

representation

for _r and _n _we obtain

F(k, q)

_m

a~[~~/(2~ -1)]

at ka < 1 and

F(k, q)

m

(2~)~/~a~/~k~~/~ cos(ka 3~/4)

at ka _» 1.

Integration

_over k and q in

equation (20) yields:

~~ ~~~°~

~

ÎÎÎÎ _ÎÎ _Îj

_~

_2(21~ 1)1'

_~~~~

The term

-1(Im w)Q(

in trie denominator in

equation (19)

for

à,

describes

broadening

of trie

plasma

mode due to random _space variations of trie

Josephson

interaction _m trie presence of

pancakes.

It may be described also _as _a

decay

of trie

homogeneous phase

collective mode into

mhomogeneous phase

collective modes. Trie rate of this

decay

_is

r)~~/2Q

_« Q at _a «

Àj.

Thus, neglecting pancake

oscillations, _we obtain the dielectric function:

)

=

)~~~

^~J2

~~~~ ~~

~~lnh'

~~

~

~~Î.

We obtained

rmh

at T

= 0. Below the

irreversibility

line

r)~~

is

anticipated

to be almost

temperature mdependent

because it is determined

mainly by

short range correlations of _cos _~gn,n+i

(r)

which

depend weakly

_on thermal

phase

fluctuations

[F(r, 0)

_m

1/2

_at small

ri.

(10)

The value

r,nh

determines trie line width of

plasma

_resonance ^r ₌

r)~~ /2Q

due to inhomogeneous

Josephson

interaction in trie _presence of

pancake

vortices. For this

mechanism;

trie line

shape

of _resonance with

respect

to B at fixed _uJ _is determined

by

trie function:

~~

À/~CiB~l ~~p

~°' ~~°

^~

~~~)~

₊

Ai~CIB-2' ^~i

^"

^2(~Î~ i) ~ÎBJ> (24)

where _~

= 0.8 and uJ~

=

Ai Bj~ (note

that Q~

=

Ai B~M).

This

expression

descnbes very well trie

experimental

data

là,16]

for line width _at ₁₌

300,

_see

Figure

1.

5.

Dissipation

of Plasmon into Vortex Oscillations

Next _we

present

trie

equation

for

pancake

oscillations

coupled

with

phase

variations. Then _we solve trie

equation

for

à(t) _taking

into account vortex motion to obtain trie

decay

of

plasma

modes into vortex oscillations. In

writing

down the

equation

of motion for

pancakes

_we will consider trie _case of

strong pinning

centers for

simphcity

and focus

mainly

of terms which describe trie interaction of vortices with

phase

collective modes. The

equation

for

pancake

deviations is:

~Unu + _OEM

in

X

Unul

+ aLUnu "

Fnu, (~à)

~~~~~~ Î _~j Î _~~~~

Î( ^~~Î~+~

^{~~~ ~~}

^~~~~~~~~+~

^~~~~

~ÎÎ~

~~

lj~cl@SÎm+i~~

^~~ ⁺

j~cl~mm+i~~>

^~~

^8£eo ~l~j ~ii

^~~ ^~~~~

Here _aM is trie

Magnus

force coefficient and _we

replace

all elastic moduli and effects of

pmning by

trie Labusch

parameter

_aL. ^Such an

approach

is _correct for

strong

^identical

pmning

centers

and it gives

only

_an order of

magnitude

estimate for

typical

^Bi-2:2:1:2

smgle crystals.

^Trie

estimate for _~ is trie

Bardeen-Stephen expression

_~ =

4l(a(~ /2~([~c~,

where

aj(~

is trie normal state

conductivity.

For _aM trie estimate is trie

hydrodynamic

_expression _aM _#

~hns,

where _ns

is trie

density

of

superconducting

electrons

[11,18].

Our focus is trie forces

Fnv acting

on

pancake

nu when trie

phase

difference deviates from that at

equihbrium. Using equation (13)

we obtain:

~""

~~~ ^"

l ~ / _~~~Y l~ _~mk _~~@lli+1(~

_~)

_~kl'

^~~

m k

~

j

~~~Î~ÎÎm+1(~>~)Î#~Îm+1(~> ~))

(~ÎÎÎm+1(~>~) l~m,m+1(r,~)j ^(~~)

j rnv

Next _we find

pancake displacements

caused

by

trie force

Fnv(uJ)

and insert trie solution for

pancake displacements

into trie

right

hand side of

equation (13)

and account for trie

homoge-

neous collective mode

only.

Trie first terni in trie

right

hand side of

equation (27)

^vanishes

for

homogeneous part

^of

plasma

mode and thus may be omitted. Trie second terni leads to

dynamic vortex-plasmon

interaction which

onginates

from

regions

near vortices

only

because trie function

çJ(~~

~~

(r, t) dm,m+i(r, t)

vamshes

exponentially

far _away from vortex centers.

This results _m

tie

_additional contribution to trie left hand side of

equation (15):

