SUMMARY

This article introduces superconductivityand BCS theory to the general public and toundergraduate physics students.

BY ANDRÉ-MARIE S. TREMBLAY

SUPERCONDUCTIVITY IN A NUTSHELL

A-M.S. Tremblay<Andre-Marie.Tremblay@USherbrooke.ca>,Département dephysique and RQMP,Université deSherbrooke,Sherbrooke, QC,J1K 2R1

and Member,Quantum MaterialsProgram, CanadianInstitute for AdvancedResearch, Toronto,ON, M5G 1Z8

made of frozen water (ice), somebody will be able to feelvery far away, on the other end of the table, that you aretrying to rock it. This is analogous to a current thatpropagates over an arbitrarily large distance withoutdissipation. Therefore, to explain the difference betweenmetals and superconductors, we should first attempt tounderstand the difference between liquids, like water, andsolids, like ice. Ordinary matter is composed of atomswhich are essentially small particles that attract each otherwhen they are a little distance apart, but repel whensqueezed together. (Feynmann once said that this phrasecontains the largest amount of scientific information withthe least words.) Atoms therefore like to sit at distanceswhere the attractive and repulsive forces are balanced. Atlow temperatures, this is what occurs in a solid. The atomsarrange themselves in a regular way and cooperate to forma rigid object, what we call a solid. Rigidity is theproperty that allows us to feel the hand hitting the table.However, increasing temperature causes the atoms tovibrate more and more. When this atomic agitationreaches a threshold level, collaboration betweenindividual atoms becomes impossible and the rigiditydisappears: the solid has melted and becomes liquid. In anormal metal, the particles that conduct electricity(electrons) move independently of one another and actmuch like water molecules in their liquid phase. Theydon't cooperate and the transport of electricity isenergetically expensive. On the other hand, in thesuperconducting state that appears at low temperatures,electrons cooperate with one another and gain a kind ofglobal rigidity, which permits electricity to be transportedeasily.” Your uncle finds himself very pleased with youranswer to his question.

SUPERCONDUCTIVITY FORUNDERGRADUATE PHYSICS STUDENTSLet's now suppose that you have been taking a firststatistical mechanics class and a first quantum mechanicsclass and that you know more about physics than youruncle. How can superconductivity be simply understood?This phenomenon was first explained in 1957, some46 years after the experimental discovery that wecelebrate this year. The theory takes the names of itsfounders, Bardeen, Cooper and Schrieffer, or BCS.

The first concept to understand is that which produces therigidity in the example above. However, solidification isnot the simplest phenomenon to understand. Instead, let'smake an analogy with a magnet. Consider a crystal lattice

PHYSICS EDUCATION

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UPERCONDUCTIVITY FOR ALLYour uncle is very proud of you, a Physics student.He has read that some metals become

superconductor at very low temperature, and he learnedthat the discovery of a room temperature superconductorwould give rise to a technological revolution. Naturally, heasks you to explain exactly what is superconductivity.Your short answer is that a superconductor allowselectricity to flow without resistance. In other words, anelectrical current flowing through a closed loop ofsuperconducting wire will continue to flow foreverwithout any need to pay a power bill. You also explain thatcurrent loops are quite useful for technologies such asmagnets and motors. And of course, the capacity totransport electricity from hydro-electric dams in the farNorth to southern cities without any energy loss is inprinciple a good idea. Already, you convinced your uncleof the usefulness and huge economic potential ofsuperconductors. Putting the nail in the coffin, you couldalso explain to him that superconductors can be used toconstruct quantum computers. That's a challenge! Youwould probably refer him to the article written byAlexandre Blais elsewhere in this issue.

Up until now everything is fine. More and more interested,your uncle now asks how it is possible for electricity toflow in a wire without heating it up, or in other words,why do some metals become superconductors. Right therethings get more complicated. You are wondering how togive a short explanation that would not be entirely false.Perhaps the best response would proceed by analogy.

