Download - PH ILI PS TECH N IC~AL REVIEW high currents, have in a short time attracted considerable interest from the de-signers of magnet coils. An article about coils which have been made inPhilips

PH I LI PS TECH N IC~AL REVIEWVOLUME 29, 1968, No. 1

'I,

Superconductivity:{1

1

J. Volger\

Although it was discovered half a century ago, the phenomenon of superconductivity had,until recently, still not yielded up many of its secrets. In the last few years the situationhas changed. Although by no means all the experimental data have been adequately ex-plained, we now have a reasonably good understanding of the nature of superconductivityand of the strange properties of superconductors of the second kind, discovered a few yearsago. These properties are in many respects the opposite of those that for decades havebeen considered characteristic of a superconductor. Some members of this group, the"hard" superconductors, which remain superconducting in strong magneticfields and whencarrying high currents, have in a short time attracted considerable interest from the de-signers of magnet coils. An article about coils which have been made in Philips ResearchLaboratories for producing very high magnetic fields will shortly appeal' in this journal.In the article below a brief survey of present knowledge of the phenomenon of super-conductivity is given.

Introduetion

ii-,

The discovery of superconductivity

The phenomenon of superconductivity, which inrecent years has attracted a great deal of attentionthrough the publication of new' experimental ortheoretical discoveries or promising technical appli-cations, was first observed in 1911 by KamerlinghOnnes and his research assistant Holst (fig. 1). At theLeyden Physics Laboratory, where Kamerlingh Onneshad in 1908 been the first to succeed in liquefyinghelium, and which was later named after him, a seriesof investigations was at that time being performed onthe electrical conductivities of metals in the newlyattained temperature range. The electrical resistance ofmetals exhibits a positive temperature coefficient, andit was considered probable that the resistance of ametal would drop to a very low level if the temperaturewere reduced to very close to absolute zero. However,in the course of some measurements with a mercurywire, it was then found that at a certain temperature(about 4 OK) the resistance, which was already verylow, suddenly became too low to measure [11. Thisdiscovery was followed by many similar ones over theyears, and it is now known that about half the metalsand metallic elements in the periodic system can be-r .

Prol Dr. J. Volger is with Philips Research Laboratories, Eind-hoven, as a Scientific Adviser.

come superconducting below a certain "transition"temperature Tc, while the same is also true for manyhundreds of compounds and alloys.

Extensive research has made it clear that super-conductivity should not be regarded as a property (thedecrease of the resistivity to zero), but rather as a state.We now, therefore, speak of the superconductingphase [21. The superconducting phase exhibits a largenumber of striking features whose interrelation hasbecome much clearer over the past few years, andwhich are also of technical interest.

The availability of low temperatures

The ease with which low temperatures can be reachedtoday (here we mean temperatures near that of liquidhelium, about 4 OK),compared with the trials and trib-ulations of the pioneering years, is a factor of con-siderable significanee in research into and applicationof superconductors.Viewed more broadly, man's ability to achieve low

temperatures is really of very recent date. The discovery

[1] H. Kamerlingh Onnes, Comm. Phys. Lab. Univ. Leiden 12,Nos. 122b and 124c, 1911. A survey of the initial period ofresearch into superconductivity will be found in C. J.Gorter, Rev. mod. Phys. 36, 3, 1964 and K. Mendelssohn,ibid. p. 7.

[2] For the "conventional" considerations on superconductivitysee F. London, Superfluids, Dover Pub!., New York 1961,and D. Shoenberg, Superconductivity, Cambridge Univ,:Press, Cambridge 1952. A survey of the recent developmentsis also given in E. A. Lynton, Superconductivity, Methuen,London 1964.

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2 PHlLlPS TECHNICAL REVIEW VOLUME 29

and mastery of fire dates back thousands of years. Theproduction of cold, on the other hand, was not success-ful until the last century, as can be seen clearly infig. 2.The points in this figure represent the principal suc-cesses achieved through the years. The liq uefactionof helium marked an important stage along this road.Further cooling would now only be possible by meansother than the liquefaction of gases. It is indeed worthyof note that, for about thirty years, research at ex-tremely low temperatures was possible at only three

An important factor in all this is the availability of helium it-self. Its concentration in the atmosphere is very low (5 X JO-6).Nevertheless, it is possible to obtain helium from air since it isone of the by-products of the large air-rectification installationsnow in use for the production of oxygen, argon, etc. The gas canalso be obtained in small quantities by heating certain radioactiveminerals. The most economical method of obtaining it is fromsources of natural gas containing helium. Here, the helium isthe end product of radioactive conversions, which has been re-tained under domed impermeable layers of rock and happensto have become mixed with natural gas trapped in the sarne geo-logical structure. As far as is known, the natural gas fields in the

Fig. I. H. Kamerlingh Onnes (right) and G. Holst, who was research assistant to Prof.Kamerlingh Onnes at the time of the discovery of superconductivity in 1911. Dr. Holst(later Prof. Holst) was to become the founder of the Philips Research Laboratories andtheir director till 1946.

laboratories in the world, the Kamerlingh Onnes Labo-ratoryat Leyden and laboratories in Toronto and Ber- 7030K

Iin. Now, however, there are many hundreds of researchand development centres where helium can be lique-fied and used as a coolant for all kinds of investigation.Among these centres are many large industrial labora-tories; this comes about chiefly because of the keeninterest at such laboratories in solid-state research, forwhich low temperatures are now essential. In additionthere is more interest than ever before in the technicalproblem of liquefaction itself. We should mentionhere the spectacular success of Collins, who has con-siderably improved the expansion machine formingpart of the Kapitza-type helium liquefier. The heliumliquefier developed by Collins is in commercial pro-duction and has been on the market now for nearlytwenty years. This has contributed considerably to-wards the post-war increase in cryogenic researchand engineering. The development of a helium lique-fier based on the Stirling cycle has also recently beensuccessful [3] and this machine offers considerable pro-mise.

Faraday.i8J4 Cat/lekt et ol.

Ul'a~;:J;~UI~~ZJ: PlCtet 1877Liq.C2Ht,. a

C2Hs. /iq. O2 LlI QewOT 1883NO. solid N2

IIIlIlIumm id

solid CC2and ether

~--._______'

freezing-mixtures etc. [J Dewar 1898

(iq. H2

liq. h'e 11 Kamer/i/1gh Onnes 1908

boiling He under{ Iii KOnnes 1921reduced pressure !Xl Keesom 1926

Gd sulphate e Giououe andMc Douga 11 1933

70-1 -

ad/ob demog(1{

• de Hoos,K-Cr alum III w.ersmc and

Kramers 7934

Cs- TI alum. de Haas c.s.1935

K -Cr alum .Gorter c.s.1949

10-3

1O-t8Lo""Co-----78~5=-cO-----7,.J90""O,--------,--,19~50-=---j

Fig. 2. A diagram which shows how recent is man's penetrationof the field of low and very low ternperatures. The dots are ineffect the milestones in the history of cryogenic research.

1968, No. 1 SUPERCONDVCTlVITY I 3

south of the United States of America are the richest in helium.Many sources there contain a small percentage of helium, andsome contain as much as 7 to 9 mol %. The helium is separatedoff by cryogenic methods. At present, annual production isestimated at 5 X 1010 litres at n.t.p. To obtain I litre of liquid he-lium about 800 litres of gas at n.t.p. are needed. The known worldreserves should be some 5 x 1012 litres of gas at n.t.p. The con-sumption of helium in cryogenic research and technology isrelatively low and amounts only to a small percentage of totalproduction. More than half the world's production is used forwelding and as a compressing gas for the liquid fuel in space craft.

