teoría de superconductores

8/10/2019 teora de superconductores

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Contents

5 Superconductivity 15.1 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Electron-phonon interaction . . . . . . . . . . . . . . . . . . . . . . 35.3 Cooper problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.4 Pair condensate & BCS Wavefctn. . . . . . . . . . . . . . . . . 115.5 BCS Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.5.1 BCS wave function, gauge invariance, and number con-servation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.5.2 Is the BCS order parameter general? . . . . . . . . . . . . . . . 165.6 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.6.1 Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . . . 185.6.2 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.6.3 Critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.6.4 Specific heat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.7 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.7.1 Linear response to vector potential . . . . . . . . . . . 255.7.2 Meissner Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.7.3 Dynamical conductivity. . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Ginzburg-Landau Theory 336.1 GL Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 Type I and Type II superconductivity . . . . . . . . . . . . . 426.3 Vortex Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.4 Properties of Single Vortex. Lower critical field Hc1 556.5 Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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5 Superconductivity

5.1 Phenomenology

Superconductivity was discovered in 1911 in the Leiden lab-oratory of Kamerlingh Onnes when a so-called blue boy(local high school student recruited for the tedious job of mon-itoring experiments) noticed that the resistivity ofHg metalvanished abruptly at about 4K. Although phenomenologicalmodels with predictive power were developed in the 30s and

40s, the microscopic mechanism underlying superconductiv-ity was not discovered until 1957 by Bardeen Cooper andSchrieffer. Superconductors have been studied intensively fortheir fundamental interest and for the promise of technologi-cal applications which would be possible if a material whichsuperconducts at room temperature were discovered. Until

1986, critical temperatures (Tcs) at which resistance disap-pears were always less than about 23K. In 1986, Bednorz andMueller published a paper, subsequently recognized with the1987 Nobel prize, for the discovery of a new class of materialswhich currently include members withTcs of about 135K.

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Figure 1: Properties of superconductors.

Superconducting materials exhibit the following unusual be-haviors:

1.Zero resistance. Below a materials Tc, the DC elec-trical resistivityis really zero, not just very small. Thisleads to the possibility of a related effect,

2.Persistent currents. If a current is set up in a super-conductor with multiply connected topology, e.g. a torus,

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it will flow forever without any driving voltage. (In prac-tice experiments have been performed in which persistentcurrents flow for several years without signs of degrading).

3.Perfect diamagnetism. A superconductor expels aweak magnetic field nearly completely from its interior(screening currents flow to compensate the field within asurface layer of a few 100 or 1000 A, and the field at thesample surface drops to zero over this layer).

4.Energy gap. Most thermodynamic properties of a su-

perconductor are found to vary as e/(kBT), indicatingthe existence of a gap, or energy interval with no allowedeigenenergies, in the energy spectrum. Idea: when thereis a gap, only an exponentially small number of particleshave enough thermal energy to be promoted to the avail-able unoccupied states above the gap. In addition, this gap

is visible in electromagnetic absorption: send in a photonat low temperatures (strictly speaking, T = 0), and noabsorption is possible until the photon energy reaches 2,i.e. until the energy required to break a pair is available.

5.2 Electron-phonon interaction

Superconductivity is due to an effective attraction betweenconduction electrons. Since two electrons experience a re-pulsive Coulomb force, there must be an additional attrac-tive force between two electrons when they are placed in ametallic environment. In classic superconductors, this force

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is known to arise from the interaction with the ionic system.In previous discussion of a normal metal, the ions were re-placed by a homogeneous positive background which enforcescharge neutrality in the system. In reality, this medium ispolarizable the number of ions per unit volume can fluctuatein time. In particular, if we imagine a snapshot of a singleelectron entering a region of the metal, it will create a netpositive charge density near itself by attracting the oppositelycharged ions. Crucial here is that a typical electron close tothe Fermi surface moves with velocity vF = hkF/mwhich is

much larger than the velocity of the ions, vI=VFm/M. Soby the time ( 2/D 1013 sec) the ions have polarizedthemselves, 1st electron is long gone (its moved a distance

vF 108cm/s 1000A, and 2nd electron can happen by

to lower its energy with the concentration of positive chargebefore the ionic fluctuation relaxes away. This gives rise to aneffective attraction between the two electrons as shown, whichmay be large enough to overcome the repulsive Coulomb in-teraction. Historically, this electron-phonon pairing mech-anism was suggested by Frolich in 1950, and confirmed bythe discovery of the isotope effect, whereinTcwas found tovary as M1/2 for materials which were identical chemicallybut which were made with different isotopes.

The simplest model for the total interaction between twoelectrons in momentum stateskandk, withq k k, in-teracting both by direct Coulomb and electron-phonon forces,is given by

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Figure 2: Effective attraction of two electrons due to phonon exchange

V(q, ) = 4e2

q2 +k2s+

4e2

q2 +k2s

2q2 2q

, (1)

in the jellium model. Here first term is Coulomb interac-tion in the presence of a medium with dielectric constant = 1 + k2s/q

2, and q are the phonon frequencies. Thescreening length k1s is 1A or so in a good metal. Secondterm is interaction due to exchange of phonons, i.e. the mech-anism pictured in the figure. Note it is frequency-dependent,

reflecting the retardednature of interaction (see figure), andin particular that the 2nd term is attractive for < q D.Something is not quite right here, however; it looks indeed asthough the two terms are of the same order as 0; indeedthey cancel each other there, givingV( 0) = 0. Further-more, V is always attractive at low frequencies, suggesting

that all metals should be superconductors, which is not thecase. These points indicate that the jellium approximationis too simple. We should probably think about a more care-ful calculation in a real system as producing two equivalentterms, which vary in approximately the same way with kT Fandq, but with prefactors which are arbitrary. In some ma-

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terials, then, the second term might win at low frequencies,depending on details.

5.3 Cooper problem

A great deal was known about the phenomenology of super-conductivity in the 1950s, and it was already suspected thatthe electron phonon interaction was responsible, but the mi-croscopic form of the wave function was unknown. A cluewas provided by Leon Cooper, who showed that the noninter-

acting Fermi sea is unstabletowards the addition of a singlepair of electrons with attractive interactions. Cooper beganby examining the wave function of this pair (r1, r2), whichcan always be written as a sum over plane waves

(r1, r2) =kq

uk(q)eikr1ei(k+q)r2 (2)

where the uk(q) are expansion coefficients and is the spinpart of the wave function, either the singlet | > /2or one of the triplet,| >, | >, | +> /2. Infact since we will demand that is the ground state of thetwo-electron system, we will assume the wave function is re-alized with zero center of mass momentum of the two elec-

trons, uk(q) = ukq,0. Here is a quick argument related tothe electron-phonon origin of the attractive interaction.1 Theelectron-phonon interaction is strongest for those electronswith single-particle energies k within D of the Fermi level.In the scattering process depicted in Fig. 3, momentum is

1Thanks to Kevin McCarthy, who forced me to think about this further

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k

p

-q

k-q p+q

k p

g g

Dp+q

k-q

Figure 3: Electrons scattered by phonon exchange are confined to shell of thicknessDabout Fermisurface.

explicitly conserved, i.e. the total momentum

k + p=K (3)

is the same in the incoming and outgoing parts of the diagram.Now look at Figure 4, and note that ifK is not 0, the phasespace for scattering (attraction) is dramatically reduced. Sothe system can always lower its energy by creating K = 0pairs. Henceforth we will make this assumption, as Cooperdid.

