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Submitted on 1 Jan 1978
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CRITICAL CURRENTS OF A ONE-DIMENSIONAL
ARRAY OF SUPERCONDUCTING VORTICES IN A
PERIODIC PINNING POTENTIAL
J. Clems, P. Martinoli
To cite this version:
CRITICAL CURRENTS OF A ONE-DIMENSIONAL ARRAY OF SUPERCONDUCTING VORTICES IN A PERIODIC
PINNING POTENTIAL.
J.R. Clems and P. Martinoli .
Ames Laboratory-DOE and Department of Physios, Iowa State University, Ames, Iowa 50011, and
Labo-ratorivm fur Festkorperphysik, ETHZ, CH-8093 Zuriah, Switzerland.
Abstract.- A theoretical study of the static interaction of a -vortex chain with a periodic pinning potential is presented. The critical currents for matching and non-matching configurations of the array are deduced from a model based on the formation of a vortex-defect superlattice. It is shown that defects (vacancies or interstitials) along the vortex chain are essential to produce a signi-ficant pinning effet.
Recent experiments on periodic pinning struc-tures /l/ have stimulated renewed interest in the problem of determining the critical currents of in-homogeneous Type II superconductors. In this letter we present preliminary results for the critical cur-rents, j , of a one-dimensional array of supercon-ducting vortices interacting with a harmonic pinning potential. A distinctive and important feature of our model is that, in contrast with existing pinning
theories 111, the discrete nature of the vortex chaii
plays an essential role in the calculation of j . We consider a vortex chain coupled to a pin-ning force, F , of the form F (x) = F sin 2TT(X/A ) ,
P' PK po g "
where A is the wavelength of the periodic pinning structure. Let us first briefly discuss the case of configurations such that a = MA , where a_ is the lattice constant and M an integer (matching configu-rations). Since all flux lines simultaneously expe-rience the same pinning potential, the restoring force, F , of the chain on a given vortex vanishes /I/. Under these conditions the macroscopic pinning
force determing j is simply the sum 111 of
(identi-cal) elementary pinning forces acting on the indi-vidual vortices of the array. As a consequence, the critical current, j , of matching configurations is the largest supercurrent the vortex array can
sus-tain before entering the dissipative flux_flow
re-gime. A straightforward calculation gives j M =
cF / A . po To
Suppose now that the array configuration de-viates from a matching one. In this case the vortex
chain lowers its energy by deforming in such a way as to take advantage of the pinning potential. The degree of deformation is controlled by the balance of F and F . It is precisely the presence of F , however, which weakens the macroscopic pinning ef-fect and thus lowers j .
To simplify the discussion _of non-matching configurations, it is convenient to temporarily ignore the elastic response of the chain. Then, the effect of turning off the interactions among the vortices amounts to relaxing an originally rigid vortex chain into the periodic potential. As shown in figure 1, if a/A = M/N, this results in the for-mation of a vortex-defect superlattice of period A = MA = Na, provided the integers M and N have no common factor. The chain defects, whose average spa-cing is A = A /|M - N|, turn out to be vacancies, if M>N, or interstitials, if M<N. The effect of tur-ning on the repulsive interaction among the vortices
is now easily understood in two extreme situations
ll/. If the resulting F ' s are much smaller than F
° I po
the superlattice is essentially unaffected by the vortex-vortex interaction, whose main effect is to slightly smooth out the defect structure. In the op-posite limit the superlattice structure is washed out, leaving a vortex chain showing a weak nearly harmonic distortion. In presence of a transport cur-rent j the elementary processes triggering the ins-tability of the whole vortex array are identified as "flux jerks" (figure 1) occurring where vortices oc-cupy interstitial sites or are located in the
imme-JOURNAL DE PHYSIQUE
Colloque
C6,
supplément au n°
8,
Tome
39,
août
1978,
page Qb-Çll
Résumé.- On présente une étude théorique de l'interaction statique d'une chaîne linéaire de vortex avec un potentiel d'ancrage périodique. Les courants critiques pour des configurations résonnantes ou non de la chaîne sont déduits à partir d'un modèle basé sur la formation d'un super réseau de vortex et défauts. On montre que les défauts (lacunes ou interstitiels) sont essentiels pour obtenir un effet d'ancrage important.
d i a t e proximity of a vacancy. a r e independent of t h e s p e c i a l c h o i c e made f o r FIm.
a RIGID VORTEX CHAIN
0 - - - - 0 - - o c : : : o
LA,
J SOFT VORTEX CHAINS I---- -;Flux Jerk Flux Jerk Flux Jerk
Fig. I : Vortex-defect s u p e r l a t t i c e f o r a/A = 413 and corresponding c r i t i c a l s t a t e f o r vanishfng vor- tex-vortex i n t e r a c t i o n .