Superconductivity and electrical technology

Industrial interest in a physical phenomenon lies in a"quality" or "outstanding feature" of that phenom-enon. And in fact, the absence of electrical resistanceunlocks a wealth of new possibilities. Although thehighest transition temperature so far found is still as lowas 20.05 "K, the possibility of using undergroundcooled superconducting cables for electricity supply,in certain densely populated areas such as London, isalready being seriously examined to see what advan-tages it might offer. There is also interest in the useof superconducting elements for logic circuits ("cryo-trons"). Some very interesting possible uses includesuperconducting magnet coils which can be used togive very high magnetic fields, sometimes with fluxdensities greater than 10 Wb/m2 (100000 gauss) overvolumes large enough to be of interest in technicalapplications [4l. One wonders how many other evenmore impressive changes in electrical technology are tobe expected if superconducting materials with a transi-tion temperature of 50 or 100 "K are ever produced.Even though insulation losses are no problem today-modern insulating materials permit the escape of lessthan IW per m2 at a temperature difference of 300 "K- the existence of such materials with higher transi-tion temperature would appreciably reduce the techni-cal difficulties and expense of cold production, both ofwhich go up as the temperature goes down. Thus inpractice the production of 1 kWh of cold at 77 "Krequires 10'kWh of electric energy, but at 4 "K I000 kWhare required.

Nevertheless, it is to be expected that, even if super-conductors with transition temperatures well above20 "K are never found, superconductivity will stillfind its applications in technology.

The superconducting state

Comparison with the normal state

The theoretical treatment of conduction in a normalmetal is fairly simple. The electrons can be regardedas independent particles which travel through thecrystal in all directions at random velocities. If there is

no electric field, the average velocity is zero. Ifthere is afield, however, the electrons move with a certainaverage velocity, the drift velocity, in a certain direc-tion. The velocity distribution of the electrons is thusnot symmetrical about zero.

In this situation the conduction electrons transfer atcollisions more energy on average to the cryftal latticethan they receive, and heat is developed. Betweencollisions, the energy which has been~iven up is re-covered from the electric field. T~~{is an irreversibleprocess, so that the current quiyly drops to zero oncethe field has been switched off/The velocity distributionthen once more becomev(mmetrical about zero.

If we are to apply quantum mechanics, we must con-sider the "cloud" of conducting electrons as a gaswhose particles obey the rules of the Fermi-Diracstatistics: in accordance with Pauli's exclusion princi-ple the various quantum states which we know fromwave mechanics may only be occupied by one electron.This means that, at absolute zero (T = 0), and zeroelectric field, all levels with energy lower than a certainvalue EF (the Fermi energy) are occupied and allhigher levels are unoccupied. At a higher temperaturethe boundary between both' regions- is not so sharp.Roughly speaking this Hikes an energy interval ofmagnitude kT distributed about the. value EF (k isBoltzmann's constant). In this scheme, the flow ofcurrent means that some electrons. have begun tooccupy higher, previously unoccupied, energy levelsby taking up energy from the field. The loss of energyon collision corresponds to falling back to a lowerenergy level. In the normal state, the flow of current ispossible only through the excitation of separate elec-trons.

In a superconductor the situation is quite different.According to Bardeen, Cooper and Schrieffer [5], exten-sive ordering has been established in the material.Electrons with an equal but opposite electromagneticmomentum (see below), particularly those whose ener-gy values lie in a narrow interval around EF, haveentered into a strong interaction in pairs, through theintermediary of the lattice vibrations. The details ofthis interaction, which can extend over a relativelylarge distance, will not be further dealt with here but weshould point out that, in the ground state of the elec-

[3] G. J. Haarhuis, Proc. 12th Int. Refrigeration Congress,Madrid 1967, p. 894.

[4] The work done in this field at Philips Research Laboratorieswill be discussed in an article by A. L. Luiten which willshortly appear in this journal.A general survey, with particular reference to large mag-

nets, will be found in C. Laverick, Superconducting magnettechnology, Adv. Electronics and Electron Phys -.23, 1967.

[5] A summary of the theory of Bardeen, Cooper and Schrieffertogether with an extensive bibliography- is to be found inJ. Bardeen and J. R. Schrieffer, Progress fh low temperaturephysics 3, 170-287, 1961.

4 PHILlPS TECHNICAL REVIEW VOLUME29

tron pairs, the total electromagnetic momentum of thetwo electrons is exactly zero. (This should not howeverbe taken to imply that their total mechanical mo-mentum or the velocity of their common centre ofgravity is also zero.) Moreover, the total energy of theelectrons forming such a pair is slightly lower than itwas before the interaction took place. Before goinginto the question of how such an ordering leads tosuperconductivity, let us have a quick look at theappearance of the band diagram in both cases, payingparticular attention to the density of states.

The two graphs of fig. 3 show the' curve of theenergy E against the momentum P of an electron onthe right, and the curve of the density of states Nagainst E on the left, all the curves referring to energyvalues near to EF. The reference point chosen for theenergy is EF. For a normal metal (fig. 3a), N graduallyrises with E; the relation between E and p can forexample be parabolic. At T = 0, there are no Evalues greater than EF and no p values greater thallPF.At T > 0, some electrons have a greater energy and,correspondingly, some levels with lower energies thanEF are unoccupied.

For a metal which is in the superconducting state(fig. 3b), there is a gap - a forbidden zone - of width2L1in the spectrum of the possible energy values of theelectrons around EF. Just above and below this gap,the density of states is much greater than in the nor-mal state. The electrons which in the normal state popu-lated the region between EF - LI and EF + LI cantake up a position in these edge regions. The quantity2L1could be said to be the "binding energy" of thepairs which is liberated when they are formed. At T= 0,LIhas its maximum value which we call Llo, and thehighest energy level which is occupied is El!' - Llo. Thevalue of Llo is a characteristic of the material. At tem-peratures above absolute zero but below Tc, there is acertain thermal excitation and as a result of this the"bond" is broken for a number of pairs. The "loose"electrons formed have a higher energy than EF + LIand thus lie in the energy band above the gap; theymay justifiably be regarded as "normal" electrons ina "sea" of "superconducting" electrons.If the temperature is increased, the percentage of

"normal" electrons increases while LI decreases.Finally, LI= 0; this happens when T has reached thetransition temperature Tc. An external magnetic fieldalso has a similar effect, since LIcan be made to fall tozero by increasing the magnetic field above the criticalvalue He. This value becomes smaller as the tempera-ture increases.We should also like to point out here that there are

limits to the current in a superconductor, even atT = ° and at zero external magnetic field. This may be

E

1-p

E

i J_).<:I

EFN(E)_ -p Pp-

.<:I

Q\ fFig.3. a) The relationship between the energy E and the momen-tum p of the free electrons in a metal in the normal state is shownon the right. The abscissa axis is drawn at the level of the Fermienergy EF. The curve of the density of states N against E, aroundE = El', is shown on the left.h) The same for a metal in the superconducting state. N(E) = 0in a zone of width 2L1around E = El'"

explained as follows: if the drift velocity of an electron- which increases with the current - exceeds a certainvalue (2L1jPF), the energy per electron becomes greaterthan EF + LI and transitions to states of individualelectrons with an energy EF + LIare possible. In otherwords, electrons forming a pair can then separate andonce more become "normal". At a higher temperatureor if there is a magnetic field, this situation occurs at alower velocity, i.e. at a lower current.Why, now, is the electrical resistance of a metal to

which fig. 3b applies zero? Although it is very difficultto answer this question properly in words, i.e. withoutbringing in the quantum-mechanical treatment of theproblem, we can at least say the following. The tran-sition from the situation of fig. 3a to that of fig. 3b(the transition of the electrons from the free to thepaired state) must be regarded as a "condensation" ofthe electron gas. Excitation of individual electrons isnot now required for a current to flow: a current must

1968, No. 1 SUPERCONDUCTIVITY 5

be regarded as a state of the condensate as a whole.This state exhibits perfect persistenee because transi-tion to an energetically lower state is impossible eitherbecause there is no such state or because the probabili-ties of transition are infinitely small; both cases willbe discussed later. The transfer of energy to the latticeis therefore impossible and there is no development ofheat and no resistance.

