Bean-Livingston barrier in uniaxial superconductors

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Bean-Livingston barrier in uniaxial superconductors

V. Damjanović, A. Simonov

To cite this version:

V. Damjanović, A. Simonov. Bean-Livingston barrier in uniaxial superconductors. Journal de

Physique I, EDP Sciences, 1991, 1 (11), pp.1639-1647. �10.1051/jp1:1991231�. �jpa-00246441�

Classificafion Physics Abstracts

74.60E 74.70V

Bean-Livingston barrier in uniaxial superconductors

V. P.

Damjanovib

_{(1, 2)} and A. Yu. SimOnOV _(I>

*)

(')

Laboratory of

High-T~ Superconductivity, Physics

Department, Moscow State

University,

117234 Moscow, U-S-S-R-

(2)

Department

of Mathematics and

Physics, Titograd University,

P-O- Box 211,

Titograd, Monte-Negro, Yugoslavia

(Received

5 April 1991, revbed 28 May 1991,

accepted

15

July

1991)

Al~stract.-The role of surface barrier _on the inclined _vortex

penetration

into uniaxial

superconductors

is

investigated.

It is shown that the field of the barrier

disappearance

is

practically independent

_{on an} angle of

magnetic

field inclination. Peculiarities of the vortex

penetration

into the

layered superconductors

are discussed. The rotation of vortex direction _near the surface is

predicted.

1. Introduction.

It is

generally recognized

that the

anisotropy plays

_an

important

role in the

properties

of

high-

7~

superconductors.

Because of the

strong anisotropy

the

screening

currents of inclined to

anisotropy

axis _c Abrikosov vortex must run in the

conducting planes

and _not in

plane perpendicular

to the vortex axis _as for _an

isotropic superconductor.

As _a

result,

the

picture

of the vortex

penetration

in the

sample

is

strongly

^different from the _one that could be observed for

isotropic superconductors.

For

example,

the direction of the extemal

magnetic

^{field and}

the Vortex axis do not coincide. For

layered superconductors,

at

H~j

^vortices

practically

for the whole range of

angles

lies in

(ab)-plane [I]

and the lower critical field

corresponds

to the

penetration

of the chains of vortices

lying

in

(cd) _{plane [2].}

All that results _was obtained for the bulk

superconductor

without

taking

into consideration surface effects. It is well-known that _near the surface of

superconductor

in the range of fields

H~j

w H _w

H~~

an energy barrier

(Bean-Livingston barrier)

arise

[3].

For the

isotropic

_case

field at which such barrier

disappear

is

Wo H~~ ₌

,

(1)

4

vhf

where

f

is the coherence

length,

A is London

penetration depth

and Wo ^{is the flux} quantum.

One _{can see} that _on the order of

magnitude H~

coincides ~vith

thermodynamic

critical field

H~.

(*)

After October 15, Permanent address _: Ames

Laboratory

and

Department

of

Physics,

Iowa State

University,

Ames, Iowa 50011, U-S-A-

1640 JOURNAL DE

PHYSIQUE

I M II

For

anisotropic superconductors

the surface effects may

play

the certain role

[4-6].

It is

pointed

out on the influence of

Bean-Livingston

barrier _on the lower critical field determination is

single crystal samples

of

TmBa~CU~O~ (see

Ref.

[4])

^and

quasi-one-

dimensional

superconductors Tl~mo~se~ [6].

^{In reference}

[5]

it is shown that the surface effects

plays

the

important

role in the _process of Tl and Bi 2212

single crystals magnetization.

The influence of surface barrier _was

theoretically investigated

in

[7],

^{but in that} work the effect of chain formation

[2]

^{and the}

anisotropy

of

magnetic

field

screening along

_a and _c axis inside the

superconductor [I]

_are

neglected.

In _our work _we consider the

penetration

of inclined Abrikosov _vortex into uniaxial

superconductor

in tilted

magnetic.

