Irreversible effects and pinning

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Irreversible effects and pinning

M. Cyrot

To cite this version:

M. Cyrot. Irreversible effects and pinning. Journal de Physique, 1972, 33 (8-9), pp.803-810.

�10.1051/jphys:01972003308-9080300�. �jpa-00207308�

1. Introduction. ^-

Probably

^{one of the most}

impor-

tant

problem

ⁱⁿ

applied superconductivity

^is transport of current without

dissipation.

^It

appeared rapidly

that a

superconductor

^cannot transport ^a

high density

of current without

loosing

^its

superconducting

pro-

perties.

Silsbee made the

hypothesis

that the critical current of a wire

just

^creates the critical

magnetic

field ^at the surface of this wire. This field is :

IRREVERSIBLE EFFECTS AND PINNING

M. CYROT (*)

Institut

Laue-Langevin,

^Cédex

156,

38-Grenoble-Gare

(Reçu

le 18 novembre

1971,

^révisé le 24 avril

1972)

Résumé. ²⁰¹⁴ Dans cet article nous détaillons les problèmes théoriques posés par l’ancrage ^des

vortex. Ceux-ci proviennent essentiellement de la difficulté de relier, ^{une force} d’ancrage ^élémen-

taire entre un défaut et un vortex, à la force mesurée expérimentalement. ^{Nous faisons une revue}

des travaux existants et montrons comment définir une force macroscopique d’ancrage qui ^se prête

à un calcul théorique. ^Des considérations énergétiques permettent de la calculer et nous montrons que ^nous obtenons un résultat analogue ^à celui de Webb.

Abstract. ²⁰¹⁴ In this _paper we review the theoretical problems ^{due to the} pinning of vortices. We

emphasize ^the difficulty ^of making ^a link between the elementary pinning force between an isolat- ed vortex and a pinning ^center with the measured pinning force. We show how to define experi- mentally ^the pinning force which permit ^a theoretical study ^{and we} calculate it. Far from Hc2,

we get Webb’s results. Finally, ^we briefly review calculations of the elementary pinning ^force.

Classification Physics abstructs :

17.24

if a is the radius of the wire. For type ^II

superconduc-

tors, ^{if H is} lower than

H,,,

the situation is stable.

When H is

higher

^than

H,,,

^{vortex lines}

begin

^to

appear :

they

^are bent in circles

following

^{the lines of}

force. Once created at the surface with radius _a,

they

tend to shrink to decrease their line _energy and

finally

annihilate near the axis of the

specimen.

^{This process}

dissipates

energy.

Thus,

^{we have zero} resistance

only

^if

Unhappily

^this

corresponds

^{to a low}

density

^of

current. If we want to carry ^a

higher density,

^{we have}

to prevent ^vortex motion and achieve a non

equili-

brium situation.

In this

article,

^{we first discuss the} concept ^{of vortex} motion and the

dissipation

^{which results from this}

motion. Then ^we introduce the concept ^of

pinning.

We

distinguish

between the measured

pinning

^force

density

^P and the individual

pinning

^force

fp

^which

prevents ^{one vortex from}

moving.

^{We first describe}

situations where these two

quantities

^are

simply

^relat-

ed

by

^{P =}

nf,,

^{where n is} the number of vortices per unit volume. Then we show

why

^{the two}

quantities

are not

simply

related in the

general

^{case and how}

to calculate one

knowing

the other.

Finally

^{we discuss}

various means to

pin

vortices and make _very elemen- tary calculations of the individual

pinning

^force.

II. Vortex motion and

dissipation.

^{- If the} concept of vortices is now well established both from a theore- tical and

experimental point

^of

^view,

the motion of vortex lines and the

dissipation

^which

results,

^{is not}

so

firmly

understood. This

point

^{stems from the fact}

that the concept of individuel vortex line is clearer when the distance between them is

large

^{i. e. at low} temperature and low field.

Unhappily,

^the

theory

^is

much easier near

H,2

when the vortices are very close

together

^{and the} concept of vortices looses its

impor-

tance. This is the reason

why

^we will review the

thêory

of vortex motion in these two limits and will try ^to

explain

^the

physical origin

^of

dissipation.

A)

^{MOTION OF AN} INDIVIDUAL VORTEX LINE H k

Hcl.

