macroscopic superconductivity

Part 6

Macroscopic superconductivity

531

533

Lecture 6.0

Overview of superconductivity

Parts 6, 7, and 8 of these Lectures are all about superconductivity. Whereas in mag-netism we progressed from the microscopic to the macroscopic, here we shall do thereverse: it would be hard to motivate the quantities computed in Part 7 without ageneral concept of superconducting behavior. 1

The fundamental property of superconductivity – or of superfluidity in general – isnot the strange transport behavior, but a strange sort of long-range order: the symmetrybreaking involves the quantum phase of a sort of wavefunction, even though (accordingto basic quantum mechanics) the value of such a phase never affects any physical ob-servables. The systems affected are quantum fluids: they escape having a solid groundstate because the particle masses are so small and the interpacticle interactions are soweak that quantum fluctuations would destroy it; this is realized (on earth) only inhelium, or the dense electron gas in metals. If those superfluids happen to be charged,they are superconductors.

Electricity and magnetism are irrelevant to the origin of superconductivity, just asthey were to (most) magnetism: as in that case, superfluidity is a consequence of thethe quantum statistics and the interparticle potentials. But once superconductivity isgiven, electromagnetism does play a more central role than in magnetic phenomenology.(Half the explanation is that the energy scale of superconductivity is smaller by a factorof up to 1000; the other half, I think, is the direct relationship between the supercurrentand the order parameter gradients.) Still, in presenting the theory of superconductivity,I will begin with the special case of a neutral superfluid, and subsequently add on themagnetic field terms.

History of superconductivity

Casimir’s marvel:

"A mile of dirty lead wire" --

Perfect transmission.

Many texts recount the same story of the stages of understanding superconductivity;it has a certain romance, because theory took some 40 years to catch up, and thisbackground helps explain why the “BCS” theory of superconductivity was the highpoint of 20th-century condensed matter theory. Briefly, the phenomenon was initiallydiscovered by Kamerlingh Onnes in 1911, who pioneered the helium techniques needed

1However, the notes are designed so you can go directly from this overview to Part 7, and return toPart 6 after that if desired.

Copyright c©2010 Christopher L. Henley

534 LECTURE 6.0. OVERVIEW OF SUPERCONDUCTIVITY

to reach 1K. Soona after, he found superconductivity was destroyed by a not-so-largemagnetic field, temporarily dashing hopes of building super electromagnets (they wererealized decades later with help of Type II superconductors, Lec. 6.6 ). In 1933, theMeissner effect – the (reversible) expulsion of flux upon entering the superconductingstate – proved this state was a thermodynamic phase. Meanwhile, in 1937 superfluiditywas discovered in 4He and was understood over the next ∼ 15 years; this provided manyhints about superconductivity, which were developed independently during the 1950s inthe Soviet Union and in the West.

The Ginzburg-Landau (GL) theory of 1950 gave a complete description of a su-perconductor from a continuum viewpoint, in the same sense that elastic theory fullycharacterizes a crystalline solid: the majority of experimental phenomena (especiallythose involving spatial dependences) can be understood in terms of GL theory: this un-derlies every lecture in Part 6. Meanwhile, the Bardeen, Cooper, and Schrieffer (BCS)pairing theory of 1957 explained why a metal goes superconducting, and how the valuesof the GL parameters are determined from microscopics; this is the basis for the lecturesin Part 7. After intense activity in the 1960s, the subject became almost dormant in the1970s – except for the discovery of superfluid helium 3, which (as anticipated) has BCSpairing but with different symmetries and origins than in the original superconductingmetals.

In the original superconductors, the electron pairs had the simplest possible sym-metries, and the mechanism was due to phonons, giving a traditional upper boundTc < 30K. Starting in the 1980s, several families of superconductors were discovered withhigher Tc’s, unusual pair symmetries, and/or non-standard mechanisms: they typicallyhave complex crystal structures and are quite poor metals in their non-superconductingstate. These exotic superconductors – not fully understood – will be covered in Part 8.At the close of the 20th century, the achievement of Bose condensation (and later paircondensation) in cold dilute gases opened up yet another avenue to realize superfluidorder.

Overview of lectures on superconductivity

Part 6 will be all about the macroscopic theory, in particular the Ginzburg-Landautheory. This is valid for all known and all imaginable superconductors, in the same waythat elasticity theory is valid for all solids (indeed, that is really the definition of a solid.)Most of practical superconductivity can be understood at this level: the expulsion ofmagnetic field, inhomogeneous states, flux quantization, Josephson junctions, Type IIsuperconductors (and their phase transitions), and flux-flow resistivity. Most of thesephenomena entail some kind of spatial variation, and so the key parameters in G-Ltheory will be two length scales, the “coherence length” ξ and the “penetration depth”λ. (These are defined in Lec. 6.2 [om. 2007])

The macroscopic level of description divides into two sublevels: somewhat moremacroscopic than the Ginzburg-Landau description is the London description. Thisamounts to assuming ξ → 0, so the superconducting order parameter magnitude hasits maximum magnitude everywhere – only the phase angle (or equivalently, the cur-rent density) are retained from the order-parameter field. London’s original approachgot amazingly far without any microscopic understanding, but is mainly of historicalinterest; we will work from Ginzburg-Landau theory from the start.

[The complex order parameter is the key element in defining superconducting or-der. In particular, the excitation gap is not essential since it is sometimes zero in asuperconducting state.

6.0 A. PHENOMENOLOGY 535

Part 6 is being condensed in 2007-2009: we will skip to the Josephson junction.Part 7 concerns the BCS pairing theory, needed for a deeper understanding of super-

conductivity. The BCS pairing wavefunction provides an explicit picture of the natureof the order in a superconductor. [There exist, in principle, forms of superconductivitywhich do not explicitly involve pairing, but all known forms do seem to have pairs,and are modeled by extensions of the BCS approach.] Given a microscopic Hamilto-nian incorporating some sort of attractive pair interaction, the pairing theory also givesquantitative parameters, in particular those that enter the Ginzburg-Landau theory.

In addition, the microscopic theory yields a dispersion relation for the fermion exci-tations (called “quasiparticles”), which dominate the low-temperature specific heat. Itfurthermore it makes specific predictions about dynamical measurements involving thequasiparticles. These are valuable diagnostics of the precise nature of the ordered state.

Finally, Part 8 concerns the microscopic mechanisms of pair attraction. There arein fact several exotic classes of superconductor besides the traditional elemental metalsand alloys, ranging from helium 3 to conducting polymers to nuclear matter. Of course,the most studied of these are the cuprate high-temperature superconductors. [Better,Part 8 is about “exotic” (non-s-wave) superconductors. For that reason, I expect toswitch the 3He lecture from part 7 to 8, and move the lecture on phonon-mediatedmicroscopics from 8 to 7.

I’ll tend to downplay the role of electricity and magnetism (but not ignore it, sinceE&M is central to many of the phenomena!), as well as that of thermodynamics. In partthis is because I’ll highlight the analogies to neutral superfluids, and in part because I’llemphasize the long range order.

References

The classic texts are (1) Tinkham’s book, with sometimes fussy experimental details(2) Schrieffer’s book for theorists; and (3) de Gennes, Superconductivity of Metals and

Alloys. The bible of the field is the Parks Superconductivity, volumes 1 and 2, whichcollect review articles on every aspect of superconductivity known in the 1960s.

6.0 A Phenomenology

The following are basic observations on superconductivity which will be explained invarious later sections. 2

1. Meissner effect and critical field

The phase diagram in Fig. 1(a) is a good orientation to the properties of super-conductors. The phase that excludes all flux is called the Meissner phase from theMeissner effect (see below). This is diamagnetism on a massive scale, χ = −1/4π, largecompared to the typical values of order 10−5 (see Lec. 4.0 ). Often this is put intothe formalism of macroscopic Maxwell equations: one ascribes a bulk magnetizationto the superconductor, just big enough to cancel the applied field. That is a sensibleway to report experiments which measure net magnetization. Furthermore, when weaddress thermodynamics (e.g. coexistence of normal and Meissner phases), this formu-lation permits us to skip the thought work of rederiving the Legendre transformationidea. However, it is unnatural to describe flux exclusion as the (cancelling) sum of twononzero field terms inside the bulk: it’s more economical to say the superconductor is a

2I apologize that this story owes too much to Ashcroft and Mermin, pp. 726-736.


cT

ρN

ρS

(T)ρ

T

Cel(T)

CN

CS

cTT

Fcond

FN

FS

cT

F(T)

T

cT

Hc2

Hc1

H

T

N

S

Hc

(a). (b). (c). (d).

Figure 1: Characteristic phenomena in a superconductor. (a). Phase diagram (Type I, solidlines) as a function of temperature and applied field. “N” and “S” denote the normal andsuperconducting phases. The double line indicates a first-order (discontinuous) transition atHc(T ); the dot marks the critical point transition at H = 0. The dot-dashed curves Hc1(T )and Hc2(T ) are the lower and upper critical fields (continuous transitions) in case the samematerial is converted into a Type II superconductor by introducing disorder (see Lec. 6.2 [om.2007]). (b). Resistivity (showing the jump to zero). (c). Specific heat (electronic). The solidcurves in (b) and (c) are observed along the T axis of (a); the dashed (normal state) datamight be measured with the cooling history represented by the horizontal dotted line in (a),at a field large enough to destroy superconductivity but not so large as to change the Fermisea appreciably. (d). Free energy obtained by integrating (c).

nonlinear medium which constrains the field to be zero in its interior. Hence I will stickwith the microscopic Maxwell equations, which means treating explicitly the screeningcurrents near the surface of the superconducting region.

If we increase magnetic field, at the critical field Hc the superconducting state goesunstable: there is a phase transition to the normal state. Notice that I write H for theapplied field. The exact statement is that the superconducting phase is in equilibriumwith an adjoining normal phase with local field Hc; since field lines are concentrateddepending on the geometry, the actual H at which superconductivity is destroyed (andthe it goes away as a function of space) depend on the sample shape and its orientationwith respect to the field. The critical field is really |H | = Hc only for a needle-shapedsample aligned with H .

2. Type II superconductors

A “type II” superconductor, rather than abruptly expel field, has a regime in whichit partially admits the field, in the form of quantized flux lines. Such a phase is calledthe mixed state, or better the Abrikosov phase, and is the subject of Lec. 6.6 . Donot confuse it with the “intermediate state” which is a coexisting mixture of normaland superconducting phase occurs in Type I superconductors in certain geometries (seeLec. 6.3 [om. 2007]). A useful (!) mnemonic is “the mixed state is intermediate, andthe intermediate state is mixed.”

the same, restated: There is a class of superconductor called “Type II” in whichflux does not penetrate as a first-order, all-or-nothing transition. Instead, between thelower critical field Hc1 and the upper critical field Hc2 is the Abrikosov phase: fluxpenetrates gradually, while superconductivity persists. In fact Hc2 > Hc, and Type IIsuperconductors are the basis of today’s superconducting magnets.

6.0 A. PHENOMENOLOGY 537

N

ScH

B

(c).

H

(b). H

SN

(a)H

χ

Figure 2: Critical field and Meissner effect. The sample geometry is a thin sliver parallel tothe field. (a). At temperature just above Tc(H). (b). T < Tc(H), showing flux expulsion(c). Internal magnetic field as a function of the applied field, taken along a vertical line inFig. 1(a) at T < Tc (solid curve) or at T > Tc (dashed curve). The arrows indicates thenegative (diamagnetic) susceptibility χ in the superconducting phase.

3. Electronic specific heat and gap

Around 1950 it was discovered that the electronic part of the specific heat, Cel(T ),has a jump at Tc.

3 At low temperatures Cel(T ) ∼ e−∆/T indicative of an excitationgap ∆(0) at T = 0. The BCS theory of Part 7 gives

Cel(T+c )

Cel(T−c )

' 1.43. (6.0.1)

Integrating Cel(T ) carefully yields the condensation energy Fcond(T ), the free energydifference between a superconducting and a normal state. Actually Fcond(0) is only∼ 10−3kBTc per electron. That means that only a small fraction of electrons are reallyinvolved in the phenomenon. Another small parameter is the ratio Tc/EF ∼ 10−5

(taking Tc ∼ 1K.)The curves for a normal metal below Tc, shown dashed in Fig. 1(c), are not just

academic: one really observes this curve if H > Hc. The normal state is the equilibriumstate at T > Tc, but can also be accessed at T < Tc by applying a field H large enoughto drive the system out of the superconducting state. (The specific heat depends onlyweakly on the magnetic field.) (In the same fashion one may observe ρN (T ).)

Also, below Tc – see Fig. 1(c) again – the specific heat goes to zero exponentially,demonstrating there is a gap in the electronic elementary excitations. I STILL NEEDTO STRENGTHEN THIS STATEMENT.

4. Resistivity

See Fig. 1(b). This was, of course, the first discovered property of superconductors(Kamerlingh Onnes, 1911). The resistivity plummets, ideally, to zero (Fig. 1(a)). Inpractice (Fig. 3) it usually falls gradually and the experimentalist must choose the fea-ture of the fall at which to place the Tc. In the normal phase, it would have behaved

3Notice that the total specific heat will be dominated by the T 3 phonon term at low temperatures,so a delicate subtraction is necessary to measure the exponential Cel(T ).


as ρ(T ) = ρ0 + O(T x), where the power x depends on the dominant (elastic or inelas-tic) scattering mechanism, e.g. electron-phonon scattering which dominates at highertemperatures.

Superconductivity is not the same as perfect conductivity. Consider a materialobeying Ohm’s Law in which resistivity ρ suddenly vanishes. The Maxwell equationstates

−1

c

(

∂B

∂t

)

= ∇×E = ∇× (ρJ) = 0 (6.0.2)

If ρ = 0, then this says B(r, t) = B(r, 0): the instantaneous value of magnetic fieldgets frozen everywhere, at all subsequent times t. But superconductors expel the field(Meissner effect, see above).

Figure 3: Resistivity in some real superconductors. (The materials illustrated here and inFig. 4 all fall in the “exotic” category, discussed further in Lec. 8.1 [2007](a). Organic super-conductors; here (TMTSF)2PF6 is measured at pressure 8 kbar while the other two salts areat atmospheric pressure (∼ 1 bar). [From D. Jerome, Science 252, 1509 (1991).] (b). High-Tc

cuprate, La2−xBaxCuO4, recorded for different current densities. [From J. G. Bednorz andK. A. Muller, Z. Phys. B 64, 189 (1986), the discovery paper.] [Missing: Heavy-fermion-typemetals. See e.g. figure in L. P. Gor’kov, Sov. Science Rev. A Phys. 9, 1 (1987).]

5. Systematics in the periodic table

Comparing Tc’s

To phonon-based resistance:

The last shall be first.

There are correlations of superconductivity with other properties, such as (for pureelements) their placement in the periodic table. 4

(Of course, they are all metals.) The most important such correlation is that badmetals (Hg, Pb) are good superconductors and good metals (Au, Cu) are bad super-conductors. This will be explained by the electron-phonon coupling. (Somewhere latein Part 7, I think.)

4See A&M tables 34.1, 34.2. [I need something like that.] A wealth of phenomenology, correlatingTc and other material parameters, may be found in R. M. White & T. H. Geballe, Long Range Order

in Solids.

6.0 B. REAL MATERIALS ARE DIRTY 539

Another observation is that magnetic moments usually destroy superconductivity.Thus the pure elements Mn, Fe, Co, and Ni show no superconductivity at any temper-ature – the superconducting ordering has been preempted by a magnetic ordering. 5

It happens that Hc ∝ Tc, which will be explained in Lec. 6.2 [om. 2007]or so, and∆ ∝ Tc too, which comes out of pairing theory (Lec. 7.4 ).

High-Tc oxides are very bad metals; undoped, they are in fact antiferromagnetic,ionic insulators. Here EF is smaller and Tc is much larger, so the Tc/EF is no longersuch a small parameter.

6.0 B Real materials are dirty

The ideal pictures in Fig. 1 are not what you see experimentally. In Fig. 3 and Fig. 4I’ve chosen a sampling of real data figures, to represent a variety of exotic superconduc-tors outside the classic elemental metals and alloys. The current doesn’t usually dropsuddenly, and the expelled flux isn’t usually 100%.

Figure 4: Magnetic susceptibility of alkali fulleride superconductor. Susceptibility χ(T ) forthree samples of KxC60 powder exhibiting shielding diamagnetism, expressed as a percentageof a niobium standard measured at 4.2K (so that 100% essentially represents 100% exclusion offlux, χ = −1/4π). “Curves A and C represent the same sample (nominal composition [x = 3])after the mixing and final stages, respectively. Curve C is a composition [x = 3.5] sampleafter the mixing stage. (From K. Holczer, O. Klein, S.-M. Huang, R. B. Kaner, K.-J. Fu,R. L. Whetten, and F. Diedrich, Science 252, 1154 (1991).)

What’s going on is that you have a mix of superconducting and normal material.What if you surround grains of normal metal by a network of superconductor thatpercolates (connects across the sample) as in Fig. 5(a)? For example, when an Alalloy forms it might exclude some pure Al which superconducts.6 (Visualize the samplemorphology as a pile of rocks, each wrapped in aluminum foil.) You will see R(T ) = 0below the Tc of Al, even though Al forms a negligible fraction of the sample. On the

5The exception is in complicated crystal structures, where the magnetism occurs mainly on one kindof site in the unit cell and the superconductivity on another kind of site.

6Even commoner, the entire sample is single-phase, but has spatial variations in the stoichiometryso that the Tc varies from point to point; consequently at a given T , part of the sample will be normaland part superconducting.


other hand, the equilibrium value of diamagnetism is proportional to the volume fractionand is negligible.

What if we invert the previous morphology, so that each superconducting grain issurrounded by normal material? (Visualize lumps of metal, each wrapped in insulatingpaper.) Then the entire sample will have a nonzero resistance, since the best currentpath consists of normal and superconductor in series. (But R will be small, sincethose paths are short.) In high-Tc cuprates, there is a finite resistance even whensuperconducting grains are touching, if they are differently oriented.

N

B

N N N

S

S

S

B

NS

S

NS

(a) (b)

Figure 5: Inhomogeneity in superconductors. In either case flux lines (arrowed) can betrapped. (a). Morphology: large normal domains separated by thin superconducting sheets.(b). The reverse morphology: superconducting domains separated by normal layers.

In any kind of inhomogeneous geometry, the magnetization shows lots of hysteresis.If a sample is cooled below Tc in a field, some flux gets trapped in the normal regions(Fig. 5(a,b)). To get out it must pass through a superconducting region, but there isa big barrier to doing this. (As mentioned above, the flux must make vortices (fluxlines) in the S regions, as will be studied in Lec. 6.4 A . As also mentioned above, theequilibrium state in a field might be the “intermediate state” with some flux crossingthe superconducting region, even in the absence of disorder: see Lec. 6.3 [om. 2007].The disorder mentioned in the figure caption, which can drive a material to Type II,is not the gross inhomogeneity, but rather is microscopic scattering centers due e.g. tosubstitution, which would cause a large residual resistivity in the normal state.

The Meissner effect is thus a more reliable indicator of superconductivity than theresistance. Specific heat is less affected by geometry than either resistivity or Meissnereffect. But a specific heat bump can be caused by many kinds of ordering – it is notparticularly diagnostic of superconductivity.

6.0 C Scorecard of facts to be explained

Here is my guide to the Lectures in which the experimental facts mentioned above (andsome other ones) will get explained:

Meissner effect, zero resistance

Meissner effect (flux expulsion) and critical field – Lec. 6.3 [om. 2007].Zero resistivity – Lec. 6.4

6.0 C. SCORECARD OF FACTS TO BE EXPLAINED 541

G-L theory from microscopics: Lec. 7.8 [omitted]From a microscopic viewpoint, these properties are a consequence of off-diagonal

long-range order (ODLRO, Lec. 7.1 ) and the “rigidity of the wavefunction.”

Thermodynamics

Specific heat jump: Lec. 6.1 Ginzburg-Landau free energy theorySpecific heat ratio Cs(T

−C )/Cn(Tc): Lec. 7.4 (Bogliubov theory can give Fcond(T ))

Ratio 2∆(0)/Tc = 3.5: Lec. 7.4 Bogoliubov approach at T > 0Low T specific heat Cs(T ) ∼ e−2∆/T : Lec. 7.4 . Follows from quasiparticle

dispersion E(k) (valid for s-wave symmetry only).

Value of Tc and systematics of occurrence

Why Tc and ∆ are so small: Lec. 7.3 Tc ∼ ∆ ∼ ~ωmaxe−(...)

Why bad metals tend to be good superconductors: Lec. 7.6 (phonon-mediatedmechanism).

Isotope effect: Lec. 7.6 (phonon-mediated mechanism).Why magnetic moments kill superconductivity – see [around] Lec. 7.4 (“pair-

breaking” effects)

Dynamics

NMR damping (peak below Tc); ultrasound damping (no peak below Tc):Lec. 7.4 or Lec. 7.5 (C, coherence factors from quasiparticle operators)

547

Lecture 6.1

Highlights of macroscopic

superconductivity

This is a condensed version of Lec. 6.1 in an earlier version of the notes, which alsosummarizes a bit of Lec. 6.2 [om. 2007]and Lec. 6.3 [om. 2007].

.... from the macroscopic, or Ginzburg-Landau (GL), theory of superconductivity.1 This theory includes the actual explanations of the most obvious phenomena, such assupercurrent. Note that for some years in the late 1950s, the macroscopic theory wasdeveloped by Russians while the microscopic theory was independently developed byAmericans. The latter approach found ways to relate those phenomena directly to themicroscopic theory, but it is a bit formal. To my taste, it is more transparent to relatethe phenomena to the GL theory, then relate GL to the microscopic theory.

I may add one or two sections to this... last time, Part 6 was truncated so severelythat I also summarized Lec. 6.4 , Lec. 6.5 , Lec. 6.6 , and Lec. 6.7 at this point. But in2007 I’ll at least hand out those chapters.

The complex order parameter is the key element in defining superconducting or-der. In particular, the excitation gap is not essential since it is sometimes zero in asuperconducting state.