In the quantum-mechanical treatment of the super-conducting state, the condensate of electrons corre-sponding to the above is described by one single wavefunction "P(r); here, r is the co-ordinate of position.Since "P(r) is a complex function, it can also be writtenin the form "P(r)= I"P(r)I exp jrp(r), i.e. as the productof amplitude and phase factors. The amplitude factoris in general independent of r and can be related to theconcentration of the superconducting electrons.The way of looking at the transition from the normal

to the superconducting state as a condensation, andtherefore as a phase transition, corresponds to the factthat it occurs at a sharply defined temperature, just asin the well-known phase transitions vapour ~ liquidand liquid -> solid.The temperature Tc and the width 2L1o of the gap at

absolute zero are proportional to each other. Bardeen,Cooper and Schrieffer, extending the theory ofFröhlich, have calculated that Tc and Llo depend on thevibration spectrum of the lattice - particularlyon theDebye temperature eD -, on the density of states Nof the electrons in the normal state close to EF and onan interaction parameter V, according to the relation:

-12L1o = 3.52 ia; ~ 3.52 keD exp ( ). (1)

NEF V

In many cases the experimental results could bedescribed by a relation of this form.An interesting implication of equation (I) is the

"isotope effect". In this effect, the transition tempera-ture of isotopes of the same element depends upon themass M of the nucleus. Since eD is inversely propor-tional to J/M, the same must apply for the transitiontemperature. This effect had in fact already been foundexperimentally in 1950 [6).

In metals which have a slightly more complex bandstructure in the normal state (e.g. the s and the d bandoverlapping), the situation can be different from thatshown in fig. 3b. Recently, cases have been found inwhich the drop in the density of states around E = EFis far less marked than the figure shows, and also theisotope effect is not found in all metals. Furthermore,it seems that the interaction between the electrons neednot necessarily take place through the lattice vibrations.Thus the question as to what direction to take in look-ing for a superconducting material, i.e. the question of

the key to the "electron engineering" of superconduc-tors, has still not been answered satisfactorily.

Two kinds of current

To assist the understanding of what follows, weshould point out here that we have to distinguish be-tween two kinds of current in superconductors, theMeissner currents and the transport currents. In thefirst case, the total charge transport through everycross-section of the superconducting body is equal tozero, just as it is for an eddy current in an ordinaryconductor; a Meissner current flows in the presenceof a magnetic field. With the transport current, there isin fact a net charge transport through a cross-section, asin a normal conductor connected to a voltage source.We shall first ofall discuss the connection between cur-rent and magnetic field.

London' s first equation

We shall now show that London's first equation,which describes the connection between current andmagnetic field for a superconductor, may be directlyderived from the basic property of a pair of electrons,the property that its electromagnetic (or generalized)momentum P is zero [7):

p _ 2 mv + 2 eA = O. (2)

Here m and e are the mass and the charge of an elec-tron, v the velocity of the centre of gravity of the pairand A the vector potential. The occurrence of theterm eA is a result of the presence of the magneticfield H whose relationship to the vector potential isgiven by:

H = curl A, (3)

which, if A is to be uniquely determined, requires that:

div A = O. . (4)

If we now reduce (2) to

v = -eA/m, (2')

we see that the local value of v is proportional to thatof the vector potential. From this we can derive thefollowing. First of all, the normal component of v atthe surface of a superconductor will be zero, and thesame must therefore apply to the normal componentof the vector potential, in agreement with the require-ment div A = O.Furthermore, using (2'), we can derivean expression for the current density j in a super-conductor by substituting the expression found for v

[6) E. Maxwell, Phys, Rev. 78, 477, 1950; C. A. Reynolds,B. Serin, W. H. Wright and L. B. Nesbitt, Phys. Rev. 78,487, 1950.

[7) For London's own calculations, see his book [2).

6 PHILlPS TECHNJCAL REVJEW VOLUME 29

In the relation j = nev (n is the concentration of theelectrons) :

j = nev = - ne2A/m,

which can be red uced by (3) to:

curl j = -ne2H/m.

This is London's first equation for a superconductorand takes the place of Ohm's law. (London's secondeq uation relates to non-stationary states and does notneed to be taken into account in this article.)

Just as (5) is a direct consequence of P = 0, so can(6) be regarded as a direct conseq uence of the eq uation:

curl P = O.

This is the 1110st general expression for London's firstequation.

The current flowing in a supercond uctor in the pres-ence of a magnetic field should be regarded as anaspect of the ground state of the condensate, which isdetermined by the condition P = O. The applicationof the magnetic field only causes a slight shift in theenergy value appropriate to the ground state, in exact-ly the same way as, say, the energy value of the elec-trons in the shells of an atom is shifted slightly whena field is applied.

We shall not deal with tile quantum-mechanicaltreatment of this, but we should like to point out thatthe condition P = 0 implies that the wave function1/)(r) describing the condensate has no spatial variation:wave mechanics show that the gradient of qJ is, in fact,proportional to the rnornentu 111P and is therefore zeroif P = O. The "wave" represented by the wave func-tion is here one of constant amplitude (see above)and an infinitely long wavelength.

The Meissner effect; the penetration depth

The current just mentioned, which flows in a super-conductor when a magnetic field is applied, has a verymarked spatial variation, and the same is true of themagnetic field inside the su percond uctor, This will beseen directly if London's equation (6) is combined withthe Maxwell equation:

curl H = j,

which also applies in a superconductor. lt is thenfound that:

whereÀ = (m/ne2)1/2. . . .

The parameter À has the dimension of length. It is a"characteristic length", which will always appear in thesolution to (9) whatever the boundary conditions, and

(5)

is generally referred to as the penetration depth. If thevalues applying in a normal metal are substituted form and n in (J 0), we f nd for À a length of the order ofonly 100 nanometres (1000 A).

The nature of the value À and the meaning of thename "penetration depth" become clear if we look atthe simple situation sketched in fig. 4. Here, a super-conductor and a non-superconductor are separatedby a plane but otherwise extend to infinity. In the non-supercond uctor there is a magnetic field of strength Ho,parallel to the plane of separation. ]f we now examinethe way in which the strength H of the magnetic field inthe superconductor varies as a function of the distancex from the plane of separation, we find:

(6)

(7)H(x) = Ho exp (-x/À). . (11 )

The magnetic field strength has thus already droppedto about a third of Ho at the depth x = À. (A solutionof (9) like that of (11) but with a positive exponentcan also be found, but this has no physical significaneehere.)

f

o

Fig. 4. The penetration depth J, characterizes the distance that amagnetic field can penetrate into a superconducting body. Thestrength H of this field has the value Ho outside the body.

(8)

In a more complicated case, e.g, that of a cylinder ofcircular cross-section, the sol ution is also more com-plicated, but it is always found that H decreases as thedepth inside the superconductor increases. The magnet-ic field inside a superconductor is reduced with respectto the external field by the action of a current. Thespatial distribution of this "screening" or Meissnercurrent satisfies an equation which, like (9), followsfrom (6) and (8) and has the same form as equation (9):

(9) (12)

(J0) Thus the Meissner current has the same depth of pen-etration as the magnetic field. The interior of a super-conductor is field-free because the external field isexactly compensated by the magnetic field of the screen-ing current.

1968, No. 1 SUPERCOND UCTIVITY 7

The state of field exclusion also sets in spontaneous-ly in a metal which is cooled in a magnetic field tobelow' the transmission temperature. This very im-portant feature was first established by Meissner andOchsenfeld in 1933. It demonstrates that a supercon-ductor differs fundamentally from a conductor of zeroresistance and has pointed the way to the understand-ing of the superconducting state as a phase.