- The concept ^{of a vortex was} introduced

by

^Abri-

kosov

[1]. Figure

¹ represents ^{a vortex} line. In the

core the order parameter

drops

^{to zero. The carac-}

teristic

length

^{for the core} is the coherence

length.

Outside the core is an

electromagnetic region

^which

spread

^{over a} distance of the

penetration depth.

^As

vortices ^are

always

^{within a distance}

Â,

the

concept

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01972003308-9080300

804

FIG. 1. ^- An individual vortex line : the order parameter and the magnetic ^{field as a} function of the distance to the core.

of individual vortex line is clearer for a

large

^ratio

of

À/(

^{i. e.}

large

value of the

Landau-Ginzburg

para- meter ^x. In the extreme

high

^{x limit one can describe}

the core

by

^a

singularity

^{in the}

electromagnetic region [2].

^This

region

^{is described}

by

^{the London}

equation

^{which stems from} the second

Ginzburg-

Landau

equation

^{for current where one can}

neglect

any

change

^{of the order} parameter. ^We get

Together

with the Maxwell

equation

we get

We must take into account the

singularities

^{due to}

the core at

point

^{i and this}

gives

The _energy associated with the

electromagnetic region

is

It is _easy to show that the London

equation just

^mini-

mizes the _energy with respect ^to the distribution of current.

Now we want to look at the effect of a transport

current

JT.

^{We first remark that the structure of a}

vortex line is

fairly rigid,

^{because of the}

quantization condition,

^{the line as a whole}

reacting

^{when a vortex} system ^is

perturbed

^from

equilibrium.

The effect of

a transport ^{current can be seen as a}

perturbation

^with

respect ^{to the}

line ;

^the

highest

transport current, known in a type ^two

superconductor,

^is of the order of 105

A/cm’.

This is small

compared

^{with the} super-

fluid flow

just

outside the core of

107 A/cm2.

^{But it}

is not

impossible

^{that the}

validity

^of this condition breaks down in the near future.

The _energy of the current

Ekin

^{is now made} up of three terms, there is a cross term in _energy between the current and the vortices which is

where

hj

^{is the}

magnetic

field due ^to the transport

current. If the current is in the x

direction, hJ

^{is in}

the ^z direction and is

given by (4 03C0lc) yJT.

Thus we get ^a force between the vortex and the current in

the y

^{direction which is}

given by

The situation we achieve is a non

equilibrium

^one.

The vortices will

begin

^{to move.}

Roughly speaking,

we can say that the flux will _vary as a function of time with the motion. From the Maxwell

equation,

^an

electric field is set up in the x direction the

intensity

of which is

1

if vL is the

velocity

^{of the flux line in}

the y direction, V ç

represents ^the

possible

contribution of

charge

created

by

the motion. The existence of an electric field

parallel

^{to the current causes a}

dissipation

This means that an effective

resistivity

appears

The

only

parameter, ^{which is not}

given by

^the

theory,

is the electric field created

by

^{the movement as a func-}

tion of the transport ^current.

In

figure 2,

^we

give

^a

typical

^{curve of}

voltage

^versus

FIG. 2. ^- Typical measurement of the voltage ^{versus current}

in a type ^II superconductor. Je defines the critical current calculated in the text.

current for a type ^II

superconductor.

^At

high

^current

density,

in the flow

regime,

^the

voltage

^is

proportional

to the current and one can define an effective resis- tance. This resistance does not

depend

^{on the criti-}

cal current so it has an intrinsic character. Further- more, neutron

experiments

show the lattice of vortex line is much more

perfect

in this flow

regime

^{than in}

the static case

[3].

This tends to prove that the lattice

moves as a whole. The

inhomogeneity

^of the materiel is

unimportant

^{and the}

resistivity

^{is intrinsic.}

It is

possible

^to get ^{an estimate of the}

resistivity,

which _agrees with

experiments, by assuming

^{that the}

current

JT

^flows

through

the normal core of

radius ç

thus

giving

^a

dissipation

where n is the number of vortex lines _per CM2 n ⁼

BI 0,,.

It follows that

If we want to estimate the

velocity

^{of the}

line,

^we

can assume a

proportionality

between the

applied

force and this

velocity defining

^a frictional

drag

So we get

for 11

The theoretical

justification

^{of this}

dissipation

^was

given by

the model of Bardeen and

Stephen [4].

