PINNING BY RANDOM DEFECTS IN AN ELASTIC VORTEX LATTICE

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PINNING BY RANDOM DEFECTS IN AN ELASTIC

VORTEX LATTICE

A. Campbell

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C6, supplt+nent au

no

8, Tome

39,

a ~ t i t

1978, page C6-617

PINNING

BY

RANDOM DEFECTS IN AN ELASTIC VORTEX LATTICE

A.M. Campbell

Dept. o f Engineering, Tmunpington S t r e e t , Cambridge, England

R6sum6.- On a prdsentd un mod~le pour l'accrochage de lignes de flux dans les supraconducteurs par des centres d'accrochage distribuds au hasard. I1 montre que les th6ories fondges sur des centres d'accrochage inddpendants dans un rdseau dlastique ne sont probablement pas valides. Une simulation sur ordinateur tenant compte des interactions entre centres d'accrochage montre que les difficultds ne peuvent pas Stre surmont6es en autorisant de telles interactions.

Abstract.- A model is presented for the pinning of flux lines in superconductors by randomly spaced pinning centres. It 'shows that theories based,on independent pinning centres in an elastic lattice are unlikely to be valid. A computer simulation which takes account of interactions between pinning centres shows that the difficulties cannot be overcome by allowing for interactions.

INTRODUCTION.- The mose commonly accepted theories models is that we consider the high field limit of pinning in Type I1 superconductors are based on

the elastic distortion of the vortex lattice by small widely spaced centres /1,2/. However, thenum- ber of practical materials to which this model can be applied is small and the only parameter which is predicted is the critical current density. Even in situations where the model might be expected to ap- ply, a recent paper 131 has shown that there are fundamental theoretical problems inherent in it. These can be summarised as follows. i) If the pinning is below a certain value, no pinning is predicted (the threshold effect)

121.

ii) If the pin- nning is only slightly greater than the threshold, we do not expect scaling laws to hold although, experimentally, scaling of J curves occurs with con- siderable accuracy 141. iii) If the pinning is much greater than the threshold, the elastic limit ofthe vortex lattice is exceeded and the model cannot be used.

In this paper, a computer simulation of the model is used to confirm these predictions of the linearised model already published 1 3 1 . This allows the effects of interactions between pinning centres to be allowed for. Attention is concentrated on the region below the threshold,since most superconductors have pinning centres in this regime and it can be shown that a number of interacting systems has a lower threshold level of stability than an isolated system. It is, therefore possible that interactions will account for the pinning observed experimentally below the threshold 1 6 1 .

THE MODEL.- The main difference from the previous

where

I$*

1,

and therefore the energy of a pinning centre, varies sinusoidally. It is also easier to visualise the variables if we invert the problem and consider a number of point pinning centres tobe pulled through a stationary rigid lattice. If we allow each pinning centre to move elastically through the same distance that the vortex lattice in its neighbourhood is distorted in the practical problem the situations are mathematically identical. The main simplification is that we consider pinning centres of equal strength which pass through the vortex cores. Off-centre pinning centres will cause a deflection of the vortices in directions.at an angle to the driving force but this will not affect the results significantly and allowance for this effect would greatly complicate the analysis.

Let the ith pinning centre be initially at x and let it be deflected to a position xi when

o i

the interaction with the vortex lattice is switched .on. If the lattice as a whole is moved by a distan- c e - ~ ~ , this is equivalent to changing xoi to-xoi+x

0'

leading to a new equilibrium position x.. The total energy of the system can now be written.

The first term is the interaction energy betweenthe pinning centre, and the lattice. The second term is the elastic energy. The term (xi-xoi-x ) is the distortion of the lattice at the ith pinning centre.

If the pinning centres are independent aij = CGij where C is the effective elastic constant of the lattice for deflection by a point force.

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the mean force, and its standard deviation for a

between centres. However, it is considerably larger

Then the condifion for equilibrium is aU/ax. = 0.

0.6

d2ydx:,

For the starting position (x = 0) this gives

I

-

k ~ S i n kxi = C(xoi-x.) or : Theory

. , ., . 5% Conf~dence

finite number of centres. The pinning force for

(by a factor d(ckk/Cs6) if the centres are on the even 90 centres shows wide fluctuations and so we

same vortex line. The computer similation was repea- have calculated the restoring force for small dis-

ted with a../aii = a/r. Results were affected by

placements from equilibrium, a2U/ax:. This can be L J

about 2 % and the difference is too small to show

measured directly from the reversible penetration

on Figure 2. It therefore seems extremely unlikely

rk

small signals.

that the inclusion of interactions between pinning

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Sin(z) = K(z

-

z )

o i

Here z = kx, K = c/&k2 and k = 2.rr/a (a is the vortex

0.2

spacing). K is a dimensionless parameter which de-

fines the stiffness of the lattice in comparison with the pinning forces. For K < 1 the solutionsare

unique and no hysteresis can occur. Figure 1 illus-

trates the solutions for K > 1. For randomly spaced

centres the values of z are uniformly spaced along

oi 0.2

Fig. 1 : Graphical solution of equilibrium conditions ---- - Computed -

__."

_.,_..'. .

--

---

--__

,

_K

...

.

'

I 2 \ \

.

\ -

centres can cause significant pinning below the threshold.

the z axis. When the interaction is switched on the

values of z will be found between P and

Q.

The pro-

bability of a solution in 6z is (1127~) Fig. 2 : Calculated and computer simulated values

of d2u/dx:

(1

+

K Cos(z))6z.This probability is also valid for K < 1 and can be used to calculate values for

centre. It is of order a/r where r is the distance

CONCLUSIONS.- Pinning of lux lines in superconductors cannot be explained by the elastic deformation of the vortex lattice and therefore attention must be directed to plastic modes of deformation in which the vortices do not preserve their relative posi-

t ions.

Figure 2 shows the restoring force as a References

function of K. The scale is such that for strong

pinning (K + m), a2IJ/ax; -t 1. The continuous line /I/ Yamafuji, K. and Irie, F., Phys. Lett.

A,

5

(1967) 387 shows the results for an infinite number of centres.

/2/ Labusch,

R.,

Crystal Lattice Defects,

1

(1969) 1 The threshold at K-1 is obvious. The dotted line

/3/ Campbell, A.M., Phil. Mag.

B,

_?L (1978) shows two standard deviations for 90 pinning cen-

/4/ Fietz, W.A. and Webb, W.W., Phys. Rev.,

178

tres. Notice that this tends to infinity at K -t 1. (1969) 657

The hatched line shows the computer simulationusing 151 Appleyard, J.R., Evetts, J.E. and Campbell, AM,

90 randomly spaced pinning centres and a relaxation Sol. State. Commun.,

14

(1974) 567

techniques

151.

The values obtained are within two

1 6 /

Kramery E'J'9 J ' To be

published standard deviations of the theoretical values. Dif-

ferent selections of 90 centres gave similar results. The off-diagonal aij can be calculated using

the method of Labusch

121.

Essentially a../a is

ii

the ratio of the displacement at the jth'ientre to