6.1 A Ginzburg-Landau free energy

The GL free energy is the macroscopic elastic theory appropriate for any superconduc-tor. The degrees of freedom are two fields, each varying smoothly in space. The firstfield is the “order parameter” field, Ψ(r), which takes on complex values. It will oftenbe convenient to separate the complex phase angle:

Ψ(r, t) ≡ |Ψ(r, t)|eiθ(r,t) ≡√

ns(r, t)eiθ(r,t) (6.1.1)

Here the arbitrary choice of θ; expresses the spontaneous breaking of the continuous(gauge) symmetry, while the magnitude |Ψ| expresses the strength of superconductivity(with |Ψ| = 0 if and only if the state is not superconducting.) In a neutral superfluid,such as 4He, the GL free energy depends only on the Ψ(r) field.

1Introduced by L. D. Landau and V. L. Ginzburg, Zh. Eksp. Teor. Fiz. 20, 1064 (1950); an Englishtranslation is printed as an appendix in Men of Physics: L. D. Landau I, by D. ter Haar (Pergamon,1965).


548 LECTURE 6.1. HIGHLIGHTS OF MACROSCOPIC SUPERCONDUCTIVITY

The London approximation is analogous to the fixed-length spin approximation instatistical mechanics of lattice models. It means we set |Ψ(r) to a fixed constant, butstill allow θ(r) to vary.

The second field is just the vector potential A(r) – a real, 3-component vector withthe usual gauge freedom. This enters the GL free energy because the superfluid ischarged and its current couples to the magnetic field.

The Landau-Ginzburg total free energy has the form

Ftot(Ψ(r),A(r)) =

∫

d3r FGL(r) (6.1.2)

where the free energy density at r is

FGL(r) = Fgrad(r) + FL(|Ψ(r)|) + Umag(B(r)) (6.1.3)

The various terms in it were originally guessed just from symmetry principles. Thephysical meaning of the terms is best understood by pretending they refer to a chargedBose superfluid with particles of mass m∗ and charge e∗ (which are, in fact, pairs.) Asuperconductor is nothing but a charged superfluid.

Terms in Ginzburg-Landau free energy

The first portion of the GL free energy density is the Landau term FL(|Ψ|), tells usthe free energy density in a spatially uniform state in which the order parameter has thevalue |Ψ|. For T > Tc, the minimum of FL(Ψ) should occur at Ψ0 = 0, expressing thelack of superconducting order. On the other hand, for T < Tc the minimum occurs fora nonzero Ψ0; more properly, on the complex plane of possible Ψ values, the minimumoccurs everywhere along a circle of radius Ψ0. The Landau free energy density wasintroduced in Lec. 1.4 C, and the critical phenmena have mean-field critical exponentsas described there.

By symmetry, the first terms of the Landau term are

FL(|Ψ|) = α|Ψ|2 +β

2|Ψ|4 + . . . (GL form) (6.1.4)

Here α > 0 or α < 0 corresponding to whether T > Tc or T < Tc. By the sameanalyticity assumptions, we expect that

α ' α′(T − Tc) (6.1.5)

in the vicinity of Tc. Away from Tc, FL(|Ψ|) does not have this exact shape, but itkeeps the qualitative property of a minimum at Ψ0.

Next, the gradient term was just guessed by Ginzburg and Landau:

Fgrad =~

2

2m∗|(∇− ie∗

~cA)Ψ|2 (6.1.6)

where A is the vector potential (of course it’s omitted in the case of a neutral superfluid).Finally, the magnetic field energy term comes straight from the elementary theory

of magnetostatics and does not depend on the material.

Gauge symmetry

Evidently the GL free energy is invariant under the same gauge transformations thatare applied to a Schrodinger wavefunction.

6.1 B. MICROSCOPIC RATIONALIZATION OF GL FREE ENERGY 549

A special case of the gauge symmetry is a uniform phase change,

Ψ(r, t) → Ψ(r, t)ei∆θ (6.1.7)

while A(r, t) is unchanged. (If we had a neutral superfluid, there is no gauge freedom(6.1.9), but we do have the global phase symmetry (6.1.7).) Eq. (6.1.7) is the continuoussymmetry which is spontaneously broken in the choice of an arbitrary phase angle θ.

More generally, FGL is invariant under the following change of gauge :

A(r, t) → A(r, t) +e∗~c

∇χ(r, t) (6.1.8)

Ψ(r, t) → Ψ(r, t)eiχ(r,t) (6.1.9)

(This is obvious since both changes affect only the Fgrad term.) Recalling (6.1.1), wesee that θ = θ + χ.

6.1 B Microscopic rationalization of GL free energy

Preview: Let’s approximate an interacting ground state of a boson system by placingall Nb bosons in the same state ψ(r), to be determined variationally by minimizing〈H − µbN〉: this is the Bose version of Hartree-Fock theory (compare Lec. 1.4X). Itturns out that Ψ(r) is proportional to ψ(r) and the GL free energy is the variationalexpectation. The relation of Fgrad ((6.1.6)) to the kinetic energy term is evident; the|Ψ|4 term comes from the interaction term.

To motivate the GL theory, I’ll justify it as a mean-field approximation of a mi-croscopic toy model of bosons. 2 Bear in mind, though: no known superconductoror superfluid is accurately pictured this way, except for cold, Bose gases in a certainrange of diluteness. In particular, it’s misleading to treat a Cooper pair in a BCSsuperconductor as a pointlike, tightly-bound boson (see Lec. 7.2).

It is justified (nowadays) through the notion of “abiabatic continuity” (see Lec. 1.4 ):since (it turns out) we can continuously change the present toy Hamiltonian into therealistic one without crossing a phase boundary, it follows that the macroscopic behaviorcan be described by the same functional form. That is, they have the same Ginzburg-Landau theory. In the case of multi-component order parameters, such as the pairedsuperfluid 3He, the boson viewpoint is very handy in visualizing how a pair’s orbitalsymmetry relates to the macroscopic symmetry of the GL free energy (see Lec. 8.2 ).

Consider Nb 1 bosons with mass m∗ interacting by a pair potential uboson(R),and placed in an external potential v(r) at zero temperature:

H = HKE +∑

r

v(r)n(r) +1

2

∑

rr′

uboson(r − r′)n(r)n(r′) (6.1.10)

which is written as a lattice model only for convenience. In (6.1.10), HKE is the kineticenergy operator, and n(r) ≡ b†(r)b(r) is the boson number operator on site r; the bosonoperators satisfy

[b(r), b†(r′)] = δr,r′ . (6.1.11)

They almost commute if n(r) ≡ 〈n(r)〉 1 – the relative error is

[b†, b]

〈b†b〉 ∼ 1

n(r). (6.1.12)

2Ginzburg and Landau, however, jumped directly to the mean-field (Landau) theory and gradientterm, by analogy to their appearance in the theory of other ordered systems.


Then b(r) and b†(r) can be treated as classical variables. (That’s exactly how wetreat harmonic oscillators, whose raising and lowering operators obey the same algebra(6.1.11).) Those classical variables will be the GL order parameter fields; observe thatsuch an approximation should be valid for a coarse-grained description such that thereare at least several bosons per unit of volume, but at shorter length scales the quantumfluctuations cannot be neglected.

What’s the (many-boson) ground state of (6.1.10)? If uboson(R) = 0, all bosonsoccupy the (normalized) single-boson ground state wavefunction ψ0(r). (Such an oc-cupation of a state by a macroscopic number of bosons is called Bose condensation.)

The creation operator for this state can be written b†0 ≡∑

r ψ0(r)b†(r), so that n(r) =

〈n(r)〉 = |ψ(r)|2〈b†0b0〉 = Nb|ψ(r)|2.Let’s approximate the interacting ground state by placing all Nb bosons in the same

state ψ(r), to be determined variationally by minimizing 〈H − µbN〉: this is the Boseversion of Hartree-Fock theory (compare Lec. 1.4X). The chemical potential of bosonsµb is, as usual, just the Lagrange multiplier to constrain Nb in the minimization. (In thecase of a superconductor or paired superfluid, you might view µb = µb(T ) as controllingthe propensity of fermions to pair: in this case, the density of “bosons”, i.e. Cooperpairs, is certainly not fixed.) We’ll show next that, after another approximation or two,the expectation (6.1.15) is the Ginzburg-Landau free energy.

To proceed further, approximate the interaction as being short-ranged compared tothe scale on which ψ(r) varies, and take

uboson(R) → UδR=0 (6.1.13)

(where U ≡∑

R U(R).) On a lattice, this interaction defines the “Bose-Hubbard”model, which has the same interaction as the Hubbard model (Lec. 5.4 ), but behavesdifferently since 〈n(r)〉 1 is assumed. The interaction expectation is approximated12U

∑

r n(r)2. This contains the factor 12Nb

2 where it ought to have 12Nb(Nb−1), since

the bosons don’t interact with themselves, 3 but the error is negligible since n(r) 1.Finally, let

α ≡ −µb (6.1.14)

(also let α absorb any uniform part of the single-particle potential v(r).)In that variational spirit, we should minimize

〈H〉 = Nb

∫

ddr

[

~2

2m∗|∇ψ(r)|2 + v(r)|ψ(r)|2

]

+

1

2Nb(Nb − 1)

∫

ddrddr′uboson(r − r′)|ψ(r)|2|ψ(r′)|2 − µbNb (6.1.15)

This equation, properly, should have been written as a discrete sum, since I haven’t(yet) switched to continuum picture.

Finally, convert to a continuum model, with∑

r → vcell−1

∫

ddr, where vcell is the

volume per unit cell. Let Ψ(r) ≡ Nb1/2vcell

−1/2ψ(r); the local number density is

ns(r) ≡ |Ψ(r)|2 = n(r)/vcell (6.1.16)

and thus Ψ(r) has a sensible limit as the system size is increased. Relation (6.1.16) willbe fundamental within the GL theory as the definition of the “superfluid density” ns.Identify Ψ(r) with the GL order parameter field, then 〈H〉 = Ftot, the total Ginzburg-Landau free energy, defined by (6.1.2) (with β = vcellU). This is the grand result. The

3For the same reason, I ought to have written n(r)n(r′) → b†(r)b†(r′)b(r)b(r′) in (6.1.10).

6.1 B. MICROSCOPIC RATIONALIZATION OF GL FREE ENERGY 551

gradient term comes from the kinetic energy, the |Ψ|4 term represents an interaction, andthe |Ψ|2 term represents the combination of true potential and (temperature-dependent)chemical potential.

Of course, at nonzero temperature, in addition to 〈H−µbNb〉, we need to include anentropy term in the minimization, and thus (6.1.2) really is a free energy. A realistic freeenergy function at lower temperatures has gradient terms like (6.1.6), but the Landauterm cannot be approximated by the simplest polynomial with a minimum at Ψ0 – itwould have more terms like (6.1.4). But right near Tc, those terms are unimportant andthe GL form becomes exact (to the extent we can ignore critical-point fluctuations).

Time-dependent Ginzburg-Landau equation

We could also derive the dynamics of the macroscopic fields, from the microscopichypothesis (Sec. 6.1 B) in which all Nb bosons were put into the same wavefunction ψ(r)and Ψ(r) =

√Nbψ(r). The result is the time-dependent Ginzburg-Landau equation:

~id

dtΨ(r, t) = − ~

2

2m∗∇2Ψ(r) +

(

α+ β|Ψ(r)|2)

Ψ(r) (6.1.17)

IMPORTANT: I have left off the magnetic field terms, so this is correct only for aneutral superfluid. For a charged superfluid, a gauge-invariant gradient appears as inthe static G-L equations.

Indeed, (6.1.17) is a natural guess. You can show that the “self-consistent” (Hartree-Fock type) eigenvalue equation is HHF Ψ(r, t) = EΨ(r, t), where

HHF = − ~2

2m∗∇2 +

(

α+ β|Ψ(r)|2)

(6.1.18)

Since (6.1.17) is related to the eigenvalue equation as the time-independent Schr”odingerequation is related to the time-dependent one, it’s natural to guess ~idΨ/dt = HHF Ψ(i.e., Eq. (6.1.17)). And (6.1.17) reduces properly to the usual Schrodinger equation aswe turn off interactions (β → 0).

I did not have time to write out the derivation of the classical equations of motion,which follow naturally from noting 〈dO/dt〉 = 〈[iH, O]〉 (valid in either Heisenberg orSchrodinger representation). Notice that Ftot plays the role of the classical Hamiltonian.and Ψ(r) and Ψ∗(r) are canonically conjugate in the classical dynamics (also θ(r) andns(r) are conjugate). There is more on this in Lec. 6.5 .

We can rewrite (6.1.17) as

~id

dtΨ(r, t) =

δ

δΨ∗(r, t)Ftot(Ψ(r, t)). (6.1.19)

I must digress to define “δ/δΨ∗” in (6.1.19). A function f(Ψ) is implicitly a functionof ReΨ and ImΨ, so we can always write

df =∂f

∂ReΨdReΨ +

∂f

∂ImΨdImΨ. (6.1.20)

It’s trivial to reexpress the r.h.s. of (6.1.20) in terms of the independent linear combi-nations dΨ and dΨ∗; then we define the coefficients of these combinations ∂f/∂Ψ and∂f/∂Ψ∗, respectively. Note ∂Ψ∗/∂Ψ ≡ 0, even though Ψ∗ is a function of Ψ.

Now, let’s try decomposing the order parameter as

Ψ(r, t) =√

ns(r, t)eiθ(r,t). (6.1.21)


Simply substituting this into (6.1.19) (and its complex conjugate), then using (6.1.20)plus the chain rule, you can obtain

~d

dtθ(r, t) = − δFtot

δns(r, t)(6.1.22)

and

~d

dtns(r, t) = +

δFtot

δθ(r, t)(6.1.23)

You see something striking: eqs. (6.1.22) and (6.1.23) are just Hamilton’s equationsof motion from classical mechanics, with θ and ns in the roles of classically conjugatecoordinate and momentum. It turns out they’re quantum-mechanically conjugate, too,and we’ll see their uncertainty relation in Lec. 7.1 .

6.1 C Phase symmetry and phase gradients

The symmetry broken in superconductivity (or superfluidity) is the phase of a pair wavefunction: the so-called gauge symmetry. What physical phenomena correspond to this?First of all, within G-L theory the supercurrent is the (gauge-invariant) gradient of thephase. (That makes sense: the equilibrium state of uniform phase is the state when nocurrents are moving.) Secondly, there should be a Goldstone mode – a kind of sound –in any neutral superfluid.

Supercurrent operator

From the same kind of correspondence as in Sec. 6.1 B the expectation of the currentoperator gives the supercurrent

Js(r) = Re

Ψ(r)∗~i

m∗[∇− ie∗

~cA(r)]Ψ(r)

(6.1.24)

I have restored the gauge-invariant derivative for a charged superfluid. In a more usefulform Was this was written elsewhere?

Js(r) =~

m∗ns

[

∇θ − e∗~c

A(r)]

. (6.1.25)

You can verify that this conservation equation is satisfied,

d

dtns(r) + ∇ · Js(r) = 0 (6.1.26)

by substituting ns ≡ ΨΨ∗ into (6.1.17).

Goldstone mode?

The continuous symmetry (6.1.7), in a neutral superfluid, implies the existence ofa gapless Goldstone mode (recall Lec. 1.5 ). As we just saw, the quantity canonicallyconjugate to the phase θ(r) turns out to be the superfluid density. Thus we have all themachinery needed to find the small (linearized) oscillation modes (within the continuumelasticity theory, which is just what GL theory is). A excitation analogous to a spinwave or a phonon consists of oscillations of phase and (90 out of phase) superfluiddensity. You can obtain

−~δθ(r) = βδns(r) (6.1.27)

6.1 D. FURTHER BRIEF NOTES 553

~δn(r) =~

2n0

m∗∇2δθ(r) (6.1.28)

with n0 = |Ψ0|2. (Actually, in (6.1.27), a ∇2δns term also appears on the right-handside, but it is negligible in the long-wavelength limit.) It is because of the continuousphase symmetry that only gradient terms can appear on the right-hand side of (6.1.28).When we put (6.1.27) and (6.1.28) together, we obtain

d2

dt2δn(r) = −v2

0∇2δn(r) (6.1.29)

withv20 = n0β/m∗ (6.1.30)

But (6.1.30) can be written to look exactly like the equation for the ordinary soundvelocity in a fluid: v2

0 = B/ρ, where B = n20β = n2

0∂2FL/∂ns

2 is simply a bulk moduluscorresponding to the condensate density, and ρ ≡ m∗n0 is the mass density of thecondensate. So our gapless mode is simply zero sound (sound which exists in a zero-temperature fluid). Such a sound mode is present with essentially the same velocity inthe normal state; in the case that the normal state is a Fermi liquid (i.e. 3He), zerosound was mentioned in Lec. 1.8. The quanta of zero sound are phonons. Such phononsare also the dominant gapless excitation in a Bose superfluid such as 4He. 4

However, in a superconductor the superfluid density fluctuations carry charge densityfluctuations, which have long-range Coulomb interactions. Just like the longitudinalphonon of a charged medium (mentioned in Lec. 1.5 ) such as the Wigner crystal(Lec. 1.4 W [omitted]), the mode goes to a nonzero frequency. In fact this mode is thefamiliar plasma oscillation of any metal.

6.1 D Further brief notes

Here I’m condensing whole lectures into paragraphs.

Critical field

It turns out the increase in energy Umag from excluding an applied field is H2/8π.(I will compute this in Lec. 6.3 [om. 2007], which is about critical fields.) On the otherhand, the free energy of the normal state (of a Type I superconductor) is higher byFcond(T ). Thus (assuming no other term is important), the free energies of normal andsuperconducting states become equal at Hc(T ) given by

Fcond(T ) =Hc(T )2

8π(6.1.31)

and there is a first-order transition at this point.

Length scales

As in transport theory (Part 2), it’s useful to convert most parameters to lengthscales, in particularly the two lengths ξ and λ which are formed from the coefficients

4At T > 0, the Goldstone excitation is properly second sound which is a distinct mode from firstsound. Second sound is well defined only at temperatures high enough that the “gas” of excitationscan keep in equilibrium; it becomes ill-defined at low termperatures when the mean-free path exceedsthe sample dimensions.


of the GL free energy density The penetration depth λ is associated with the vectorpotentials field A(r) and is relevant only in superconductors. It gives the distance overwhich a magnetic field (present outside a superconducting region) decays exponentiallywith the distance into that region. In a traditional metal superconductor, λ ≈ 102 –103 Angstrom; or ∼ 3000Angstrom in high Tc cuprates.

The coherence length ξ is associated with the order parameter field Ψ(r) and isdefined even for a neutral superfluid. It is the dimension of a region in which the orderparameter phase θ is essentially locked (i.e. coherent). Within the GL theory (butnot generally), it equals the healing length, the distance over which |Ψ(r) returns toits equilibrium value, if it is made to deviate at some place. In a traditional metalsuperconductor, ξ ≈ 102 – 104 Angstrom; or about ∼ 15Angstrom in a high-Tc cuprate.

6.1 X Table of useful formulas

6.1 X. TABLE OF USEFUL FORMULAS 555

Name Symbol Formulas LectureEffective (pair) mass m∗ 2me? (Lec. 6.1 V [om.])Pair charge e∗ 2e above

Ginzburg-Landau coefficients α, β

α ≈ α′(T − Tc)

β > 0above

Critical temperature Tc – Lec. 6.0

Condensation (free) energy Fcond

α2/2β

H2c /8π (cgs units?)

? Lec. 6.0

[Order parameter magnitude] Ψ0 (−α/β)1/2 Lec. 6.2 [om. 2007]?

Superfluid density ns |Ψ0|2 = −α/β Lec. 6.2 [om. 2007]

[Gradient stiffness in FGL][

ns~2

m∗

]

Fcondξ2 Lec. 6.2 [om. 2007]

Penetration depth λ(

m∗c2

4πe∗2 · 1

ns

)1/2

Lec. 6.2 [om. 2007]

(GL) Coherence length ξ(

~2

2m∗

· 1|α|

)1/2

Lec. 6.2 [om. 2007]

Ginzburg-Landau parameter κ λ/ξ Lec. 6.2 [om. 2007](and Lec. 6.6 )

(pair) flux quantum Φ∗0 2π~c/e∗ Lec. 6.4

(Thermodynamic) Critical field Hc

√2

4π

(

Φ∗

0

λξ

)

Lec. 6.0 (?), Lec. 6.3 [om. 2007]

Lower critical field Hc1lnκ4π

(

Φ∗

0

λ2

)

= ln κ√2κHc Lec. 6.6

Upper critical field Hc224π

(

Φ∗

0

ξ2

)

=√

2κHc Lec. 6.6

GL critical current JGLc

23√

3nse∗~

m∗ξ Lec. 6.3 [om. 2007]

vortex line tension εv 4π|Fcond|ξ2 lnκ Lec. 6.6

Table 6.1.1: Ginzburg-Landau parameters and useful formulas. Values of α′, Tc and β arematerial dependent; formulas for them can be derived only within the microscopic (BCS)theory, see Lec. 7.8 [omitted]. Numerically Φ∗

0 = 2. − 7 × 10−7gauss − cm2.

579

Lecture 6.4

Supercurrent

All about critical currents, zero resistance, and flux quantization

This lecture gathers several diverse answers to the question “When (or why) doesa superconductor superconduct?” We previously (re)defined “superconductivity” as“the existence of long-range order of the order parameter Ψ(r), and we (re)defined“supercurrent” Js(r) as “the collective current of the condensate” described by Ψ(r).So, it’s not self-evident that the zero-resistance current is Js; or that it’s related to thelong-range order of the phase field θ(r).

To start, Sec. 6.4 A explains why the magnetic flux through any closed ring ofsuperconductor must be quantized. (This will be the basis of the SQUID device, tobe discussed in Lec. 6.5 , and of the vortices in Type II superconductors discussed inLec. 6.6 and Lec. 6.7 .) The “stiffness” of the phase, as evidenced in flux quantization,is ultimately the explanation of zero resistivity. Specifically, we consider a “persistentcurrent” in a superconducting ring, with the philosophy that understanding zero re-sistivity really means understanding when it breaks down. Following that philosophy,the Sec. 6.4 C addresses the critical current, the maximum sustainable supercurrent.There are two scenarios for a maximum, depending on the sample size and geometry,corresponding to the two fields of G-L theory (magnetic field B(r) and order param-eter Ψ(r).) One of the cases forces us to (briefly) face the microscopic arguments forLandau’s critical velocity: namely, the critical current has a nonzero value because theexcitations in a superconductor have a dispersion relation which (at smaller currents)does not allow any gapless excitations.