They

studied the movement of one flux line which

they

schematized

by

^{a normal core of}

diameter 03BE

surrounded

by superfluid

^{currents. As can be}

^shown,

the value

of 11

^is

larger

^{than can} be accounted for

by eddy

^currents

resulting

from the electric

field,

E ⁼

(1/c)

VL

/B H, generated by

^the

moving magnetic

field of the vortex line. One must assume that the cur-

rent flows

through

the normal core of the vortex.

Bardeen and

Stephen

^{showed that there is}

charge density developed

^{at the} surface of the vortex core

by

the motion. This

charge

^creates within the core a

uniform field which drives the lines of current inside the core. If the vortex were

pinned,

^the transport

current would flow around the core and there would be no

dissipation.

According

^{to London’s}

theory [5],

^the

charge

^den-

sity

^{is due to} the electric field which is needed to

change locally

^the

superfluid velocity

vs ^as the vor-

tices move. An additional electric field

beyond

^that

generated by

^the

moving magnetic

^{field is}

required

to

change

vs with time. This additional field _{is respon-} sible for the

major

part ^{of the} energy loss. The _equa- tion of motion

for Vs

^{is :}

The field may be

expressed

^{in the form}

with the electrostatic

potential ç

outside the core

given by

This

potential

^leads

to

^a

charge’distribution

^around

This

potential

^{leads to a}

charge

distribution around the core such that the field within the core is uniform.

B)

COLLECTIVE MOTION OF THE LATTICE H

HC1.

^·

- We have

discussed,

^so

far,

the motion of a free vortex line due to a transport ^{current and}

analysed

the

physical origin

^of

dissipation.

^In the flux flow

regime,

^we mentioned that the lattice of vortices moves as a whole. We will now discuss this effect near

Hc2

where ^a

theory

^is

possible

^{due to} the existence of a

time

dependent Ginzburg-Landau equation.

^{It is}

well known that a

Ginzburg-Landau equation

^is

obtained

by

minimization of a free _energy

F(4).

We will assume

that,

^if the order parameter ^{is not} the

equilibrium

^{one, then it} will relax to the

equili-

brium value at a rate

governed by

^the

slope

^{of F in}

function _space

The parameter y is to be obtained from the micro-

scopic theory.

^The gauge invariant time

dependent Ginzburg-Landau equation

^{will be}

la B

where 0 and A are scalar and vector

potentials

^res-

pectively,

^D the diffusion constant and _eo ⁼

2nDHcl/ tPo.

In the absence of an electric field i. e. no motion of vortex

lines,

^{A is}

given by

the Abrikosov solution with a

regular triangular

^lattice structure. In the _pre-

sence of a scalar notential

the solution has been

given

^near

T, by

^Schmid

[6]

who showed that

A(r, t)

^is

given by

+-

with u ⁼

E/H.

This solution satisfies

which

implies

that the order parameter ^{moves in}

the y

direction with a uniform

velocity

^{-u. The current is}

806

given by

^{the second}

Ginzburg-Landau equation :

^one

finds :

where M is the

magnetization. J2

consists of two terms.

Only

^{the term}

proportional

^{to u} contributes to the _average transport ^{current while the}

remaining

part contributes to the

magnetization

^{of the} system and represents ^the

superfluids

^currents around the vortex lines. Near

H,,2,

Maki derives the

following

result

[7] :

Comparison

^{of this}

formula,

^{valid near}

Hc2,

^with

experiments

^has

recently appeared [8].

III.

Concept

^of

pinning.

^{- In the}

preceding

part,

we have shown that no current can flow in the mixed state without movement of vortices. This movement creates

dissipation.

^The

problem

^{is to know if it is}

possible

^{to make a current} flow in the mixed state

by preventing

any vortex motion. This seems

impos-

sible in _any infinite

homogeneous

^{system but it turns}

out that it is

possible

^{for an}

inhomogeneous

system.

The

inhomogeneity

^can be created

by

the finite dimen- sion of the system ^or

by inhomogeneity

in the bulk.

We will describe in this part ^{the two} situations.

