Layered superconductors in a parallel field: on the mixed state at equilibrium

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Submitted on 1 Jan 1991

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mixed state at equilibrium

J. Carton

To cite this version:

J. Carton. Layered superconductors in a parallel field: on the mixed state at equilibrium. Journal de

Physique I, EDP Sciences, 1991, 1 (1), pp.113-123. �10.1051/jp1:1991118�. �jpa-00246299�

J

Phys.

I 1

(1991)

^{113-123 JANVIER 1991, PAGE l13}

Classification

PhysicsAbswacis

74.60E

Layered superconductors in

_a

parallel ^field:

_on

the mixed state at equilibrium

J-P Carton

Service de

Physique

du Solide et de Resonance

Magn6tique, CEN-Saclay,

91191 Gif-sur-Yvette, France

(Received

23

July

199f

accepted

18

September1990)

R£sum£. Le mod61e d6crit _{un ensemble} de

plans supraconducteurs

faiblement

couples

_par eflet

Josephson.

Quand _un

champ

ma

gn6tique

est

applique paral161ement

aux couches et que ^la

tempdra-

ture est assez basso pour que la

longueur

de coh6rence _entre

plans

^{wit inf6rieure} 1 leur distance _a,

les coeurs de vortex

s'ajustent

entre deux

plans

cons6cutilh. L'6tat mixte est 6tud16 dans _{ce cas} pour

des

champs

forts et des

champs

faibles. Los r6sultats _sent

compatibles

avec un

triangle

isocdle _comme cellule de base du r6seau de vortex. On trouve

H~i

jj ⁼ ~

fl

~ lnA

la

ou et

ii

sent les deux

longueurs

« j

de

p6n6tration.

Abstract. The model describes _a set of

superconducting planes weakly coupled by Josephson

tunnel

ling.

When a

magnetic

field is

applied parallel

to the

layers

and the temperature is low

enough

so that the

interplane

coherence

length

is smaller than the

corresponding spacing

_a, vortex cores fit in between two

adjacent planes.

In this _case the mixed state is studied for

high

and low fields. The

results _are consistent with _an isosceles

triangle picture

for the unit cell of the vortex lattice.

H~ijj

^is

found to be

~

fl

~ ln

~

where and

ii

_are the two

penetration lengths.

r j a

This paper ^{is devoted to the}

study

^of

magnetic properties

of _a model of

superconducting lay-

ers. The model describes _a

stacking

of

superconducting planes

as shown in

figure

I. The inter-

plane

_space is

insulating

but

tunnelling provides

a

Josephson coupling

between two

neighbouring planes.

The

superconducting

sheets

(taken

^of zero

thickness)

are assumed to be well described

by

a continuous

(s-state Cooper pair)

wave function _or order

parameter.

^Such a model is realistic in situations where:

I)

^{the coherence}

length along

the direction normal to the

planes

is of the same order of

magnitude

as the

interlayer spacing, it)

the coherence

lengths along

the other directions

(parallel

to the

planes)

remain

larger

than any

microscopic

scale.

This model has been introduced

[I]

to describe intercalated

compounds [3,

4] These

display alternatively superconducting (e.g. lbS2)

and

insulating layers (e.g. pyridine).

It also turtls out

that the

present

^{model is} more suitable than the standard

l~andau-Ginzburg theory

to account for the low

temperature properties

of

Copper-oxyde

based

superconductors.

It is

commonly accepted

a

/

~~

^~

~x

~

/ /H

Fig.

I.

that the _zero

temperature

^coherence

length

in the

c direction

(normal

to the Cu-O

planes)

is

comparable

to the cell

parameter.

lb be _more

precise

in

YBaCUO, f~ ~

a~ for

temperatures

T _<

0.95Tc

and in BiSrCaCUO for T _<

0.9997Tc! _[12]

The coherence

lengths fa, fb

seem to be

larger (although

not much

larger)

than _aa and _{ab so} that the continuum

approximation

for the Cu-O

planes

should

yield qualitatively

valid results.

Concerning

the

electromagnetic properties

of such

systems,

two

typical

^situations have to be

distinguished: applied

field

parallel

or normal to the

planes.