The transport current through a superconductor; thepersistent current

The currents discussed above were reaction currents- or magnetization currents - brought about by theexternal field. It was tacitly assumed that we weredealing with a singly-connected body, i.e. one inwhich each path through its interior from a point Pto a point Q can be transferred continuously by meansof infinitely small changes into any other path betweenPand Q. If such a body is cut by a plane, the integralof these currents across the cut surface is equal to zero.

In order to prepare for situations which can occur ina multiply-connected body, let us now discuss thecase of a superconducting rod with its ends connectedto a battery by copper wires. A transport current canthus be passed through this rod, and the integral of thecurrent across a cross-section is now not equal to zero.The forces which form the electron pairs remaineffective, but the value of the generalized momentumis, in this case, no longer equal to zero:

P = 2mv + 2eA :f=0,

as we shall explain in a moment.A superconductor through which a transport

current is flowing does not remain in the ground state.In this case the superconducting condensate enters astate of collective excitation without there being anyquestion of excitation by decou pling of the pairs.

Here, too, the starting point for the treatment of theelectrodynamic behaviour is London's equation of theform curl P = O. It follows from this equation thateven though the field is now due to the flow of injectedcurrent, the current and field distribution are stilldetermined by (9) and (12), and that, in particular, theconcept of penetration depth retains its significance.Like the Meissner current, the transport current flowsin a thin skin at the surface of a superconducting rod.There is neither current nor field within it but the vectorpotential A now has a value different from zero. Theintegral from which A can be calculated by means ofMaxwell's theory:

A = f (jIr) d'r.

(where j is the current density in the volume elementdr at a distance r from the point where A is required),

now gives a value different from zero. This explainswhy P cannot be zero. .

A special case of the collective excitation of the su-perconducting condensate noted above is the persistentcurrent in a superconducting ring, i.e. a current whichis induced in the ring and continues to flow unattenuat-ed as long as the ring remains superconducting. Hereagain, the basis for an electrodynamic approach is oneof the equations curl P = 0 or curl j = -ne2H/m, but,at a given external magnetic field, there are now severalsolutions for the currents in the superconductor, andnot just one, as was the case for a singly-connected'superconducting body.

As with the flow of a transport current in a super-conducting rod connected to a current source, the flowof a persistent current in a ring must be regarded as asituation in which P:f= O. In the wave-mechanicalapproach given above, this means that the collectivewave function will in this case have a spatial wave-likevariation: if P is not zero, cp varies uniformly with theco-ordinates and 'tjJ does have the character of a period-ic function. We shall now see that this has rather re-markable consequences for the magnetic flux enclosedby the ring.

(13)

Flux quantization

A ring, like any other doubly-connected body, ischaracterized for superconductivity by the centralregion (the "hole") in which 'tjJ = O. The superconduct-ing body has been pierced by either a hole, or a tube ofnormal material. In the latter case, the normal materialneed not be chemically different from the superconeductor, a magnetic field in the central region could hav-been applied to remove the superconductivity. Withthis topological structure it is possible that P =1= 0without there being any external current source, butthe related spatial dependence of cp now poses a prob-lem connected with the uniqueness of 'tjJ. If we assumethat 'tjJ extends over macroscopie distances, the ringmust be a whole number of wavelengths along itscircumference, just as it is for the path of an electronin an atom. As in atomic physics (Bohr, Sommerfeld),this is the case if the "phase integral" p Pdl is a wholemultiple of h, Planck's constant:

pPdl = nh, (15)

(14)

Here, dl is an element of the integration contour and nis an integer. It can be shown directly from (15) and(3) (see below) that the magnetic flux cp passingthrough the surface of a cross-section is therefore then-times multiple of a flux quantum (/Jo, equal to h/2e:

(/J = nh/2e = n(/Jo. • • • . • (16)

The flux quantum (/Jo is about2x 10-7 gauss cm'', This

8 PHILlPS TECHNICAL REVIEW VOLUME 29

is also true for the flux enclosed in a normal patchor cylinder within a superconducting object.The persistent current in a doubly-connected body,

which' we have come to recognize as the form inwhich the excitation of the superconducting collectiveappears, is a quantum state which apparently makesthe transition to the ground state only with great diffi-culty. No cases are known of transitions under inter-action with an electromagnetic radiation field, i.e.witha single photon, like those which occur in atomic sys-tems. Such a photon would in fact have to possess tre-mendous energy in a real superconducting system:with a persistent current of 100A, a quantum jumpzl (/J = (/Jo corresponds to an energy jump of about1MeV. The extremely remote probability of such atransition is undoubtedly partly connected with theenormous disparity between the dimensions of thering and the wavelength of the radiation (about10-3 nm, or 10-2 A).Flux quantization is not a purely theoretical concept,

but has also been demonstrated experimentally [81.

However, it is still, of course, an open question whetherthe relation (15) for the phase integral also applies ifthe integration contour is very long, as, for instance,in a short-circuited coil consisting of a superconductingwire a few miles long.Later we shall discuss how flux can disappear from

a superconducting ring; as far as we know, this is al-ways a process of escape at the surface of the super-conductor, in which in point of fact separate fluxquanta are involved and where energy is converted intomagnetic field energy elsewhere or into vibrationalenergy of the lattice.

Experimental determination of the width of the forbidden zone

The change in the energy spectrum that the material undergoeson cooling to below Tc, as discussed above, may be found experi-mentally in various ways. We should like to discuss two of themhere ..A fairly direct method is the study of the absorption of a high-

frequency electromagnetic field directed against a supercon-ducting plate: for example in a microwave resonant cavity [9J.

As the metal cools the normal resistance drops and so thereforedoes the depth of penetration of the electromagnetic field [lOJ,

and the absorption, but electromagnetic energy will always bedissipated as long as the electrical component ofthe electromag-netic fie1d can interact with normal electrons. Even in the super-conducting state this is still possible, because the alternatingfield still meets electrons that are normal - i.e. electrons un-paired because of thermal excitation - in the thin surface layerinto which it penetrates. In the extreme case, however, of in-finitely low ternperature, this absorption would finally have todecrease to zero unless the energy /ZV of the wave packets of thealternating field is itself high enough to decouple a pair of elec-trons and .bring them across the forbidden zone to an excitedstate. This effect gives us the chance of accurate spectroscopiedetermination of the energy gap. The method is therefore quite

comparable to the one used in semiconductor research, althoughthere we find the fundamental absorption at wavelengths whichare a few orders of magnitude smaller. Fig. 5 gives the result ofthe classical experiments of Biondi and Garfunkel: the surfaceimpedance (as a measure of the absorption) of a small plate ofaluminium is plotted here in relative co-ordinates as a functionof the value of /ZV of the energy quanta of the microwave field (inunits of kTc) with the temperature as a parameter. It can clearlybe seen, particularly at a relatively low temperature (T« Tc),that absorption sets in as soon as the quantum energy is higherthan about 3.5 kTc. A different value is found for some materials.

z

r 0.8

r/Tc=0.9

Fig. 5. The width 2<1of the forbidden zone in the energy -spec-trum of the "superconducting" electrons can be deterrninedby finding out how the absorption of microwaves in the surfacelayer of the superconductor depends on the quantum energy ofthe incident radiation. The surface impedance z (a measure oftheabsorption) is plotted against the quantum energy in units ofk'I'«. At very low temperatures absorption suddenly sets in athvjk T; = 3.5.