6.4 A Flux quantization

Consider a ring of superconductor (Fig. 6.4.1), pierced by a net flux ΦB . In the super-conductor,

Js =~nse∗m∗

(

∇θ − e∗

~cA

)

(6.4.1)

(repeat of the First London equation from Lec. 6.2 [om. 2007]).But, deep within the superconductor’s bulk – farther than ∼ λ from its surface –

Js = 0 (recall LecHc on critical fields). Hence

∇θ =e∗

~cA. (6.4.2)


580 LECTURE 6.4. SUPERCURRENT

sJ

vortexdl

a) b) c)

B

BB

Figure 6.4.1: (a). A cylinder of superconductor pierced by flux. Loop integrals will be donealong the dashed curve, which is deep within the superconducting bulk. (b). Top (end-on)view. Supercurrents Js (indicated by arrows) flow only along inner surface; dl marks thecontour for the loop integral of ∇θ. (c) A normal region contains one flux quantum, i.e. avortex. The path integral would be smaller by 2π for a loop passing inside the vortex than fora loop passing outside the vortex.

Now let’s do the loop integral of both sides of (6.4.2) along the curve l indicated inFig. 6.4.1. On the one hand,

∮

dl · ∇θ = 2πn (6.4.3)

for some integer n. That’s because the phase factor eiθ must take the same values ateither end of the loop, of course, otherwise the order parameter would have a jumpsomewhere.

On the other hand, by a Stokes identity

e∗

~c

∮

dl ·A =e∗

~cΦB (6.4.4)

Combining (6.4.2),(6.4.3) and (6.4.4), we get

ΦB = nΦ∗0 (6.4.5)

where

Φ∗0 ≡ 2π~c

e∗= 2.07 × 10−7gauss-cm2 (6.4.6)

The superconducting flux quantum, because it contains e∗, is half as big as the fluxquantum considered in mesoscopic transport (Lec. 2.2).

Vortices

Remember (from Lec. 6.3 [om. 2007]) the domains of Normal phase (containingflux) in the intermediate state of Type I superconductors? A corollary of (6.4.5) is thateach such domain must contain an integral number of flux quanta. (We can simply takethe loop integral around the normal domain instead of a physical hole.)

Consequently, too, there is a minimum value (n = 1 flux quanta) of the total fluxthrough any normal domain. That smallest domain is a quantized vortex or flux line, atube-like region in which the order parameter and magnetic field take on values which(according to Lec. 6.3 [om. 2007]would be excluded in the equilibrium bulk of thesuperconductor.

Notice that if θ winds through 2π along a loop around a vortex, then there mustbe some curve along which θ is singular. That is the centerline of the vortex. In order

6.4 B. WHY ZERO RESISTANCE? 581

for Ψ(r) to be continuous while having an indeterminate phase, we must have Ψ(r) = 0along this line. The region in which |Ψ(r)| Ψ0 is called the vortex core. (Vorticeswill be discussed in more detail in the first two sections of Lec. 6.6 .)

Vortices and vortex lattice

Preview of Lec. 6.6A Type II superconductor is characterized by λ/ξ > 1/

√2. In that case, magnetic

field can enter the superconducting state in the form of a vortex line, a topologicaldefect akin to a dislocation around which θ(r) changes by 2π. Roughly speaking, theexternal magnetic field penetrates within a radius ∼ λ of the vortex line, and the netflux associated with it is exactly a flux quantum. High-Tc cuprates are extreme TypeII superconductors and the flux lattice has several regimes depending on temperatureand defects that pin the lattice (similar to pinning of a charge-density wave.)

The Abrikosov flux lattice phase consists of a regular triangular lattice of flux lines.It turns out that a transverse current exerts a force on the lines (normal to B and tothe current) and if they move – which they will, unless pinned – then there is a finiteresistivity.

6.4 B Why zero resistance?

Later sections 6.4 C and 6.4 D will ask how increasing the current could destroy thesuperconducting phase coherence by driving the order parameter to zero.

Although Js was called the supercurrent, we have not yet seen why it flows withzero resistance: this is a slippery point in basic superconductivity. We’ll address thistwo ways. One way to parse “zero resistance” is via Ohm’s law: that is, we want toshow the voltage drop is V = 0 while the current is nonzero.

First take on zero resistance

A superconductor cannot maintain a voltage difference. Recall (see the Londonequations in Lec. 6.2 [om. 2007]) that, in the presence of an electric field, the super-current accelerates, just like an undamped charged particle. If the superconductor is inseries with an ordinary resistance, that suffices to show that the equilibrium state musthave zero voltage drop across the superconductor.

The same argument can be restated in the language of the phase function θ(r, t).As a special corollary of the Time-Dependent Ginzburg-Landau equation introduced inLec. 6.1X , I claim

dθ(r, t)

dt= −1

~µ(r, t) (6.4.7)

where µ is the (electro)chemical potential at r. [You might guess (6.4.7) from thesingle-particle Schrodinger wavefunction, in which the phase angle rotates in time asdθ/dt = E/~, where E is the eigenenergy.]

So, if you had a fixed electric potential drop between points r1 and r2, the phasedifference θ1−θ2 grows linearly with time, as must the phase gradient – which is entirelyequivalent to saying the supercurrent Js ∝ ∇θ accelerates ballistically. If nothing elseintervened, it would quickly reach the critical current (see below): thus, voltage isinconsistent with a steady state 1

1But AC V (t) will be possible, as in the Josephson effect (Lec. 6.5 ).


Persistent current in a ring

A second way to parse “resistance” is: given a current, by what process does itsvelocity get damped; and its kinetic energy get dissipated?

Next, consider a ring (or hollow cylinder) at T < Tc, with a (super)current circu-lating, as in Sec. 6.4 A: how can this current decay? A lower energy state is alwaysavailable (superconducting with Js = 0). But we’ll find there’s a large barrier to thatlower energy state, so the persistent current is (very) metastable. Still, in any finite

ring there is, in fact, a nonzero but very small energy dissipation, via quantized eventsrelated to flux quantization. (For a standard type I superconductor, an limit on resis-tivity ρ ≤ 3 × 10−23Ωcm was measured 2 Another measurement (runs of ∼ 30 days)found time constants of roughly 105 years. 3 )

The current is on the inner surface of the cylinder. Since there’s no field deep inthe bulk of the cylinder, there must be a uniform field in the hollow center which isbeing screened by the surface current, and therefore proportional to it. As argued inthe Sec. 6.4 A, the net flux of that field must be a multiple nΦ∗

0 of the flux quantum.We showed in Sec. 6.4 A too that

∮

(∇θ · dl) = 2πn. The only way to decrease thecurrent is to decrease the encircled flux, such that n → n − 1, called a “phase slip” 4

since∮

(∇θ) · dl changes by 2π. That requires moving a quantum of flux from inside tooutside the cylinder.

The energy cost of making a vortex line is proportional to its length, so the energybarrier is related to the thickness of the ring or cylinder; hence, thermal activation ispossible and important in sufficiently small samples. More realistically, the barrier isagainst nucleating a single small closed loop of vortex. Once the loop is nucleated, everybit of it feels a “Magnus” force from ythe supercurrent (see Lec. 6.7 ) pushing the loop’sdiameter to expand – which it can do without any further barriers 5 until it stretchesacross the thickness of our ring. However, in a large sample, the expected frequencyof phase slips can exceed than the age of the universe: in effect we have a persistent

current that flows without dissipation.In view of (6.4.7), the rate of phase slips is the voltage difference, in a supercon-

ducting type material:

~d

dt(θ2 − θ1) = −(µ2 − µ1). (6.4.8)

[I believe µi here denotes the pair chemical potential. Possibly this equation and thesurrounding ideas belongs after Lec. 6.5 .] So if the flux decay rate happens to be propor-tional to the gauge-invariant gradient, or equivalently if the dissipation is proportionalto J2

s , it means V ∝ I with I2R Joule heating, and we have an Ohmic resistance. Inthat case, the current must decay exponentially with time, as in an ordinary RL circuit.Contrariwise, if the decay rate and dissipation scale more rapidly to zero when Js → 0– indeed we expect an activated form exp(−B/T ) where B is the barrier mentionedabove – we say the resistance is zero.

A current-carrying wire (or sheet) is no different – locally – from a cylinder or ring.In place of the multiple-connected topology, some boundary condition fixes the (gauge-invariant) phase difference between the two ends. Therefore, to change the current, westill have to pass a flux quantum’s worth of flux across the wire, a phase-slip process

2D. J. Quinn III and W. B. Ittner III, J. Appl. Phys. 33, 748 (1962)3J. File and R. G. Mills, PRL 10, 93 (1963). I need to check if this is the record.4Phase slips also occur in charge-density waves [Lec. 3.4 ] where the order parameter also has a

phase, but they have a different relation to the current in that case.5This probably occurs at special sites where the nucleation barrier is lowered, completely analogous

to a Frank-Read source that nucleates dislocation loops repeatedly in crystalline solids.

6.4 C. LANDAU’S CRITICAL VELOCITY 583

which has the same huge barrier in any reasonable-sized samples, so we observe zeroresistance.

In summary, whereas a normal current can be degraded just by scattering one in-dependent electron, a supercurrent can be degraded only by a process which involvesa large chunk of superconductor. It is due to the topological properties of the phaseangle, and so it is possible only in a system which has undergone a broken symmetry.As Anderson emphasizes in Basic Notions, the rigidity of ordinary solids is completelyanalogous. 6 [As we approach the normal state continuously by letting |Ψ| vanish –see next sections – the cost of making a vortex also vanishes. Thus it becomes easy tonucleate phase slips and dissipation will be seen.]

At T = 0 quantum tunneling replaces activation over the barrier.

6.4 C Landau’s critical velocity

Landau considered the instability of superflow with respect to creating elementary ex-citations. This fundamental mechanism for a critical current does not involve the mag-netic field, so it applies in principle to neutral superfluids. It certainly applies in anysuperconducting slab or wire sufficiently thin compared to the penetration depth, socurrent doesn’t make an appreciable magnetic field.

[Sec. 6.4 X works out a distinct mechanism, related to electromagnetism: by Ampere’slaw, a supercurrent necesssarily creates a magnetic field, but this tends to destroy su-perconductivity. The mechanism of the present section is more general since it appliesto neutral superfluids too.

Basic idea

First note that by Galilean relativity, a neutral superfluid in vacuum can be boostedas fast as we like without affecting its superfluidity. We have a critical current onlywhen our system is in contact with a stationary external world, which serves as amomentum reservoir: (i) the container walls (or sometimes a porous medium), in thecase of a neutral superfluid (ii) the solid lattice, in the case of a superconductor. So|vs|, implicitly, is always measured relative to an environment with v = 0.

Let’s imagine we constrain the phase difference as a boundary condition. So long asthe superfluid order parameter is nonzero (Sec. 6.4 B) we have a nonzero phase gradientkθ and a nonzero supercurrent. The alternative is a normal state, with zero order pa-rameter; here, phase difference ∆θ is undefined and we can have zero current. Comparethe respective total energies: the SC state is higher by the KE of the supercurrent, butlower by the condensation energy. Hence, once the former exceeds the latter, the systemgoes normal.

Within the Ginzburg-Landau picture (see Sec. 6.4 D), as |vs| increases (alwaysmeasured relative to the environment), the order parameter is reduced, until supercon-ductivity disappears at a critical |vs|, or equivalently at a critical current Jc.

Doppler effect for elementary excitations

Consider any elementary excitation in any medium with velocity v. (It’s a general-ized quasiparticle as described in Lec. 1.7 , which might be either a boson or a fermion.)Let its energy dispersion be ε(q) as a function of the wavevector, in a comoving frame

6See N. P. Ong’s website for a popular-level explanation of the rigidity (2007).


(in which the superfluid appears at rest). Claim: measured in the “lab frame” (in whichthe metal ions or channel walls are at rest), the effective dispersion

εeff (q) = ε(q) + v · ~q. (6.4.9)

This is the Galilean transformation of the dispersion law, or equivalently the Dopplereffect, as we shall see.

Let’s see how (6.4.9) is justified. According to Galilean relativity, an object withenergy E and momentum p in a frame moving at velocity v, relative to the lab istransformed to E′ = E+v ·p in the stationary reference frame. Transcribing E → ε(q)and p → ~q, we claim the effective dispersion relation is (6.4.9). 7

A second approach to rationalize (6.4.9) could be called the “Doppler shift” ar-gument. Write the wavefunction, for the quasiparticle in the comoving frame of thefluid.

ψ(r′, t) = e−iω′t+iq·r′ (6.4.10)

where r′ is measured in the comoving frame, and ~ω ≡ ε(q), Just substituting r = r′+vtinto (6.4.10), we obtain ψ(r, t) = exp(−iωt+ iq · r), with

ω = ω′ − v · q. (6.4.11)

When ψ(x) represents the amplitude of a sound wave, (6.4.11) is precisely the Dopplereffect; when it is a Schrodinger amplitude in quantum mechanics, we identify ~ω′ =ε(q) and ~ω = εeff (q), and (6.4.11) becomes (6.4.9). RESTATING: We obtain ψ =ei(~q·r−εeff (q)t/~) which is just (6.4.10) with ε(q) → εeff (q) as defined by (6.4.9).

8

[This justification has some analogies to the argument in Lec. 6.3 [om. 2007] , as towhy our effective field energy given a background field H, reduced to |B−H|2.]

Landau’s critical velocity

Now, if εeff (q) ≤ 0 for some q, then the system is unstable to emitting excitationsat that wavevector. This always happens for large enough v, as is clear when (6.4.9)is represented graphically, in the (q, ε) plane by the difference between the ε(q) curveand a line at slope ~v as in Fig. 6.4.2). Then Landau’s critical velocity vc is the firstvelocity where this happens. It is simply 1/~ times the slope of a line from the origintangent to the bottom of the dispersion curve (see Fig. 6.4.2.)

If you had the dispersion relation ~2q2/m of an ordinary particle (e.g. one free atom

of 4He), the line would be tangent at q = 0 so vc = 0. For an ordinary metal vc = 0too, since we have ε(kF ) = 0 for electron or hole excitations. (It is proper to measurefrom the Fermi level (as introduced in Lec. 1.7 .)

However, in a superconductor the Landau critical slope is nonzero because a gapdevelops in the dispersion relation, ε(kF ) = ∆.9

7Another version from (classical) Newtonian mechanics, in terms of the excitation’s momentum p.If that is changed by ∆p in a time interval ∆t, then a momentum −∆p is transferred to the stationaryreservoir and exerts a force f = −∆p/∆t during that interval. The site where this force acts is displacedδr = v∆t, so the work done is ∆W = f · ∆r = −∆p · v. By integrating this, one obtains an energyterm −v · p which (after identifying p = ~q) is the second term of (6.4.9).

8An alternative way to frame the “Doppler” argument starts with the definition of group velocityin the comoving frame vg = ~−1∇qkε(q) (as derived in basic solid state theory, for example). Thenthe group velocity in the stationary frame, determined from εeff (q) in the same fashion, ought to bejust v + vg , and (6.4.9) is the only formula that gives this.

9M. Tinkham (Introduction to Superconductivity, p. 119 (1st ed.), says this vc is called the “depairingvelocity” since (as just noted) the condensate of pairs will emit quasiparticles until it decays to zero.He refers us to J. Bardeen, Rev. Mod. Phys. 34, 667 (1962) for a review.

6.4 C. LANDAU’S CRITICAL VELOCITY 585

q

(q)ε

q

(q)ε

q

(q)ε(b). (c).(a).

k F

q0

∆

Figure 6.4.2: The bold line is the dispersion curve and the dashed line, tangent to it, hasslope ~vc where vc is the critical velocity. In (b), the roton minimum occurs at |q| = q0; in c,the heavy dashed line shows the dispersion in the absence of superconductivity, and ∆0 is thesuperconducting gap. I should add a fourth picture, ~

2k2/2m for a free particle.

In a neutral superfluid, the low-lying excitations are phonons with dispersion ω(q) =vsq, with vs the speed of sound, so the Landau criterion would say vc = vs due tophonons, as in Fig. 6.4.2(a). In 4He, as is well known, the curve ω(q) bends downwardsagain at larger q and has the so-called “roton” minimum around q = q0 ≈ 2π/(3nm),shown in Fig. 6.4.2(b); the line would actually be tangent near to this minimum yieldinga much smaller Landau vc around 60 m/s. The real critical velocity is about 1/100smaller than that; it is believed to be due to not-so-elementary excitations such asnucleation of vortex loops at irregularities along the surface.

The main point of the Landau criterion, then, is that (i) it provides a strict upperbound on vc, and (ii) it extends the idea that superflow persists because the systemmust overcome an energy barrier to reach a state of no superflow.

Landau’s critical velocity and T > 0

At finite temperatures, the superfluid/cuperconductor contains a gas of thermalexcitations (fermions called quasiparticles in a superconductor, or phonons in superfluidhelium) in equilibrium with the unmoving walls of the container, or (in a solid) with thelattice (and perhaps the defects which are fixed in it). In view of (6.4.9) the excitationswith wavevector antiparallel to the flow are favored over those with wavevector alignedwith it, indeed it turns out the excitations carry a current contribution opposite to thesuperflow. Thus J is decreased while the phase gradient ∇θ is unchanged, so that n∗

s isreduced from its T = 0 value. [One can profitably develop this picture into a “two-fluid”model, with some transport due to a “normal” fluid of quasiparticles that behave muchlike ordinary carriers in (say) a semiconductor.]

Now let’s note what happens to the order parameter for a velocity close to Landau’scritical velocity, vc. Recall that by definition, at v = vc the effective dispersion curveεeff (q) has a zero-energy excitation at a certain wavevector qc. Such excitations willhave a large thermal population, if T > 0, and even in the ground state there will belarge zero-point fluctuations. Consequently, as v → vc and εeff (qc) → 0, the orderparameter magnitude |Ψ| → 0.

Having εeff (qc) = 0 is much like having a soft phonon mode in an elastic lattice(see Lec. 3.0 and Lec. 3.4 ). The reduction of |Ψ| by fluctuations in a superfluid orsuperconductor is just like Debye-Waller factor reduction of the harmonic crystal’s order


parameter due to fluctuations (Lec. 1.6 ). 10

6.4 D Ginzburg-Landau picture of critical current

How does Landau’s microscopic vc fit into the macroscopic Ginzburg-Landau picture?The only way to represent instability of the superconducting state is that ns, or equiv-alently the order parameter magnitude, is driven to zero. The GL picture doesn’texplicitly include any microscopic elementary excitations, but thermal excitations areimplicit in the reduction of the order parameter amplitude |Ψ| at T > 0. 11 To rephrasethe ending of the preceding section, as we bring εeff (q) closer to zero at the critical q

point, more such excitations appear (as thermal excitations, or as quantum zero-pointtype fluctuations at T = 0) and |Ψ| gets reduced. (For a general discussion of therelationship of GL and BCS – for equilibrium statics only – see Lec. 7.8 [omitted].)

I will next show that, all by themselves, the GL equations imply a critical current.The Landau and GL critical velocities are respectively the microscopic and macroscopicformulations of the same phenomenon; and the following derivation shows that the twoanswers are same to within a factor of order unity.

Digression: boundary condition subtleties

Let’s assume space-independent fields; the external constraint is the phase gradient,so we take that fixed and write kθ ≡ |∇θ| for short. (Naively we might have triedinstead to constrain the total current. But physically, we’re considering the stabilityagainst a very local fluctuation: in that case, certainly the external phase difference isthe constraint.)

[the rest of this subsection expands the above statements, in perhaps overmuchdetail]

The question we will set up first is, given a boundary condition with a net phasechange ∆θ = θ(L) − θ(0) across our sample in (say) the x direction, what is the orderparameter reduction and the current? We’ll proceed by minimizing the free energy

given this boundary condition,∫ L

0 dxkθ(x) = ∆θ, where kθ ≡ |∇θ| for short. If thesample is not too thick, we can have a strong current density without making a largemagnetic field, so we neglect the magnetic field energy as well as the vector potentialin the gauge-invariant gradients. As a preliminary note, it can be verified that theminimum free energy solution is always to have kθ(x) uniform.

Assume kθ varies; to conserve current, ns(x) must vary correspondingly. ExpandFL() to second order as a function of ns. The first order term will cancel because∫

δns(x) ∝ −∫

δkθ(x) = 0 due to the ∆θ boundary condition. The second order termis proportional to d2FL/dns

2[δkθ(x)]2, which is positive definite since d2FL/dns

2 = β >0.

Since v = ~kθ/m∗, the assumption of fixed and uniform kθ is evidently proper whenwe want a critical velocity, e.g. for a superfluid put in motion relative to a channelcontaining it. If on the other hand we want a critical current, it’s a rather subtle pieceof thermodynamics that this is correct thermodynamically, and that it’s wrong to fix

10This is a preview of calculations in Lec. 7.5 , which includes an exercise finding the order parameterreduction via BCS/Bogoliubov theory. Note that even at T = 0, I think there’s a Ψ reduction (in aneutral superfluid): when εeff (q) → 0 for such a phonon/roton mode, then its zero-point motiondiverges, hence Ψ → 0. Something similar must with Bogoliubov quasiparticles in that exercise.

11Fluctuations reduce order parameters, as we first computed in the case of phonon fluctuations andcrystal order (see Lec. 1.6 , Debye-Waller factor.) For a general discussion of the relationship of GLand BCS – for equilibrium statics only – see Lec. 7.8 [omitted].

6.4 D. GINZBURG-LANDAU PICTURE OF CRITICAL CURRENT 587

J = ns~kθ/m∗. See Tinkham, Sec. 4.4. The physical reason is that, to change ∆θ (whilekeeping a fixed J), we’d transiently have a nonzero time derivative (θ(L)− θ(0)). It canbe shown (Lec. 6.5 ) that θ(x, t) = e∗V (x, t)/~, so we’d have a transient voltage acrossthe superconductor, meaning that work would be done on it by our current source.

Energy minimization

The gradient free energy under these assumptions can be written

Fgrad =~

2

2m∗|∇Ψ|2 =

~2

2m∗ns|∇θ|2 = |α| ξ2kθ

2 |Ψ|2 (6.4.12)

where we used ns ≡ |Ψ|2 and rewrote the coefficient using the definition of ξ (seeLec. 6.0 and Lec. 6.1 ). (We’ve assumed uniformity in space, so ∇|Ψ| = 0.) The totalfree energy density is

Fgrad + FL = −|α|(1 − kθ2ξ2)|Ψ|2 +

1

2β|Ψ|4. (6.4.13)

Minimizing the free energy (6.4.13), with respect to ns ≡ |Ψ|2 (as done in Lec. 6.1 forthe gradient-free case) ,

ns = |Ψ|2 = (1 − kθ2ξ2)ns0 (6.4.14)

where ns0 = |Ψ0|2. Thus

Js = e∗~kθ

m∗(1 − kθ

2ξ2)ns0. (6.4.15)

At kθ = 1/√

3ξ this has its maximum

Jc =2

3√

3e∗

~

m∗ξns0 (6.4.16)

I’ve checked that numbers from Landau’s critical velocity agrees with the last result, tofactors of order unity (see Ex. 6.4.2).