Then (r1, r2) becomes

kukeik

(r1

r2)

. Note that ifuk isevenink, the wave function has only terms cos k (r1r2),whereas if it is odd, only the sin k (r1 r2) will contribute.This is an important distinction, because only in the formercase is there an amplitude for the two electrons to live ontop of each other at the origin. Note further that in order to

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-pKk

D

Figure 4: To get (attractive) scattering with finite cm momentum K, need both electron energiesto be within D of Fermi level very little phase space.

preserve the proper overall antisymmetry of the wave function,uk even (odd) in k implies the wave function must be spinsinglet (triplet). Let us assume further that there is a general

two-body interaction between the two electrons (the rest ofthe Fermi sea is noninteracting in the model!) V(r1, r2), sothat the Hamiltonian for the system is

H= 21

2m

22

2m+V(r1, r2). (4)

Inserting the assumed form ofinto the Schrodinger equationH=E, and Fourier transforming both sides with respectto the relative coordinate,r=r1 r2, we find

(E 2k)uk = k>kF

Vkkuk, (5)

wherek =k2/2mand the Vkk =

d3rV(r)ei(k

k)r are the

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matrix elements of the two-body interaction.

Recallk, k correspond to energies at the Fermi level F inthe absence ofV. The question was posed by Cooper, is it

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positive quantity, e.g. negligible compared to 2c. We thenarrive at the pair binding energy

Cooper 2F E 2ce2/N0V. (10)There are several remarks to be made about this result.

1. Note (for your own informationCooper didnt know thisat the time!) that the dependence of the bound state en-ergy on both the interactionVand the cutoff frequencycstrongly resembles the famous BCS transition temperature

dependence, with c identified as the phonon frequencyD, as given in equation (I.1).

2. the dependence on V is that of an essential singularity,i.e. a nonanalytic function of the parameter. Thus wemay expect never to arrive at this result at any order inperturbation theory, an unexpected problem which hin-

dered theoretical progress for a long time.

3. The solution found has isotropic or s-symmetry, since itdoesnt depend on thekon the Fermi surface. (How wouldan angular dependence arise? Look back over the calcula-tion.)

4. Note the integrand (2k E)1

= (2k + Cooper)1

peaksat the Fermi level with energy spread Cooper of statesinvolved in the pairing. The weak-coupling (N0V 1)solution therefore provides a bit ofa posteriorijustifica-tion for its own existence, since the fact that Cooper cimplies that the dependence ofVkk on energies out near

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the cutoff and beyond is in fact not terribly important, sothe cutoff procedure used was ok.

5. The spread in momentum is therefore roughly Cooper/vF,

and the characteristic size of the pair (using Heisenbergsuncertainty relation) about vF/Tc. This is about 100-1000A in metals, so since there is of order 1 electron/ unitcell in a metal, and if this toy calculation has anythingto do with superconductivity, there are certainly manyelectron pairs overlapping each other in real space in a

superconductor.

5.4 Pair condensate & BCS Wavefctn.

Obviously one thing is missing from Coopers picture: if it isenergetically favorable for two electrons in the presence of a

noninteracting Fermi sea topair

, i.e. to form a bound state,why not have the other electrons pair too, and lower the en-ergy of the system still further? This is an instabilityof thenormal state, just like magnetism or charge density wave for-mation, where a ground state of completely different charac-ter (and symmetry) than the Fermi liquid is stabilized. TheCooper calculation is a T=0 problem, but we expect that asone lowers the temperature, it will become at some criticaltemperatureTcenergetically favorable forallthe electrons topair. Although this picture is appealing, many things aboutit are unclear: does the pairing of many other electrons alterthe attractive interaction which led to the pairing in the first

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place? Does the bound state energy per pair change? Do allof the electrons in the Fermi sea participate? And most im-portantly, how does the critical temperature actually dependon the parameters and can we calculate it?

5.5 BCS Model.

A consistent theory of superconductivity may be constructedeither using the full effective interaction or our approxima-tion V(q, ) to it. However almost all interesting questions

can be answered by the even simpler model used by BCS.The essential point is to have an attractive interaction forelectrons in a shell near the Fermi surface; retardation is sec-ondary. Therefore BCS proposed starting from a phenomeno-logical Hamiltonian describing free electrons scattering via aneffective instantaneous interaction a la Cooper:

H=H0 Vkkq

ckck+q ck+q ck, (11)

where the prime on the sum indicates that the energies of thestateskandkmust lie in the shell of thicknessD. Note theinteraction term is just the Fourier transform of a completelylocal4-Fermi interaction(r)(r)(r)(r).2

Recall that in our discussion of the instability of the nor-mal state, we suggested that an infinitesimal pair field couldproduce a finite amplitude for pairing. That amplitude wasthe expectation value ckck. We ignore for the moment

2Note this is not the most general form leading to superconductivity. Pairing in higher angular momentumchannels requires a bilocalmodel Hamiltonian, as we shall see later.

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the problems with number conservation, and ask if we cansimplify the Hamiltonian still further with a mean field ap-proximation, again to be justified a posteriori. We proceedalong the lines of generalized Hartree-Fock theory, and rewritethe interaction as

ckck+q ck+q ck = [ckck+q +(cc)]

[ck+q ck +(cc)], (12)where, e.g. (cc) =ck+q ck ck+qck is the fluctu-ationof this operator about its expectation value. If a mean

field description is to be valid, we should be able to neglectterms quadratic in the fluctuations when we expand Eq (20).If we furthermore make the assumption that pairing will takeplace in a uniform state (zero pair center of mass momentum),then we putck+qck =ckckq,0. The effectiveHamiltonian then becomes (check!)

H H0 ( k

ckck+h.c.) + kckck, (13)

where =V

kckck. (14)

What BCS (actually Bogoliubov, after BCS) did was then to

treat the order parameter as a (complex) number, andcalculate expectation values in the approximate Hamiltonian(13), insisting that be determined self-consistently via Eq.(14) at the same time.

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5.5.1 BCS wave function, gauge invariance, and number con-

servation.

What BCS actually did in their original paper is to treat the

Hamiltonian (11) variationally. Their ansatz for the groundstate of (11) is a trial state with the pairs k, k occupiedwith amplitude vk and unoccupied with amplitude uk, suchthat |uk|2 + |vk|2 = 1:

| >= k(uk+vkckck)|0> . (15)This is a variational wave function, so the energy is to beminimized over the space of uk, vk. Alternatively, one candiagonalize the Hartree-Fock (BCS) Hamiltonian directly, to-gether with the self-consistency equation for the order param-eter; the two methods turn out to be equivalent. I will followthe latter procedure, but first make a few remarks on the formof the wave function. First, note the explicit violation of par-ticle number conservation:| > is a superposition of statesdescribing 0, 2, 4 , N-particle systems.3 In general a quan-tum mechanical system with fixed particle number N (like,e.g. a real superconductor!) manifests a global U(1) gaugesymmetry, because H is invariant under ck eick. Thestate

| > is characterized by a set of coefficients

{uk, vk

},

which becomes{uk, e2ivk} after the gauge transformation.The two states| > and(), where = 2, are inequiva-lent, mutually orthogonal quantum states, since they are not

3What happened to the odd numbers? In mesoscopic superconductors, there are actually differences in theproperties of even and odd-number particle systems, but for bulk systems the distinction is irrelevant.