~~~°~ ÎÎÎ2 ~~ÎÎÎ~ Î ~'~~

~~~°~

(2/~ÎÎÀj

s

~~ÎÎÎ~

_' ^~~~~

(11)

where

S(uJ)

₌

(iuJ~

+

aL)~ uJ~a[

and C

depends

_on trie difference

dn,n+i ç~~Î+i

^and ^is

of trie order

unity.

This additional term in

equation (13)

bas _an

imaginary

part

duè

_to

vortex

viscosity

and it determines the

dissipation

rate

rv

of

plasma

modes into vortex oscillations:

rv ~CfOQO

11 B

~/~

Î 2(2~ 1)sÀj a[ Bje2

^' ^~~~~

Here _we _assume that

pinning

is

strong, aL/Q

_» _~, _aM.

rv depends weakly

_on

B,

and

rv/Q

increases with B in contrast with

inhomogeneous broadening.

There is also shift of

plasma frequency

due to

dynamic plasmon-pancake

interaction.

For trie contribution

rv

due to

dissipation

of _a

plasmon

mto vortex

oscillations,

_we

anticipate

a

significant

increase with

magnetic field, r/Q

_c~

BM/2;

_see

_equation (29).

We

anticipate

also

significant temperature dependence

due to that of trie

viscosity

coefficient _~.

Up

to _now _we can not estimate this contribution to trie line width because _~ _is unknown.

However, predicted

field and temperature

dependence

is in obvious

disagreement

with

experimental

data

[5, 6].

We conclude that this mechanism _is meffective in

magnetic

fields used to

study

trie c~axis

plasmon

_so

far;

B < 7 T. We note that in

higher magnetic

fields _or weaker

pinning couphng

of

plasmon

with vortex oscillations _may become _more effective. _{If so,}

plasma frequency

shift will

change

_a _power law

dependence

of

Q(B)

and relative line width will increase with

magnetic

field. Thus expenments _in

high magnetic

fields _may _uncover _a _vortex _mechamsm _of

plasma

dissipation

and

provide

information _on _vortex

dynamics.

To conclude discussion of

dynamic plasmon-pancake coupling

_we consider trie

possibility

of

exciting

_a

plasma

mode via vortex oscillations

by applying

an AC

magnetic

field. Let _us _assume that vortices oscillate

homogeneously, unv(t)

=

u(t),

around their

equilibrium positions

under trie effect of such _a field. Then

by

_use of

equation (11)

_we obtain _on trie

right

hand side of

equation (13)

for

çJ$(~~,

trie term

j Î~°~ §~Î~+1(~)Î§~Î~+1 j

Î~°~

§~Î~+1(~)Î ~ _Î~(~ ~ÎÎ _~(~ _~ÎÎI,u)Î ~(~)> (~Ù)

J J

~

and _a similar contribution

originates

from trie term with

dn,n+1

Both terms vanish after

averaging

_over _space. Thus

homogeneous plasma

mode _cannot be excited

by homogeneous

coherent _vortex

oscillations; only inhomogeneous phase

collective _modes _may _be _excited

by

_an AC external

magnetic

field.

Similarly,

_a

plasma

mode _is

coupled

with

inhomogeneous

vortex

oscillations

only,

and _an AC electric field excites such vortex oscillations but not

homogeneous

ones.

6. Conclusions

We derived

equations

which descnbe oscillations of the

phase

difference

coupled

with

pancake

oscillations in trie vortex

glass phase. Using

these

equations

_we

proved

that

plasma frequency

is determmed

by equation (1)

when

magnetic

field is

apphed along

trie _c-axis.

We derived also hne width of trie

plasma

_resonance due to

interlayer tunneling

of

quasiparti- des,

_ansmg from

mixing

ofthe

homogeneous plasma

mode with

inhomogeneous phase

collective modes _m trie _presence of

randomly positioned pancakes (inhomogeneous broadening),

and due to trie

decay

of trie

plasma

mode _into _vortex oscillations.

We attributed trie observed hne width of trie c-axis

plasma

_resonance _m

magnetic fields

below 7 T to trie

mhomogeneous broadenmg.

Trie

positional

disorder of

pancake

vortices

induced

by pinmng

leads _to trie

possibility

of

exciting

_many

phase

collective modes

by

_an

(12)

homogeneous

external AC electric field. This mechanism results _m trie

practically temperature mdependent

line

broadening

which decreases _as _a

power-law

with

magnetic

field in

agreement

with

experimental

data [5].

Acknowledgments

The author thanks A.R.

Bishop,

D.

Dominguez,

M.

Maley,

N-P-.

Ong,

Vi.

Pokrovsky,

M.

Tachiki,

and O.K.C. Tsui for valuable contributions to this work. for useful discussions. This research _was

supported by

the U-S-

Department

of

Energy.

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