In samples larger than λ, the magnetic field mechanism of Sec. 6.4 X comes intoplay and we must consider some sort of “intermediate state.”

Another viewpoint on the GL critical current – Since kθ is analogous to the strain ina solid, the kθ maximizing (6.4.15) is analogous to the elastic limit of a pure solid: if we“stretch” the phase variation too much, the superconducting state “breaks” by goingnormal. Like the mechanical limits of real solids, the critical currents of real superfluidsare usually determined by defects or specially weak places. For classic superconductors,a typical value12 is

Jc ∼ 104 Amp/cm2. (6.4.17)

Digression on Galilean invariance

In his celebrated book on superconductivity, de Gennes asserted the choice of m∗ inthe GL theory is arbitrary; if so, the superfluid velocity vs is not a physical observablebut just a convenient way to parametrize a current. However, the velocity is physicallymeaningful as is clear from microscopic formulas such as that (above) for the εeff (k).

12Lifted from W. A. Harrison, Solid State Physics.


Of course, Galilean invariance isn’t exact in a solid: the electron dispersion relation notexactly of the form ~

2k2/2me and hence is changed by a Galilean boost. 13

The GL theory, which applies even to neutral superfluids said in (6.4.14) that |Ψ|gets reduced in a moving superfluid. That’s an apparent violation of Galilean invariance!To address this, we need to use a two-fluid model: the superfluid velocity represents anunderlying superfluid state with a nonzero phase gradient; the normal fluid velocity vn

accounts for the effects of a gas of elementary excitations in this background, which arein equilibrium with some other degrees of freedom at velocity vn. The actual statement,then, is the order parameter reduction depends on vs−vn; the GL theory had implicitlytaken vn = 0.

Josephson critical current: preview

High Tc or granular superconductors may consist of many small grains, with thesupercurrent propagated from grain to grain by coherent (pair) tunneling: this is aJosephson junction, which is the subject of the next lecture (Lec. 6.5 ). It will be shownthere that the junction’s current is

Ic sin ∆θ (6.4.18)

where ∆θ is the jump in phase (of Ψ) between the two sides; clearly Ic is the maxi-mum supercurrent of that junction. This is closely analogous to the order parametercritical current, since ∆θ is a discrete analog of kθ ≡ |∇θ|. (Notice how (6.4.15) and(6.4.18) have similar dependences on the phase difference, beginning linear and showinga maximum.)

6.4 X Critical current due to magnetic field

There is a second route to critical currents. Consider the following paradox: supercon-ductivity and magnetic field are mutually exclusive (Meissner effect); but supercurrentsmake magnetic fields; ergo there are no supercurrents in superconductors! In fact this isessentially true: the paradox’s resolution is that supercurrents only flow on the surface.More precisely they are the screening currents (see Lec. 6.2 [om. 2007]) of the magneticfields they create. and decay in the same exponential fashion, so they are essentiallyconfined to a layer that is about a penetration depth (λ) thick. If the current is sogreat as to produce a field that exceeds the critical field Hc, then it must drive thesample normal at that point, which might disconnect the domain of material in thesuperconducting phase.

Consider a wire of radius R λ at T < Tc carrying a current I ; Now, the fieldjust outside the surface is B(R) = 2I/cR from Ampere’s law. If B(R) > Hc, then thewire can no longer (all) stay superconducting, and a resistivity appears. (the “Silsbeeeffect”). Thus Ic = c

2HcRUsing the formulas for λ, Φ∗

0, and Hc in Table 6.1.1, Eq. (6.4.16) can be massagedinto the form

Jc =

√2c

16πλHc (6.4.19)

Eq. (6.4.19) shows that the order-parameter mechanism of Sec. 6.4 D dominates forsample thicknesses small compared to λ, where the current density can be high while

13In addition, k really means the crystal momentum, not the real momentum, so our use of Newton’sconservation laws is valid only when we can neglect umklapp, e.g. at low temperatures when thermalphonons all have small wavevectors.

6.4 X. CRITICAL CURRENT DUE TO MAGNETIC FIELD 589

the total field produced is small compared to Hc. In a sample thicker than λ, we knowthe current is confined within ∼ λ of the surface; then the magnetic field limit of thissection appears at the same Jc as (6.4.19) (within factors of order unity).

Partly restated the last paragraph: In the form (6.4.19), you see that along a domainwall (which always adjoins a normal region with the critical field Hc), the screeningcurrents necessarily approach the critical current. It’s also apparent that in a samplecarrying the critical current density, the magnetic field produced (according to Ampere’slaw) becomes comparable to Hc only when the sample thickness is at least λ.

Digression: intermediate state due to current flow

Tinkham (1st ed.) 3-5 discusses what happens for I > Ic. We encounter a paradox:for any cylindrically symmetrical current distribution, B(r = 0) = 0 so there mustbe some superconducting core along the wire’s central axis. This core is a cylinderextending (say) to a radius R′ < R. This core must carry all the current (it wouldhave no longitudinal voltage drop, so there is nothing to drive a current in the normal-state shell surrounding the core.) But B(R′) > B(R) > Hc, so we face the same oldcontradiction at the smaller radius R′.

The only way out is that there is a longitudinal voltage drop, and the supercon-ducting core region must be broken up into disconnected domains (in the longitudinaldirection). I believe the current passing from one domain to the next has such a highdensity that the metal gets driven normal by the current density, even though B = 0along the central axis.

This is another form of the “intermediate state” made up of coexisting domains ofsuperconductor and normal material (at field Hc.) The resistivity has been describedby a sort of effective medium theory; that is, one coarse-grains to a scale bigger thanthe domains but smaller than the sample size, and finds an effective uniform resistivityfor the mixture. Such theories are suspect, since the conductivity certainly depends onthe spatial arrangement of conducting regions.

Similar contradictions appear when a slab geometry is considered.

I am somewhat puzzled whether the whole notion is well-posed. If the field adjacentto the surface is nearly Hc, then Ampere’s law says ∇× B = (4π/c)J; since the fieldsare exponentially decaying this implies Js = cHc/4πλ is the current density adjacentto the surface. On the other hand, we see from (6.4.19) that (within G-L theory) afundamental limit on the current density anywhere is

√2cHc/16πλ, which is certainly

smaller.

Exercises

Ex. 6.4.1 Landau’s critical velocity in a superconductor I (T)

Try the dispersion of a particle in free space, ε(q) = |~q|2/2m∗, in (6.4.9): checkyou get |~q −m∗v|2 + const. Is this sensible?

Ex. 6.4.2 Landau’s critical velocity in a superconductor II (T)

The dispersion relation of a quasiparticle in a BCS superconductor at T = 0 has adispersion law with a sharp minimum at ε(kF ) = ∆0.

(a) Write the critical velocity vc this implies according to Landau’s argument.


(b) Note also that the microscopic theory gives a BCS coherence length as ξ0 =~vF /π∆0. Thus show that your answer from (a) agrees with the Ginzburg-Landau an-swer (within a factor of order unity!) provided we identify the G.-L. and BCS coherencelengths, ξ ∼ ξ0.

595

Lecture 6.5

Josephson junctions

A Josephson junction is one across which superconducting pairs can tunnel. The tun-neling current depends on the difference of the order parameter phase θ between the twosides, highlighting the physical significance of θ. One striking consequence is the Shapiro

steps, in which the DC (I, V ) curve shows steps in the current at voltages proportionalto the frequency of a perturbing rf field. Another one is the interference between twoJosephson junctions in parallel as a function of the flux between them, which is of greatutility since it is the basis of SQUID magnetometers.

The Josephson effect is not just relevant to artificial devices; imperfect superconduc-tors naturally contain Josephson junctions. In particular, granular superconductors (asin thin deposited metal films) can be modeled as a random array of superconductingislands connected by Josephson links. In high-temperature cuprate superconductors,every twinning grain boundary is a Josephson coupling; while in a perfect untwinned

crystal, the intrinsic coupling between CuO2 layers also has the form of Josephsontunneling. Sec. 6.5 X treats a REGULAR array of junctions.

The Josephson effect should be distinguished from single-electron (or quasiparticle)tunneling. In superconductors, that is a probe of the density of states of quasiparticles,which are modeled only at the BCS level (see Lec. 7.5 ). The microscopic calculationof Ic itself is also touched in Lec. 7.5 .

Sorry, a more complete overview of this chapter could be written.Unfinished topic no. 1: Josephson junctions as wide (rather than pointlike), and

effect of external flux which threads between. If a SQUID is the analog of 2-slit interfer-ence in elementary optics, then the wide junction is the analog of wide-slit diffraction. 1

Unfinished topic no. 2. Macroscopic quantum tunneling experiments, and recentapplications to qubits. 2

6.5 A Josephson equations

Consider two pieces of superconductor 1, 2 with a tunnel junction between them. Therespective voltages are V1 and V2 and the phases are θ1 and θ2. The Josephson junctionequations say

dθ

dt=e∗~V (t) (6.5.1)

1Tinkham, 2nd edition, has a very extensive treatment of such Josephson effects2See papers of S. Girvin and M. Devoret.


596 LECTURE 6.5. JOSEPHSON JUNCTIONS

I = Ic sin θ (6.5.2)

[written IJ some places below?] with the phase difference θ = θ1 − θ2, voltage V =V1 − V2, and I the current from 1 to 2. It is in fact a supercurrent, since it flows whenθ is constant i.e. when the voltage is zero. The maximum value of the supercurrentaccording to (6.5.1) is Ic, which therefore is called the critical current: if you tried topass a larger current, the junction would go normal (or part of the current would haveto follow a parallel, resistive path)

If you work out the differential AC response from (6.5.1) and (6.5.2) you see that theJosephson junction behaves roughly as an inductor. In circuit diagrams it is indicatedby the symbol ×. [I should add this to Fig. 6.5.3.]

Justification of Josephson equations

Let’s return to the idea of a single one-particle wavefunction ψ occupied by a macro-scopic numberNb of (Cooper pair) bosons, each with charge −e∗ ≡ −2e. This derivationis essentially a rerun of the “derivation” of time-dependent Ginzburg-Landau equationin Lec. 6.1X , except the bosons’ motion in continuous space is reduced here to hoppingon just two sites, representing the two pieces of superconductor. 3 The wavefunction is

|ψ〉 = ψ1|1〉 + ψ2|2〉 (6.5.3)

with normalization |ψ1|2 + |ψ2|2 = 1.The Hamiltonian has the form of a hopping model:

Hhop = −e∗φ1N1 − e∗φ2N2 − w(

b†1b2 + b†2b1)

. (6.5.4)

Since we’re modeling this with a noninteracting wavefunction, we can get the wavefunc-tion ψ1 from the more elementary version of (6.5.4),

Hhop = −e∗φ1|1〉〈1| − e∗φ2|2〉〈2|) − w(

|1〉〈2| + |2〉〈1|)

(6.5.5)

here φi are respective external potentials due (say) to gate electrodes. The hoppingcoefficient −w being negative ensures ψ1 and ψ2 have the same phase in the groundstate. [Should I call hopping −w12 instead?] The Schrodinger equation of motion forHamiltonian (6.5.5) is

d

dt

(

ψ1

ψ2

)

= − i

~

(

−e∗V1 −w−w −e∗V2

) (

ψ1

ψ2

)

(6.5.6)

This could also be writtendψi(t)

dt=i

~

∑

j

Hijψj(t). (6.5.7)

with hii = −2e∗φi and h12 = h21 = −w.The first part of the trick for obtaining the Josephson equations is to substitute in

Eqs. (6.5.7) the change of variables

ψi(t) = e−iθi(t)|ψi(t)|. (6.5.8)

3This section is adapted from the concluding chapter (chapter 21) in The Feynman Lectures in

Physics, vol. III (R. P. Feynman, R. B. Leighton, and M. Sands, Addison-Wesley, 1965).

6.5 A. JOSEPHSON EQUATIONS 597

For example

d

dt|ψ2|2 = −w

~(iψ∗

1ψ2 − iψ∗2ψ1) = |ψ1||ψ2| sin(θ1 − θ2) (6.5.9)

(I am now omitting the time arguments in the formulas.)The second part of the trick is to reinterpret the variables are representing macro-

scopic quantities. Thus, the total number on each site is Ni ≡ Nb|ψi|2 The numbercurrent from 1 to 2 is −dN1/dt = dN2/dt = Nbd|ψ2|2/dt; multiply this by −e∗ to getthe electric current,

I(t) ≡ I1→2 = −e∗2w

~|ψ1ψ2|Nb sin(θ1 − θ2) ≡ Ic sin θ(t) (6.5.10)

which is (6.5.2). (The actual variations in Ni are small compared to Ni, even in ananoparticle, so we can treat the factor |ψ1ψ2| as a constant.)

From (6.5.7), it also follows that

θi(t)

dt = −e∗

~Vi(t) (6.5.11)

– and (6.5.1) follows – provided we define

−e∗Vi(t) =d

d|ψi|2〈Hhop〉. (6.5.12)

The physical meaning is that what we call the “voltage” Vi is the (electro)chemicalpotential – the cost of adding or removing a charge. In the (usual) limit of nearlydecoupled islands, when w e∗|V |, the voltage simply is −e∗φi(t). In this case,|1〉 and |2〉 are practically eigenfrequencies, and then the familiar e−iEit/~ dependenceimplies the phase winds according to dθi/dt = −e∗φi(t), giving (6.5.1) trivially.

Microsopic formula for Josephson critical current

It can be shown (Lec. 7.5 ) from the microscopic theory 4 that the Josephson criticalcurrent is related to the resistance Rn the same junction would have in the normal state,by

IcRn =π∆(0)

e∗(6.5.13)

where ∆(0) is the gap at T = 0, a property of the material not the geometry. I emphasizethis holds for the Ic and Rn of each particular junction. 5

The l.h.s. of (6.5.13) is technically pertinent: the frequency e∗IcRn/~ ∼ π∆(0)/~turns out to be the junction’s maximum switching rate. Beyond the BCS theory, thel.h.s. need not equal the gap; e.g., it may be increased by tuning the system near to itsmetal-insulator transition. 6

4V. Ambegaokar and A. Baratoff, Phys. Rev. Lett. 10, 486 (1963); erratum, 11, 104 (1963).Actually they derived the T > 0 dependence; the basic formula was already derived by Josephsonand by Anderson, I think. Here’s an elementary-language explanation: Ic is proportional to w, a pair

hopping amplitude; it consequently is proportional to |A|2, where A is the hopping amplitude for asingle electron and the A∗ factor is for the other electron. On the other hand, as elucidated e.g. inLec. 2.1the normal conductance R−1

n is a probability rate for hopping of one electron, a process withamplitude A. Then by Fermi’s golden rule, Rn ∝ |A|2 for that process.

5A first step to seeing the plausibility of a relation of form (6.5.13) is to imagine the junction’ssurface area is varied while tunnel barrier is kept constant: then Ic scales as the area while Rn scalesinversely, so the product indeed would remain constant.

6J. M. Freericks et al


Alternate derivation

In the spirit of a future chapter on Bose condensation (Lec. 7.2), we could take a dif-ferent approach to (6.5.4): we can say the operators develop macroscopic expectations,and consider the small fluctuations around those. This depends on the factorization

bi = exp(iθi)

√

Ni; (6.5.14)

(The order of factors matters, since they don’t commute.) What are the commutators

of θi and Ni? Well, from [bi, b†i ] = 1, we have

bib†i = exp(iθ)Ni exp(−iθ) = b

†i bi + 1 = Ni + 1 (6.5.15)

i.e.[Ni, e

iθi ] = eiθi ] (6.5.16)

Now suppose for a moment that [Ni, θi] = A, a c-number (it could be anything that

commutes with θ). Then it’s easy to show

[Ni, θni ] = nθn−1

i A, (6.5.17)

for any power, and hence[Ni, f(θi)] = f ′(θ)A, (6.5.18)

where f(θ) is any analytic function and f ′(θ) is its derivative. Inspecting (6.5.16), wesee it works for f(θ) = eiθ, provided

[Ni, θi] = i (6.5.19)

– which means these are canonically conjugate quantum operators. 7 If 〈Ni〉 1,we are safe in approxmating Ni → Ni, a classical object; then the correspondenceprinciple assures us the quantum dynamics must reduce to Hamilton’s equations ofmotion. (Compare discussion of time-dependent GL equation in Lec. 6.1X .)

6.5 B Statics and dynamics of a junction

We next study the statics and dynamics resulting from these – classical! – equationsof motion. We have a (classical) Hamiltonian in terms of θi’s. It turns out the bosonpair numbers Ni are canonically conjugate to the θi. drives in the point that the phaseangle θi which turn out to be the pair numbers ni.

[Do not trust all signs in this section!]

Potential energy of a junction

The work done by the current at a rate IV must go into a potential energy U(θ):we have

d

dtU(θ) = IV = (Ic sin θ)

(

~

e∗

dθ

dt

)

(6.5.20)

whence the “potential energy”

U(θ) = −JJ cos θ (6.5.21)

7This all is not rigorous, due to the unpleasantness of dividing by zero when Ni = 0.

6.5 B. STATICS AND DYNAMICS OF A JUNCTION 599

with JJ = ~Ic/e∗ as the coupling constant. This cos(θ1 − θ2) coupling looks just likethe exchange coupling of two spins that happen to be confined to the XY plane, if θi

were the spin orientation. (See Sec. Fig. Ex. 6.5.3 for an array of such junctions as theanalog of a lattice of spins). The JJ equations of motion can be derived from (6.5.21)using the general “Hamilton’s equations” similar to Lec. 6.1X , or justified from (6.5.19)above.

Mechanical analogy of motion

In an isolated pair, the voltage is related to the charge by

Q(t) = CV (t) (6.5.22)

where C is a capacitance [I should call it C12] between the two superconducting islands;Q(t) and −Q(t) are the deviations of islands 1 and 2 from neutrality.

This dynamical equations from inserting (6.5.22) are

dθ

dt=

e∗~C

Q(t) (6.5.23)

dQ

dt= −Ic sin θ = −e∗

~

dU(θ)

dθ(6.5.24)

This is exactly the same as the Hamilton’s equations if we added a “kinetic” energyQ2/2C to (6.5.21). Here θ is proportional to a generalized coordinate, dU(θ)/dθ isanalogous to a force, and Q ∝ N1 − N2, is the momentum conjugate to θ, so thislooks analogous to the KE of a relative coordinate.) (WARNING: signs may be inerror.) Indeed, these are exactly Newton’s equations for the motion of a pendulum(parametrized so the pendulum shaft points along θ); it equally well describes theclassical motion of a particle in a sinusoidal potential. To round out the analogy, V (t)is a velocity, C is a mass, and Ic is like the force of gravity mg.

There is also an analogy to an LC circuit, since the Josephson junction looks likean effective inductor. It’s just that the present equations are nonlinear rather than aharmonic oscillator.

MOVE TO AC Jos. effect [This motion on time is mathematically equivalent to thespatial dependence of the displacement field in the Frenkel-Kontorova model, Lec. 3.3 .This can be extended to an analogy of spatially and temporally quasiperiodic motions,as well as commensurate phase-locking.]

It is interesting to note the resulting frequency of small oscillations, ω2J = e∗Ic/~C.

This is called the Josephson plasma frequency, because the charge is sloshing back andforth as in a plasma mode. That is, the Josephson plasma oscillation is analogous to thebulk plasma oscillation in the same fashion that the Josephson coupling is analogous tothe phase stiffness (superfluid density) and the Josephson current is analogous to theG-L supercurrent. Indeed, the plasma frequency is nonzero in a metal on account ofthe long-range Coulomb interactions, which are expressed here by 1/C. 8

Note too that there is usually an Ohmic shunt resistance R‖ in parallel with theJosephson tunneling, at least when T > 0. (Not to be confused with Rn.) It is dueto single-electron tunneling across the barrier (which in turn comes from Bogoliubov

8Do not confuse the “Josephson plasma frequency” with the “Josephson frequency.” The formermeans θ(t) oscillates back and forth in the bottom of a well in Fig. 6.5.1, dependent on the effectiveinductance Leff and some capacitance. The latter means θ(t) is sliding forever (with some damping)down the modulated slope of the washboard potential.


(θ)U

θ

(θ)U

θ

(θ)U

(c).(b).(a).

θ

Figure 6.5.1: Washboard potential. (a) zero Iext (b) small Iext (c) large Iext. The dotrepresents the instantaneous θ(t).

quasiparticle excitations founda at T > 0. This serves as viscous damping of the“pendulum” motion. [I should write it in the equation. It would also be nice to havea figure of a pendulum – perhaps in water, to represent the damping – add this to thechart of analogies.

Analogy to spin resonance

We can make an analogy of Josephson effect to spin resonance: θ(t) is analogous tothe angle of the spin around the field axis, and V (t) is analogous to the constant field.

The Josephson coupling is quite analogous to the exchange coupling of two spins

Uexch = −J12s1 · sj (6.5.25)

(see Lec. 4.0 (?)). When the spins are confined to the XY plane (in spin space), so~si = (cos θi, sin θi, 0), then Uexch = −J12 cos(θ1 − θ2), the same functional form as theJosephson effect. The closest analogy would be the exchange coupling of two ferro-magnetic particles in contact, which has form (6.5.25). Indeed, we can push the spinanalogy further: when the respective moments are not collinear, there is necessarilya spin current from 1 to 2, proportional (it turns out) to s1 × s2

9 In the continuumdescription of any bulk magnet, a twist (gradient) in the order parameter directionsimilarly corresponds to a spin current, analogous to the GL supercurrent.

6.5 C The Josephson effects

Attach leads to the two islands. The current is given by the Josephson formula, but itno longer needs to pile up on the islands so steady (and other) states are possible.

My explanations are a bit confused since I did not sort out the role of resistantnce.

DC Josephson effect

The equations allow solutions with nonzero IDC , but only with zero VDC ; the phaseadjusts itself. If you drive with a current bias, there’s a solution up as long as |IDC | < Ic,but not beyond.