In the _case of _a normal field the

picture

of the mixed state at

equilibrium

is similar _to that of a 3-D continuum since the _z varhble

plays

no role: it is the Abrikosov

trhngular

lattice of vortices. Thin _has been verified in

high Tc compounds by

decoration

[14]

However Nelson

[~

has shown

that,

out of

equilibrium,

the

weakness of the

interlayer coupling

should lead to

peculiar

behaviour. lvhen the field is

parallel

to the

layers,

the situation is far from

booing

conventional when the

temperature

^{is small}

enough.

In

particular

it has been

long

known

[2,

6] ^{that the} upper ^{critical field} Hc~jj ^{is infinite for T < T*}

if

paramagnetic depairing

is not taken into account

(otherwise Hc2

₌

Hp

^{for T} < T*

). Precisely

T*

corresponds

to a

point

where

f~

₌

ii

=

a/vi.

The

divergence

of

Hc2jj

^is

interpreted

as the

possibility

of

fitting

the _cores of the vortices

exactly

in between two

neighbouring planes. Actually

there _{are no}

longer

holes in the

superconducting

wave function to

prevent

^the

packing

of flux lines.

The

present paper

^{aims at}

providing

_{a more} accurate

description

of the mixed state at

equi-

librium in the

parallel

field _case. Further

topics

such _as out of

equilibrium

or

thermally

activated

phenomena,

^{effect of}

^disorder,

^will not be discussed here.

I. Basic

equations.

Following

notations of reference

[2]

we write the

thermodynamic potential

for an external field H _{as a sum} of four contributiotls:

G=Gjj+Gi+Gh+GH

I) superconducting planes

contribution:

Gi

_{= a}

l~ / _dP ^a _l~n

l~ +

)fl ^l~n

l~ + 7

1(vp 2~iai ~iAi) ~n ~j

n

it) Josephson _type coupling

between

planes:

na (n+Ua ^~

Gi

₌ aJ

~j _{dp ~nexp 2~i~Jp Azdz ~n+iexp 2~i~Jp Azdz}

~

N° I LAYERED SUPERCONDUCTORS IN A PARALLEL fIELD lls

iii) magnetic

^field contributions

(internal

and

external)

Gh

=

/ _dpdz ^~) _GH

=

/ _{dpdz (H~ 2h.H)}

8~ 8~

In the above

expressions

_a is tile

interlayer spacing,

_~Jo the

superconducting

flux

quantum

_{(~Jo =}

~~

and the order

parameter

is taken as

~n p)

where _n labels the

plane

and _p is _a two-dbnetlsional 2e

coordinate _{p =}

(z, y). Imposing stationarity

^{of G with}

respect

to _%l and A leads to the fundamental

equatiotls

of the model

~n(U

₊

fl ~n

_l~)

7(v

p

2~i§2i~Ajj)~%in+

(n+I)a (n-I )a

+J 2§§n

%~n+leXp 2~i§2(~ Azdz §§n-leXp 2~i§2(~ Azdz

= 0

la la

~

~jjjj6(z na)

=

(curl h)jj

c

~

~

jz

₌

~(curl h)z

c ~

~~~~~

jjj

=

2~i~p~7(%l~vp~In %lnVP~~)

+

(2~h'0

~)~ ~

~7 ~~

_'~

~ll

and

1 _~ _~

(n+i)a

jz

₌

2~aJ~2o

x

2Im(~l( ~i#nexp2~1~2o Azdz)

C

~a

jjj

^and

jz

are

respectively

tile current

density flowing

in the

plane

and

tunnelling

bebveen two

layers.

2. Low

temperature

^{limit and reduction} to

phase

variables.

We will consider the _case when the

applied

field is

parallel

to -say- ^tile y direction. The

following

j ~2 ^1f2

simplified equations

Mill be useful when

ii

₌ < a

(I.e.

^T < T*

).

^{In this}

^limit,

^{if there}

a ^{'* '*}

~

lf2 is _no normal

region,

the modulus of tile order

parameter

^becomes a constant

#o

₌

fl

and

)ill

^"

j _()V0

Ajj)

is the

magnetic penetration length originating

from in

plane

currents:

4~A~

=

(8~~7~2i~lfi()

These currents cost an energy:

G>jj ⁼

~$~ ~ /dPj( (1)

~

~

It is convenient to introduce the

electromagnetic phase

difference between two

planes:

~ (n+I)a

41+~(p)

₌

°n+i(pJ _@n(p) ) / _Az(p)dz

na

so that

jz

₌

jj

sin

#(+~

for _na < _z <

(n

+

I)a

The maximum

junction

current b

jj

₌

4~ac~2p~ J~I(

The

following

fundamental relation _are derived

(using

h

= curl

A)

_:

0~#(+~

₌

0~@n+1 0~0n

^~~

