5 \,

IC/72/73

ERNATIONAL CENTRE FORTHEORETICAL PHYSICS

INTERNATIONALATOMIC ENERGY

AGENCY

MAGNETIC PROPERTIES OP SUPERCONDUCTORS

J . Che la -F lo roa

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1972 MIRAMARE-TRIESTE

/ . IC/72/78

-̂ International Atomic Energy Agency

and' • " • •

United Nations Eduoational S c i e n t i f i c and Cul tura l Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHISICS

MAGNETIC PROPERTIES OP SUPERCONDUCTORS*

J« Chela-Flores**

International Centre for Theoretical Physiosi Trieste, Italy.

ABSTRACT

We study i n i t i a l pene t ra t ion of magnetic f lux in a type I I

superconductor wi th equat ions from the Lagrangian theory whose main

feature is i t s non-locality; we restriot our attention to the

neighbourhood of a vortex filament and study this problem with the

Euler-Lagrange equations supplemented by reasonable boundary conditions,

We find a small departure from the linear variation of the macrowave

function which is found from the (local) Ginzburg-Landau equations.

Our equations are valid at T = TQ , while the linear variation is

found for T £ T .o

MIRAMARE - TRIESTE

July 1972

* To be submitted for publication.

** On leave of absence from Centro de Fisica, Instituto Venezolano de

Investigaoiones Cientifioas (iVTc), Apartado 1827, Caraoas, Venezuela,

I . ON THE EQUATIONS OF SOTERCOITO'CJCTIVITY

I t has "been f e l t f o r some time t h a t some of the more important

p r o p e r t i e s of superconductors ought t o be pred ic tab le i r r e s p e c t i v e of

the form of a microscopio t heo ry . We have attempted t o approach, the

Question of superconduct ivi ty us ing the Lagrangian method . We

succeeded in e s t a b l i s h i n g the Euler-Lagrange equat ions which coincide

with the aero- temperature Vala t in BCS equat ions for the macrowave

function <j> and dens i ty matr ix SI , ' On the other hand ( t h i s

motivated our work), Frohl ioh and Taylor have derived such equat ions from

the f a o t o r i z a t i o n of the seoond reduced dens i ty matr ix S2.o , in

conjunction with the exact equation of motion s a t i s f i e d by iftp}

i t is further advocated by Frohlich that $ should satisfy an equation

of motion from whioh all general features of currents and magnetic fields

of superoonduotors can be derived.

In the present work we shall take one such general feature of

a magnetic fieldt the imperfect Meissner effect in type I I super-

oonduotors. We expect, with Blatt^', that in order to get a satisfactory

theory of the vortex oore i t is neoessary to introduce funotions for

pairsj a theory treating <£ as a function of two spaoe-points is

expected to yield more information than a theory, suoh as the one due to

Ginzburg and Landau (GL), where <|> is treated as a function of a

single space-point*

A further reason for looking at the core is that, so far, there

has been no explioit evaluation of the variation of >̂ close to the

vortex filament for a magnetic field of strength of the order of the

first critical value HOi at very low temperatures, to wit, at

absolute zero. The problem of the vortex oore has been studied only in

the framework of OL and summarized in Ref»5« Although i t is true that

Tewordt, Werthamer, Zumino and TJhlenbrook ' have derived some very

general equations whioh, as T~->T , coincide with the already proved

(in BCS) equations of Ginzburg and Landau, i t is not those general

equations whioh have been solved for the vortex oore. Therefore, to the

best of our knowledge, ours is a fresh attempt to understand vortex

filaments close to Ht , at absolute zero temperature.

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However, our success is limitedj we are unable to study the

problem entirely with the Euler-Lagrange equations themselves. We

formulate a boundary value problem and require to prescribe reasonable

functions for the Fourier transforms of <2> and Q in a region of

the superconducting sample where >̂ and iX are approximately equal

to their values if the magnetio field were turned off. We feel i t is

reasonable to assume these to take the values found in the simple

Hartree-Fock approach of BCS , where the momentum of the pairs is

oonstrained within a.small shell in momentum space. We further find that

our final oonolusion is independent of whether we view the vortex core

as a tube of normal region with radius c, (coherence range) embedded in7 )

a superconducting matrix or whether the other extreme case is

maintained, viewing the core as a tube of material where the density is

approximately that of a BCS sample. To illustrate this point we recall

thai

where y is a fermion operator and x.. , x2 are space-points together

with a spin label. It is more oonvenient to refer to the co-ordinates

R , r where

"R = - A

The Goodman point of view is that

where SlJf represents the normal sample density. We remark, however,

that our conclusion is

• «

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Since &* -;&*( | r | ) and £l*CS - &*CS( | r_ | ) , we conclude that

( l ) is independent of the particular form chosen for SI , "We choose

to vork on the boundary, with ol — ^c .

I I . ASSUMPTIONS AKD RANGE OP VALID ITT

Throughout t h i s work we use u n i t s such tha t -if a c - 1 .

