A SIMPLIFIED MODEL OF THE INTERMEDIATE STATE OF THIN TYPE I SUPERCONDUCTING SLABS

(1)

HAL Id: jpa-00217751

https://hal.archives-ouvertes.fr/jpa-00217751

Submitted on 1 Jan 1978

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

A SIMPLIFIED MODEL OF THE INTERMEDIATE

STATE OF THIN TYPE I SUPERCONDUCTING

SLABS

E. Paumier, J. Girard

To cite this version:

(2)

JOURNAL DE PHYSIQUE _{Colloque C6, supplPment au}

no

8, Tome 39, aofit 1978, page C6-687

A SIMPLIFIED MODEL OF THE INTERMEDIATE STATE OF THIN TYPE

I

SUPERCONDUCTING SLABS

E. Paumier and J.P. Girard

Laboratoire de Physique du- SoZide de ZfUniversitB de Caen, 14032 Caen Cedex, F m c e

Rdsum6.- On propose un modzle simplifid de 1'Btat intermddiaire d'dchantillons dont l'dpaisseur n'est pas grande devant la pdriode de la structure. Ce modsle confirme la ndcessitd de tenir compte de l'dnergie de surface 3 l'interface supraconducteur vide.

Abstract.- A simplified model of the intermediate state is proposed for samples the thickness of which is not much larger than the periodicity of the structure. The model confirms the necessity of taking into account the surface energy of the superconducting

-

vacuum boundaries.

The first model of the intermediate state of type I superconducting slabs was the non branching model of Landau/]/. Some more recent models were121 based on the assumption that the thickness

R

of the sample is much larger than the periodicity a of the intermediate state structure. It is difficult to test the validity of this assumption mainly because non branching intermediate state experimentally oc- curs/3/ for R1<R<R2 ; Rl(%5.000

i)

is the upper limit of,the mixed state of very thin slabs and R2 (%100~ m) is the limit between the non branching intermediate state and the branching one. For these low values, 9. cannot be considered as 1arge.when compared with a. In this paper we propose a model in which the assumption R>> a is not made. This has consequences not only on the calculation of the magnetic moment, but also on the necessary introduction of the surface energy 0 SVS of the wall area SVS between superconducting (S) material and vacuum (V). This energy is usually neglected because

the area SVS is much smaller than the area SNS bet- wen superconducting (S) and normal (N) regions. If R is not much larger than a, this assumption is no longer valid. This fact was noticed by Marchenko/4/ who recently proposed a model of the intermediate state of thin slabs in an inclined magnetic field. He was interested by the influence of the inclina- tion angle on the shape of the N-S boundaries and on the transition field HIN from the intermediate to the normal state. We propose here a simplified model for a thin slab in a perpendicular magnetic

field.

The sample is a slab infinite along Ox and Oy and of thickness R along Oz which is the direction of the applied field

Sa

(figure 1). The superconducting volume fraction is called s. The figure 2a

Fig. 1 : Sketch of the periodic structure of a slab in the intermediate state.

Fig. 2 : Representation of a quarter of a period in the real plane u = x + iz (a), in the complex potential plane J, = (J + iA/po(b), and in the plane

5 =

5

+ in.

represents the section by the xOz plane of a quarter of a period. In this simplified model the exact shape of the N-S wall is not taken into account and the wall is replaced by a plane perpendicular to the sample faces. The field derives from a scalar potential (J(x,z) or from a vector potential reduced here to its y component A(x,z). The region (figure 2a) limited by the lines of force 1234 (A=O) and

1

1'5 (A = A 12 _o = -J _{2 0 a}aH ) and by the equipotential

(3)

45($=0) is reprekented by a half-hand in the plane of the complex potential J, = I$

+

iA/po(figure 2b) and J, is an analytic function of the variable u = x

+

iz. The conformal mapping of these regions

onto the half-plane q>o (figure 2c) of the intermediate variable 5 =

5 +

iq is obtained by the trans- formations :

1 1

The conditions ul,-u =

-

a and A5

-

A = -11 aH ha-

1 2 4 2 0 a

ve been used and we'have chosen 52 = 1, 5 3 = k2 and

1

E1,

= p2. The condition u3 - u2 =

-

sa and u4 - u =

1 2 3

-

i R give :

2

and

The sample is made of a material characteri- qed by its critical field H and its surface energies

'1 1

+

H

'

A

and

+I

H '6 by unit area of N-S walls and 2 0 c 2 0 c

V-S walls respectively. The parameter A is positive gnd 6 is negative for type-I materials/5/. The state iof the slab of thickness R in the field Ha is deter- knined by the periodicity a and the superconducting iolume fraction s. These parameters are given arbi- ltrary values and the thermodynamic potential G is calculated as a function of a and s. The equilibrium ivalues are obtained by minimization of G with res-

1

pect to a and s. For the volume a1 formed by a ,quarter of period and unit length along Oy, G is the sum :

,G = EC + EW + E V W + GM 1

in which EC

-

1.1 H a1 s is the condensation ener-

1 8 0 c 1

gy, EW $.ioHc2~k is the N-S wall energy, Ei =

7

po s 6a is the V-S wall energy, and

GM

-

-po \Hl(;b a )dH a is the magnetic contribution due to the magnetic moment M(Ha) =

-Iz3(

dx of the volume under consideration. The total reduced thermodynamic potential for the unit volume is :

in which ha = Ha/HC is the reduced applied field and

The e-quation (1),(2) and (4) show that the minimization of g with respect to s and a leads to te- dious calculations. Exact calculations would be use- ful for a more realistic model with a curved N-S wall. As an introduction to this model, we simplify

the equations (1),(2) and (4) by using the following argument : as the points 3 and 4 (figure 2) are near each other we suppose that the difference between k 2 and p 2 is small. This approxima ion gives :

71s lrs R s

,A

k = cos

-,

p = cos --(l--tg71?) and

= - -

2

a.

' r

n 42 (sin-'k

an

-

+

-

Rn p).

712 P 2

For h near hIN = HIN/I$, we have s<<l and g

R

m

may be written as : gm = (1 + 2 ); s2.

his

ap-. proximation leads to the following equilibrium value of a :

but gives an unreliable value of hIN. However, the expression (5) shows the usual dependence of a in (

A R ) '

, and one additional term comes from the introduction of 6. The comparison with experimental results requires a more realistic model with curved N-S wall. This leads to more complicated calculations which must be treated by numerical methods.

References

/I/ Landau,L.D., Phys. Z. Sowjet.

11

(1937) 129 /2/ Lifschit2,E.M. and Sharvin,Yu.V., Dokl. Akad.

Nauk (SSSR) 79 (1951) 783

Fortini,A. and Paumier,E., Phys. Rev. s(1972) 1850

Krempasky,J.J. and Farrell,D.E., Phys. Rev. (1974) 2894

Landau,~.l.and Sharvin,Yu.V., Phys. Rev.

B13

(1976)1359

/3/ Dolan,G.J., J. Low Temp. Phys.

2

(1974) 133 Kiend1,A. and Kirchner,H., J. Low Temp. Phys.

16

(1974) 349

/4/ Marchenko,V.I., Zh. Eksp. Teor. Fiz.

2

(1976) 2194. Sov. Phys. JETP

5

(1976) 1156