~~~~~~

dz(0zA~ hy)

i~o na

=

0~@n+i ~~A~(z

=

(n

+

I)a) 0~0n

₊

~~A~(z

=

na)

+

~~ah(+~

i~o i~o i2o

=

~~~ ~

lJj(n

+

iJ Jj(n)1

+

)~

hi+~

h(+I (z)

is the constant value of tile field for _na < _z <

(n

+

I)a.

From the Maxwell

equations

one has

f jjj(n)

₌

) _{(h(_i h(+~)}

_and

_jz

=

)0~h(+~ ⁽²⁾

c ~ c ~

This leads to

~

jn+1

^2~a

~n+1 ^~~

~hn+2

~ ~n

hn+i~j

~~j

z n n 2 n+I n-I _n

§20 a

and

Al al #(+

= sin

#(+

^~

_(sin #(((

₊ sin

#(_1

2 sin

#(+~ (4)

a

~

defining

the

junction penetration deptll

Aj :

Ai~

=

8~~ajj/~2oc

W1tll these definitions:

Gi

₌ ^~'°

jj ~j _dp(I

_cos

#(+~)

2~c

/

l~3a'

^~

~ / _~~~

~°~

~~~~

In all

physical

situations > _a so that _a z-continuous version of

equation (4)

_can be

convenient,

I-e-_:

Aj0j#(z, _y)

₊

_{A~0)(sin #)}

= sin

#(z, z)

Some further

expressions

^{for the different}

^energies

^{will be useful: from}

(I)

and

(2)

we can write

~'~~ ^~

~~ ~~ ~ /

~~ ~~~~~~

^~

~

(~~~~

~-I)~j

/~ ~ / _~~

~~~~ ~~~~ ~

(h~~(

+

h(-1 2h(~~ )j

^{+ surface terms}

N° I LAYERED SUPERCONDUCTORS IN A PARALLEL fIELD l1?

Using (3)

^for a

given equilibrium

state, ^the

potentials

can be written _{as a} function of

#(+I only:

~Jll

^~

~~ ii~2 ~ /

~~ ~~~~~~~~~~

~~~

n

=

~'( ~j _dp #(+ ~0~h(+~

+ surface _terms 16~

~

/

=

~'°

jj ~j _dp #(+I

_sin

_#(+~

4~c

/

G>jj +

Gi

₊

Gh

₌

fIJJ lO / _{dP (i}

CCS

4t+~ ~4t+~Sin 1+~)

(6)

n

l~3a'