In section 3 the results of numerical calculations for the

case of

layered

and

quasi-one dimensional-superconductors

_are

presented,

the

peculiarities

of the situation in

layered superconductor

_are discussed.

2. General formulae.

We consider the vortex

penetration

into the

anisotropic type-II superconductors

in the London

approach,

where the Gibbs

potential

_can be written _as

(see,

for

example [1, 8])

G

=

Id

V

[i~

₊

~(curl I) _(curl I)

₂

hi ₍₂₎

8 ~

where A

= A

~~ is the London

penetration depth

where

screening

currents flows in

(ab)-plane, I

_{is the local field and is reduced the effective}

mass tensor. We _use the

following

notation for the

principal

values of the _mass tensor v, =

m,/m~ (m,

is the electron effective _mass

along I-axis,

I ₌ a,

b, c)

and _v~

= v~ =

I,

_p~

=

m~/m~

_{= e} + I. For

layered superconductors

m~ ~ m~

(e

_~ 0

),

but for

quasi-one-dimensional

type of

anisotropy

_{m~ ~ m~} and

~ e ~ 0.

Local field

I

can be determine from London

equation _{[1, 8]}

_:

- - -

h ₊ curl [@

(curl h)]

_

=

Wo

^I

~j

³

^(r I,). (3)

Here

I

_{is the unit}

vector

along

vortex axis.

According

to the condition A »

f

_we

neglect

in

(2, 3)

the structure of the vortex _core and _assume that vortex _cores do not

overlap,

I-e- distance between

neighboring

vortices is d »

f.

In the range of fields H _w

H~

that condition is fulfilled.

For

analysis

of the _vortex

penetration

^into the

sample

it is convenient _{to use} the _new coordinate system

(X,

Y, Z

)

rotated with respect to the

principal

^axis

(a, b,

_c

by

the 8

angle (see Fig. I).

In that _new

system

vortex directed

along

Z-axis and the components of effective

mass tensor _are

p_~~ = l _{+ e}

sin~

₈

,

Myy =

i

,

p~~ =

I _{+ e}

cos~

₈

,

(4)

p~z = p~~ = e sin 8 _cos 8 Mxy = Myx = Myz ^{= M}_zy ^{= °}

Although sufficiently simple analytic

solution

I(X, _Y)

_for

a

single

vortex is not

available,

the Fourier transformation in

(X, Y) plane

it

=

I (I )

_exp

(- it

_I

d~i

,

v fi

~ _z,I

~

b, Y

Fig.

I. Axes

(a,

_l~, c) are the

crystallographic

axes of the uniaxial

superconductor, I

_{axis is directed}

along

vortex axis.

(I

_{is the}

reciprocal-lattice

vector in

(X, Y) plane)

is

readly

obtained for _a vortex oriented

arbitrary

within the uniaxial

crystal [8]

h~j

=

h~j _k~/k~

₌ ^W_o v

~=

k)/d

hzi

~

lPo('

+ l~zz

k~)/d (5)

d=

(I+k~)(I+v~~kj+v~k)).

Here _{we use} units of

length

reduced _to the London

penetration depth

A.

We consider the situation when the

superconducting sample

_occupy the

half-space

Y~ 0.

Boundary

conditions

require

that the normal _to the surface

component

^{of the} vortex

screening

current turns to _zero. We _can

comply

with this condition for the vortex situated in Y= Yo,

having placed

a mirror

image

of the vortex in

point

Y= Yo, ^with

opposite

directions of fields and currents

[3].

Local field

I

can be

presented

_as a sum of

decreasing

external field

ii

_Y~

= H

i~

_{cos w exp}

_Y/A

₊

i~

sin _w exp v

j

^~/~

Y/A (6)

and the field of the

pair

vortex-anti vortex

h~

₌

h~~

+

h~_

^On the surface of the

sample

h~ 0

Let us Gibbs potential _(2), where the _integral is taken ver the whole

the

sample

without the

vortex

cores. Using (3),

one

can transform ₍₂₎ to the surface ntegral

~

2

_ _ _ _

G⁼ ~ j (dS .

