Antisuperconductors: Properties of Layered Compounds with Coupling

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Antisuperconductors: Properties of Layered Compounds with Coupling

J.-P. Carton, P. Lammert, Jacques Prost

To cite this version:

J.-P. Carton, P. Lammert, Jacques Prost. Antisuperconductors: Properties of Layered Com- pounds with Coupling. Journal de Physique I, EDP Sciences, 1995, 5 (11), pp.1379-1384.

�10.1051/jp1:1995204�. �jpa-00247143�

Classification

Physics

Abstracts

74.20-z 74.50+r 74.62Dh

Short Communication

Antisuperconductors: Properties of Layered Compounds with

~r-3unction Coupling

J.-P.

Carton(~),

P. E.

Lammert(~)

and J.

Prost(~)

(~) Groupe

de

Physico-Chimie Théorique, ESPCI,

10 _rue

Vauquehn,

75231 Paris Cedex OS, France

(~)

CEA/Service

de

Physique

de L'Etat Condensé,

CE-Saclay,

91191 Gif-sur-Yvette

Cedex,

France

(Receivedll Apri11995,

revisedl

September1995, accepted

19

September1995)

Résumé. Dans cette note, _nous considérons les

propriétés

d'un

supraconducteur hypothé-

tique

composé

de couches

microscopiques, couplées

_par effet

Josephson,

mais dont

l'énergie

de

couplage

est minimisée _pour

une différence de

phase

de _m. L'état de base _a des

propriétés

fasci- nantes dues à l'effet combiné de l'ordre supraconducteur et des défauts structuraux du cristal.

Dans le _cas de cristaux très

désordonnés,

_on attend des

propriétés magnétiques exceptionnelles,

qui ^sont

compatibles

_avec les observations dans

quelques supraconducteurs

cuprate haute-Tc d'un effet "Meissner

paramagnétique"

_ou "Wohlleben."

Abstract. In this note, _we consider

properties

of _a

hypothetical superconductor composed

of

Josephson-coupled microscopic layers

witl~

tunnehng

_energy minimized _at

a

phase

diiference

of _m. The _non-zero

phase

offset in the

ground

state

engenders

_an

intriguing interplay

between

the

superconductive ordenng

and structural lattice defects. Unusual

magnetic

properties _are

expected

in the _case of l~igl~ly ^disordered

crystals,

wl~icl~ _are consistent with observations of _a

'~paramagnetic

Meissner" _or "Wol~lleben" effect in

higl~-Tc

cuprate

superconductors.

Layered superconductors composed

of

essentially

two-dimensional

layers coupled

_to each other

by Josephson pair tunneling through

normal

layers,

have been known for _some

time,

the earliest

examples being

intercalation

compounds

_[1]. The

widely

believed

[2,

3] ^{idea that trie}

high-Tc cuprate superconductors

also

belong

to this

dass,

^{of which there} is some

expenmental

evidence

[4, 5],

^{has renewed interest in such materials.}

The dassic theoretical framework for

descnbing

these

systems

is the Lawrence-Doniach model

[3, 6].

The free energy contains _a

magnetic contribution,

and contributions from intra-

layer ordering

and

inter-layer Josephson couphng.