A particularly elegant method of spectroscopy - this is thesecond method that we shall discuss here- is that due to Giaeverin which a tunnel diode is used [l1J. Two metals are separated bya very thin non-metallic layer (a few nm thick). Electrons canpass through this skin from one metal to the other by a tunnellingprocess, that is to say, they do not need to make use of thehigh, thermally inaccessible conduction band of the intermediatematerial (usually an oxide); on the contrary, during the tran-sition, they are located in the forbidden zone of the energy-leveldiagram of that material. Now, with this kind of contact thesame thing happens as with a true contact: the Fermi levels ofboth plates reach exactly the same value after the exchange of afew electrons. If, however, one ofthe metals is a superconductor,the electrons on opposite sides are not "on speaking terms",that is to say, electrons that would be prepared to cross cannotfind an available energy level on the other side (fig. 6), againparticularly in the extreme case of infinitely low ternperatures.A relative displacement of the level diagrams can be broughtabout by applying an electric voltage V, and this is possible be-cause of the comparatively good insulation of the non-metallic


layer. A crossover of charge is then possible if V ;;;; á]«. The ener-gy gap may therefore be read off immediately from the 1- Vcharacteristic of this Giaever diode as the double-thresholdvoltage. Experiments of this type also show that, in most cases,the energy gap is about 3.5 IcT«;

a

.// / r>

b

. - (xt~è'x

)<

xX'

"x.ol'I

.//

c

V=Aje

Fig. 6. Characteristic (a) and energy band diagrams (b, c) of aGiaever tunnel diode for determining LI. A superconductingplate (on the left) and a normal plate of the same metal areseparated by a layer of oxide so thin that the electrons can crossfrom one plate to the other by means of the tunnel effect. If theapplied voltage V is lower than A]e, no current can flow becausethere are no energy levels (b) at the required height in the su-perconductor; if V;;;; A]« such levels are present (c).

Magnetization; superconductors of the second kind

,.

Disruption of the superconducting ordering by a mag-neticfield

Up till now we have not dealt with the question ofwhat exactly happens when the strength of an externalmagnetic field in which a superconductor is located isallowed to increase until it exceeds the value He atwhich superconductivity ceases. Neither have we en-quired into the exact situation in the surface layer ofthe superconductor into which the field has penetrated.

If a magnetic field penetrates a superconductor,even if only to the penetration depth, this signifies anattack on the superconducting state. If the field is toohigh, the transition to the normal state takes place but,even where the superconducting state is retained, there

is already an effect, in the form of a local reduction inthe concentration n« of the superconducting electrons,i.e. a reduction in the width 2,1 of the gap in the energydiagram. As we mentioned earlier, an increase in tem-perature also does this and it is known that both effectsact on the penetration depth À. There is therefore aninteraction between I1s and H (or A), which means thatLondon's equation does not hold in the regionspenetrated by the magnetic field. A solution in thisregion can be found from the equations due to Landauand Ginzburg [12]. We shall not deal in detail-with theseequations but simply point out that they are two equa-tions which can be used to find n« and the vector po-tential A as functions of the position co-ordinates, pro-vided that the boundary conditions are not too compli-cated. (Strictly speaking, the equations do not containns, but a complex order parameter which can be iden-tified with the wave function "P mentioned earlier.) Thecoefficients which appear in the equations are charac-teristic of the material. The fact that there are now twoequations leads directly to the possibility of more thanone characteristic penetration distance. And in fact,besides the penetration depth À for the magnetic fieldand the current (already given by London's theory), wenow also find a characteristic penetration depth for thedisturbance in the configurations, i.e. for the variationin I"PI. This characteristic penetration depth is calledthe coherence length, and is indicated by ç. The termcoherence length is perhaps a little confusing as "co-herence" does not relate here to the maintenance ofthe phase coherence of the wave function, which as wehave seen is maintained over a considerable distance,but to the "stiffness" of the wave function, in the sensethat I"PI cannot vary to any great extent within thedistance ç. Like À, ç is generally small: we find valuesfor ç of the order of 10 to 1000 nm.It will appear presently that it is of fundamental

importance for the behaviour of a superconductingbody in a magnetic field whether the penetration depthÀ or the coherence length ç is the greater. More precise-ly, the important question is whether Äjç is greater orsmaller than 1/1/2. This can easily be shown to bereasonable from considerations of the surface energy.We shall therefore begin by discussing the main prin-

[8J R. Doll and M. Näbauer, Phys, Rev. Letters 7, 51, 1961;Ho S. Deaver, Jr. and W. M. Fairbank, Phys. Rev. Letters7,43, 1961.

[9J M. A. Biondi and M. P. Garfunkel, Phys. Rev. 116, 853 and862, 1959.

[lOJ See H. B. G. Casimir and J. Ubbink, The skin effect I,Philips tech. Rev. 28, 271-283, 1967 (No.9), particularlyequation (10).

[l1J 1. Giaever, Phys. Rev. Letters 5, 147, 1960.[12J V. L. Ginzburg and L. D. Landau, Zh. eksper. teor. Fiz,

20, 1064, 1950. Important contributions have been made tothe elaboration of this theory by L. P. Gorkov and A. A.Abrikosov. For complete bibliography see Lynton's book [2J.

10 PHILIPS TECHNICAL REVIEW VOLUME29

ciples of the thermodynamic treatment of the phasetransition superconducting +±: normal.

Thermodynamics of the phase transition superconduct-ing +±: normal

In the thermodynamic treatment of the phase tran-sition superconducting +t normal [13] use is made ofthe Gibbs' free energy G. Let us suppose that this hasthe value Gs(H) in the superconducting phase withthe value Gn(H) in the normal phase. Now there isalways some freedom of choice in the expression forthe thermodynamic quantities when a magnetic fieldis present, as it is a question of taste whether all oronly a part of the field energy is to be assigned to thematerial in which the field exists. The choice that weshall make is such that for an isothermal change:

dG=-MdH.

Here M is the magnetization of the body in the externalfield H, assumed to remain undisturbed. If the field isgradually increased in strength, the field intensity He atwhich the phase transition takes place must satisfy:

He

Gs(He) = Gs(O) - SM dH = Gn(He). (18)o

We now assume that the superconducting body isperfectly diamagnetic, i.e. B is zero (according toMaxwell's theory, B actually has the significanee oftheaverage microscopie magnetic field intensity). Themacroscopie magnetization M due to the screeningcurrent is then equal to -H, since B = H + M. If,further, we assume that the magnetic susceptibility ofthe material in the normal state can be neglected, wecan reduce (18) directly to:

He'

Gn(O) - Gs(O) = S H dH = tHe2. . (19)o

To summarize, in the absence of a magnetic field, thefree energy in the superconducting state is lower thanthat in the normal state; the value of He is determinedby the difference.

The reduction ofthe free energy in the superconduct-ing state with respect to that in the normal state canalso be derived from the change in the energy spectrumof the electrons, in the following way. As we haveseen, in the superconducting state, the electrons in theenergy interval Lt just below the Fermi level EF areaccommodated at lower levels where there is a suffi-ciently high density of states available (fig.3b). IfN(EF) is the density of states at the Fermi level of themetal in the normal state, then we must have:

. . (20)

(17)

Here, a is a coefficient of the order of unity, whichdepends on the precise form of the density of states inthe superconducting state. Ifwe compare (19) with (20),we see that He is proportional to Llo and from (1) thatHe must therefore also be proportional to Te.

When dealing with a phase transition, it is necessaryto take into account the surface energy. In our case,this means the supplementary energy which resultsfrom the presence of a boundary between the super-conducting and normal parts of the metal. Two contri-butions can be distinguished. First, there is a correctionto the magnetic energy term SMdH as a result of thefact that the magnetic field penetrates the supercon-ducting material to a distance A. The field-free volumeis thus slightly smaller than the volume of the super-conductor itself by an amount AS in which S is thearea of the boundary surface and the free energy istherefore smaller by an amount tASH2 than the valuetaken into account above. This correction is thereforenegative.

