THE INFLUENCE OF DISLOCATIONS ON SUPERCONDUCTIVITY

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THE INFLUENCE OF DISLOCATIONS ON

SUPERCONDUCTIVITY

G. van Gurp, D. van Ooijen

To cite this version:

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THE INFLUENCE OF DISLOCATIONS ON SUPERCONDUCTIVITY

by

G. J. VAN GURP and D. J. VAN OOIJEN Philips Research Laboratories N. V. Philips' Gloeilampenfabrieken

Eindhoven

-

Pays-Bas.

RBsumB.

-

L'article passe en revue les effets des dislocations sur les propriktks des supraconduc- teurs. On ktablit une distinction entre les effets du libre parcours moyen et les effets hors d'kquilibre. Les premiers donnent lieu & une diminution ou & une augmentation de la tedpQature critique, selon l'ampleur de l'anisotropie du (c gap n d'knergie ou des contraintes internes prksentes dans le supraconducteur. Un autre effet du libre parcours moyen est l'augmentation des champs critiques superieurs.

Des effets hors d'kquilibre sont engendr6s dans un champ magnktique par l'interaction entre let. dkfauts et les lignes de force dans l'6tat mixte du supraconducteur de deuxieme espkce. Les lignes de force peuvent dtre bloquks aux dkfauts de structure dans le supraconducteur. L'article expose les propriktks de ce blocage pour trois types de structures : 1) une rkpartition uniforme des dislocations, 2) une rkpartition hetkrogene, comme dans le cas des parois de polygonisation ou d'une structure cellulaire et 3) d'importantes heterog6nCitks bidimensionnelles comme dans les structures fibreuses ou les joints de grains. Les auteurs sont d'avis que le blocage croft en importance dans cet ordre.

Abstract.

-

A review is given on the effect which dislocations have on superconducting properties. A distinction is made between mean free path effects and non-equilibrium effects. Mean free path effects result in a decrease or an increase of the critical temperature depending on the relative importance of the anisotropy of the energy gap or internal stresses present in the superconductor. A second mean free path effect is the increase of the upper critical fields.

Non-equilibrium effects arise by the interaction between defects and flux lines in the mixed state of a type I1 superconductor. The flux lines can be pinned by defects in the superconductor. The pinning properties of three types of structure are discussed : 1) a uniform dislocation distribution,

2) an inhomogeneous distribution, such as polygonization walls, or a cell structure and 3) large two dimensional inhomogeneities such as a fiber structure or grain boundaries. It is suggested that pinning becomes more effective in this sequence.

1 . Introduction. - Superconductivity has become of technical importance now the application of it in the production of high magnetic fields is widely used. Since for this application defects in the materials appear to be essential, much work has been done on superconductors containing various sorts of defects. The role of each type of defect is not very clear although in a recent review of structural effects on superconductivity [l] in the discussion of an unexpected experimental result, Livingston and Schadler remarked :

As is usual in superconductivity when a strange effect is observed, dislocations were suggested as the origin. However, no evidence showing the connection was presented. )) This may illustrate the present situation.

The precise role played by dislocations is often obscured by other effects such as precipitation or

segregation. We will however confine ourselves as much as possible to the effects which dislocations have or .might have.

2. Superconductivity.

-

In this section we will treat some concepts of superconductivity that will be used in our discussion of the influence of dislocations. For a detailed introduction into superconductivity we refer to textbooks, such as those by Shoen- berg [2], Lynton [3], Blatt [4] and to lecture notes by De Gennes [5] or review articles [6].

Below a certain temperature Tc and in the absence of strong magnetic fields, a superconductor is charac- terized by two equations

E = O a n d B = O .

The first equation means that no electric field can

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C 3 - 5 2 G. J. VAN GURP AND D. 3. VAN OOIJEN

exist in a superconductor : the resistance is unmea- surably small and the second equation, which is called the Meissner effect, means that a superconductor is a perfect diamagnet. The bulk is shielded by a surface current from the external field. The Londons' phenomenological theory [42] shows that the field H and the current density J die out from the surface over Londons' penetration depth

A,,

defined as

iL =

JL

4 zn, e2

where n, is the number of superconducting electrons per unit volume. We consider here the case where demagnetizing effects can be neglected. The condition B = 0 means that the magnetization can be written as :

-

4 nM = H. The Gibbs free energy in a field is given as

with V, = superconducting volume.

In one class of superconductors, which is called type I, there is a critical field H, where the magnetization drops to zero. There is a first order phase transi- tion to the normal state. At this field G, = G,(H,), so that the free energy difference between the superconducting and the normal state is given by

The magnetization curve is for a type I superconductor given in figure l . The temperature dependence of H, obeys closely the law

where H, is the critical field near T = 0.

If there is demagnetization there will be places on the surface where the effective field is larger than the

external field and will reach the critical value when the external field H

<

H,. The superconductor now splits up into superconducting and normal domains ; this is called the intermediate state. At H = H, the super- conducting domains have disappeared. The magnetization curve for such a superconductor is also shown in figure 1. The area under the curve is the same as for a superconductor without demagnetization.