^~

~ / _~~ ^~

~°~

~~~~ ~~~~~~~ ~~~~

3. The mixed state for

parallel

fields.

For

temperatures

T*

~

^T

~ ^Tc

^the

anisotropic

^continuous 3D model is valid and the mixed state

(Hci

< H <

Hc2 displays

a

triangular

^lattice of vortices distorted

by anisotropy

effects

[7, 13]

The situation for T < T* will be

investigated by cotlsidering

two limits:

I)

very

high applied

fields since

Hc~jj

= co for

diamagnetic effects, it)

low field I.e. H

~-

Hcijj

3, I HIGH HELD LIMIT. Tills calculation does not take into account the

paramagnetic

limi-

tation

(if any)

and b

physically

valid for H _<

Hp.

^{We cotlsider that the}

magnetic

_energy

^aH2 /8~

is much

larger

than the

Josephson coupling ~2ojJ/c

_{so as} to

regard

the latter _{as a}

perturbation.

When

jj

₌ 0 the field

penetrates perfectly (h

₌

H)

and from

(3)

#(+~(z)

=

2~a~2j~H(z zn)

The field H defines _a characteristic

length AH

~ PO

~ 2~aH

Because of the absence of

tunnelling

current

jz

^there is _a full

degeneracy

of

phase

difference be- 1ween

layers

which is

expressed by introducing

_zn. The methods amounts to

finding

the structure

by removing

this

degeneracy

for small

jj.

Let B be the average value of the internal field and write

#(+

_as the

superposition

of _a linear function

if 4f(z)

=

2~aiai~B(z zn)

=

~

~

~~

B

and a

correcting

term

~ wn(z)

_:

j

~

it+~ _(z)

=

if _(z)

₊

())

^~

_wn(z)

Our calculation turns out to be _an

expansion

in the small

parameter lB /lj (regarding

this it bears some

similarity

vith that done

by Campbell

et al.

[13]

^{in the London}

approximation). Equation (4) gives

to lowest order

Al _fi(wn

= sin

if ~a~

^{~ ~}

_(sin #f~i

₊ sin

#f_1

2 sin

if wn(z)

₌

-sin~'

~

~'"

+

~

sin ^~'

£"~~

^{+ sin ~'}

£"~~

^{2 sin ~'}

~

~'"

B a B B B

The

thermodynamic potential,

after

using (5)

_can be written G

(condensation energy)

₌

"

i~ / _{~~~~ ~~~}

~

~c~~ ~ / _~~

~~ ^~°~

~~~~

_~

^~~2

^~

~ / ~~ ~~~~~~~°"

16 order

~

the _terms in G

depending

on the

phases

_zn are

J

~

_

' '

-

' '

z ~

~

N° I LAYERED SUPERCONDUCTORS IN A PARALLEL FIELD l19

and vortices

appear

as

solitons,

each of them

increasing

tile

phase by

2~ [8] In the

present problem

flux

quantization

occurs in the

following

_way. From

(3)

_we

_get

£ _(j(+~ (+cc) #(+~(-cc))

₌ ^~~ _x

(total magnetic flux)

~~

i~o

The condition of _zero current at

infinity gives #[+~ (~cc)

=

2~p (Vn)

so that the minimum _en- closed flux is achieved in tliis way:

#((+cc)

= 2~

#((-cc)

= 0

#(+~(~cc)

= 0

(Vn # 0)

These

requirements

_are the

boundary

conditions for an isolated vortex

bearing

one flux

quantum.

In the outer

part

^{the current is small}

#(+~

_ci sin

#(+~

except

^for n = 0 in which _case

#(

~i sin

#o

+

2~9(z)

(9

is the

step function)

whence the

equation

derived from

(4)

and valid at

large

distances:

I

)fij (sin #]+~

+

~

(sin #](

_{+ sin}

#]_

2 sin

#]+~

sin

#]+~

₌

-2~l]b'(z)bno

a

~

Or

(ljfij

₊

l~fij _I)

Sin

#

₌

-2~aljb'(z)b(z)

The Solution iS

~~ ~2

xl lj

(7)

sin

#(z, z)

₌

)Iii _~

+

@ _~2

z2

~~

jj

+ 12

~

whence

~2 z2

h(z,z)

_#

)KU

j2

^~

j2

J

~~

Thus the section Of _a vortex in the

(z, z) plane

is

elliptical

^of

long (resp. short)

axis

lj (resp.

I).

This has been _seen

experimentally

in YBaCUO

[lfl

The linear

approximation

breaks down in the inner

part

Or nucleus since

by

definition sin

#[

reaches the values ~l for some z. The non linear character Of the

physics

here involved is concentrated in the nucleus _so called to avoid the

misleading

use Of the word "core" which

suggests

^{the existence of} a normal

region.