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simply related by a multiplicative phase factor.4 Since His independent of, however, all states|() > are contin-uously degenerate, i.e. the ground state has a U(1) gauge(phase) symmetry. Any state

|() > is said to be a bro-

ken symmetry state, becaue it is not invariant under a U(1)transformation, i.e. the system has chosen a particular out of the degenerate range 0< = 2

0 deiN/2

|()> . (16)[The integration over gives zero unless there are in the ex-pansion of the product contained in| >precisely N/2 paircreation terms, each with factor exp i.] Note while this statehas maximal uncertainty in thevalueof the phase, the rigidityof the system to phase fluctuations is retained.5

It is now straightforward to see why BCS theory works.The BCS wave function| > may be expressed as a sum| >=NaN|(N)>[Convince yourself of this by calculat-

4In the normal state, | >and() differ by a global multiplicative phaseei, which has no physical consequences,and the ground state is nondegenerate.

5The phase and number are in fact canonically conjugate variables, [ N/2, ] = i, where N = 2i/ in the representation.

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ing theaNexplicitly!]. IF we can show that the distribution ofcoefficientsaNis sharply peaked about its mean value< N >,then we will get essentially the same answers as working witha state of definite number N =< N >. Using the explicitform (23), it is easy to show

N = | k

nk| = 2 k

|vk|2 ; (NN)2 =k

u2kv2k.

(17)Now the uk and vk will typically be numbers of order 1, sosince the numbers of allowed k-states appearing in the k sums

scale with the volume of the system, we have < N > V,and < (N < N >)2 > V. Therefore the widthof thedistribution of numbers in the BCS state is < (N < N >)2 >1/2 / < N > N1/2. As N 1023 particles, thisrelative error implied by the number nonconservation in theBCS state becomes negligible.

5.5.2 Is the BCS order parameter general?

Before leaving the subject of the phase in this section, it isworthwhile asking again why we decided to pair states withopposite momenta and spin, k andk. The BCS argu-ment had to do 1) with minimizing the energy of the entire

system by giving the Cooper pairs zero center of mass mo-mentum, and 2) insisting on a spin singlet state because thephonon mechanism leads to electron attraction when the elec-trons are at the same spatial position (because it is retardedin time!), and a spatially symmetric wavefunction with large

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amplitude at the origin demands an antisymmetric spin part.Can we relax these assumptions at all? The first require-ment seems fairly general, but it should be recalled that onecan couple to the pair center of mass with an external mag-netic field, so that one will create spatially inhomogeneous(finite-q) states with current flow in the presence of a mag-netic field. Even in zero external field, it has been proposedthat systems with coexisting antiferromagnetic correlationscould have pairing with finite antiferromagnetic nesting vec-tor Q [Baltensberger and Strassler 196?]. The requirement

for singlet pairing can clearly be relaxed if there is a pair-ing mechanism which disfavors close approach of the pairedparticles. This is the case in superfluid 3He, where the hardcore repulsion of two 3Heatoms supresses Tc for s-wave, sin-glet pairing and enhancesTcfor p-wave, triplet pairing wherethe amplitude for two particles to be together at the origin is

always zero.In general, pairing is possible for some pair mechanism if

the single particle energies corresponding to the stateskandk are degenerate, since in this case the pairing interactionis most attractive. In the BCS case, a guarantee of this degen-eracy fork

and

k

in zero field is provided by Kramers

theorem, which says these states must be degenerate becausethey are connected by time reversal symmetry. However,there are other symmetries: in a system with inversion sym-metry, paritywill provide another type of degeneracy, so k,k,k andk are all degenerate and may be paired

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with one another if allowed by the pair interaction.

5.6 Thermodynamics

5.6.1 Bogoliubov transformation

We now return to (13) and discuss the solution by canonicaltransformation given by Bogoliubov. After our drastic ap-proximation, we are left with a quadratic Hamiltonian in thecs, but withccandccterms in addition toccs. We can di-agonalize it easily, however, by introducing the quasiparticleoperatorsk0andk1by

ck = ukk0+vkk1

ck =vkk0+ukk1. (18)You may check that this transformation is canonical (preservesfermion comm. rels.) if

|uk

|2 +

|vk

|2 = 1. Substituting into

(13) and using the commutation relations we get

HBC S =

kk[(|uk|2 |vk|2)(k0k0+k1k1) + 2|vk|2

+2ukvkk1k0+ 2ukvk

k1

k0]

+k

[(kukvk+

ku

kvk)(

k0k0+

k1k1 1)

+(kv2k ku2k)k1k0+ (kv2k ku2k)k0k1+kckck, (19)

which does not seem to be enormous progress, to say the least.But the game is to eliminate the terms which are not of theform, so to be left with a sum of independent number-type

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terms whose eigenvalues we can write down. The coefficientsof theandtype terms are seen to vanish if we choose

2kukvk+ kv

2k ku2k= 0. (20)

This condition and the normalization condition |uk|2+|vk|2 =1 are both satisfied by the solutions

|uk|2|vk|2 =

1

2

1 k

Ek

, (21)

where I defined the Bogolibov quasiparticle energy

Ek=

2k+ |k|2. (22)The BCS Hamiltonian has now been diagonalized:

HBC S=

kEk

k0k0+

k1k1

+k

k Ek+ kckck

.

(23)

Note the second term is just a constant, which will be impor-tant for calculating the ground state energy accurately. Thefirst term, however, just describes a set offree fermionexci-tations above the ground state, with spectrum Ek.

5.6.2 Density of states

The BCS spectrum is easily seen to have a minimum k fora given direction k on the Fermi surface; k therefore, inaddition to playing the role of order parameter for the su-perconducting transition, is also the energy gap in the 1-particle spectrum. To see this explicitly, we can simply do

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a change of variables in all energy integrals from the normalmetal eigenenergiesk to the quasiparticle energies Ek:

N(E)dE=NN()d. (24)

If we are interested in the standard case where the gap ismuch smaller than the energy over which the normal state dosNN() varies near the Fermi level, we can make the replace-ment

NN() NN(0) N0, (25)so using the form ofEk from (22) we find

N(E)

N0=

EE22 E >

0 E


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First note that atTc, the gap equation becomes

1

N0V =

D0 dk

1

ktanh

k2Tc

. (29)

This integral can be approximated carefully, but it is usefulto get a sense of what is going on by doing a crude treatment.Note that since Tc D generally, most of the integrandweight occurs for > T, so we can estimate the tanh factorby 1. The integral is log divergent, which is why the cutoffD is so important. We find

1N0V0

= log Tc

Tc De1/N0V (30)The more accurate analysis of the integral gives the BCS result

Tc= 1.14De1/N0V (31)

We can do the same calculationnearTc, expanding to lead-ing order in the small quantity (T)/T, to find (T)/Tc3.06(1 T /Tc)1/2. AtT= 0 we have

1

N0V =

D0 dk

1

Ek=

D dEN(E)/E (32)

=D dE

1E

2

2

ln(2d/), (33)

so that (0) 2Dexp 1/N0V , or (0)/Tc 1.76. Thefull temperature dependence of (T) is sketched in Figure 6).In the halcyon days of superconductivity theory, comparisonswith the theory had to be compared with a careful table of/Tc painstakingly calculated and compiled by Muhlschlegl.