It was shown by Pippard [7] that one needs a second parameter with the dimension of a length to describe a superconductor. He called this the coherence distance

<,

which was introduced as the distance over which a disturbance of the superconducting wave function is extended. This new parameter was needed to explain the mean free path dependence of the penetration depth and Pippard also showed that it could explain the positive superconducting-normal interface energy. The coherence distance for a pure metal is also given by the microscopic theory of superconductivity (BCS theory) E81 as

A general treatment of superconductivity was given by Ginzburg and Landau [9] in terms of a superconducting order parameter Y which was treated as an electron wave function. They derived an expression for the surface energy and found that the sign depended on the value of a dimensionless parameter K defined for T E

T,

as

with

mc' 4 ze2

1

Yo

l2

where ?Po is the order parameter in zero field.

The interface energy - is positive for K < l/& and negative for K > l / J2.

Below a field H,, = KH, 1/2 the normal state is unstable. For K

<

1/45

is H,,

<

H,. This class of superconductors which was introduced before is called type I. For K > 1/42, H,, > H, and is then the upper critical field below which the normal state is unstable. This class is called type I1 superconductors.

Hc It has been shown by Saint James and De Gennes [10],

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The distance

5

over which the Ginzburg-Landau order parameter Y varies can be written as AlKso that

K = A/(

.

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The value of K for a pure type I1 superconductor was derived by Gor'kov [l11 as

KO &/to

.

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For impure materials Goodman [l21 showed that one may write to a good approximation

K = K , -t- '7.5 X 103 y + p ₍₁₀₎ where y is the electronic specific heat coefficient in erg cm-3 OK-' and p is the residual resistivity in C2 cm. The mean free path dependence of

5

and A as calculated for 1 4

5,

is given as [l 1, 131

Abrikosov [l41 has calculated with the G. L. theory the magnetization curve for type I1 superconductors which is shown in figure 2. Above a field H,, < H,

-4 7ZM

FIG. 2. -Magnetization curve of a type I1 superconductor.

a regular lattice of quantized flux units is formed in the bulk of the superconductor. This lattice is triangu- lar [15, 161 and is shown in figure 3. The density of these flux lines or vortices increases with the external field. This state, which is called the mixed state, extends up to H,, where a second order phase transition to the normal state occurs. The thermodynamical critical field H, can be found from

This model of a flux line lattice has been confirmed by neutron diffraction by Cribier et a1 [17]. The flux quantum qo that is enclosed in a vortex is equal to

FIG. 3. - Cross-section through vortex lattice in the mixed state of a type I1 superconductor. Contour diagram of

I

iy 12

close to H,, with the applied field perpendicular to the drawing.

After Kleiner et a1 [l S].

The lower critical field H,, is given as the field at which the energy per unit length of a flux line E, is equal to the decrease of magnetic energy due to the penetration of a single flux line

For large K a flux line may be considered as a core of normal material with radius

5

and a superconducting ring in which supercurrents flow over a distance A from the core.

The flux line energy which is composed of the kinetic energy of the electrons and the magnetic field energy outside the core can be written for

l

<

A as

The magnetic field energy and the superconducting condensation energy in the core have been neglected in this derivation, which is justified for K 9 1.

The value of H,, near T, as a function of K was calculated by Abrikosov for K 9 1 and by Harden and Arp [l81 for lower values of K. Figure 4 gives H,,/H, as a function of K for type I1 superconductors.

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C 3 - 5 4 G. J. VAN GURP AND D. J. VAN OOIJEN

FIG. 4. - K-dependence of Hc,/Hc. After Harden and Arp 1181.

tronic energy spectrum : the breaking up of an elec- tron pair costs a minimum amount of energy A . The value of this energy gap at T = 0 is proportional to T,. It decreases with temperature and vanishes at Tc. The gap has been shown by Gor'kov to be proportional to

/

Y

12,

where Y is the Ginzburg-Landau order parameter. The critical temperature T, is in the BCS theory given as

Tc = 0.85 OD exp

-

---

[ N i l

v1

*

In this expression 8, = Debye temperature

N(0) = density of states at the Fermi surface

V = Parameter for attractive electron-electron interaction via phonons, corrected for Cou- lomb repulsion.

In the following discussion of the influence of defects we will distinguish between two cases :

1. The influence of the defects is averaged and is manifested in mean free path effects.

2. The influence of the defects is to give rise to non-equilibrium properties, due to local interaction with flux lines (pinning effects).

3. Mean free path effects.

-

Mean free path effects arise when solute atoms, interstitials, vacancies or small clusters of these with random distribution are introduced into the lattice. It will apply to dislocations too for a material with low K, i. e. a type I superconductor. The effect of these defects is a shortening of the mean free path of the normal electrons. It turns out experimentally that the influence on the superconducting properties can often be described in terms of the mean free path irrespective of the type of defect.

3.1 CRITICAL TEMPERATURE. - One of the main effects of a shorter mean free path l is the change of critical temperature. It has been found in many superconductors that introduction of small concentrations of impurities decreases T,, often linearly with 111. This decrease of T, by alloying has been explained by Anderson [19]. In a pure superconductor the value of the energy gap is anisotropic with respect to different crystallographic directions. This has been confirmed on a number of superconductors by different experimental methods [20]. If one introduces scatterers, states of different K values are mixed and averaged over the Fermi surface so that the electrons which form a pair now have an energy gap that is averaged. The anisotropy of the gap is smoothed out and disap- pears when l becomes smaller than

to.

Since superconductivity is measured as soon as a gap appears one always measures a transition temperature which depends on the largest gap value. Scattering will cause averaging out the gap anisotropy so that Tc decreases. From this consideration one would expect that T, becomes independent of l for l

<

to.

