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Solid State Communications Vol. 3, Pp. 127-130, 1965. Pergamon Press Ltd. Printed in Great Britain.

VORTEX NUCLEATIONB~TYPE U SUPERCONDUCTORS

P. G. de Gennes

Facult des Sciences, Orsay (Seine-et-Oise), France

(Received 7 April 1965)

The barrier field1 1 s above which vortices can enter in a type U superconductor

is derived from the Landau-Ginzbarg equations in the limit ~.>> 1. For a macro-scopic superconductor with a plane surface H,~=H~exactly, In agreement withexperiment.1 For thin films in parallel fields, the results are very different.

THE surface barrier2 opposing the entry of parameter. We consider only the limit ~t >> 1flux lines in ~ tjpe U superconductor Is common- and replace equations (1) and (2) by:6

ly discussed ~in terms of a London approxi-mation: The London equation (with a fixed f2 - - A2 3penetration depth X T) is applied everywhereexcept in the hard core of a vortex (hard coreradius ~T) The merit of this approximation is (1 2 A 3to permit simple calculations involving image ~- =A - A (4)forces, etc. Numerically, however, It is not d zsatisfactory for two reasons;

Equation (4) is the equation of motion ofa par-

(a) the results for B~are sensitive to the tide of abcisaa A and mass unity in a potentialexact value taken for the core radius,

(b) as we shall see; for applied fields H0 A

2 A4oforder H~,the order parameter 1 i I is strong- U(A) = -y- +ly reduced near the surface, even when the vor-tex has not yet appeared.

z being the time.This differs strikingly from the situation ob-tained in type II superconductors where magne- The corresponding energy Integral is:tic fields of orderH~(the thermodynamic field)are known to distort 1 4 I only very slightly. 1 d A2

~ (i) + U(A) = E = constant (5)In fact, the fieldH~can be derived from

a study of the solutions dl the one-dimensionalLandau-Ginsburg equations:5 We now apply equation (5) to some specific

examples:1~2f 2 2

~ f = f(-1 +r+A ) (1) (a) Semi-infinite superconductor. Ifthe super-conductor occupies the half apace z

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12 8 VORTEX NUCLEATION IN TYPE II SUPERCONDUCTORS Vol. 3 , N o . 6

where ~ is related to the applied field H0 by the 2 3/2e

condition: = ~& - r (e~X,~, it ~ 1) (10)c Tt2sthhC

* ~ za~- (7) Note thut In this particular case the current den-CO B city is uniform within the film thickness and

a~eeswith the conventional critical current.8The penetration depth x (In units of A T) is:

For more general values of e, Hg(e) Is

A rta~~1-1 8 derivedas the limiting curve of the various H z 0 solutions H(s) obtained whenA (O) Is varied (see

FIg. 1). Note that, for finite e, this limiting

The maximum permissible value H8 of the ap- curve corresponds to values of A(s) which are

plied field is obtained when sinh C 1, and Is: smaller than 1. Thus, by equation (3), the orderparameter does not fall to 0 on the outer surface

- = (9) when H0 a

(c) Thin film in symmetrical fields. Taking theas announced. This is to be compared with the film limits as ~ a *fz, we1~ve:London typi apprcaimatlon according to

which H

H~= reference 2.A(s) is oddandA(O).0. HereE>0andH5>H~.

Experimentally, for the NbTa alloy of The limiting curve of the H(s) nwMoldref. 1:

0C 3lOoe, H5 3 2 0 oe.

7 nowcorreepondstovah*eoIA(Z)whicharelarger than 1 andi~thus ui~ihysical(f2

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Vol. 3, No. 6 VORTEX NUCLEATION D~TYPE IfSUPERCONDUCTORS 129

1 .0

C

H H5

.5

(~)

00 1 .~ 2

xl.FIG. 1

Field distributions in thin films with 0 field on one side (z = 0). The curve (r)Is one particular solution. The heavy curve is the limit of the region spannedby the rcurves, and gives the entrance barrier field Hg as a function of filmthickness e = z.

2 N

H~ S

10 2

FIG. 2

Field distributions in films with symmetrical fields (origin at mid-point of thefilm). (r) Is one partIcular solution, and is physically meaningful up to thepoint M where dH/dz 0 and the order parameter vanishes. The curve for Mwhen r is varied gives the entrance barrier field H8 as a function of the filmthickness e = 2z.

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130 VORT EX NUCLEATION IN TYPE II SUPERCONDUCTORS Vol. 3, No. 6

in cases (b ) an d (c), it is important to avoid any source of trapped flux in the initial state. 10

References

1 . D E BLOIS R.W. and DE SORBO W., Phys. Rev. Letters 12, 49 9 (1964).

2. JOSEPH A. S. and TOMASCH W . J., Phys. Rev. Letters 12, 219 (1964).

3. BEAN C. P. and LIVINGSTON J. D., Phys. Rev. Letters 12, 14 (1964); DE GENNES P. G., Cours:

M4taux at Aulages Supraconducteurs, Orsay (lN3~.

4. BOBEL G. a nd R AT T O C. F., to be published.

5. LA NDA U L.D. and GINZBURG V. L., J. Exp. Theor. Phys. U.S.S.R. 20, 1064 (1950).

6. The order parameter f as given by equation (3) does no t satisfy the boundary condition duds = 0 atthe sample surface. The deviations from (3), however, are restricted to a region of thickness ~Tnear the boundary, and in the limit it >1 they are negligible.

7. Similar results are quoted in ref. 1 for pure Nb : Hc = 1360 oe, H 8 ~ 1400 oe. Bu t here it 1 an d we

cannot apply the present analysis.

8. GINZBURG V. L., J. E~.Theor. Phys. U.S.S.R. 34, 313; Soviet Phys. - JETP 7~78.

9 . BOATO G., GALLINARO 0. and RIZZUTO C., to be published. W e would like to thank ProfessorBoato for both preprints ref. 4 and 9.

10. W e wish to thank J. Friedel and Y.A. Rocher for a discussion of this point.

Le champ de barribre Hg au-dessus duquel les lignes de tourbillon peuventpdntrer dana un mat~rIausupraconducteur de 2e espbce eat ca1cuI1~kpartirdes dquations de Landau-Ginsburg dane In limlte oti it ~ 1. Pour un mat-rlau massif lImit4 par u ne surface plane, Hg eat exactement gal au champthermodynamique Hc en accord avec lexp4rlence. 1 Pour des films minces

(en champ parailbie ~ Ia surface) lea rdsultats sont tree diffrents.