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TTc

(T)1.76Tc

Figure 6: BCS order parameter as fctn. ofT.

Nowadays the numerical solution requires a few seconds ona PC. It is frequently even easier to use a phenomenologicalapproximate closed form of the gap, which is correct nearT= 0 and =Tc:

(T) =scTctanh{

sc

aC

CN

(Tc

T1)}, (34)

where sc = (0)/Tc = 1.76, a = 2/3, and C/CN = 1.43is the normalized specific heat jump.6 This is another of theuniversal ratios which the BCS theory predicted and whichhelped confirm the theory in classic superconductors.

5.6.4 Specific heat.

The gap in the density of states is reflected in all thermody-namic quantities as an activated behavior e/T, at low T,due to the exponentially small number of Bogoliubov quasi-

6Note to evaluate the last quantity, we need only use the calculated temperature dependence of near Tc, andinsert into Eq. (47).

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particles with thermal energy sufficient to be excited over thegap at low temperatures T . The electronic specificheat is particularly easy to calculate, since the entropy of theBCS superconductor is once again the entropy of a freegasof noninteracting quasiparticles, with modified spectrum Ek.The expression (II.6) then gives the entropy directly, and maybe rewritten

S= kB0 dEN(E){f(E)lnf(E)+[1f(E)]ln[1f(E)]},

(35)

wheref(E) is the Fermi function. The constant volume spe-cific heat is justCel,V =T[dS/dT]V, which after a little alge-bra may be written

Cel,V = 2

T

dEN(E)(

fE

)[E2 12

Td2

dT ]. (36)

A sketch of the result of a full numerical evaluation is shown

in Figure 1. Note the discontinuity atTc and the very rapidfalloff at low temperatures.

It is instructive to actually calculate the entropy and specificheat both at low temperatures and near Tc. For T ,f(E) eE/T and the density of states factor N(E) in theintegral cuts off the integration at the lower limit , giving

C (N05/2/T3/2)e/T.77To obtain this, try the following:

replace the derivative of Fermi function by exp-E/T do integral by parts to remove singularity at Delta expand around Delta E = Delta + delta E change integration variables from E to delta E

Somebody please check my answer!

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Note the first term in Eq. (47)is continuous through thetransition 0 (and reduces to the normal state specificheat (22/3)N0T above Tc), but the second one gives a dis-continuity atTcof (CN

CS)/CN= 1.43, whereCS=C(T

c )

and CN = C(T+c ). To evaluate (36), we need the T depen-dence of the order parameter from a general solution of (28).

5.7 Electrodynamics

5.7.1 Linear response to vector potential

The existence of an energy gap is not a sufficient condition forsuperconductivity (actually, it is not even a necessary one!).Insulators, for example, do not possess the phase rigiditywhich leads to perfect conductivity and perfect diamagnetismwhich are the defining characteristics of superconductivity.

We can understand both properties by calculating the cur-rent response of the system to an applied magnetic or electricfield. The kinetic energy in the presence of an applied vectorpotentialAis just

H0= 1

2m

d3r(r)[i (

e

c)A]2(r), (37)

and the second quantized current density operator is given by

j(r) = e

2m{(r)(i e

cA)(r) + [(i e

cA)(r)](r)} (38)

= jpara e2

mc(r)(r)A, (39)

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where

jpara(r) =ie

2m{(r)(r) ((r))(r)}, (40)

or in Fourier space,jpara(q) =

e

m

k

kckqck (41)

We would like to do a calculation of the linear current re-sponse j(q, ) to the application of an external field A(q, )to the system a long time after the perturbation is turned

on. Expanding the Hamiltonian to first order in Agives theinteraction

H=d3rjpara A= e

mc

k

k A(q)ckqcq. (42)The expectation value< j>may now be calculated to linearorder via the Kubo formula, yielding

j(q, ) =K(q, )A(q, ) (43)with

K(q, ) = ne2

mc+ [jpara,jpara](q, ). (44)

Note the first term in the current

jdia(q, ) ne2

mc A(q, ) (45)

is purelydiagmagnetic, i.e. these currents tend to screen theexternal field (note sign). The second, paramagneticterm isformally the Fourier transform of the current-current corre-lation function (correlation function used in the sense of our

26


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discussion of the Kubo formula).8 Here are a few remarks onthis expression:

Note the simple product structure of (43) in momentumspace implies a nonlocalrelationship in general between jand A., i.e. j(r) depends on the A(r) at many points r

aroundr.9

Note also that the electric field in a gauge where the elec-trostatic potential is set to zero may be written E(q, ) =

iA(q, ), so that thecomplexconductivity of the sys-

tem defined by j=Emay be written

(q, ) = i

K(q, ) (47)

What happens in a normal metal? The paramagnetic sec-ond term cancels the diamagnetic response at = 0,leaving no real part ofK(Im part of), i.e. the conduc-tivity is purely dissipative and not inductive at , q = 0in the normal metal.

5.7.2 Meissner Effect.

There is a theorem of classical physics proved by Bohr10 which

states that the ground state of a system of charged particles8We will see that the first term gives the diamagnetic response of the system, and the second the temperature-

dependentparamagneticresponse.9If we transformed back, wed get the convolution

j(r) =

d3rK(r, r)A(r) (46)

10See The development of the quantum-mechanical electron theory of metals: 1928-33. L. Hoddeson and G.Baym, Rev. Mod. Phys., 59, 287 (1987).

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in an external magnetic field carries zero current. The essen-tial element in the proof of this theorem is the fact that themagnetic forces on the particles are always perpendicular totheir velocities. In a quantum mechanical system, the threecomponents of the velocity do not commute in the presenceof the field, allowing for a finite current to be created in theground state. Thus the existence of the Meissner effect insuperconductors, wherein magnetic flux is expelled from theinterior of a sample below its critical temperature, is a clearproof that superconductivity is a manifestation of quantum

mechanics.