Experi- mentally it is found however that Tc either starts to increase or goes down further with decreasing I, though with a smaller slope. This may be understood from the BCS relation (15) for T,. The Debye, tempe- rature 8, and the density of states N(0) are changed by alloying in such a way that Tc increases or decreases depending on whether the alloying elements are electronegative or electropositive with respect to the matrix.

This is reflected in the relation [21]

ATc

- = a

+

b l n p P

with P = R 4 , 2 I R z 7 3

-

R4,z

where a includes effects due to alloying other than scattering and is different for electropositive and electronegative alloying elements and b is determined by the anisotropy of the pure material.

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FIG. 5. - Shift in transition temperature due to cold work as a function of resistivity.

a) AI and AI-Zn b) AI-Ge and AI-Si. After Joiner [22].

behaviour of point defects and dislocations. No decrease but a very small increase of T, (0.002 OK) was found [24] in pure Ta after room temperature deformation by which the resistance ratio decreased from 104 to 200, whereas interstitial impurities cause a large decrease in T,. Torsion at liquid helium temperature of lead (resistance ratio changed from 1 200 to 45) was found by Liiders [25] to increase T, by only 0.004 OK. Annealing at about 200 OK caused T, to

recover. It was also shown that an elastic stress of the order of 300 kg/cm2 gave the same AT, as after torsion. The variation of T, with elastic stress was in agreement with known variation of T, with hydro- static pressure. Values of aT,/dP which are of the order 10-5 "K/atm have been tabulated by Olsen and Rohrer [26] for a number of elements. It is caused by pressure dependence of the Debye temperature and the N(0) V product.

The absence of a decrease of T, after cold work in

Pb and Ta may be attributed to the fact that the anisotropy of the energy gap there is probably small [27, 281.

Large T, increases were found after rolling at 4.2 "K of foils and evaporated films by Von Minni- gerode [29]. An increase of up to 10

%

was found for T1, Sn, Nb, In, A1 and a large increase (80

%)

for Ga, but not for Pb or Ta and a decrease was reported for Hg. For some materials AT, was shown to anneal out below room temperature. This author also found that for films evaporated on to liquid helium cooled substrates,

T,

increased by even greater amounts. If these highly disordered films were cold-worked, in some metals Tc was lowered to the same value as was found after cold work of an annealed film. A decrease of the resistance after cold work showed that some lattice defects were mobile and disappeared due to the deformation.

The explanation of the effects should possibly be sought partly in the presence of elastic stresses as was illustrated on Pb. Stress by differential contraction of evaporated Sn or In films on glass substrates have been shown to lead to increased T, [30, 311.

Effects of cold work on T, have also been studied on In and Hg [32] where, as in T1 and Al, an initial decrease of T, was found.

Increases of T, after cold work have also been found in Re [33, 341 (1 "K) where annealing gives the ori- ginal value back. Neutron irradiation at 78 "K L351 also gives an increase in T, of Re. On annealing

T,

is found to go through a maximum before it goes to zero, and may be caused by vacancy migration and subsequent building up of a dislocation network. A similar behaviour was found for cold-worked Pb [25] where it was explained in terms of recrystallization. These radiation effects are of the order 0.01 to 0.03 "K. By a-irradiation [36] AT, of

-

0.2 "K was found on Sn, which annealed out at room temperature.

In the interpretation of T, measurements one should be aware of the inhomogeneous state of cold-worked materials, dislocation concentration and internal stresses usually not being homogeneous, so that different parts of the sample may have different T,.

One usually defines T, as the temperature where the resistance is restored to half its normal value without taking into account the width of the transition, which may be indicative of the state of the material.

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C3-56 G. J. V A N GURP A N D D. J. VAN OOIJEN

electron density of states, superconducting interaction parameter or Debye temperature.

3.2 UPPER CRITICAL FIELDS.

-

A second effect of a decreased mean free path is an increase in the upper critical fields. It was shown that the upper critical fields are given as H,, = K <2 H, and H,, = 1 . 7 H,,. Using Goodman's expression (10) one can calculate H,, and H,, as a function of resistivity. This relation has been confirmed experimentally for many materials. It has been found that type I superconductors (K < 1/42) may become type I1 superconductors (K

>

l/&) through an increase of p. Bonnin et a1 [37] found this changeover to occur at the right value of p after cold working at 78 OKand after subsequent annealing

of AlMg alloys.

The effect of cold work on the H,, of type I1 superconductors is difficult to establish as the determination of H,, on cold-worked materials is often not unambiguous. Cold work usually introduces a statistical distribution of values of H,, so that a magnetization curve exhibits a tail at higher fields and approaches zero magnetization asymptotically so that measurement of H,, is difficult. The same applies to the value of H,, as found from critical current measurements. In homogeneous samples the critical current (defined as the current that causes a small voltage to appear across the specimen) falls off rapi- dly at H,,. In cold-worked materials however this fall off is often much more gradual.

Type I superconductors with K < 0 . 4 have a surface critical field H,, = 1.7 K

&

H, that is smaller than H,. Deformation may increase H,, to values higher than H,. By electrical measurements one then finds critical fields that are related to the surface only. Probably because of inhomogeneous dislocation distribution critical fields are found to be higher than one would expect using expression (10). As zero resistance is measured as long as a continuous superconducting path can be found, one tends to measure the higher transition fields. Critical field changes after cold work or neutron irradiation of Pb (K

x

0.4) have been reported [38, 39, 251 which annealed out below room temperature.