As a start, ^we will now show that a condition for

a current to flow without vortex

motion,

^{is that the}

energy of the vortex lattice will increase for _any dis-

placement.

^For

^this,

^we

investigate

the conditions under which the _array of vortex lines can be in

equi-

librium. We use a

thermodynamic

argument ^which has been

given by Josephson [9].

^{Consider a} type ^two

superconductor

in the mixed state and _suppose that the

applied

^{field is}

produced by

^{an external current}

distribution

Jext.

^The

thermodynamic

^{condition of}

equilibrium

is that under a virtual

displacement

^of

the field which ^we

specify by

^ôA

where ôF is the

change

in the free _energy of the system and ô W is the work done

by

^{the sources of the field}

We have

Inside the

superconductor

^{bA can} be related ^to the

displacement

^{of the vortex}

lines,

^while

outside,

^it

can be

arbitrary.

^Let

ç(r)

^{be a vector field}

giving

^the

virtual

displacement

^{of the vortex lines}

perpendicular

to its axes and not

leading

^to any

cutting

^{of the}

lines. We have inside the

superconductor

So we must have for an

arbitrary e(r)

Therefore no current can flow in the

superconductor.

In the

particular

^case

J 11

^B the critical current

density

is infinite but of course some form of

instability

^other

than vortex line motion determines the critical current

density

^{in this case.}

When the

superconductor

^is

inhomogeneous,

^we

must include an

inhomogeneity

^term

F,

in the free energy of the vortex lines and define a

pinning

^{force P}

given by

^the functional derivative

Thus P is related to the

change

ⁱⁿ

Fi

^{when the vor-}

tex lines are

displaced ;

^{in this case we obtain}

The

interpretation

^of

(1/c)

^{J A} ço ^as the force on a

single

^vortex

line,

^{is not}

immediately justifiable

^from

this

equation since,

^{both J and B are} average of micro-

copic quantities.

^{J related to H which is}

given by

the derivative of the free energy with respect ^{to B} is a rather obscure _average.

We will now examine two

special

^{cases. In the}

first one,

everything happens

^{in the same} way ^on each vortex line. So the concept ^{of Lorentz force} and

pinning

^{force is here} very

simple

^{and exact cal-}

culation can be done. This case

corresponds

^to

pin- ning by

surfaces. The second _case, which

corresponds

^to

point pinning

^{in the}

bulk,

^{will add some} difficulties due to statistical

problem

and will introduce some new concepts.

A)

^PURE PINNING PROCESSES. ^- As an

example

of this _case, we calculate the critical current of a

thin film whose thickness

permits

^the entry ^of

just

one line of vortices

[10] (Fig. 3).

In the absence of current, the cores are located in the middle of the film.

They

^are in fact in a

potential

well. This well is due to the

magnetic

pressure which

pushes

^{the vor-}

tices inside the film and to the attractive

image

^force

which tends to

push

them outside. The _energy of the vortices located at xo in the film is

H(xa)

is the field when there is no vortex line and is

given by

---0 ---;>8

FIG. 3. ^- A chain of vortices in the film. The external field is perpendicular ^{to the} plane ^{of the} drawing.

Hl

^{is the} field due to the

image

force. The field created

by

^{one vortex is}

So we obtain

The

potential F(x)

^is

given

ⁱⁿ

figure

^4.

In the _presence of a current,

owing

^{to the}

linearity

of the London

equation,

^we

just

^{add a term}

FiG. 4. ^- The potential ^seen by ^{a vortex line inside the film} for various values of the external field.

where

HJ

^{is the field due to the} transport ^current

This term

gives

the interaction between current and vortices. Now we have to look at the minimum of

F(x)

which can be written

The sum of the forces which act on any vortex

line is zero. So we get

P(dl2Â)

^cannot be calculated from this formula because

(d/2) - x > ç

^{in our} treatment. However

making

^use of the definition of the

superheating

^field

one has

The solution of the

equation

is illustrated in

figure

⁵

and the critical current we obtain is

given by figure

^6.

FIG. 5. ^- The function P(x).

FIG. 6. ^- The dependence of the critical current on the external

magnetic ^field.

808

We see that a

peak

effect should be observed. The

essence of this

phenomenon

^{reduces to} the fact that with

increasing magnetic ^field,

^{there is an increase}

in the

depth

^{of the}

potential

well in which the vortices

are located. One must therefore increase the current to

destroy

^the

stability

^{of the} vortex structure.