The typical theorists geometry for calculating the penetra-tion of an electromagnetic field into a superconductor is thehalf-space shown in Figure 7, and compared to schematics ofpractical experimental setups involving resonant coils and mi-crowave cavities in Figs. 7 a)-c). In the gedankenexperiment

H0

a)

L (2 r)

b)

Icavity

c)

Figure 7: a) Half-space geometry for penetration depth calculation; b) Resonant coil setup; c)Microwave cavity

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case, a DC field is applied parallel to the sample surface, andcurrents and fields are therefore functions only of the coordi-nate perpendicular to the surface, A = A(z), etc. Since weare interested in an external electromagnetic wave of very longwavelength compared to the sample size, and zero frequency,we need the limit = 0, q of the response. We willassume that in this limitK(0, 0) const, which we will call(c/4)2 for reasons which will become clear! Equation(63) then has the form

j= c

42

A, (48)

This is sometimes called Londons equation, which must besolved in conjunction with Maxwells equation

B= 2A=4c

j= 2A, (49)

which immediately gives A ez/, and B=B0ez/. Thecurrents evidently screen the fields for distances below thesurface greater than about . This is precisely the Meissnereffect, which therefore follows onlyfrom our assumption thatK(0, 0) = const. A BCS calculation will now tell us how thepenetration depthdepends on temperature.

Evaluating the expressions in (44) in full generality is te-dious and is usually done with standard many-body methodsbeyond the scope of this course. However for q = 0 the cal-culation is simple enough to do without resorting to Greensfunctions. First note that the perturbing HamiltonianHmay

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be written in terms of the quasiparticle operators (18) as

H = (5

e

mc

k

k

A(q)

(ukuk+q+vkvk+q)(

k+q0k0

k+q1k1

+(vkuk+q ukvk+q)(k+q0k1 k+q1k0)

q0

emc

k

k A(0)(k0k0 k1k1) (5

If you compare with theA =0Hamiltonian (23), we see that

the new excitations of the system are

Ek0 Ek emc

k A(0)Ek1 Ek+ e

mck A(0) (52)

We may similarly insert the quasiparticle operators (18) into

the expression for the expectation value of the paramagneticcurrent operator(41):

jpara(q= 0) = em

k(k0k0 k1k1)

= e

m

k

(f(Ek0) f(Ek1)) . (53)We are interested in thelinear responseA

0, so that when

we expand wrt A, the paramagnetic contribution becomes

jpara(q= 0) = 2e2

m2c

k

[k A(0)] k f

Ek

. (54)

Combining now with the definition of the response functionK and the diamagnetic current (45), and recallingk

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N0dk(d/4), with N0 = 3n/(2F) and

(d/4)kk =

1/3, we get for the static homogeneous response is therefore

K(0, 0) =ne2

mc{1

dk(

fEk

)

}1 (55)

ns(T)e2

mc 1 (56)

where in the last step, I defined the superfluid densityto bens(T) n nn(T), with normal fluid density

nn(T) n dk f

Ek

. (57)

Note atT = 0, f/Ek 0, [Not a delta function, as inthe normal state casedo you see why?], while at T =Tc theintegralnn 1.11 Correspondingly, the superfluid density asdefined varies between natT = 0 and 0 at Tc. This is the BCSmicroscopic justification for the rather successful phenomeno-logical two-fluid model of superconductivity: the normal fluidconsists of the thermally excited Bogoliubov quasiparticle gas,and the superfluid is the condensate of Cooper pairs.12

Now lets relate the BCS microscopic result for the statichomogeneous response to the penetration depth appearing inthe macroscopic electrodynamics calculation above. We findimmediately

(T) = ( mc2

4ns(T)e2)1/2. (58)

11The dimensioness function nn(T /Tc)/n is sometimes called the Yoshida function, Y(T), and is plotted in Fig.8.12The BCS theory and subsequent extensions also allow one to understand the limitations of the two-fluid picture:

for example, when one probes the system at sufficiently high frequencies , the normal fluid and superfluidfractions are no longer distinct.

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Y(T)

~ exp-/ T

1

1

0

n /nn

T/Tc

a)

T/Tc 1

1

0 10

T/Tc

b) c)

n /ns (T)

Figure 8: a) Yoshida function; b) superfluid density ; c) penetration depth

AtT= 0, the supercurrent screening excludes the field from

all of the sample except a sheath of thickness (0). At smallbut finite temperatures, an exponentially small number ofquasiparticles will be excited out of the condensate, depeletingthe supercurrent and allowing the field to penetrate further.Both nn(T) and (T) (0) may therefore be expected tovary as e/T for T Tc, as may be confirmed by explicitexpansion of Eq. (57). [See homework.] Close toTc, the pen-etration depth diverges as it must, since in the normal statethe field penetrates the sample completely.

5.7.3 Dynamical conductivity.

The calculation of the full, frequency dependent conductivity

is beyond the scope of this course. If you would like to read anold-fashioned derivation, I refer you to Tinkhams book. Themain point to absorb here is that, as in a semiconductor witha gap, at T= 0 there is no process by which a photon canbe absorbed in a superconductor until its energy exceeds 2,

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the binding energy of the pair. This threshold for opticalabsorption is one of the most direct measurements of the gapsof the old superconductors.

6 Ginzburg-Landau Theory

6.1 GL Free Energy

While the BCS weak-coupling theory we looked at the lasttwo weeks is very powerful, and provides at least a qualita-

tively correct description of most aspects of classic supercon-ductors,13 there is a complementary theory which a) is simplerand more physically transparent, although valid only near thetransition; and b) provides exact results under certain circum-stances. This is the Ginzburg-Landau theory [ V.L. Ginzburgand L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950)],

which received remarkably little attention in the west untilGorkov showed it was derivable from the BCS theory. [L.P.Gorkov, Zh. Eksp. Teor Fiz. 36, 1918 (1959)]. The the-ory simply postulated the existence of a macrosopic quantumwave function(r) which was equivalent to an order param-eter, and proposed that on symmetry grounds alone, the free

energy density of a superconductor should be expressible interms of an expansion in this quantity:

fs fnV

=a||2 +b||4 + 12m

|( +ie

cA)|2, (59)

13In fact one could make a case that the BCS theory is the most quantitatively accurate theory in all of condensedmatter physics

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where the subscripts n and s refer to the normal and super-conducting states, respectively.

Lets see why GL might have been led to make such a

guess. The superconducting-normal transition was empiri-cally known to be second order in zero field, so it was naturalto write down a theory analogous to the Landau theory ofa ferromagnet, which is an expansion in powers of the mag-netization, M. The choice of order parameter for the super-conductor corresponding to M for the ferromagnet was notobvious, but a complex scalar field was a natural choicebecause of the analogy with liquid He, where||2 is knownto represent the superfluid densityns;

14 a quantum mechani-cal density should be a complex wave function squared. Thebiggest leap of GL was to specify correctly how electromag-netic fields (which had no analog in superfluid He) wouldcouple to the system. They exploited in this case the simi-

larity of the formalism to ordinary quantum mechanics, andcoupled the fields in the usual way to chargeseassociatedwith particles of mass m. Recall for a real charge in amagnetic field, the kinetic energy is:

=

1

2m

d3r(

+

ie

cA)2 (60)

= 12m

d3r|( +ie

cA)|2, (61)

after an integration by parts in the second step. GL just re-placede,mwithe,mto obtain the kinetic part of Eq. (59);

14 in theH e case has the microscopic interpretation as the Bose condensate amplitude.

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they expected that e and m were the elementary electroncharge and mass, respectively, but did not assume so.

T>T

T0. The value offin equilibrium will be fn = f[= 0].

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This symmetry is broken when the system chooses one of theground states (phases) upon condensation (Fig 1.).