4. Pinning effects.

-

Apart from an averaged effect of cold work, deformation also gives rise to interaction of the Abrikosov vortex lattice with inhomogeneities introduced by the cold work. Before this was recognized the properties of cold-worked superconductors were generally explained in terms of superconducting filaments, due to dislocations, as will be reviewed in the following section.

4.1 HISTORICAL INTRODUCTION.

-

The considera- ble attention that has been devoted to inhomogeneous superconductors is due to their ability to carry high currents in high magnetic fields, without measurable dissipation. These superconductors can be applied to the production of high magnetic fields in solenoids. Most of the recent literature on superconductors is concerned with the role of inhomogeneities in these so called cc hard )) superconductors. High current

densities of the order of 105 A/cm2 can be carried by some superconductors in fields up to 100 kOe which is very much higher than the thermodynamic critical field H, 1401. The effect which dislocations have has long been explained in terms of superconducting threads or filaments in a normal matrix. This was first pointed out by Mendelssohn 1411 who suggested that superconducting alloys have a sponge- like structure with meshes having higher critical field than the matrix. The mechanism by which critical fields can be increased is making the superconducting dimensions smaller than the penetration depth [42]. This was demonstrated by pressing mercury into porous Vycorglass [43]. It was therefore assumed that high field superconductors contained thin filaments. As cold working was found to increase the current carrying capacity, it was suggested 1441 that dislocations would be the superconducting filaments. This was explained [45,46] by the free energy difference between the stress field of a dislocation in the superconducting and in the normal state. This meant that only a small fraction of the volume was superconducting in high fields.

However, measurements of the electronic specific heat at high fields showed that the greater part of the volume still contained superconducting electrons [47]. Explanation in terms of the filament model required extremely high dislocation densities.

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now go into detail as to the role played by dislocations, we want to treat more generally what inhomogeneities may do when introduced into superconductors and more specifically type I1 materials. The influence on type I superconductors is much less pronounced as has for example been shown on the amount of flux trapped in the material after turning off a high magnetic field. This trapped flux is much less if the superconductor is type I than if it is type I1 [37, 501 probably because the structure of the intermediate state in type I superconductors is on a much coarser scale than that of the type 11-mixed state. In a constant external field the vortex lattice does not give a net current flow in the specimen. If one imposes a transport current density J on the superconductor the Abrikosov lattice is not stable and a Lorentz force

per unit length of vortex line, where

ii

= unit vector in field direction, causes the lattice to cross the specimen in the direction of this force : the time independent solution of the Abrikosov model gives way for one which is time dependent. The description in terms of a Lorentz force perpendicular to the current has been found to be correct'to a first approximation. This is indicated by very small Hall angles (lop' to 10-3 rad) in some type I1 superconductors [51, 521.

It has been shown experimentally and theoretically [53, 541 that in the mixed state no appreciable lossless current can be carried. The motion of vortices gives rise to dissipation, so that a voltage is measured across the specimen. If one uses the appearance of a finite voltage (of the order of 10-' V) across the superconductor as the criterion for a critical current I,, this means that above H,,, I, = 0.

By these arguments one may conclude that it is only possible to pass a lossless current through a type I1 superconductor in the mixed state if one fixes the Abrikosov vortex pattern by pinning the vortices in some way so that they cannot move. If there are places in the superconductor with free energy minima

out of equilibrium with the others. Therefore whole regions will move together. At higher current densities a viscous flux flow takes place [56]. The friction due to this motion is primarily caused by interaction with the normal electrons [57]. In the case of viscous flow the flux also moves in bundles. In cold rolled vanadium foils, the size of the bundles as determined by noise measurements [58] was found to decrease with increasing J. An explanation of this, which is somewhat analogous to the motion of dislocations, is that a pile-up of flux lines can overcome a barrier more easily than an isolated line. When J is increased, the Lorentz force increases too, so that now smaller bundles may jump over. .

I t may be clear that one can increase the critical current above H,, by introducing pinning centres. This may be brought about by locally lowering the line energy of the flux lines. In the case of a superconductor with high value of K, where the line energy E, is concentrated outside the core, eq. (14) shows that E, may be changed by modifying K = A/c. I n the case of a low K superconductor where the energy inside the core cannot be neglected, E, may be changed too by modifying the condensation energy per unit length of the core 7c12 ~ : / 8 E .

Pinning manifests itself not only by enhancing the critical current density but it also found to have a marked influence on the magnetization curve. The driving force can also be described in term3 of a flux-line gradient. The critical condition for the begin- ning of fluxmotion can be written [l] :

In this expression aB/dx is the derivative of the inter- nal field B(x) and F, is the pinning force on a flux line per unit length exerted by inhomogeneities. Accor- ding to Friedel et al. [S91 ,U has the value dB/dH, i. e. the derivative of the ideal B(H) curve.

The critical flux gradient can be related to the critical transport current density J, by the relation

for a vortex then vortices will be pinned by such dB 4 n

inhomogeneities and only start to move when the - = - J , . _ax _c (18) driving force is larger than the pinning force. At low

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C 3 - 5 8 G. J. VAN GURP AND D. J. VAN OOIJEN are shown in figure 6. The magnetization

-

4 nM

in increasing fields is higher than the reversible value :

flux penetration is delayed and obstructed by pinning defects. The curve is also hysteretic because not all the flux is able to leave the specimen due to the pinning. Correspondingly, the critical current is high. Annea- ling removes the defects so that the magnetization curve is now nearly reversible and Jc is very small.