The f i r s t two Euler-Lagrange equat ions are obtained ' by making

v a r i a t i o n s of the t o t a l Lagrangian with respec t t o a> and ,J<L ;

whe re we have

We recall that the second term on the right-hand side of (3a)

arises from our assumption of a strongly peaked interpartiole potential

We shall consider a model solution of the vortex core of a type

I I superoonductor for temperatures very close to absolute zero. We view

the core itself as a tube.of radius c inside which

i i ) on the boundary, Q> = JP and

whereas inside the core a solution of the coupled Eqs.(2a) and (2b)

with the appropriate boundary values for ^ and £1 will lead us to

the oorreot behaviour of the roaorowave funotioixj

i i i ) for H — Hf4^ HCi the contribution from A is neglected.

In order to compare our results with those of GL we work in the

almost local limit ( | X. | « 1 ) and olose to the vortex core

( | B 1 « 1 )•In the more convenient ( JL , jr_ ) co-ordinates (in which the

a-axis of the cylindrical t r iad R , ZP' , z coincides with the singlevortex l i n e ) some simple re la t ions follow:

(A)

0)

In writing the last equality we drop the spin variable:

(C)

A,

(4)

(5)

I I I . COS'VENIEM' GAUGE FOR THE FIRST EULER-LAGRAtfGE EQUATION

We shal l usef for our convenience, some of the gauge freedom we

enjoy* Reoall that if

Athen we have

(6)

In the new oo—ordinates we make a Taylor expansion of <A

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Henoe, under a gauge tranafonnation, Eq..(6) Implies (to first order)

Now we axe in a position to exploit the symmetry of the problem,

theThus, we choose to work in a gauge <f> » - — QI , in which

azimuthal angle v coincides with the gauge function p . Hence (7a)

implies

With the full symmetry of the problem we may rewrite (5) as

where, for small | X. | , we have used a Taylor expansion

and later imposed the symmetry in (5^,z) ,

In the almost local case we can therefore write (2a) very

close to the vortex line as

(9a)

where our assumption (iii) nas been used. Eq.(9a) becomes,for our

choice of gauge (7b),and after simple differentiations and cancellation

of the exponential faotor exp( - P

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This equation may be written more compactly as

• (9c)

where

with

b =

IV. SEARCH FOR THE GREEN'S FUNCTION

L i s not a se l f -adjoint operator . Let us define the operator

M f adjoint to L , operating on a Green's function G ,

(10a)

where £ i 0 ^ e Dirac de l t a funotion. ffe reoa l l Green's formula

= JVY

•where is the component along the normal to the surface B ;

is a volume element. Using the defining Eq..( 10a), we find

-7 -

We shall introduce a rather useful notation in Eq,.(lla),

where the volume and surface integrals have "been abbreviated for

convenience; in Appx.I we diacuss which terms contribute to

Hence, the problem of studying <5" is reduced to the study of the volume

integral <£>. and the three surface integrals <b . , i • 1 , 2 , 3

The crux of the matter is to evaluate the Green's function G

In detail, the defining Eq..(lOa) is

where pi • 4(il ~ ^ - r , where we have taken the Fourier transform

with respect to the .r-v&riables; G is the Fourier transform of G .

Then, we consider the above differential equation in a single variable

R since, momentarily, we suppose lc to be constant.

By the standard method of solution ' we find (cf. Appx.Il)

(-(

(12)

where o is a constant, and where, I. , K, are Bessel functions of f irst

order.

V. FOURIER TRANSFORM OF THE MACEOWAVS FUNCTION

I f we choose the source point a t the origin* then

and we can write the surface integrals in k-space (with respect to the

r-variables) as,

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~2K

/ N /

where the integral i s defined by means of ( l l a ) and ( l i b ) ' ' ,

But b*n » m,. J J / V ? RQ

and, "by construction, nun - 1 ; hence

Prom our assumption (ii) of Sec,II we have

IL

O otherwise,

A ia the energy gap, W_ is the Debye (maximum phonon) energy and

Thus, if we use the above value for (£, and G we find

fX Vi

In the almost looal limit ( J r_ ) « l) we have

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Thus,

C frrr

We perform the above integration in Appx.III and find

where G^ IB independent of R variations. Similarly, ve find

where Op i s independent of R var ia t ions . On the other hand,

- I GJUft) •= J

(13a)

(13b)

In k-space,

where, in BCS, «i££ is the oooupation number. Performing the integration

over the spherioal angular variables, we find olose to the vortex (H « l)

1 , V O L « - S L B ! [^ oik A k su{^) &%O

• (14

o

Hence, whatever c]j> turns out to he, the volume contribution i s

VOL —> X

X

- 1 0 -

VI. BEHAVIOUR CLOSE TO THE VORTEX

Prom Green's theorem we have found that close to the vortex

filament the macrowave function (5 i s given in the almost local

approximation ( [ r_ ) « l ) by

(15)

Fe note tha t for a d i f ferent physical s i t ua t ion , namely, at

temperatures close to the t r ans i t i on temperatures and from the ( s t r i c t l y

l o c a l ) Ginzburg-Landau theory

in a small neighbourhood of a single vortex line.