9This is pertinent to SO(5) theory of high-Tc superconductivity, see Lec. 8.4 .

6.5 C. THE JOSEPHSON EFFECTS 601

I (t)J

θ (t)

h*eω

I(V)

extI

VRextI −

V(t)

(b).

(a).

(c).

(d).

V

V

(e).

t

t

t

2π

Figure 6.5.2: Mode-locking and the Shapiro steps. of mean DC current. The dashed linesare with no AC modulation; Thick curves are in the presence of voltage modulation v1 cos ωt,where ω was chosen to match the frequency of θ(t) when there is just a DC voltage. (a).voltage across junction (b). phase difference across junction θ(t) (c). IJ(t) = Ic sin θ(t), theJosephson current. The mean height difference between the curves is the mean DC current I.(d). Heavy lines – the horizontal axis, and spikes at special values – shows the (possible) meanDC supercurrent I(V ) as a function of voltage; nonzero values are possible when V = n~ω/e∗.In the presence of a shunt resistance R, and a fixed total current Iext, the actual voltage isat the intersection of the I-V curve and the dashed linees, so that V is pinned at n~ω/e∗ fora range of Iext values. (e). Schematic of observed Shapiro steps (usually one plots V as afunction of Iext.)


Basic AC Josephson effect

When you apply VDC , you get out IAC at a frequency (f = ω/2π) proportional tovoltage: f/V =0.48 GHz/µV. This is so high, it’s hard to detect! so the next effect isthe most commonly observed.

This means the I-V curve is weird: for all applied VDC , the resulting IDC is zero,except at VDC = 0 where any |IDC | < Ic is possible.

Current driveWe can imagine driving the system with a constant current source Iext so the equa-

tion becomesdQ

dt= Iext − Ic sin θ (6.5.26)

This is the equation of motion that would be obtained from an effective potential mod-ified as follows:

U(θ) → U(θ) − ~

e∗Iextθ (6.5.27)

i.e., a net downwards tilt is added to the potential; the result is called a “washboardpotential” (Fig. 6.5.1(b,c). The damping – which I omitted from the equations of motion– ensures that for small Iext the system finds steady-state equlilibrium in one of theminima of Fig. 6.5.1(b), which means there is zero voltage drop. For Iext > Ic, there isno minimum (Fig. 6.5.1(c)), which means the system rolls “downhill” at a fixed meandrift rate (owing to damping) ω0 ≡ dθ/dt. Now recall that dθ/dt is the voltage; hencethis state has a nonzero mean voltage, which by (6.5.1) is simply

VDC =~ω0

e∗. (6.5.28)

There are also high-frequency voltage oscillations at frequency ω0 (which come out asradiation). Thus we find that a junction with DC bias produces an AC voltage, at afrequency related to the DC voltage by (6.5.28).

Inverse AC Josephson effect: Shapiro steps

The “DC” Josephson effect is with V12 ≡ 0.What about the voltage driven case? A constant DC voltage VDC gives θ = ω0t

where ω0 satisfies (6.5.28). Then I(t) = Ic sinω0t which oscillates too fast to observeand averages to exactly zero: a DC voltage causes no DC current! However, if wesuperimpose a small AC modulation at this frequency, namely V (t) = V0 − v sinω0t,then integrating (6.5.1) gives

θ(t) = ω0t−e∗~

v

ω0cosω0t (6.5.29)

Now as seen in Fig. 6.5.2(c), this distorts the curve sin θ(t) so that it has a nonzeroaverage value I0. The way this shows up practically is that, when the junction is bathedin rf radiation at ω0 (realizing the AC modulation), then the measured I0 − V0 curveshows steps whenever V0 has the value (6.5.28) (or an integer multiple). (Fig. 6.5.2(e).

I should add for Fig. 6.5.2(d) that, at V = n~ω/e∗, just as at V = 0, there is acritical current, corresponding to a shift of θ(t) in Fig. 6.5.2(b); the shift actually shownin (b) is the optimal one.

The key physics here is locking, as a function of time, which is analogous to lockingas a function of space (see Lec. 3.1 ).

6.5 D. SQUID DEVICE 603

θ1b θ2b

θ2aθ1a

Imax(Φ)

2Ic

0Φ∗Φ

(b).(a).

B

Figure 6.5.3: (a). SQUID geometry. There are two leads 1 and 2 and two Josephson junctionsa and b. The superconducting ring is threaded with a magnetic field B (oriented out of thepage) giving a total flux Φ. The phase values at the junctions (θa1, etc.) are indicated. Wantedto add a part showing the circuit diagram with the symbol × for a junction. (b). Graph ofmaximum current Imax(Φ) as a function of flux Φ, where each junction has critical current Ic

and Φ∗0 is the Cooper pair flux quantum.

Alternative text for this effect –(No DC bias current) Now, let’s imagine we again consider the I-V curve, but we add

an AC modulation, by irradiating with microwaves at frequency ω (current modulation,but I think it’s equally valid if the effect is voltage modulation). Result: for mostapplied VDC , the resulting IDC is still zero. However, there are special voltages –not only VDC = 0, but at other discrete values of VDC – where a range of |IDC | is

possible. The fundamental VDC has the same coefficient V(1)DC = ~ω/e∗, and the others

are multiples, V(n)DC = nV

(1)DC .

These are called (Shapiro steps), I think because that’s the I-V curve when thejunction is in series with a resistance. However, the intrinsic junction I-V characteristic(described above) would better be called “Shapiro spikes.” 10

Since the frequency/voltage ratio is given exactly by fundamental constants, andsince frequencies are the most accurately measured quantities, the AC Josephson effectsprovide the currently most accurate physical standards for the volt (not its definition,which is derived). 11

Reverse AC Josephson effectApply AC voltage (letting DC float): get out oscillating current, no DC current,

and constant DC voltage appears.Fractional AC Josephson effectSubharmonic gap structure: I. K. Yanson et al, Zh. Eksp. Teor. Fiz. 47, 2091

(1964) [Sov. Phys. JETP 20, 1404 (1965)].

6.5 D SQUID device

The name stands for “Superconducting Quantum Interference Device”. Consider twoidentical junctions in parallel as shown in Fig. 6.5.3(a). The current is

I = Ic(sin θa + sin θb) = 2Ic sin1

2(θa + θb) cos

1

2(θa − θb). (6.5.30)

The mean phase difference (θa+θb)/2 can adjust itself to the supplied current, but θa−θb

is fixed by an extension of the flux-quantization argument of Sec. [6.4 C]. Namely, as we

10I was greatly helped by the website of Christoph Bergemann (Cambridge Univ. lectures, 2006):“www-qm.phy.cam.ac.uk/bergemann/SQClecture 5 ink.pdf”.

11One reference: /www.lne.eu/en/r and d/electrical metrology/josephson-effect-ej.asp


Figure 6.5.4: A 2D array of Josephson junctions. The black objects are pieces of supercon-ducting film on a substrate; each has a prong which almost touches a neighbor, thereby makinga weak Josephson link.

take a loop integral∫

A(r) ·dr, along the path shown, the supercurrents within the bulk(on either side) are negligible so that ∇θ(r) = e∗

~cA(r). We can break up the integralinto two parts [just like the derivation of flux quantization, in Lec. 6.4 ] obtaining

(θ2a − θ2b) + (θ1b − θ1a) =e∗~c

∮

dr ·A(r), (6.5.31)

that is,

θa − θb = 2πΦ/Φ∗0 + 2πn (6.5.32)

where Φ is the flux between the junctions, and Φ∗0 is the flux quantum (see Lec. 6.4 ).

Hence

|I | ≤ Imax(Φ) ≡ 2Ic| cos(πΦ/Φ∗0)| (6.5.33)

which is plotted in Fig. 6.5.3(b).Assume that there is a “shunt” resistance R‖ in parallel with the junction, and drive

it with a constant current a bit larger than 2Ic, then the voltage observed will be astrong periodic function of Φ/Φ∗

0. It is possible, for example, to measure the amount offlux which have entered or left the loop by counting how many oscillations V has madeand multiplying this number by Φ∗

0.Statically consider the potential E, a sum of the Josephson energy (6.5.21) and the

magnetic field energy. If JJ is sufficiently large, the system pays the cost of field energyso as to make the flux an exact multiple of Φ∗

0 and minimize the Josephson term. Thisis the basis of devices such as the “flux transformer”.

6.5 X Josephson junction array

Artificial arrays have been made of regular lattices of blobs of superconductor (“islands”)deposited on a two-dimensional substrate. We always assume that the phase is uniformwithin each island. They are connected by various kinds of weak links so there is apossibility of interaction of the phases and of currents between islands.

Typically we have a two-dimensional artificial square array of identical islands ofsuperconductor, with lattice constant a. (Of course, this is orders of magnitude longerthan the interatomic distances.) The islands couple to their four nearest neighbors(see Fig. 6.5.4 by weak links which are by Josephson junctions with critical current Ic,implying a coupling J as in (6.5.21).

6.5 Y. MACROSCOPIC QUANTUM PHENOMENA 605

6.5 Y Macroscopic Quantum phenomena

We’re treating the the phase θi and boson (pair) number Ni on particle i of super-conductor as conjugate coordinate and momentum, classically; then this should holdquantum mechanically as well. Our potential energy U(θi), in particular, should en-ter a Schrodinger equation for a wavefunction ψθ(θi). Then tunneling over potentialbarriers is possible as usual, but seems more surprising because θi is a collective degreeof freedom of an object with (say) 1010 electrons; hence the name macroscopic quantum

phenomena. The situations in which it actually works to use a single coordinate for anparticle have been discussed by Leggett; see Jim Sethna’s guest lecture about tunnelingand the environment (after Lec. 3.5 ).

Simple two state system?

The charge can slosh back and forth between islands 1 and 2, connected by a Joseph-son junction. It’s much like the expectation of any quantum particle in any two statesystem – except that the prefactor Nb makes it a macroscopic and deterministic cur-rent. Such behavior is known as macroscopic quantum coherence and the Josephsonsystem was a prime candidate for its observation; unfortunately, the dissipation fromR‖ normally destroys the coherence.

Bloch oscillations

Consider a single Josephson junction, which is above is described by the differencecoordinate θ (conjugate to N1 −N2). In the absence of external current, the potentialU(θ) (6.5.21) is periodic in θ, just like the potential felt by an electron in a crystal.From basic solid state physics, the eigenstates obey Bloch’s theorem; their eigenenergiesε(Kθ) are functions of a “wavevector” Kθ which lives in the Brillouin zone (−1/2, 1/2).

Now apply a small external current Iext. We noted this tilts the potential as inFig. 6.5.1(b), which is analogous to applying an electric field to the electron in a crystal.From the “semiclassical” equations of motion of basic solid state, such an electron has asteadily increasing wavevector. 12 The group velocity, as a function of the wavevector,has the period of the Brillouin zone, so the electron’s motion in coordinate space isoscillating rather than uniformly accelerating. Such “Bloch oscillations” are almostimpossible to realize because real electrons get scattered much faster than the Blochoscillation frequency.

In the Josephson junction, the Bloch oscillation 13 would occur with frequency

ωB/2π = Iext/e∗ (6.5.34)

The Bloch oscillations are dual to the usual Josephson effects. The usual effects areobserved when the Josephson tunneling JJ is large – specifically, the junction resistanceis small compared to the quantum of resistance for pairs, 2π~/e∗2 = 6.5kΩ. Then θ iswell defined but Ni is uncertain. The Bloch effects are seen when Ni is well definedbut θi is uncertain. Kuzmin and Haviland 14 say these effects are strongest near theborderline, i.e. when the when the charging energy of the particle is comparable to theJosephson coupling JJ . To quote them, this “results in a Coulomb blockade of Cooper

12In our analogy, remember, the wavevector Kθ is just N1−N2; in this light it may seem tautologicalthat this changes monotonically as current is driven from 1 to 2!

13K. K. Likharev and A. B. Zorin, J. Low Temp. Phys. 59, 347 (1985).14L. S. Kuzmin and D. B. Haviland, Physica Scripta T42, 171 (1992).


pair [=Josephson] tunneling and a time correlation of individual Copper pair tunnelingevents” at frequency ωB .

The Bloch oscillations are observed experimentally 15 as the usual Josephson oscilla-tions are, by the analog of the Shapiro steps. That is, you apply external AC field at ω,vary Iext, and see that the voltage locks for an interval of Iext whenever Iext = e∗nωB/2πfor integer n.

Uncertainty relation of θi and Ni

A first consequence is that we should have commutators (also derived above)

[θi, Nj ] = −iδij (6.5.35)

When 〈Nj〉 1 – you can check (6.5.35) by writing the boson creation operator

bi ≡ (〈Ni〉 + δNi)1/2eiθi . (6.5.36)

Assume [θi, δNj ] = c, insert into (6.5.36), and solve for c such that [bi, b†i ] = 1 as it

should. (This is a favorite trick in field theory approaches.)From (6.5.35), we get an uncertainty relation ∆θi∆Ni ≥ [2π] CHECK Coefficient

for WAVEPACKET.Now, the analog of the “kinetic energy” term is quadratic in the conjugate momen-

tum Ni. But Nie∗ is the charge on particle i, so this is none other than the chargingenergy, Q2/2Ci where Ci is the capacitance of particle i to a ground plane; of coursethere will also be capacitances Cij between particles i and j. Our equations (up tohere) have assumed a well-defined θi and hence an ill-defined number Ni. On the otherhand, a small Ci – as in a small enough particle – favors a fixed charge Ni, the so-called“Coulomb blockade”.

This has been demonstrated experimentally. 16

More recently, these quantum effects have been used for a possible implementationof a “qubit” as in a quantum computer.

6.5 Z Josephson effects in superfluids

Josephson oscillations have been observed in the neutral superfluid 3He, too. Theanalog of a tunnel junction is a “weak link” – a very small hole in a wall separating tworeservoirs of the same superfluid. This is the analog of the Dayem bridge mentionedabove; to work as a weak link, its diameter should be no larger than the coherence length(healing length) ξ. This is achievable in 3He rather than 4He, because ξ0 = 650Angstrom(at zero pressure) in 3He, whereas ξ0 is a few Angstrom in 4He. See S. Backhaus,S. V. Pereverzev, A. Loshak, J. C. Davis, and R. E. Packard, Science 278, 1435 (1997).

Sorry, I didn’t properly say what was actually done in the experiment.

Exercises

Ex. 6.5.1 Effective inductance of Josephson junction (T)

Consider a Josephson junction operated at a fixed phase difference γ (notice thisimplies a fixed DC current, and zero voltage). Now add a small modulation to the

15L. S. Kuzmin and D. B. Haviland, Phys. Rev. Lett. 67, 2890 (1991)16W. J. Elion et al, “Direct demonstration of Heisenberg’s uncertainty principle in a superconductor”,

Nature 371, 594 (1994).

6.5 Z. JOSEPHSON EFFECTS IN SUPERFLUIDS 607

current. The linearized behavior of the circuit is described by an effective (differential)inductance, Leff .

i) What is the formula for Leff , in terms of Ic, γ, and fundamental constants?ii) Consider the corresponding impedance at a finite frequency ω. What is its (di-

mensionless) ratio to the resistance quantum 2π~/e2 = 25kΩ?[For now, the only physical point of (ii) is to massage it into a form where you can

check that the units make sense. But it’s almost certain to mean something physically...maybe in 7.5, which touches the microscopic derivation of the Josephson tunneling.]

Ex. 6.5.2 Weak link as a wire

Consider two islands of superconductor connected, not by a tunnel junction, but bya thin wire of superconductor with length l and cross-section A. (This is known as a“Dayem bridge.”) Assume the phase θi is uniform within each island, but that it varieslinearly along the thin wire.

(a) What equation replaces Josephson’s equation (6.5.2)? (Eq. (6.5.1) will be justthe same.)

(b). Show that the potential energy analogous to (6.5.21) can be written

U(θ1, θ2) = −Kwire

2(θ1 − θ2)

2 (6.5.37)

reminiscent of a spring.

Ex. 6.5.3 Josephson junction array

Consider a Josephson junction array as in Fig. 6.5.4 with critical current Ic for eachjunction. Say θ(~r) is slowly varying from one side of the lattice to the next. Assumethere is no magnetic field. 17 Show that we can write the total potential energy as

U =

∫

d2~r~

2

2mns|∇θ(~r)|2, (6.5.38)

just like the gradient term of the (2D) Ginzburg-Landau free energy. What is the (2D)effective superfluid density ns?

The point of this exercise is to show how continuum elastic behavior emerges atlarge scales from discrete microscopic behavior. What’s amusing in Ex. 6.5.3 is thatit’s also continuum (that is, GL) behavior on the scale smaller than the island size.The situation is closely analogous to taking an array of balls on springs and finding itscoarse-grained elastic constants.

17If there were, we’d need to replace ∇θ → ∇θ − (e∗/~c) ~A).

609

Lecture 6.6

Vortex lattice state

In a “Type II” superconductor, defined by λ/ξ > 1/√

2 there is an interval of appliedfields H ∈ (Hc1, Hc2) for which the flux partially penetrates the sample, yet a kind ofsuperconducting phase order persists. This flux doesn’t form the so-called “intermediatestate” characteristic of Type I superconductors, see Lec. 6.3 C): since the domain wallenergy is negative (see Lec. 6.3 B), the domains “want” to maximize their total surfacearea by minimizing the domain size, and the flux dissolves in the superconducting back-ground. Due to flux quantization, the minimal domain has Φ∗

0 of flux – a vortex. Dueto mutual repulsion, the vortices form a regular lattice, with lattice constant enormouscompared to atomic spacings. This state is called the “Abrikosov phase” (also knownas the “mixed state”).

This lecture describes the equilibrium behavior of the Abrikosov phase. (The equallyimportant transport properties are left to Lec. 6.7 .) We start with one vortex, abeautiful example of a topological defect (as introduced in Sec. 1.5X), working outthe radial dependence of Ψ(r), B(r), and Js(r) around it. Then the vortex lattice(Sec. 6.6 C) is a crystal of line-like objects in place of (point-like) atoms It exhibits aspontaneous symmetry breaking of the vortex positions [also a symmetry breaking bythe phase of the superconducting order parameter Ψ(r).] Following the crystal analogy,Sec. 6.6 D obtains the parameters of an effective Hamiltonian as a function of the vortexpositions – analogous to the interatomic interactions that govern the physics of ordinarysolids – and macroscopic elastic constants.

Finally, we survey the phase diagram. Fig. 6.6.1(a) shows the (mean-field) phasediagram of a Type II superconductor, without disorder. (Compare the similar figure inLec. 6.0 .) In applied fields H < Hc1, the superconductor has a “Meissner” phase inwhich flux is perfectly excluded, just like a Type I superconductor below Hc. AboveHc2, it enters the normal metal phase in which flux penetrates completely, just like aType I superconductor with H > Hc. (The calculations of Hc1 and Hc2 – as instabilitiesof the superconducting and normal states respectively – are left to Secs. 6.6 W and6.6 X.) Sec. 6.6 Y also reviews phases with thermal fluctuations and/or disorder, whichare especially pertinent to high-Tc cuprates.

As Lec. 6.7 will explain, vortex motion controls the electrical conductivity and themagnetic relaxation. Indeed, a superconductor in the Abrikosov state is truly super-conducting only if the flux lines are pinned.

London limit

Sorry – this is partly built on material from Lec. 6.2 [om. 2007]which I didn’t hand


610 LECTURE 6.6. VORTEX LATTICE STATE

Abr.

Abr.

cT

Hc1

Hc2H

TMeiss.

N

B

Hc1

Hc2

HMeiss. N

(b). (a).

Figure 6.6.1: a. Phase diagram as a function of temperature and applied field. The tran-sitions are continuous. Dashed line shows the “thermodynamic” critical field Hc. (There isno transition at this line, but would be in the Type I case.) “Abr” and “Meiss” indicate theAbrikosov and Meissner phases. Dotted line shows an isotherm (line of constant temperature),which is plotted in (b). b. The mean field |B| inside the sample, as a function of applied field.Dashed line shows the behavior of a Type I superconductor with the same thermodynamic Hc,and dotted line the behavior of a normal metal.

out so far.We’ll also massage the phase gradient term in the GL free energy from Lec. 6.1 ,

Fgrad =1

2m∗ns|vs|2. (6.6.1)

Here (recall)

vs(r) =~

m∗∇Aθ(r), ∇A ≡ ∇θ − e∗A

~c. (6.6.2)

As advised in Lec. 6.2 [om. 2007], let’s convert every quantity possible to a length,so form dimensionless (physically meaningful) ratios can be formed λ and ξ. For ex-ample, ∇Aθ is a handy way to rewrite vs with dimensions of inverse length. Manipu-late Ginzburg-Landau coefficients using ~

2/2m∗ = |α|ξ2, then ns = |α|/β, and finallyns|α| = |α|2/β = 2|Fcond|, to obtain:

Fcond = α2/β = |α| ·∣

∣

∣

∣

α

β

∣

∣

∣

∣

=~

2

2m∗ξ2ns (6.6.3)

andFgrad = 2|Fcond|ξ2|∇Aθ|2. (6.6.4)

Eq. (6.6.4) is valid only in the London limit (Lec. 6.2 B), the approximation in whichns = Ψ2

0 everywhere.What we can derive: take the curl of (6.6.2); on the right hand-side ∇ × A → B.

After moving some constants around, you get

4πλ2

c∇× Js(r) = −B(r), (6.6.5)

the London equation. Meanwhile, from Maxwell, since Js is the only current source,

∇×B(r) =4π

cJs(r). (6.6.6)

6.6 A. ONE VORTEX 611

Combining (6.6.5) and (6.6.6), we get

∇2B(r) +1

λ2B(r) = 0 (6.6.7)

That was the reason, in Lec. 6.2 [om. 2007], we found an exponential decay B ∼ e−z/λinto the bulk; that’s why λ is called the “penetration depth.”

6.6 A One vortex

Superconductors are classified, as “Type I” or “type II” according to whether the (thenormal/superconductor domain wall surface tension) σW is positive or negative, re-spectively. In terms of the Ginzburg parameter defined by κ ≡ λ/ξ (valid within theGinzburg-Landau model, see Lec. 6.2 [om. 2007]and Lec. 6.3 [om. 2007]) Type I isκ < 1/

√2 and Type II is κ > 1/

√2. If a Type I superconductor is placed in a magnetic

field, we get the intermediate state, dicussed in Lec. 6.3 [om. 2007], made of coexistingdomains of superconducting phase (B = 0) and of normal phase (with B = Hc); thesize of those domains was a tradeoff between the field energy and surface tension σW ,depending on the macroscopic sample geometry.