So far we have not treated the specific nature of the pinning centres but it seems that a great variety of defects have analogous macroscopic effects on critical current and magnetization. Pinning has been found to have increased after cold work, introduction of oxygen, precipitation of a second phase, segregation, grain boundaries, neutron or particle irradiation, surface treatments, etc. In general it is found [67,80] that the presence of a statistical distribution of point defects has no influence on the pinning. This suggests that the size of (or the distance between) the defects with respect to the coherence distance is important. This was found [63] for precipitates where the pinning increa- sed with decreasing particle size, down to 0 . 2 p. It is sometimes difficult to find out what the effect of dislocations is, as in many cases other pinning centres

are active at the same time.

We will confine ourselves mainly to the effect of dislocations and describe the mechanisms that cause

the pinning.

4.3' PINNING BY DISLOCATIONS. - The earlier theoretical work on dislocations in superconductors was done when thin superconducting filaments were held responsible for the high field - high current properties of superconductors. Calculations were made on the interaction of a superconducting filament with a screw dislocation. This interaction is caused by the fact that the elastic constants are modified by the superconducting to normal transition. These modifi- cations can be expressed in terms of change of thermodynamic critical field with stress o.

From eq (3) for the Gibb's free energy and using

where E is the strain and V is the volume, one can

derive [64] for the change of strain with stress oi

for a type I superconductor at the critical field H,

.E? and E; denote the strain in the normal state and in

the superconducting state in the absence of a magnetic

field respectively. In this expression the change of strain with magnetic field H

<

H, has been neglected. For type I1 superconductors where the magnetization in the mixed state may also be a function of H this approximation is probably not justified and A E ~ is there an upper limit. Writing the elastic compliance coefficients

a E i S . . = -

5J

adaj

one can calculate the change in Sij at the transition. To second order one finds

The change in S , , and S,, is negative. The change in S,, has been found to be negative for most materials.

The change in elastic constants was measured for Nb, V, Pb and Pb alloys [65] and is of the order 10 to 100 ppm. In an isotropic material S44 = 1/G where G is the shear modulus.

Fleischer [45] considered the decrease in elastic energy due to the decrease in shear modulus G when the material around a screw dislocation goes superconduc- ting. This has also been worked out by Webb [46] who gave a general expression for the change in free energy per unit volume A(g,

-

g,) of a stress field as a result of the change of strain and the change of elastic constants at the transition. For a screw dislocation, where only shear stresses appear and no volume change, the stress field in the isotropic case is

where b = Burgers vector and r = distance from the dislocation core. The interaction energy between a screw dislocation and a volume element dv is U = -

5

A(g,, - g,) dv =

(10)

spacing where the integration is carried out over the normal volume. In this treatment a flux line is considered as a normal cylinder in a superconducting matrix. From the interaction energy a force can be estimated which is attractive for a superconducting filament and repulsive for a normal filament. By this reasoning a flux line is thus repelled by one dislocation, but is pinned if it is in between two dislocations. The pinning force exerted by a lattice of screw dislocations with spacing d on a perpendicular flux line of unit length is found as

For niobium with dislocation density of 10'' cm-2 so that d = 10-5 cm this results in a force of the order 10-3 dynelcm. The pinning force is a maximum when the dislocation spacing equals the size of the vortex core.

The interaction between a screw dislocation and a coaxial superconducting filament gives a force of the same order of magnitude. In the case of a flux line parallel to the dislocation the repulsive force will be smaller as the distance to the dislocation core is larger than for a superconducting filament.

The case of an edge dislocation has not been treated by Webb, but will give roughly the same result, with the difference, that the interaction force is different on both sides of the slip plane, due to lattice expansion on one side and contraction on the other side.

Similar calculations have been made for the stress field due to composition fluctuation in alloys by Toth and Pratt [66] who arrive at values for a pinning force comparable to Webb's or somewhat larger. It has been pointed out 1671 that one should not consider the interaction of a dislocation with one flux line, but with a somewhat rigid lattice of flux lines, thus taking into account their mutual interaction. Preliminary calculations by Labusch [68] show that the Webb force is decreased by an order of magnitude, if one takes this into account.

When at enhanced magnetic fields the fiux line separation becomes less than 2

A,

the vortices begin to overlap and the field in between them is no longer zero. As the elastic constants change continuously with field in the mixed state [65] the value of AS,, to be used in the expression for the pinning force may have to be reduced accordingly [68].

In the foregoing treatment the difference in stress energy between the superconducting and the normal state is worked out for the normal vortex core which contains only a fraction of the flux line energy, as

was mentioned before. This fraction is higher, the lower the value of K. It is likely that this treatment, which ignores the effect of the stressfield on the flux line energy outside the core will give an order of magnitude for the pinning force only for low values of K.

An alternative approach to the problem of pinning by dislocations is to consider the pinning due to a decrease of the fluxline energy outside the core. This can be done by local variation of K by varying the electron mean free path. This was proposed by Nar- likar and Dew-Hughes [69, 701. They suggest that dislocation tangles are more effective in flux trapping than a homogeneous dislocation distribution. The electron mean free path l may be reduced in regions of high dislocation density and cause the line energy of a flux line to go down, so that the flux line is attracted to the dislocated region. This can be seen as fol- lows. By substituting the mean free path dependence of

t

and 1 (equ. 11 and 12) into the expression (14) for the line energy E, one finds

Constant E,

--

l In ---

l (26)

so that to first approximation E , is proportional to I. Many experiments on pinning have been done on pure Nb. It is however difficult to work out analy- tically a pinning force by this mechanism in Nb, as here

5

z

A

and l >

5 ,

so that the approximations for eqs. (11) (12) and (14) are not justified.