“Let's suppose we splash the palm of our hand into a poolof water, like a karate blow but in very slow motion. Smallwaves will be generated on the surface, but they will bequickly absorbed and our hand will completely penetratethe water. Somebody situated at the far end of the pool willnot feel any effect. For a metal, splashing our hand in thepool is analogous to applying a voltage for a moment andobserving the dissipation of the electrical current, just likethe small waves. Now, if we move our hand in much thesame way against the end of a solid object, such as a table

S

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of spins that can only point up or down. The energy associatedwith a given spin configuration can be described by

(1)

where Ji,j is a positive number that is non-zero if the spins arenot too far apart and Si = ±1. In a disordered phase, namelynon-magnetized, spins point randomly up or down. Bycontrast, in a magnetized phase, a large fraction of the spinswill point in the same direction. More precisely, the thermalaverage +Sj, will be non-zero. It is then reasonable to make thefollowing approximation

(2)

The problem is now one of independent spins in the effectivemagnetic field due to all the other spins. It is easy to computethe free energy of this system with the usual methods ofstatistical mechanics. We find that at high temperatures, theentropy wins and the spins are disordered. At low temperatures,the energy wins and the spins are ordered. We also say that thespins break the symmetry since there is now a preferred spindirection where there was none initially. The spin system hasbecome ‘rigid’ in its orientation. From one spin to itsneighbours, to the neighbours of those and so on, spins assumethe same direction with respect to one another. Whatdetermines this global spin direction is any tiny externalmagnetic field that is present by chance.

In the transition from normal to superconducting phase, it is theCooper pairs that play the role of Si. A Cooper pair is composedof two electrons with correlated motions. How is it produced?Electrons, negatively charged, interact with the lattice ofpositive ions. A given electron will attract the ion latticetowards itself. As the electron moves more rapidly than theions, it will leave an excess of positive charge in its wake,which will attract another electron. Therefore, lattice vibrationsare mediating an attractive force between two electrons.Because of the time-lag, also called retardation, the pairedelectrons are never at the same place at the same time, and thedirect repulsive force between them is balanced by the delayedattractive force of the ions. This phenomenon is illustrated inFig. 1. This is where your knowledge of quantum mechanicscomes into play. Superconductivity is a fundamentallyquantum phenomenon. The wave function of the Cooper pair,ψp8,-p9, which has total momentum equal to zero, plays the roleof Si.

Now how can we create a global rigidity? This is the secondingredient that gives rise to superconductivity. We have to findthe expression for the energy. Roughly speaking, an electronwith momentum p and spin up and another with momentum -pand spin down will be scattered by crystal lattice vibrations andend up in states pNN up and -pNN down, which conserves linearmomentum of the center of mass and the total angularmomentum. The potential energy can be written as

(3)

Here Up-pNN is a negative number, whose p - pNN dependence canbe neglected to leading order. It represents the energy gain dueto the attraction caused by the lattice vibrations. The time-lagshows up in this model only in the fact that we allow Up-pNN tobe negative. The complex conjugate comes from the fact thatψp8,-p9 is a complex number while the energy is a real number.(When computing the transition matrix element, the final stateis the complex conjugate.) Kinetic energy must also be takeninto account in the computation. To minimize the potentialenergy, we must make sure that the phase θ defined by ψp8,-p9 = ei2θ*ψp8,-p9* is independent of p. Otherwise, wewould add numbers with random signs and energy would notbe minimized. The superconducting state is called a coherentstate. It is the analogue for matter, of the laser for light. All thepairs have precisely the same phase.

Fig. 1 Schematic illustration of the mechanism of formation of aCooper pair. Positive ions are in blue. The dotted circleindicates the region where the lattice is polarized (in (b)and (c)). Red lines represent a wave packet for one electronamong all others that ensure electro-neutrality. (a) Twoelectrons of opposite momentum move toward each other.(b) One of the electrons polarizes the medium within thedotted circle and scatters. (c) The second electron isattracted by the excess of positive charge and also scatters.(d) Both electrons move apart with opposite momentum.It would be more accurate to represent the lattice vibrationsas a wave whose inverse wave length scales as p - pNN. Thiskind of scattering occurs everywhere in the crystal andresults in the creation of Cooper pairs below the transitiontemperature.

E J S Si j

i j i j=−,

,

E J S S S Si j

i j i j i j=− +( ),

, .Σ

Σ

E UP =′

− ′ − ′ − ′p p

p p p p p p,

, ,* .ψ ψΣ 9 988

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Since ψp8,-p9 is the order parameter, analogous to Si in thepreceding example, the potential energy in the superconductingstate can be written

(4)

and we obtain a problem similar to the one with the magnet. Itis a bit more complicated because we must also take intoaccount the kinetic energy and Pauli principle when computingthe free energy, but the main idea remains the same. Bit by bitin momentum space, the phases of all pairs order themselvesand become identical. This is again analogous to spins allpointing in the same direction and so is the analog of rigidity.