Here the

integral

is taken to the vortex _core surface S' and to the surface of the

sample

S".

Integral

to the

sample boundary

_can be taken

analytically. By using

the Fourier

-

_ - -

transformation of

h(r ) (see Eq. (5))

and

bearing

in mind that _on the surface h

=

H,

_{one can}

find for the energy

connecting

with vortex

I ² _{_}

_

~p~ H

Gl'= (dS [H

_x curl

h]

=

dk~ k~ exp(ik~ Yo)

x

8 _w

~,, 4 _w

cos(8-w)(Mxx+Mckj)+~~~ ~~~~~"~~= ~°~(iii).

~

(l +k))(I

₊

v~kj)

^~~

1642 JOURNAL DE PHYSIQUE I M II

In the limit

f

- 0 the

integral

_over the vortex core surface is

equal

to

Wo-

- ~

G(=-I.H-E;~~+E~,

where

E(

is the _energy of

solitary

vortex

El

=

~°

_v

)(~In

_«

(

8

(8)

16 w A

~~ A

~

~here «(8)

=

(A~/f~)[v~~/v~]~/~

is the

angular-dependent Ginzburg-Landau parameter, f~~

^{is the} correlation

length

in

(ab) plane

and term E;~~ =

i. i~~ (2 Yo)

describes the

2

interaction between vortex and its

image. Using

the Fourier transformation in

(X, Y) plane (5)

_one can rewrite the

expression

_{for E,~~} as

al _(I

_{+ v~~}

_k~)

_exp

_{(2 ik~ Yo)}

~-

al

E;~~ =

©D

~ ~ ~ ~ ~

d k

= ~ ~

F(k~) dk~, (9)

32 _w I ₊ k I _{+ v}

~

k~

+ v

=~

k~)

16 _w A _o

where

F(q)

=

('

_{+ q}^~

sin~

₈

)~

_(COS~ 8 ' ₊

q~)~

^~/~

exP1-

2

Yo('

₊

_q~)~/~l

₊

+

sin~

₈ ^{~ ~~~}

^{~~ ~~exp}

₂

Yo

~ ~~~

~~

~~) (9.I)

"

c "

r

Integral (9)

is well convergent.

The result for the energy

connecting

with vortex is

G~

₌

G(

+

Gl'

=

~° i

(l~j

#)

E;~~ +

El. ₍₁₀₎

If _we consider not a

solitary

vortex, but the chain of vortices with

period

_a, oriented in the

(ci _)-plane,

_{it is necessary to consider the} interaction of the chain with its

image. Owing

to the

analysis, analogous

to the _case of isolated vortex, we obtain the _same

expression

for the vortex energy

G~, expressed by equation (10),

but

(9)

is

replaced by

E,~~ =

~)

~

jj

F ~ ~~~

+

j~ ^dq

^~°~~

^~~~~

^~

^{)~~~~~~ ~~}

16 _qr

~ ~_~

a o

(I

_{+ q}

)

^/~

q2

^coth

(a(

I ₊

q~)~/~/2

A

q~ cotg~

8

~

(l

₊

q~)~/~

~ ~

iii

i

~~

C°th

A

^{" ~}

Ill ^~l _{~i ')}

Integral

term in

(11)

is the energy

gain

from the vortex chain formation. For

YBa~Cu307 (anisotropy

constant _e

m 25

[9])

^this

gain

is maximal at 8

60(

and the

period

of such chains is _a 2 A

[2].