This

particularly simple

^model

permits

^{to extract}

some

simple

^{facts which can} be useful in more

compli-

cated system.

First,

^we remark that there is no

problem

in

defining

^{a Lorentz} force and a

pinning

^{force on}

a

single

^{vortex line. In the}

model,

the force which

corresponds

^to the critical current is

proportional

^to

the maximum

pinning

^{force of one vortex line. This}

will be _compare to our second

exemple.

^{We want}

also to remark that the Lorentz force is

given by

^the

current at the core but the current cannot be homo- geneous. It has _{to vary} over a distance Â. The interac- tion _energy is such that the minimum of current occurs

at the core to minimize the interaction energy. This is very

important

^{since much of the}

experimenta

work assumes an

homogeneous

^current.

To our

knowledge,

^no

experiment

^{has been done}

on this

simple

system. ^{But a} very similar situation has been studied

by

^H.

Raffy [11 ].

^{Instead of}

having

^one

line of vortices in a

well,

she achieved a two dimen- sional situation where each line of vortices is in its

own well. This has been obtained

by modulating

^the

concentration of a

binary alloy

^{with a}

period

^which

fits the distance between vortices. In this manner one

also obtained ^a situation where

everything

^{is identical}

on any line. The

principal

^{feature of the film must}

be

preserved.

^{If the}

magnetic

^field is such that the distance between lines of vortices

exactly

^{fits the}

period

of

modulation,

^the

pinning

force will be maximum and for this

field,

^we will obtain a maximum in the critical current.

Preliminary experiments

have shown

peak

^effects.

B)

COOPERATIVE EFFECTS IN PINNING PROCESSES.

- Now we want to deal with a more

complicated

system where there is no

identity

between all the vortex lines. We assume random

inhomogeneities

in the bulk of the

specimen,

^{which we call}

pinning points.

^These

objects

will be described

by

^an interaction

potential

with a vortex line. We leave to the

following

^section

examples

^of

pinning points

^and calculation of the

potential.

^The

pinning points

^{are then}

inhomogeneities

which either favor or inhibit the

pair

condensation

responsible

^for

superconductivity

^{i. e. which}

repell

or attrack the flux lines. The central

problem,

ⁱⁿ

understanding

the role of

pinning

ⁱⁿ

determining

^the

critical current, is the

analysis

^{of the means}

by

^which

the

point

interactions between flux lines and

pinning points

^{add to} determine the

pinning

^force

density

^P.

The critical current is then

given by

Let us

call fp

the interaction force between a

pinning

point

^{and an} individual vortex line. A

simple approach

to P is to assume linear

superposition

where N is the number of interaction between

pin- ning points

and vortices.

However,

^{this is} not correct for this case because it is _{easy to} show that this

simple

model of a

rigid

^{lattice of vortices} and random inho-

mogeneities,

^{lead to no}

pinning

^{at all. The reason is}

that the

pinning

^{forces are}

randomly

^{oriented and}

statistically

cancelled. One can also understand this result

by considering

the fact that the interaction energy for this model in an infinite medium _array, would be

independent

^of the relative

position

^{of the}

rigid

^lattice and the random _array. As

pinning

^effecti-

vely

occurs, ^an

explanation

^{is needed. This has been}

provided by

^Labusch

[12]

^{who shows that it is essential}

to

pinning

processes that the lattice be deformed. In this case, the total _energy of the system ^{is lowered}

by

deformation of the lattice and

pinning

may ^occur if

an energy increase is

required

^{to move the lattice}

with respect ^{to the}

pinning

^{array. So it is necessary}

to describe the

rigidity

^of the fluxoid lattice

against

distortions due to

point

forces. The fluxoid lattice stiffness is described

by

^{elastic constants whose} pro-

perties

^{are well known and}

prescriptions

^{are avai-}

lable for

experimental

determinations. The central

quantity

^{to calculate is the} local deformation of the lattice

by

^a

point

^{force. It can be}

represented by

^the

displacement ô

of the flux line from the

periodic

lattice. The result is

where

fp

^{is the localized}

^pinning

^force.