The relative phases of two pieces of superconductor is verywell defined, and in quantum mechanics it is associated withthe conservation of the number of particles, as we can see fromwriting the expression for the current: a phase gradient leads toa variation in time of the charge density. Therefore, if we puttwo pieces of superconductor in contact, a current is generatedto exchange particles and eventually equalize the phases of thetwo pieces. This occurs even when the contact is throughtunnelling. This is the Josephson effect, which is discussed byPatrick Fournier and Alexandre Blais in this issue. You havealready understood where the rigidity is coming from and itsconsequences. If the phase is changed very gradually at oneend of the material, it will tend to equalize elsewhere and acurrent will circulate.

The choice of phase for a pair, which will be the same for allpairs, is analogous to the choice of direction for the spins. Thefact that all phases are identical is imposed by the energy. Howis a particular phase chosen? There again, we can imagine anappropriate infinitesimal external field acting on the system.But let's be more specific concerning the broken symmetry.Normally, the system conserves the number of particles. Let*N , be that state of the system. When adding two particles, thatstate can be multiplied by ei2θ with any θ to obtain ei2θ*N + 2,.This factor ei2θ will disappear for all expectation values+N + 2*O*N + 2,. But in fact, the BCS state does not conservethe total number of particles, even if charge is conservedlocally. The BCS state is a linear combination of stateswith different numbers of particles: *BCS(θ), = ...+eiNθ*N,+ ei(N+2)θ*N + 2, + ..... Otherwise, +ψp8,-p9, vanishes. In asuperconductor, the number of particles must not be fixed if wewish to know the phase θ. For a fixed number of particles, thephase remains unknown since *N, = I2π

0 e-iNθ*BCS(θ),. Phaseand number of particles are conjugate variables obeying anuncertainty relation. In practice, for a macroscopic system,ignoring the precise number of particles is not a problem, inparticular if some can be exchanged with a neighbouringsystem. Furthermore, with a relatively small uncertainty on thenumber of particles, the phase can be known rather well. Evenif the BCS state does not conserve the number of particles, it

also very surprising to realize that in our spin problem, wherethe system is rotationally invariant, the ground state chooses aparticular direction. What is important in both cases is therigidity, namely, the fact that correlations are long ranged.Broken symmetry states are convenient for representing thisphysical result.

There are a few other interesting results of the BCS theory thatwe can explain in a few words. First, the wave function of theCooper pair for an interaction mediated by the lattice vibrationsis of type s. We borrow from the terminology of the hydrogenatom: there is no preferred direction since at large wavelengths, vibrations are insensitive to details of the lattice. Theexistence of Cooper pairs can also be observed experimentallyby a forbidden range of energy, i.e. an energy gap. Absorptionof a photon at zero temperature requires the minimal energythat is needed to break a pair. This energy ∆ is of courseproportional to the critical temperature Tc below whichsuperconductivity exists. For a temperature larger than Tc,thermal excitations (entropy) are sufficient to destroy allCooper pairs and therefore superconductivity itself. BCStheory predicts 2∆/kBTc = 3.53... where kB is the Boltzmannconstant. Tom Timusk discusses the energy gap in this issue.Typical Tc values in conventional superconductors vary from4.2 K for mercury, as Kammerling Onnes discovered 100 yearsago, up to 39 K for MgB2 for example.

Another spectacular property of superconductors is revealed ina magnetic field. Surface currents develop to prevent themagnetic field from penetrating inside the volume of thesuperconductor. The distance over which surface currents existis called the penetration depth. The expulsion of the magneticfield has a cost in magnetic energy. When this exceeds the freeenergy associated with the creation of a Cooper pair, thematerial becomes normal. The field at which this happens iscalled Hc1. In many superconductors, the penetration depth λ islarger than the distance ξ needed to restore superconductivity ifit is destroyed at one location. The latter distance is called thecorrelation length. When λ > ξ/ , the superconductor iscalled type-II, while the other case, described above, is type-I.In a type-II, the magnetic field penetrates through vorticeswhose core of radius ξ are not superconducting. The article byDavis & Stamp elaborates on this phenomenon. The magneticflux in a vortex is quantized. This is required for the wavefunction of the pair to have an unique value everywhere outsidethe vortex. A larger applied magnetic field then leads to a largerdensity of vortices, and when vortices are a distance of order ξapart, superconductivity vanishes. This is the definition of thecritical field Hc2, the more relevant one for applications. Thiscritical field can be huge for high temperature superconductors.