For a uniform system in zero field, the total free energy

F =

d

3

rf is minimized whenf is, so one find for the orderparameter at the minimum,

||eq = [a2b ]1/2, a 0. (63)

When a changes sign, a minimum with a nonzero value be-comes possible. For a second order transition as one lowers thetemperature, we assume thata andbare smooth functions ofT nearTc. Since we are only interested in the region near Tc,we take only the leading terms in the Taylor series expansionsin this region: a(T, H) =a0(T Tc) andb= constant. Eqs.(62) and (63) take the form:

|(T)|eq = [a

0(T

cT)

2b ]1/2, T < T c (64)|(T)|eq = 0, T > T c. (65)Substituting back into Eqs.59, we find:

fs(T) fn(T) = a20

4b(Tc T)2, T < T c (66)

fs(T)

fn(T) = 0, T > T c. (67)

The idea now is to calculate various observables, and de-termine the GL coefficients for a given system. Once theyare determined, the theory acquires predictive power due toits extreme simplicity. It should be noted that GL theory isapplied to many systems, but it is in classic superconductors

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that it is most accurate, since the critical region, where de-viations from mean field theory are large, is of order 104 orless. Near the transition it may be taken to be exact for allpractical purposes. This is not the case for the HTSC, wherethe size of the critical region has been claimed to be as muchas 10-20K in some samples.

Supercurrents. Lets now focus our attention on theterm in the GL free energy which leads to supercurrents, thekinetic energy part:

Fkin =d3r

1

2m|( +ie

c A)|2 (68)=

d3r

1

2m[(||)2 + ( e/cA)2||2]. (69)

These expressions deserve several remarks. First, note thatthe free energy is gauge invariant, if we make the transforma-tion A

A +

, where is any scalar function of position,

while at the same time changing exp(ie/c) . Sec-ond, note that in the last step above I have split the kineticpart offinto a term dependent on the gradients of the orderparameter magnitude|| and on the gradients of the phase. Let us use a little intuition to guess what these termsmean. The energy of the superconducting state below Tc is

lower than that of the normal state by an amount called thecondensation energy.16 From Eq. (59) in zero field this is oforder||2 very close to the transition. To make spatial vari-ations of the magnitudeof must cost a significant fractionof the condensation energy in the region of space in which it

16We will see below from the Gorkov derivation of GL from BCS that it is of order N(0)2.

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occurs.17 On the other hand, the zero-field free energy is ac-tually invariant with respect to changes in , so fluctuationsofalone actually cost no energy.

With this in mind, lets ask what will happen if we applya weak magnetic field described by A to the system. Sinceit is a small perturbation, we dont expect it to couple to ||but rather to the phase. The kinetic energy density shouldthen reduce to the second term in Eq. (69), and furthermorewe expect that it should reduce to the intuitive two-fluid ex-pression for the kinetic energy due to supercurrents, 12mnsv

2s .

Recall from the superfluid He analogy, we expect||2

nsto be a kind of density of superconducting electrons, but thatwe arent certain of the charge or mass of the particles. Solets put

fkin 12m

|( +ie

cA)|2.=

d3r

1

2m( e/cA)2||2 1

2mnsv

2s.

(70)

Comparing with Eq. (xx), we find that thesuperfluid veloc-itymust be

vs= 1

m(+e

cA). (71)

Thus the gradient of the phase is related to the superfluidvelocity, but the vector potential also appears to keep theentire formalism gauge-invariant.

Meissner effect. The Meissner effect now follows imme-diately from the two-fluid identifications we have made. The

17We can make an analogy with a ferromagnet, where if we have a domain wallthe magnetization must go to zeroat the domain boundary, costing lots of surface energy.

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supercurrent density will certainly be just

js= ens vs= ensm

(+e

cA). (72)

Taking the curl of this equation, the phase drops out, and wefind the magnetic field:

js= e2ns

mcB. (73)

Now recall the Maxwell equation

js=

c

4 B, (74)

which, when combined with (14), gives

c

4 B = c

42 B = e

2nsmc

B, (75)

or2

2 B = B, (76)

where

= mc2

4e2ns

1/2

. (77)

Notice now that if we use what we know about Cooper pairs,this expression reduces to the BCS/London penetration depth.

We assumee is the charge of the pair, namely e = 2e, andsimilarly m = 2m, and||2 = ns = ns/2 since ns is thedensity of pairs.

Flux quantization. If we look at the flux quantizationdescribed in Part 1 of these notes, it is clear from our sub-sequent discussion of the Meissner effect, that the currents

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which lead to flux quantization will only flow in a small partof the cross section, a layer of thickness . This layer enclosesthe flux passing through the toroid. Draw a contourCin theinterior of the toroid, as shown in Figure 10. Then vs = 0everywhere onC. It follows that

js

C

Figure 10: Quantization of flux in a toroid.

0 =Cd

vs= 1m

Cd

(+e

cA). (78)

The last integral may be evaluated usingCd

= 2 integer, (79)and

e

c

Cd

A =

e

c

Sd

S

A (80)

= e

c

Sd

S B (81)

= e

c. (82)

HereSis a surface spanning the hole and the flux through

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the hole. Combining these results,

= 2hc

2en=n

hc

2e=n0, (83)

where n is a integer, 0 is the flux quantum, and Ive rein-serted the correct factor of hin the first step to make the unitsright. Flux quantization indeed follows from the fact that thecurrent is the result of a phase gradient.18

Derivation from Microscopic Theory. One of thereasons the GL theory did not enjoy much success at first

was the fact that it is purely phenomenological, in the sensethat the parametersa0, b , m

are not given within any micro-scopic framework. The BCS theory is such a framework, andgives values for these coefficients which set the scale for allquantities calculable from the GL free energy. The GL theoryis more general, however, since, e.g. for strong coupling su-

perconductors the weak coupling values of the coefficients aresimply replaced by different ones of the same order of mag-nitude, without changing the formof the GL free energy. Inconsequence, the dependence of observables on temperature,field, etc, will obey the same universal forms.

The GL theory was derived from BCS by Gorkov. The

calculation is beyond the scope of this course, but can befound in many texts.

18It is important to note, however, that a phase gradient doesnt guarantee that a current is flowing. For example,in the interior of the system depicted in Fig. 2, both and Aare nonzero in the most convenient gauge, and canceleach other!

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6.2 Type I and Type II superconductivity

Now lets look at the problem of the instability of the normalstate to superconductivity in finite magnetic field H. A what

magnetic field to we expect superconductivity to be destroyed,for a given T < Tc.

19 Well, overall energy is conserved, sothe total condensation energy of the system in zero field,fs fn(T) of the system must be equal to the magnetic field energyd3rH2/8 the system would havecontained at the critical

fieldHcin the absence of the Meissner effect. For a completely

homogeneous system I then havefs(T) fn(T) = H2c /8, (84)

and from Eq. (8) this means that, nearTc,

Hc=

2a20

b (Tc T). (85)

Whether this thermodynamic critical fieldHc actually rep-resents the applied field at which flux penetrates the sampledepends on geometry. We assumed in the simplified treat-ment above that the field at the sample surface was the sameas the applied field. Clearly for any realistic sample placedin a field, the lines of field will have to follow the contour

of the sample if it excludes the field from its interior. Thismeans the value ofH at different points on the surface willbe different: the homogeneity assumption we made will notquite hold. If we imagine ramping up the applied field from

19Clearly it willdestroy superconductivity since it breaks the degenerace of between the two componenets of aCooper pair.