There is a consistency chock in the integral expression for

(g. (of. Eq.14). Having obtained a solution for <̂> , we performed

the integrations indicated in (14) and have verified that no divergent

integrals arise when our solution (15) is used.

ACKNOWLEDGMENTS

I would like to thank Prof. J,G. Valatin for discussions on the

vortex problem and Profs. Abdus Salam and P, Budini as well as the

International Atomic Energy Agency and UMESCO for hospitality at the

International Centre for Theoretical Physics, Trieste., I am

also indebted to the IAEA for an Associateship at the ICTP which allowed

me to conclude this work.

- 1 1 -

APPEMDIX I

I t i s useful to realize that the surface over which we are

integrating can be "pictured" as the surface of Eins te in ' s s t a t i c model

of the universe. The R-axis of our hypereylinder would, in our analogy,

coincide with the time axis , whereas the three-dimensional curved surface

of the hypereylinder ( r » p ) would correspond to three-dimensional

space; D , in the case of superconductivity, represents the extension

of the correlated superconducting pai rs ; in the case of Eins te in ' s

model, p corresponds to the "radius of the universe".

The analogy breaks down when we consider the "ends" of thehypercylinder: R -* 0 , R —=> {? ; the Einstein universe has nobeginning or end, whereas the boundary of our problem has a f in i te lengthalong the R-axis. The boundary conditions are

( i ) 5 * o ,

IE -First we notioe that the \ \ integral does not contribute on any of

the three surfaces (the two "flat" surfaces "being defined by R = 0

R - ^ and the "curved" surface being defined by r » p ), since

Next,-the <^-. integral does not contribute on the curved surface

since i t has £ as a faotor in the integrand and vanishes on the curvedsurfaoe (of. condition I I I ) , Further, since G has a factor R , theintegral cannot contribute on the surface R - 0 . Finally, the cj5 integraonly contributes on the surface R —> £ for the same reasons as the$ ^ R integral.

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APPENDIX I I

Before we can claim that G is the correct Green's function veo \

must remark that Courant and Hilbert ' restrict their discussion to self

adjoint equations in dealing with the characteristic behaviour

= - I

where p(R) is the usual function in the notation of Ref.8. We can treat

the present case as follows: we first integrate Eq.(lOb) to find

^n I,In the self-adjoint case the third integral is shown to vanish and the

first is integrable.

In the present case we have to discuss the further integration

o* 6

f AJO

and the right-hand side is seen to vanish in view of assumed continuity

of the function G with respect to L , k_ being assumed fixed at

present. Hence, G iB indeed the required Green's funotion for the

adjoint Eq.(lOb) with homogeneous boundary conditions.

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APPEKDI3C I I I

We oonsider the integral in Eq..(l3a). F i r s t , we rewrite the

square of the denominator of the integrand :

However, we have that' k £ / 4m • — Ej, , where E_ i s the Fermi energy( r 5 eV), On the other hand, A ~t 1® eV ; hence, i t i s evidentthat the square of the denominator in (13a) can he approximated as(k - k - j j / ^ m . Hence, (13a) implies

f K

2 2 ?Changing to the /A variable (|i « 4(k - kp) ) we find

We may split the range of integration as

then i t i s known that the integral 1L converges . On the other hand,l 2 '

the range of the M2 integral can be taken from zero to plus 4krp since,given K Q ( ^ R Q ) as a solution of Bessel 's equation, then KQ( -/JL"RQ)

will be a solution. We next note that we need iuu Yio and, by defini t ion,

we expect q to be very large on a microscopic scale; asymptoticallyKQ(/AZ) i s exponentially decreasing and therefore, since ^ i ssufficiently large, we expect K, >> 1U • Finally, the contribution of^ . can be written as in (13b) (with respect to variat ions in R ,G. in a constant) .

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REFERENCES

1) J . Chela-Flores, Collective Phenomena 1 (1972), in p r e s s .

2) J .G. Valat in, Phys. Rev. 122, 1012 (1961).

3) H. Fr'ohlich, Nature 228, 1145 (1970).

4) J J I . Blatt, Rev. Mod. Phys. 36, 30 (1964).

5) D. St.James, 0. Sarma and E.J . Thomas, Type I I Superconductivity

(Pergamon Press, London 1969) p»46.

6) L. Tevordt, Phys. Rev. 132, 595 (1963);

K.R. Uerthamer, Phys. Rev. 132, 663 (1963);

B. Zumino and D.A. Uhlenbrock, Nuovo Cimento 33,, 446 (1964).

7) B.B. Goodman, Rep. Prog. Phys. £9, 459 (1966).

8) R. Courant and D, Hilbert, Methods of Mathematical Physics, Vol.1

(interscierice Publishers Inc . , New York 1953) p.355«

9) J . Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev, 108,

1175 (1957).

10) A. Erdelyi, Tables of Integral Transforms, Vol.2 (McGraw-Hill

Book Co., Ino. 1954) p.28.

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