The size Of the nucleus _can be estimated from

(7), using I(o(r)

_~ ln for _r

- 0. It is Of order _a in

~ r

the _z direction

(transverse

to the

planes)

^and Of Order

~lj

^{in the} z direction. The nucleus is thus very ^small

([Or example

in

YBaCUO,

~

j

^<

^10~~)

^as

^compared

^{vith the whole vortex. The}

variation Of

#

and sin

#

is

schematically

shown in

figure

3. We _now turn to the evaluation Of

the energy sv of _an isolated vortex. In the nucleus the absolute value of the current

jz

has two maxima such that

(sin

₁₁ ~ l. From

(6)

it is seen that the nucleus contributes _an energy ^{of order}

~pn+1

~

Xj

=0

2n

n=0

0

^'~

,

-~' _''n=I

-

J

--

Fig.

3. Variation of

#(+~

_and

sin#(+~

_{as a} function _of

z for _{n =} 0

(full line)

and

n = 1, 2...

(dashed lines).

) _jjlj)

=

) £.

_{In the}

outer

region, except

^for the middle

interlayer

_{n =}

0, #

^{is small.}

~c j

Then I _cos

# )#

sin

#

_~

#~

~

(~)

^~ and the

corresponding

energy ^is

negligible.

^{Now for}

I

n = 0 a similar

argument

^{holds but}

#

_~ 2~ for _{z >} 0. Thb

yields

_a contribution

§7o

j~~

^d Sin

Ii

^~~

~i _{1~ ~~l}

2c^~~

jAj

~ ~

~~ ~

Since

h( ()lj)

~ ~ ~

ln~,

_thin

term is much

larger

^than the nucleus contribution and _{as a}

~ j a

N°I LAYERED SUPERCONDUCTORS IN A PARALLEL FIELD 121

~CS~l~

~(

l

~~ ^~

16~21jl~~a

The first field

penetration

occurs at

4~sv

_iao 1

~~~"

~ao

4~ljl~a

It is worth

comparing

this formula to that obtained for _an

anisotropic

continuum [2] :

~~ j

j)~~~

H(i

"

fi~~ _[fjj

lvhen T « T* there b _no In

~

term because in the absence of _a normal core, the cutoff at

fjj

short distances is

only governed by

the

spacing

_a, whence the term

In~ Numerically, taking

for

a

YBaCUO 1 ₌

10~~

_{cm and}

lj

₌ 4 _x

10~~

_cm,

gives _Hcijj

~ 180 G which is in

quite

reasonable

agreement

^with measurements

[16]

Note that the

expression

^of

Hci

ii

given

_{in [2] is} incorrect.

3.3 REMARKS ON OTHER POSSIBILITIES FOR MELD PENETRATION. it is not obvious _a

pried

that the mixed state consists of

quantized

flux lines _or vortices. The first

picture proposed

for the mixed _state of

ordinary _type

II

superconductors

_was that of laminae

[9,10] Indeed,

since the surface tension between normal and

superconducting regions

is

negative (It

>

I/vi),

_a

regular

_array of

parallel alternately

normal and

superconducting

laminae may ^{have an} energy less than that of the whole

superconducting sample

if _a

magnetic

field is

applied.

In classical

superconductors

it _{turns out} that flux lines are more favourable. But in the situation of very ^short coherence

length,

the

possibility

of laminar

phases

is worth

considering.

3.3. I Laminae

parallel

to the

superconducting planes.

The situation is

quite

similar to that

originally

^studied

by

Goodman [9] ^{and so is} the calculation. In _{our case} the normal

regions

_are

reduced _to isolated

single planes separated by

^D

superconducting layers.

The

gain

^{in electro-}

magnetic

_energy is

~

j _H~tanh ))

_and

has to be

compared

vith the loss in condensation

~

~ a

energie

I.e.

).

This leads to a first

penetration

at

Hi

~

~ ~ao I

jajl/2

^{~o I}

_(1)~'~

~~~ 2~V5ljfi

1 ^~

2~V5l jl

_a

3.3.2 Laminae

parallel

_to the c-axis- Because of

phase quantization

this

problem

is

quite

different from the latter. In _a

z-independent situation, (4)

reduces to the classical Sine-Gordon

equation

I)fi~#

= sin

#

The

equivalent

ofa lamina if thus _a

"macroscopic

soliton"

(#(-cc)

₌ 0 and

#(+cc)

₌

2~)

of some

length

L

along

the _z axis

(obviously

currents _are

flowing

in the outer

region

_{or on} the boundaries

of the

sample). Contrary

to the

previous

_case and

similarly

to a vortex there is _no normal

region although

the field

penetrates periodically along

_z. The energy ^{of this} state is [8]

L _i~o

~~

L

i~(

~~

a

2~c~

^~ ~

a 2~3

lja

and it contains ~

flux

quanta.