The second correction, on the other hand, which isa result of the disturbance of configuration at the sur-face, is positive. This correction term varies monotoni-cally with e. the coherence length, and is dominant if ~is sufficiently large. The surface energy is then positiveand the boundary surface will become as small aspossible or disappear entirely. If, on the other hand, ~is relatively small, the magnetic term in the surfaceenergy is dominant. When a magnetic field is applied,the boundary surface between superconducting andnon-superconducting material then attempts to in-creasein area. In a body that is already totally supercon-ducting, this can only happen through the onset of dis-persion: a state arises in which the superconductorcontains a large number of normal regions, whose totalvolume is very small, but whose total area is very large,and in which there is a magnetic field, as in the border-ing penetration layers.

Two kinds of superconductors

The above thermodynamic considerations show thatthe picture of a superconductor given in the previoussection is true only for materials where ~ is relativelylarge. In materials with a small ~, the Meissner effect(i.e. almost perfect diamagnetism) does not occur.According to the theory of Landau, Ginzburg andAbrikosov, the first case is encountered when", = A.f~is smaller than 1/V2, and the other if '" is larger than1/V2. We speak of superconductors of the first kind(with Meissner effect) and superconductors of thesecond kind. The elements lead, tin and mercury belongto the first group and niobium, together with severalalloys, to the second.The superconductors of the second kind are the


ones which have received most attention in the lastfew years and for the moment offer most promise fortechnical applications. In alloys, ~ is usually small asa result of the effect of the mean free path of the elec-trons on the "stiffness" of the collective wave function(or order parameter) '!jJ. Gorkov has derived:

u = Uo + bç",

Here, Uo is the u value of the perfect lattice, i.e. thelattice ofthe appropriate material at T = 0 and with nocrystal imperfections, and b is a coefficient which de-pends on the density of states of the electrons at theFermi level in the normal state. The quantity e" isthe residual resistance, i.e. the value that the resistivityof the material would have in theory at T = 0 if nosuperconductivity occurred. The val ue of e" increaseswith the number ofirregularities in occupancy or otherlattice defects in the metal crystal, which reduce themean free path.

A supercond uctor of the second kind has a magneti-zation curve like the one shown infig. 7. At small valuesof the applied field there is still an effective screening

-/VI/

//

//

//

---H Hc2

Fig.7. Magnetization curve of a superconductor of the secondkind. If the field H is increased, the magnetization M also in-creases at first, but begins to drop at the value Hel. At H = He2,M has become equal to zero and the material reverts to thenormal state.

current, but when the field exceeds a characteristicvalue HCl, flux begins to penetrate. As we noted, thereis then no longer any Meissner effect and we speak ofthemixed state. The flux invasion takes place in the formof millions of very small flux "threads" or vortices.These vortices are very interesting entities, their mainfeatures being (see also fig. 8):1) They can terminate only at a surface, or be closed

on themselves.2) They carry persistent and circular currents flowing

over their entire length as current walls.3) Inside them, and coupled to them, there is a mag-

netic field; current and magnetic field extend to adistance À from the cylinder.

[13J C. J. Garter and H. B. G. Casimir, Physica 1, 306, 1934.This treatment will also be found in the book by Shoen-berg [2J.

(21 )

4) The flux threads have a non-superconducting core,a very thin cylinder of radius r

5) The vortex is in fact an excited state of the persistentring current type (P =1= 0) that has already beendiscussed, and carries only a single flux quantumhl'Ie.

In the mixed state a part of the material - a verylarge part just above Hel - is still in the superconduct-ing state. Because of the flux penetration, this situationcan be retained to a much higher field intensity thanever could be the case if the pure Meissner effect weremaintained. The completely normal state is finallyattained when the concentration of flux threads is sogreat that the entire volume of the material is filledby the normal cores. The value of H at which thishappens is indicated by He2; this value increases withdecreasing ~. In various niobium-containing supercon-ductors, values between 100000 and 200 000 gauss arefound for Hc2.

We mayalso note that for superconductors of thesecond kind, the area beneath the magnetization curve(fig. 7) is equal to the condensation energy (cf. eq. 19).

Fig. 8. Current and field curve in a longitudinal cross-section ofa flux thread. The flux is trapped in a normal zone (of radius ~)which bas a circular current flowing at the wall. The magneticfield penetrates in the normal way into the surrounding materialto a distance À.

Induction effects

In the light of what has just been said, we shouldnow like to take another look at the behaviour of asuperconducting body in amagneticfield, this time allow-ing the body to be multiply-connected and the flux tobe displaced. We shall deal with various cases in thissection.


Let us first take a ring or hollow cylinder of lead orother superconducting materialof the first kind. Whenplaced in a magnetic field, the ring will persist in theinitial state: as long as it remains completely super-conducting, the enclosed flux will not change (e.g. itwill remain zero). In the interior of a cylinder which islong enough, thefield, too, will not change (e.g. it willremain zero). Once the critical field strength is exceededthe field will, of course, be able to penetrate everywherebut now removal of the field will no longer result in theinitial state. A quantity of magnetic flux has now beenenclosed, and this can be supported by a persistentcurrent.A variation on this theme is found in Buckingham's

"persistatrön". A superconducting ring is asymmetri-cally connected to two current wires (fig. 9). If acurrent !, which is high enough to cause the ring torevert to the normal state at some point, is made toflow through the wires, the flux through the ring willbe maintained when the current! is switched off, andsupported by a persistent current in the ring.

I•

Fig.9. Buckingham "persistatron". A current I strong enoughto remove superconductivity locally is passed through part ofa superconducting ring. The magnetic flux contained by the ringduring the flow of I is enclosed when I is switched off.

In an apparently singly-connected body, too, flux can betrapped in such a way that an irreversible magnetization curve isproduced. This will be the case if, for example, because of an un-favourable shape, part of the body becomes superconductingagain when the magnetic field decreases and if that part is doublyconnected. Now, flux can no longer escape from the supercon-ducting ring thus formed and, when the field is switched off,there remains a normal central zone in which there is flux, thuseffectively making the body no longer singly-connected.

We shall now direct our attention towards the pro-cess oî fiux creep. In the discussion ofthe magnetizationof a superconductor of the second kind, we have al-ready referred to the penetration of flux threads deepinto the material. Without going in too great detail in-to the question of how these ar~ released from thesurface - they cannot arise spontaneously within thebody - we shall now, in a very general manner, exam-ine the significanee of the transfer of flux across thesuperconducting current path in terms of electromag-netic induction. Let us therefore consider the situationin fig. lOa. We assume that there is a ring RwithazoneP where the material is not superconducting, this zonebeing of such a size that it does not cut right through

the superconducting ring. We also assume that thisnormal zone P encloses a flux tP. This flux can beshifted, together with the zone, with the aid of a mov-able external auxiliary magnetic field (the" displace-ment field"). The flux ifJ can thus be moved across thesuperconducting ring and, when the edge has beenpassed, the flux is enclosed within the hole in the ring.If the force which has taken the flux inwards is removed(i.e. the auxiliary field switched off), the flux is main-tained by a persistent current in R. We may now con-sider the zone P with the moving flux ifJ as being thesource of a voltage V which has brought about an in-crease I in the persistent current in the ring of in-ductance L:

fVdt = LI = tP. (22)

If there is a regular quasi-continuous transfer of alarge number of such flux tubes, then

V = dtP/dt (23)

is the average flux creep per second. If flux begins tomove in the opposite direction, the current willdecrease.