To _a

good approximation, except

very near

Tc

_or in the _cores of

vortices,

the

magnitude

lbo of the

superconducting

order

parameter

is

@

Les Editions de Physique 1995

1380 JOURNAL DE PHYSIQUE I N°11

constant,

^{and in that} case, one can write the latter _two

pieces

of the free _energy

simply

_m

terms of the order

parameter phase

_as

F =

j~~ £ /d~x(

Vjj@n

~~Ajj

8x À~b

n

'o

~

Îd2 ~°~~~~~~ °~~

~

lx / _~~~~~~~'

~~~

where _@n is the order

parameter phase

in

superconducting layer

_n, and

~~ ~n+1)d

@]+~ =

@n+i

@n

/

_A

dl

(2)

4io nd

is the

gauge-invariant phase

difference _across

junction layer

_n. The

penetration lengths

asso-

ciated with currents in the a-b

planes

and between them

(Josephson current)

_are denoted Àab and

Àj respectively,

and _{~ %}

Àj /Àab

is the

anisotropy

ratio.

The form of this

expression

is dictated

by

_gauge invariance

plus

translations in the a-b

plane

and rotations about the _axis

perpendicular

to the

planes (z

_or

c). Imposing

also time-reversal invariance

requires

F to be

unchanged

under _{@n -}

-@n,

A _-

-A,

and

parity

under _{n - -n,}

Az

_-

-Az.

In either case, @o is then constrained to be

0,

the usual

value,

_{or x.} The

Josephson

term has its

origm

in _a form

cxlb(~~lbn

+ c-c-, is

quadratic

in the

Ginzburg-Landau

order

parameter

^field _~b. ^{If the} coefficient _cx is

small, (which requires

_some

explanation iii) higher

order _{terms can} shift the minimum of F to

arbitrary

_{(@n+i @n(, even} in the _presence of the discrete

symmetries. However,

_@o and

-@o

would be

degenerate.

Mechanisms which could

produce

_@o

#

0 in the

ground

state have been descnbed in dis- cussions of

single Josephson junctions, originating

in

strong

Hubbard interactions [8] ^or local

paramagnetic

moments [9] in the

junction,

_or Andreev reflection in _a normal metal

junction iii.

Although

discussed in the _context of

single macroscopic junctions

_{or grain} boundaries

[10, iii,

the

possibility

of

x-coupling

_{as an} intrinsic feature of bulk

systems

appears ^{not to} have been considered. In this note, we discuss

implications

of _@o

= x, which _{we propose} to call "antisu-

perconductor."

For _a

perfect crystal,

most

properties

_are

completely

insensitive to the value of @o.

Unique

features which _occur for _an

imperfect crystal having

_{@o = x} are the

subject

of this letter. In the

case of _a

perfect crystal,

_a

simple

redefinition of the

phase

elimmates _@o

" x: @n - @n + nx,

without

altering

the _vector

potential

A. An alternative method is to

replace

A

by

A _{+ a} in the

inter-layer coupling

term of

equation il),

where _a is _an additional fictitious _gauge

potential

such that

~~ <n+i)d

4lo

Id

~ ~~ _~' ~~~

For instance, one may take

~~

ÎÎ~'

_~~~

where iî is _a unit vector

along

the local normal to the

layers. Gauge

transformations with _a

taking

the

place

of A _are then

possible.

This latter

approach

is _more convenient in the presence of dislocations.

Dislocations

Suppose

that _our

antisuperconductor

contains _a dislocation of the

layered

structure with

Burg-

ers vector b

parallel

to the average

layer

normal 1

[12],

b =

(bd)à,

b _E

Z, (5)

and with

arbitrary

line director.

The definition of _a introduced in the

previous

section is _now

generalized by imposing

Î /~

^{a dl = nC2r}

¹⁶⁾

where _ne is the number of

insulating layers through

which

path

C passes in the

positive

_sense minus the number

passed through

in the

negative

_sense. In that _{case, we} have

ia.dl=-)b _Ii)

for _a dosed contour

encircling

the dislocation line. This _gauge formulation is

analogous

to that introduced

by

de Gennes

[13]

^for dislocations in smectics-A.

Generally,

_even when

Jz

is small

compared

to the critical current

Jo

_"

4xàc(cx(~b(/4lo

"

