THE INFLUENCE OF DISLOCATIONS ON
SUPERCONDUCTIVITY
G. Van Gurp, D. Van Ooijen
To cite this version:
G. Van Gurp, D. Van Ooijen.
THE INFLUENCE OF DISLOCATIONS ON SUPERCONDUCTIVITY. Journal de Physique Colloques, 1966, 27 (C3), pp.C3-51-C3-67.
<10.1051/jphyscol:1966307>. <jpa-00213117>
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JOURNAL DE PHYSIQUE
Colloque C 3, Supplhent au no 7-8, Tome 27, juillet-aozit 1966, page C 3-51
THE INFLUENCE OF DISLOCATIONS ON SUPERCONDUCTIVITY
by
G. J. VAN GURPand 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 supraconducteurs. 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 Shoenberg [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 characterized by two equations
E=OandB=O.
The first equation means that no electric field can
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1966307
G. J. VAN GURP AND D. 3. VAN OOIJEN
C3-52
exist in a superconductor : the resistance is unmeasurably 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=
JLzn, e2
4
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 transition 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
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 superconducting 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
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
Hc
-H
FIG. 1. - Magnetization of a type I superconductor without
and with demagnetization (dashed curve).
mc'
4 ze2 1 Yol2
where ?Pois 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.
It has been shown by Saint James and De Gennes [10],
that at the surface of the superconductor there exists
a surface upper critical field Hc3 =
which
is the field below which the surface is superconducting.
THE INFLUENCE O F DISLOCATIONS ON SUPERCONDUCTIVITY
C3-53
The distance 5 over which the Ginzburg-Landau
order parameter Y varies can be written as AlKso that
.
K = A/(
(8)
The value of K for a pure type I1 superconductor was
derived by Gor'kov [l11 as
KO
&/to
.
(9)
For impure materials Goodman [l21 showed that one
may write to a good approximation
K = K , -t- '7.5
X
103y + 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 triangular [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 [lS].
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.
The origin of superconductivity has not been treated
so far. We only want to mention here that it is
found [8] in an electron-electron interaction via virtual
phonons. The interaction by which electrons with
opposite spin and momentum over a distance of the
order are correlated gives rise to a gap in the elec-
G. J. VAN GURP AND D. J. VAN OOIJEN
C3-54
FIG. 4. - K-dependence of Hc,/Hc. After Harden and Arp 1181.
tronic energy spectrum : the breaking up of an electron 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 *
= Debye temperature
In this expression 8,
N(0) = density of states at the
Fermi surface
V
= Parameter for attractive electron-electron interaction via
phonons, corrected for Coulomb 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 disappears 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. Experimentally 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, temperature 8, and the density of states N(0) are changed
by alloying in such a way that Tcincreases 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
P
with
+ blnp
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.
The effect is shown in figure 5 for cold-worked and
annealed A1 alloys. It was shown by Joiner [22] that
quenching, cold working in various ways (twisting,
stretching, rolling), annealing of pure Al, A1 with Zn,
AI with Ge or Si all give results that for a particular
type of impurity can be represented by one curve.
Hasse and Liiders [23] also measured T, in thallium
after deformation by torsion at liquid helium temperature and found T, first to go down by 0.03 OK with
increasing and then to go up again. At different temperatures annealed material gave a slightly different
AT, versus p curve which they attributed to different
THE INFLUENCE OF DISLOCATIONS ON SUPERCONDUCTIVITY
FIG. 5. - Shift in transition temperature due to cold work as
a function of resistivity.
b) AI-Ge and AI-Si.
a) AI and AI-Zn
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 hydrostatic 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
C3-55
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 Minnigerode [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 original 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.
In summarizing this section we conclude qualitatively that introduction of dislocations gives rise
to two effects : a decrease of T, through destruction
of the energy gap anisotropy and an increase (or
decrease) of T, by the effect of internal stresses on
C3-56
G. J. VAN GURP A N D D. J. VAN OOIJEN
electron density of states, superconducting interaction parameter or Debye temperature.
3.2 UPPERCRITICAL 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 OK and 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 rapidly 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 considerable 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 spongelike 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.
A break-through was given by Goodman and
co-workers [49] who showed that high field superconductors were in fact type I1 superconductors, as described in the Ginzburg-Landau [9] and Abrikosov
theories [14, 121. In these superconductors the upper
critical field H,, can be very much higher than the
thermodynamic critical field H, so that one does
not need a filament model to explain the high critical
field, which is in fact an equilibrium property of the
entire material. The high critical currents in deformed
superconductors could be interpreted in terms of
interaction between flux lines and inhomogeneities.
The filamentary model has now generally been given
up for the explanation of high critical current densities.
4.2 PINNINGBY
INHOMOGENEITIES.
- Before we
THE INFLUENCE OF DISLOCATIONS ON SUPERCONDUCTIVITY
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
for a vortex then vortices will be pinned by such
inhomogeneities and only start to move when the
driving force is larger than the pinning force. At low
values of J the force is small and motion may be aided
by thermal activation : this is the regime of what has
been called flux creep : thermally activated hopping
of flux [55]. This takes place in the case of strong
pinning and the flux moves in bundles of flux lines
rather than as single units. One flux line, when
unpinned, cannot move independently, it would get
C3-57
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. In
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 ~ : / 8E .