The field of the surface barrier

disappearance

_H~~ can be found from the condition

(3G/3Y~y=

~~ ⁼ 0

[3]

^{both for}

solitary

vortex

~fi~ ^CD

Hj(~

"

~ ~ ^~

Fl (q) dq (12)

and for the chain of vortices

where

Fj (q)

=

(I

₊

q~ ^sin~

8

)~ (cos~

⁸ exp

[- 2(1

+

q~)~/~/«~~]

+

~

I _{+ v}

zz

q~

I _{+ v}

zz

q~

~/~

+ sin 8 exp 2

/«

~ _,

MC MC

D

= cos 8 _{cos w} exp

[- 1/«~]

+

Ml

^{~/~ sin 8 sin} _w exp

[- Ml

~/~/«~~]

In the

general

_case of

arbitrary

directed field the

problem

_can be solved

only numerically,

but

we can find the value of H~~

analytically

in the first

approximation

for _{some cases.} First of

all,

we can solve the

problem

of H~~ determination if the extemal field direction coincides with the symmetry axes a and _c. For such orientation chains _are not formed

[2].

It field directed

along

_c axis

(w

=

0)

_{one can} find

~

Wo

Kj(2/x~)

_Wo

~~

2

«AS eXp(- I/K~b)

^{4 vi} _~

f~b

^~~~~

where

Kj(x)

is the modified Bessel function and _{we use} the limit

Kj(x

_-

0)

m

I/x

and the

condition _{«~ »} l.

Note,

that

expression (14)

coincide with the

expression

for the

isotropic

case

(I).

If the field is directed

parallel

to the

(ab)-plane,

~

wo Ki (2/«~)

_~Po

~~ (15)

2 VA

~ A~ exp

(- 1/«~)

^{4 "A} _ah

tab

So,

the values of field H~~ in ^the directions

along

and

perpendicular

to the

conducting plane

differ

slightly

on the factor

exp(-1/«~)

^of order of

unity.

We _can find the estimation for H~~ for the

solitary

vortex

by using

the

expression

for the _z- component ^{of the}

single

_vortex field

I iv (°, Y)

= ~

~ ~

~

~ v

zz

K

i

( Yo/v]/~)

The correction to this

expression along

the line

(0, Y)

is small

[11].

We also consider that

« » I and

exp(- I/«)

m I. In this _case

~~ ~~'

~ 4

"~

tab (v]/~cos

₈

o~j~~~

sin 8 sin _p

)

^~~~~

The minimization of that

expression

over 8

gives

the

angle

of vortex inclination in

H=H~~

tan 8

= v )/~ ^tan w

(17)

Note that for such field the

dependence

of 8 _{on e} is not so

strong

as the

dependence

of the

angle

of vortex inclination in the lover critical field

[I]

tan 8

= v

~

tan w

(18)

1644 JOURNAL DE PHYSIQUE I M II

So,

in H

= H~~

angle

of _vortex inclination 8 is

sufficiently higher

that the

angle

of field inclination _w. It is

important

to note, that inside the bulk

superconductor

in such

high

field H

= H~~ »

H~j

that two

angles

_are

practically

coinside 8

m w

[10].

As _a result _near the surface the

equilibrium angle

of _vortex inclination must be

changed comparing

with its value inside the

sample.

If the

equation (17)

is

fulfilled,

the value of the field of surface barrier

disappearance

is

Hen

_"

1fi0/4 "hub tab

and in the first

approximation (solitary

vortex

penetration

and

« »

I)

is

angular independent.

That result is

quite unusual,

because the lower and _upper critical fields _are

strongly angular- dependent.

It is the consequence of the

taking

into consideration the

anisotropy

of the

external field

screening (6).

That effect _was not considered in

[7],

that is

why

the

H~~ ^{value in}

[7] depends

_on field orientation.

3. Numerical results.

To find the true value of the field H~~ ^{it is} necessary ^to minimize

(12)

_over 8 and

(13)

_over 8

and _a and to compare obtained values. The minimal value of the field of

penetration

corresponds

to the real situation. Such _a

problem

_can be solved

only numerically.