(c4lo/8x)(1/à]d),

1e. the system ^is

only slightly

disturbed from the local _energy

minimum,

neither _{a non} the

gradient

^V@ of _a smooth

interpolation

of

through

_a

junction layer

is small within the

junction. However,

the combination

Vi

_e V@

Îa ₍₈₎

lbo

is

small,

if _we

appropriately

resolve the 2x

ambiguity

_m the

change

of _across

junction layers.

Although Vi

is then

always well-defined, #

is not _a

single-valued

function in the _presence of dismclinations.

In regions where the _current

density j

is

small,

it _can be

expressed

_as

~~

A~ j

=

~° Vi

A.

(9)

where A is the tensor of

penetration lengths:

A~

=

À(iîé

_{+ À(~}

_{jr éil). (10)}

Quantization

of the total

magnetic

flux 4l borne

by

_a dislocation reads _in

gel~eral

u%)=)jÎV#.dl=~+n,

nez.

iii)

o ^X 2

The

coarse-grail~ed magl~etic

field

obeys

h ₊ V _x

(A~

V x

h)

_{= u}

4lo162(x), i12)

where 62 is _a delta-ful~ctiol~ distribution with

support alol~g

the dislocation

inormalized by arc-length).

The

corresponding electromagnetic

_{energy is}

proportional

to

u~.

_For

screw dis-

locations,

_an additional

n-dependent

condensation

penalty

associated with _a normal _{cote may}

occur.

Two _{cases are now} to be

distinguished:

ii

b odd. The minimum energy is obtained for _u

=

+1/2.

A

magnetic

^{flux hue} _carrying flux

4lo/2

is bound _to the dislocation. The

sign

of the flux _is

arbitrary, reflecting

time-reversal

invariance of the Hamiltonian.

(see

also Ref.

[14])

ii)

b _even. The

system

is not frustrated and the

ground

_state has _v

= 0.

The formalism admits excited state solutions with additional

integer multiples

of

4lo along

the dislocation. Just _as in

ordinary superconductors, however,

such

configurations

should be

1382 JOURNAL DE

PHYSIQUE

I N°11

ul~stable to

splittil~g

into

multiple

vortices of minimum allowed flux. In

conclusion,

_a dislocation will

produce

_a

spontaneous

^local

magnetization

if the

Burgers

vector is of odd

strength.

The

magnetization

is

parallel

to the direction of the dislocation line:

parallel

to the

layers

for

edge

dislocations and

perpendicular

for _screw dislocations.

The

binding

_energy between the

4lo/2

_vortex and _an odd

Burgers

vector dislocation is in- finite in the _sense that the relative _energy of the

superconducting

_state

containing

the naked dislocation _is

divergent

in the

system

^radius. Under the action of _a

Magnus

force

resulting

from _an electrical current, the _vortex could be

pulled

off the

defect;

_{a new}

ihalf)

_vortex would be created

spontaneously

to

replace

it. These _are

essentially linear,

rather than

point like, pins,

_so a

single

^defect _can

compete

^with the Lorentz force.

Unfortunately,

it appears

quite

diilicult _to calculate the

pinning

force.

The

foregoing applies

to vortices associated to either _an

edge

_{or screw} dislocation. Their

shapes

and _core structures _are

quite different, however,

and

consequently,

_{so too are} their

energies.

We consider them

separately.

i) Edge dislocation,

b

= 1.

The

large-scale

structure of _currents and

magnetic

field for this vortex is very similar to that

of _{a vortex} oriented

parallel

to the

layers

of _an

ordinary superconductor [15,16],

aside from

rescaling by

_a factor of _u

i= 1/2). Again

_as in that _{case, smce}

(c $ d,

no normal _{core is}

required.

The vortex is

elhptical

in

cross-section,

with

major-to-minor

axis ratio _~. The _core

of the vortex

iwhich

is not in the normal

phase)

has width d in the c-direction and

~d along

the

planes.

This _is the

region

m which the

nonlinearity

of the _governmg

equations

_comes into

play.

The _{energy pet} unit

length

is evaluated _{as m} references

[15]

^and

[16]:

~_{~~g~ ~}

jij~ ^40~

~ ~~ Cil(

~Î)~ ~ÎÎ)

^~

^a~Àj

^~~

_~Î~'