On the other hand, in a Type II material, σW < 0 favors the domains to be asfragmented as possible. How far can this go? The minimum size domain turns out tobe a vortex.1 – a line-like object consisting of a topological defect in the order parameterphase, to which is bound one quantum of flux Φ∗

0 ≡ hc/e∗ = hc/2e (as introduced inLec. 6.4 .) Vortex lines are the basis of the class of the Type II superconductors whichincludes most of the technologically important ones.

Fields and currents around a vortex

Consider a normal domain with magnetic field through it. We can integrate thevector potential along a loop within the bulk of the superconducting portion that sur-rounds the normal domain, just as we did for a ring with an empty center. So the totalflux through the domain is an integer multiple of the flux quantum Φ∗

0. The vortexis a domain with just one Φ∗

0: vortices are quantized. Since magnetic field lines can’tterminate, the vortex must be a line-like object. The minimum energy configurationof fields and currents has cylindrical symmetry around the singular line, as shown inFig. 6.6.2(a).

The spatial dependence of the order parameter is

Ψ(r, φ, z) = f(r)ei(φ+χ) (6.6.8)

where (r, φ, z) is r in cylindrical coordinates (with vortex is centered at r = 0), and χis an arbitary phase. The boundary conditions are f(0) = 0 and f(∞) = Ψ0.

2 Thedetailed shape of f(r) depends on λ/ξ. The radial dependences, shown in Fig. 6.6.2(b),look qualitatively like those of the normal-superconductor domain wall as a functionof x (see Fig. [6.3.3 - omitted). A vortex looks just like a wall that has been wrappedaround an axis!

1The alternate term “flux line” is somewhat misleading since the vortex is defined by the order

parameter field. A vortex has a precisely defined line along which |Ψ(r)| vanishes and θ(r) is undefined.Similar quantized lines and vortex lattices exist in neutral superfluids, but nothing like flux is boundto them.

2If f(0) were nonzero, then |∇Ψ| ∼ 2πf(0)/r near r = 0 due to the angle dependence in (6.6.8),and the Fgrad energy term would diverge.


Ψ (r)

B(r),

r

(b).

ξ

λ

J (r)sJ (r)s

B(r)

r

θ

(a).

Figure 6.6.2: (a). Field B(r) and current Js(r) around a flux line. Recall that Js(r) isessentially the (gauge-invariant) gradient of the order parameter phase angle. (b). Radialdependence of order parameter amplitude |Ψ| and magnetic field B as a function of radialdistance from a flux line.

The tube-like region in which —Ψ(r)| ≈ 0 is the vortex “core” and has radius O(ξ),according to the Ginzburg-Landau theory. 3 Supercurrent Js and B decay over a length∼ λ. In the type II case with λ ξ, these exist much farther out from the singularitythan the core does; they each have a radial dependence ∼ df(r)/dr (see Fig. 6.6.2(a)again.) The magnetic field attached to a vortex, in the charged superfluid, is thusa secondary consequence of the currents (by Ampere’s law). The associated vectorpotential A(r) grows with radius from the vortex line. Since the current is actuallyproportional to the gauge-invariant derivative ∇Aθ, it follows that A(r) = ∇θ(r) atlarge r: that is, the currents around a vortex get screened for r > λ [As mentioned inLec. 6.2 [om. 2007], this is analogous to the screening of a charge in a metal at distancesbeyond the [Debye] screening length.]

Vortex charge

The “vortex charge” n is defined by a dependence einφ replacing the function in(6.6.8), which was the special case n = +1. (∆θ = 2πn when we traverse it coun-terclockwise). An antivortex (relative to these axes) is equally possible, with a phasefactor ei(−φ+α), that is n = −1. The name “charge” is used because (i) it is conserved– the only way to get a vortex out of your system is to push it to a boundary, orannihilate it with an antivortex; (ii) it acts as a source for other fields: indeed, thevortex charge n, phase field θ(r), current field Js(r), and vortex-vortex interaction arerespectively analogous to the electric charge, potential, field, and Coulomb interactionin electrostatics. 4

A vortex is a “topological defect” (see Lec. 1.5 and the comparative table there). Itsexistence can be inferred from measurements along a loop encircling the vortex withoutprobing the region around the vortex core. There is no way to eliminate it, either, exceptby annihilation with a vortex of the opposite sense. (In this and some other particulars,it is reminiscent of a pole of an analytic function in complex variable theory.)

3Which gets this wrong when T Tc. See, perhaps, Lec. 7.8 [omitted]about the relation of GL toBCS theory.

4Are there vortices with n = 2? Not usually: the current (gradient) energy in (6.6.14) scales asn2. Thus, if a single vortex with n = 2 broke up into two vortices, each with n = 1, the total costwould get reduced by a factor of roughly 2. That’s consistent with the negative domain wall energyσW of our Type II superconductor (see Lec. 6.3 B) – driving any normal domains to fragment – or withthe repulsion between vortices of the same sign. Vortices of opposite signs will attract and eventuallyannihilate each other. However, in the usual situation that vortices are induced by an external appliedfield, all vortices present have the same sign.

6.6 B. VORTEX FIELD CONFIGURATION AND LINE ENERGY 613

6.6 B Vortex field configuration and line energy

To continue the analogy to a domain wall, which had a surface energy σW , a vortexhas a line energy (per unit length) εv. Its importance is (i) it enters the tradeoff in freeenergy costs determining Hc1; (ii) it gives the line tension which determines an elasticconstant of the Abrikosov phase; (iii) the energy barrier to pass one vortex line acrossanother can be guessed 5 as ∼ εvξ: this is an activation energy in vortex dynamics atnonzero temperature. (See Lec. 6.7 ; it is probably the nucleation barrier for makingvortex line in the phase slip argument of Lec. 6.4 .)

The mathematical structure of the fields around a vortex is exactly analogous to theN-S interface problem (in Lec. 6.3 [om. 2007]). 6 Part of the vortex line cost comesfrom the “core”. a cylindrical region of radius ∼ ξ, in which the system is basicallynormal, |Ψ(r)| ≈ 0. The cost for doing this is the condensation energy |Fcond| times the“normal volume” ∼ πξ2L : thus

εcorev ≈ ccore|Fcond|ξ2 (6.6.9)

where ccore is a constant. 7 In general ccore is a function of λ/ξ; however, we’ll beinterested in the extreme type II limit λ/ξ 1 – where the vortex lattice is best defined– in which limit ccore doesn’t depend on λa, since the field is essentially uniform acrossthe core portion.

Beyond the core, where |Ψ| ≈ Ψ0, we can adopt the London approximation. Thephase gradient energy is written (6.6.4). In the vortex geometry, (6.6.2) reduces to

|∇Aθ(r)| =Φout(r)

Φ∗0

1

r(6.6.10)

where Φout(r) is the flux of the field outside radius r. (At large r, the vector potentialexactly cancels the gradient in (6.6.2) so the gradients are zero.) Now, most of the B

field is inside radius λ; hence, at r > λ, most of the gradient in (6.6.2) is canceled bythe vector potential – the first ratio in (6.6.10) is small and, roughly speaking,

∇Aθ ≈

1/r for r < λ

0 for r > λ(6.6.11)

What was undone in this subsection: (1) It should be emphaseized that the Londonlimit is just the extreme type II limit. (2) What’s nice about the London limit is there’sonly one length scale λ rather than two (3) Possibly some steps in the derivation wereskipped; in particular, I should have been explicit about Φout.

Deriving B(r) dependence

incomplete derivationEq. (6.6.7) governs the decay of mag. field away from the origin. In cyl. coordinates,

the divergence is

∇2 → d2

dr2+

1

r

f

dr(6.6.12)

5Seeing that the affected length of one line would be comparable to the core diameter of the otherline. [This could be a T.Q.].

6In the wall (free) energy (Lec. 6.3 [om. 2007]), the result was a multiple of the condensation energy|Fcond| times O(λ) − O(ξ). Here, in the line energy, λ and ξ appear not to the first but to the zero-th(as in logarithmic) order: namely, our result comes out proportional to |Fcond|(ln λ − ln ξ).

7Due to the variation of |Ψ| with r, there is a contribution from the ∇|Ψ| term in Fgrad which alsois O(|Fcond|ξ

2).


you can check that if f(r) = r−1/2e−r/λ, then ∇2f(r)/f(r) = (1/λa) plus correctionsof order 1/r2, so this is the correct asymptotic form. The exact result for B(r) (in theLondon approximation) is proportional to K0(r/λ), where K0() is a Hankel function (avariety of Bessel function), with

K(r/λ) ≈

const ln(λ/r) r . λ;√

πλ2r e

−r/λ r λ.(6.6.13)

at large distances. 8 (Which gives the matching of short and long coefficients, that Icould not have obtained from my asymptotics.) So if we considered only the gradientterm, dividing the total cost by the vortex line’s length L, we get

εv =F tot

grad

L= Fcond ξ

2

∫ λ

ξ

(2πrdr)1

r2∼ |Fcond| ξ2 ln(λ/ξ) (6.6.14)

for λ ξ; the log divergence was cut off by λ. (In a neutral superfluid, the superflowis just ∇θ (rather than ∇Aθ), and so a vortex has Js(r) ∝ 1/r at large r; the vortexenergy per unit length actually diverges in this case.) In a charged superfluid, of course,we must add the field energy contribution: it comes out equal to the gradient term,merely multiplying our estimate (6.6.14) by two.

I need to insert! See calculation in Tinkham Sec. 5-1; it’s a bit less straightforwardto see where the log comes from in this term.

The total formula with the correct coefficient is

εv ≈(

Φ∗0

4πλ

)2

ln

(

λ

ξ

)

= 4π|Fcond|ξ2 ln(λ/ξ) = 4π~

2

2m∗ns ln(λ/ξ). (6.6.15)

I used (6.6.3) to obtain the second formula on the right hand side of (6.6.15); it isobviously consistent with the estimate (6.6.14).

Even in high-Tc’s with λ/ξ > 100, the log factor in (6.6.15) is only ≈ 5. Modulothe log, (6.6.15) has a crude interpretation that εv is the condensation energy cost ofdriving the core to the normal state. 9

Numerical estimate of vortex line cost εv

We can numerically estimate the vortex line energy in terms of fundamental con-stants. Notice the final form in (6.6.15) didn’t involve ξ or Ψ or (to anticipate Part 7)the gap ∆; it only depends on ns/m∗ which is (roughly) the ratio of the free electrondensity and effective mass.

Now recollect the energy scales of a typical metal as treated in the simplest free-electron approximation. The Fermi wavelength is 2π/kF ∼ 0.1nm; the Fermi energy isEF ∼ ~

2k2F /2me, assuming m∗ ∼ me; and the electron density is n ∼ k3

F . Inserting allthis into (6.6.15) we get

εv ∼ ~2k2

F

2mekF ∼ EF kF ∼ 10eV/nm (6.6.16)

(ignoring the logarithm factor in (6.6.15), which will typically be of order unity). That’sa large energy. 10

8de Gennes, page 63.9For the Type I case (ξ λ), beyond r ∼ ξ, the currents have already dropped to zero negligible:

in this case term (6.6.9) dominates [there isn’t a log term, right?].10We’d expect the vortex line energy to vanish at Tc, since that is the point where there is no

6.6 C. VORTEX LATTICE: STRUCTURE, SYMMETRY BREAKING 615

6.6 C Vortex lattice: structure, symmetry breaking

Imagine it: an ordering not of material particles, but of topological defects in theabstract field Ψ(r) representing the subtle kinds of electron correlations that characterizesuperconductivity. Yet the vortices are as real as the tiles on your bathroom floor, andthey were imaged decades before the advent of scanned-probe microscopy. The latticesymmetry is triangular, like most other 2D problems (e.g. the Wigner crystal), sincethat’s the best packed lattice; in the present case, for a given density of vortices, thetriangular lattice has the largest inter-vortex distance.

Position order

As usual for a new kind of ordered state, we will discuss its order parameter, sym-metry, and elasticity. There are two symmetry breakings in the ideal Abrikosov lattice:vortex position order and phase order.

Position order means the locations of the vortex lines, like atoms in an ordinarysolid, and is the more important kind of order in a vortex lattice. (see Fig. 6.6.4 (c)).This is a symmetry breaking: the vortices could be displaced by any vector u withoutany change in energy, just like the atomic positions in a crystal. Solid order of this kindis characterized by a shear rigidity. As in Lec. 1.4 , we can define an order parameter

Φg(ρρρi(z), z) ≡ eig·ρρρi(z) (6.6.17)

I need a figure for these three experiments.

The position order has been observed experimentally in (at least) three ways. First,the magnetic moment of neutrons couples to electron magnetic moments, not just ofspin origin (Lec. 5.1 ) but also of orbital origin. Hence, the vortex array shows upas Bragg peaks in the neutron structure factor. (From small-angle neutron scattering,as the reciprocal lattice vectors are small ∼ 1/10 nm−1, since a ∼ 10 nm in classicsuperconductors.) 11 A second classic method to detect the vortex lattice is to decoratethe surface with a nanoscale magnetic powder, which (just like more macroscopic ironfilings) adheres preferentially near the vortex cores where the field is largest. Thepowder, once deposited, adheres even after the vortex lattice changes (e.g. as the fieldis turned off, or the sample warmed up.) It in effect develops a photograph of thepattern, which is read by scanning electron microscopy. Finally, vortices are observedby tunneling microscopy. (The tunneling rate is sensitive to the local value of the gapin the quasiparticle dispersion relation, which vanishes at the vortex center which is ineffect normal; in effect it is a probe of the pattern of ns(r).)

12

Phase order

A second kind of order is the symmetry-breaking of the phase θ(r) of the Ginzburg-Landau order parameter Ψ(r) = |Ψ(r)|eiθ(r) in the (ideal!) Abrikosov phase, just as in

difference between the superconductor and the normal metal. Well, Ψ0 ∼ (Tc − T )1/2 hence εv(T ) ∼ns(T ) ∼ (T − Tc) So εv indeed vanishes as T → Tc. However, we’d have to get extremely close to Tc

in order for the vortex-line-crossing barrier (∼ εvξ) to be less than the temperature T .11The original neutron observations by D. Cribier, B. Jacrot, L. Madhav Rao, and B. Farnoux,

Physics Letters 9, 106 (1964). are summarized by B. Serin in Superconductivity, ed. Parks. For theneutron experiments in high-Tc’s, see D. M. Paul and E. M. Forgan, in Physical and Material Properties

of High-Temperature Superconductors, ed. S. K. Malik (Nova Science, New York, 1992).12For articles see Physics Today, p. 17 (June 1990), and H. F. Hess et al, Phys. Rev. Lett. 62, 214

(1989).


Happ

triangularlattice

(a) (b).

Figure 6.6.3: Abrikosov vortex lattice. (a). viewed down the magnetic field. Arrow showscirculation of supercurrent, and star marks the singularity in the vortex core where |Ψ| → 0.Dashed line shows Wigner-Seitz cells, and an integration contour is marked (heavy) aroundone of them; the current is zero along these lines. (b). side view; the lines indicate vortices,and the arrows indicate the sense of flux which they carry. NOTE: Sorry, the little disks heredon’t really mean anything.

the ordinary Meissner phase, See Fig. 6.6.4 (a). 13 The order parameter field could bemultiplied by any uniform phase factor without changing the energy or anything drawnin (a), so this is a phase (i.e. gauge) symmetry breaking just as in the ordinary Meissnerstate in which Ψ(r) would simply be constant. 14

Since θ(r) winds through every possible angle around each vortex, it’s evident thatthe average value of Ψ(r) comes out to zero. This situation is analogous to that of anantiferromagnet, where the total moment cancels. The proper correct order parameteris a “staggered” order parameter, that is, a Fourier component of Ψ(g) evaluated at thefirst reciprocal lattice vector of the (2D) Bravais lattice.

[THE SAME IDEA , AMPLIFIED:] The Abrikosov state’s phase order is to theMeissner state’s phase order the same way an antiferromagnet’s spin order is relatedto the ferromagnet’s spin order. Indeed, just as the system-averaged spin of an an-tiferromagnet is zero, the system-averaged phase of the Abrkikosov state is zero (forthat matter, so is the system-averaged Js(r) which is more directly observable.) Theanalogy is that Js(r) or Ψ(r) in the superconductor, or S(r) in an antiferromagnet, haveno uniform Fourier component – the ordering occurs at a nonzero wavevector, You cansketch the analogous figure for the Meissner phase, in which Ψ has a uniform phase:that’s analogous to ferromagnetic spin order.

[In Fig. 6.6.4(b) and (c), of course the order parameter defined everywhere, not justthe lattice where I marked its values.

The continuous Goldstone symmetry is θ′(r) = θ0(r) +φ. As in the Meissner phase,there is no gapless Goldstone mode since it involves charge oscillations which lift itsfrequency to the plasma frequency. There is, however, a gradient-squared elasticity:that is, if you let the relative angle φ vary slowly in space, it corresponds to a netsupercurrent with local velocity v′

s(r) ≡ ~

m∇φ(r) superimposed on the ground statevortex currents, and its additional cost is 1

2m∗n∗s |v′

s(r)|2 where n∗s is a kind of average

of the superfluid density over the vortex array. This is the current that flows when you

13The unlikely case of a square lattice was used in order to simplify the figure. It is assumed thevector potential has the symmetry of the flux lattice and that it is nonsingular at the vortex cores.

14However, in order to define the aforementioned order parameter Ψ(g) or its correlation function,the phase θ(r) must be well-defined, but in view of gauge invariance this could take arbitrary values.The natural, non-arbitrary criterion is to pick the London gauge, ∇ · A = 0. See M. A. Moore, Phys.Rev. B 45, 7336 (1992). This article shows that thermal fluctuations of the vortex positions destroylong-range phase order in Ψ(r), although only over a correlation length scale of order millimeters.

6.6 D. EFFECTIVE HAMILTONIAN AND ELASTIC THEORY 617

attach a sample to an external current source.

Vortex lattice as soft matter: analogies

Note: In 4He, a similar vortex array develops in a sample in a rotating bucket;putting it in a non-inertial frame is analogous to putting the metal in a magnetic field.

Several other kinds of inhomogeneous system, can be viewed as periodic arrays ofsome kind of object living in a medium. Probably the closest examples are the modula-tions in an incommensurate crystal, such as charge-density waves (Lec. 3.4 ), since theycarry no mass with them. In particular, the Frenkel-Kontorova model (Lec. 3.3 ) formsa one-dimensional lattice of discommensurations (a kind of topological defects); the the-ory of the onset of this lattice (the commensurate-incommensurate transition) is quitesimilar mathematically to that of the lower critical transition in the Abrikosov lattice(Sec. 6.6.21). The vortex lattice is also reminiscent of the Wigner crystal (Lec. 1.4 W[omitted]) of charged carriers (electrons in a semiconductor, or held on the surface ofliquid helium). Later on (Lec. 8.3 ) we’ll encounter arrays of charged stripes in cupratessimilar to the high-Tc superconductors. All these lattices typically have periods muchlarger than the interatomic spacing; they can slide smoothly in principle, but tend toget pinned by disorder (inhomogeneities due to non-topological defects in the medium).

Another related example is a colloidal solid, formed in liquids by particles with µmdiameter (and comparable lattice constants), but there is no fixed pinning potential inthis case. (There are many such examples in the world of soft matter: see Chaikin &Lubensky, Principles of condensed matter physics.) These arrays have an elastic theoryarising out of interactions which are (typically) mediated by the underlying medium’sown elasticity, e.g. the wall-wall interactions in the Frenkel-Kontorova model. Most ofthe solids mentioned have overdamped rather than propagating sound modes, since thedefects (or particles) experience viscous drag from the surrounding fluid.

* *

**

*

* *

*i −i

−i

1

−1

−i i

−1

1

i

−1

1

* *

**

J (r)s

(c). (b). (a).

Figure 6.6.4: Symmetry breakings in the Abrikosov phase, looking down the vortex lines Asquare vortex lattice has been shown only because it’s simpler to show; actual vortex latticesonly rarely have square symmetry. (a). Positional order: locations of vortices shown by stars(magnetic field strength B(r) is maximum in these places); supercurrent Js(r) shown by arrows.(b). Phase order of the order parameter Ψ(r): pattern of exp(iθ(r)) is shown. (c). The samephases, indicated by arrows.

6.6 D Effective Hamiltonian and elastic theory

Sorry, this section is not quite integrated with Sec. 6.6 C.If a continuous symmetry is broken, then one has an elastic theory. This is the analog

of the elastic theory of an ordinary solid, if we take a flux line as being analogous to


an atom. This exists at a coarser-grained scale than the elastic theory of the Ψ(r) fielditself. In that sense it is a quite independent theory from the GL elastic theory; it isdependent, in the sense that the flux lattice elastic constants are functions of the GLparameters and of the lattice constant a.

Bending energy: vortex wandering term

The framework from Sec. 6.6 B can also be deployed to compute an effective Hamil-

tonian for a vortex configuration. Unlike an atom in a crystal, a single vortex line is anextended object with an internal degree of freedom:

The path of each vortex is given, to an excellent approximation, by a single-valued 15

function ρρρ(z), consisting of the two coordinates transverse to z. Whereas the state spaceof an ordinary 3D crystal of N atoms is parametrized by 3N arbitrary coordinates, herethe state space of N vortices is parametrized by 2N arbitrary functions of z, so thissystem is somewhat more complicated to describe.

Our first step must be to determine the one-vortex wandering term. In an isotropicsuperconductor, it is simply the arc length times the energy per unit length (stringtension) εv of a vortex, a term which resists bending the vortex lines away from vertical,

Ubend =

∫

dzεv

√

1 +

∣

∣

∣

∣

dρρρ

dz

∣

∣

∣

∣

2

≈ εvL+

∫

dz1

2εv

∣

∣

∣

∣

dρρρ

dz

∣

∣

∣

∣

2

(6.6.18)

We define l oriented along the average path of the vortex. (The sense is chosen by theright hand rule, such that the supercurrent Js(r) runs counterclockwise, when viewed

down l.)