The s u p e r l a t t i c e model c o n s i d e r a b l y s i m p l i f i e s t h e c a l c u l a t i o n of j
.
The problem i s now reduced t o t h a t of determining t h e " c r i t i c a l " p o s i t i o n s of t h e N v o r t i c e s forming t h e b a s i s of an elementary super- l a t t i c e c e l l from a s e t of N coupled non-linear e q u i l i b r i u m e q u a t i o n s of t h e formFL + Fp (xm)
+
FIm = 0where FL = jQo/c i s the Lorentz d r i v i n g f o r c e . I n f i g u r e 2 we show r e s u l t s of a numerical c a l c u l a t i o n of j c based on a Hooke's law i n t e r a c t i o n f o r c e of t h e form FIm
-
- k ( ~ , + ~-
2x +x
) , where k i s t h em m-1 f o r c e c o n s t a n t . A more r e a l i s t i c c a l c u l a t i o n based on t h e Coulomb-like i n t e r a c t i o n c h a r a c t e r i z i n g t h e mixed s t a t e i n t h i n superconducting f i l m s w i l l be published elsewhere. R e f e r r i n g t o f i g u r e 2 , we f i r s t n o t e t h a t j shows a d i s c o n t i n u i t y a t t h e o n s e t of a matching c o n f i g u r a t i o n . This f e a t u r e r e f l e c t s t h e phase t r a n s i t i o n o c c u r r i n g w h e n t h e s u p e r l a t t i c e s t r u c t u r e changes i n t o a commensurate c o n f i g u r a t i o n of t h e c h a i n . ~ o r e o v e r , o u r c a l c u l a t i o n s show t h a t , f o r a given v a l u e of a/A a marked decrease of j c
g'
s e t s i n when t h e parameter
f3
= kX /F becomes l a r - g POg e r than m i ty
.
R e c a l l i n g our previous d i s c u s s i o n , we a r e thus l e d t o t h e important conclusion t h a t t h e formation of d e f e c t s along t h e v o r t e x chain i s es- s e n t i a l f o r t h e o c c u r r e n c e of a s i g n i f i c a n t pinning e f f e c t . Another i n t e r e s t i n g f e a t u r e emerges from o u r model. As f3 i n c r e a s e s , t h e symmetry of a given c h a i n c o n f i g u r a t i o n becomes a c r u c i a l f a c t o r i n de- termining t h e s t r e n g t h of j c . This accounts, f o r i n s t a n c e , f o r t h e presence of t h e jc-peaks a t a/Ag =
M/2,
a r a t i o d e f i n i n g a s e t of h i g h l y symmetric c o n f i g u r a t i o n s denoted a s Bragg c o n f i g u r a t i o n s i n Ref / I / . We emphasize t h a t , i n t h e one-dimensional c a s e considered h e r e , t h e conclusions l i s t e d aboveF i g . 2 : C r i t i c a l c u r r e n t s of a v o r t e x c h a i n i n a harmonic pinning p o t e n t i a l f o r a Hooke's law i n t e r - a c t i o n f o r c e .
f3
= kA IFg P O '
As a f i n a l p o i n t , we would l i k e t o mention t h a t t h e p r e s e n t model i n c o n t r a s t w i t h o t h e r pin- ning t h e o r i e s , does n o t p r e d i c t a pinning t h r e s h o l d
/ 2 / f o r t h e occurence of f i n i t e c r i t i c a l c u r r e n t s . This work was supported by t h e U.S. Department of Energy, Basic Energy Sciences, and t h e Swiss Na- t i o n a l Science Foundation.
References
/ 1 / M a r t i n o l i , P., Phys. Rev. B