~~~~

The

prefactor

of

1/4

reflects the

magnitude

of the flux.

ii)

Screw

dislocation,

b

= 1.

The distortion of the lattice which _occurs here leads to additional

coupling

between the

in-plane

and

Josephson

currents since

they

_{are no}

longer purely perpendicular

_or

along1, particularly

close to the dislocation.

However,

this turns out to be _a minor

perturbation [17]

on the limit

d/Àab

_" 0 with

Jo

fixed. In this latter

situation,

it is dear that _one has _a vortex with

purely in-plane

currents. It possesses a normal _core of radius

(ab,

and is unremarkable

except

^{for the value of}

magnetic

flux which it carries. The _vortex has

inearly) cylindrical

symmetry ^and a radius

Àab.

Its

energy/length

is

simply

~~~~~~'

~

~

^~

Îb

~~

~Î

Il 4)

One

might

think that the

system

^would

prefer

to

replace

the normal _core with _{one m} which the

Josephson coupling

is

completely

frustrated. _In

fact,

such _a deviation of the

inter-plane phase

difference from

x can

only

heal _{over a} distance

mJ

Àj

and this would _{cost an energy or} order

(4lo/4x)~il/d~)

A similar remark

applies

to the

proposal

of

having

_a

phase

difference

of _{zero across} the extra

half-plane insulating layer

of the

edge

dislocation. This would result

m zero current

everywhere,

but would also have _an energy which

diverges linearly

with the

system size.

The choice of _sign for 4l

=

+4lo/2

for _an isolated dislocation _m the absence of extemal field entails

spontaneous breaking

of the time-reversal

symmetry;

^the two

signs

_are

equally probable.

However,

the vortices associated with two dioEerent dislocations interact. The range of the

pair

interaction is _as usual the

penetration depths

of the associated field. For _{a screw}

dislocation,

this is

Àab,

and for _an

edge dislocation,

it is

àab

in the z-direction and

àj parallel

to the

layers.

Two

roughly parallel

dislocation lines within interaction range of each

other,

but distant

enough

that the _{cores are} distinct will _carry

anti-parallel

flux. Dislocations fines which _are

separated

by

much less that the interaction range should behave _{as a}

single

dislocation of _even

Burgers

vector and therefore not carry any

magnetic

^flux at all. A dislocation fine which spans the entire

sample

_can

provide

return flux for another

such,

and since the

magnetic

field is not screened outside the

sample,

the

corresponding

interaction is

long-ranged.

Under zero-field

conditions,

a

sample containing

defects will therefore

display randomly

oriented frozen

magnetic

moments what _one may call _an

Ising

hne

glass.

These lines could be observed

by

standard decoration

techniques

since

they

_are insensitive to the direction of the

magnetization.

An external field will be able to orient the flux lines

only

if it exceeds

Hc,

in

strength.

Full vortices with the favorable

sign

will then

penetrate

^the

sample

and annihilate

"misaligned" 4lo/2 vortices, leaving

_a net

alignment.

This behavior does not lead _to observations

strikingly

dioEerent from conventional

superconductors. However,

under field-cooled conditions _a

paramagnetic

_response of the moments associated with

sample-spanning

dislocations should be observable. Such _a

"paramagnetic

^Meissner effect"

[10,11,18,19] (PME)

is

expected

for _any field direction in _a

single crystal having

all

types

^of dislocations. To _overcome the

interaction,

the external field must still exceed _some coercive threshold which will

depend

_on its direction _as well _as the distribution of defects. For

example,

_screw dislocations will become

aligned

for

Hz

>

Hj°~~,

where _{we can} estimate

jfcoer _~

~0

XÀab ^~~~

~-D

/À~j,

(~ ~)

~

4xà(~

2D ^'

where D is the _average distance between defects. This estimate is

simply

the field of _a

4lo/2

vortex at _a distance D. Note that it is

vanishingly

small close to the

normal-antisuperconductor

transition.

Discussion

Symmetry

considerations have led _us to

postulate

the existence of

"antisuperconductors"

which differ from conventional