In order to get a numerical estimate of pinning forces by variation of K we consider the pinning by a surface or a grain boundary. It has been found that these form effective pinning barriers [71, 721. One can estimate the decrease of the mean free path near such a discontinuity and do an order of magnitude calculation of the pinning force on a flux line parallel to such a boundary.

Consider a layer of thickness d e 1 at the surface or at a grain boundary. The effective electronic mean free path in such a surface layer is reduced by scattering with one wall and may be written approximately as 1731

where 1, is the mean free path in the bulk.

One can derive [74] a value for K from this effective mean free path, by using equ. (12) and find the value of H,,. Since the flux line energy E, is related to H,, through equ. (13) it can be calculated.

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C 3 - 6 0 G. J. VAN GURP AND D. J. VAN OOIJEN

of the flux line the pinning force F, can be calculated. Experiments by Heaton and Rose-Innes [62] on This is approximated by determining E, a t two NbTa showed that drawn wire had higher Jc and more slightly different values of d as magnetic hysteresis than annealed wires, as was AE shown in figure 6. It is not likely that the effect is

F

= - L

. , . (28)

A d

Nb -45 Ta

We did this for N b with 1, = 5 000

A

(resistance Cold

-

worked ratio of 40), taking d w ,l= 500

A

and found

F,, w 0.05 dynelcm. In purer Nb this force is roughly

WO

the same, and in less pure Nb (resistance ratio 10)

it is about 10 times smaller. In order to obtain a ₁ ₂ ₃ ₄ ₅ similar force by the Webb mechanism one needs an -200 -H(koe)

extremely high dislocation density.

In the case of external surfaces there is an extra pinning due to the attraction of a fluxline by its image [75]. This force has been shown to have an effect in magnetization measurements if the surface is perfect [76]. If no special care is taken to make a smooth surface, the effect is probably small.

The foregoing calculation supports the idea that pinning by dislocation tangles, boundaries, etc. or in general large scale inhomogeneities gives rise to more effective pinning than a homogeneous defect distribution.

4 . 4 EXPERIMENTAL RESULTS. - In general it is found that cold working a material increases the critical current density J, and causes hysteresis to appear in the magnetization curve. This is usually ascribed to the introduction of dislocations. Most of the materials that have been studied are transition metals or transition metal alloys, which often contain some amount of the elements C, N and 0. These elements may be present in various ways : interstitially, in precipitates, segregated along dislocation lines and have also an influence on critical current and magnetization. Their influence varies with the arrangement in which they are present and depends on the mecha- nical and thermal history of the specimen so that they may interfere with the effects of dislocations. This makes an interpretation of experimental results often difficult.

Increases of Jc have been found after cold drawing of Nb and Re single crystals [33, 34, 771. The increase of critical current, (which was then considered as increase of critical field) disappeared after annealing. In rhenium anisotropy of the critical current was found, Jc being highest when H was parallel to the slip plane. The deformation was supposed to result in an anisotropic dislocation distribution. Anisotropies have also been reported for Nb [77] but as the experiments were not done on one and the same specimen, the conclusions may be somewhat doubtful.

FIG. 6. - Critical current density and magnetization of Nb-45 Ta before and after annealing. After Heaton and Rose- Innes [62].

caused by dislocations only, as the wires probably contained gaseous impurities. It has been shown [78, 801 that not only deformation but also introduction of oxygen or nitrogen causes the critical current to go up.

Evidence for flux pinning and trapping was also found in the magnetization curves of cold-worked Re and Ru [33].

Neutron irradiation [35] gives rise to trapped flux in Re which increases at low annealing temperatures, but disappears after annealing a t about 8000C. This rise was attributed to the formation of dislocations by vacancy diffusion. Influence of neutron irradiation [81, 821 increased Jc in Nb,Sn, possibly by disordering. In the case of cold-worked Nb and NbZr no irradiation effect was found probably because the materials already contained many defects. A large amount of experimental evidence for increased critical current and magnetic hysteresis after cold work of alloys can be found in the literature. We will not discuss this in detail but merely give a list of some relevant experiments in table I. In the case of alloys, changes in superconducting parameters after cold work are often caused not only by the dislocations, but also by precipitation e. g. along the dislocation lines.

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Ref. - 37 62 78 83 84 85 86 87,88 89 90 System

-

A1

-

Mg Nb

-

Ta Nb

-

0, Nb - Nb

-

Zr MO

-

Re various Ti - M O Nb

-

Zr Nb

-

Zr Pb alloys Property measured

-

Jc Jc Jc, el. micL Jc

M

By equating the Friedel expression (17) to the pinning force per unit length of flux line and using an approximated H(B) curve to yield dH/dB he got aB/ax as a function of the pinning force. Equating the calculated remanent magnetization due to trapped flux

J

B(x) dx to the experimental values gave an estimate of F,. Figure 8 gives this remanence as a

are produced of this material.

A

high Jc can be obtai- ned by annealing cold-worked NbZr in a temperature range (600-8000C) where precipitation of P-Zr occurs and the dislocation network is rearranged [88, 891. Annealing out the dislocations causes Jc to go down. Apparently both dislocations and precipitation are needed for strong pinning. Figure 7 shows J, as a function of annealing temperature.