Effective vortex-vortex Hamiltonian

The next step is the pair interaction of two vortex lines. Two straight vortex linesoriented along the field (z) axis, and separated by a distance r ξ, feel an effectiverepulsive potential

Veff (r) =Φ∗

02

8π2λ2K0(r/λ) (6.6.19)

(free energy per unit of length in the z direction). The effective interaction (6.6.19)could be derived by constraining a system to have vortices at (say) x = ±r/2, and thenfinding the configurations Φ(r) and A(r) which minimize the total Ginzburg-Landaufree energy, in the same spirit as the calculations that found the free energy of the N-Sdomain wall, or the core energy εv of one vortex. 16 The vortex-vortex interaction,mediated by the elasticity of the superconducting order parameter between vortices,is completely analogous like the elastically mediated interaction of crystal defects inSec. 1.5Y.

The short distance logarithmic behavior is the same form as the repulsion betweenlines of electric charge. In the case of a neutral superfluid, the logarithm is validout to infinity; the exponential factor in (6.6.19). (It was due to screening of the

15What about a vortex running in the sense opposite to the external field? It has an additional termΦ l ·H per unit length in the z direction. This strongly disfavors even a bit of back-tracking in a vortex.

16Indeed, K0() in (6.6.19) is the same function appearing in the profile of a single vortex. Theinteraction is essentially the product of the fields due to one vortex with the sources of the othervortex’s fields, as is typical of all mediated interactions: we first same that for interactions mediatedby the Fermi sea, Lec. 1.3 X .

6.6 E. MEAN-FIELD PHASE DIAGRAM 619

field by supercurrent, and the cancellation of A(r) and ∇θ(r) terms in the gauge-invariant gradient, so the circulation decays with distance: these are specific to chargedsuperfluids.)

In the dilute regime near Hc1, at least, we can approximate the vortex-vortex in-teraction by (6.6.19). Then we can write an effective Hamiltonian of the entire vortexarray as

Heff =∑

i

∫

dz εv

√

1 +

(

dρρρ

dz

)2

+∑

i<j

∫

dz Veff (|ρρρi(z) − ρρρj(z)|)

≡∫

dzH2D(ρρρi(z)) (6.6.20)

The first term in (6.6.20) is just (6.6.18) and the second term is (6.6.19). Eq. (6.6.20) isa good model description, but it is inaccurate that it includes only interactions betweenparts of different vortices at the same z coordinate. (Really, there are even interactionsbetween one line at one z and other bits of the same line at different z).

Elasticity

Imagine deforming the vortex lattice on scales much larger than its lattice constant.These are shears and compressions of the vortex array and have their own continuumelastic theory, analogous to that of an ordinary solid. Note, however, that the dis-

placement field u = ρρρi(z) − ρρρ(0)i (z) in the present case has just two components – a

lengthwise displacement of a vortex line does not make a different state. Still the fieldis a function of 3 dimensions, u(x, y, z). The first term of (6.6.20) gives rise to z shearelasticity (terms proportional to |∂u/∂z|2, while the second term gives rise to shear andcompression elastic terms ∂u/∂x and ∂u/∂y.

What about the Goldstone mode, associated with the continuous symmetry break-ing of the vortex positions? In real life, the motion of a vortex in a superconductingmetal is highly overdamped, not only due to random pinning, but also to Ohmic-likedissipation by the electrons. 17 Hence the vortex lattice has no sound mode, but onlyan overdamped mode is expected.

6.6 E Mean-field phase diagram

The Abrikosov phase (see Fig. 6.6.3) consists of a regular array of parallel vortex lines,which (viewed down the axis) form a triangular lattice, 18 Each line is associated withexactly one flux quantum, so the mean magnetic field B is related to the lattice constantby Φ∗

0/Ca2, where Ca2 is the unit cell area and the coefficient is C =

√3/2 for a

triangular lattice.

The “thermodynamic” critical field Hc is defined by Fcond = H2c /8π, as in Type I

17See Lec. 6.7 for more about this. Note also that although an s-wave superconductor has a gap toelectronic excitations in its bulk, the vortex core is like a one-dimensional metal and supports gaplessexcitations. This [should be] discussed in Lec. 7.4 on the Bogoliubov-de Gennes equations Even in 4He,an utterly pure material, vortex lines are damped by emitting phonons, I believe, as they move relativeto the fluid.

18Anisotropic Type II superconductors are more complicated in that (i) the flux lattice is distortedaway from triangular symmetry (ii) if H is not aligned to a symmetry direction, then B will not bealigned with H either (even though the sample is needle-shaped and aligned with H.)


Ψ

(r)Ψ

Ψ , B

B(r)

r

(r)Ψ

Ψ , B

B(r)

r

(b).

λ

ξ

λ

0

a > a

(a).

Figure 6.6.5: Superconducting order parameter Ψ and magnetic field B as a function ofposition in the vortex lattice. (a). For H & Hc1, with a ≥ λ; (b). for H . Hc2, with a ∼ ξ.OOPS: A vertical dashed line in (a) marks the places where |Ψ| → 0, but was meant to be cutout. Note that case (a) only occurs just above Hc1. The Meissner phase has (|Ψ|, B) = (Ψ0, 0)and the Normal phase has (|Ψ|, B) = (0, H).

superconductors, but nothing happens at this value of applied field. It turns out that

Hc1 ≈ lnκ√2κHc (6.6.21)

andHc2 =

√2κHc (6.6.22)

where κ ≡ λ/ξ is the Ginzburg-Landau parameter so Hc is roughly the geometric meanof the two critical fields. [Hc1 is the field at which flux first enters the superconductorin equilibrium, and Hc2 the point at which it goes completely normal.]

The relation to ξ and λ is derived simply by using the identities relating Ginzburg-Landau parameters (from Lec. 6.1 ). It’s easy to remember the result:

Hc1 ∼ lnκΦ∗0/λ

2 (6.6.23)

andHc2 ∼ Φ∗

0/ξ2. (6.6.24)

The |Tc −T | critical behavior shown in Fig. 6.6.1(a) simply reflects that of Hc(T ); sinceξ and λ both behave as |T − Tc|−1/2, κ remains essentially temperature-independent.

We can also notice that (if we forget the logarithm in (6.6.23)!) Hc1 = Hc2 = Hc

according to the above formulas, exactly when κ = 1/√

2; for smaller κ values, Hc1

would be larger than Hc2, so what happens instead is the first-order transition directlyfrom the Meissner to the normal state (at Hc). This criterion for the critical value ofκ agrees exactly with that one (Lec. 6.3 [om. 2007]) that the interface free energy σW

vanishes – that seems rather deep to me, since there are not any walls at all in thecalculations of either Hc1 or Hc2 (see sections below).

Both transitions are continuous: the penetrating flux B (averaged across a unit cell)increases from zero to H between Hc1 and Hc2. Just above Hc1 (Fig. 6.6.5(a)), thevortices are far apart and have the shape described in Sec. 6.6 A – in particular theperturbations of Ψ and B from their equilibrium values decays with r to a negligiblevalue well within the Wigner-Seitz cell. On the other hand, just before Hc2, the spacingbetween the vortex lines (defined by the places where Ψ = 0) is only a coherence length,

6.6 W. LOWER CRITICAL FIELD HC1 621

which is much shorter than the penetration depth (Fig. 6.6.5(b)). 19 The Abrikosovlattice consists merely of small (almost sinusoidal) modulations around an average statewhich barely differs from the normal state: everywhere, the field B(r) is only slightlyreduced from the external H , and the order parameter |Ψ(r)| is never more than a smallfraction of Ψ0. (As it were, the single-vortex configuration of Fig. 6.6.2 gets truncated,before it even gets started, by the unit-cell boundary of Fig. 6.6.3.)

So it is not quite evident why the flux per unit cell Φ is exactly Φ∗0, even if the lattice

constant is much smaller than the natural radius of a lone vortex. To see this, let firstnote that, by symmetry, the component of supercurrent Js along the Wigner-Seitz celledge vanishes. 20 Thus the loop integral

∮

Js(r) · dr along the cell boundary vanishesjust as surely (though for different reasons) as the integral along a loop along the interiorof a solid ring (see Lec. 6.4 A ); from this it follows that Φ = nΦ∗

0. Since vortices repel,any case with n > 2 would split up into individual vortices, so Φ = Φ∗

0 as asserted.

Mathematical details of the estimation of Hc1 and Hc2 are left to Secs. 6.6 W and6.6 X, where they are analyzed as instabilities of the superconducting and normal states,respectively. There are actually several phases of On closer examination (with statisticalmechanics), the Abrikosov phase breaks up into several phases (Sec. 6.6 Y).

6.6 W Lower critical field Hc1

Now we turn to some details of computing the equation of state shown in Fig. 6.6.1(b).).If we think of the vortices (in a 2D projection) as being particles, then the externalmagnetic field looks like a “pressure” term. The story of the vortices here is veryanalogous to that of the domain walls (discommensurations) in the Frenkel-Kontorovamodel of the commensurate-incommensurate transition (Lec. 3.3 ).

This analogy is detailed in a figure and table which were included 636/1999, butomitted in 2003.

There is a certain field, above which the free energy for a single vortex to come ingoes negative. That free energy, per unit length, is a combination of the vortex coreenergy and the “magnetic pressure” energy.

Beyond that field, the single-vortex energy tells us merely that the system is unstableto the entry of vortices, but does not tell us what is the stable state – i.e., how manyvortics enter, or how they arrange themselves. We can, however, answer both thosequestions using the vortex-vortex effective interaction found in (6.6.19); that will sufficefor finding how the ground state lattice constant a is related to the external field.

Useful conversions

For use below, note the following handy manipulations (mostly repeated from thetable in Lec. 6.1Y ).

Note that

Hc =

√2

4π

(

Φ∗0

λξ

)

(6.6.25)

19NMR can measure the profile of B(r) as shown in Fig. 6.6.5.20This is easily checked when you write, as an approximation, the linear superposition of the current

fields of the vortices. To show it generally, note that a mirror reflection in the vertical plane containingthe cell edge reverses all the velocities (time-reversal) while on the other hand it leaves fixed the velocitycomponent within the plane.


and so

|Fcond| =H2

c

8π≡ 1

(4π)3

(

Φ∗0

λξ

)2

. (6.6.26)

Details

I showed in Lec. 6.2 V [omitted] that given your sample has a field H on theoutside 21 the effective magnetic energy density is 22

1

8π|B(r) −Happ|2. (6.6.27)

This term competes with the vortex-vortex repulsion (6.6.19).Sorry, I should write out the integral for the total free energy. This integral can

be rearranged in the same fashion as we did with the continuum model energy of theFrenkel-Kontorova model (Lec. 3.3 ). The result is that H couples like a chemicalpotential (per unit length) favoring vortices, thus competing with the vortex core energyestimated in Sec. 6.6 B.

The resulting energy is

(Ev − 1

4πΦ∗

0H)NvZ + const (6.6.28)

where Ev is the energy per unit length of the vortex, Nv is the total number of vortices,and Z is the size of the system in the direction the vortices run.

The transition at Hc1 is an instability of the vortex-free state to the entry of a fewvortices. Thus, it occurs precisely when the coefficient in (6.6.28) goes negative:

Hc1 = 4πEv/Φ∗0 (6.6.29)

Now in Sec. 6.6 B we estimated that Ev = CFcondξ2 for some prefactor C of order

unity (which probably contains a logarithm, but never mind), then Ev = CH2c /8π. We

replace one factor of Hc using the the relation Hc =√

24π

(

Φ∗

0

λξ

)

. 23 The end result of this

manipulation isHc1 =

√2C(ξ/λ)Hc = Cκ−1Hc (6.6.30)

as was claimed.

6.6 X Estimation of upper critical field Hc2

We ask about the instability of the normal state to developing a small Ψ(r). (RecallFig. 6.6.5(b).) We assume the field is along the z axis.

Notice that

(i). In this limit, B(r) ' H and is virtually uniform within the superconductor.Thus we needn’t bother with the field energy (6.6.27); it’s just a constant. (SeeFig. 6.6.5.)

(ii). Since the order parameter Ψ(r) is small everywhere when H is near Hc2, itwill be OK to linearize; we needn’t bother with the 1

2β|Ψ(r)|2 term in the Landaufree energy.

21or in standard electromagnetic parlance, the magnetic induction is H on the inside.22This is completely analogous to the mismatch energy in the Frenkel-Kontorova model.23These handy identities are from the table in Lec. 6.1Y .

6.6 X. ESTIMATION OF UPPER CRITICAL FIELD HC2 623

Thanks to (i) and (ii), our total free energy is simply a quadratic form in Ψ(r):

Ftot ≈∫

d3r

~2

2m∗

∣

∣

∣

∣

(∇− ie∗

~cA(r)Ψ(r)

∣

∣

∣

∣

2

+ α|Ψ(r)|2

(6.6.31)

(Here ∇ × A = B = H . Recall α = −|α|, since certainly T < Tc(0), where Tc(0) iscritical temperature in zero field.

In principle, then, we should completely diagonalize the quadratic form (6.6.31).At B > Hc2, every eigenvalue will be positive, and Hc2 will be defined as the largestvalue of B at which any eigenvalue crosses through zero and becomes negative. Now,if Ψ(r) were an ordinary (normalized) single-particle wavefunction, then (6.6.31) wouldbe simply the energy expectation 〈H〉 of a certain Hamiltonian

H ≡ − ~2

2m∗

(

∇− ie∗

~cA(r)

)2

− |α| (6.6.32)

We want to see where the lowest eigenvalue E0 crosses zero; this is just the ground state

eigenvalue E0 of H|ψ〉 = E0|ψ〉.But – apart from the constant term added at the end – (6.6.32) is simply the Hamil-

tonian for a free charged particle in a magnetic field H . The solution is thus a LandauLevel wavefunction (see Lec. 9.1 [omitted]). The quantum numbers are kz, the wavevec-tor in the direction of the field, and n, the Landau level index. The energy is

E(n, kz) = (n+1

2)~ωc +

~2

2m∗k2

z − |α|. (6.6.33)

whereωc(H) ≡ e∗H/m∗c (6.6.34)

would be the cyclotron frequency of the free charged particle. The minimum energy isE(0, 0). The critical field is where this passes through zero, i.e. it is given by

1

2ωc(Hc2) = |α| (6.6.35)

Inserting the definition (6.6.34) and rewriting |α|, (6.6.35) becomes 12~(e∗Hc2/m

∗c) =~

2/2m∗ξ2, or

Hc2 =1

2π

Φ∗0

ξ2(6.6.36)

where I inserted the definition of the flux quantum Φ∗0 ≡ ~c/e∗. From (6.6.36), it’s clear

that the unit cell of the vortex lattice is indeed ∼ ξ at Hc2. Finally (using Hc ≡√

24π

Φ∗

0

λξwe find

Hc2 =√

2λ

ξHc (6.6.37)

as in (6.6.22).What this section has done is, conceptually, just like the Landau theory, within

which the critical point was exactly the place where the coefficient of the quadratic formFL(Φ) went negative. 24 Once that coefficient was negative, the state Φ = 0 becameunstable; if we are curious to what state (e.g. what value of Φ) Φ is unstable, we mustwork harder (e.g. must take account of and understand the nonlinear terms). (Thepresent case was a bit more complicated than Landau theory because the instabilityoccurs to a spatially modulated state, not a uniform one. )

24That was discussed in Lec. 1.4 C.


6.6 Y More phases: thermal fluctuations and pin-

ning

[Since flux lattice is a kind of solid, it melts when the temperature is raised sufficiently.]Such thermal disorder effects began to be studied mainly in the 1980’s, as they are wereunimportant in the classic superconductors but are overwhelming in high Tc cuprates.Depending on the temperature, and the degree of pinning disorder, there are several

possible phases for the vortex array, with different degrees of long-range order (seeFig. 6.6.6, in Sec. 6.6 Y.) The phase diagram is relevant technologically: as Lec. 6.7 willshow, the Abrikosov state has nonzero intrinsic resistivity – but can be made zero ifvortices are prevented from moving. That in turn depends on the thermodynamic phase.

To understand high-Tc superconductors at temperatures comparable to Tc, we needto do the statistical physics of the Abrikosov state. To see why, look back at Sec. 6.6 Aand note that (crudely) the line energy εv – which governs the bending stiffness – is in-dependent of the gap ∆ (which is normally comparable to Tc). Thus, in standard metalsuperconductors – except in a critical regime very near to Tc – εv/T was very large andthermal fluctuations were unimportant. On the other hand, in high-Tc superconductors,the phase diagram is dominated by partially disordered phases with fluctuations: notonly because εv/T is smaller, but (in cuprates) the extreme anisotropy of the layeredstructure makes the bending rigidity very small for vortices normal to the planes. (In-deed, in the “pancake vortex” regime, the vortices in one layer are totally decoupledfrom the next one, except to the extent that the magnetic flux must be conserved fromone layer to the next.)

Thermally excited fluctuations

The simplest configuration is a perfect (triangular) lattice extended in the z direc-tion; at T > 0 this will include small thermal fluctuations of each vortex line aroundits average position. This can be described within the common elastic models, in whicheach vortex line i has its assigned position Ri in the 2D plane, and all energies arewritten in terms of the displacements ρρρi(z) −Ri (assumed to be small).

But larger deviations are possible which are more radical, as they affect the vortexlattice’s topology; different phases can be classified according to whether such defectsare present in the limit of large scales. First, dislocation in the vortex lattice appearsas a dislocation of the 2D lattice in any constant z slice; a dislocation may glide fromone location in the ρρρ plane to another, as a function of z. Second, it’s possible for twovortices to switch places occasionally; this won’t disturb the long-range crystal order ofthe vortex positions. Such a switch is called an entanglement since the world lines of thetwo vortices are twisted as in a braid. (See Fig. 6.6.6(b)). This has major consequencesfor the dynamics (Lec. 6.7 ).

Pinning

The situation grows more complicated when one introduces pinning disorder. Apinning site is a place (in real materials, it is usually bigger than one site) at which thevortex core energy is lower than the average value, so it is energetically favorable forvortices to pass through such sites. Pins are caused by, perhaps, impurity atoms, or –more likely – inclusions of different composition and crystal defects (such as dislocationsor crystallite domain walls.) These are places where it already isn’t superconducting –say defects, grain boundaries, or tiny inclusions of a non-superconducting phase. Even in

6.6 Y. MORE PHASES: THERMAL FLUCTUATIONS AND PINNING 625

its T = 0 ground state, an isolated vortex line deviates from being straight – fluctuatesas a function of z – in order to pass through pinning sites; this has to be balancedagainst the cost of its line tension energy (εv).

At higher external fields, forcing a the flux lines to be closer, the repulsive forcesbetween them are strong and it’s wrong to picture individual lines as following thedisorder. Instead, all the lines in a region much larger than a will deviate together ina direction for which the net pinning energy (summed over all these lines) is favorable.I don’t mean that many lines go through the same pinning site. Recall that, due totheir repulsive interactions, many nearby vortex lines must all deviate together. So, thedeviation in a given portion of the sample is determined by the resultant of the pinningforces acting on all the vortices in that region. These forces will mostly cancel, but aswith any sum of independent random terms, there will be a O(±1/

√N) fraction that

doesn’t cancel.

Pinning is central to vortex dynamics (Lec. 6.7 ), but our concern in this section isthe static equilibrium statistical mechanics.

Phase diagram

[2003 – Sorry, this subsection is not very coherent. See the literature cited, or readthe 2003 term papers by some of the class members!

Long ago, it was shown that – in the elastic model at T = 0 – pinning disorder issufficient to destroy the long-range order, i.e. the lines will wander arbitrarily far.

It is presently believed that in the presence of random site pinning, there are threephases (see Fig. 6.6.6). At the lowest temperatures/fields, one has the so-called “Braggglass” in which there are no dislocations; this lacks true Bragg peaks but it has power-law correlations, much like a 1D Luttinger liquid (Lec. 1.9 ) or a 2D classical crystal(Lec. 1.6 ).

In a vortex array, a vortex wandering due to pinning pays an additional cost becauseit must push against the repulsive potentials of its neighbors in order to deviate. If thiseffect is strong enough, free dislocations appear and it is called a vortex glass phase.[TO CHECK THIS]

Another effect is that the vortices will wander transversely due to thermal fluctu-ations; this can cause a melting of the lattice into the vortex liquid; entanglementscharacterize this phase. 25

References: An accessible article, mainly about the vortex glass, is “Are supercon-ductors really superconducting” by D. A. Huse, M, P. A. Fisher, and D. S. Fisher, Nature358, 553 (1992). Some review articles, mainly about the vortex fluid, are G. W. Crab-tree and D. R. Nelson, Physics Today, April 1997, p. 38 (the most accessible); “Lineliquids” by D. R. Nelson, Physica A 177, 20 (1991); “Dynamics of the flux-line liquidin high-Tc superconductors” by M. C. Marchetti and D. R. Nelson, Physica C 174,40 (1991); “Correlations and transport in vortex liquids” by David R. Nelson, in Phe-

nomenology and Applications of High Temperature Superconductors (Proceedings of LosAlamos Symposium, 1991). (The latter two are more technical reviews.)

25Melting to the vortex liquid is governed by the Lindemann criterion – when the thermal wanderingof each vortex around its mean position grows to 10% or so of the lattice constant – much like ordinarycrystals.


Figure 6.6.6: (a). Flux lattice with pinning: phase diagram. Here BG = “Bragg glass”, VG =“vortex glass”, and VL = “vortex liquid”. [From A. van Otterlo et al, cond-mat/9803021. Tobe compared with experimental data of K. Delgiannis et al, Phys. Rev. Lett. 79, 2121 (1997)].(b). The entangled flux liquid.

6.6 Z Mapping of quantum Bose system in d = 2 to

Flux lines

See Lec. 1.4 E and Lec. 1.4 Zabout quantum phase transitions.

One interesting example of a quantum system mapping to a classical one is bosonsin d = 2 mapping to an array of directed lines. A configuration of the quantum systemis a set of particle positions in d = 2. Thus, the corresponding classical system is a“spaghetti” consisting of world-lines which are stretched out in the stacking direction(that is, a given line cuts a given layer only once – it can’t double back on itself).

At Tq > 0, the classical system is finite in the stacking direction. When a world-linereaches the system’s end, it must (due to periodic boundary conditions) connect up toa line in the starting layer. However, since bosons are identical, it need not be the sameworld-line we started on! , The lines are free to braid around each other: hence, whenone follows the world lines from 0 to τ , a permutation may be induced.

As we raise Tq, we are shortening the extent of the system in the stacking direction,and it is harder for the world lines to wander far enough away from their initial locationin the first layer to create a nontrivial permutation. This is one picture of why, atsufficiently large Tq , the system undergoes a transition out of the Bose-condensed stateto the normal state, where each world line is a trivial straight strut in the stackingdirection, connecting to itself.