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 beginning of fluxmotion can be written [l] :
In this expression aB/dx is the derivative of the internal field B(x) and F, is the pinning force on a flux line
per unit length exerted by inhomogeneities. According 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
dB 4 n
-=-J,.
ax
c
(18)
In this way one can calculate the magnetization curve
from measured J,(H). This agrees qualitatively with
experimentally found curves [60]. This model has
also been used to explain the influence of a transport
current on the magnetization curve [61]. Magnetization and corresponding critical current curves for a
cold worked and an annealed superconductor [62]
C3-58
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. Annealing 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 increased 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' PINNINGBY 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 modifications 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
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
aEi
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 superconducting. 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 =
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,
and E; denote the strain in the normal state and in
the superconducting state in the absence of a magnetic
.E?
This is integrated for two different cases : a coaxial
superconducting filament around a dislocation line
where the integration is carried out over the superconducting volume and for the case of one flux line
perpendicular to a forest of dislocations with constant
THE INFLUENCE OF DISLOCATIONS ON SUPERCONDUCTIVITY
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
C3-59
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 Narlikar 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 follows. 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 analytically a pinning force by this mechanism in Nb,
as here 5 zA 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.
By differentiating E, with respect to d at the place
G. J. VAN GURP AND D. J. VAN OOIJEN
C3-60
of the flux line the pinning force F, can be calculated.
This is approximated by determining E, a t two
slightly different values of d as
AE
F =. , .L
Ad
Experiments by Heaton and Rose-Innes [62] on
NbTa showed that drawn wire had higher Jc and more
magnetic hysteresis than annealed wires, as was
shown in figure 6. It is not likely that the effect is
(28)
We did this for Nb with 1, = 5 000 A (resistance
ratio of 40), taking d w ,
l= 500 A and found
F,, w 0.05 dynelcm. In purer Nb this force is roughly
the same, and in less pure Nb (resistance ratio 10)
it is about 10 times smaller. In order to obtain a
similar force by the Webb mechanism one needs an
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 mechanical 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.
Nb -45 Ta
Cold - worked
WO
1
-200
2
3
4
5
-H(koe)
FIG.6. - Critical current density and magnetization of
Nb-45 Ta before and after annealing. After Heaton and RoseInnes [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 at 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.
The NbZr system has been studied intensively
because very high critical currents and fields can be
achieved in NbZr alloys. Most of the present soIenoids
THE INFLUENCE OF DISLOCATIONS ON SUPERCONDUCTIVITY
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
-
C3-61
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
Jc
Jc
Jc, el. micL
Jc
M
are produced of this material. A high Jc can be obtained 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.
Inhomogeneous distributions. - It was suggested
in section 4.3 that pinning by K variations may be
stronger than by a homogeneous dislocation distribution. There is some experimental evidence that
flux pinning is increased when the dislocation network
is made inhomogeneous. Tedmon et a1 [80]measured
critical currents of an annealed Nb crystal. Bending
increased Jc7 but annealing at 900 0C increased J,
even more as is shown in figure 9. The increase in J,
on bending was attributed to the formation of a
dislocation network. The heat treatment caused
polygonization of the dislocation structure which
increased Jc again. These authors also found that
when the substructure was introduced by increasing
the deformation by small amounts and annealing
at 900 OC in between, the increase of critical current
by strain increments was always larger than without
G. J. VAN GURP AND D. J. VAN OOJJEN
C3-62
9
a
5
5
t
Nb Crystal
bla
T=k2
O K V=~O-~VOI~
,04
FIG. 10. - Magnetization curve of cold-worked
and polygonized NbTa. After Narlikar 1941.
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 substructured 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.
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.
density and found that it levels off at a dislocations
density where the cell structure has been formed.
They attributed the pinning to the interaction of the
cell walls with the flux lines through local variation
of K. At larger deformations the dislocation density
in the cell walls is increased but the cell size remains
constant so that the pinning will not increase very
much anymore. It is not clear whether all the specimens had the same thickness. The levelling off of the
THE INFLUENCE O F DISLOCATIONS ON SUPERCONDUCTIVITY
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-
C 3 - 63
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.
-
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. 14. -Cross
section of electrolytic Nb foil.
The critical current was found to be a maximum
when the magnetic field is parallel to these grain
boundaries. Figure 15 shows J, as a function of the
G. J. VAN GURP AND D. J. VAN OOIJEN
C3-64
line spacing to dislocation spacing [86], a non uniform
K distribution 1701. Swartz and Hart [71] have shown
that suppression of surface superconductivity by
5
Jc
1
(~/cm3
2!
Cold-rolled Vanadium 3,g
lo3
T=4.2%
v=lC7vdt
\
5 \
ix
2
I
\X.
102
0
30
60
90
120
----t
l50
,X.
I
.l'l
S-,'
\
180
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 roughened 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
-
cbd mlled vdudium 3J,g
l l -T
FIG. 16a. - Critical current density and b. resistance transitions of a colled-rolled V foil in a field perpendicular to the
plane of the foil.
THE INFLUENCE OF DISLOCATIONS ON SUPERCONDUCTIVITY
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
C3-65
large fields. It has been pointed out [70] that the filamentary 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 coldworked 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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