FIG. 8. - Trapped flux in a twisted Nb single crystal as a function of angle of torsion. After Nembach [67].

function of the angle of torsion. The experimental values of the pinning force per dislocation do not differ by a great amount for various dislocation densities but are about 20 times less than the Webb value. This is not surprising as this theoretical value is too large as was discussed in section 4.3.

FIG. 7. - Critical current density at 20 kOe versus annealing temperature for Nb-25 Zr wire. After Chandrasekhar et a1 [91].

We now want to discuss the pinning by various dislocation structures.

Homogeneous distribution.

-

The previously described model by Webb which considered pinning of a flux line by a homogeneous distribution of perpendicular screw dislocations has been tested experimentally by Nembach [67]. He measured the remanent magnetization due to trapped flux as a function of dislocation density on pure Nb [l101 single crystals after torsion about their axes.

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C 3 - 6 2 G. J. VAN GURP AND D. J. VAN OOJJEN

Nb Crystal bla

9

T=k2 O K V = ~ O - ~ V O I ~

a

5

1

t

,04

FIG. 9.

-

Influence of plastic deformation E by bending, subsequent annealing at 900° and charging with oxygen on crjtical current density of an outgassed Nb single crystal. After Tedmon et a1 [SO].

intermediate annealing. Introduction of oxygen at 900 O C in strained material was found to increase Jc

even more. It is not impossible that the high J, in substructured Nb is partly caused by oxygen precipitation along the substructure. The anisotropy of Jc with respect to the field orientation is for substruc- tured Nb the same as for not strained 0-containing Nb, as was also shown by these authors.

The results of Van Ooijen and Van der Goot [92] may be explained in a similar way. The critical current of cold-worked Nb wires containing oxygen was found to have increased after annealing at 1 000 D C .

Internal friction measurements and electron microscopy [93] show that the annealing results in a decrease in dislocation density and precipitation along dislocations.

Polygonization effects were also found on NbTa by Narlikar [94] who measured the magnetization after cold rolling and after subsequent anneal at

1 000 O C . The pinning was very much stronger in the

latter case where the dislocation distribution was much less homogeneous as was shown by electron microscopy. This effect is shown in figure 10.

FIG. 10.

-

Magnetization curve of cold-worked and polygonized NbTa. After Narlikar 1941.

In heavily cold-worked materials, the initially homogeneous dislocation network in general grows with large deformations into a three dimensional cell structure, the cell walls having high dislocation density, surrounding low dislocation density areas [95] (Fig. 11). Narlikar and Dew-Hughes [69] measured the remanent magnetization due to trapped flux in cold-rolled Nb foils as a function of dislocation

FIG. l l .

-

Dislocation cell structure in cold-rolled Nb.

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remanent magnetization has therefore been attributed to a size effect [67] (*).

After formation of the cell structure a further increase of flux trapping was found after repeated bending of the foil which produced elongated dislocation walls.

These authors conclude from these and related experiments [71] that dislocation tangles rather than individual dislocations are necessary for flux trapping. Grain boundaries. - It is difficult to see how the cell structure can explain the often found anisotropy in critical current density when the magnetic field is rotated from a direction parallel to the rolling plane to a direction perpendicular to it [86, 87, 961. The

_-

-

three dimensional cell structure ii either isotropic (see figure 11) or elongated in a direction at 450 to the rolling direction [88, 961. The critical current with a transverse field parallel to the rolling plane is hardly dependent on the angle between the field and the rolling direction. The critical current varies strongly however with the angle between the field and the rolling plane, as can be seen in figure 12. This aniso-

-

d (degrees)

FIG. 12. - Critical current density at 5 kOe for Nb-30 Zr foil with square cross section versus the angle 6 between a transverse field and the rolling plane. Specimens were cut at various angles cc to the rolling direction R. D. After Walker and Fraser 1871

tropy can be satisfactorily explained by the Jiber structure present in these cold-rolled materials as shown in figure 13. This is a two dimensional structure on a larger scale than the dislocation cell structure, the fiber thickness being of the order 5 p as compared to about 0.5 y for the dislocation cell size. The fibers are somewhat elongated in the rolling direction. The critical current is always very much higher when

(*) The kink in the remanence versus deformation curve is still present after correction for the thickness dependence. [log].

FIG. 13. - Fiber structure in cold-rolled Nb foir.

the flux lines are parallel to the fiber boundaries than when they are perpendicular to them. The anisotropy disappears after recrystallization. It is likely that this anisotropy is caused by the fact that the pinning force on a flux line is a maximum when it is parallel to a fiber boundary because then the line is pinned over its entire length. It was shown in section 4.3 that the line energy is a minimum close to a parallel surface. This is further supported by experiments on electrolytic Nb foil by one of the authors 1721. This material which contains only few dislocations is electrodeposited at about 800 O C and contains grain boundaries perpendicular to the plane of the foil, which is shown in figure 14.

FIG. 14. -Cross section of electrolytic Nb foil.

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0 30 60 90 120 l50 180

----t d (degrees)

FIG. 15. -Critical current density of electrolytic Nb at 4 kOe versus the angle 6 between a transverse field and the plane of the foil.

angle of rotation 6 of the field with respect to the plane of the foil. The maxima at 6 = 0 and 180° are caused by the fact that now the flux lines are parallel to the external surface, so that then surface pinning is a maximum. Grain growth by annealing or altering the direction of the grain boundaries by rolling removes the maximum at 6 = 900.