superconductors

_{m a} subtle but

significant respect.

We _now consider the likelihood of

finding

_a matenal with such _a

phase,

_or whether

they

have

perhaps already

been observed.

i)

^The existence of

spontaneous

currents due to the presence of

x-Josephson junctions

^has been invoked

by

several authors _to

explain

the

paramagnetic

Meissner effect observed in _some CUO based

high-Tc superconductors.

These

x-junctions

_were assumed to dwell in

intergranular

links

[10,11,18,19].

We stress that such _an effect _may also arise from _more intrinsic features such

as

topological

defects of the

layered

structure common to all these

cuprate compounds.

If _so, the

microscopic tunneling

mechanism between CUO

planes

_as trie ongm of trie

x-coupling

would prove ^to be fundamental. In _view of trie

complexity

of the

interlayer regions,

the

possibility

of _x

couphng

cannot be ruled out at

present.

Dislocation-bound vortices could reconcile the

contradictory

remarks of Braunisch et ai.

[10]

^"The persistence of trie PME after

powdering

mdicates...that these weak links _are

intragranular."

The

density

of dislocations

(or B)

_can be estimated from the value of the field at which _{x =} 0: H _cf 0.5 Oe

[loi yields

_{an average} inter- defect distance of _a fe~. _microns. The

sample

obtained after

powdenng ito

a size of 2 3

~tm)

would contain about _one defect _{per grain,} which

gives

the

strongest

PME. The most condusive evidence for these

spontaneously

created vortices would be

provided by

^decoration

experiments

on zero-field cooled

samples.

1384 JOURNAL DE

PHYSIQUE

I N°11

ii)

Such

antisuperconductors might

be

deliberately engineered,

_smce the

technology

_now

exists to build _up

layered compounds layer-by-layer _[20].

In this _{case, one or} several atomic

layers

_are intercalated between the

superconducting

sheets.

Josephson coupling

arises

through hopping

_on these intermediate sites. While the

sign

of _o

is

positive

for direct

tunneling,

it _can have either

sign

for indirect

tunneling.

In

particular, tunneling

ma localized

magnetic impurity

sites _was studied

long

_ago

by

Bulaevskii et ai.

[6,17].

They

showed that the

junction

current is the difference

Jo Js

of _two terms, the latter

being

related to the

spin-dependent tunneling amplitude.

The

difference,

hence _{cx, can} be

negative

if the second term is

large enough:

the reader is referred to the

original

_papers for _a discussion of the

required

conditions. A

transparent

formulation of these

phenomena

_was

given by Spivak

and Kivelson

[5]. They

^{observed that}

transfernng

_a

pair

via _a

singly occupied

site results in

a

sign change

of the

pair

_wave

function, giving

_a <

0, provided

that the intra-site Coulomb interaction is

strong enough

to enforce

single

_occupancy. These observations indicate _a direction to follow in

attempting

to fabricate

insulating x-junctions.

The most

interesting

behavior of

staggered superconductors

is the _very unusual interaction of the

superconducting

order with lattice defects. This

provides

_one

experimental signature, precisely

the

paramagnetic

Meissner effect

IPME)

which has been

reported

for _some

high-Tc

materials

[loi. Josephson junctions

between such a material and _an

ordinary superconductor behave, perhaps

rather

surprisingly,

in _an

entirely

conventional _manner, and therefore do not

provide good experimental

tests unless it is

possible

to observe the

complicated

counter-

propagating

current

pattem.

It may be that _some

currently

known

layered superconductor belongs

to the dass described here.

Antisuperconductors

could also be

engineered

at the

mesoscopic level,

once a way were found _to

produce

_x

Josephson junctions.

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Chakravarty

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In-plane

components of tl~e

Burgers

vector bave been

ignored

here

smce

tl~ey

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P.-G.,

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