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zero, there will inevitably come a point Happl=Happl,cwherethe field at certain points on the sample surface exceeds thecritical field, but at other points does not. For applied fieldsHappl,c < Happl< Hc, part of the sample will then be normal,with local field penetration, and other parts will still excludefield and be superconducting. This is the intermediate stateof a type I superconductor. The structure of this state for areal sample is typically a complicated striped pattern of su-perconducting and normal phases. Even for small fields, edgesand corners of samples typically go normal because the field

lines bunch up there; these are called demagnetizing effects,and must be accounted for in a quantitatively accurate mea-surement of, say, the penetration depth. It is important tonote that these patterns are completely geometry dependent,and have no intrinsic length scale associated with them.

In the 50s, there were several materials known, however,

in which the flux in sufficiently large fields penetrated thesample in a manner which didnotappear to be geometry de-pendent. For example, samples of these so-called type IIsuperconductors with nearly zero demagnetizing factors (longthin plates placed parallel to the field) also showed flux pen-etration in the superconducting state. The type-II materials

exhibit a second-order transition at finite field and the fluxB through the sample varies continuously in the supercon-ducting state. Therefore the mixed state must have currentsflowing, and yet the Meissner effect is not realized, so that theLondon equation somehow does not hold.

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The answer was provided by Abrikosov in 1957 [A.A.A.,Sov. Phys. JETP 5, 1174 (1957).] in a paper which Landauapparently held up for several years because he did not be-lieve it. Let us neglect the effects of geometry again, and goback to our theorists sample with zero demagnetizing factor.Can we relax any of the assumptions that led to the Lon-don equation (72)? Only one is potentially problematic, thatns(r) =|(r)|2 = constant independent of position. Letsexamineas Abrikosov didthe energy cost of making spatialvariations of the order parameter. The free energy in zero

field isF =

d3r[a||2 + 1

2m||2 +b||4], (86)

or1

aF =d3r[||2 +2||2 + ba||

4], (87)

where Ive put

= [ 1

2ma]1/2 = [

1

2ma0(Tc T)]1/2. (88)

Clearly the length represents some kind ofstiffnessof thequantitiy ||2, the superfluid density. [Check that it does in-deed have dimensions of length!] If, the so-calledcoherence

length, is small, the energy cost ofns varying from place toplace will be small. If the order parameter is somehow changedfrom its homogeneous equilibrium value at one point in spaceby an external force, specifies the length scale over whichit heals. We can then investigate the possibility that, asthe kinetic energy of superfluid flow increases with increasing

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field, ifis small enough it might eventually become favorableto bend ||2 instead. In typical type I materials, (T = 0)is of order several hundreds or even thousands of Angstrom,but in heavy fermion superconductors, for example, coher-ence lengths are of order 50-100A. The smallest coherencelengths are attained in the HTSC, where ab is of order 12-15A, whereasc is only 2-3A.

The general problem of minimizing F when depends onposition is extremely difficult. However, we are mainly in-terested in the phase boundary where is small, so life is abit simpler. Lets recall our quantum mechanics analogy oncemore so as to writeFin the form:

F =d3r[a||2 +b||4]+< |Hkin| >, (89)

where Hkin is the operator

12m( +iec A)2. (90)Now note 1) sufficiently close to the transition, we may

always neglect the 4th-order term, which is much smaller;2) to minimize F, it suffices to minimize < Hkin >, sincethe||2 term will simply fix the overall normalization. Thevariational principle of quantum mechanics states that theminimum value of < H > over all possible is achievedwhen is the ground state (for a given normalization of).So we need only solve the eigenvalue problem

Hkinj =Ejj (91)

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for the lowest eigenvalue,Ej, and corresponding eigenfunction

j. For the given form ofHkin, this reduces to the classicquantum mechanics problem of a charged particle moving inan applied magnetic field. The applied fieldH is essentiallythe same as the microscopic fieldB since is so small (at thephase boundary only!). Ill remind you of the solution, due toto Landau, in order to fix notation. We choose a convenientgauge,

A= Hyx, (92)in which Eq. 44 becomes

1

2m[(i

x+

y

2M)2

2

y2

2

z2]j =Ejj, (93)

whereM= (c/eH)1/2 is themagnetic length. Since the co-

ordinatesxandzdont appear explicitly, we make the ansatzof a plane wave along those directions:

=(y)eikxx+ikzz, (94)

yielding

1

2m[(kx+

y

2M)2

2

y2+k2z](y) =E(y). (95)

But this you will recognize as just the equation for a one-dimensional harmonic oscillator centered at the point y =kx2M with an additional additive constant k2z/2m in theenergy. Recall the standard harmonic oscillator equation

( 12m

2

x2+

1

2kx2) =E, (96)

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with ground state energy

E0=0

2 =

1

2(k/m)1/2, (97)

where k is the spring constant, and ground state wavefunc-tion corresponding to the lowest order Hermite polynomial,

0 exp[(mk/4)1/2x2]. (98)Lets just take over these results, identifying

Hkinkx,kz= eH2mc

kx,kz. (99)

The ground state eigenfunction may then be chosen as

kx,kz=0(2ML2y

)1/4eikxx+ikzzexp[(y+kx2M)2/22M)],(100)

where Lyis the size of the sample in the ydirection (LxLyLz=V= 1). The wave functions are normalized such that

d3r|kx,kz|2 =20 (101)

(since I set the volume of the system to 1). The prefactorsare chosen such that20represents the average superfluid den-stity. One important aspect of the particle in a field problemseen from the above solution is the large degeneracy of the

ground state: the energy is independent of the variable kx,for example, corresponding to many possible orbit centers.

We have now found the wavefunction which minimizes . Substituting back into (89), we find using (99)

F = [a0(T Tc) + eH

2mc]d3r||2 +b d3r||4. (102)

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When the coefficient of the quadratic term changes sign, wehave a transition. The field at which this occurs is called theupper critical fieldHc2,

Hc2(T) =2mca0e (Tc T). (103)What is the criterion which distinguishes type-I and type IImaterials? Start in the normal state for T < Tc as shown inFigure 3, and reduce the field. Eventually one crosses eitherHc or Hc2 first. Whichever is crossed first determines the

nature of the instability in finite field, i.e. whether the sampleexpels all the field, or allows flux (vortex) penetration (seesection C).

c1H

H

H

Meissner phase

T

H

Normal statec2

c

Figure 11: Phase boundaries for classic type II superconductor.