In

figure

2 _{one can see} the

angular dependence

of the H~~ ^{value for various values of}

anisotropy _parameter

_e and

Ginzburg-Landau

parameter _«~~. ^One can see, that the H~~ ^is

poorly angular-dependent-

The small difference of H~~ ^{from constant} is the result of the

taking

into consideration

angular dependence

of _« and the effect of vortex chain formation. It is

interesting

to note, that for

layered superconductor H)(~~~H)(~

for all

angles

except w =

0° and _w

= 90[ The

angular dependence

for such chains parameters

~period

of chains and

angle

8 of vortex inclination at H

=

H~~)

^{for the} case of

(Re) Ba2Cu~O~

_are

presented

in

figure

3. One _{can see,} that the

magnetic

flux

penetration

is realized _{as a}

quite

dense chains with _a~ A. The

angular dependence

^of the

angle

of vortex incline is not far from the

dependence (17).

For

quasi-one-dimensional superconductors

vortex chains _are not formed

[10]

and the

magnetic

flux penetrate as

single

vortex. The

angular dependence

of the

angle

of

1.oos

Kab"14000 e=-168/169

1.000

4 n

ld

' ",

S-0.995 ',, K,b"80

~i ",,e=z5

tc ',,,

0.990 e=z5

0.985

0 15 30 45 60 75 90

P

Fig.

2.

-Angular dependence

of H~~ for various values of

anisotropy

_{parameter s} and

Ginzburg-

Landau parameter _K~.

u

le '

i

I

i e~

I / / / /

II -% -' -'

0 15 30 45 60 p ^{75 90}

Fig.

3.

Angle

of the vortex

penetration

and

equilibrium period

of chain at H

= H~~ as function of the

angle

of

magnetic

field incline for

layered superconductor (e

= 25, _K~b

= 80, full

lines)

and

quasi-Id superconductor (s

=

-168/169,

_K~~

= 14 000, dashed

line).

h=0

h=5

Ya/A

Fig.

4.

Energy

of vortex _vs. distance between vortex and surface Y for various values of extemal field (s

= 25, _K~b

= 80, _~

=

10].

vortex incline for the _case of

Tl~mo~se~ [6]

is shown in

figure

3 _too

(see

dashed line in

Fig. 3).

In

figure

4 _{one can see} the

dependence

of vortex energy

G~

on the distance from the surface to the vortex. It is

interesting

to note, that these

dependencies

_{are very} similar to these _ones for the

isotropic

_case

[3]. According

to

(9),

the minimal energy of the _vortex in

layered

superconductor

is reached for 8

= 90° and 8

=

0° for

quasi-one-dimensional superconductor.

As _a

result, for

barrier value at

#

= 0

(that

is maximal value of the

barrier)

_{one can} obtained

- ~

for

layered superconductors AG~(H=0)

=

E~(90°)

and for

quasi-one-dimensional

one

- ~

AG~(H

=

0)

=

E~(0°).

The difference of London

penetration depth along crystallographic

_{axes a} and _c leads _to the

magnetic

field rotation inside the

anisotropic superconductor (see Eq. (5)).

Near the surface

1646 JOURNAL DE

PHYSIQUE

I M II

that effect is

important

and in

strong

^{fields it} can reduce to the rotation of vortex direction while the _{vortex comes to} the surface. It is

interesting

to note, that _near the surface

angle

between the vortex and

anisotropy

axis _can be smaller than that between _c and external field

# (see Fig. 5).

We can obtained that

phenomenon

if _we consider _vortex in effective field

it* _{( Yo)}

=

ii _{( Yo)} it(

_Yo)

,

(19)

u

h=5

h=4nA~H/#a=10

y~/x

a)

h=14000(hoc)

~

~H/#o=l100

~

i if

%°

h=200

Y~/A b)

Fig.

5. -Angle between vortex and

anisotropy

axis

vs. distance between vortex and surface Y for various values of extemal field for

layered superconductor

(s

=

25, _K~

= 80, _~

= 10( Fig. 5a) and

quasi-Id superconductor (s

= 168/169, _K~

=

14 000, _~

= 30(

Fig. 5b).