A very interesting case is the one in which the fluxis set in motion under the influence of a current IR al-ready flowing in the superconductor, which "washesaround" the patch of flux (fig. lOb). As acareful analysisof the total magnetic energy of the system as a functionof a lateral shift of the zone has shown, such a currentexerts a lateral force Fcton the tube of flux. This forceis proportional to ifJ and the current density. The neg-ative induced voltage connected with the movement ofthe flux can be best interpreted in this case as resistancein the circuit. The situation is fully comparable to thatin an ordinary type of d.c. dynamo; this can, ofcourse, be operated as a current generator or as a motoraccording to the direction of energy flow, and when

Fig. 10. a) Diagrammatic representation of a doubly-connectedbody (R) in which there is a non-superconducting zone (P)which does not disturb the doubly-connected nature of the super-conducting region and encloses the flux W. If P is shifted from theouter to the inner edge of R, the flux contained by R can be in-creased by W.b) If the location of P is not fixed and a current IR flows in R,P will move across the direction of flow because the flow exertsa lateral force Fd on the tube of flux. Fd is proportional to Wandto the current density at that point.

J968, No. I SUPERCONDUCTIVITY J3

operated as a motor it appears in the circuit as a posi-ti ve resistance.

The lateral force on a tube of flux is extremely im-portant in all kinds of processes in superconductors. Itis sometimes referred to as a Magnus force, as theeffect shows a certain similarity to the Magnus effectin aerodynamics, but the force exerted on the flux canequally well be regarded as an electrodynamic force asdescribed by Ampère or as an effect of the Lorentz forceon the electrons. It is this force that helps to bringabout the mixed state when a superconductor of thesecond kind is magnetized. When, as soon as H hasbecome greater than Hei, vortices begin to split offfrom the screening current (Meissner current) initiallyexcited at the surface ofthe body, this force ensures thatthese vortices are evenly distributed through thesuperconductor. This follows because with an unevendistribution of the vortices the total current in theinterior is not zero everywhere, as may be seen fromfig. J 1, and they will thus be subject to a restoringforce. If the vortices are evenly distributed, this totalcurrent is zero and no change in the distribution canoccur.

Fig. 11. What happens in a superconductor of the second kind asthe magnetic field H increases, if Hel < H < He2 (mixed state).Vortices containing a certain flux split off from the screeningcurrent at the surface. The free vortices attempt to distributethemselves as evenly as possible over the available space and thusinitially move from the surface to the interior. The movement isproduced through the agency of a force which is the result of thetotal current which flows locally where there is an uneven distri-bution; at the broken line, for instance, the current Rows in fivevortices to the left and only in three to the right.

Superconducting d.c, dynamos

The induction effect discussed above gives the basis for thedesign of superconducting dynamos [14]. These current generatorsconsist essentially of a fixed system of conductors - there aretherefore no brushes or oommutators - in which both the Fara-day induction and the periodic changes in the circuit are producedby an alternating field having the character of a periodic dis-placement or rotation field. The periodic changes in the circuit areeffected by the field periodically bringing certain parts of thesuperconductor into the normal state - in effect this is a kind ofcommutator - thus producing a rectifying or unipolar effect.One of the first designs of the superconducting dynamo, whichis very closely related to the conventional unipolar dynamo, hasalready been discussed in this journal [151. Fig. J 2 gives a quicklook at the family of superconducting dynamos, showing theway in which the flux is displaced. If a certain flux rp is carriedround in the dynamo at a frequency j; the e.m.f. is:

@Ib

Fig. 12. Examples of the four most important types of design ofthe superconducting dynamo for the generation of a persistentcurrent in a superconducting circuit.a) With a rotating field provided by a rotating permanent magnet.b) With a displacement field. c) and d) Variations of (b) with asuperconductor divided into two strips; in (d) the displacementfield is obtained by means of coils.

v= rpf (23')

There are certain attractive features in superconducting dyna-mos. They can, for instance, energize superconducting coilswhere they form closed circuits with these coils, the whole beingkept in the bath of liquid helium. This means that no externalsupply wires are needed and there is therefore no inherent leak-age of heat to the helium bath. As long as the dynamo is in opera-tion, the current in the coil increases with time t :

J = V//L. (24)

Here, L is the inductance of the coil. Once the desired currentintensity has been attained, the dynamo is simply stopped, atwhich it becomes a passive conducting element and the currentremains circulating on its own. A persistent current is far moreconstant than that obtainable even with the best current stabili-zers, and this is particularly attractive for precision work. A greatadvantage is that it is easy to generate very high currents up tothousands of amperes, with a relatively small dynamo. Smallvariations in the current are obtained simply by rotating thedynamo shaft a few turns [16J. The e.m.f. V with simple dynamodesigns like that of jig. 13 is of the order of 1-10 mV. Whensuperconducting cable of very high current-carrying capacity(e.g. 1000 A) has to be used, this generally implies that the in-ductance of the circuit will be reasonably small, and so the timerequired to attain a high current will certainly be acceptable. Adisadvantage of these generators is that they are not entirelyloss-free. This is because the movement of the flux induces a

[141 J. Volger and P. S. Admiraal, Physics Letters 2, 257, 1962.[15J J. Volger, A dynamo for generating a persistent current in a

superconducting circuit, Philips tech. Rev. 25, 16-19, 1963/64.A survey of various possible designs will be found in J. vanSuchtelen, J. Volger and D. van Houwelingen, Cryogenics5, 256, 1965.

[16] An application where use is made of th is property has recent-ly been described in this journal: F. W. Smith, P. L. Boothand E. L. Hent ley, Masers for a radio astronomy interfer-ometer. Philips tech. Rev. 27, 313-321, 1966.

14 PHTLIPS TECHNICAL REVIEW VOLUME 29

small (local) current in the normal zone, and this of coursedevelops heat. In dynamos for use in large installations, specialmeasures may have to be taken to get over this difficulty. How-ever, the dissipation of perhaps a few watts which occurs indynamos for energizing laboratory-type superconducting coilswill never be prohibitive.

Fig. 13. A superconducting magnet coil (below) complete withsuperconducting dynamo (above).

Response of a superconductor of the second kind to thepassage of a current

As discussed in one of the previous sections, acurrent flowing in a superconductor exerts a lateralforce on a tube of flux passing through it, and thesame applies to the elementary flux threads in thesuperconductor of the second kind when in themixed state. If there are no opposing forces, i.e. if theflux threads are not fixed at some "anchor point",they will be set in motion. We then have the case ofthe macroscopically observable voltage deterrnined byinduction effects, i.e. the case where the supercond uctorexhibits a certain resistance. In agreement with the

foregoing, we obtain for the voltage drop per unitlength:

E= uB. (25)

Here, u is the velocity of the transverse movement ofthe flux threads and B the flux density they produce inthe specimen.

There are one or two very convincing experimentalindications of the reality of this flux movement in su-perconductors. Let us note first the measurements ofthe noise component of E in vanadium which weremade by Van Ooijen and Van Gurp [17] at PhilipsResearch Laboratories. These workers found a cut-offfrequency in the noise spectrum which was equal to thereci procal val ue of the crossing time of the flux threadscalculated from (25) and the width of the measuringplate. A spectrum ofthis type is characteristic ofa shot-noise effect, which would be expected here on accountof the discrete values of the moving flux. The magni-tude of the tubes of flux can be derived from the inten-sity of the noise signal. Calculation shows that thismagnitude must depend upon the value of the primarycurrent, but in the limit of an extremely high primarycurrent, the flux crosses in separate elementary quantawithout the formation of groups.

A second and very direct experimental proof hasbeen given by Giaever [18]. He has examined themovement of the flux by bringing two very thin metalplates very close to each other. There was no metalliccontact, but the magnetic patterns of the mixed statein both plates were coupled. If a current was then madeto flow in one of the plates, it was found that a voltagewas generated not only in this plate but also in the otherone. The only way in which this voltage could havebeen excited was by the magnetic coupling between thetwo plates. This proves that the voltages must be dueto the movement of the flux pattern.