In the figure, I have displayed the situation where Hc2 is

higher, meaning it is encountered first. The criterion for thedividing line between type 1 and type II is simply

|dHcdT

| = |dHc2dT

| (104)

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or, using the results (38) and (56),

(m)2c2b(e)2

=1

2. (105)

This criterion is a bit difficult to extract information from inits current form. Lets define the GL parameterto be theratio of the two fundamental lengths we have identified so far,the penetration depth and the coherence length:

=

. (106)

Recalling that

2 = mc2

4e2ns= m

c2b2e2a

(107)

and

2 =

1

2ma. (108)

The criterion (58) now becomes

2 =mc2b/2e2a

1/2ma =

(m)2c2be2

=1

2. (109)

Therefore a material is type I (II) if is less than (greater

than) 12. In type-I superconductors, the coherence length is

large compared to the penetration depth, and the system isstiff with respect to changes in the superfluid density. Thisgives rise to the Meissner effect, where ns is nearly constantover the screened part of the sample. Type-II systems cantscreen out the field close to Hc2 since their stiffness is too

49


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small. The screening is incomplete, and the system must de-cide what pattern of spatial variation and flux penetration ismost favorable from an energetic point of view. The result isthevortex latticefirst described by Abrikosov.

6.3 Vortex Lattice

I commented above on the huge degeneracy of the wave func-tions (53). In particular, for fixed kz= 0, there are as manyground states as allowed values of the parameterkx. AtHc2it

didnt matter, since we could use the eigenvalue alone to de-termine the phase boundary. BelowHc2the fourth order termbecomes important, and the minimization offis no longer aneigenvalue problem. Lets make the plausible assumption thatif some spatially varying order parameter structure is goingto form below Hc2, it will be periodic with period 2/q, i.e.

the system selectssome wave vector qfor energetic reasons.Thex-dependence of the wave functions

kx,kz=0=0(2ML2y

)1/4eikxx exp[(y+kx2M)2/22M)]. (110)

is given through plane wave phases, eikxx. If we choose kx =

qnx, with nx =integer, all such functions will be invariantunderx x+ 2/q. Not all nxs are allowed, however: thecenter of the orbit,kx

2Mshould be inside the sample:

Ly/2< kx2M=q2Mnx< Ly/2, (111)

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53/60

dz

dx eiq(nx1nx2+nx3+nx4)x (118)

dy e

{ 122M

[(y+qnx12M)

2+(y+qnx22M)

2+(y+qnx32M)

2+(y+qnx42M)

2]}.(119)

The formoff[0] is now the same as in zero field, so we

immediately find that in equilibrium,

0|eq= (a2b

)1/2. (120)

and

f=a2

4b . (121)

This expression depends on the variational parametersCnx, qonly through the quantitysappearing inb. Thus if we mini-mize s, we will minimize f (remember b >0, sof


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remain the same since the integrand is the gradient of a scalarfield. This is unphysical because it would imply a finite phasechange along an infinitesimal path (and a divergence of thekinetic energy!) The only way out of the paradox is to havethe system introduce its own hole in itself, i.e. have the am-plitude of the order parameter density||2 go to zero at thecenter of each cell. Intuitively, the magnetic field will have anaccompanying maximum here, since the screening tendencywill be minimized. This reduction in order parameter ampli-tude, magnetic flux bundle, and winding of the phase once by

2 constitute a magnetic vortex, which Ill discuss in moredetail next time.

AssumingCn= constant, which leads to the square latticedoes give a relatively good (small) value for the dimensionlessquantitys, which turns out to be 1.18. This was Abrikosovsclaim for the absolute minimum of f[

|

|2]. But his paper

contained a (now famous) numerical error, and it turns outthat the actual minimums= 1.16 is attained for another setof theCns, to wit

Cn= n1/2max , n = even (128)

Cn= in1/2max , n = odd. (129)

This turns out to be atriangularlattice (Fig. 12b), for whichthe optimal value ofqis found to be

q=31/41/2

, (130)

Again the area of the unit cell is 22, and there is one flux

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quantum per unit cell.

6.4 Properties of Single Vortex. Lower critical

field Hc1

Given that the flux per unit cell is quantized, it is very easy tosee that the lattice spacingdis actually of order the coherencelength nearHc2. Using (103) and (88) we have

Hc2= c

e1

2=

022

. (131)

On the other hand, asHis reduced,dmust increase. To seethis, note that the area of the triangular lattice unit cell isA=

3d2/2, and that there is one quantum of flux per cell,

A= 0/H. Then the lattice constant may be expressed as

d= 4

3(

Hc2

H

)1/2. (132)

Since is the length scale on which supercurrents andmagnetic fields vary, we expect the size of a magnetic vor-tex to be about . This means at Hc2 vortices are stronglyoverlapping, but as the field is lowered, the vortices separate,according to (126), and may eventually be shown to influence

each other only weakly. To find the structure of an isolatedvortex is a straightforward but tedious exercise in minimizingthe GL free energy, and in fact can only be done numericallyin full detail. But lets exploit the fact that we are inter-ested primarily in strongly type-II systems, and therefore goback to the London equation we solved to find the penetration

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1. But lets simplify things even further, by using the factthat : lets treat the core as having negligible size,which means it is just a line singularity. We therefore putg(r) = (r). Then the modified London equation with linesingularity acting as an inhomogeneous source term reads

22 B+ B = 02(r)z (137)2 1

(

Bz

) +Bz = 02(r), (138)

where is the radial cylindrical coordinate. Equation (91)

has the form of a modified Bessels equation with solution:

Bz= 022

K0(

). (139)

The other components of B vanish. If you go to Abramowitz& Stegun you can look up the asymptotic limits:

Bz =

022 [log(

) + 0.116], < (140)Bz =

022

2e/, . (141)

Note the form (93) is actually the correct asymptotic solutionto (91) all the way down to = 0, but the fact that the

solution diverges logarithmically merely reflects the fact thatwe didnt minimize the free energy properly, and allow theorder parameter to relax as it liked within the core. So thedomain of validity of the solution is only down to roughly thecore size, , as stated. In Figure 5 I show schematicallythe structure of the magnetic and order parameter profiles in

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an isolated vortex. The solution may now be inserted into the

core

0

supercurrents

||

Bz

Figure 13: Isolated vortex

free energy and the spatial integrals performed, with some

interesting results:

F =Lz20

1622log(). (142)

It is easy to get an intuitive feel for what this means, sinceif we assume the field is uniform and just ask what is themagnetic energy, we get roughly

Fv = 18

vortex volume B2 (143)

18

(2Lz) (0/2)2 (144)

= Lz20

822, (145)

the same result up to a slowly varying factor. Now the lowercritical fieldHc1 is determined by requiring the Gibbs freeenergies of the Meissner phase with no vortex be equal to theGibbs free energy of the phase with a vortex.20 Gdiffers from

20We havent talked about the Gibbs vs. Helmholtz free energy, but recall the Gibbs is the appropriate potentialto use when the external field H is held fixed, which is the situation we always have, with a generator supplyingwork to maintain H.

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Fthrough a term BH/4. In the Meissner stateG=F,so we may put

F = F+ElineLz

1

4

Hc1Bd3r (146)

= F+ElineLz 14

0Lz, (147)

where Eline is the free energy per unit length of the vortexitself. Therefore

Hc1=4Eline

0(148)

is the upper critical field. But the line energy is given preciselyby Eq. (95),Eline=

201622

log(), so

Hc1(T) = 042

log(). (149)

6.5 Josephson Effect

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teoría de superconductores