^{The lower critical field associated with this solution is tlius}

a

~s ^2iao

^{2i~o 1}

~~ ~2

lja

~2

ljl

_a

Now,

#

~

is

sufficiently large,

it is clear that

a

Hcijj (vortex)

_<

Hi

_<

Hi

and that the mixed state in

parallel

field

probably

^{consists of vortices.}

4. Conclusion.

The

study

which is

reported

here

strongly suggests

^{tliat there b} no

qualitative change,

when go-

ing

from

Tc

to low

temperatures

^{in the mixed state}

which,

_{as a} consequence, appears as a

unique phase.

^{However tile}

problem

of

describing

tliis

phase

is not

completely

^elucitated for H _>

Hci

ii

and tile

complicated

non linear

equation(4)

would

probably require

numerical treatments.

But,

a

priori

it is clear that

increasing

the field and thus

changing

the vortex array, ^involves a lattice lock-in

problem.

In

particular

_a

stepvise

evolution

(staircase function)

of the _z

-projected

distance

between vortices is

possible,

whence

irreversibility.

This would be controlled

by

the condensation energy involved

by

a core

crossing

a

layer.

Thin

question

^is connected with _non

equilibrium phe-

nomena such as flux

creep, pinning

and vortex undulation. This

topics

are now

receiving

a

great

deal of attention

ill, _12]

and _{so are} also the

possibility

of _a

glass phase

^due to disorder and of _a melted

phase

of vortices. It b

hoped

tliat the

present understanding

of the basic and

underlying physics

of the

equilibrium

state win

help

in further

developments.

References

ill

LAWRENCE WE., DONIACH s., hoc.

Con?

Low

limp. Phys.

LT12

Kyoto (1970) p.361.

[2] ^BoccARA N., CARTON

J.P,

SARMA

G., Phys.

Lett. 49A

(1974)

165.

[3] GAMBLE

ER.,

DI SALVO

FJ.,

KLEMM

R-A-,

GEBALLE

TH.,

Science la

(1970)

^568.

[4] MORRIS

R-C-,

COLEMAN

R-V, Phys.

Rev B 7

(1973)

991.

[5j ^NELSON D-R-,

Phys.

Rev LeL 60

(1988)

1973.

[6j ^BULAYEVSKr

L.N.,

Zh

Eksp.

riot Fk 64

(1973)

2241.

[7j ^BALAJSKII

A~V,

BURLACHKOV

L.I.,

GOR'KOV

L-P,

Sov

Phys.

JETP 63

(1986)

^866.

[8] see for instance SOLYMAR

L., superconductive tunnelling

and

applications, Chapman

ed.

[9] GOODMAN

B-B-, Phys.

^{Rev LetL} 6

(1961)

597.

[10] SAINT-JAMES

D.,

SARMA

s.,

THOMAS

E-J-, Type

II

superconductivity (Pergaman Press) 1969, Chap.2.

[ll]

CHAKMVARrY s., IVLEV

B.I.,

OVCHINNIKOV

Y.N., Preprint.

[12] DONIACH

s., Preprint, Report

from the Los Alamos

symposium (1989).

N°1 LAYERED SUPERCONDUCTORS IN A PARALLEL FIELD 123

[13] ^CAMPBELL

LJ.,

DORm

M.M,

KOGAN

VG., Phys.

Rev. B 3ll

(1988)

2439.

[14] ^DOLAN

GJ.,

CHANDRASHEKHAR

G.V,

DINGER

TR.,

FEILD

C.,

HOLTzBERG

E, Phys.

Rev Lett. 62

(1989)

827.

[1~

DOLAN

GJ.,

HOLTzBERG

E,

FEILD

C.,

DINGER

TR., Phys.

^{Rev Lett. 62}

(1989)

2184.

[16j

SENOUSSI s., AGUILLON C.,

Earophys.

Lett. 12

(19W)

273.

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