The movement of flux has recently been made visible in ex-periments with very thin lead foil at Philips Research Laborato-ries [191. A state of dispersion can in fact also occur in super-conductors of the first kind, to which lead belongs, and in thiscase the dispersion is associated with the fact that the internalfield strength in a body is in general not uniform, because of de-magnetization [20]. This means that when the external field in-creases, not all the parts of the body change to the superconduct-ing state at the same time. When the body is partly in the normalstate and partly in the superconducting state - this situation isknown as the interrnediate state - the normal regions carry aflux which is much greater than one flux quantum. However,in the movement of flux under the inAuence of a current theseregions are very similar to the flux threads of the mixed state.Their movement has been made visible with the aid of smallgrains of niobium: when the bundles of flux move through thelead foil, the niobium grains follow them. This has been 01;>-served and recorded with the aid of a specially designed micro-scope and a closed TV circuit (fig. 14).

1968, No. 1 SUPERCONDUCTIViTY 15

Fig. 14. Experirnental arrangement for visual demonstration of the movement of magnetic flux in leadin the intermediate state. Inside the helium cryostat, which can be seen in the middle of the picture,there is a special microscope, focused onto a lead foil which has been coated with niobium powder andwhich is placed in a variable magnetic field. (This field is provided by a superconducting coil.) A tele-vision camera is mounted above the cryostat, and part of the illuminating system of the microscope canbe seen just to the left of the operator's hand. In the left background there is a television recorder, usedfor recording the movement of the niobium grains. The dark patches on the TV screen are niobium grainsor small groups of niobium grains (magnified 10000 diameters). The crazing in the lighter areas of thepattern is due la a lacquer coating applied to the lead to prevent oxidation.

The resistance in the mixed state

The voltage drop in a speci men of a supercond uctorof the second kind in the mixed state with a currentflowing in it is thus deter mined by the rate of fluxdrift. The question of the factors determining this ratehas been examined in detail in the last two years. Thepicture that we now have, which to some extent hasbeen derived from the work at Philips Research Lab-oratories [21l, is rather as follows. The flux threads areaffected by a driving force with a value Fd per unitlength, given by:

Fd = lc[J.

In the stationary state, the flux threads have a constantvelocity, i.e. there is a frictional force Fr per unit

length which exactly compensates Fd. As we mentionedabove, the force Fd can be regarded as an electrodynam-ic force as described by Ampère. This also applies tothe frictional force, at least where the braking is causedby the eddy currents generated by and at the movingflux threads. Looked at locally, dB/df =F 0 because ofthe movement of the flux thread, and there is thus a ro-tary electric field generating small eddy currents. It is

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[17J D. J. van Ooijen and G. J. van Gurp, Physics Letters 17,230, 1965.

[18J I. Giaever, Phys. Rev. Letters 15, 825,1965 and 16,460,1966.[19] J. van Suchtelen and A. P. Severijns, not published.[20] See for example the books by London and Lynton quoted

under [2].

[21] F"~,, survey of this with bibliography see the article by J.Volger in Quantum Fluids, Proc. Sussex Univ. Syrnp.1965, North Holland Publ. Co., Amsterdam 1966, pp.128-135.


these eddy currents, which also pass through thenormal core of every flux thread, which provide thefrictional force and in fact lead to energy dissipationmentioned earlier. If a flux thread is held fixed (by.whatever mechanism), then the current will pass com-pletely outside its normal core but, if the flux thread ismoving, the current "corrected" by the eddy currentswill pass through the vortex core. This picture thusimmediately makes it clear that the normal or residualresistance of the material determines the resistance be-haviour in the mixed state. These ideas have beenexamined theoretically by various authors, who havesuccessfully demonstrated a connection with Kim'sempirical relation:

em = eB/Hc2.

Here em is the resistivity in the mixed state and ethe resistivity in the normal state. The flux density Bis the average internal magnetic field intensity in thespecimen; B/Hc2 is nothng more than the fraction ofthe volume taken up by the normal cores of thevortices.

Yet another interesting consequenceof this theory is the exis-tence of a Hall effect: the magnetic field in the vortex core initiatesa Hall effect there. Because of this, the local current loops men-tioned, which cause the frictional force, are rotated slightly withrespect to their electric field through the influence of the magneticfield. This rotation is in fact through an angle which must be ofthe order of magnitude of the Hall angle that must apply in thecore region. We do not propose here to discuss the difficultieswhich have to be tackled in an adequate treatment of thisproblem. The idea of a rotation in the frictional force Fr withrespect to -11 (cf. 25) through such an angle is however undoubt-edly correct. It leads to a deviation in the flux drift with respect tothe ideal lateral movement and therefore to a transverse voltagewith all the symmetry properties of the Hall voltage, which cansimply be referred to as the Hall voltage of the superconductorin the mixed state [211. This Hall voltage has been observed inthese laboratories in samples of NbTa etc., by Niessen andStaas [221. Their observations, however, and those published later.by other authors, show that a good quantitative agreement be-tween theory and experiment has by no means yet been achieved.

Since a normal flux vortex core represents a certain quantityof entropy, it is to be expected that thermal effects will also beconnected with the flux movement. These have indeed been foundand superconductors of the second kind, in the mixed state,also exhibit the whole family of thermo-galvanomagnetic effectsof the second order, i.e., the Nernst, Ettingshausen and Righi-le Due effects, besides the Hall effect.

[221 A. K. Niessen and F. A. Staas, Physics Letters 15, 26, 1965.[231 See for example: G. J. van Gurp and D. J. van Ooijen,

J. Physique 27, C3-51, 1966.

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"Hard" superconductorsThe superconducting materials which for a few

years now have been attracting a great deal of attentionbecause of possible applications belong basically tothe group of superconductors of the second kind .They differ however from the materials discussedabove in that the resistance remains zero right up tovery high values of the external magnetic field strengthor up to very high currents. This extremely favourableproperty is a result of the fact that, in such materials,the flux threads are held tightly in place and cannot,therefore, move at all in the first instance. They can beheld fast by all kinds of structural faults, like disloca-tions, crystal boundaries, precipitations of a secondphase, etc. [23].As yet there is no proper quantitativeunderstanding of the mechanism by which the fluxthreads are held in place.

Superconductors of the second kind belonging tothis group are aptly enough known as "hard" super-conductors - most of them are indeed mechanicallyhard due to the presence of many crystal imperfections- or as superconductors of the third kind. Examplesof these are NbZr and other niobium alloys and theintermetallic compound Nb3Sn, which is used formaking cables for superconducting magnet coils.The properties of these materials will be discussed inthe article by A. L. Luiten on superconducting mag-nets mentioned under [4], which will appear shortly inthis journal.

Summary. Superconductivity, discovered in 1911 by KamerlinghOnnes and Holst, remained for half a century a phenomenonwhich was not satisfactorily explained and which could not beused in any technical application. In the last ten years the situa-tion has changed drastically in both respects. It is now possibleto construct superconducting magnet coils for fields of 100 kilo-gauss or more; other technical applications are under considera-tion. A theoretical understanding of superconductivity has beenobtained by treating it as a "condensation" of conductionelectrons into pairs (Bardeen, Cooper and Schrieffer). The con-densate assumes a macroscopie quantum state which cannotalter to an energetically lower state. There is therefore no de-velopment of heat and no resistance. If the temperature or anexternal magnetic field is increased, then there is an instant atwhich the superconducting state disappears. The theory of Lan-dau and Ginzburg has made clear why superconductors have tobe divided into two groups with regard to their behaviour in anexternal magnetic field: one group in which the interior of thesuperconductor remains field-free and another in which thefield can penetrate the superconductor in the form of billions offlux threads if the field strength exceeds a certain value. Mostmembers of this group exhibit a certain resistance to the passageof a current (superconductors of the second kind) while othersdo not do so and are capable of carrying very heavy currents(third kind). The superconducting materials of the third kind areused in the construction of superconducting magnet coils.