The addition of coupling to the electromagnetic field raises thequestion of gauge invariance and changes the notion of orderparameter, as discussed by Catherine Kallin in this issue.Another consequence is the Higgs-Anderson phenomenon,related to elementary particles physics. This last subject isdiscussed by David Sénéchal in this issue.

E UP = (

+ )′

− ′ − ′ − ′

− ′ − ′

p pp p p p p p

p p p p

,, ,

*

, ,*

ψ ψ

ψ ψ

2

Σ 9 988

9 988

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UNCONVENTIONAL SUPERCONDUCTORSSuperconductors based, for example, on organic compounds,on iron, strontium, uranium, or on copper, can sometimes havehigh transition temperatures, like iron-based compounds, 55 K,or copper based ones, 164 K. In addition, these are special inthat some aspects of BCS theory fail. Kirill Samokhindiscusses those “unconventional” superconductors. See alsothe discussion of organic conductors (Claude Bourbonnais) andiron-based compounds (Johnpierre Paglione). The ideas ofCooper pairs and phase coherence, the basis of the theory,survive in these compounds. It is a property that Anderson calls“emerging”, or in other words, one that characterizes the kindof order or rigidity, but does not uniquely determine itsphysical microscopic origins.

There are several indications that the physics of these“unconventional” superconductors is different. For example,lattice vibrations do not explain the Cooper pair formation.Indeed, a pair can involve electrons of the same spin instead ofopposite spin and/or have a wave function that depends on themomentum direction with respect to the lattice axis. For many“unconventional” superconductors, we can build a simplisticidea of the pairing mechanism, in the presence of a strongCoulomb repulsion, in a state where the wave function ofthe pair changes sign when rotating the crystal. A priori, if Up-pNN > 0, there is no potential energy advantage to choosingidentical phases. We can circumvent this problem in thefollowing way. Suppose Up-pNN is large for the right value of p - pNN. Then, if +ψp8,-p9, changes sign under rotation of a givenangle consistent with the lattice symmetry, there is an energygain because the product Up-pNNψp8, -p9ψ

∗pNN8, -pNN9 is negative even

if forces are repulsive. The potential Up-pNN can be more

important for some value of p - pNN because this wave vector isrelated to that of magnetic order dominating the system.Magnetic properties are crucial for unconventionalsuperconductors. They are discussed by Buyers-Yamani andSonier-Luke in this issue. In strontium ruthenate, the symmetryof the order parameter is type p and so it changes sign under180o rotation. In high temperature superconductors, a simplemodel suggests that electrons scatter on antiferromagneticfluctuations and so the symmetry of Cooper pairs is naturallytype d i.e. it changes sign under 90o rotation. But this time, thepair wave function is written ei2θdp8,-p9 where dp8,-p9 is a realfunction that can depend on the orientation of p, whereas θ iscompletely independent of p.

BCS theory and its generalization that includes time-lageffects, the Eliashberg theory discussed by Jules Carbotte andFrank Marsiglio in this issue, lead to quantitative explanationsof experiments on conventional superconductors. However,quantitative methods have only begun to be developed forunconventional superconductors. They have not yet attainedthe many successes of BCS theory, but indeed are heading inthis direction. They must take into account some effects whichI have not discussed here, such as Mott physics and criticality.They are described in many of the articles found in this issueof Physics in Canada, including those of Carbotte, Nicol,Paramekanti, Taillefer, and myself.

ACKNOWLEDGEMENTSI thank David Sénéchal, Alexandre Day and Guylaine Séguinfor judicious comments on the manuscript and FrancisLaliberté and Paul Calvert for help with the English translationof the original French.

BIBLIOGRAPHY1. Failed theories of superconductivity by Jörg Schmalian, in Bardeen Cooper and Schrieffer: 50 YEARS, edited by Leon N. Cooper and

Dmitri Feldman, arXiv:1008.0447.2. “Emergence, reductionism and the seamless web, when and why is science right”, P.W. Anderson, Current Science, 78,1 (2000).3. The path of no resistance: The story of the revolution in Superconductivity, Bruce Schechter, Simon and Schuster, NY (1989).4. On the web at http://www.superconductivity.eu/en/index.php

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