Surface qfects. - That the surface is also effective in pinning was shown [71] in experiments on bars with triangular cross section. When the external transverse field was parallel to a surface, the critical current was a maximum. Recent experiments [97] show that if the surface of a cylindrical rod is roughe- ned on two opposite sides the critical current is a maximum when the field is parallel to those areas. Surface irregularities due to the rolling process have been found [98] to give rise to a preferred motion of fluxlines in the rolling direction.

A peculiar effect which is often found after cold working type I1 superconductors is the so called peak qfect : a peak in the field dependence of the critical current near H,,. Related to it is a minimum in the resistance transition with magnetic field. Both effects are shown in figure 16. It has been found in Nb [78, 80, 991 and many of its alloys [85, 861, V [72], PbTl [71], PbIn after deformation at 40K [loo], in NbC/NbN whiskers [l011 and also after introduction of oxygen in Nb [78]. Generally annealing at a temperature where dislocation movement takes place, destroys the effect.

The effect has been explained in various ways as enhanced pinning by mechanisms such as : increased rigidity of the flux line pattern [55], fitting of flux

line spacing to dislocation spacing [86], a non uniform K distribution 1701. Swartz and Hart [71] have shown that suppression of surface superconductivity by

cbd

-

mlled vdudium 3J,g

l T-l

5 Jc ( ~ / c m 3 2 ! lo3 2 102

FIG. 16a. - Critical current density and b. resistance tran- sitions of a colled-rolled V foil in a field perpendicular to the plane of the foil.

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coating the superconductor with a normal metal also depresses the peak effect.

We found the effect to occur after minor deformation of ultrapure Nb (resistance ratio 4 loo), which changed the resistivity by only 10-9 Q cm. This may also explain the occurence of the effect in annealed PbTl [71]. To get the effect in impure Nb or Nb alloys one needs heavier deformation. This is consistent with the idea that a given defect produces more pinning in a low K material than in a high K material, as in the former case AK/K and consequently AE,/E, is greater.

We believe that the resistance minimum may be caused by an inhomogeneous current distribution in the sample. If the surface is not coated with a normal metal, the order parameter

II/

is higher at the surface than in the bulk [l021 not only at fields above H,, but also slightly below H,,. The current density may therefore be higher near the surface, so that the total current density, which is the sum of the transport current and internal currents has a minimum in the bulk. At the place of this minimum the driving force J q O / c on the vortices may then be too small to unpin the flux lines, so that the flux motion is obstructed. This effect takes place when H approaches H,, and causes the voltage which is due to flux motion to decrease and the critical current to increase.

Annealing removes the pinning centres so that the effect is then not observed.

4.5 SURFACE SUPERCONDUCTIVITY, FILAMENTS.

-

So far we have discussed the superconducting properties at fields below H,,. The effect of cold work above H,,, where surface superconductivity exists, is not very well understood.

Tedmon et a1 [80] showed that deformation, but especially polygonization and introduction of oxygen increased J, above H,,. As there is no flux line lattice these effects cannot be explained in terms of pinning. Calculated values [103, 1041 for the surface layer critical current are one or two orders of magnitude higher than experimentally found values.

One also finds after cold work that superconductivity may persist up to very high fields. Values of 3.7 times H,, as compared to H,, = 1.7 H,, were reported [78] for Nb. Experiments on PbIn [l001 gave values of 2.2 H,,.

As was remarked before, precise determination of H,, and H,, is not always possible in cold-worked materials because of the inhomogeneous nature of the material. Small, continuous regions may have high values of K so that with very small measuring currents superconductivity may be detected up t o

large fields. It has been pointed out [70] that the fila- mentary model which was rejected in favour of the flux pinning model is making a re-entry in this way. It seems however that it is needed here only to expIain the last traces of superconductivity. A different case is that where one has a continuous network of a second phase as was found [l051 in impure cold- worked Sn, where segregation of impurities along dislocation lines gives rise to a network with higher T, than the matrix.

5. Conclusions.

-

The influence of dislocations on critical temperature and critical fields can be explained in terms of mean free path effects and internal stress in the material.

Dislocations are also found to provide effective pinning barriers to flux motion. Pinning by dislocations is possible by either one of the following mechanisms : a homogeneous distribution, a polygonized configuration, a cell structure, fiber or grain boundaries. Which is the most important depends on the particular type of cold work. Experiments on simple defined dislocation structures as was done on twisted Nb crystals should be extended to other homogeneous dislocation structures. The interaction between flux lines and parallel dislocation lines is also of interest.

Pinning by grain boundaries has not been treated theoretically and experiments on well defined grain boundary patterns have hardly been carried out. Surfaces have attracted much more attention and have been shown to be very effective pinners. Also here theoretical consideration of the interaction force with vortices, apart from the image force, is needed. Electron microscopical investigation of flux pinning has also not yet been carried out, although attempts have been made [l061 and optical microscope studies [107, 1081 have yielded interesting results.

Acknowledgements.

-

The authors wish to thank Prof. Dr P. Haasen, Dr C . W. Berghout and Dr A. G. van Vijfeijken for helpful discussions. They are much indebted to Dr E. Nembach and Dr R. Labusch and to Dr D. Dew-Hughes and Dr A. V. Narlikar for providing results prior to publication.

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