Dashed lines _are the

equation

(20).

One _{can see} that the

angle

between

#*(Yo)

_and

_anisotropy

_{axis is}

I exp

[- ('

₊

~)

^~~~

~?

t ~~~~

tan p ^* =

j exp

[- Yo/Aj

^~~

In

higher fields,

where _p

m

8,

that effect is strong. ^{For H} H~~ the

equilibrium

value of 8 and

w ^*

(see Eq. (20))

_are

practically

coincide

except

the small

vicinity

of the surface

(see

dashed line in

Fig. 5).

At H

=

H~j

vortices lies

practically along (ab)-plane

for

layered superconduc-

tors and

parallel

to c-axis for

quasi-Id superconductors (see Eq. (17)).

We _{can see} such behavior for small values of fields in

figure

5.

E,~~. As

a

result, for such ForYo _vortices

rotates to the (ab) plane for

layered

the

c-axisfor quasi-ld one and the extreme _on the ependencies8(Yo)appear.

4. Conclusion.

So,

in the uniaxial

superconductors

surface barrier is

poorly angular-dependent.

Field of

disappearance

of that barrier H~~ ^is

practically

constant and coincides with the

thermodynamic

critical field

H~~ by

the order of

magnitude.

Near the surface of

superconductor

considerable distortion of vortex

orientation,

which connected with different field

screening along

_a and _{c axes, as} well _as the intrinsic _energy of vortex and the energy of interaction between vortex and its

image

takes

place.

For

layered superconductor

the

penetration

of vortex chains is

preferable,

but for

quasi-

ld _one

magnetic

flux

penetrate

as

single

vortex.

Such surface barrier _can influence _on the

H~j

determination. It is necessary ^to take surface effects into consideration when the

sample

size is small.

Acknowledgements.

Authors wish _{to express} their

gratitude

to Prof. A. I. Buzdin for his kind attention and

stimulating

discussions. One of the author

(A.

Yu.

S.) gratefully acknowledges Titograd University

for the

hospitality during

the time spent ⁱⁿ

Jugoslavia,

when the part ^{of this} work

was done. This research _was

supported by

the USSR State

Program High Temperature

Superconductivity

» under grant 90062

Magloc

».

References

[1] BALATSKII A. V., BURLACHKOV L. I. and GoR'Kov L. P., Zh.

Eksp.

Tear. Fiz. 90

(1986)

1478 (Sov.

Phys.

JETP 63

(1986) 866).

[2] BuzDiN A. I. and SimoNov A. Yu., Pisma Zh.

Eksp.

Tear. Fiz. 51

(1990)

168

(JETP

Lett. 51

(1990) 191)

_;

Physica

C168 (1990) 421.

[3] ^DE GENNES P. G.,

Superconductivity

of Metals and Alloys

(Benjanfin,

New York,

1966).

[4] MOSHCHALKOV V. V., ZHuKov A. A., PETROV D. K., VORONKOVA V. I. and YANovsKii V. K.,

Physica

C166

(1990)

185.

[5] KOPYLOV V. N., KOSHELEV A. E., SCHEGOLEV I. F., TOGONIDzE T. G.,

Physica

C170

(1990)

291.

[6] BRUSETTi R., MONCEAU P., POTEL M. et

al.,

Solid State Commun. 66

(1988)

181.

[7] KRzYSTON T.,

Phys.

Status Solidi l~158

(1990)

K21.

[8] KOGAN V. G.,

Phys.

Rev. 824

(1981)

1572.

[9] ^{FARRELL D.} E., WILLIAMS C. M., WOLF S. A., BANSAL N. P. and KOGAN V. G.,

Phys.

Rev. Lett.

61

(1988)

2805.

[10] BuzDiN A. I. and SimoNov A. Yu.,

Physica

C175

(1991)

143.

[I Ii

KOGAN V. G., NAKAGAVA N. and THIEMANN S. L.,

Phys.

Rev. 842

(1991)

2631.