(Blo(»112

^is

the inverse of the fluxoid lattice

spacing, 1 jc

^{is an}

effective fluxoid

compliance expressed

^{in terms of}

an

appropriate

combination of flux lattice elastic constants

(expressed

in Labusch _eq.

(8)),

^{and a a}

numerical constant.

The _energy

gained by

the lattice under deformation is _per unit

volume,

This energy is

proportional

^{to the} average of the square of the

pinning

^{force. So we see that}

repulsive

force can also

pin

^an array of vortices. The

pinning

force

density

^P will be such that ô W ⁼ Pd if we

define the critical current as in

figure

^{2. d is a}

length characterizing

^{the width of the well}

describing

^the

local

pinning

force. So we get :

This result is ^not identical to Labusch’s one but agrees with Webb’s modification

[14]

of Labusch’s derivation. To first order Labusch gave ^a rather

general

criterion for

pinning : namely

^{that the maxi-}

mum

displacement

^{due to the}

pinning point

^exceeds

the width of the

pinning potential,

that is £5 > d.

In the case of

dislocations,

Webb has shown that this deformation is

always

^small

compared

^{to the}

effective width d which should be

comparable

^{to the}

coherence

length.

Webb found it _necessary to intro- duce

cooperative

processes due to the strong

coupling

between vortices. He used the

phenomenological

concept ^{of flux} bundle introduced

by

^Anderson

[15]

in his discussion of flux _creep. So he considered clusters of

pinning points

^{that combine to}

give

^a

local effective

pinning strength

In a random _array a

simple

statistical calculation shows that these _groups occur with a

frequency

Webb assumed that m is such that the

pinning

force

fm produces

^a deformation

equal

^{to d}

Then he

supposed

^{that the}

pinning

^force

density

is still

expressible

^{as a linear}

superposition

which is

exactly

^{our result. This result has several}

interesting

^{features :}

(1)

^the

pinning

^force

density

^{has the} square law

dependence

^on

fp

^{that Labusch}

^{found ;}

(2)

the fluxoid lattice

compliance

appears

explici- tely ;

(3)

^the

pinning point density

appears

only

^{to first}

order.

Fietz and Webb

[16]

^have

applied

^this

analysis

^to

cold-worked

alloys

^{of niobium with various amount}

of titanium.

Using

^{value of} the elastic constant cal- culated

by

^Labusch and the local

pinning

^{force cal-}

culated

by Webb, they

^{show that}

They

^{obtained a} remarkable agreement ^{with this}

prediction

^{of the} temperature

dependence :

^{P decreases}

as the temperature ^is

increasing.

We will _{not go}

deeper

^{in this}

problem

^because

we wanted

just

^to

emphasize

^one

point.

^{There is an}

intervening difficulty

^{that has}

commonly

^been

ignor-

ed and that

happens

ⁱⁿ many ^cases

although

^{not in}

our first

example.

^This is the need for a

theory

^connect-

ing

the measured continuum

quantity

^{P and the}

array of local interactions between

pinning points

and flux lines that sum to P. The

only existing theory is,

up ^to now, Labusch’s one. However we have seen

that some difficulties occur when

applying

^the

theory

to

experiments.

Modifications were introduced

by

Webb but

they

^cannot

simply

^be

justified.

^{In our}

derivation of Webb’s result we were led to define the critical current as the

limiting

^current in the flux flow

regime

^{when the}

voltage

^{tends to zero}

(Fig. 2).

This seems to us the

only

way of

defining theoretically

the critical current

although

^{we do not take into}

account the _creep

region.

The whole

analysis given

^{in this} part ^is

only

^valid

for field near

H,,.

^For

magnetic

field close to

Hc2,

the

theory

breaks down because the concept ^{of indi-} vidual vortex lines cannot be used. An attempt ^to describe the lattice in the presence of random inhomo-

geneities

^near

Hc2

^{has been made}

by

^{Larkin. He used}

a

phenomenological Ginzburg-Landau equation

^with

random coefficients. This is the way ^to calculate the critical current in this domain of field.