Mapping of “entangled” phase of lines and Bose-Einstein (superfluid)phase of bosons.

It turns out that the Bose-condensed phase of the quantum system corresponds tothe case where the world lines don’t connect up and, indeed, the cyclic permutations(into which the total permutation factors) diverge in length as the system size does. 26

It turns out that some other physical properties, such as the superfluid density, canbe extracted in simulations from expectations of geometrical properties of the world-

26These are precisely the long permutation loops occurring in Feynman’s picture of the wavefunctionof superfluid 4He. See R. P. Feynman, Statistical Physics, at the very end.

6.6 Z. MAPPING OF QUANTUM BOSE SYSTEM IN D = 2 TO FLUX LINES 627

lines 27 This relationship, of course, also carries a d = 3 quantum system to a set oflines in d = 4.

We can make this more precise. Let b(r) denote the annihilation operator for thebosons at a point r in (2D) space. Sorry – the 2D coordinate should be called ρρρ. Theorder parameter correlation for this 2D superfluid is 〈b(r)†b(0)〉 (see Lec. 7.1 ). So, when

limr→∞

〈b(r)†b(0)〉 6= 0 (6.6.38)

we say superfluid LRO exists 28

To explain the mapping, we must invoke the transfer-matrix technique to calculatethe classical partition function, worked out in Lec. 1.4 Z. A quantum particle in d dimen-sions became a line in a d + 1 dimensional classical model. The dominant eigenvectorof the transfer-matrix is the same as the ground state wavefunction |Ψ〉 of the quantumsystem.

Adopt the formalism where we calculate 〈exp(−τ Hhop)〉 by slices, where τ is asmall time step. What does an expectation with b(r) correspond to in this picture? Ithas a matrix elements between states which are identical except the second one has aboson removed at r; thus, in the classical system, b(r; t) acting at (imaginary) time tcorresponds to a world line that suddenly stops at positoin r and time t. Conversely, afactor b(r′; t)† is a world line that starts at r′, at time t.

Given the ground-state wavefunction |Ψ〉,

〈b(r′)†b(r)〉 = 〈Ψ|b(r′)†b(r)|Ψ〉. (6.6.39)

However, this can also be interpreted as a correlation function in the classical ensemble,which (it can be shown) is formed by the trace of same product of transfer-matrices forthe whole set of slices, except inserting a matrix representing this operator. Geomet-rically, the creation/annihilation operator means we’ve cut one strand of the spaghettiin the slice at time t, such that the world line jumps there from r to r′ (while all theother world-lines still obey periodic b.c.’s).

Now, in the un-entangled phase, that world line is likely to keep vertical anyhow dueto its neighbors, so the expectation (6.6.39) is negligible unless r is close to r′. But inthe entangled phase, that world line wanders (like all the others) a macroscopic numberof times from the top slice through the bottom slice. In so doing it can make its waygradually from position r′ to position r, no matter how far apart they are. 29 Thus, thecorrelation does not vanish as r → ∞.

Applications of the mapping

Of course, the “spaghetti” phase precisely describes the flux-lattice found in theAbrikosov phase of a Type II superconductor. 30 In high-Tc’s (extreme case of verylarge κ), these flux lines can wander. There may be a transition from a “straight” phase(at low Tcl), in which permutations are trivial, to an “entangled” phase (at higher Tcl)in which the flux lines are randomly braided. The entanglings are physically significant

27Karl Runge, personal communication, ca. 1990; this goes back at least to D. Ceperley, c. 1980.28The physical significance of (6.6.38) is, roughly speaking, wherever in the system you remove a

boson, the change in the wavefunction’s cphase factor is the same. (See Lec. 7.1 ).29For the path-integral picture of the superfluid phase, see Pollock and Ceperley, PRL 56, 351 (1987)

and Phys. Rev. B 36, 8343 (1987); they are inspired by Feynman’s picture of superfluid helium assketched in RP Feynman and AR Hibbs, QM and Path Integrals, 1965, pp. 287-294, and in a coupleof other places as well.

30D. R. Nelson, Phys. Rev. Lett. 60, 1973 (1988).


since they affect the ability of flux lines to flow in response to a magnetic field (whichis what causes a nonzero resistance below the superconducting Tc.)

The flux-line mapping has been fruitful in suggesting some new physics. For exam-ple, a random external potential in a quantum-mechanical system can make localizedstates, so that a boson might become tied to the vicinity of an attractive impurity. Butan impurity at one point in Hq maps to a line of impurities (in the stacking direction)in Hcl. This led to the suggestion to study the effect in high-Tc’s of introducing randomcolumnar pinning defects (which are made by shooting ions vertically from an accel-erator); these indeed cause stronger pinning (meaning more robust superconductivity)than random pointlike pinning defects. 31

Exercises

Ex. 6.6.1 Vortex interaction in Type I? (T)

Thought question: do two vortex lines attract, or repel, in a Type I superconductor?

Ex. 6.6.2 Variational calculation for vortex

CORRECTED Eq. (6.6.41) on 4/13/09

The purpose of this exercise is to check the statements about the radial dependenceand line energy of a vortex, using the crudest possible variational form.

Try

Ψ(r, θ, z) = Ψ0eiθ ×

(r/a) 0 < r < a;

1 r > a.(6.6.40)

A(r, θ, z) =Φ∗

0

2πθθθ ×

r/b2 0 < r < b;

(1/r) r > b.(6.6.41)

(The coefficients must be the values given in order that the energy density goes tozero far from the vortex’s axis; then (6.6.41) ensures the flux is the flux quantumΦ∗

0. The second crudest form would have radial dependences e−r/a and (1 − e−r/b)/r,respectively.)

(a) Compute each of the three terms in the total G-L free energy difference (comparedto the uniform state Ψ = Ψ0), in the form of FcondLz times a function of a, b, ξ, λ.(The sooner you can eliminate α, β, m∗ in favor of the G-L length scales, the better.)

Some hints (you should still work them, not assume them) (i) FL = Fcond

(

1 −

|Ψ|2/|Ψ0|2)

. (ii) Fgrad has a factor (1/r − r/b2)2 both in the (0, a) and in the (a, b)

ranges. (iii) ξ should appear only in the Fgrad term, and λ only in the magnetic fieldenergy term.

(b) Then, write the equations to minimize it with respect to a and b. Intuition (andthe assertions in the text) suggest that a ∝ ξ and b ∝ λ. If necessary to simplify theresult, you may assume ξ λ (extreme type II case).

31See R. D. Kamien, P. Le Doussal, and D. R. Nelson, Phys. Rev. A 45, 8727 (1992), about theentangled phase.

6.6 Z. MAPPING OF QUANTUM BOSE SYSTEM IN D = 2 TO FLUX LINES 629

Ex. 6.6.3 Anisotropic case with vortex

Imagine you have different effective masses mx and my in the x and y directions.

(b) Show you can rescale the system with variables x′ = γx, y′ = y/gamma so thatevery term in the G-L free energy looks isotropic again. (Notice that you must rescaleAx and Ay correspondingly so that

∮

rA · d – the flux – is unchanged.)

(b) The isotropic G-L system always has a triangular vortex lattice, which can beconsidered centered rectangular whose lattice constants have a ratio b/a =

√3. What

is the lattice for the anisotropic case?

635

Lecture 6.7

Flux-flow conductivity

This lecture is not too well edited, and may not be fully covered in class.Kamerlingh Onnes, as soon as he discovered superconductivity, visualized making

high-field superconducting magnets, but his hopes were dashed by the discovery of thecritical field Hc (he was working with elemental metals which are usually Type I). Thegood news about Type II superconductors is that their superconductivity extends tohigher fields Hc2 = κHc (much higher, in the extreme Type II limit κ 1 as in high-Tc

cuprates). The bad news is that these superconductors don’t generally superconduct.The key points take only three sentences: (i) a vortex, in the presence of a supercur-

rent, feels a transverse force; (ii) sometimes the vortices develop a drift velocity in thedirection of that force; if they do, there is dissipation; (iii) if that velocity is proportionalto the force, then the superconductor obeys Ohm’s Law and has a nonzero resistivity. 1

The entire art of making practical superconductors, then, consists of preventing (ii)by providing ample pinning sites.

So this is one of the rare places where we consider DISORDER. The others were (i)all of unit II, transport; (ii) charge-density-wave pinning in Lec 3.5 (skipped in 2009);(iii) spin glasses, mentioned in Lec. 5.1.

Duality

If you get only a little from this lecture, I hope you’ll get some sense of the dual

relationship between the behavior of electric charge and of magnetic vortex (flux) lines.In two dimensions (as in a thin film of superconductor) the vortex is a pointlike objectand this duality is a rigorous mathematical mapping within many models. Here, how-ever, I prefer to be more qualitative and just to point out that the ways that you mustinvert your thinking when you turn from tranport in an ordinary metal to tranportin a superconductor. Ordinarily, charge comes in discrete particles while the magneticfield is continuous; but supercurrent is collective – it’s basically wrong to visualize it asthe totality of separate trajectories – while the field is quantized into flux lines (tied tovortex singularities). You were told that a magnetic field exerts a Lorentz force on acurrent; now the current exerts a Lorentz force on the flux lines2 In an ordinary metal,

1The original references about flux flow are by P. Nozieres and W. F. Vinen, Philos. Mag. 14, 667(1966), and by J. Bardeen & M. J. Stephen, Phys. Rev. 140, A1197 (1965). In the Parks compendium,you have “Flux flow and irreversible effects” by Y. B. Kim & M. J. Stephen, in Superconductivy, ed.Parks (1969); and “A comparison of the properties of superconductors and superfluid helium” by W.F. Vinen, in Superconductivity, v. II, ed. Parks (1969).

2Actually, by Newton’s First Law the charges experiencing the Lorentz force always must exert anequal and opposite force on the magnetic field; but we focus on the discrete localized object. That’s what


636 LECTURE 6.7. FLUX-FLOW CONDUCTIVITY

a voltage drives a drift of point charges i.e. a current; now, a (super)current drives adrift of discrete vortices, which as we shall show generates a voltage difference. Thus,in an ordinary metal, impeding the charge motion increases the resistivity (with infiniteresistivity when the charges are pinned, as in a charge-density wave; 3 in the Abrikosovphase, pinning the vortices restores the perfect conductivity. Finally, in mesoscopicinterference effects, if the quantum diffusion of an electron around a ring threaded byflux introduces an Aharonov-Bohm phase factor into the amplitude of that contributionto the electron’s wavefunction, then the quantum motion of a vortex around a ring con-taining electric charge picks up an analogous Aharonov-Casher phase; 4 this has beenobserved in experiments.

6.7 A Magnus force on a vortex

The Magnus force exists in both neutral and charged superfluids, and is the most im-portant contribution for understanding resistivity in the Abrikosov phase. 5 You obtain

fMagnus = nse∗(vs − vn) × Φ0 l (6.7.1)

per unit length, per vortex.Sorry, somewhat repetitive: Our derivation is related to the discussion (Lec. 6.4 ) of

how a superflow around a ring might be degraded if a vortex line can slip across the ring(the phase slip process). Imagine a vortex in the backround of a uniform supercurrentJs = −e∗vs, giving a net phase shift 2πp in a distance Lx Fig. 6.7.1 Now, the phase θ(r)changes by 2π along a path that encircles the vortex. Hence, if the net phase differencefrom x = 0 to x = Lx is ∆θ = 2πp along a path with y > yv, then it is 2π(p+ 1) takenalong a path with y < yv. In other words, ∇θ is bigger on the y < yv side of the samplethan on the y > yv side, and so is the gradient energy. Move a vortex across it in the xdirection, now the net phase shift is 2π(p−1). This decreases the total gradient energy,by an amount proportional to LyLz. Dividing by Ly, we get the energy change per unitdistance moved by the vortex – i.e. the net force; dividing by Lz, you get the net forceper unit length, and it comes out to be the same coefficient as in (6.7.1).

The Magnus force on a vortex explains the mechanism of Frank-Read source men-tioned in Lec. 6.4 .

I wanted to explain how this force is dual to the Aharonov-Bohm effect. Or better,is related to the Lorentz force as an action/reaction pair. One is the force on electroncurrent due to flux lattice; the other is vice versa.

Let’s compute the energy change upon moving the vortex from yv → yv + ∆y. Thegradient energy density is

Fgrad =~

2

2m∗n∗

s

[

2πn

Lx

]2

for y > yv (6.7.2)

has a well-defined position, obeys a semiclassical dynamics, and interacts with the disorder (scatterersor pinning) sites in the metal.

3Charge-density waves (CDWs) are described in Lec. 3.4 . In that lecture, I also showed a tableshowing the duality between the equations describing the respective phase fields θ(r) which describethe order parameter in both the CDW and the superconducting cases. That duality is not preciselythe same as the one I’m describing here, because among other things the CDW current flows only inone dimension, also vortices and magnetic field do not seem to appear in the equations. I’m sure thereis some relation to the duality discussed here, but I haven’t quite pinned it down.

4See Lec. 2.2on “weak localization”5The force in (6.7.1) is called the “Magnus force” from a very similar force that acts on vortices in

fluids obeying the Navier-Stokes equations (in the limit of vanishing viscosity).

6.7 B. DAMPING FORCES ON A VORTEX 637

Lx

Ly

Lz

(x ,y )v v

vs

∆θ=2π (n+1)

∆θ=2π n

Figure 6.7.1: Oops! The n in the figure should be “p”. A sample of dimensions Lx ×Ly ×Lz

contains a vortex running in the z direction, at location (xv, yv), in the background of asuperfluid velocity vs. The phase change ∆φ is shown along paths above and below the vortexline.

and the same with n→ (n+ 1) for y < yv. Therefore, the total energy difference is

∆A = LxLz∆y~

2

2m∗n∗

s

(

2π

Lx

)2

[(n+ 1)2 − n2] (6.7.3)

where [(n+ 1)2 − n2] ≈ 2n, assuming n large.Finally the force on the vortex, per unit length, can be defined as

fMagnus =(∆E/∆y)

Lz≈ (2π~)2ns

m∗· nLx

=

(

−2π~

2e

)

Js (6.7.4)

The coefficient above can be identified as the flux quantum, and so we finally obtain

fMagnus = Φ0 l× Js (6.7.5)

when vv = 0. This agrees with (6.7.1). Here l is a unit vector directed along the fluxline, and I have not checked the right-hand-rule sign in (6.7.5)

In retrospect, (6.7.5) should be obvious. It is nothing but the Lorentz force, fromthe magnetic field’s point of view. (When a particle is deflected it always exerts anequal and opposite force on the field, but usually the field is nebulous and its motion ishard to visualize).

6.7 B Damping forces on a vortex

The Magnus force would accelerate the vortices without limit, but other forces put onthe brakes.

Bardeen-Stephen drag

An intrinsic viscous drag force dependent on vs and related to the normal-stateresistivity is called the “Bardeen-Stephen” friction or drag. The source is excitations ofthe normal electrons, or more precisely quasiparticles, which live in vortex cores (SeeLec. 7.5 ). (Because the superconducting gap vanishes at the vortex core’s midline, theseexcitations turn out to be gapless.) This drag literally has a viscous form (meaning theforce is linear in velocity). A second source of drag is the lines being snagged by the


pinning sites. This is likely to be nonlinear in its velocity dependence , and hystereticin its time dependence. But either kind of drag is manifestly a function of the velocityof the flux line relative to the crystal lattice. The form is a viscous drag

fv = −η(vs − vv) (6.7.6)

per unit length. The vs − vv form is obviously correct in a superfluid which obeysGalileian invariance, but I don’t see why you can’t have terms in vs or vv when thecurrent is in a lattice.

(Do not confuse this with the viscosity due to inter-vortex forces in a “vortex liquid”(see Lec. 6.6 ) under shear.)

To see where it comes from, assume for simplicity that the vortex is fixed (vv = 0)and the supercurrent moving. When you solve the equations properly, a portion of thesupercurrent gets converted to normal on one side of the vortex, passes across the core,and is converted back to supercurrent on the other side. While in the core, it is subjectto more or less ordinary scattering. Hence it is proportional to velocity, like viscousdamping.

This all may explain the experimental fact: as H → Hc2, the flux-flow resistivityapproaches the normal state resistivity. It is not too surprising in this picture, since thefraction of a Type II system in vortex cores approaches 1 as H → Hc2,

Pinning force

A second kind of force is fp due to pinning. The pinning is much like pinning ofCDW’s (Lec. 3.4 ) or Bloch walls in ferromagnets (Lec. 5.3 Z(?)) The potential energyhas the form

∑

i

∫

dzV (rxy(z), z) (6.7.7)

where V (r) is a random potential, and rxy(z) parametrizes vortex line. A finite force isrequired to detach a vortex from a given pin. Thus vortex motion in presence of pinninghas the nature of dry friction, with complicated stick-and-slip motion that is difficultto understand. Clearly it is not generally linear in the velocities.

Collective (weak) vs. strong pinning

I don’t have time to write a proper explanation. Note that Lec. 3.4 F also mentionsweak versus strong pinning.

Fig. 6.7.2 shows different situations of pinning. In the case of strong pinning(Fig. 6.7.2a the vortex line moves by separating from one pin at a time, like a netpassing over rocks which snag it. In the case of collective pinning (Fig. 6.7.2b, the inter-vortex vortex repulsion is strong enough to keep them almost uniformly spaced. Theeffect of pinning is more subtle: it causes long-wavelength distortions (of less than theinter-vortex spacing) so that more lines go through favorable than unfavorable places.In a local minimum, then, some vortex lines pass to the right and some to the left ofnearby pins, so that the resultant force on the bundle of vortices is zero.

The motion of vortex lines is particularly sensitive to the equilibrium phase, in theregime of collective pinning. (see Lec. 6.6 Y). Whether the cause is pinning disorder orthermal fluctuations, if the vortices wander far enough from regular lattice positions,they will actually entangle: that is, one vortex will wind around others. In particular,the vortex lines in an entangled flux liquid (see Fig. 6.6.6 (b)) strengthen the couplingbetween threads, as in braided hair: instead of the mediated force, it’s the line-crossing

6.7 C. DISSIPATION AND DRIFT VELOCITY 639

barrier. Thus, the effective pinning force is summed over a larger region and is aug-mented.

Actually, it may well happen that most vortices are pinned, but there are pathsthrough this background on which some vortices can move, like dental floss passingthrough a crack in a solid (Fig. 6.7.2c).

Figure 6.7.2: Kinds of pinning, acting on a vortex lattice. Dashed lines show an alternativeconfiguration of the vortex lines. (a) Strong pinning. Each impurity strongly attracts a vortexline; the barrier to a different metastable state is detaching from one pin and attaching toanother. (b). Weak (collective) pinning. The position of the vortex lines is the resultant ofany small pinning forces. (c). Real flux flow may occur in channels

6.7 C Dissipation and drift velocity

There are three Galileian frames in the problem: (i) the background lattice (possiblyincluding pinning site) which defines the reference frame; (ii) the supercurrent withvelocity vs; and (iii) the vortices with a possible drift velocity vv . There are forces onthe vortices due to the supercurrent and also due to the background lattice.

Whenever the vortices feel a force in their direction of motion, energy is beingdissipated and it can only be due to the superflow. The forces might, in principle, bedue either to (i) an interaction with the lattice pinning potential, or (ii) an interactionwith the superflow, which can excite quasiparticles in the vortex cores thereby causinga viscous damping.

We expect the lines will reach a steady state in which vv and vs are constant,implying there must be a net zero force on both vortices and supercurrent, i.e. the forceexerted by the lattice on the vortices must equal the force exerted by the vortices onthe supercurrent.

This last force must be balanced by an electric field E. We’ll find that Ohm’s lawis indeed obeyed: i.e., E = ρJs.

You have three velocities – that of the supercurrent, that of the vortex core, and thatof the solid lattice (taken to be zero); so when you speak of velocities, you must be clearwhat they are relative to, and must worry whether your result is Galileian invariant.

Vortex mobility

The simplest assumption for the drag is the Bardeen-Stephen viscous drag, (6.7.6).That implies, in a steady state,

vv = µvfdrag (6.7.8)

where µv = 1/η is the vortex mobility. [More generally, µv might phenomenologicallyinclude an (averaged) pinniung force as well.] The total power dissipated by one moving


vortex is(fvLz) · vv = µv |fdrag|2Lz = µvΦ

20|J2

s |Lz (6.7.9)

Now consider not one vortex but an array, with a density nv per unit area transverseto the vortex lines, so there are nvLxLy vortices. Putting it all together, the total powerbeing dissipated is

P = ρ|Js|2LxLyLz. (6.7.10)

This is identical to the power dissipation in a material with resistivity

ρ = µvΦ20nv. (6.7.11)

Notice that resistivity is proportional to the vortex mobility. It’s amusing to recall that,in the ordinary Drude theory, resistivity is proportional to the carrier mobility. This isan aspect of the duality.

The (perhaps surprising) moral of this story is that, to give the Abrikosov state agenuine zero resistivity the material should be “dirty” so the pinning is strong and µv

is small.The pinning strength gives us a third kind of critical current (in addition to those

of Lec. 6.4 ). It is the current needed so that the Magnus force (6.7.1) can overcomethe pinning force (6.7.7), and detach the vortices so they can start drifting.

Another note: to handle the case where the net force is zero; we didn’t need tounderstand the pinning force or the controversial details of the magnus force law, andwe don’t need to worry whether vortices have an effective mass. The non-steady statewith nonzero force would be much harder to get correctly.

This potential energy ought to be harmonized with Heff in Lec. 6.6D .

6.7 X Controversies and confusions on vortex motion

In the 1990s, it became clear that the original understanding was deficient; in particular,there was a long-standing confusion whether or not the force on a vortex should beconsidered a Magnus force. That’s a term from fluid dynamics, for the sideways forcefelt by a vortex in the presence of a uniform flow. The question is whether it’s themotion of the vortex with respect to the lattice, or with respect to the superflow, thatdetermines this force.

It became critical to understand this, because it was found in high-Tc’s (and subse-quently in classic Type II metals as well) that the sign of the Hall coefficient reversed

after cooling below Tc. See R. Ferrell, Phys. Rev. Lett. 68, 2524 (1992), and S. J.Hagen et al (C. J. Lobb group), Phys Rev. B 47, 1064 (1993), as well as more recentexperiments from the Lobb group. There have been recent (1993-97) articles by P. Aoand D. Thouless working out the microscopic principles for a proper theory of vortexmotion.

macroscopic superconductivity

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