IV. Local

pinning

^{forces. - It is now}

interesting

to

investigate

the various mechanism which can

prevent ^{vortex motion. We have}

already

^described

pinning by

^the

potential

^{barrier near a} surface and

we will _{not say}

anything

further. Surfaces can also

pin

^vortices

perpendicular

^{to it. For}

instance,

^{if there}

is _any scratch on the surface it will decrease the

length

of the vortex line and so decrease its _energy and

pin

the line. This has been considered

by Maldy.

^However,

in this section we will concentrate on

pinning points.

The first

example

considered in the literature is

pin- ning by

^{cavities. This was}

investigated by

^Friedel

et al.

[17]

and Silcox and Rollins

[18].

^{Much work}

has also been done on

pinning by

dislocations.

Webb

[19]

^{and Kramer and Bauer}

[20]

^{have calculated}

the interaction between a flux line and a screw dislo- cations. Labusch has considered the more

general problem

^of

pinning by

^a field strain. A third case that has been considered

by

Willis et al.

[21] ]

^is

pinning

due to

region

^of

disparate

^{K. We will} not go too much in the details of the calculation of the

strength

^{of the}

interaction between individual flux line and

parti-

cular kinds of

pinning points. They

^are

mainly pheno- menological

^and

give essentially

^{order of}

magnitude

results. We will

only

sketch the derivation.

We first consider the case of cavities of radius b of the order of the interline distance d. Thus one

line

only

^is

pinned

^{on one}

cavity.

^{We assume}

Hc1

^{H «}

H,2.

^{In that case the free energy is}

The

pinning

force is of order

If the

cavity

^{has a diameter b} greater

than d,

810

M ⁼

nb2

lines are

pinned

^{on the same}

cavity and fp

is

roughly

^reduced

by

^{a factor M. When b}

^d, fp

is also reduced so b - d

corresponds

^to the maximum

pinning

^forces.

The _case, considered

by

^{Willis et}

^al.,

^of

pinning

^to

the

region

^of

disparate

^{x can} be calculated

easily.

The

binding

energy is

roughly

the difference in _energy between material of different K

Near

Hc2

^{we have}

So we get

Much more work has been done on

pinning by

dislocations. This is because one can

imagine

many

pinning

mechanisms. As an

example,

^{we will} quote three mechanisms of interaction between a flux line and a strain field of a dislocation. In the

first,

^{the dis-}

location reduces the mean free

path

^of normal elec- trons and so

reduces ç

^{and increases K. The second}

mechanism,

considered

by

^Kramer and Bauer and Kronmüller and

Seeger [22],

is called the linear inter- action : the lattice in the normal core of the flux line is contrasted with respect ^to that of the

superconduct- ing

^{matrix. This stress} interacts with the one

produced by

dislocations and

gives

^an interaction _energy linear in the dislocation stress field. The third mechanism is considered much more often. The normal core has

higher

^{elastic constants} than the

superconducting

matrix : thus there is a

repulsive

interaction _energy between the flux line and _any

region

^of

high

^strain

in the dislocation network. The reason is that the dislocation has

always

^an energy

proportional

^{to the}

elastic constants. It minimizes its _energy in a super-

conducting

^{matrix. We}

emphasize that,

^{with the}

theory

of

Labusch, repulsive

forces also lead to

pinning.

We will _{not go} into _any calculation because

good

definition of the

pinning

^{process due to} dislocations

requires complete

measurements of the elastic coeffl- cients that enter into calculation of the

properties

of the

pinning points.

^{It is}

quite likely

that the results would _prove to be too

complex

^{for more than}

quali-

tative

understanding

^{like many other}

properties

^of

dislocations. All the calculations done _up to now

give

^{an order of}

magnitude

^of

^10-’ dyn.

This is the

most

significant

^result.

We want to conclude with some remarks.

First,

the

difflculty

^{one has to link in a} theoretical manner

the measured

pinning

^force

density

^{and the local}

pinning force. This complicates

any

interpretation

of critical current.

Second,

^we

emphasize

^{the advan-}

tage ^{one has to}

study pinning

^{on such}

simple

^cases

as those described at the

beginning

^{of the second} part.

Third,

^{we do not} discuss flux _{creep but} one must take into account Webb’s detailed articles on the interest to

study

^flux creep ^to

gain

^an

insight

ⁱⁿ

pin- ning problems [14].

Acknowledgments.

^- The author has benefited from _many discussions with members of Laboratoire de la CGE in Marcoussis.

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